Financial intermediation in the theory of the risk-free rate

Journal of Banking & Finance 35 (2011) 1663–1668

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Financial intermediation in the theory of the risk-free rate François Marini ⇑ Université Paris-Dauphine, Place du Maréchal de Lattre de Tassigny, 75 775 Paris cedex 16, France

a r t i c l e

i n f o

Article history: Received 15 December 2009 Accepted 13 November 2010 Available online 20 November 2010 JEL classification: D53 G12

a b s t r a c t This paper constructs a general equilibrium model of the interaction between financial intermediaries and financial markets that sheds some light on the short-term volatility of real interest rates. The main findings of the paper are as follows. When financial intermediaries issue contingent (non-contingent) liabilities, an increase in the consumers’ relative risk aversion coefficient decreases (increases) the interest rate. Also, the interest rate rises when capitalists are less risk-averse and financial intermediaries are hit by a liquidity shock. Ó 2010 Elsevier B.V. All rights reserved.

Keywords: Financial intermediation Financial markets Liquidity preference Risk aversion Risk-free rate Risk sharing

1. Introduction In 2008, long-term real interest rates have been highly volatile. Fig. 1 shows the yield on 5-year inflation-adjusted government bonds. At the beginning of 2008, the yield fell to near 0%. Then, in the fall of 2008, the yield has been rising sharply. The dynamics of 10-year and 30-year inflation-indexed government bonds have been identical. The volatility of real interest rates in 2008 is a puzzle. This because one would expect that during a financial crisis there is a flight to quality that drives the prices of safe assets up, and therefore their yields down. Hence, it is not easy to understand the volatility of real interest rates in 2008.1 Investors who regard long-term Treasury bonds as a safe long-term investment must consider the short-term volatility of their yields, especially if they can be hit by liquidity shocks. This paper aims at constructing a theoretical model that sheds some light on this puzzle, and more generally that allows to understand better the short-term volatility of real interest rates. Clearly, we cannot achieve this goal if we do not take into account the role of financial institutions. Consequently, our framework cannot be the standard asset pricing theory, i.e., the ICAPM, for two reasons. First, the ICAPM deals with asset pricing at low frequencies,

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E-mail address: [email protected] On this puzzle, see Campbell et al. (2009).

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whereas we want to deal with asset pricing at high frequencies. Second, as Allen (2001) points out, the ICAPM is based on the assumption that financial intermediaries (FIs hereafter) can be ignored. The standard theory assumes that savers directly invest their savings in financial markets. Financial institutions are a veil. They do not affect asset prices and the allocation of risks and resources. However, Allen and Santomero (1998) and Rajan (2006) show that recent changes in the financial intermediation sector invalidate this assumption. In the USA, despite the growth of equity markets, direct participation of households in financial markets has declined over the period 1969–2005.2 In other countries such as Japan, Germany, the United Kingdom and France, direct holdings are much lower. In other words, the bulk of individual savers do not directly participate in financial markets. Instead, they invest in financial markets through FIs such as mutual funds, insurance companies, and banks.3 In this paper, we construct a model which is in the spirit of Allen and Gale (2004). Our economy comprises a FI, i.e., a coalition of risk-averse consumers who are subject to liquidity shocks, and a capitalist who is not subject to liquidity shocks. The presence of the capitalist is an important departure from the Allen and Gale (2004) framework: in comparison with consumers, he has more

2

See Fig. 3 in Rajan (2006). For an analysis of the determinants of household risky asset holdings in Australia, see Cardak and Wilkins (2009). 3

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F. Marini / Journal of Banking & Finance 35 (2011) 1663–1668

consumers’ relative risk aversion coefficient decreases (increases) the interest rate on the risk-free long asset. Second, an increase in the capitalist’s risk aversion always decreases the risk-free long rate. Third, a liquidity shock increases the risk-free rate. Overall, the model sheds some light on the short-term volatility of real interest rates. The paper is organized as follows. Section 2 describes the environment. Section 3 describes financial markets and FIs. Section 4 derives the general equilibrium of the model. Section 5 presents numerical simulations and analyzes how financial intermediation and the risk-free long rate are articulated. Section 6 concludes.

2. The basic economy

Fig. 1. Yield on 5-year inflation-adjusted government bonds in 2008.

