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Securitized banking, procyclical bank leverage, and financial instability
Accepted Manuscript Title: Securitized Banking, Procyclical Bank Leverage, and Financial Instability Author: Hyun Woong Park PII: DOI: Reference:
S0954-349X(17)30182-0 https://doi.org/doi:10.1016/j.strueco.2018.10.001 STRECO 756
To appear in:
Structural
Received date: Revised date: Accepted date:
16-6-2017 4-10-2018 4-10-2018
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Please cite this article as: Hyun Woong Park, Securitized Banking, Procyclical Bank Leverage, and Financial Instability, (2018), https://doi.org/10.1016/j.strueco.2018.10.001 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Highlights
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– This paper provides a theoretical explanation of procyclical bank leverage and examines its macroeconomic consequences within the context of the securitized banking system characterized by loan securitization and collateralized short–term borrowing.
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– From a model of repo transaction, it is shown that a broker–dealer (repo borrower)’s optimal leverage is positively related to asset prices. The main mechanism underlying this relation is a movement of repo rate, which reflects counter–party risks and collateral value risks repo lenders are facing.
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– Based on the above result, I develop a macroeconomic model with a securitized banking system characterized with procyclical bank leverage and identify conditions under which the model exhibits a limit cycle behavior. A vulnerability of the repo market to asset price fluctuations leads to procyclical bank leverage, which aggravates asset price cycles driven by the supply–led credit expansion and collapse.
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– It is also found that as the procyclicality of bank leverage becomes stronger the financial cycles also get more intense and severe.
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Hyun Woong Park∗
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Financial Instability
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Securitized Banking, Procyclical Bank Leverage, and
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Denison University, Department of Economics
Abstract
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This paper provides a theoretical explanation of procyclical leverage of the broker– dealer sector and examines its macroeconomic consequences within the context of the securitized banking system characterized by loan securitization and collateralized
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short–term borrowing. First, from a model of repo transaction, it is shown that a
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broker–dealer (repo borrower)’s optimal leverage is positively related to asset prices. The main mechanism underlying this relation is a movement of repo rate, which reflects
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counter–party risks and collateral value risks repo lenders are facing. Second, based on this result, I develop a macroeconomic model with a securitized banking system characterized by procyclical bank leverage and identify conditions under which the model exhibits a limit cycle behavior. A vulnerability of the repo market to asset price fluctuations leads to procyclical bank leverage, which aggravates asset price cycles driven by the supply–led credit expansion and collapse. It is also found that as the procyclicality of bank leverage becomes stronger the financial cycles also get more intense and severe.
Keywords: securitized banking, procyclical bank leverage, credit supply, financial instability, limit cycle
JEL Classication: E32, E44, E51, G21, G24 ∗
Permanent address: 100 W College St, Granville, OH 43023; [email protected]
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1
Introduction
Adrian and Shin (2010a) have recently made an empirical observation that banking sector leverage is procyclical; a financial intermediary leverages up its balance sheet during upturns
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of cycles marked by asset price appreciation while shedding risk exposure by deleveraging during downturns. As the banking sector’s total balance sheet size implies credit flows
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through the real economy, a direct consequence of procyclical bank leverage is an amplification of an asset price cycle and hence an aggravation of financial instability. Motivated by
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this finding, the literature is now growing in two directions. Research in the first direction provides a theoretical explanation for procyclical bank leverage, and research in the second
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direction examines the macroeconomic consequences.1 The main purpose of this paper is to contribute to this emerging body of literature by addressing both of these issues. Underlying the procyclical pattern of financial intermediaries is the transformation of
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banking to the ‘originate and distribute’ model, which characterizes the run–up period to the 2007-2008 disruption and has decisively aggravated the financial fragility of the system
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(Brunnermeier, 2009, e.g.,). In contrast to the traditional banking model, where bank loan
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creation is financed primarily by FDIC–insured deposits and where the issuing banks hold loans until maturity, in the new intermediation model, banks offload risk via securitization,
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and security broker–dealers finance their portfolios through collateralized short-term borrowing, such as repurchase agreements (repo) or asset–backed commercial paper (ABCP). Gorton and Metrick (2012) label this type of banking business model ‘securitized banking’. While macro–models incorporating the banking sector have been growing in number since the recent financial turmoil, those that distinguish between commercial banks and investment banks are relatively few. As a way to fill this gap, the model presented in this paper explicitly incorporates the securitized banking system. In particular, I focus on the repo as a funding source for the securitized banking system and explore the underlying mechanism of its contribution to the endogenous development of boom–bust cycles of asset price, bank leverage, and liquidity, which have become more severe in the securitized banking system. In repo transactions, collateral replaces govern1
I will discuss this literature in greater detail below.
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ment insurance. As the prices of underlying assets are subject to wide fluctuations, repo creditors are typically exposed to collateral value risk. On the other hand, in most cases, repo maturities are overnight, which creates a liquidity risk, as borrowers have to continuously roll over debt. When asset prices are on the rise, repo investors have little concern
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about such risks and allow broker–dealers, i.e., repo borrowers, to borrow not only at a lower rate but also with a lower haircut,2 , which enables borrowers to gain leverage up further.
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However, in times of stress, investors quickly pull back from the market as the probability of the borrower’s default increases. Both the repo rate and haircut rate increase, which forces
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securitized banks to deleverage sharply.3
This paper reveals the connection between such unstable repo market dynamics and the
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procyclicality of securitized banks’ leverage. For this purpose, I first provide a theoretical explanation for the procyclical leverage of the securitized bank by presenting a one–period
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model of repo transactions. In doing so, I compare, among others, Adrian and Shin (2014)’s contracting model, where banks follow the Value–at–Risk rule and where the main friction is risk–shifting moral hazard. A distinctive aspect of the Adrian and Shin model is that it
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adopts Extreme Value Theory to study extreme and rare events that match the repo market
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disruption. In contrast, the repo model presented in this paper adopts a more generalized framework, where the bank is simply characterized by standard profit–maximizing behavior
model.
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without necessarily invoking the extreme outcome specification, as in the Adrian and Shin
In particular, the model focuses on the repo rate, instead of haircuts, as in Geanakoplos (2009) and Adrian and Shin (2014), as a channel through which an asset price cycle is transmitted to the bank leverage cycle. This reflects Smith (2012)’s finding that for the 2008 crisis episode, the repo rate is a superior measure of market stress in the form of collateral value risk. Accordingly, in the model, the repo rate moves in a way that reflects risks 2
A haircut is the value of equity measured against the market value of an asset that is being used as
collateral; it is the ‘skin in the game’. A haircut is an inverse of the leverage ratio. 3 This is a situation comparable to the tradition bank run. Repo investors’ rejection of rollover is analogous to retail depositors’ cash withdrawal. When herd behavior emerges, each of these can place commercial banks and investment banks, respectively, into a liquidity crisis. This is the run on repo as analyzed in Gorton and Metrick (2012) and Martin et al. (2014).
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faced by risk–neutral repo lenders. That is, when asset prices rise, which strengthens the collateral value, the counter–party risk and collateral value risk of the repo lenders will fall, and consequently, the repo rate will also fall. In this environment, the profit–maximizing borrowers, i.e., security broker–dealers, will prefer a higher leverage ratio. On the contrary,
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in times of market stress when assets serving as collateral experience loss, the repo rate will rise. Consequently, the profit–maximizing repo borrowers must lower the leverage. In this
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respect, the optimal leverage ratio of the securitized bank is shown to be positively related to asset prices.
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By comparison, the model also examines a leveraged investor who issues a more standard uncollateralized debt security instead of the repo. It is found that the riskier the asset of the
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investor, the stronger the correlation of her optimal leverage ratio to the asset price. From this, it follows that the correlation becomes close to zero as long as the asset of the leveraged
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investor is significantly safe, as compared with the case of repo borrowers investing in highly risky assets, such as subprime mortgage–backed securities. To examine the macroeconomic implications of the procyclical bank leverage, I then build
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a macroeconomic model consisting of a nonfinancial sector and a financial intermediary
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sector. The intermediary sector is subdivided into three subsectors, including traditional banks that extend loans to the real economy and two other non–bank financial firms, i.e.,
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broker–dealers that purchase securitized bank loans by issuing repos and asset–managing firms that provide funds to the repo market. The main analytical result of the model derives from the Hopf bifurcation theorem, and accordingly, it is shown that under plausible conditions, the model exhibits limit cycle behavior, generating a persistent cycle of asset prices and bank leverage.
More specifically, the key motivations of the macroeconomic modeling is, first, theoretical that the nonfinancial sector’s leverage behavior shapes the demand for lending, while the financial sector’s leverage behavior shapes the supply of lending since the financial sector borrows in order to lend; second, empirical evidence indicates that while the nonfinancial sector and the commercial sector’s leverage ratios are not procyclical in relation to asset prices, the securitized banking sector’s leverage is procyclical. Hence, I first carefully examine all the items of each sector’s balance sheet in a highly 4 Page 5 of 54
simplified economy and construct balance sheet identities for each sector, from which the total demand for bank loans is derived as a positive function of the nonfinancial sector’s leverage and the total supply of bank loans is derived as a positive function of the securitized bank’s leverage. The central behavioral equations of the model concern specifications of each
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sector’s leverage ratio. The securitized banking sector’s leverage ratio is specified as clearly positively related to the price of its assets (asset–backed securities), whereas the nonfinancial
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sector’s leverage ratio is specified as uncorrelated with the price of its assets (real assets). These specifications are based on the empirical findings in section 2, where the broker–
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dealer sector’s leverage ratio is positively related to asset prices, while the correlations of the household and nonfinancial firm sectors’ leverage ratios with the asset prices are not quite as
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strong over time. They are also supported by the above–discussed theoretical results from the model of repo transactions, which are in line with the empirical findings.
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In this setup, the procyclical leverage ratio of the securitized banking sector is defined as the coefficient that measures the marginal response of the securitized bank’s leverage ratio to a change in the asset price, being sufficiently large that in effect, the total supply of bank loan
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is more responsive than the total demand for bank loan is to asset price fluctuations. This
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definition is compared with the one suggested in Adrian and Shin (2010a), where leverage is said to be procyclical in the sense that the growth rate of total assets is equal to that of the
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leverage ratio.
The analytical definition of procyclical bank leverage presented in this paper allows us to find that underlying the dynamics of leverage cycle and asset price cycle is the supply–led credit expansion and collapse. To be more specific, the securitized banking sector’s leverage ratio being procyclical implies in this paper that in the case of asset price appreciation, securitized banks will sufficiently increase borrowing to purchase more of the securities backed by various kinds of bank loans, which will allow banks to issue more loans. However, since the real sector’s leverage ratio is not procyclical, the asset price increase will not affect the real sector’s borrowing; thus, the demand for bank loans will remain the same. The overall result is an excess supply of bank loans, which lowers the loan interest rate and, in turn, further boosts the asset price rise. In contrast, in the case of asset price depreciation, by the same logic, the supply of bank loans decreases more than the demand for bank loans; hence, 5 Page 6 of 54
excess demand for bank loans, which will tend to raise the loan interest rate, will, in turn, further aggravate the decrease in asset prices. Overall, the securitized banking sector’s leverage ratio being procyclical implies that the response of the loan supply to a change in asset prices is stronger than that of the loan
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demand, which eventually makes the dynamic relation between asset price and interest rate destabilizing, where each explode permanently. It is also shown that when the securitized
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banking sector is negligibly small, as in the traditional commercial banking system, the response of the loan demand to a change in asset prices is stronger than that of the loan
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supply, such that the dynamic relation between the asset price and interest rate is stabilizing. Another interesting analytical result of the model is that the procyclical bank leverage
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amplifies the boom–bust cycle. The larger the value of the coefficient that measures the marginal response of the securitized banking sector’s leverage to asset prices, the more severe
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and more intense the upturns and downturns of the cycles of asset prices and bank leverage will be. In this way, another merit of the analytical definition of procyclical bank leverage adopted in this paper is to allow the quantification of the degree of procyclicality of leverage.
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This paper relates to two strands of the contemporary macro-finance literature. First,
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there are a group of papers that incorporate the financial intermediary sector in the dynamic stochastic general equilibrium model (Christiano et al., 2010, Gerali et al., 2010, Gertler and
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Karadi, 2011, e.g.,). While in these papers leverage constraints which banks face develop endogenously due to various forms of agency problems, they require exogenous shocks to actually generate leverage cycles. For instance, one of the main findings of Nuño and Thomas (2017) — which is one of the few recent papers to explicitly formalize procyclical bank leverage in the DSGE framework — is that cross–sectional volatility shocks, instead of the standard total factor productivity shocks, are responsible for bank leverage fluctuations. Additionally, in most of the models in this literature, the household sector does not borrow but only lends to firms through intermediation, and there is no securitization. In contrast, my model does not require exogenous disturbances to generate a cycle; it is an entirely endogenous phenomenon. In addition, firms and households both borrow and lend via the banking sector, which securitizes loans. Second, there is a small but growing literature that studies bank behavior using the 6 Page 7 of 54
macrodynamic modeling and stock–flow consistent approach (Chiarella et al., 2012, Hartmann and Flaschel, 2013, Nikolaidi, 2014, Ryoo, 2013, e.g.,). A key result commonly emerging from these papers is that the endogenous development of the bank’s financial position tends to destabilize the system. However, in most cases, little attention is paid to the lia-
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bility side of the bank balance sheet. In my model, in contrast, the liability structure of the intermediaries plays an important role in generating the key results of the model.
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To understand the limitations of this paper, it should be noted that in contrast to Gorton and Metrick (2012), who highlight the central role played by the repo market for the 2008
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financial crisis, Krishnamurthy et al. (2014) find that as for the funding source for securitized assets, the repo was much smaller than ABCP or direct holdings of the securitized assets.
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However, this finding does not necessarily contradict the focus of this paper on repos, as the essential aspect of a repo that drives the dynamics of the model is the collateralized short–
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term borrowing, where the stability depends on the rise and fall of the collateral value. The ABCP is another type of collateralized short–term borrowing, and thus, similar dynamics can possibly be obtained even when the repo market is replaced by the ABCP market in the
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model. That said, the model should not be taken as a one–to–one mapping of what actually
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happened in the U.S. during the run–up period to 2008. Rather, it is an analysis of the unstable nature of collateralized short–term borrowing, using the repo market as a specific
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example.
The rest of the paper is organized as follows. Section 2 presents empirical data on sectoral leverage to see in what sense the securitized banking sector’s leverage is procyclical. In section 3, by relying on a model of repo transactions, a theoretical explanation of procyclical bank leverage is provided. Using the result of section 3 as a microfoundation, section 4 develops a macrodynamic model to examine the consequence of procylical bank leverage within a broader macroeconomic context. The model is analyzed in section 5. I conclude the paper in section 6.
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2
Empirical observation on leverage and asset prices
The main empirical finding in Adrian and Shin (2010a) is that in the case of the broker– dealer sector, total assets and leverage ratio — measured by total assets over equity — grow
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at the same rate, while equity is constant. Asset growth is entirely driven by growth of debt, not by growth of equity, indicating procyclical leverage and what they call ‘sticky’
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equity. Figure 1 visualizes the mechanics underlying Adrian and Shin (2010a)’s observation. It shows how balance sheet variables change in response to changes in asset prices in the
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case of a marking–to–market accounting system.
In panel (a), a rise in asset prices initially increases the book value of equity along with
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the total asset size, thereby lowering the leverage ratio. The leverage is counter–cyclical. In comparison, panel (b) illustrates the subsequent case where the consequent financial slack is used to issue more debts, thereby purchasing more assets and hence increasing the exposure
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to risks. Moreover, equity is maintained at its initial level through capital policies such as dividend payments and share buybacks. Consequently, the leverage ratio more than restores
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its initial level. In this case, the leverage is procyclical, and equity is sticky.
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Figure 1: Distinctive leverage management
(A: total assets, D: total debts, E: equity)
Adrian and Shin (2010a) illustrate the relation between total assets and the leverage ratio in growth terms. However, since the change in total asset and leverage ratio is in response to changes in asset prices, I suggest directly examining the relation between the leverage ratio
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and asset prices and defining the cyclicality of the leverage ratio accordingly.4 Furthermore, these variables are examined in level terms, not growth terms, as this makes the modeling in the following sections easier. As a way to motivate this approach, data are collected from Z.1 Financial Accounts of the United States released by the Federal Reserve, and the results
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are reported in the figures below.
Figure 2 plots the asset price–leverage relation for the broker–dealer sector. The asset
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prices that matter for broker–dealers may be tradable assets, such as asset–backed securities (ABS) or mortgage–backed securities (MBS). However, since the latter tend to be positively
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correlated with the prices of underlying assets, the housing price is used in constructing the asset price–leverage ratio in figure 2.5 It is clear that the leverage ratio rises with the asset
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price. The case for the commercial banking sector is depicted in figure 3. This figure plots the leverage in relation to both the housing price and the stock market index. In either case,
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while the asset prices change widely, the leverage trend is quite stable despite a few outliers. Figure 4a describes the positive relation between the asset price and leverage for the household sector. However, when compared with the case of securitized banks as shown in
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figure 4b, we can see that the change in the household sector’s leverage ratio is much less
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pronounced; indeed, it is almost negligible. The case for nonfinancial corporate businesses is presented in figure 5. The first panel covers the entire period of 1975–2012, and it shows
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that the asset price–leverage ratio is nonlinear; the leverage is procyclical when the stock market index is relatively low and becomes counter–cyclical when the stock market index is relatively high. The turning point occurs at about 1990. The asset price–leverage relation for the period of 1990–2012 is separately plotted in the second panel. 4
It should be noted that the impact of asset price changes on the bank leverage ratio goes through bank
assets. 5 The chain relation running from the underlying asset price through the price of ABS finally to the broker-dealer sector leverage ratio is formalized in the model presented in section 4.
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Figure 2: Asset prices and leverage of securitized banks (1975-2012)
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(Source: Z.1 Financial Accounts of the United States released by the Federal Reserve)
Figure 3: Asset prices and leverage of commercial banks (1975–2012) (b) Stock price and leverage
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(a) Housing price and leverage
(Source: Z.1 Financial Accounts of the United States released by the Federal Reserve)
Figure 4: Asset prices and leverage of households (1975–2012) (a) Households
(b) Households and securitized bank
(Source: Z.1 Financial Accounts of the United States released by the Federal Reserve)
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Figure 5: Asset prices and leverage of nonfinancial firms (1975-2012) (a) 1975-2012
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(b) 1990-2012
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(Source: Z.1 Financial Accounts of the United States released by the Federal Reserve)
A notable observation from these descriptive data is that for the broker–dealer sector, the
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positive correlation between asset prices and the leverage ratio is quite strong and consistent over time, while for the other sectors, the correlation is either weak or inconsistent. This indicates that the broker–dealer sector borrows more in the case of asset price appreciation
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this paper.
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and deleverages when asset prices fall. This is what is implied by ‘procyclical leverage’ in
A model of repo transaction
This section develops a model that explains the procyclical leverage of a broker–dealer, or securitized bank, that finances its asset positions through repo. A repo contract stipulates that a borrower sells security to lenders with an agreement whereby she will repurchase it at a later date. In effect, the difference between the repurchase price and original price constitutes interest, called the repo rate, and the borrower’s security acts as collateral. In the event of the borrower’s default on her obligation, the lender keeps the security and liquidates it to recover her initially lent cash. However, liquidation will be difficult, possible only at a fire-sale price, if the securities market is under stress. Accordingly, the repo rate reflects a credit risk premium over and above the interest rate for insured deposits. In addition, to mitigate such credit risk, repos are often overcollaterized
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to a degree that is called a haircut. Overall, the haircut and the repo rate are two key variables that reflect the conditions of the repo market. On the one hand, empirical research on the movement of haircuts during the recent financial crisis provides mixed evidence depending on the type of repo market and quality of
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collateral. In the bi–lateral repo market for low–grade collateral, haircuts surged during the financial crisis (Gorton and Metrick, 2012). In the tri–party repo market, however, haircuts
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changed very little during the same period (Krishnamurthy et al., 2014). On the other hand, Smith (2012) finds that the repo rate gives a consistent measure of market stress in the form
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of collateral value risk. This paper adheres to Smith (2012)’s finding, and accordingly, the analysis will focus on the repo rate as reflecting the repo market dynamics. Since a haircut
to be implicitly reflected in the leverage ratio.
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is an inverse of the leverage ratio, I will take a simplifying approach that considers a haircut
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Consider a repo contract between a representative borrower and lender, both being risk neutral. The borrower finances her asset position in securities, denoted by A; however, she owns equity E and has repo borrowing D at a repo rate rp > 1, posting the securities as
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collateral.6 At the repo maturity date when the repo borrower is obliged to repurchase the
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securities, the rate of return of the security can be high at rH > 1 with probability θ or low at rL < 1 with probability 1 − θ.
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As repo transactions involve counter–party risk, the repo rate includes a risk premium over the risk–free rate rf > 1. Let us further suppose that the repo rate is lower than the high return on the security. Accordingly, the order of magnitude of interest rates is assumed to be as follows.
Assumption 1. rL < 1 < rf < rp < rH Since the expected return of the security is θrH + (1 − θ)rL , the security’s expected
value at the maturity date is θrH + (1 − θ)rL A. Ignoring time preference, the security’s
fundamental value in the current period is θrH + (1 − θ)rL A. Intuitively, a rise in θ, which implies an improvement in the prospect of the economy, has a positive impact on the security price due to the assumption 1. 6
The rate of return and interest rate in this paper all refer to, more precisely, the interest rate factor,
which includes both the principal term and interest term.
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Suppose, on the one hand, rH is high enough for the repo borrower to honor the contract, i.e., rH A > rp D. In this case, the repo borrower’s net revenue is rH A − rp D. On the other hand, rL < 1 is too low for the repo borrower to carry out the obligation, i.e., rL A < rp D. In this case, the repo borrower defaults and the repo lender keeps the collateral. Accordingly,
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the repo borrower’s net revenue is simply zero. As a consequence, the repo borrower’s expected return is obtained as follows. θ(rH A − rp D) E
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rb =
(1)
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In the event of the default of the repo borrower and the consequent transfer of the collateral ownership to the repo lender, the lender will most probably attempt to sell it to
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recover the initially invested cash. However, when the default rate is high and the number of repo lenders attempting to liquidate collateral is large, liquidation may take place only
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at a loss reflecting a fire sale discount. Let us use q to denote the discount coefficient with 0 6 q 6 1.7 For instance, note that the fundamental value of the collateral confiscated by the repo lender from the defaulted repo borrower is rL A; depending on the repo market
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condition, the collateral may or may not be sold at its fundamental value; that is, the
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liquidation value of the collateral is rL Aq.
A greater indebtedness of borrowers in the economy makes their financial burden more
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intense, and therefore, the probability of debt default is more likely. Hence, q can be specified as a decreasing function of the borrower’s leverage ratio. Denoting the leverage ratio by λ, we have
q = q(λ),
q0 < 0
(2)
By considering the effect of the liquidation, we can construct a revised expected return of the security as θrH + (1 − θ)rL q, where the low return is discounted by a factor of q. Note that the revised return is compared with the original expected return of the security θrH + (1 − θ)rL . A position in the security is viable only when the revised return of the security is larger than the return on the alternative investment, i.e., rf . In this respect, we adopt the following assumption. Assumption 2. θrH + (1 − θ)rL q > rf 7
Asset price bubbles are ignored in this model; hence, q > 1 is ruled out.
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Next, the repo lender’s expected return, denoted by rc (superscript c indicating ‘creditor’), is measured as rc =
θrp D + (1 − θ)rL Aq D
(3)
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With probability θ, the repo lender’s gross payoff is rp D, and with probability 1 − θ, it is the collateral’s liquidation value rL Aq. As the repo lender is risk neutral, rc is required to be equal to the alternative risk–free rate rf for the lender to be at least indifferent between the
cr
repo and the riskless asset. The repo rate rp should be determined at a level that guarantees
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rc = rf , which is a participation constraint of the repo lender. As a consequence, using the definition of the leverage ratio λ = A/E, the equilibrium repo rate is obtained from rc = rf as
rf 1−θ λ L − r q θ θ λ−1
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rp =
(4)
Proposition 1 summarizes the relations.
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Equation (4) demonstrates that given rf and rL , the repo rate depends on λ and θ.
Proposition 1. The repo rate responds to a change in the repo borrower’s leverage ratio
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and the collateral value in the following ways. (i) rp rises (falls) as λ increases (decreases).
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(ii) rp rises (falls) as θ decreases (increases). (iii) The impact λ has on rp as described in (i) becomes weaker (stronger) as θ gets larger (smaller).
On the one hand, since borrowers’ leverage λ incurs risks on the part of lenders, its rise will increase the repo rate. This is stated in (i). On the other hand, θ affects the repo rate in two ways. First, it is positively related to the security price that serves as collateral. Consequently, when θ is higher, the repo lender will allow the securitized bank to borrow at a lower rate. This is stated in (ii). Second, while a rise in λ will lead to an increase in rp as in (i), the increase will be smaller when the asset value improves with a rise in θ. This is because the enhancement of collateral value weakens the lender’s counter–party risk. This is stated in (iii).
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Substituting the equilibrium repo rate obtained in equation (4) and using the definition of the leverage ratio, the repo borrower’s rate of return in equation (1) can be rearranged as follows: rb = θrH λ + (1 − θ)rL λq − rf (λ − 1)
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(5)
Now, the optimal leverage of the repo borrower that maximizes rb is obtained as the
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solution to the following optimization problem. max θrH λ + (1 − θ)rL λq − rf (λ − 1) λ
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From the first–order condition, the optimal leverage is yielded as 1 rf − θrH λ = 0 −q q (1 − θ)rL
!
(6)
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∗
Due to assumption 2 and q 0 < 0, the optimal leverage being negative is ruled out; hence,
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λ∗ > 0 always holds.
It can be verified how the optimal leverage changes in relation to a change in the asset price reflected in a change in θ. Proposition 1 states this result. ∂λ∗ ∂θ
=
rf −rH q 0 rL (1−θ)2
> 0 holds.
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Proposition 2. For a repo borrower,
The proof of proposition (2) is immediate due to rH > rf from assumption 1 and q 0 < 0
Ac ce p
from (2). A change in θ, which affects the asset value, imparts an impact on the repo borrower’s leverage behavior in two ways. First, a rise in θ lowers rp , as demonstrated in (i) of proposition 1. The lower borrowing cost makes a debt–financed investment more profitable, leading to a larger optimal leverage ratio. Second, the consequent rise in the leverage ratio pushes the repo rate up, as shown in (ii) of proposition 1. However, this adverse feedback effect of λ on rp becomes weaker as the asset value strengthens, as shown in (iii) of proposition 1; thus, it does not fully offset the initial reduction of rp . Through these two processes, a rise (fall) in θ leads to a fall (rise) in rp , which, in turn, increases (decreases) λ∗ . Proposition 2 shows that the leverage ratio of an investor financed by collateralized short– term borrowing is procyclical. This supports the empirical observation in figure 2, which displays the positive correlation between the asset price and the leverage behavior of the broker–dealer sector, which mainly relies on collateralized short–term borrowings. 15 Page 16 of 54
As a comparison with a repo borrower, let us now consider a leveraged investor who relies on more standard uncollateralized debt, denoted by L, with the interest rate of rl . The uncollateralized debt is backed by the debtor’s creditworthiness instead of collateral; thus, the credit risk of debtor default is directly reflected in the interest rate charged. Hence,
ip t
in the model, the loan interest rate can be considered a positive function of the investor’s indebtedness: f0 > 0
cr
rl = f (λ),
(7)
borrower.
rb =
an
In this setup, the nonrepo borrower’s return is
us
To distinguish the investor from the repo borrower considered so far, let us call her a nonrepo
θ(rH A − rl L) + (1 − θ)(rL A − rl L) E
M
which is simplified to
rb = θλ(rH − rL ) + rL λ − rl (λ − 1)
(8)
(9)
d
Now, the profit–maximizing nonrepo borrower’s optimization problem is as follows.
te
max θλ(rH − rL ) + rL λ − rl (λ − 1) λ
Ac ce p
from which the optimal leverage ratio is obtained as λ∗ =
θ(rH − rL ) + rL − rl +1 f0
(10)
How does the nonrepo borrower’s optimal leverage ratio respond to a change in asset prices? The answer is summarized in the following proposition. Proposition 3. For a leveraged investor who relies on a standard unsecured debt contract in financing her asset position,
∂λ∗ ∂θ
=
rH −rL f0
> 0 holds.
The proof of proposition 3 is immediate owing to rH > rL from assumption 1 and f 0 > 0 from (7). It can be seen that the positive correlation between the leverage ratio and the asset price holds for both the repo borrower and the nonrepo borrower, as shown in propositions 2 and 3, respectively.
16 Page 17 of 54
On the other hand, another interesting result coming from proposition 3 is that
∂λ∗ ∂θ
is a positive function of rH − rL . That is, the larger (smaller) the difference between rH and rL , the stronger (weaker) the response of the optimal leverage ratio to a change in asset prices. Consider the 2008 subprime mortgage crisis, which has demonstrated that
ip t
securitized banks relied on collateralized short–term borrowings, including repos to finance their positions in mortgage–backed securities, which turned out to be highly risky. Using
cr
the model terminology, rH rL due to rL is extremely low. This was possible thanks to the loose regulations. In contrast, investors borrowing from debt markets other than the
us
collateralized short–term debt market were largely prevented from such risky investments because they were faced with different types of regulatory and institutional environments.
rH ≈ rL holds,
∂λ∗ ∂θ
an
In this context, if the nonrepo borrower seeks significantly safe investments such that ≈ 0 will ensue. This result, on the one hand, supports the patterns of
M
commercial banks and households displayed in figures 3 and 4, respectively. The former is flat, whereas the latter exhibits a correlation that is positive but very weak, especially when compared with that of the securitized banking sector.
d
On the other hand, the correlation for nonfinancial firms during more recent periods,
te
as displayed in figure 5b, is negative. This can be understood within the framework of proposition 3 only when f 0 < 0, i.e., when the correlation between the leverage ratio and
Ac ce p
interest rate is negative. This relation is in line with the fact explained in corporate finance that by definition the leverage ratio and cost of capital are negatively correlated because debt is tax–advantaged and hence is cheaper than equity.
4
A macroeconomic model of procyclical bank leverage
In this section, I build a macroeconomic model to examine the consequence of procyclical bank leverage within a macro context. The model is built by first specifying distinctive leverage behaviors of financial and nonfinancial sectors in response to asset price developments; then, from these, I derive the demand and supply of bank loans. Note that the nonfinancial sector’s leverage shapes the demand for credit, while the financial sector’s leverage shapes the supply of credit. Finally, the model describes how the procyclical bank leverage gen17 Page 18 of 54
erates supply–led boom and bust of credit and, consequently, determines the credit market interest rate in a way that amplifies asset price cycles. A central behavioral specification of the model is founded upon the empirical and theoretical findings, which are presented in section 2 and 3, respectively. I find that securitized banks’ leverage has a sufficiently strong
ip t
correlation with asset prices.
The nonfinancial sector includes nonfinancial firms and households that are consolidated
cr
into one sector, which I call the real sector. The financial sector is divided into three subsectors, i.e., commercial banks, investment banks — interchangeably called securitized
us
banks or broker–dealers — and asset–managing firms. Notations for the real sector and the three financial sectors are R, B, X, and N , respectively. Notations for the balance sheet
an
variables are used in the same way as in the previous sections, with A for assets, D for debt, E for equity, and λ for the leverage ratio. The sectors associated with these balance sheet
M
variables will be indicated as a subscript. For instance, debt of the real sector is denoted by DR , and the leverage ratio of the securitized banking sector, by λX .
d
Real sector: firms and households
te
4.1
The real sector’s asset, AR , consists of both real and financial assets. First, there are two
Ac ce p
types of financial assets: deposit accounts, denoted by M B , at the commercial bank, and nondeposit accounts, denoted by M N , at insurance companies, pension funds, mutual funds, etc., where clients’ funds are managed by asset managers. Real assets, denoted by K, include the firms’ various equipment, machines, and the households’ housing. To make things simple, let us use some consolidated price index of the real assets and denote it by PR , which will be a key variable in the model.
The real sector’s assets are financed by a mix of own funds, i.e., equity, ER , and debts, DR . It is assumed that the real sector’s debt exclusively consists of bank loans, denoted by L, i.e., DR = L, and particularly that the bank loans finance the real sector’s positions in the real assets only, but not the financial assets. In sum, the real sector’s balance sheet identity (AR ≡ DR + ER ), as expressed in PR K + M B + M N ≡ L + E R
(11)
18 Page 19 of 54
The first behavioral assumption of the model regards the real sector’s leverage ratio in relation to asset prices. Reflecting the empirical observations in section 2 on the relation between leverage and asset prices for each sector, the leverage ratio of each sector will be
ip t
modeled as a simple linear function of the asset price. For the real sector, λ R = α 0 + α 1 PR
(12)
1 , 1−vB m∗
α1 = 0
us
Assumption 3. α0 >
cr
As for the parameters, α0 and α1 , the following assumption is adopted.
The second part of assumption 3, α1 = 0, reflects figures 4 and 5 and proposition 3 and the
an
discussion that followed regarding nonrepo borrowers’ investment being less risky; that is, the correlation of the real sector leverage to asset prices is close to zero. Since it will be 1 , 1−vB m∗
along with
M
shown later that 0 < vB < 1 and 0 < m∗ < 18 , the first part, α0 >
α1 = 0, implies that the real sector leverage λR is constant at α0 , which is greater than
in section 5.1.
d
some positive threshold. The first part of the assumption will be useful for a later analysis
te
On the other hand, the real sector’s equity value will change, by the accounting definition, in line with asset price fluctuations. Therefore, the real sector’s equity can also be specified
Ac ce p
as a simple linear function of PR as
ER = δ0 + δ1 PR ,
δ0 , δ1 > 0
(13)
The consolidated price of the real asset PR is modeled with regard to two channels, i.e., the expectation channel and funding cost channel. First, when asset prices are expected to rise, investors will increase their demand for assets, while the supply will fall.9 Second, obviously, a funding cost negatively affects the demand for assets, while the supply of assets is not affected significantly. In this setting, PR is an increasing function of its expected future value, denoted by PRe , and a decreasing function of the bank lending interest rate, denoted 8 9
These notations will be introduced later From the perspective of sellers, assets can be sold at a higher price in subsequent periods.
19 Page 20 of 54
by rl .10 PR = 0 + 1 PRe − 2 rl ,
0 , 1 , 2 > 0
(14)
1 and 2 reflect the strength of the expectation channel and the funding cost channel,
ip t
respectively. In modeling the expected asset price PRe , suppose that investors reckon that asset prices cannot rise permanently. There is a certain level of the asset price beyond which the asset
cr
price is perceived by the market as unsustainable. When the asset price is below this thresh-
market sharply adjusts its expectation downward.
us
old, investors expect the future value to rise, but as the asset price reaches this level, the
This nonlinear relation can be formalized by modeling the change in the expected asset
g 0 < 0,
PR ≶ PR∗ ⇔ g ≷ 0
(15)
M
P˙Re = g(PR ),
an
price as a negative nonlinear function of the current asset price as follows.
where the nonlinearity of g is displayed in figure 6.11 PR∗ is the threshold beyond which the
d
asset price is perceived by the market to be unsustainable. When PR is below this level,
te
investors expect it to increase further but at a slower pace as PR becomes higher.12 As PR reaches the threshold, investors expect it to undergo a free fall in the near future.
Ac ce p
This specification of asset price expectations, along with equation (14), excludes a degenerate case where asset prices permanently explode. The dynamics of the asset price expectation channel represent a stabilizing factor of the model. On the other hand, as will be shown in detail below, the contribution of the interest rate channel to the asset price stability depends on the specific type of the banking system. More specifically, the interest 10
An alternative way to motivate equation (14) is to rely on the present valuation asset pricing, where
the interest rate is negatively correlated with the asset price, while the expected asset price has a positive impact on the latter. 11 The nonlinear curve in figure 6 can be approximated by the following function, which is adopted from Ryoo (2010) with a slight change: g(x) = −(g0 + g4 ) +
g1 + g0 1 + e−g2 (x−g3 )
where g0 , g1 , g2 , g3 , g4 > 0. 12 The flat segments of the curve are slightly downward sloping, reflecting g 0 < 0.
20 Page 21 of 54
Figure 6: Changes in the expected asset price as a nonlinear function of the current asset price;
us
cr
ip t
g = P˙Re .
an
rate channel operates as a stabilizing factor in the traditional banking system as well as a destabilizing factor in the securitized banking system. Consequently, in the securitized
M
banking system, the model will avoid a degenerate case of permanent explosion only when
Commercial bank
te
4.2
d
the destabilizing channel, reflected in 2 , is sufficiently small.
The commercial bank’s total asset, AB , includes reserves, denoted by HB , and loans, denoted
Ac ce p
by L. Bank loans are extended to the real sector. Part of such loans are packaged into ABS and unloaded from the bank balance sheet. Let us call this share the securitization ratio and denote by ω with 0 6 ω 6 1. The ABS is purchased only by the securitized banking sector. Therefore, ω is determined by securitized bank’s activity, which will be discussed shortly.13 Since the bank loans finance real assets, such as housing and firms’ factories and machines, securitized loans are backed by the performance of such real assets. Hence, the real assets are underlying assets for ABS.
Commercial banks’ debt, DB , represents FDIC–insured deposits held by the real sector; hence, DB ≡ M B . The commercial banking sector’s balance sheet identity (AB ≡ DB + EB ) can be expressed as L(1 − ω) + HB ≡ M B + EB 13
(16)
The unspecified assumption here is that ω positively affects the profitability of commercial banks, and
the latter is therefore willing to securitize loans to the extent possible.
21 Page 22 of 54
The reserve ratio or liquidity ratio, denoted by hB , is measured as hB ≡
HB . DB
Reflecting the
fact that commercial banks are highly regulated, their liquidity ratio and leverage ratio, λB , are taken as fixed at a policy level, i.e., λB = λ∗B and hB = h∗B .14
Securitized bank
ip t
4.3
cr
Suppose ABS is the only asset of the securitized banking sector. The number of the security and its price are denoted by X and PX , respectively. The securitized bank’s asset position
us
is financed by issuing debts in the repo market, using ABS as collateral. Accordingly, the securitized banks’ debt, DX , consists of repos, denoted by Q. This sector’s equity is taken as ∗ constant at EX , reflecting Adrian and Shin (2010a)’s finding of ‘sticky equity’. The balance
an
sheet identity, (AX ≡ DX + EX ), can be expressed as
∗ PX X ≡ Q + EX
M
(17)
As investment banks operate outside of regulatory control, it is supposed that they do
d
not hold any reserves. Therefore, the total asset size of this sector equals the total volume
te
of outstanding bank loans that are packaged into ABS. Denoting the securitized banking ∗ = ωL. From this, an expression for ω sector’s leverage ratio by λX , it holds that λX EX
Ac ce p
follows as
∗ λX EX ω= L
(18)
which shows that the securitization ratio is determined by the total bank loans, that is, the balance sheet policy of the securitized bank, i.e., its equity and leverage. Intuitively, the rise in securitized banks’ leverage, given their sticky equity, enables commercial banks to securitize more of their existing loans.15 14
The standard supply curve of bank loans is increasing with respect to the lending interest rate. This is
confirmed by proving
∂L ∂r l
> 0. However, in the case of the ‘originate and distribute’ model, bank revenues
are produced mostly out of fees incurred in various stages across the loan securitization process rather than interest earnings. Consequently, the more the loans are securitized, as measured by a higher ω, the less concerned banks will be about the level of the loan interest rate. The standard credit supply curve will 2
∂ L become flatter as the securitization rate rises. This is confirmed by proving ∂ω∂r l < 0. 15 As will be shown in section 4.5, ω has a positive impact on commercial banks’ lending capacity.
22 Page 23 of 54
In parallel with the behavioral specification of the real sector leverage in equation (12), the leverage behavior of the securitized banking sector can be specified as a simple linear function of the ABS price, in support of the empirical observations in section 2. β0 , β1 > 0
(19)
ip t
λX = β0 + β1 PX ,
The key parameter is β1 , the marginal response of the leverage ratio of the securitized
cr
banking sector λX to the ABS price PX . It reflects how securitized banks actively manage their balance sheet in response to changes in asset prices, which are immediately reflected in
us
their equity value through the marked–to–market accounting rule. The central behavioral assumption underpinning the main analytical results of the model concerns the value of β1 .
an
On the one hand, according to the positive correlation between the securitized banking sector’s leverage and the security price evidenced in figure 2, section 2, we should have
M
β1 > 0. In addition, this was the result of the model of repo transaction summarized in proposition 2, section 3. On the other hand, the full macroeconomic model developed in this
d
section requires β1 to be not only positive but also sufficiently large so that the procyclical bank leverage has meaningful macroeconomic consequences; see section 5.1.
te
The next step is to model PX . Determining the value of ABS is not an easy task, as its
Ac ce p
value is ultimately backed by underlying assets as collateral. The initial aim of designing these derivatives so as to spread and mitigate risks through pooling and trenching has made it extremely difficult to reveal the full information on the properties of the underlying assets. This is particularly true in the case of the mortgage–backed securities market during the run–up to 2008, where the major portion were subprime loans (Fender and Scheicher, 2008). I take a simpler and more intuitive approach to specify the price of ABS as an increasing function of the underlying asset value16 and a decreasing function of the repo rate. PX = µ0 + µ1 PR − µ2 rp ,
µ0 , µ1 , µ2 > 0
(20)
The behavioral specification in (20) reflects the recent observation regarding the housing market boom that the economic environment with low rates of interest led to an appreciation 16
This assumption is based on the recent experience of the correlation between housing prices and the
price of mortgage–related securities during the subprime mortgage turmoil.
23 Page 24 of 54
in the underlying assets market and subsequently in the ABS market by drastically increasing the demand for these assets while the asset supply was relatively stable (Perraudin, 2008). The correlation between the leverage ratio and asset price for the securitized banking sector can be examined with either type of asset, i.e., the ABS or the underlying assets.
ip t
The securitized bank’s leverage behavior in response to the ABS price is reflected in β1 , while that in response to the underlying asset price is reflected in β1 µ1 , since
∂λX ∂PR
=
∂λX ∂PX ∂PX ∂PR
cr
from equations (19) and (20). A specific level of these parameters will be discussed more
Asset managing firm
an
4.4
us
rigorously in section 5.1.
An asset managing firm is an unleveraged entity. Such an entity manages the real sector’s nondeposit funds at insurance companies, pension funds, mutual funds, hedge funds, etc.,
M
without any leverage. Therefore, the asset managing sector’s balance sheet identity is AN ≡ EN . These institutional investors’ funds are voluminous much beyond the deposit insurance
d
limit. Thus, they are invested in the repo market, where funds are secured by collateral.
te
The asset managing sector also holds liquid assets. Suppose a share, denoted by hN , of this sector’s total asset, AN , is held in liquid assets, and the rest in reverse–repo, denoted by
Ac ce p
Q.17 Accordingly, the balance sheet identity is expressed as Q + hN AN ≡ EN
(21)
One of the key drivers of the model result regards the repo market, where the securitized banks are debtors and asset managing firms are creditors. In addition, the determination of the repo rate, the key variable in the repo market, is specified as follows. r˙p = κ1 (rp − rp∗ ) − κ2 P˙R ,
κ1 , κ2 > 0
(22)
First, recall proposition 1 (ii), according to which asset price appreciation (depreciation), by enhancing (undermining) collateral value, leads to a reduction (rise) in the repo rate. In line with this result in relation to the collateral value risk, the second term of (22) captures 17
In this model, the run on repo would be reflected in a surge in hN .
24 Page 25 of 54
a change in the price of the underlying asset, which acts as collateral, having a negative impact on a change in the repo rate. In addition, there is a feedback effect from the repo rate back to the price of the underlying asset. According to equation (14), the price of the real asset PR is a negative function of the
ip t
lending interest rate rl . However, rl , in turn, is modeled in section 4.5, with equation (27) as a spread over rp ; hence, rl is a positive function of rp . Overall, a fall (rise) in rp ultimately
cr
leads to a rise (fall) in PR . That is, a lower (higher) repo rate weakens (strengthens) a funding cost burden on the part of the securitized bank, and this will raise (reduce) the
us
demand for assets and hence their prices.
When this feedback effect is taken into consideration in relation to the second term in
an
equation (22), it follows that a fall (rise) in rp will increase (decrease) the asset price, which, in turn, can possibly reduce (raise) rp further, and so on. Both rp and PR can possibly
M
explode in either direction. Such destabilizing dynamics of the repo rate are captured by the first term of equation (22). It is supposed that there is some normal level of the repo rate, denoted by rp∗ . When the repo rate is lower than this threshold, it tends to fall further,
d
whereas when it is higher, it tends to increase further.
te
Such unstable dynamics of the repo rate can be justified from the recent disruption of the repo market — so–called the repo run — especially related to low quality collateral.
Ac ce p
Before the subprime mortgage crisis emerged, the spread in the repo market over the federal funds rate was close to 1% points even when the leverage of borrowers, i.e., broker–dealers, was exceptionally high. However, with the rising mortgage default rate and the collapse of Lehman Brothers, the spread surged and liquidity dried, which pushed up the repo rate even further.
However, the permanent explosion of rp and PR was prevented by the dynamics of PR and PRe , a discussed in section 4.1. According to equation (15), as displayed in figure 6, the market did not expect the asset price to increase forever; when it reached a certain threshold beyond which the asset price was considered as unsustainable, investors abruptly and sharply revised down their expectations about the asset price, which, according to equation (14), then led to a plunge in the asset price. This mechanism acts as a stabilizing factor in the repo rate dynamics. For instance, 25 Page 26 of 54
along the exploding process of decreasing rp and increasing PR , at some point PR will reach the threshold and subsequently will make a huge drop according to the dynamics displayed in figure 6. Then, due to the second term of the repo rate equation (22), this will reverse the decreasing course of rp .
ip t
For this scenario to be effective, the response of the repo rate to the underlying asset price fluctuations should be sufficiently sensitive so as to reverse the course of the destabilizing
characterizing the full dynamics of the model; see section 5.2.
cr
force. That is, κ2 should be sufficiently large. This condition will play an important role in
us
Thus far, the balance sheet description — asset allocation and leverage decision (capital structure) — of the sectors of the model and behavioral equations of the key variables were
an
noted; most importantly, they were the sectoral leverage ratio and asset prices. Remember that the nonfinancial sector’s leverage shapes the demand for credit, while the financial
M
sector’s leverage shapes the supply of credit. Now that each sector’s leverage behavior is specified, we are ready to analyze how the demand and supply of credit is formed and
Bank credit market
te
4.5
d
consequently how interest rates are affected.