endowments, a broader access to financial markets, and he faces less uncertainty. The capitalist is not subjected to liquidity shocks. Following Allen and Gale (2004), we distinguish between general financial intermediaries (GFIs hereafter) and banks. GFIs are able to issue contracts that are contingent on the aggregate risk, whereas banks issue non-contingent contracts. So GFIs resemble more mutual funds and life insurance companies. This interpretation is also justified by the fact that life insurance companies may be hit by liquidity shocks.4 Table 1 shows that despite the fall in the share of assets managed by banks, they remain the dominant FIs. Nevertheless, the share of mutual funds has risen sharply. Our economy has three assets: a risk-free short asset, a risk-free long asset, and a risky long asset. The FI and the capitalist exchange Arrow–Debreu securities contingent on the aggregate risk. So unlike consumers, the capitalist has the capacity to bear losses without trying to hedge them. In the real world, the representative capitalist can be thought of as individual investors with deep pockets, pension funds, or sovereign wealth funds; i.e., investors that do not provide liquidity insurance and that are willing to receive an expected long return higher than the risk-free long rate in exchange for a positive probability of incurring a loss. When financial markets are complete (in our model we have complete markets for hedging the aggregate risk), if we know the prices of the Arrow–Debreu securities, we can price any financial instrument. Hence, our framework allows us to analyze the impact on the risk-free long rate of liquidity shocks, shocks on the consumers’ risk aversion, and shocks on the capitalist’s risk aversion. We are motivated to consider shocks on risk aversion because several empirical studies have shown that during financial crises risk appetite decreases and risk aversion increases.5 Kumar and Persaud (2002), Scheicher (2003), Deutsche Bundesbank (2005), Gai and Vause (2004, 2006), and Coudert and Gex (2006, 2008), show that risk appetite falls during financial crises. The Goldman Sachs risk aversion index measures fluctuations of the Arrow–Pratt coefficient of risk aversion.6 Illing and Aaron (2005) show that between 1996 and 2004, the index displays a high volatility. They also show that risk aversion increased sharply after the Russian debt default in 1998.7 The main findings of the paper are as follows. First, when FIs issue contingent (non-contingent) liabilities, an increase in the 4 For example, the life insurance companies First Executive Corporation, First Capital Holdings Corporation, and Mutual Benefit Life experienced runs in 1990– 1991. 5 Risk appetite is the willingness of investors to bear risk. It depends on the Arrow– Pratt coefficient of risk aversion and the level of risk that investors think they are faced with. 6 See Goldman Sachs (2003). 7 See chart 1 in Illing and Aaron (2005).

There are three periods T = 0, 1, 2. The good is used for consumption and investment. There are N + 1 agents. One agent is not subject to liquidity shocks and is called the capitalist: at T = 0, he knows that he will want to consume only at T = 2. The capitalist maximizes his expected utility at T = 2. His von-Neumann– Morgenstern utility function in state s = H, L is v(cK(s)) = (cK(s))1r/ (1  r), where r is his relative risk aversion coefficient. There are also N agents who are called the consumers and are subject to idiosyncratic preference shocks. Their preferences are given by:

 uðcT Þ ¼

uðc1 Þ with probability k; uðc2 Þ with probability 1  k;

ð1Þ

where u() is a von-Neumann–Morgenstern utility function and ct denotes consumption at period T. The utility function u() is twice continuously differentiable, increasing, strictly concave, and satisfies Inada conditions u0 (0) = 1 and u0 (1) = 0. We assume that u(cT) = (cT)1a/(1  a) with a > 1.8 A consumer who wants to consume at T = 1 is subject to a liquidity shock and is designated a type 1 agent, while a consumer who wants to consume at T = 2 is not subject to a liquidity shock and is designated a type 2 agent. At T = 0, consumers do not know their type. Each consumer learns his type at T = 1, and this information is private. We also assume that there is no uncertainty on the proportion of type 1 agents in the population: at T = 0, it is common knowledge that kN consumers will be early consumers, and (1  k)N consumers will be late consumers. Each agent has an endowment of the good at T = 0 and none at T = 1 and T = 2. Each consumer is endowed with one unit of the good, and the capitalist is endowed with NK units of the good. In other words, he is endowed with K units of the good per consumer. The capitalist can be thought of as a representative agent: it is equivalent to assume that there are N capitalists, each endowed with K units of capital, or one capitalist endowed with NK units of capital. Since agents have an endowment at T = 0 and want to consume at T = 1 or at T = 2, they have to transfer the good from T = 0 to T = 1 and T = 2. They can do this by investing in assets. There are two physical assets: a risk-free short asset and a risky long asset. The short asset is a storage technology: one unit invested at T yields one unit at T + 1. The long asset yields a return after two periods. One unit invested at T = 0 yields a nil return at T = 1, and a random return R(s) at T = 2, where s 2 S is a state of nature which is realized at T = 2. So the investment in the long asset is completely irreversible because it cannot be liquidated at T = 1. We assume that there are only two possible states of nature at T = 2 denoted H and L, with R(H) > R(L) P 1. At T = 0, all agents have a common prior probability density P(s) over the states of nature H 8 Weinbaum (2010) also analyzes how preference heterogeneity affects asset prices.

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F. Marini / Journal of Banking & Finance 35 (2011) 1663–1668 Table 1 Relative shares of total financial intermediary assets in the USA. Source: Mester (2007).