Ac ce p
As is assumed, real sector debt consists solely of bank loans, demand, and supply of bank credit, which can be derived from the balance sheet specifications of each sector. In addition, the interest rate and the interest rate spread in the bank credit market can be determined by examining the demand and supply of bank credit. In the model, the real sector’s outstanding debt constitutes the total demand for bank loan, which is denoted by LD ; hence, DR ≡ LD18 Using the balance sheet identity of the real sector, AR ≡ DR + ER , and the definition of leverage ratio, the following can be easily obtained. LD ≡ (λR − 1)ER
(23)
On the other hand, the outstanding loans in the balance sheets of commercial banks shape the total supply of bank credit, which is denoted by LS . This can be derived from the 18
Note that the demand and supply of bank lending are considered in stock terms.
26 Page 27 of 54
balance sheet identity of commercial banks in equation (16) and the definition of its leverage and liquidity ratios as follows: λ∗ B
− h∗B
1−ω
M B
(24)
ip t
LS ≡
λ∗B −1
Equation (24) demonstrates the loan supply as a product between deposit fund M B — which
cr
is determined by the real sector’s portfolio decision — and the bracketed term consisting of the commercial banking sector’s leverage λB , required reserve ratio (or liquidity ratio) hB ,
us
and securitization ratio ω. This shows the determination of the commercial banking sector’s lending capacity.
an
The bracketed term can be taken as a bank loan multiplier, which differs from the conventional money multiplier. The latter is the concept that captures a conversion of central bank money into commercial bank money through the reserve requirement reflected in hB .
M
In contrast, the former describes a conversion of commercial bank money (bank debt) into bank lending (bank assets). In this process, not only the reserve requirement but also secu-
d
ritization and bank leverage behavior, each reflected in ω and λB , are the key mechanisms.
te
Among the variables that affect the bank loan multiplier, i.e., those in the bracketed term, the most interesting for our purpose is ω. Denoting the multiplier by m, we have
∂m ∂ω
> 0.
Ac ce p
That is, securitization enhances commercial banks’ lending capacity, which is the greatest concern of financial innovations. A traditional commercial bank without securitization will have ω = 0.19
Let us introduce a new parameter vB , which denotes a portfolio coefficient of the real sector, reflecting the share of its total asset held in bank deposits. Then, by definition, it holds that vB AR ≡ M B . Using this and the expression of ω in equation (18) and the identity AR ≡ λR ER , equation (24) can be rearranged into ∗ LS = m∗ vB λR ER + λX EX 19
For the other two variables constituting m, we have
∂m ∂h∗ B
< 0 and
(25) ∂m ∂λ∗ B
< 0. The former is as expected,
but the latter requires a brief comment. Since what is taken as given in equation (24) is not equity but liability, i.e., deposits, a higher λ∗B reduces the asset size, and given the required reserve ratio, it thus reduces the loan supply.
27 Page 28 of 54
where m∗ =
λ∗B ∗ λB −1
− h∗B ; note that this differs from the bank loan multiplier m =
1 1−ω
λ∗B ∗ λB −1
−
h∗B . In comparison with equation (24), the above equation shows how the total outstanding bank loans, originally extended by commercial banks, are divided between commercial banks and securitized banks through loan securitization. The first term on the right–hand side
ip t
represents the loans kept in the originators’ loan book, while the second term represents the loans packaged and sold to the securitized banking sector.
cr
Now, using the demand and supply of bank loans derived above, we can define the excess
bank loans by Φ, we have Φ ≡ LD − LS
us
demand for bank loans as the difference between the two. Denoting the excess demand for
(26)
an
Using the excess demand for bank loans, we can determine the interest rate in the bank loan market. Since the bank loan interest rate is a long–term rate, while the repo rate is a
M
short–term rate in the model, let us adopt a simple approach to model rl as determined by the interest rate spread over the repo rate.
(27)
te
d
rl = rp + ξ
where ξ denotes the interest rate spread in the bank loan market.
Ac ce p
The spread is determined by the interaction between the demand and supply forces in the bank loan market. It can be modeled as an increasing function of the excess demand for bank loans.
ξ = τ0 + τ1 Φ,
τ0 , τ1 > 0
(28)
τ0 is the constant equilibrium level of the spread, where the demand and supply of bank loans equilibrates. When the bank loan market is in excess demand, i.e., Φ > 0, the spread will rise above τ0 , and in case of excess supply, it will fall below τ0 . The spread will be at its equilibrium level, τ0 , only when Φ = 0. Introducing the constant level of equilibrium spread can be justified within the context of this paper, which describes a short–run fluctuation without a long–run trend.20 20
A more economically intuitive way to model the spread would be to have ξ˙ = τ2 +τ3 Φ in place of equation
(28). In this case, the model will be reduced to a three–dimensional differential equations system. As will
28 Page 29 of 54
Figure 7: Causal links of the model (a) Dynamics of the underlying asset price PR in relation to its expected future value PRe and to
Summary of the model
M
4.6
an
us
(b) Details of the PR → rl link
cr
ip t
the bank lending interest rate (funding cost) rl
The model consists of twelve endogenous variables and twelve equations, which are repro-
d
duced in appendix A. As the basic aim of the model is to explain a persistent boom–bust
te
cycle of asset prices in the securitized banking system, the price PR of real assets, which is an underlying asset for asset–backed securities, is first modeled as a function of its expected
Ac ce p
future value PRe — expectation channel — and loan interest rate rl — funding cost channel — as in equation (14). Second, there are feedback mechanisms from PR back into PRe and rl , thereby ultimately generating endogenous loops, as depicted in figure 7a. The feedback link of PR → PRe on the one hand is modeled simply as a nonlinear relation as in figure 6. On the other hand, the other feedback link of PR → rl is more complicated. Figure 7b illustrates the key causal relations of this link. The main thrust is the underlying asset price PR that affects the leverage behavior of the real sector λR and that of the securitized banking sector λX , which, in turn, shapes the demand and supply of bank loans LD and LS , respectively, which, ultimately affect the loan rate rl . Section 5.1 discusses this link be seen below, by adopting equation (28), it becomes possible to reduce the model to a two–dimensional differential equation system, which is much more convenient to deal with. My approach can be compared with the existing models of cycles where the bench mark rate, such as the federal funds rate, is taken as a constant.
29 Page 30 of 54
in greater detail. The model’s full dynamics as a combined result of all of the four links in figure 7a are presented in section 5.2. The twelve equations of the model are reduced to two differential equations in PRe and rp , which are two state variables of the model. Once the model is solved,
ip t
all of the twelve variables will be expressed as a function of PRe and rp , which reflect the two channels of the asset price determination, i.e., expectations channel and funding cost
cr
channel. The full solution of the model is presented in appendix B.21 The main analytical result of the model, discussed in section 5.2, regards the conditions under which the system
Model analysis
Analytics of procyclical bank leverage
M
5.1
an
5
us
exhibits limit cycle behavior.
Let us formalize the PR → rl link. It will be shown that this link exhibits two opposite cases
d
depending on the procyclicality of bank leverage. The key mechanism here is that each of
te
the nonfinancial and financial sector’s leverage behavior shapes the demand and supply of bank credit, respectively. To see this, expand the identities of LD and LS in equations (23)
Ac ce p
and (25) by using behavioral equations (12), (13), (19), and (20). LD = δ0 (α0 − 1) + δ1 (α0 − 1)PR
(29)
LS = vB m∗ α0 δ0 + β0 + β1 µ0 − β1 µ2 rp + (vB m∗ α0 δ1 + β1 µ1 )PR
∗ In obtaining the above expressions of LD and LS , EX = 1 is assumed. This is equivalent to ∗ that where all the balance sheet variables are normalized by EX , as indicated in appendix
A. In any case, now the demand and supply of bank loans are expressed as a linear function of PR . The marginal responses of the loan demand and loan supply to the underlying asset price 21
All the mathematics involved in solving and analyzing the model in the rest of the paper are obtained
with the help of Mathematica, as they are somewhat messy to be obtained simply by hand. Mathematica codes will be available directly from the author upon request.
30 Page 31 of 54
are measured as, respectively, ∂LD = δ1 (α0 − 1) ∂PR ∂LS = vB m∗ α0 δ1 + β1 µ1 ∂PR
ip t
(30)
By comparing the two, we can identify the condition under which the demand response
cr
is stronger than the supply response and the condition under which the opposite holds. The condition concerns the value of β1 , i.e., the marginal response of the securitized bank’s
us
leverage to the ABS price. More specifically, if β1 is sufficiently large, the supply response is stronger than the demand response, while if β1 is sufficiently small, the demand response
an
is stronger than the supply response. This result is stated in lemma 1.
Lemma 1. The level of β1 and the relative response of the demand and supply of the bank
M
loan to the asset price are related in the following way:
te
d
∂LS δ1 ∂LD < ⇐⇒ β1 > α0 (1 − vB m∗ ) − 1 ⇐⇒ W1 > 0 ∂PR ∂PR µ1 D S ∂L ∂L δ1 α0 (1 − vB m∗ ) − 1 ⇐⇒ W1 < 0 > ⇐⇒ β1 < ∂PR ∂PR µ1
Ac ce p
where W1 = β1 µ1 − α0 (1 − vB m∗ ) − 1 δ1 . Lemma 1 states that β1 being larger than a certain level,
δ1 µ1
α0 (1 − vB m∗ ) − 1 is an-
alytically equivalent to the loan supply responding more strongly to the underlying asset price than the loan demand does; in contrast, β1 being smaller than the threshold is analytically equivalent to the loan demand responding more strongly to the underlying asset price than the loan supply does. Furthermore, lemma 1 introduces an additional notation
W1 ≡ β1 µ1 − δ1 α0 (1 − vB m∗ ) − 1 and demonstrates an analytical equivalence between the former case and W1 > 0 and that between the latter case and W1 < 0. This result is related to the sign of W1 and will be useful for the rest of the analysis. Assumption 3, which assumes that the real sector leverage is higher than a certain threshold, is useful here. It guarantees
δ1 µ1
α0 (1 − vB m∗ ) − 1 > 0 so that the distinction between
the two cases in lemma 1 is meaningful. Without the assumption 3,
δ1 µ1
α0 (1−vB m∗ )−1 < 0
becomes possible, in which case, since β1 > 0, the second case of lemma 1 will be precluded. 31 Page 32 of 54
As will be shown shortly, the relative strength of the demand and supply responses to the underlying asset price as described in lemma 1 generate two entirely different results in terms of the stability properties of the model. More specifically, when W1 > 0, or β1 >
δ1 µ1
α0 (1 −
vB m∗ )−1 , the funding cost channel of the asset price determination becomes a destabilizing
W1 < 0, or β1 <
δ1 µ1
ip t
factor that can make the system permanently exploding under certain conditions, while when
α0 (1 − vB m∗ ) − 1 , the funding cost channel becomes a stabilizing
cr
factor.22 Note that in Adrian and Shin (2010a)’s original discussion of procyclical leverage, the implication is that there is a feedback loop between the asset price boom and leverage δ1 µ1
α0 (1 − vB m∗ ) − 1 , which has an
us
growth. To highlight the destabilizing effect of β1 >
implication similar to Adrian and Shin (2010a)’s feedback loop, I use the inequality condition
an
of β1 in defining the procyclical leverage.23
Definition 1 (Procyclical bank leverage). A leverage ratio of the securitized banking sector
α0 (1 − vB m∗ ) − 1 , or W1 > 0, holds.
M
δ1 µ1
is said to be procyclical when β1 >
When the value of β1 is at least as high as in definition 1, we may say that the securitized
d
banking sector is actively managing its balance sheet in response to asset price changes.
te
Overall, the first consequence of the procyclical leverage ratio of the securitized banking sector is the stronger response of the loan supply to changes in asset prices than that of the
Ac ce p
loan demand. Figure 8 illustrates this. By implication, if the securitized banking sector’s leverage is procyclical in the sense of the definition 1, when the asset price rises, the loan supply will increase more than the loan demand, whereas when the asset price falls the loan supply will decrease more than the loan demand. The relative response of the loan demand and supply to a change in the asset price is the opposite if bank leverage is not procyclical. 22
Remember from equation (14) that in addition to the funding cost channel, there is the expectation
channel for determining the asset price. 23 Note that in the repo transaction model presented in section 3, a simple positive relation between the leverage ratio of the repo borrower — usually the securitized banking sector — and the asset price was characterized as procyclical leverage. This is compared with the macroeconomic model in this section where not simply the sign of the relation but also the strength of the relation matters. In contrast, Adrian and Shin (2010a)’s original concept of procyclical leverage was based on the observation that total assets and the leverage ratio grow at the same rate.
32 Page 33 of 54
Figure 8: The first consequence of the procyclical bank leverage: the loan supply curve is steeper than the loan demand curve with respect to the asset price. (a) Bank loan market with procyclical bank (b) Bank loan market without procyclical leverage
d
M
an
us
cr
ip t
bank leverage
Then, what is the ultimate consequence of a loan supply curve that is steeper than the
te
loan demand curve with respect to the asset price, as in figure 8a? To answer this question,
Ac ce p
let us use the expressions for demand and supply of bank loans in (29) to obtain the excess demand for bank loans, Φ, as a function of PR .
Φ = −β0 + α0 (1 − vB m∗ ) − 1 δ0 − β1 µ0 + β1 µ2 rp − W1 PR
(31)
By examining the marginal response of the excess demand for bank loans to the asset price, i.e.,
∂Φ ∂PR
= −W1 , the following lemma is easily yielded.
Lemma 2.
∂Φ ∂PR
< 0 if and only if W1 > 0;
∂Φ ∂PR
> 0 if and only if W1 < 0.
According to lemma 2, when the securitized banking sector’s leverage is not procyclical (W1 < 0), the excess demand for bank loans rises (falls), when the underlying asset price rises (falls), which is normally expected. In contrast, the excess demand moves in the opposite direction when it is procyclical (W1 > 0); that is, if the asset price increase (decrease) when the excess demand for bank loans falls (rises). 33 Page 34 of 54
The result in lemma 2, in turn, has a direct implication for the dynamics of the bank loan interest rate, as the latter is definitely affected by the excess demand for bank loans. The marginal impact of Φ on rl can be obtained from equations (27) and (28) as
Lemma 3.
∂rl ∂PR
< 0 if and only if W1 > 0;
∂rl ∂PR
∂rl ∂PR
=
∂rl ∂Φ , ∂Φ ∂PR
= τ1 > 0.
the following lemma is given.
ip t
Combining this result with lemma 2 and using
∂rl ∂Φ
> 0 if and only if W1 < 0.
cr
Lemma 3 characterizes the PR → rl link in figure 7a, clarifying the main goal of this section. This shows that when the securitized banking sector’s leverage ratio is not procyclical
us
the loan interest rate moves in the same direction with the underlying asset price movement, which is normally expected. In contrast, in the case of procyclical bank leverage, it moves in
an
the opposite direction; i.e., the loan interest rate fluctuates counter–cyclically, rising in the case of asset price depreciation and falling in the case of appreciation.
M
When this result is combined with the rl → PR link described in equation (14), it can be seen that while the funding cost channel operates as a stabilizing factor in the banking system without procyclical bank leverage, it operates as a destabilizing factor in the case of
d
procyclical bank leverage. This is in contrast to the expectation channel. The specifications
te
for the PR → PRe link and the PRe → PR link, described in equations (14) and (15), respectively, ensure that the expectations channel always operates as a stabilizing factor, regardless
Ac ce p
of the cyclicality of bank leverage. Consequently, the stability properties of the securitized banking system with procyclical bank leverage will depend on the strength of the funding cost channel of asset price determination, while for a banking system without procyclical bank leverage, the two channels of asset will always be stable.
5.2
Full dynamics: limit cycle
To understand the full consequence of the procyclical leverage ratio of the securitized banking sector by analyzing the complete dynamics of the model, let us reproduce the model in the two–dimensional system in P˙Re and r˙p . P˙Re = g(PR ) r˙p =
g1 κ2 + (rp − rp∗ )κ1 W2 W3
(32)
34 Page 35 of 54
where from appendix B, we know that PR is solved as U2 + 1 PRe − (2 + β1 2 µ2 τ1 )rp PR = − W2 with24
ip t
W1 = β1 µ1 − δ1 α0 (1 − vB m∗ ) − 1
W3 = W2 + 2 κ2 (1 + β1 µ2 τ1 )
(33)
cr
W2 = −1 + 2 τ1 W1
The Jacobian matrix evaluated at the steady–state, denoted with an upper bar, is ob-
J=
where g¯0 =
∂g ∂PR PR =P¯R
e ∂ P˙R e ∂PR
e ∂ P˙R ∂rp
∂ r˙p e ∂PR
∂ r˙p ∂rp
¯0
¯0
2 (1+β1 µ2 τ1 )g 1 g W − W2 2 = 21 κ2 g¯0 W2 0 − W2 W3 κ1 W3 + 1 g¯ W12 −
an
us
tained as
1 W3
< 0, the sign of which is due to the shape of the function g, i.e., g 0 < 0.
M
These expressions show that the stability of the system depends on the signs of W1 , W2 , and W3 , since all the other parameters are a positive constant. It is already known from lemma 1 and definition 1 that W1 is related to the securitized bank’s leverage behavior. It
d
will be further shown that W2 is related to the strength of the funding cost channel and that
te
W3 concerns repo market behavior. By implication, we can identify conditions — associated with these three aspects — under which the model exhibits limit cycle behavior.
Ac ce p
In studying the limit cycle behavior of the model, I rely on the Hopf bifurcation theorem (Gandolfo, 2010). According to this theorem, a bifurcation occurs when the system shifts from a stable fixed point to a stable cycle as a system parameter gradually changes. The change in the system parameter underpins the shift in the dynamic stability properties of the model. In the model, κ1 will be used as the system parameter, or bifurcation parameter, that is responsible for a cycle. Remember κ1 is the coefficient that captures the destabilizing force in the repo market; see equation (22). The central result of the model is presented in the following proposition. Proposition 4 (Limit cycle). Under the conditions of (i) W2 < 0, (ii) W3 > 0, and (iii) κ1 = κ∗1 = 24
1 g¯0 , W2
the system undergoes a bifurcation. That is, in the case of W2 < 0 and
See appendix B for the expression of U2 , which is omitted here since it is inessential for the following
discussion.
35 Page 36 of 54
W3 > 0, when the bifurcation parameter κ1 passes through its critical value κ∗1 , a limit cycle emerges by way of Hopf bifurcation theorem. Let us first consider condition (iii), which concerns the Hopf bifurcation parameter κ1 .
cr
r˙p = κ1 (rp − rp∗ ) − κ2 P˙R
ip t
For this, equation (22) is reproduced:
Remember that the first term on the right–hand side reflects a destabilizing force character-
us
istic in the repo rate market, and the second term captures the sensitivity of the repo market to changes in the underlying asset price, i.e., collateral value. Condition (iii) requires that
an
for the system to reach the bifurcation point and generate a permanent cycle, the destabilizing force in the repo market has to be neither too small nor too large but be near the neighborhood of a critical value κ∗1 =
1 g¯0 . W2
If κ1 is larger than this, κ1 > κ∗1 , the system may
M
cyclically diverge (unstable focus) under certain conditions; if κ1 is smaller than the critical value, κ1 < κ∗1 , the system may cyclically converge to the steady–state (stable focus).
d
Next, regarding condition (i) of proposition 4, consider the following sign relations.
1. in case W1 > 0,
te
Lemma 4 (Sign of W2 ). Since, by definition from (33), W2 = −1 + 2 τ1 W1 , 1 ; τ1 W1
(b) W2 < 0 if and only if 2 <
1 ; τ1 W1
Ac ce p
(a) W2 > 0 if and only if 2 >
2. in case W1 < 0, it always holds that W2 < 0. Remember that 2 from equation (14) is the marginal response of the price of real assets to the bank loan interest rate, thus reflecting the strength of the funding cost channel. According to lemma 4, when the securitized banking sector’s leverage is procyclical (W1 > 0), on the one hand , the sign of W2 depends on the strength of the funding cost channel, i.e., the latter has to be sufficiently weak, being lower than a certain threshold
1 . τ1 W 1
On the
other hand, W2 is always negative when the bank leverage is not procyclical (W1 < 0). This difference is obvious from the earlier discussion in section 5.1 since in the case of W1 > 0, the funding cost channel is a destabilizing factor, and therefore, if it is too strong, it may make the cycle from condition (iii) permanently exploding. In contrast, in the case of W1 < 0, the 36 Page 37 of 54
funding cost channel is a stabilizing factor, and therefore, its strength does not matter in generating a stable cycle. Finally, using the expression for W3 in (33), it can be seen that condition (ii) of proposition
κ2 > −
ip t
4 imposes, given the other parameters, a certain restriction on κ2 :25 W2 2 (1 + β1 µ2 τ1 )
cr
This condition requires that repo investors are sufficiently sensitive to changes in collateral value; see the behavioral specification of the repo rate reproduced above. This is another
us
mechanism that prevents the system at the bifurcation point due to condition (iii) from diverging, thereby contributing to generating a stable cycle. Condition (ii) does so by enabling
an
drastic changes in the underlying asset price displayed in figure 6 to be sufficiently reflected in the repo rate, thereby making an adjustment in the repo rate.
M
For instance, a sufficiently large κ2 will ensure that in an environment of an ever–falling repo rate — due to the repo rate being below rp∗ — a substantial drop in PR , caused by an expectation adjustment as in figure 6, will change the course of rp by pushing it above
d
rp∗ , which will now force rp to keep rising. If, on the contrary, κ2 is relatively small so that
te
W3 < 0 and, accordingly, the repo investors are not sufficiently sensitive to collateral value,
Ac ce p
then there would be no such mechanism that reverses the destabilizing tendency in the repo market, and thus, the repo will keep rising or keep falling, in which case the stability property of the system will be an unstable node. In sum, proposition 4 states that the securitized banking system will experience persistent boom–bust cycles of asset prices, bank leverage, and liquidity under three conditions. First, the funding cost channel that drives the market for the underlying assets is sufficiently weak. That is, the price of real assets is not much affected by funding costs, i.e., the loan interest rate. Second, a change in the repo rate is sufficiently sensitive to a change in the price of underlying assets, i.e., a change in the collateral value. Third, there exist a certain degree of a destabilizing tendency in repo rate movement. With the signs of W1 , W2 , and W3 , required for the existence of a limit cycle in the case of procyclical bank leverage, and the critical value of the bifurcation point κ∗1 = 25
1 g¯0 , W2
the
Due to condition (i) W2 < 0, the right–hand side is positive.
37 Page 38 of 54
us
cr
ip t
Figure 9: Simulation of the limit cycle of the underlying asset price and repo rate
J=
e ∂ P˙R e ∂PR
e ∂ P˙R ∂rp
∂ r˙p e ∂PR
∂ r˙p ∂rp
=
− +
M
an
Jacobian matrix of the system now obtains definite signs:
− +
These signs clearly reflect the underlying mechanisms for the limit cycle discussed above.
d
While the repo rate is behaving in a destabilizing way, with
te
price expectations operate as a stabilizing mechanism
e ∂ P˙R e ∂PR
∂ r˙p ∂rp
> 0, the dynamics of asset
< 0. Due to the collateral value
Ac ce p
risk, the repo market is susceptible to widely fluctuating asset price expectations, which has thus strengthened the rise in the repo rate when asset price expectations deteriorate; ∂ r˙p e ∂PR
< 0. On the other hand, the asset price appreciation reinforced by a fall in the repo
rate26 cannot continue without bounds, due to the nonlinearity in the asset price expectation dynamics as displayed in figure 6; the decrease in the repo rate, at some point, will ultimately lower the change in the expected asset price even to a negative territory;
e ∂ P˙R ∂rp
> 0.
Figure 9 depicts the simulation of the limit cycle behavior of the full model linearized around the steady–state, using the parameter values listed in table 1. While the 2D system is in PRe and rp , since PR is what is actually observed in the market, the limit cycle has been simulated in PR and rp . The simulation result confirms that starting from some arbitrary initial values for PR and rl near the steady–state, the model eventually generates a limit cycle under the parameter values that satisfies the three 26
This can be seen by verifying the solution for PR that
∂PR ∂r p
< 0 holds.
38 Page 39 of 54
Table 1: Parameter values for the simulation of the limit cycle
α1
δ0
δ1
β0
β1
h∗B
λ∗B
vB
0
1
2
µ0
1.5
0
600
5.4
-5
1
0.1
10
0.1
-5
1.2
2
75
µ1
µ2
κ1
κ2
τ0
τ1
rp∗
g0
g1
g2
g3
g4
2.3
6.66
5
4
1
0.001 5
-35
2
2
100
32
cr
ip t
α0
conditions in proposition 4. The two episodes that are characteristic of the repo market cycle
us
are immediately observable: the one in which the repo rate falls with asset price appreciation, and the other where the repo rate rises with asset price depreciation.
an
Since all the other variables are solved as a function of the two state variables, PRe and rp , which exhibit cyclical behavior, they will also evolve cyclically. Their complete descriptions
M
are illustrated in appendix D. To clarify the intuition, let us use the solution of the model listed in appendix B and examine the partial derivative of each with respect to the two state variables. With the conditions W2 < 0 and W3 > 0, which are required for the existence
d
of a limit cycle, these partial derivatives obtain definite signs that otherwise would have
te
been ambiguous. Therefore, the relations that emerge from this exercise provide important insights regarding the mechanisms underlying the limit cycle. The result is demonstrated in
Ac ce p
table 2.
The first two columns correspond to the signs of the Jacobian matrix. It can be easily verified that the rest of the signs are exactly as expected. A central element that drives these results is the leverage behavior of the securitized banking sector. A rise in the expected asset ∂PR price, through raising the current underlying asset price ( ∂P e > 0) and hence asset–backed
securities price
X ( ∂P e ∂PR
R
> 0), ultimately induces the securitized banks to increase their leverage
X ( ∂λ > 0). This type of procyclical movement of bank leverage shapes the total bank credit ∂P e R
supply to be more responsive to asset prices than the total bank credit demand. As a result, the improvement in the market expectation will increase the bank credit supply more than ∂Φ the demand, resulting in a fall in the excess demand ( ∂P e < 0), which, in turn, will suppress R
the interest rate spread
∂ξ ( ∂P e R
l
∂r < 0) and thus the loan interest rate ( ∂P e < 0). R
These partial derivative results are visualized in figure 10, which reports the simulations 39 Page 40 of 54
Table 2: Comparative dynamic analysis: partial derivatives of the endogenous variables with respect to the two state variables, PRe and rp .
r˙p
PR
PX
λX
Φ
ξ
rl
PRe
–
–
+
+
+
–
–
–
rp
+
+
–
–
–
+
+
+
cr
ip t
P˙Re
(Note: These partial derivatives are obtained by using the solutions for the model listed in appendix
us
B.)
of the securitized banking sector’s leverage λX in relation to some key variables. Panel 10a
an
displays that the underlying asset price PR drives and hence leads the securitized bank’s leverage λX , with the two variables moving in close step. However, the securitized bank’s
M
leverage λX moves in the opposite direction from the funding cost, with the repo rate rp as shown in panel 10b.
The impacts of the procyclicality of the securitized bank’s leverage on the bank loan
d
market are depicted in the lower two panels. A rise in the securitized banking sector’s leverage
te
λX enhances the bank loan market condition by increasing the loan supply more than the loan demand, while the decrease undermines the bank loan market condition by decreasing
Ac ce p
the loan supply more than the loan demand. As a result, the excess demand for bank loans moves in the opposite direction from the securitized bank’s leverage, as depicted in panel 10c. Such changes in the demand and supply conditions are reflected in the changes in the bank lending interest rate, and consequently, the latter also moves in the opposite direction from the securitized banking sector’s leverage, as illustrated in panel 10d. Ultimately, the changes in the lending rate will impart an impact on the asset price in a way that reinforces the cyclical movement.
Considering the main goal of the model analysis in this section of identifying the conditions under which a banking system experiences a permanent boom–bust cycle, it is important to note that the procyclical bank leverage as defined in definition 1 is neither a sufficient nor necessary condition for the existence of a limit cycle. This is obvious from the three conditions in proposition 4. On the one hand, procyclical bank leverage (W1 > 0) itself does 40 Page 41 of 54
Figure 10: The relation of securitized bank leverage with (b) repo rate
(c) excess demand for bank loans
(d) bank loan interest rate
Ac ce p
te
d
M
an
us
cr
ip t
(a) the underlying asset price
not necessarily generate a limit cycle; it alone is not sufficient but requires three additional conditions, i.e., (ii) and (iii) of proposition 4 and 1–(b) of lemma 4; hence, procyclical bank leverage is not a sufficient condition. On the other hand, a limit cycle may emerge even though procyclical bank leverage is absent, i.e., even in the case of W1 < 0, as long as (ii) and (iii) of proposition 4 are satisfied; hence, procyclical bank leverage is not a necessary condition.27 Then, what is the role of procyclical bank leverage in relation to the existence of a limit cycle? A close look at the specific expressions of the solution of the model laid out in appendix B shows that an increase in β1 reinforces the effect of interactions among the endogenous 27
Remember from 1–(a) of lemma 4 that condition (i), W2 < 0, of proposition 4 is automatically satisfied
when W1 < 0.
41 Page 42 of 54
us
cr
ip t
Figure 11: Amplifying effect of procyclical bank leverage
variables, thereby amplifying the ups and downs of the cycle.28 Numerically, this will be
an
reflected in an intensification of the result of the comparative dynamic analysis in table 2. For instance, from the solution for PR , we can easily see that the impact of PRe on PR becomes
M
stronger when β1 is larger, by verifying that the absolute value of β1 rises.
∂PR e ∂PR
= − W12 increases when
d
Figure 11 illustrates the amplifying effect of procyclical bank leverage on asset prices.
te
Note here that procyclical bank leverage is measured by the marginal impact of PR on λX , i.e., β1 µ1 , instead of by that of PX on λX , i.e., β1 . It is clear that the procyclicality of
Ac ce p
securitized bank’s leverage with larger β1 µ1 magnifies the asset price cycle. Finally, what if the securitized banking sector is very small, regardless of whether the sector’s leverage ratio is procyclical, or even entirely absent from the model, as in the traditional commercial banking system? In addressing this question, let us take equity as a proxy for the size of the sector. Since the securitized banking sector’s equity is taken as constant due to Adrian and Shin (2010b)’s notion of ‘sticky equity’ of broker–dealers, all the ∗ balance sheet variables so far have been normalized by EX to be further simplified. As of ∗ now, we aim to examine the consequence of EX being sufficiently small or even zero, and
the normalization has to be undone. 28
This result resonates with the main thesis of the financial accelerator model, where financial frictions
amplify external shocks and thus aggravate the impact on the real economy (Bernanke and Gertler, 1989, e.g.).
42 Page 43 of 54
Accordingly, the expanded identities of LD and LS in (29) are rewritten as LD = δ0 (α0 − 1) + δ1 (α0 − 1)PR
(34)
∗ ∗ ∗ ∗ LS = vB m∗ α0 δ0 + EX β0 + E X β1 µ0 − EX β1 µ2 rp + (vB m∗ α0 δ1 + EX β1 µ1 )PR
are
cr
∂LD = δ1 (α0 − 1) ∂PR ∂LS ∗ = vB m∗ α0 δ1 + EX β1 µ1 ∂PR from which the relation corresponding to lemma 1 can be obtained as
ip t
The marginal responses of the loan demand and loan supply to the underlying asset price
us
(35)
M
an
∂LD ∂LS δ1 < ⇐⇒ β1 > ∗ α0 (1 − vB m∗ ) − 1 ∂PR ∂PR EX µ1 D S ∂L ∂L δ1 α0 (1 − vB m∗ ) − 1 > ⇐⇒ β1 < ∗ ∂PR ∂PR EX µ1
The consequence of the securitized banking sector being negligible is stated in the fol-
d
lowing corollary. ∂LD ∂PR
>
te
∗ is smaller, Corollary 1. When EX
will certainly be the case. As long as
∂LD ∂PR
>
∂LS ∂PR
∗ ≈ 0, is more likely; or when EX
∂LS , ∂PR
it will hold that
∂Φ ∂PR
> 0 and
∂LD ∂PR
∂rl ∂PR
>
∂LS ∂PR
> 0 due
Ac ce p
to lemmas 2 and 3, respectively.
That is, when the securitized banking sector is negligibly small as in the traditional commercial banking system, the loan demand will respond more strongly than the loan supply to a change in asset prices, as illustrated in figure 8b. Consequently, when the asset price increases, there will be an excess demand for loans, which, in turn, will raise the loan interest rate; furthermore, due to the behavioral specification of the asset price in (14), the initial asset price appreciation will eventually be curbed. When the asset price falls, the same logic will bring it back up. In sum, in an economic system with a negligibly small securitized banking sector, any rise or fall of asset prices will be contained because of the response of the loan demand to the change in asset prices being stronger than that of the loan supply. The funding cost channel of asset price determination is stabilizing. 43 Page 44 of 54
6
Conclusion
I have presented a macrodynamic balance sheet model that replicates the high drama of 2008 marked by securitization, housing market trouble, the run on money markets, and so
ip t
forth. The main drivers of the persistent cycles of asset prices, bank leverage, and liquidity are the dynamics of asset price expectations and repo rate behavior. First, while the market
cr
for underlying assets exhibits an inflationary spiral led by self–realizing expectations, asset prices do not explode without bounds but finally collapse as investors at some point reckoned
us
that the current asset price is too high, thereby sharply adjusting their expectations. Second, as the repo transactions are secured by collateral and as the collateral value
an
is directly affected by the underlying asset market, repo investors were typically exposed to counter–party risk and collateral value risk. Consequently, the repo rate was set in a highly unstable way, falling when asset prices appreciated and rising when they depreciated.
M
More importantly, this provided an incentive for securitized banks to actively manage their leverage so that the leverage ratio exhibits procyclical behavior and ultimately so that the
d
asset price cycle is intensified.
te
While the model contains a number of crucial features of the recent financial crisis and replicates them analytically, it simplifies certain other related aspects. First, the real sector
Ac ce p
dynamics have not been dealt with sufficiently. Nonfinancial firms and households demand bank loans to finance their portfolios and hence shape the total demand for bank loans. However, the income generation of the real sector is not considered in the model. Second, while the asset price expectation changes endogenously to the current asset price, a specific degree of the change is exogenous to the model, given by the specific form of function g (see figure 6). This makes the asset price expectation not sufficiently endogenous. One possible way to address these two issues simultaneously is to make the expectation formation a function of not only the current price but also the profitability and cash flow in the real sector. Concerning the specific shape of function g, parameters g3 and g4 are particularly crucial.29 On the one hand , g3 determines PR∗ , which is the threshold of PR beyond which the asset price is perceived by the market as unsustainable. On the other hand, 29
See footnote 11 for the mathematical form of g.
44 Page 45 of 54
g4 shapes the vertical segment of the curve, which determines the speed of the expectation adjustment. To model the connection between the financial cycle and the economic performance of the real sector, it may be desirable to endogenize these two parameters, g3 and g4 , in relation to, for instance, income share, debt–to–income ratio, and corporate profitability.
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Regarding the repo market dynamics, my result suggests that the vulnerability to asset price fluctuations make it a key causal factor regarding the severity of the financial cycle.
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Since repo funds are not supported by government guarantees, investors in this market tend to pull back in times of market stress, either requiring higher returns or even withdrawing
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funds. This raises a question for regulating this segment of financial markets.
Recently, macroprudential regulations such as minimum capital and liquidity require-
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ments have made short–term funding more expensive. As a result, tri–party repo contracts in the U.S. have fallen from their peak of $2,800bn in early 2008 to about $1,600bn in
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October 2014. However, not only is repo borrowing still the largest source of funding for broker–dealers, but the core vulnerability has not gone away. The key problem of repo transactions is that there is no third–party that guarantees that the deal is credible during the
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market stress. Accordingly, the possibility of investors not being able to receive their cash
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back or to sell collateral even at a fire sale price still lingers. Hence, it is highly probable that lenders will panic again and run on repos at the first sign of disaster, thereby bringing
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about another crisis.
A fundamental way to address this problem, other than directly limiting repo activity, would be to extend implicit and explicit public backstops, such as FDIC insurance or low– cost loans from the Fed discount window to the short–term money market. In fact, these measures were what the Fed had adopted temporarily in 2008. In terms of my model, they will lower κ2 , which captures the degree of the repo market’s susceptibility to asset price movements. Consequently, in terms of stability properties of the model, it becomes possible that the system exhibits stable focus, instead of a limit cycle. While the significant role played by repo transactions in the fall–down of two giant investment banks in 2008 has been well recognized, Rosengren, the Boston Fed president, has recently commented “[u]nfortunately that potential for problems has not been fully addressed since the crisis” (Rosengren, 2014). The key result of my model draws attention 45 Page 46 of 54
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to grave implications of this observation.
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References Adrian, T. and Shin, H. S. (2010a). Liquidity and leverage. Journal of Financial Intermediation, 19(3):418–437.
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Adrian, T. and Shin, H. S. (2010b). The Changing Nature of Financial Intermediation and
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the Financial Crisis of 2007–2009. Annual Review of Economics, 2(1):603–618.
Adrian, T. and Shin, H. S. (2014). Procyclical leverage and value-at-risk. Review of Financial
us
Studies, 27(2):373–403.
Bernanke, B. and Gertler, M. (1989). Agency costs, net worth, and business fluctuations.
an
The American Economic Review, 79(1):14–31.
Brunnermeier, M. K. (2009). Deciphering the Liquidity and Credit Crunch 2007-2008. Jour-
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nal of Economic Perspectives, 23(1):77–100.
Chiarella, C., Flaschel, P., Hartmann, F., and Proaño, C. R. (2012). Stock market booms,
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endogenous credit creation and the implications of broad and narrow banking for macroe-
te
conomic stability. Journal of Economic Behavior & Organization, 83(3):410–423.
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Christiano, L., Motto, R., and Rostagno, M. (2010). Financial Factors in Economic Fluctuations. Working Paper Series 1192, European Central Bank. Fender, I. and Scheicher, M. (2008). The ABX : how do the markets price subprime. BIS Quarterly Review, (December 2007):67–81. Gandolfo, G. (2010). Economic Dynamics. Springer, 4th edition. Geanakoplos, J. (2009). The Leverage Cycle. Cowles Foundation Discussion Papers, (1715). Gerali, A., Neri, S., Sessa, L., and Signoretti, F. M. (2010). Credit and Banking in a DSGE Model of the Euro Area. Journal of Money, Credit and Banking, 42:107–141. Gertler, M. and Karadi, P. (2011). A model of unconventional monetary policy. Journal of Monetary Economics, 58(1):17–34.
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Gorton, G. and Metrick, A. (2012). Securitized banking and the run on repo. Journal of Financial Economics, 104(3):425–451. Hartmann, F. and Flaschel, P. (2013). Financial markets, banking and the design of monetary
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policy: A stable baseline scenario. Economies, 2(1):1–19. Krishnamurthy, A., Nagel, S., and Orlov, D. (2014). Sizing Up Repo. The Journal of
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Finance.
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Martin, a., Skeie, D., and Thadden, E.-L. V. (2014). Repo Runs. Review of Financial Studies, 27(4):957–989.
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Nikolaidi, M. (2014). Margins of safety and instability in a macrodynamic model with Minskyan insights. Structural Change and Economic Dynamics, 31:1–16.
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Nuño, G. and Thomas, C. (2017). Bank leverage cycles. American Economic Journal: Macroeconomics, 9(2):32–72.
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Perraudin, W. (2008). Determinants of asset-backed security prices in crisis periods.
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Rosengren, E. (2014). Broker-dealer finance and financial stability. Keynote Remarks: Con-
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ference on the Risks of Wholesale Funding sponsored by the Federal Reserve Banks of Boston and New York.
Ryoo, S. (2010). Long waves and short cycles in a model of endogenous financial fragility. Journal of Economic Behavior & Organization, 74(3):163–186. Ryoo, S. (2013). Bank profitability, leverage and financial instability: a Minsky-Harrod model. Cambridge Journal of Economics, 37(5):1127–1160. Smith, J. (2012). Collateral Value Risk in the Repo Market. Working Paper. Retrived from http://people.stern.nyu.edu/jsmith/RepoRiskPremia.pdf.
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Appendix A
Equations of the model
The model consists of twelve variables and twelve equations — including both behavioral equations (=) and identities (≡). In reproducing them below, all the balance sheet variables
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∗ are normalized by the securitized banking sector’s sticky equity EX . However, the same
notations are used in order to avoid introducing another set of notations. For example, Φ, Φ ∗ . EX
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which originally denotes the excess bank loan demand, here refers to
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LD ≡ (λR − 1)ER LS ≡ m∗ vB λR ER + λX
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Φ ≡ LD − LS λR = α0 + α1 PR
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ER = δ0 + δ1 PR
λX = β0 + β1 PX
(A.1)
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PR = 0 + 1 PRe − 2 rl PX = µ 0 + µ 1 PR − µ 2 r p
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P˙Re = g(PR )
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rl = rp + ξ ξ = τ0 + τ1 Φ
r˙p = κ1 (rp − rp∗ ) − κ2 P˙R
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Appendix B
Solutions of the model
λR = α0
W2 β1 U 1µ1 + 1 µ1 PRe − (2 µ1 − W2 µ2 + β1 2 µ1 µ2 τ1 )rp )
λX = β0 + β1 µ0 −
(B.1)
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g1 κ2 + (rp − rp∗ )κ1 W2 W3
where
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r˙p =
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cr
W2 U2 + 1 PRe − (2 + β1 2 µ2 τ1 )rp PR = − W2 U1 µ1 + 1 µ1 PRe − (2 µ1 − W2 µ2 + β1 2 µ1 µ2 τ1 )rp PX = µ 0 − W2 e + W − (W + β µ )τ rp U τ + W τ P 2 1 2 1 2 1 2 1 1 1 1 R rl = τ0 + W2 τ1 U2 + W1 1 PRe − (W 12 + β1 µ2 )rp ξ = τ0 + W2 e U1 + 1 W1 PR − (2 W1 + β1 µ2 )rp Φ= W2 P˙Re = g(PR ), g 0 < 0
ip t
ER = δ0 −
δ1 U1 + 1 PRe − (2 + β1 2 µ2 τ1 )rp
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W1 = β1 µ1 − δ1 α0 (1 − vB m∗ ) − 1 W2 = −1 + 2 τ1 W1
(B.2)
W3 = W2 + 2 κ2 (1 + β1 µ2 τ1 )
U1 = (1 − α0 + α0 vBm∗ )(δ0 + 0 δ1 − 2 δ1 τ0 ) + β0 + β1 (µ0 + 0 µ1 − 2 µ1 τ0 ) U2 = δ0 + 0 δ1 − 2 δ1 τ0 + 2 (β0 δ1 + β1 δ1 µ0 − β1 δ0 µ1 )τ1
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Appendix C
Proofs
Proof of Proposition 1. Each of the three statements can be proved by examining the relevant partial derivative as follows: ∂rp ∂λ
q (λ−1)2
λ = − 1−θ rL q 0 λ−1 − θ
> 0 due to 0 < θ < 1, q 0 < 0, and λ > 1. The last
ip t
1.
condition reflects a simple fact that the broker–dealer sector (repo borrower) is leveraged. ∂rp ∂θ
=
1 θ2
λ rL q λ−1
− rf < 0 due to 0 ≤ q ≤ 1, rL < rf , and
λ λ−1
≈ 1 with λ being
cr
2.
substantially large, which is supported by the empirical findings in section 2. ∂ 2 rp ∂θ∂λ
=
rL θ2
q (λ−1)2
λ − q 0 λ−1
< 0.
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3.
Proof of Proposition 2. It is enough to examine the sign of the partial derivative of λ∗ ,
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expressed in equation (6), with respect to θ.
∂λ∗ −rH (1 − θ) + (rf − θrH ) = >0 ∂θ q 0 (1 − θ)2 rL ∂q . ∂λ
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where q 0 =
(C.1)
The inequality holds since rH > rf and q 0 < 0.
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Proof of Proposition 4. According to Hopf bifurcation theorem, if the differential equa-
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tions system with a system parameter ρ obtains the Jacobian matrix, which, evaluated at the steady state, has the following properties: i), it possesses a pair of simple complex conjugate
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eigenvalues x(ρ) ± y(ρ)i that become pure imaginary at the critical value ρ∗ of the parameter — i.e. x(ρ∗ ) = 0, while y(ρ∗ ) > 0 — and no other eigenvalues with zero real part exist at the steady state at ρ∗ ; ii)
dx(ρ) dρ ρ=ρ∗
6= 0; then the system has a family of periodic solutions.
Regarding the first property, on the one hand, it needs to be shown that there exists ρ∗ > 0 such that Tr J|ρ=ρ∗ = 0 and det J|ρ=ρ∗ > 0. Trace and determinant of the Jacobian matrix are:
1 g 0 W2 1 1 Tr J = − + κ1 + 1 g¯0 − W2 W3 W2 W3 0 1 κ1 g det J = − W3
κ1 is the system parameter. Its critical value can be obtained from setting Tr J = 0 and solving it for κ1 . The result is κ∗1 =
g 0 1 W2
which is the Hopf bifurcation point. In this g 02 2
way, we have Tr J|κ1 =κ∗1 = 0 automatically. Using κ∗1 generates det J|κ1 =κ∗1 = − W2 W13 . Since 51 Page 52 of 54
W3 > W2 by definition of W3 , the only conditions for det J|κ1 =κ∗1 > 0 to hold are W2 < 0 and W3 > 0. This addresses the first property of Hopf bifurcation theorem. Moreover, this result generates κ∗1 > 0, remembering g 0 < 0, which satisfies the basic assumption of the model that that all the parameters are positive.
W2 W3
∂ Tr J ∂κ1 κ1 =κ∗
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Regarding the second property, on the other hand, it can be easily verified that 6= 0 as long as W2 6= 0 and W3 6= 0.
=
1
cr
Consequently, W2 < 0 and W3 > 0 guarantee that the two conditions of Hopf bifurcation
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theorem are satisfied.
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Cycles of the endogenous variables
(a) expected price of underly(b) underlying asset price
(c) ABS price
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ing asset
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Appendix D
(e) excess demand for bank loan
(f) equity of the real sector
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(d) securitized bank leverage
(g) bank lending interest rate
(i) interest rate spread
(h) repo rate
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S0954-349X(17)30182-0 https://doi.org/doi:10.1016/j.strueco.2018.10.001 STRECO 756
To appear in:
Structural
Received date: Revised date: Accepted date:
16-6-2017 4-10-2018 4-10-2018
Change
and
Economic
Dynamics
Please cite this article as: Hyun Woong Park, Securitized Banking, Procyclical Bank Leverage, and Financial Instability, (2018), https://doi.org/10.1016/j.strueco.2018.10.001 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Highlights
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– This paper provides a theoretical explanation of procyclical bank leverage and examines its macroeconomic consequences within the context of the securitized banking system characterized by loan securitization and collateralized short–term borrowing.
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– From a model of repo transaction, it is shown that a broker–dealer (repo borrower)’s optimal leverage is positively related to asset prices. The main mechanism underlying this relation is a movement of repo rate, which reflects counter–party risks and collateral value risks repo lenders are facing.
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– Based on the above result, I develop a macroeconomic model with a securitized banking system characterized with procyclical bank leverage and identify conditions under which the model exhibits a limit cycle behavior. A vulnerability of the repo market to asset price fluctuations leads to procyclical bank leverage, which aggravates asset price cycles driven by the supply–led credit expansion and collapse.
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– It is also found that as the procyclicality of bank leverage becomes stronger the financial cycles also get more intense and severe.
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Hyun Woong Park∗
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Financial Instability
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Securitized Banking, Procyclical Bank Leverage, and
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Denison University, Department of Economics
Abstract
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This paper provides a theoretical explanation of procyclical leverage of the broker– dealer sector and examines its macroeconomic consequences within the context of the securitized banking system characterized by loan securitization and collateralized
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short–term borrowing. First, from a model of repo transaction, it is shown that a
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broker–dealer (repo borrower)’s optimal leverage is positively related to asset prices. The main mechanism underlying this relation is a movement of repo rate, which reflects
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counter–party risks and collateral value risks repo lenders are facing. Second, based on this result, I develop a macroeconomic model with a securitized banking system characterized by procyclical bank leverage and identify conditions under which the model exhibits a limit cycle behavior. A vulnerability of the repo market to asset price fluctuations leads to procyclical bank leverage, which aggravates asset price cycles driven by the supply–led credit expansion and collapse. It is also found that as the procyclicality of bank leverage becomes stronger the financial cycles also get more intense and severe.
Keywords: securitized banking, procyclical bank leverage, credit supply, financial instability, limit cycle
JEL Classication: E32, E44, E51, G21, G24 ∗
Permanent address: 100 W College St, Granville, OH 43023; [email protected]
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1
Introduction
Adrian and Shin (2010a) have recently made an empirical observation that banking sector leverage is procyclical; a financial intermediary leverages up its balance sheet during upturns
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of cycles marked by asset price appreciation while shedding risk exposure by deleveraging during downturns. As the banking sector’s total balance sheet size implies credit flows
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through the real economy, a direct consequence of procyclical bank leverage is an amplification of an asset price cycle and hence an aggravation of financial instability. Motivated by
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this finding, the literature is now growing in two directions. Research in the first direction provides a theoretical explanation for procyclical bank leverage, and research in the second
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direction examines the macroeconomic consequences.1 The main purpose of this paper is to contribute to this emerging body of literature by addressing both of these issues. Underlying the procyclical pattern of financial intermediaries is the transformation of
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banking to the ‘originate and distribute’ model, which characterizes the run–up period to the 2007-2008 disruption and has decisively aggravated the financial fragility of the system
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(Brunnermeier, 2009, e.g.,). In contrast to the traditional banking model, where bank loan
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creation is financed primarily by FDIC–insured deposits and where the issuing banks hold loans until maturity, in the new intermediation model, banks offload risk via securitization,
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and security broker–dealers finance their portfolios through collateralized short-term borrowing, such as repurchase agreements (repo) or asset–backed commercial paper (ABCP). Gorton and Metrick (2012) label this type of banking business model ‘securitized banking’. While macro–models incorporating the banking sector have been growing in number since the recent financial turmoil, those that distinguish between commercial banks and investment banks are relatively few. As a way to fill this gap, the model presented in this paper explicitly incorporates the securitized banking system. In particular, I focus on the repo as a funding source for the securitized banking system and explore the underlying mechanism of its contribution to the endogenous development of boom–bust cycles of asset price, bank leverage, and liquidity, which have become more severe in the securitized banking system. In repo transactions, collateral replaces govern1
I will discuss this literature in greater detail below.
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ment insurance. As the prices of underlying assets are subject to wide fluctuations, repo creditors are typically exposed to collateral value risk. On the other hand, in most cases, repo maturities are overnight, which creates a liquidity risk, as borrowers have to continuously roll over debt. When asset prices are on the rise, repo investors have little concern
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about such risks and allow broker–dealers, i.e., repo borrowers, to borrow not only at a lower rate but also with a lower haircut,2 , which enables borrowers to gain leverage up further.
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However, in times of stress, investors quickly pull back from the market as the probability of the borrower’s default increases. Both the repo rate and haircut rate increase, which forces
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securitized banks to deleverage sharply.3
This paper reveals the connection between such unstable repo market dynamics and the
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procyclicality of securitized banks’ leverage. For this purpose, I first provide a theoretical explanation for the procyclical leverage of the securitized bank by presenting a one–period
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model of repo transactions. In doing so, I compare, among others, Adrian and Shin (2014)’s contracting model, where banks follow the Value–at–Risk rule and where the main friction is risk–shifting moral hazard. A distinctive aspect of the Adrian and Shin model is that it
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adopts Extreme Value Theory to study extreme and rare events that match the repo market
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disruption. In contrast, the repo model presented in this paper adopts a more generalized framework, where the bank is simply characterized by standard profit–maximizing behavior
model.
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without necessarily invoking the extreme outcome specification, as in the Adrian and Shin
In particular, the model focuses on the repo rate, instead of haircuts, as in Geanakoplos (2009) and Adrian and Shin (2014), as a channel through which an asset price cycle is transmitted to the bank leverage cycle. This reflects Smith (2012)’s finding that for the 2008 crisis episode, the repo rate is a superior measure of market stress in the form of collateral value risk. Accordingly, in the model, the repo rate moves in a way that reflects risks 2
A haircut is the value of equity measured against the market value of an asset that is being used as
collateral; it is the ‘skin in the game’. A haircut is an inverse of the leverage ratio. 3 This is a situation comparable to the tradition bank run. Repo investors’ rejection of rollover is analogous to retail depositors’ cash withdrawal. When herd behavior emerges, each of these can place commercial banks and investment banks, respectively, into a liquidity crisis. This is the run on repo as analyzed in Gorton and Metrick (2012) and Martin et al. (2014).
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faced by risk–neutral repo lenders. That is, when asset prices rise, which strengthens the collateral value, the counter–party risk and collateral value risk of the repo lenders will fall, and consequently, the repo rate will also fall. In this environment, the profit–maximizing borrowers, i.e., security broker–dealers, will prefer a higher leverage ratio. On the contrary,
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in times of market stress when assets serving as collateral experience loss, the repo rate will rise. Consequently, the profit–maximizing repo borrowers must lower the leverage. In this
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respect, the optimal leverage ratio of the securitized bank is shown to be positively related to asset prices.
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By comparison, the model also examines a leveraged investor who issues a more standard uncollateralized debt security instead of the repo. It is found that the riskier the asset of the
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investor, the stronger the correlation of her optimal leverage ratio to the asset price. From this, it follows that the correlation becomes close to zero as long as the asset of the leveraged
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investor is significantly safe, as compared with the case of repo borrowers investing in highly risky assets, such as subprime mortgage–backed securities. To examine the macroeconomic implications of the procyclical bank leverage, I then build
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a macroeconomic model consisting of a nonfinancial sector and a financial intermediary
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sector. The intermediary sector is subdivided into three subsectors, including traditional banks that extend loans to the real economy and two other non–bank financial firms, i.e.,
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broker–dealers that purchase securitized bank loans by issuing repos and asset–managing firms that provide funds to the repo market. The main analytical result of the model derives from the Hopf bifurcation theorem, and accordingly, it is shown that under plausible conditions, the model exhibits limit cycle behavior, generating a persistent cycle of asset prices and bank leverage.
More specifically, the key motivations of the macroeconomic modeling is, first, theoretical that the nonfinancial sector’s leverage behavior shapes the demand for lending, while the financial sector’s leverage behavior shapes the supply of lending since the financial sector borrows in order to lend; second, empirical evidence indicates that while the nonfinancial sector and the commercial sector’s leverage ratios are not procyclical in relation to asset prices, the securitized banking sector’s leverage is procyclical. Hence, I first carefully examine all the items of each sector’s balance sheet in a highly 4 Page 5 of 54
simplified economy and construct balance sheet identities for each sector, from which the total demand for bank loans is derived as a positive function of the nonfinancial sector’s leverage and the total supply of bank loans is derived as a positive function of the securitized bank’s leverage. The central behavioral equations of the model concern specifications of each
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sector’s leverage ratio. The securitized banking sector’s leverage ratio is specified as clearly positively related to the price of its assets (asset–backed securities), whereas the nonfinancial
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sector’s leverage ratio is specified as uncorrelated with the price of its assets (real assets). These specifications are based on the empirical findings in section 2, where the broker–
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dealer sector’s leverage ratio is positively related to asset prices, while the correlations of the household and nonfinancial firm sectors’ leverage ratios with the asset prices are not quite as
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strong over time. They are also supported by the above–discussed theoretical results from the model of repo transactions, which are in line with the empirical findings.
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In this setup, the procyclical leverage ratio of the securitized banking sector is defined as the coefficient that measures the marginal response of the securitized bank’s leverage ratio to a change in the asset price, being sufficiently large that in effect, the total supply of bank loan
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is more responsive than the total demand for bank loan is to asset price fluctuations. This
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definition is compared with the one suggested in Adrian and Shin (2010a), where leverage is said to be procyclical in the sense that the growth rate of total assets is equal to that of the
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leverage ratio.
The analytical definition of procyclical bank leverage presented in this paper allows us to find that underlying the dynamics of leverage cycle and asset price cycle is the supply–led credit expansion and collapse. To be more specific, the securitized banking sector’s leverage ratio being procyclical implies in this paper that in the case of asset price appreciation, securitized banks will sufficiently increase borrowing to purchase more of the securities backed by various kinds of bank loans, which will allow banks to issue more loans. However, since the real sector’s leverage ratio is not procyclical, the asset price increase will not affect the real sector’s borrowing; thus, the demand for bank loans will remain the same. The overall result is an excess supply of bank loans, which lowers the loan interest rate and, in turn, further boosts the asset price rise. In contrast, in the case of asset price depreciation, by the same logic, the supply of bank loans decreases more than the demand for bank loans; hence, 5 Page 6 of 54
excess demand for bank loans, which will tend to raise the loan interest rate, will, in turn, further aggravate the decrease in asset prices. Overall, the securitized banking sector’s leverage ratio being procyclical implies that the response of the loan supply to a change in asset prices is stronger than that of the loan
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demand, which eventually makes the dynamic relation between asset price and interest rate destabilizing, where each explode permanently. It is also shown that when the securitized
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banking sector is negligibly small, as in the traditional commercial banking system, the response of the loan demand to a change in asset prices is stronger than that of the loan
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supply, such that the dynamic relation between the asset price and interest rate is stabilizing. Another interesting analytical result of the model is that the procyclical bank leverage
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amplifies the boom–bust cycle. The larger the value of the coefficient that measures the marginal response of the securitized banking sector’s leverage to asset prices, the more severe
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and more intense the upturns and downturns of the cycles of asset prices and bank leverage will be. In this way, another merit of the analytical definition of procyclical bank leverage adopted in this paper is to allow the quantification of the degree of procyclicality of leverage.
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This paper relates to two strands of the contemporary macro-finance literature. First,
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there are a group of papers that incorporate the financial intermediary sector in the dynamic stochastic general equilibrium model (Christiano et al., 2010, Gerali et al., 2010, Gertler and
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Karadi, 2011, e.g.,). While in these papers leverage constraints which banks face develop endogenously due to various forms of agency problems, they require exogenous shocks to actually generate leverage cycles. For instance, one of the main findings of Nuño and Thomas (2017) — which is one of the few recent papers to explicitly formalize procyclical bank leverage in the DSGE framework — is that cross–sectional volatility shocks, instead of the standard total factor productivity shocks, are responsible for bank leverage fluctuations. Additionally, in most of the models in this literature, the household sector does not borrow but only lends to firms through intermediation, and there is no securitization. In contrast, my model does not require exogenous disturbances to generate a cycle; it is an entirely endogenous phenomenon. In addition, firms and households both borrow and lend via the banking sector, which securitizes loans. Second, there is a small but growing literature that studies bank behavior using the 6 Page 7 of 54
macrodynamic modeling and stock–flow consistent approach (Chiarella et al., 2012, Hartmann and Flaschel, 2013, Nikolaidi, 2014, Ryoo, 2013, e.g.,). A key result commonly emerging from these papers is that the endogenous development of the bank’s financial position tends to destabilize the system. However, in most cases, little attention is paid to the lia-
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bility side of the bank balance sheet. In my model, in contrast, the liability structure of the intermediaries plays an important role in generating the key results of the model.
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To understand the limitations of this paper, it should be noted that in contrast to Gorton and Metrick (2012), who highlight the central role played by the repo market for the 2008
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financial crisis, Krishnamurthy et al. (2014) find that as for the funding source for securitized assets, the repo was much smaller than ABCP or direct holdings of the securitized assets.
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However, this finding does not necessarily contradict the focus of this paper on repos, as the essential aspect of a repo that drives the dynamics of the model is the collateralized short–
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term borrowing, where the stability depends on the rise and fall of the collateral value. The ABCP is another type of collateralized short–term borrowing, and thus, similar dynamics can possibly be obtained even when the repo market is replaced by the ABCP market in the
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model. That said, the model should not be taken as a one–to–one mapping of what actually
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happened in the U.S. during the run–up period to 2008. Rather, it is an analysis of the unstable nature of collateralized short–term borrowing, using the repo market as a specific
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example.
The rest of the paper is organized as follows. Section 2 presents empirical data on sectoral leverage to see in what sense the securitized banking sector’s leverage is procyclical. In section 3, by relying on a model of repo transactions, a theoretical explanation of procyclical bank leverage is provided. Using the result of section 3 as a microfoundation, section 4 develops a macrodynamic model to examine the consequence of procylical bank leverage within a broader macroeconomic context. The model is analyzed in section 5. I conclude the paper in section 6.
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2
Empirical observation on leverage and asset prices
The main empirical finding in Adrian and Shin (2010a) is that in the case of the broker– dealer sector, total assets and leverage ratio — measured by total assets over equity — grow
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at the same rate, while equity is constant. Asset growth is entirely driven by growth of debt, not by growth of equity, indicating procyclical leverage and what they call ‘sticky’
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equity. Figure 1 visualizes the mechanics underlying Adrian and Shin (2010a)’s observation. It shows how balance sheet variables change in response to changes in asset prices in the
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case of a marking–to–market accounting system.
In panel (a), a rise in asset prices initially increases the book value of equity along with
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the total asset size, thereby lowering the leverage ratio. The leverage is counter–cyclical. In comparison, panel (b) illustrates the subsequent case where the consequent financial slack is used to issue more debts, thereby purchasing more assets and hence increasing the exposure
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to risks. Moreover, equity is maintained at its initial level through capital policies such as dividend payments and share buybacks. Consequently, the leverage ratio more than restores
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its initial level. In this case, the leverage is procyclical, and equity is sticky.
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Figure 1: Distinctive leverage management
(A: total assets, D: total debts, E: equity)
Adrian and Shin (2010a) illustrate the relation between total assets and the leverage ratio in growth terms. However, since the change in total asset and leverage ratio is in response to changes in asset prices, I suggest directly examining the relation between the leverage ratio
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and asset prices and defining the cyclicality of the leverage ratio accordingly.4 Furthermore, these variables are examined in level terms, not growth terms, as this makes the modeling in the following sections easier. As a way to motivate this approach, data are collected from Z.1 Financial Accounts of the United States released by the Federal Reserve, and the results
ip t
are reported in the figures below.
Figure 2 plots the asset price–leverage relation for the broker–dealer sector. The asset
cr
prices that matter for broker–dealers may be tradable assets, such as asset–backed securities (ABS) or mortgage–backed securities (MBS). However, since the latter tend to be positively
us
correlated with the prices of underlying assets, the housing price is used in constructing the asset price–leverage ratio in figure 2.5 It is clear that the leverage ratio rises with the asset
an
price. The case for the commercial banking sector is depicted in figure 3. This figure plots the leverage in relation to both the housing price and the stock market index. In either case,
M
while the asset prices change widely, the leverage trend is quite stable despite a few outliers. Figure 4a describes the positive relation between the asset price and leverage for the household sector. However, when compared with the case of securitized banks as shown in
d
figure 4b, we can see that the change in the household sector’s leverage ratio is much less
te
pronounced; indeed, it is almost negligible. The case for nonfinancial corporate businesses is presented in figure 5. The first panel covers the entire period of 1975–2012, and it shows
Ac ce p
that the asset price–leverage ratio is nonlinear; the leverage is procyclical when the stock market index is relatively low and becomes counter–cyclical when the stock market index is relatively high. The turning point occurs at about 1990. The asset price–leverage relation for the period of 1990–2012 is separately plotted in the second panel. 4
It should be noted that the impact of asset price changes on the bank leverage ratio goes through bank
assets. 5 The chain relation running from the underlying asset price through the price of ABS finally to the broker-dealer sector leverage ratio is formalized in the model presented in section 4.
9 Page 10 of 54
us
cr
ip t
Figure 2: Asset prices and leverage of securitized banks (1975-2012)
an
(Source: Z.1 Financial Accounts of the United States released by the Federal Reserve)
Figure 3: Asset prices and leverage of commercial banks (1975–2012) (b) Stock price and leverage
Ac ce p
te
d
M
(a) Housing price and leverage
(Source: Z.1 Financial Accounts of the United States released by the Federal Reserve)
Figure 4: Asset prices and leverage of households (1975–2012) (a) Households
(b) Households and securitized bank
(Source: Z.1 Financial Accounts of the United States released by the Federal Reserve)
10 Page 11 of 54
Figure 5: Asset prices and leverage of nonfinancial firms (1975-2012) (a) 1975-2012
us
cr
ip t
(b) 1990-2012
an
(Source: Z.1 Financial Accounts of the United States released by the Federal Reserve)
A notable observation from these descriptive data is that for the broker–dealer sector, the
M
positive correlation between asset prices and the leverage ratio is quite strong and consistent over time, while for the other sectors, the correlation is either weak or inconsistent. This indicates that the broker–dealer sector borrows more in the case of asset price appreciation
3
te
Ac ce p
this paper.
d
and deleverages when asset prices fall. This is what is implied by ‘procyclical leverage’ in
A model of repo transaction
This section develops a model that explains the procyclical leverage of a broker–dealer, or securitized bank, that finances its asset positions through repo. A repo contract stipulates that a borrower sells security to lenders with an agreement whereby she will repurchase it at a later date. In effect, the difference between the repurchase price and original price constitutes interest, called the repo rate, and the borrower’s security acts as collateral. In the event of the borrower’s default on her obligation, the lender keeps the security and liquidates it to recover her initially lent cash. However, liquidation will be difficult, possible only at a fire-sale price, if the securities market is under stress. Accordingly, the repo rate reflects a credit risk premium over and above the interest rate for insured deposits. In addition, to mitigate such credit risk, repos are often overcollaterized
11 Page 12 of 54
to a degree that is called a haircut. Overall, the haircut and the repo rate are two key variables that reflect the conditions of the repo market. On the one hand, empirical research on the movement of haircuts during the recent financial crisis provides mixed evidence depending on the type of repo market and quality of
ip t
collateral. In the bi–lateral repo market for low–grade collateral, haircuts surged during the financial crisis (Gorton and Metrick, 2012). In the tri–party repo market, however, haircuts
cr
changed very little during the same period (Krishnamurthy et al., 2014). On the other hand, Smith (2012) finds that the repo rate gives a consistent measure of market stress in the form
us
of collateral value risk. This paper adheres to Smith (2012)’s finding, and accordingly, the analysis will focus on the repo rate as reflecting the repo market dynamics. Since a haircut
to be implicitly reflected in the leverage ratio.
an
is an inverse of the leverage ratio, I will take a simplifying approach that considers a haircut
M
Consider a repo contract between a representative borrower and lender, both being risk neutral. The borrower finances her asset position in securities, denoted by A; however, she owns equity E and has repo borrowing D at a repo rate rp > 1, posting the securities as
d
collateral.6 At the repo maturity date when the repo borrower is obliged to repurchase the
te
securities, the rate of return of the security can be high at rH > 1 with probability θ or low at rL < 1 with probability 1 − θ.
Ac ce p
As repo transactions involve counter–party risk, the repo rate includes a risk premium over the risk–free rate rf > 1. Let us further suppose that the repo rate is lower than the high return on the security. Accordingly, the order of magnitude of interest rates is assumed to be as follows.
Assumption 1. rL < 1 < rf < rp < rH Since the expected return of the security is θrH + (1 − θ)rL , the security’s expected
value at the maturity date is θrH + (1 − θ)rL A. Ignoring time preference, the security’s
fundamental value in the current period is θrH + (1 − θ)rL A. Intuitively, a rise in θ, which implies an improvement in the prospect of the economy, has a positive impact on the security price due to the assumption 1. 6
The rate of return and interest rate in this paper all refer to, more precisely, the interest rate factor,
which includes both the principal term and interest term.
12 Page 13 of 54
Suppose, on the one hand, rH is high enough for the repo borrower to honor the contract, i.e., rH A > rp D. In this case, the repo borrower’s net revenue is rH A − rp D. On the other hand, rL < 1 is too low for the repo borrower to carry out the obligation, i.e., rL A < rp D. In this case, the repo borrower defaults and the repo lender keeps the collateral. Accordingly,
ip t
the repo borrower’s net revenue is simply zero. As a consequence, the repo borrower’s expected return is obtained as follows. θ(rH A − rp D) E
cr
rb =
(1)
us
In the event of the default of the repo borrower and the consequent transfer of the collateral ownership to the repo lender, the lender will most probably attempt to sell it to
an
recover the initially invested cash. However, when the default rate is high and the number of repo lenders attempting to liquidate collateral is large, liquidation may take place only
M
at a loss reflecting a fire sale discount. Let us use q to denote the discount coefficient with 0 6 q 6 1.7 For instance, note that the fundamental value of the collateral confiscated by the repo lender from the defaulted repo borrower is rL A; depending on the repo market
d
condition, the collateral may or may not be sold at its fundamental value; that is, the
te
liquidation value of the collateral is rL Aq.
A greater indebtedness of borrowers in the economy makes their financial burden more
Ac ce p
intense, and therefore, the probability of debt default is more likely. Hence, q can be specified as a decreasing function of the borrower’s leverage ratio. Denoting the leverage ratio by λ, we have
q = q(λ),
q0 < 0
(2)
By considering the effect of the liquidation, we can construct a revised expected return of the security as θrH + (1 − θ)rL q, where the low return is discounted by a factor of q. Note that the revised return is compared with the original expected return of the security θrH + (1 − θ)rL . A position in the security is viable only when the revised return of the security is larger than the return on the alternative investment, i.e., rf . In this respect, we adopt the following assumption. Assumption 2. θrH + (1 − θ)rL q > rf 7
Asset price bubbles are ignored in this model; hence, q > 1 is ruled out.
13 Page 14 of 54
Next, the repo lender’s expected return, denoted by rc (superscript c indicating ‘creditor’), is measured as rc =
θrp D + (1 − θ)rL Aq D
(3)
ip t
With probability θ, the repo lender’s gross payoff is rp D, and with probability 1 − θ, it is the collateral’s liquidation value rL Aq. As the repo lender is risk neutral, rc is required to be equal to the alternative risk–free rate rf for the lender to be at least indifferent between the
cr
repo and the riskless asset. The repo rate rp should be determined at a level that guarantees
us
rc = rf , which is a participation constraint of the repo lender. As a consequence, using the definition of the leverage ratio λ = A/E, the equilibrium repo rate is obtained from rc = rf as
rf 1−θ λ L − r q θ θ λ−1
an
rp =
(4)
Proposition 1 summarizes the relations.
M
Equation (4) demonstrates that given rf and rL , the repo rate depends on λ and θ.
Proposition 1. The repo rate responds to a change in the repo borrower’s leverage ratio
te
d
and the collateral value in the following ways. (i) rp rises (falls) as λ increases (decreases).
Ac ce p
(ii) rp rises (falls) as θ decreases (increases). (iii) The impact λ has on rp as described in (i) becomes weaker (stronger) as θ gets larger (smaller).
On the one hand, since borrowers’ leverage λ incurs risks on the part of lenders, its rise will increase the repo rate. This is stated in (i). On the other hand, θ affects the repo rate in two ways. First, it is positively related to the security price that serves as collateral. Consequently, when θ is higher, the repo lender will allow the securitized bank to borrow at a lower rate. This is stated in (ii). Second, while a rise in λ will lead to an increase in rp as in (i), the increase will be smaller when the asset value improves with a rise in θ. This is because the enhancement of collateral value weakens the lender’s counter–party risk. This is stated in (iii).
14 Page 15 of 54
Substituting the equilibrium repo rate obtained in equation (4) and using the definition of the leverage ratio, the repo borrower’s rate of return in equation (1) can be rearranged as follows: rb = θrH λ + (1 − θ)rL λq − rf (λ − 1)
ip t
(5)
Now, the optimal leverage of the repo borrower that maximizes rb is obtained as the
cr
solution to the following optimization problem. max θrH λ + (1 − θ)rL λq − rf (λ − 1) λ
us
From the first–order condition, the optimal leverage is yielded as 1 rf − θrH λ = 0 −q q (1 − θ)rL
!
(6)
an
∗
Due to assumption 2 and q 0 < 0, the optimal leverage being negative is ruled out; hence,
M
λ∗ > 0 always holds.
It can be verified how the optimal leverage changes in relation to a change in the asset price reflected in a change in θ. Proposition 1 states this result. ∂λ∗ ∂θ
=
rf −rH q 0 rL (1−θ)2
> 0 holds.
te
d
Proposition 2. For a repo borrower,
The proof of proposition (2) is immediate due to rH > rf from assumption 1 and q 0 < 0
Ac ce p
from (2). A change in θ, which affects the asset value, imparts an impact on the repo borrower’s leverage behavior in two ways. First, a rise in θ lowers rp , as demonstrated in (i) of proposition 1. The lower borrowing cost makes a debt–financed investment more profitable, leading to a larger optimal leverage ratio. Second, the consequent rise in the leverage ratio pushes the repo rate up, as shown in (ii) of proposition 1. However, this adverse feedback effect of λ on rp becomes weaker as the asset value strengthens, as shown in (iii) of proposition 1; thus, it does not fully offset the initial reduction of rp . Through these two processes, a rise (fall) in θ leads to a fall (rise) in rp , which, in turn, increases (decreases) λ∗ . Proposition 2 shows that the leverage ratio of an investor financed by collateralized short– term borrowing is procyclical. This supports the empirical observation in figure 2, which displays the positive correlation between the asset price and the leverage behavior of the broker–dealer sector, which mainly relies on collateralized short–term borrowings. 15 Page 16 of 54
As a comparison with a repo borrower, let us now consider a leveraged investor who relies on more standard uncollateralized debt, denoted by L, with the interest rate of rl . The uncollateralized debt is backed by the debtor’s creditworthiness instead of collateral; thus, the credit risk of debtor default is directly reflected in the interest rate charged. Hence,
ip t
in the model, the loan interest rate can be considered a positive function of the investor’s indebtedness: f0 > 0
cr
rl = f (λ),
(7)
borrower.
rb =
an
In this setup, the nonrepo borrower’s return is
us
To distinguish the investor from the repo borrower considered so far, let us call her a nonrepo
θ(rH A − rl L) + (1 − θ)(rL A − rl L) E
M
which is simplified to
rb = θλ(rH − rL ) + rL λ − rl (λ − 1)
(8)
(9)
d
Now, the profit–maximizing nonrepo borrower’s optimization problem is as follows.
te
max θλ(rH − rL ) + rL λ − rl (λ − 1) λ
Ac ce p
from which the optimal leverage ratio is obtained as λ∗ =
θ(rH − rL ) + rL − rl +1 f0
(10)
How does the nonrepo borrower’s optimal leverage ratio respond to a change in asset prices? The answer is summarized in the following proposition. Proposition 3. For a leveraged investor who relies on a standard unsecured debt contract in financing her asset position,
∂λ∗ ∂θ
=
rH −rL f0
> 0 holds.
The proof of proposition 3 is immediate owing to rH > rL from assumption 1 and f 0 > 0 from (7). It can be seen that the positive correlation between the leverage ratio and the asset price holds for both the repo borrower and the nonrepo borrower, as shown in propositions 2 and 3, respectively.
16 Page 17 of 54
On the other hand, another interesting result coming from proposition 3 is that
∂λ∗ ∂θ
is a positive function of rH − rL . That is, the larger (smaller) the difference between rH and rL , the stronger (weaker) the response of the optimal leverage ratio to a change in asset prices. Consider the 2008 subprime mortgage crisis, which has demonstrated that
ip t
securitized banks relied on collateralized short–term borrowings, including repos to finance their positions in mortgage–backed securities, which turned out to be highly risky. Using
cr
the model terminology, rH rL due to rL is extremely low. This was possible thanks to the loose regulations. In contrast, investors borrowing from debt markets other than the
us
collateralized short–term debt market were largely prevented from such risky investments because they were faced with different types of regulatory and institutional environments.
rH ≈ rL holds,
∂λ∗ ∂θ
an
In this context, if the nonrepo borrower seeks significantly safe investments such that ≈ 0 will ensue. This result, on the one hand, supports the patterns of
M
commercial banks and households displayed in figures 3 and 4, respectively. The former is flat, whereas the latter exhibits a correlation that is positive but very weak, especially when compared with that of the securitized banking sector.
d
On the other hand, the correlation for nonfinancial firms during more recent periods,
te
as displayed in figure 5b, is negative. This can be understood within the framework of proposition 3 only when f 0 < 0, i.e., when the correlation between the leverage ratio and
Ac ce p
interest rate is negative. This relation is in line with the fact explained in corporate finance that by definition the leverage ratio and cost of capital are negatively correlated because debt is tax–advantaged and hence is cheaper than equity.
4
A macroeconomic model of procyclical bank leverage
In this section, I build a macroeconomic model to examine the consequence of procyclical bank leverage within a macro context. The model is built by first specifying distinctive leverage behaviors of financial and nonfinancial sectors in response to asset price developments; then, from these, I derive the demand and supply of bank loans. Note that the nonfinancial sector’s leverage shapes the demand for credit, while the financial sector’s leverage shapes the supply of credit. Finally, the model describes how the procyclical bank leverage gen17 Page 18 of 54
erates supply–led boom and bust of credit and, consequently, determines the credit market interest rate in a way that amplifies asset price cycles. A central behavioral specification of the model is founded upon the empirical and theoretical findings, which are presented in section 2 and 3, respectively. I find that securitized banks’ leverage has a sufficiently strong
ip t
correlation with asset prices.
The nonfinancial sector includes nonfinancial firms and households that are consolidated
cr
into one sector, which I call the real sector. The financial sector is divided into three subsectors, i.e., commercial banks, investment banks — interchangeably called securitized
us
banks or broker–dealers — and asset–managing firms. Notations for the real sector and the three financial sectors are R, B, X, and N , respectively. Notations for the balance sheet
an
variables are used in the same way as in the previous sections, with A for assets, D for debt, E for equity, and λ for the leverage ratio. The sectors associated with these balance sheet
M
variables will be indicated as a subscript. For instance, debt of the real sector is denoted by DR , and the leverage ratio of the securitized banking sector, by λX .
d
Real sector: firms and households
te
4.1
The real sector’s asset, AR , consists of both real and financial assets. First, there are two
Ac ce p
types of financial assets: deposit accounts, denoted by M B , at the commercial bank, and nondeposit accounts, denoted by M N , at insurance companies, pension funds, mutual funds, etc., where clients’ funds are managed by asset managers. Real assets, denoted by K, include the firms’ various equipment, machines, and the households’ housing. To make things simple, let us use some consolidated price index of the real assets and denote it by PR , which will be a key variable in the model.
The real sector’s assets are financed by a mix of own funds, i.e., equity, ER , and debts, DR . It is assumed that the real sector’s debt exclusively consists of bank loans, denoted by L, i.e., DR = L, and particularly that the bank loans finance the real sector’s positions in the real assets only, but not the financial assets. In sum, the real sector’s balance sheet identity (AR ≡ DR + ER ), as expressed in PR K + M B + M N ≡ L + E R
(11)
18 Page 19 of 54
The first behavioral assumption of the model regards the real sector’s leverage ratio in relation to asset prices. Reflecting the empirical observations in section 2 on the relation between leverage and asset prices for each sector, the leverage ratio of each sector will be
ip t
modeled as a simple linear function of the asset price. For the real sector, λ R = α 0 + α 1 PR
(12)
1 , 1−vB m∗
α1 = 0
us
Assumption 3. α0 >
cr
As for the parameters, α0 and α1 , the following assumption is adopted.
The second part of assumption 3, α1 = 0, reflects figures 4 and 5 and proposition 3 and the
an
discussion that followed regarding nonrepo borrowers’ investment being less risky; that is, the correlation of the real sector leverage to asset prices is close to zero. Since it will be 1 , 1−vB m∗
along with
M
shown later that 0 < vB < 1 and 0 < m∗ < 18 , the first part, α0 >
α1 = 0, implies that the real sector leverage λR is constant at α0 , which is greater than
in section 5.1.
d
some positive threshold. The first part of the assumption will be useful for a later analysis
te
On the other hand, the real sector’s equity value will change, by the accounting definition, in line with asset price fluctuations. Therefore, the real sector’s equity can also be specified
Ac ce p
as a simple linear function of PR as
ER = δ0 + δ1 PR ,
δ0 , δ1 > 0
(13)
The consolidated price of the real asset PR is modeled with regard to two channels, i.e., the expectation channel and funding cost channel. First, when asset prices are expected to rise, investors will increase their demand for assets, while the supply will fall.9 Second, obviously, a funding cost negatively affects the demand for assets, while the supply of assets is not affected significantly. In this setting, PR is an increasing function of its expected future value, denoted by PRe , and a decreasing function of the bank lending interest rate, denoted 8 9
These notations will be introduced later From the perspective of sellers, assets can be sold at a higher price in subsequent periods.
19 Page 20 of 54
by rl .10 PR = 0 + 1 PRe − 2 rl ,
0 , 1 , 2 > 0
(14)
1 and 2 reflect the strength of the expectation channel and the funding cost channel,
ip t
respectively. In modeling the expected asset price PRe , suppose that investors reckon that asset prices cannot rise permanently. There is a certain level of the asset price beyond which the asset
cr
price is perceived by the market as unsustainable. When the asset price is below this thresh-
market sharply adjusts its expectation downward.
us
old, investors expect the future value to rise, but as the asset price reaches this level, the
This nonlinear relation can be formalized by modeling the change in the expected asset
g 0 < 0,
PR ≶ PR∗ ⇔ g ≷ 0
(15)
M
P˙Re = g(PR ),
an
price as a negative nonlinear function of the current asset price as follows.
where the nonlinearity of g is displayed in figure 6.11 PR∗ is the threshold beyond which the
d
asset price is perceived by the market to be unsustainable. When PR is below this level,
te
investors expect it to increase further but at a slower pace as PR becomes higher.12 As PR reaches the threshold, investors expect it to undergo a free fall in the near future.
Ac ce p
This specification of asset price expectations, along with equation (14), excludes a degenerate case where asset prices permanently explode. The dynamics of the asset price expectation channel represent a stabilizing factor of the model. On the other hand, as will be shown in detail below, the contribution of the interest rate channel to the asset price stability depends on the specific type of the banking system. More specifically, the interest 10
An alternative way to motivate equation (14) is to rely on the present valuation asset pricing, where
the interest rate is negatively correlated with the asset price, while the expected asset price has a positive impact on the latter. 11 The nonlinear curve in figure 6 can be approximated by the following function, which is adopted from Ryoo (2010) with a slight change: g(x) = −(g0 + g4 ) +
g1 + g0 1 + e−g2 (x−g3 )
where g0 , g1 , g2 , g3 , g4 > 0. 12 The flat segments of the curve are slightly downward sloping, reflecting g 0 < 0.
20 Page 21 of 54
Figure 6: Changes in the expected asset price as a nonlinear function of the current asset price;
us
cr
ip t
g = P˙Re .
an
rate channel operates as a stabilizing factor in the traditional banking system as well as a destabilizing factor in the securitized banking system. Consequently, in the securitized
M
banking system, the model will avoid a degenerate case of permanent explosion only when
Commercial bank
te
4.2
d
the destabilizing channel, reflected in 2 , is sufficiently small.
The commercial bank’s total asset, AB , includes reserves, denoted by HB , and loans, denoted
Ac ce p
by L. Bank loans are extended to the real sector. Part of such loans are packaged into ABS and unloaded from the bank balance sheet. Let us call this share the securitization ratio and denote by ω with 0 6 ω 6 1. The ABS is purchased only by the securitized banking sector. Therefore, ω is determined by securitized bank’s activity, which will be discussed shortly.13 Since the bank loans finance real assets, such as housing and firms’ factories and machines, securitized loans are backed by the performance of such real assets. Hence, the real assets are underlying assets for ABS.
Commercial banks’ debt, DB , represents FDIC–insured deposits held by the real sector; hence, DB ≡ M B . The commercial banking sector’s balance sheet identity (AB ≡ DB + EB ) can be expressed as L(1 − ω) + HB ≡ M B + EB 13
(16)
The unspecified assumption here is that ω positively affects the profitability of commercial banks, and
the latter is therefore willing to securitize loans to the extent possible.
21 Page 22 of 54
The reserve ratio or liquidity ratio, denoted by hB , is measured as hB ≡
HB . DB
Reflecting the
fact that commercial banks are highly regulated, their liquidity ratio and leverage ratio, λB , are taken as fixed at a policy level, i.e., λB = λ∗B and hB = h∗B .14
Securitized bank
ip t
4.3
cr
Suppose ABS is the only asset of the securitized banking sector. The number of the security and its price are denoted by X and PX , respectively. The securitized bank’s asset position
us
is financed by issuing debts in the repo market, using ABS as collateral. Accordingly, the securitized banks’ debt, DX , consists of repos, denoted by Q. This sector’s equity is taken as ∗ constant at EX , reflecting Adrian and Shin (2010a)’s finding of ‘sticky equity’. The balance
an
sheet identity, (AX ≡ DX + EX ), can be expressed as
∗ PX X ≡ Q + EX
M
(17)
As investment banks operate outside of regulatory control, it is supposed that they do
d
not hold any reserves. Therefore, the total asset size of this sector equals the total volume
te
of outstanding bank loans that are packaged into ABS. Denoting the securitized banking ∗ = ωL. From this, an expression for ω sector’s leverage ratio by λX , it holds that λX EX
Ac ce p
follows as
∗ λX EX ω= L
(18)
which shows that the securitization ratio is determined by the total bank loans, that is, the balance sheet policy of the securitized bank, i.e., its equity and leverage. Intuitively, the rise in securitized banks’ leverage, given their sticky equity, enables commercial banks to securitize more of their existing loans.15 14
The standard supply curve of bank loans is increasing with respect to the lending interest rate. This is
confirmed by proving
∂L ∂r l
> 0. However, in the case of the ‘originate and distribute’ model, bank revenues
are produced mostly out of fees incurred in various stages across the loan securitization process rather than interest earnings. Consequently, the more the loans are securitized, as measured by a higher ω, the less concerned banks will be about the level of the loan interest rate. The standard credit supply curve will 2
∂ L become flatter as the securitization rate rises. This is confirmed by proving ∂ω∂r l < 0. 15 As will be shown in section 4.5, ω has a positive impact on commercial banks’ lending capacity.
22 Page 23 of 54
In parallel with the behavioral specification of the real sector leverage in equation (12), the leverage behavior of the securitized banking sector can be specified as a simple linear function of the ABS price, in support of the empirical observations in section 2. β0 , β1 > 0
(19)
ip t
λX = β0 + β1 PX ,
The key parameter is β1 , the marginal response of the leverage ratio of the securitized
cr
banking sector λX to the ABS price PX . It reflects how securitized banks actively manage their balance sheet in response to changes in asset prices, which are immediately reflected in
us
their equity value through the marked–to–market accounting rule. The central behavioral assumption underpinning the main analytical results of the model concerns the value of β1 .
an
On the one hand, according to the positive correlation between the securitized banking sector’s leverage and the security price evidenced in figure 2, section 2, we should have
M
β1 > 0. In addition, this was the result of the model of repo transaction summarized in proposition 2, section 3. On the other hand, the full macroeconomic model developed in this
d
section requires β1 to be not only positive but also sufficiently large so that the procyclical bank leverage has meaningful macroeconomic consequences; see section 5.1.
te
The next step is to model PX . Determining the value of ABS is not an easy task, as its
Ac ce p
value is ultimately backed by underlying assets as collateral. The initial aim of designing these derivatives so as to spread and mitigate risks through pooling and trenching has made it extremely difficult to reveal the full information on the properties of the underlying assets. This is particularly true in the case of the mortgage–backed securities market during the run–up to 2008, where the major portion were subprime loans (Fender and Scheicher, 2008). I take a simpler and more intuitive approach to specify the price of ABS as an increasing function of the underlying asset value16 and a decreasing function of the repo rate. PX = µ0 + µ1 PR − µ2 rp ,
µ0 , µ1 , µ2 > 0
(20)
The behavioral specification in (20) reflects the recent observation regarding the housing market boom that the economic environment with low rates of interest led to an appreciation 16
This assumption is based on the recent experience of the correlation between housing prices and the
price of mortgage–related securities during the subprime mortgage turmoil.
23 Page 24 of 54
in the underlying assets market and subsequently in the ABS market by drastically increasing the demand for these assets while the asset supply was relatively stable (Perraudin, 2008). The correlation between the leverage ratio and asset price for the securitized banking sector can be examined with either type of asset, i.e., the ABS or the underlying assets.
ip t
The securitized bank’s leverage behavior in response to the ABS price is reflected in β1 , while that in response to the underlying asset price is reflected in β1 µ1 , since
∂λX ∂PR
=
∂λX ∂PX ∂PX ∂PR
cr
from equations (19) and (20). A specific level of these parameters will be discussed more
Asset managing firm
an
4.4
us
rigorously in section 5.1.
An asset managing firm is an unleveraged entity. Such an entity manages the real sector’s nondeposit funds at insurance companies, pension funds, mutual funds, hedge funds, etc.,
M
without any leverage. Therefore, the asset managing sector’s balance sheet identity is AN ≡ EN . These institutional investors’ funds are voluminous much beyond the deposit insurance
d
limit. Thus, they are invested in the repo market, where funds are secured by collateral.
te
The asset managing sector also holds liquid assets. Suppose a share, denoted by hN , of this sector’s total asset, AN , is held in liquid assets, and the rest in reverse–repo, denoted by
Ac ce p
Q.17 Accordingly, the balance sheet identity is expressed as Q + hN AN ≡ EN
(21)
One of the key drivers of the model result regards the repo market, where the securitized banks are debtors and asset managing firms are creditors. In addition, the determination of the repo rate, the key variable in the repo market, is specified as follows. r˙p = κ1 (rp − rp∗ ) − κ2 P˙R ,
κ1 , κ2 > 0
(22)
First, recall proposition 1 (ii), according to which asset price appreciation (depreciation), by enhancing (undermining) collateral value, leads to a reduction (rise) in the repo rate. In line with this result in relation to the collateral value risk, the second term of (22) captures 17
In this model, the run on repo would be reflected in a surge in hN .
24 Page 25 of 54
a change in the price of the underlying asset, which acts as collateral, having a negative impact on a change in the repo rate. In addition, there is a feedback effect from the repo rate back to the price of the underlying asset. According to equation (14), the price of the real asset PR is a negative function of the
ip t
lending interest rate rl . However, rl , in turn, is modeled in section 4.5, with equation (27) as a spread over rp ; hence, rl is a positive function of rp . Overall, a fall (rise) in rp ultimately
cr
leads to a rise (fall) in PR . That is, a lower (higher) repo rate weakens (strengthens) a funding cost burden on the part of the securitized bank, and this will raise (reduce) the
us
demand for assets and hence their prices.
When this feedback effect is taken into consideration in relation to the second term in
an
equation (22), it follows that a fall (rise) in rp will increase (decrease) the asset price, which, in turn, can possibly reduce (raise) rp further, and so on. Both rp and PR can possibly
M
explode in either direction. Such destabilizing dynamics of the repo rate are captured by the first term of equation (22). It is supposed that there is some normal level of the repo rate, denoted by rp∗ . When the repo rate is lower than this threshold, it tends to fall further,
d
whereas when it is higher, it tends to increase further.
te
Such unstable dynamics of the repo rate can be justified from the recent disruption of the repo market — so–called the repo run — especially related to low quality collateral.
Ac ce p
Before the subprime mortgage crisis emerged, the spread in the repo market over the federal funds rate was close to 1% points even when the leverage of borrowers, i.e., broker–dealers, was exceptionally high. However, with the rising mortgage default rate and the collapse of Lehman Brothers, the spread surged and liquidity dried, which pushed up the repo rate even further.
However, the permanent explosion of rp and PR was prevented by the dynamics of PR and PRe , a discussed in section 4.1. According to equation (15), as displayed in figure 6, the market did not expect the asset price to increase forever; when it reached a certain threshold beyond which the asset price was considered as unsustainable, investors abruptly and sharply revised down their expectations about the asset price, which, according to equation (14), then led to a plunge in the asset price. This mechanism acts as a stabilizing factor in the repo rate dynamics. For instance, 25 Page 26 of 54
along the exploding process of decreasing rp and increasing PR , at some point PR will reach the threshold and subsequently will make a huge drop according to the dynamics displayed in figure 6. Then, due to the second term of the repo rate equation (22), this will reverse the decreasing course of rp .
ip t
For this scenario to be effective, the response of the repo rate to the underlying asset price fluctuations should be sufficiently sensitive so as to reverse the course of the destabilizing
characterizing the full dynamics of the model; see section 5.2.
cr
force. That is, κ2 should be sufficiently large. This condition will play an important role in
us
Thus far, the balance sheet description — asset allocation and leverage decision (capital structure) — of the sectors of the model and behavioral equations of the key variables were
an
noted; most importantly, they were the sectoral leverage ratio and asset prices. Remember that the nonfinancial sector’s leverage shapes the demand for credit, while the financial
M
sector’s leverage shapes the supply of credit. Now that each sector’s leverage behavior is specified, we are ready to analyze how the demand and supply of credit is formed and
Bank credit market
te
4.5
d
consequently how interest rates are affected.
Ac ce p
As is assumed, real sector debt consists solely of bank loans, demand, and supply of bank credit, which can be derived from the balance sheet specifications of each sector. In addition, the interest rate and the interest rate spread in the bank credit market can be determined by examining the demand and supply of bank credit. In the model, the real sector’s outstanding debt constitutes the total demand for bank loan, which is denoted by LD ; hence, DR ≡ LD18 Using the balance sheet identity of the real sector, AR ≡ DR + ER , and the definition of leverage ratio, the following can be easily obtained. LD ≡ (λR − 1)ER
(23)
On the other hand, the outstanding loans in the balance sheets of commercial banks shape the total supply of bank credit, which is denoted by LS . This can be derived from the 18
Note that the demand and supply of bank lending are considered in stock terms.
26 Page 27 of 54
balance sheet identity of commercial banks in equation (16) and the definition of its leverage and liquidity ratios as follows: λ∗ B
− h∗B
1−ω
M B
(24)
ip t
LS ≡
λ∗B −1
Equation (24) demonstrates the loan supply as a product between deposit fund M B — which
cr
is determined by the real sector’s portfolio decision — and the bracketed term consisting of the commercial banking sector’s leverage λB , required reserve ratio (or liquidity ratio) hB ,
us
and securitization ratio ω. This shows the determination of the commercial banking sector’s lending capacity.
an
The bracketed term can be taken as a bank loan multiplier, which differs from the conventional money multiplier. The latter is the concept that captures a conversion of central bank money into commercial bank money through the reserve requirement reflected in hB .
M
In contrast, the former describes a conversion of commercial bank money (bank debt) into bank lending (bank assets). In this process, not only the reserve requirement but also secu-
d
ritization and bank leverage behavior, each reflected in ω and λB , are the key mechanisms.
te
Among the variables that affect the bank loan multiplier, i.e., those in the bracketed term, the most interesting for our purpose is ω. Denoting the multiplier by m, we have
∂m ∂ω
> 0.
Ac ce p
That is, securitization enhances commercial banks’ lending capacity, which is the greatest concern of financial innovations. A traditional commercial bank without securitization will have ω = 0.19
Let us introduce a new parameter vB , which denotes a portfolio coefficient of the real sector, reflecting the share of its total asset held in bank deposits. Then, by definition, it holds that vB AR ≡ M B . Using this and the expression of ω in equation (18) and the identity AR ≡ λR ER , equation (24) can be rearranged into ∗ LS = m∗ vB λR ER + λX EX 19
For the other two variables constituting m, we have
∂m ∂h∗ B
< 0 and
(25) ∂m ∂λ∗ B
< 0. The former is as expected,
but the latter requires a brief comment. Since what is taken as given in equation (24) is not equity but liability, i.e., deposits, a higher λ∗B reduces the asset size, and given the required reserve ratio, it thus reduces the loan supply.
27 Page 28 of 54
where m∗ =
λ∗B ∗ λB −1
− h∗B ; note that this differs from the bank loan multiplier m =
1 1−ω
λ∗B ∗ λB −1
−
h∗B . In comparison with equation (24), the above equation shows how the total outstanding bank loans, originally extended by commercial banks, are divided between commercial banks and securitized banks through loan securitization. The first term on the right–hand side
ip t
represents the loans kept in the originators’ loan book, while the second term represents the loans packaged and sold to the securitized banking sector.
cr
Now, using the demand and supply of bank loans derived above, we can define the excess
bank loans by Φ, we have Φ ≡ LD − LS
us
demand for bank loans as the difference between the two. Denoting the excess demand for
(26)
an
Using the excess demand for bank loans, we can determine the interest rate in the bank loan market. Since the bank loan interest rate is a long–term rate, while the repo rate is a
M
short–term rate in the model, let us adopt a simple approach to model rl as determined by the interest rate spread over the repo rate.
(27)
te
d
rl = rp + ξ
where ξ denotes the interest rate spread in the bank loan market.
Ac ce p
The spread is determined by the interaction between the demand and supply forces in the bank loan market. It can be modeled as an increasing function of the excess demand for bank loans.
ξ = τ0 + τ1 Φ,
τ0 , τ1 > 0
(28)
τ0 is the constant equilibrium level of the spread, where the demand and supply of bank loans equilibrates. When the bank loan market is in excess demand, i.e., Φ > 0, the spread will rise above τ0 , and in case of excess supply, it will fall below τ0 . The spread will be at its equilibrium level, τ0 , only when Φ = 0. Introducing the constant level of equilibrium spread can be justified within the context of this paper, which describes a short–run fluctuation without a long–run trend.20 20
A more economically intuitive way to model the spread would be to have ξ˙ = τ2 +τ3 Φ in place of equation
(28). In this case, the model will be reduced to a three–dimensional differential equations system. As will
28 Page 29 of 54
Figure 7: Causal links of the model (a) Dynamics of the underlying asset price PR in relation to its expected future value PRe and to
Summary of the model
M
4.6
an
us
(b) Details of the PR → rl link
cr
ip t
the bank lending interest rate (funding cost) rl
The model consists of twelve endogenous variables and twelve equations, which are repro-
d
duced in appendix A. As the basic aim of the model is to explain a persistent boom–bust
te
cycle of asset prices in the securitized banking system, the price PR of real assets, which is an underlying asset for asset–backed securities, is first modeled as a function of its expected
Ac ce p
future value PRe — expectation channel — and loan interest rate rl — funding cost channel — as in equation (14). Second, there are feedback mechanisms from PR back into PRe and rl , thereby ultimately generating endogenous loops, as depicted in figure 7a. The feedback link of PR → PRe on the one hand is modeled simply as a nonlinear relation as in figure 6. On the other hand, the other feedback link of PR → rl is more complicated. Figure 7b illustrates the key causal relations of this link. The main thrust is the underlying asset price PR that affects the leverage behavior of the real sector λR and that of the securitized banking sector λX , which, in turn, shapes the demand and supply of bank loans LD and LS , respectively, which, ultimately affect the loan rate rl . Section 5.1 discusses this link be seen below, by adopting equation (28), it becomes possible to reduce the model to a two–dimensional differential equation system, which is much more convenient to deal with. My approach can be compared with the existing models of cycles where the bench mark rate, such as the federal funds rate, is taken as a constant.
29 Page 30 of 54
in greater detail. The model’s full dynamics as a combined result of all of the four links in figure 7a are presented in section 5.2. The twelve equations of the model are reduced to two differential equations in PRe and rp , which are two state variables of the model. Once the model is solved,
ip t
all of the twelve variables will be expressed as a function of PRe and rp , which reflect the two channels of the asset price determination, i.e., expectations channel and funding cost
cr
channel. The full solution of the model is presented in appendix B.21 The main analytical result of the model, discussed in section 5.2, regards the conditions under which the system
Model analysis
Analytics of procyclical bank leverage
M
5.1
an
5
us
exhibits limit cycle behavior.
Let us formalize the PR → rl link. It will be shown that this link exhibits two opposite cases
d
depending on the procyclicality of bank leverage. The key mechanism here is that each of
te
the nonfinancial and financial sector’s leverage behavior shapes the demand and supply of bank credit, respectively. To see this, expand the identities of LD and LS in equations (23)
Ac ce p
and (25) by using behavioral equations (12), (13), (19), and (20). LD = δ0 (α0 − 1) + δ1 (α0 − 1)PR
(29)
LS = vB m∗ α0 δ0 + β0 + β1 µ0 − β1 µ2 rp + (vB m∗ α0 δ1 + β1 µ1 )PR
∗ In obtaining the above expressions of LD and LS , EX = 1 is assumed. This is equivalent to ∗ that where all the balance sheet variables are normalized by EX , as indicated in appendix
A. In any case, now the demand and supply of bank loans are expressed as a linear function of PR . The marginal responses of the loan demand and loan supply to the underlying asset price 21
All the mathematics involved in solving and analyzing the model in the rest of the paper are obtained
with the help of Mathematica, as they are somewhat messy to be obtained simply by hand. Mathematica codes will be available directly from the author upon request.
30 Page 31 of 54
are measured as, respectively, ∂LD = δ1 (α0 − 1) ∂PR ∂LS = vB m∗ α0 δ1 + β1 µ1 ∂PR
ip t
(30)
By comparing the two, we can identify the condition under which the demand response
cr
is stronger than the supply response and the condition under which the opposite holds. The condition concerns the value of β1 , i.e., the marginal response of the securitized bank’s
us
leverage to the ABS price. More specifically, if β1 is sufficiently large, the supply response is stronger than the demand response, while if β1 is sufficiently small, the demand response
an
is stronger than the supply response. This result is stated in lemma 1.
Lemma 1. The level of β1 and the relative response of the demand and supply of the bank
M
loan to the asset price are related in the following way:
te
d
∂LS δ1 ∂LD < ⇐⇒ β1 > α0 (1 − vB m∗ ) − 1 ⇐⇒ W1 > 0 ∂PR ∂PR µ1 D S ∂L ∂L δ1 α0 (1 − vB m∗ ) − 1 ⇐⇒ W1 < 0 > ⇐⇒ β1 < ∂PR ∂PR µ1
Ac ce p
where W1 = β1 µ1 − α0 (1 − vB m∗ ) − 1 δ1 . Lemma 1 states that β1 being larger than a certain level,
δ1 µ1
α0 (1 − vB m∗ ) − 1 is an-
alytically equivalent to the loan supply responding more strongly to the underlying asset price than the loan demand does; in contrast, β1 being smaller than the threshold is analytically equivalent to the loan demand responding more strongly to the underlying asset price than the loan supply does. Furthermore, lemma 1 introduces an additional notation
W1 ≡ β1 µ1 − δ1 α0 (1 − vB m∗ ) − 1 and demonstrates an analytical equivalence between the former case and W1 > 0 and that between the latter case and W1 < 0. This result is related to the sign of W1 and will be useful for the rest of the analysis. Assumption 3, which assumes that the real sector leverage is higher than a certain threshold, is useful here. It guarantees
δ1 µ1
α0 (1 − vB m∗ ) − 1 > 0 so that the distinction between
the two cases in lemma 1 is meaningful. Without the assumption 3,
δ1 µ1
α0 (1−vB m∗ )−1 < 0
becomes possible, in which case, since β1 > 0, the second case of lemma 1 will be precluded. 31 Page 32 of 54
As will be shown shortly, the relative strength of the demand and supply responses to the underlying asset price as described in lemma 1 generate two entirely different results in terms of the stability properties of the model. More specifically, when W1 > 0, or β1 >
δ1 µ1
α0 (1 −
vB m∗ )−1 , the funding cost channel of the asset price determination becomes a destabilizing
W1 < 0, or β1 <
δ1 µ1
ip t
factor that can make the system permanently exploding under certain conditions, while when
α0 (1 − vB m∗ ) − 1 , the funding cost channel becomes a stabilizing
cr
factor.22 Note that in Adrian and Shin (2010a)’s original discussion of procyclical leverage, the implication is that there is a feedback loop between the asset price boom and leverage δ1 µ1
α0 (1 − vB m∗ ) − 1 , which has an
us
growth. To highlight the destabilizing effect of β1 >
implication similar to Adrian and Shin (2010a)’s feedback loop, I use the inequality condition
an
of β1 in defining the procyclical leverage.23
Definition 1 (Procyclical bank leverage). A leverage ratio of the securitized banking sector
α0 (1 − vB m∗ ) − 1 , or W1 > 0, holds.
M
δ1 µ1
is said to be procyclical when β1 >
When the value of β1 is at least as high as in definition 1, we may say that the securitized
d
banking sector is actively managing its balance sheet in response to asset price changes.
te
Overall, the first consequence of the procyclical leverage ratio of the securitized banking sector is the stronger response of the loan supply to changes in asset prices than that of the
Ac ce p
loan demand. Figure 8 illustrates this. By implication, if the securitized banking sector’s leverage is procyclical in the sense of the definition 1, when the asset price rises, the loan supply will increase more than the loan demand, whereas when the asset price falls the loan supply will decrease more than the loan demand. The relative response of the loan demand and supply to a change in the asset price is the opposite if bank leverage is not procyclical. 22
Remember from equation (14) that in addition to the funding cost channel, there is the expectation
channel for determining the asset price. 23 Note that in the repo transaction model presented in section 3, a simple positive relation between the leverage ratio of the repo borrower — usually the securitized banking sector — and the asset price was characterized as procyclical leverage. This is compared with the macroeconomic model in this section where not simply the sign of the relation but also the strength of the relation matters. In contrast, Adrian and Shin (2010a)’s original concept of procyclical leverage was based on the observation that total assets and the leverage ratio grow at the same rate.
32 Page 33 of 54
Figure 8: The first consequence of the procyclical bank leverage: the loan supply curve is steeper than the loan demand curve with respect to the asset price. (a) Bank loan market with procyclical bank (b) Bank loan market without procyclical leverage
d
M
an
us
cr
ip t
bank leverage
Then, what is the ultimate consequence of a loan supply curve that is steeper than the
te
loan demand curve with respect to the asset price, as in figure 8a? To answer this question,
Ac ce p
let us use the expressions for demand and supply of bank loans in (29) to obtain the excess demand for bank loans, Φ, as a function of PR .
Φ = −β0 + α0 (1 − vB m∗ ) − 1 δ0 − β1 µ0 + β1 µ2 rp − W1 PR
(31)
By examining the marginal response of the excess demand for bank loans to the asset price, i.e.,
∂Φ ∂PR
= −W1 , the following lemma is easily yielded.
Lemma 2.
∂Φ ∂PR
< 0 if and only if W1 > 0;
∂Φ ∂PR
> 0 if and only if W1 < 0.
According to lemma 2, when the securitized banking sector’s leverage is not procyclical (W1 < 0), the excess demand for bank loans rises (falls), when the underlying asset price rises (falls), which is normally expected. In contrast, the excess demand moves in the opposite direction when it is procyclical (W1 > 0); that is, if the asset price increase (decrease) when the excess demand for bank loans falls (rises). 33 Page 34 of 54
The result in lemma 2, in turn, has a direct implication for the dynamics of the bank loan interest rate, as the latter is definitely affected by the excess demand for bank loans. The marginal impact of Φ on rl can be obtained from equations (27) and (28) as
Lemma 3.
∂rl ∂PR
< 0 if and only if W1 > 0;
∂rl ∂PR
∂rl ∂PR
=
∂rl ∂Φ , ∂Φ ∂PR
= τ1 > 0.
the following lemma is given.
ip t
Combining this result with lemma 2 and using
∂rl ∂Φ
> 0 if and only if W1 < 0.
cr
Lemma 3 characterizes the PR → rl link in figure 7a, clarifying the main goal of this section. This shows that when the securitized banking sector’s leverage ratio is not procyclical
us
the loan interest rate moves in the same direction with the underlying asset price movement, which is normally expected. In contrast, in the case of procyclical bank leverage, it moves in
an
the opposite direction; i.e., the loan interest rate fluctuates counter–cyclically, rising in the case of asset price depreciation and falling in the case of appreciation.
M
When this result is combined with the rl → PR link described in equation (14), it can be seen that while the funding cost channel operates as a stabilizing factor in the banking system without procyclical bank leverage, it operates as a destabilizing factor in the case of
d
procyclical bank leverage. This is in contrast to the expectation channel. The specifications
te
for the PR → PRe link and the PRe → PR link, described in equations (14) and (15), respectively, ensure that the expectations channel always operates as a stabilizing factor, regardless
Ac ce p
of the cyclicality of bank leverage. Consequently, the stability properties of the securitized banking system with procyclical bank leverage will depend on the strength of the funding cost channel of asset price determination, while for a banking system without procyclical bank leverage, the two channels of asset will always be stable.
5.2
Full dynamics: limit cycle
To understand the full consequence of the procyclical leverage ratio of the securitized banking sector by analyzing the complete dynamics of the model, let us reproduce the model in the two–dimensional system in P˙Re and r˙p . P˙Re = g(PR ) r˙p =
g1 κ2 + (rp − rp∗ )κ1 W2 W3
(32)
34 Page 35 of 54
where from appendix B, we know that PR is solved as U2 + 1 PRe − (2 + β1 2 µ2 τ1 )rp PR = − W2 with24
ip t
W1 = β1 µ1 − δ1 α0 (1 − vB m∗ ) − 1
W3 = W2 + 2 κ2 (1 + β1 µ2 τ1 )
(33)
cr
W2 = −1 + 2 τ1 W1
The Jacobian matrix evaluated at the steady–state, denoted with an upper bar, is ob-
J=
where g¯0 =
∂g ∂PR PR =P¯R
e ∂ P˙R e ∂PR
e ∂ P˙R ∂rp
∂ r˙p e ∂PR
∂ r˙p ∂rp
¯0
¯0
2 (1+β1 µ2 τ1 )g 1 g W − W2 2 = 21 κ2 g¯0 W2 0 − W2 W3 κ1 W3 + 1 g¯ W12 −
an
us
tained as
1 W3
< 0, the sign of which is due to the shape of the function g, i.e., g 0 < 0.
M
These expressions show that the stability of the system depends on the signs of W1 , W2 , and W3 , since all the other parameters are a positive constant. It is already known from lemma 1 and definition 1 that W1 is related to the securitized bank’s leverage behavior. It
d
will be further shown that W2 is related to the strength of the funding cost channel and that
te
W3 concerns repo market behavior. By implication, we can identify conditions — associated with these three aspects — under which the model exhibits limit cycle behavior.
Ac ce p
In studying the limit cycle behavior of the model, I rely on the Hopf bifurcation theorem (Gandolfo, 2010). According to this theorem, a bifurcation occurs when the system shifts from a stable fixed point to a stable cycle as a system parameter gradually changes. The change in the system parameter underpins the shift in the dynamic stability properties of the model. In the model, κ1 will be used as the system parameter, or bifurcation parameter, that is responsible for a cycle. Remember κ1 is the coefficient that captures the destabilizing force in the repo market; see equation (22). The central result of the model is presented in the following proposition. Proposition 4 (Limit cycle). Under the conditions of (i) W2 < 0, (ii) W3 > 0, and (iii) κ1 = κ∗1 = 24
1 g¯0 , W2
the system undergoes a bifurcation. That is, in the case of W2 < 0 and
See appendix B for the expression of U2 , which is omitted here since it is inessential for the following
discussion.
35 Page 36 of 54
W3 > 0, when the bifurcation parameter κ1 passes through its critical value κ∗1 , a limit cycle emerges by way of Hopf bifurcation theorem. Let us first consider condition (iii), which concerns the Hopf bifurcation parameter κ1 .
cr
r˙p = κ1 (rp − rp∗ ) − κ2 P˙R
ip t
For this, equation (22) is reproduced:
Remember that the first term on the right–hand side reflects a destabilizing force character-
us
istic in the repo rate market, and the second term captures the sensitivity of the repo market to changes in the underlying asset price, i.e., collateral value. Condition (iii) requires that
an
for the system to reach the bifurcation point and generate a permanent cycle, the destabilizing force in the repo market has to be neither too small nor too large but be near the neighborhood of a critical value κ∗1 =
1 g¯0 . W2
If κ1 is larger than this, κ1 > κ∗1 , the system may
M
cyclically diverge (unstable focus) under certain conditions; if κ1 is smaller than the critical value, κ1 < κ∗1 , the system may cyclically converge to the steady–state (stable focus).
d
Next, regarding condition (i) of proposition 4, consider the following sign relations.
1. in case W1 > 0,
te
Lemma 4 (Sign of W2 ). Since, by definition from (33), W2 = −1 + 2 τ1 W1 , 1 ; τ1 W1
(b) W2 < 0 if and only if 2 <
1 ; τ1 W1
Ac ce p
(a) W2 > 0 if and only if 2 >
2. in case W1 < 0, it always holds that W2 < 0. Remember that 2 from equation (14) is the marginal response of the price of real assets to the bank loan interest rate, thus reflecting the strength of the funding cost channel. According to lemma 4, when the securitized banking sector’s leverage is procyclical (W1 > 0), on the one hand , the sign of W2 depends on the strength of the funding cost channel, i.e., the latter has to be sufficiently weak, being lower than a certain threshold
1 . τ1 W 1
On the
other hand, W2 is always negative when the bank leverage is not procyclical (W1 < 0). This difference is obvious from the earlier discussion in section 5.1 since in the case of W1 > 0, the funding cost channel is a destabilizing factor, and therefore, if it is too strong, it may make the cycle from condition (iii) permanently exploding. In contrast, in the case of W1 < 0, the 36 Page 37 of 54
funding cost channel is a stabilizing factor, and therefore, its strength does not matter in generating a stable cycle. Finally, using the expression for W3 in (33), it can be seen that condition (ii) of proposition
κ2 > −
ip t
4 imposes, given the other parameters, a certain restriction on κ2 :25 W2 2 (1 + β1 µ2 τ1 )
cr
This condition requires that repo investors are sufficiently sensitive to changes in collateral value; see the behavioral specification of the repo rate reproduced above. This is another
us
mechanism that prevents the system at the bifurcation point due to condition (iii) from diverging, thereby contributing to generating a stable cycle. Condition (ii) does so by enabling
an
drastic changes in the underlying asset price displayed in figure 6 to be sufficiently reflected in the repo rate, thereby making an adjustment in the repo rate.
M
For instance, a sufficiently large κ2 will ensure that in an environment of an ever–falling repo rate — due to the repo rate being below rp∗ — a substantial drop in PR , caused by an expectation adjustment as in figure 6, will change the course of rp by pushing it above
d
rp∗ , which will now force rp to keep rising. If, on the contrary, κ2 is relatively small so that
te
W3 < 0 and, accordingly, the repo investors are not sufficiently sensitive to collateral value,
Ac ce p
then there would be no such mechanism that reverses the destabilizing tendency in the repo market, and thus, the repo will keep rising or keep falling, in which case the stability property of the system will be an unstable node. In sum, proposition 4 states that the securitized banking system will experience persistent boom–bust cycles of asset prices, bank leverage, and liquidity under three conditions. First, the funding cost channel that drives the market for the underlying assets is sufficiently weak. That is, the price of real assets is not much affected by funding costs, i.e., the loan interest rate. Second, a change in the repo rate is sufficiently sensitive to a change in the price of underlying assets, i.e., a change in the collateral value. Third, there exist a certain degree of a destabilizing tendency in repo rate movement. With the signs of W1 , W2 , and W3 , required for the existence of a limit cycle in the case of procyclical bank leverage, and the critical value of the bifurcation point κ∗1 = 25
1 g¯0 , W2
the
Due to condition (i) W2 < 0, the right–hand side is positive.
37 Page 38 of 54
us
cr
ip t
Figure 9: Simulation of the limit cycle of the underlying asset price and repo rate
J=
e ∂ P˙R e ∂PR
e ∂ P˙R ∂rp
∂ r˙p e ∂PR
∂ r˙p ∂rp
=
− +
M
an
Jacobian matrix of the system now obtains definite signs:
− +
These signs clearly reflect the underlying mechanisms for the limit cycle discussed above.
d
While the repo rate is behaving in a destabilizing way, with
te
price expectations operate as a stabilizing mechanism
e ∂ P˙R e ∂PR
∂ r˙p ∂rp
> 0, the dynamics of asset
< 0. Due to the collateral value
Ac ce p
risk, the repo market is susceptible to widely fluctuating asset price expectations, which has thus strengthened the rise in the repo rate when asset price expectations deteriorate; ∂ r˙p e ∂PR
< 0. On the other hand, the asset price appreciation reinforced by a fall in the repo
rate26 cannot continue without bounds, due to the nonlinearity in the asset price expectation dynamics as displayed in figure 6; the decrease in the repo rate, at some point, will ultimately lower the change in the expected asset price even to a negative territory;
e ∂ P˙R ∂rp
> 0.
Figure 9 depicts the simulation of the limit cycle behavior of the full model linearized around the steady–state, using the parameter values listed in table 1. While the 2D system is in PRe and rp , since PR is what is actually observed in the market, the limit cycle has been simulated in PR and rp . The simulation result confirms that starting from some arbitrary initial values for PR and rl near the steady–state, the model eventually generates a limit cycle under the parameter values that satisfies the three 26
This can be seen by verifying the solution for PR that
∂PR ∂r p
< 0 holds.
38 Page 39 of 54
Table 1: Parameter values for the simulation of the limit cycle
α1
δ0
δ1
β0
β1
h∗B
λ∗B
vB
0
1
2
µ0
1.5
0
600
5.4
-5
1
0.1
10
0.1
-5
1.2
2
75
µ1
µ2
κ1
κ2
τ0
τ1
rp∗
g0
g1
g2
g3
g4
2.3
6.66
5
4
1
0.001 5
-35
2
2
100
32
cr
ip t
α0
conditions in proposition 4. The two episodes that are characteristic of the repo market cycle
us
are immediately observable: the one in which the repo rate falls with asset price appreciation, and the other where the repo rate rises with asset price depreciation.
an
Since all the other variables are solved as a function of the two state variables, PRe and rp , which exhibit cyclical behavior, they will also evolve cyclically. Their complete descriptions
M
are illustrated in appendix D. To clarify the intuition, let us use the solution of the model listed in appendix B and examine the partial derivative of each with respect to the two state variables. With the conditions W2 < 0 and W3 > 0, which are required for the existence
d
of a limit cycle, these partial derivatives obtain definite signs that otherwise would have
te
been ambiguous. Therefore, the relations that emerge from this exercise provide important insights regarding the mechanisms underlying the limit cycle. The result is demonstrated in
Ac ce p
table 2.
The first two columns correspond to the signs of the Jacobian matrix. It can be easily verified that the rest of the signs are exactly as expected. A central element that drives these results is the leverage behavior of the securitized banking sector. A rise in the expected asset ∂PR price, through raising the current underlying asset price ( ∂P e > 0) and hence asset–backed
securities price
X ( ∂P e ∂PR
R
> 0), ultimately induces the securitized banks to increase their leverage
X ( ∂λ > 0). This type of procyclical movement of bank leverage shapes the total bank credit ∂P e R
supply to be more responsive to asset prices than the total bank credit demand. As a result, the improvement in the market expectation will increase the bank credit supply more than ∂Φ the demand, resulting in a fall in the excess demand ( ∂P e < 0), which, in turn, will suppress R
the interest rate spread
∂ξ ( ∂P e R
l
∂r < 0) and thus the loan interest rate ( ∂P e < 0). R
These partial derivative results are visualized in figure 10, which reports the simulations 39 Page 40 of 54
Table 2: Comparative dynamic analysis: partial derivatives of the endogenous variables with respect to the two state variables, PRe and rp .
r˙p
PR
PX
λX
Φ
ξ
rl
PRe
–
–
+
+
+
–
–
–
rp
+
+
–
–
–
+
+
+
cr
ip t
P˙Re
(Note: These partial derivatives are obtained by using the solutions for the model listed in appendix
us
B.)
of the securitized banking sector’s leverage λX in relation to some key variables. Panel 10a
an
displays that the underlying asset price PR drives and hence leads the securitized bank’s leverage λX , with the two variables moving in close step. However, the securitized bank’s
M
leverage λX moves in the opposite direction from the funding cost, with the repo rate rp as shown in panel 10b.
The impacts of the procyclicality of the securitized bank’s leverage on the bank loan
d
market are depicted in the lower two panels. A rise in the securitized banking sector’s leverage
te
λX enhances the bank loan market condition by increasing the loan supply more than the loan demand, while the decrease undermines the bank loan market condition by decreasing
Ac ce p
the loan supply more than the loan demand. As a result, the excess demand for bank loans moves in the opposite direction from the securitized bank’s leverage, as depicted in panel 10c. Such changes in the demand and supply conditions are reflected in the changes in the bank lending interest rate, and consequently, the latter also moves in the opposite direction from the securitized banking sector’s leverage, as illustrated in panel 10d. Ultimately, the changes in the lending rate will impart an impact on the asset price in a way that reinforces the cyclical movement.
Considering the main goal of the model analysis in this section of identifying the conditions under which a banking system experiences a permanent boom–bust cycle, it is important to note that the procyclical bank leverage as defined in definition 1 is neither a sufficient nor necessary condition for the existence of a limit cycle. This is obvious from the three conditions in proposition 4. On the one hand, procyclical bank leverage (W1 > 0) itself does 40 Page 41 of 54
Figure 10: The relation of securitized bank leverage with (b) repo rate
(c) excess demand for bank loans
(d) bank loan interest rate
Ac ce p
te
d
M
an
us
cr
ip t
(a) the underlying asset price
not necessarily generate a limit cycle; it alone is not sufficient but requires three additional conditions, i.e., (ii) and (iii) of proposition 4 and 1–(b) of lemma 4; hence, procyclical bank leverage is not a sufficient condition. On the other hand, a limit cycle may emerge even though procyclical bank leverage is absent, i.e., even in the case of W1 < 0, as long as (ii) and (iii) of proposition 4 are satisfied; hence, procyclical bank leverage is not a necessary condition.27 Then, what is the role of procyclical bank leverage in relation to the existence of a limit cycle? A close look at the specific expressions of the solution of the model laid out in appendix B shows that an increase in β1 reinforces the effect of interactions among the endogenous 27
Remember from 1–(a) of lemma 4 that condition (i), W2 < 0, of proposition 4 is automatically satisfied
when W1 < 0.
41 Page 42 of 54
us
cr
ip t
Figure 11: Amplifying effect of procyclical bank leverage
variables, thereby amplifying the ups and downs of the cycle.28 Numerically, this will be
an
reflected in an intensification of the result of the comparative dynamic analysis in table 2. For instance, from the solution for PR , we can easily see that the impact of PRe on PR becomes
M
stronger when β1 is larger, by verifying that the absolute value of β1 rises.
∂PR e ∂PR
= − W12 increases when
d
Figure 11 illustrates the amplifying effect of procyclical bank leverage on asset prices.
te
Note here that procyclical bank leverage is measured by the marginal impact of PR on λX , i.e., β1 µ1 , instead of by that of PX on λX , i.e., β1 . It is clear that the procyclicality of
Ac ce p
securitized bank’s leverage with larger β1 µ1 magnifies the asset price cycle. Finally, what if the securitized banking sector is very small, regardless of whether the sector’s leverage ratio is procyclical, or even entirely absent from the model, as in the traditional commercial banking system? In addressing this question, let us take equity as a proxy for the size of the sector. Since the securitized banking sector’s equity is taken as constant due to Adrian and Shin (2010b)’s notion of ‘sticky equity’ of broker–dealers, all the ∗ balance sheet variables so far have been normalized by EX to be further simplified. As of ∗ now, we aim to examine the consequence of EX being sufficiently small or even zero, and
the normalization has to be undone. 28
This result resonates with the main thesis of the financial accelerator model, where financial frictions
amplify external shocks and thus aggravate the impact on the real economy (Bernanke and Gertler, 1989, e.g.).
42 Page 43 of 54
Accordingly, the expanded identities of LD and LS in (29) are rewritten as LD = δ0 (α0 − 1) + δ1 (α0 − 1)PR
(34)
∗ ∗ ∗ ∗ LS = vB m∗ α0 δ0 + EX β0 + E X β1 µ0 − EX β1 µ2 rp + (vB m∗ α0 δ1 + EX β1 µ1 )PR
are
cr
∂LD = δ1 (α0 − 1) ∂PR ∂LS ∗ = vB m∗ α0 δ1 + EX β1 µ1 ∂PR from which the relation corresponding to lemma 1 can be obtained as
ip t
The marginal responses of the loan demand and loan supply to the underlying asset price
us
(35)
M
an
∂LD ∂LS δ1 < ⇐⇒ β1 > ∗ α0 (1 − vB m∗ ) − 1 ∂PR ∂PR EX µ1 D S ∂L ∂L δ1 α0 (1 − vB m∗ ) − 1 > ⇐⇒ β1 < ∗ ∂PR ∂PR EX µ1
The consequence of the securitized banking sector being negligible is stated in the fol-
d
lowing corollary. ∂LD ∂PR
>
te
∗ is smaller, Corollary 1. When EX
will certainly be the case. As long as
∂LD ∂PR
>
∂LS ∂PR
∗ ≈ 0, is more likely; or when EX
∂LS , ∂PR
it will hold that
∂Φ ∂PR
> 0 and
∂LD ∂PR
∂rl ∂PR
>
∂LS ∂PR
> 0 due
Ac ce p
to lemmas 2 and 3, respectively.
That is, when the securitized banking sector is negligibly small as in the traditional commercial banking system, the loan demand will respond more strongly than the loan supply to a change in asset prices, as illustrated in figure 8b. Consequently, when the asset price increases, there will be an excess demand for loans, which, in turn, will raise the loan interest rate; furthermore, due to the behavioral specification of the asset price in (14), the initial asset price appreciation will eventually be curbed. When the asset price falls, the same logic will bring it back up. In sum, in an economic system with a negligibly small securitized banking sector, any rise or fall of asset prices will be contained because of the response of the loan demand to the change in asset prices being stronger than that of the loan supply. The funding cost channel of asset price determination is stabilizing. 43 Page 44 of 54
6
Conclusion
I have presented a macrodynamic balance sheet model that replicates the high drama of 2008 marked by securitization, housing market trouble, the run on money markets, and so
ip t
forth. The main drivers of the persistent cycles of asset prices, bank leverage, and liquidity are the dynamics of asset price expectations and repo rate behavior. First, while the market
cr
for underlying assets exhibits an inflationary spiral led by self–realizing expectations, asset prices do not explode without bounds but finally collapse as investors at some point reckoned
us
that the current asset price is too high, thereby sharply adjusting their expectations. Second, as the repo transactions are secured by collateral and as the collateral value
an
is directly affected by the underlying asset market, repo investors were typically exposed to counter–party risk and collateral value risk. Consequently, the repo rate was set in a highly unstable way, falling when asset prices appreciated and rising when they depreciated.
M
More importantly, this provided an incentive for securitized banks to actively manage their leverage so that the leverage ratio exhibits procyclical behavior and ultimately so that the
d
asset price cycle is intensified.
te
While the model contains a number of crucial features of the recent financial crisis and replicates them analytically, it simplifies certain other related aspects. First, the real sector
Ac ce p
dynamics have not been dealt with sufficiently. Nonfinancial firms and households demand bank loans to finance their portfolios and hence shape the total demand for bank loans. However, the income generation of the real sector is not considered in the model. Second, while the asset price expectation changes endogenously to the current asset price, a specific degree of the change is exogenous to the model, given by the specific form of function g (see figure 6). This makes the asset price expectation not sufficiently endogenous. One possible way to address these two issues simultaneously is to make the expectation formation a function of not only the current price but also the profitability and cash flow in the real sector. Concerning the specific shape of function g, parameters g3 and g4 are particularly crucial.29 On the one hand , g3 determines PR∗ , which is the threshold of PR beyond which the asset price is perceived by the market as unsustainable. On the other hand, 29
See footnote 11 for the mathematical form of g.
44 Page 45 of 54
g4 shapes the vertical segment of the curve, which determines the speed of the expectation adjustment. To model the connection between the financial cycle and the economic performance of the real sector, it may be desirable to endogenize these two parameters, g3 and g4 , in relation to, for instance, income share, debt–to–income ratio, and corporate profitability.
ip t
Regarding the repo market dynamics, my result suggests that the vulnerability to asset price fluctuations make it a key causal factor regarding the severity of the financial cycle.
cr
Since repo funds are not supported by government guarantees, investors in this market tend to pull back in times of market stress, either requiring higher returns or even withdrawing
us
funds. This raises a question for regulating this segment of financial markets.
Recently, macroprudential regulations such as minimum capital and liquidity require-
an
ments have made short–term funding more expensive. As a result, tri–party repo contracts in the U.S. have fallen from their peak of $2,800bn in early 2008 to about $1,600bn in
M
October 2014. However, not only is repo borrowing still the largest source of funding for broker–dealers, but the core vulnerability has not gone away. The key problem of repo transactions is that there is no third–party that guarantees that the deal is credible during the
d
market stress. Accordingly, the possibility of investors not being able to receive their cash
te
back or to sell collateral even at a fire sale price still lingers. Hence, it is highly probable that lenders will panic again and run on repos at the first sign of disaster, thereby bringing
Ac ce p
about another crisis.
A fundamental way to address this problem, other than directly limiting repo activity, would be to extend implicit and explicit public backstops, such as FDIC insurance or low– cost loans from the Fed discount window to the short–term money market. In fact, these measures were what the Fed had adopted temporarily in 2008. In terms of my model, they will lower κ2 , which captures the degree of the repo market’s susceptibility to asset price movements. Consequently, in terms of stability properties of the model, it becomes possible that the system exhibits stable focus, instead of a limit cycle. While the significant role played by repo transactions in the fall–down of two giant investment banks in 2008 has been well recognized, Rosengren, the Boston Fed president, has recently commented “[u]nfortunately that potential for problems has not been fully addressed since the crisis” (Rosengren, 2014). The key result of my model draws attention 45 Page 46 of 54
Ac ce p
te
d
M
an
us
cr
ip t
to grave implications of this observation.
46 Page 47 of 54
References Adrian, T. and Shin, H. S. (2010a). Liquidity and leverage. Journal of Financial Intermediation, 19(3):418–437.
ip t
Adrian, T. and Shin, H. S. (2010b). The Changing Nature of Financial Intermediation and
cr
the Financial Crisis of 2007–2009. Annual Review of Economics, 2(1):603–618.
Adrian, T. and Shin, H. S. (2014). Procyclical leverage and value-at-risk. Review of Financial
us
Studies, 27(2):373–403.
Bernanke, B. and Gertler, M. (1989). Agency costs, net worth, and business fluctuations.
an
The American Economic Review, 79(1):14–31.
Brunnermeier, M. K. (2009). Deciphering the Liquidity and Credit Crunch 2007-2008. Jour-
M
nal of Economic Perspectives, 23(1):77–100.
Chiarella, C., Flaschel, P., Hartmann, F., and Proaño, C. R. (2012). Stock market booms,
d
endogenous credit creation and the implications of broad and narrow banking for macroe-
te
conomic stability. Journal of Economic Behavior & Organization, 83(3):410–423.
Ac ce p
Christiano, L., Motto, R., and Rostagno, M. (2010). Financial Factors in Economic Fluctuations. Working Paper Series 1192, European Central Bank. Fender, I. and Scheicher, M. (2008). The ABX : how do the markets price subprime. BIS Quarterly Review, (December 2007):67–81. Gandolfo, G. (2010). Economic Dynamics. Springer, 4th edition. Geanakoplos, J. (2009). The Leverage Cycle. Cowles Foundation Discussion Papers, (1715). Gerali, A., Neri, S., Sessa, L., and Signoretti, F. M. (2010). Credit and Banking in a DSGE Model of the Euro Area. Journal of Money, Credit and Banking, 42:107–141. Gertler, M. and Karadi, P. (2011). A model of unconventional monetary policy. Journal of Monetary Economics, 58(1):17–34.
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Gorton, G. and Metrick, A. (2012). Securitized banking and the run on repo. Journal of Financial Economics, 104(3):425–451. Hartmann, F. and Flaschel, P. (2013). Financial markets, banking and the design of monetary
ip t
policy: A stable baseline scenario. Economies, 2(1):1–19. Krishnamurthy, A., Nagel, S., and Orlov, D. (2014). Sizing Up Repo. The Journal of
cr
Finance.
us
Martin, a., Skeie, D., and Thadden, E.-L. V. (2014). Repo Runs. Review of Financial Studies, 27(4):957–989.
an
Nikolaidi, M. (2014). Margins of safety and instability in a macrodynamic model with Minskyan insights. Structural Change and Economic Dynamics, 31:1–16.
M
Nuño, G. and Thomas, C. (2017). Bank leverage cycles. American Economic Journal: Macroeconomics, 9(2):32–72.
d
Perraudin, W. (2008). Determinants of asset-backed security prices in crisis periods.
te
Rosengren, E. (2014). Broker-dealer finance and financial stability. Keynote Remarks: Con-
Ac ce p
ference on the Risks of Wholesale Funding sponsored by the Federal Reserve Banks of Boston and New York.
Ryoo, S. (2010). Long waves and short cycles in a model of endogenous financial fragility. Journal of Economic Behavior & Organization, 74(3):163–186. Ryoo, S. (2013). Bank profitability, leverage and financial instability: a Minsky-Harrod model. Cambridge Journal of Economics, 37(5):1127–1160. Smith, J. (2012). Collateral Value Risk in the Repo Market. Working Paper. Retrived from http://people.stern.nyu.edu/jsmith/RepoRiskPremia.pdf.
48 Page 49 of 54
Appendix A
Equations of the model
The model consists of twelve variables and twelve equations — including both behavioral equations (=) and identities (≡). In reproducing them below, all the balance sheet variables
ip t
∗ are normalized by the securitized banking sector’s sticky equity EX . However, the same
notations are used in order to avoid introducing another set of notations. For example, Φ, Φ ∗ . EX
cr
which originally denotes the excess bank loan demand, here refers to
us
LD ≡ (λR − 1)ER LS ≡ m∗ vB λR ER + λX
an
Φ ≡ LD − LS λR = α0 + α1 PR
M
ER = δ0 + δ1 PR
λX = β0 + β1 PX
(A.1)
d
PR = 0 + 1 PRe − 2 rl PX = µ 0 + µ 1 PR − µ 2 r p
te
P˙Re = g(PR )
Ac ce p
rl = rp + ξ ξ = τ0 + τ1 Φ
r˙p = κ1 (rp − rp∗ ) − κ2 P˙R
49 Page 50 of 54
Appendix B
Solutions of the model
λR = α0
W2 β1 U 1µ1 + 1 µ1 PRe − (2 µ1 − W2 µ2 + β1 2 µ1 µ2 τ1 )rp )
λX = β0 + β1 µ0 −
(B.1)
d
g1 κ2 + (rp − rp∗ )κ1 W2 W3
where
te
r˙p =
M
an
us
cr
W2 U2 + 1 PRe − (2 + β1 2 µ2 τ1 )rp PR = − W2 U1 µ1 + 1 µ1 PRe − (2 µ1 − W2 µ2 + β1 2 µ1 µ2 τ1 )rp PX = µ 0 − W2 e + W − (W + β µ )τ rp U τ + W τ P 2 1 2 1 2 1 2 1 1 1 1 R rl = τ0 + W2 τ1 U2 + W1 1 PRe − (W 12 + β1 µ2 )rp ξ = τ0 + W2 e U1 + 1 W1 PR − (2 W1 + β1 µ2 )rp Φ= W2 P˙Re = g(PR ), g 0 < 0
ip t
ER = δ0 −
δ1 U1 + 1 PRe − (2 + β1 2 µ2 τ1 )rp
Ac ce p
W1 = β1 µ1 − δ1 α0 (1 − vB m∗ ) − 1 W2 = −1 + 2 τ1 W1
(B.2)
W3 = W2 + 2 κ2 (1 + β1 µ2 τ1 )
U1 = (1 − α0 + α0 vBm∗ )(δ0 + 0 δ1 − 2 δ1 τ0 ) + β0 + β1 (µ0 + 0 µ1 − 2 µ1 τ0 ) U2 = δ0 + 0 δ1 − 2 δ1 τ0 + 2 (β0 δ1 + β1 δ1 µ0 − β1 δ0 µ1 )τ1
50 Page 51 of 54
Appendix C
Proofs
Proof of Proposition 1. Each of the three statements can be proved by examining the relevant partial derivative as follows: ∂rp ∂λ
q (λ−1)2
λ = − 1−θ rL q 0 λ−1 − θ
> 0 due to 0 < θ < 1, q 0 < 0, and λ > 1. The last
ip t
1.
condition reflects a simple fact that the broker–dealer sector (repo borrower) is leveraged. ∂rp ∂θ
=
1 θ2
λ rL q λ−1
− rf < 0 due to 0 ≤ q ≤ 1, rL < rf , and
λ λ−1
≈ 1 with λ being
cr
2.
substantially large, which is supported by the empirical findings in section 2. ∂ 2 rp ∂θ∂λ
=
rL θ2
q (λ−1)2
λ − q 0 λ−1
< 0.
us
3.
Proof of Proposition 2. It is enough to examine the sign of the partial derivative of λ∗ ,
an
expressed in equation (6), with respect to θ.
∂λ∗ −rH (1 − θ) + (rf − θrH ) = >0 ∂θ q 0 (1 − θ)2 rL ∂q . ∂λ
M
where q 0 =
(C.1)
The inequality holds since rH > rf and q 0 < 0.
d
Proof of Proposition 4. According to Hopf bifurcation theorem, if the differential equa-
te
tions system with a system parameter ρ obtains the Jacobian matrix, which, evaluated at the steady state, has the following properties: i), it possesses a pair of simple complex conjugate
Ac ce p
eigenvalues x(ρ) ± y(ρ)i that become pure imaginary at the critical value ρ∗ of the parameter — i.e. x(ρ∗ ) = 0, while y(ρ∗ ) > 0 — and no other eigenvalues with zero real part exist at the steady state at ρ∗ ; ii)
dx(ρ) dρ ρ=ρ∗
6= 0; then the system has a family of periodic solutions.
Regarding the first property, on the one hand, it needs to be shown that there exists ρ∗ > 0 such that Tr J|ρ=ρ∗ = 0 and det J|ρ=ρ∗ > 0. Trace and determinant of the Jacobian matrix are:
1 g 0 W2 1 1 Tr J = − + κ1 + 1 g¯0 − W2 W3 W2 W3 0 1 κ1 g det J = − W3
κ1 is the system parameter. Its critical value can be obtained from setting Tr J = 0 and solving it for κ1 . The result is κ∗1 =
g 0 1 W2
which is the Hopf bifurcation point. In this g 02 2
way, we have Tr J|κ1 =κ∗1 = 0 automatically. Using κ∗1 generates det J|κ1 =κ∗1 = − W2 W13 . Since 51 Page 52 of 54
W3 > W2 by definition of W3 , the only conditions for det J|κ1 =κ∗1 > 0 to hold are W2 < 0 and W3 > 0. This addresses the first property of Hopf bifurcation theorem. Moreover, this result generates κ∗1 > 0, remembering g 0 < 0, which satisfies the basic assumption of the model that that all the parameters are positive.
W2 W3
∂ Tr J ∂κ1 κ1 =κ∗
ip t
Regarding the second property, on the other hand, it can be easily verified that 6= 0 as long as W2 6= 0 and W3 6= 0.
=
1
cr
Consequently, W2 < 0 and W3 > 0 guarantee that the two conditions of Hopf bifurcation
Ac ce p
te
d
M
an
us
theorem are satisfied.
52 Page 53 of 54
Cycles of the endogenous variables
(a) expected price of underly(b) underlying asset price
(c) ABS price
M
an
us
ing asset
cr
ip t
Appendix D
(e) excess demand for bank loan
(f) equity of the real sector
Ac ce p
te
d
(d) securitized bank leverage
(g) bank lending interest rate
(i) interest rate spread
(h) repo rate
53 Page 54 of 54