Insurance companies Life insurance Property and casualty Pension funds Private Public Finance companies and ABS issuers Mutual funds Stock and bond Money market GSEs, REITs, and mortgage companies

1960

1970

1980

1990

2000

2003

2005

20.6 3

15 2.7

10.9 3.5

11.5 3.5

9.4 2.5

9.6 2.4

9.4 2.6

3.9 3.7

3.2 4.3

4.3 4.2

4.7 4.1

3.2 4.1

2.7 3.4

2.2 2.6

4.8

4.7

5.1

7.5

12.2

12.9

14

1.1 0

1.3 0

0.8 1.2

6.4 3.8

8.5 6.3

9.2 5.4

10.2 4.6

2.5

5.1

9

14.8

21.2

23.8

22.2

Depository institutions (banks) Commercial banks Savings and loans and mutual savings banks Credit unions

39.1

42.1

39.1

30.2

25.4

23.7

24.6

20.4 0.9

20.4 1.3

20.5 1.5

11.9 1.7

5.3 1.8

5 2

5.5 2

Total

100

100

100

100

100

100

100

Note: Shares are shown in percentage. Totals may not sum to 100% because of rounding.

and L. The uncertainty about the realization of s is resolved at the beginning of T = 2. 3. Financial intermediaries and financial markets 3.1. Financial markets In our model, the economy is populated by N consumers and one capitalist who have different relative risk aversion coefficients. Hence, a financial system increases social welfare in comparison with autarky by allowing risk sharing. Since the realization of the return on the long asset at T = 2 is a public information, a complete set of Arrow–Debreu securities contingent on the states of nature H and L can optimally share the aggregate risk at T = 0. A complete set of Arrow–Debreu securities markets for hedging the aggregate risk is defined as follows. For the state of nature H, there is a security traded at T = 0 that promises one unit of the good at T = 2 if state H is observed, and nothing if state L is observed. Let p2(H) be the price of one unit of the Arrow–Debreu security contingent on state H. This price is the number of units of the good which is needed at T = 0 to buy the promise that one unit of the good will be delivered at T = 2 if, and only if, the state of nature is H. Also, there is a security traded at T = 0 that promises one unit of the good at T = 2 if state L is observed. The price of this security is denoted p2(L). 3.2. The capitalist Since the capitalist derives satisfaction only from a consumption at T = 2, he solves:

max PðHÞv ðcK ðHÞÞ þ PðLÞv ðcK ðLÞÞ;

ð2Þ

s.t:

p2 ðHÞcK ðHÞ þ p2 ðLÞcK ðLÞ ¼ NK;

ð3Þ

PðHÞv ðcK ðHÞÞ þ PðLÞv ðcK ðLÞÞ P PðHÞv ðRðHÞKÞ þ PðLÞv ðRðLÞKÞ; ð4Þ where cK(H) is the consumption at T = 2 in the state of nature H, and cK(L) is the consumption at T = 2 in the state of nature L. Eq. (2) is the expected utility. Eq. (3) is the budget constraint which can be derived as follows. Let zC be the proportion of wealth that the capitalist invests in the long asset, and nC(s) be the quantity of Arrow–Debreu securi-

ties that he buys, with s = H, L. At T = 0, the budget constraint of the capitalist is NK = zC + p2(H)nC(H) + p2(L)nC(L). At T = 2, his consumptions are given by cK(H) = nC(H) + R(H)zC and cK(L) = nC(L) + R(L)zC. Substituting these two budget constraints at T = 2 into the budget constraint at T = 0 yields the intertemporal budget constraint:

NK ¼ zC þ p2 ðHÞ½cK ðHÞ  RðHÞzC  þ p2 ðLÞ½cK ðLÞ  RðLÞzC ; which can be rewritten:

NK ¼ p2 ðHÞcK ðHÞ þ p2 ðLÞcK ðLÞ þ zC  ½p2 ðHÞRðHÞ þ p2 ðLÞRðLÞzC : Since p2(H)R(H) + p2(L)R(L) = 1 in equilibrium, the budget constraint of the capitalist is given by (3). In Eq. (4), the left-hand side is his expected utility when he participates in financial markets, and the right-hand side is his expected utility in autarky. This constraint says that the capitalist participates in financial markets if this participation does not decrease his expected utility. We assume that while the capitalist can trade directly on securities markets, consumers do not have access to financial markets. They can only invest in FIs that have access to financial markets. So consumers have an indirect access to financial markets through FIs. This is the assumption of limited market participation made for example by Diamond (1997) and Allen and Gale (2004). This assumption is motivated by participation costs, i.e., the costs of participation in financial markets. This assumption is crucial for the results. This because a common theme in banking theory is that financial markets conflict with the optimal provision of liquidity insurance by banks. The Jacklin’s (1987) critique of the Diamond–Dybvig model is that financial markets allow to arbitrage the difference between the yield curve of the deposit contract and the technological yield curve. Hence, if consumers can participate directly in financial markets, banks and markets cannot coexist. Diamond (1997) studies an intermediate case between complete participation (i.e., Jacklin, 1987) and no participation (i.e., Diamond and Dybvig, 1983). He shows that if the proportion of consumers who participate directly is high (low), the yield curve of the deposit contract is close to (far from) the technological yield curve. Hence, there is little (a lot of) cross-subsidization between early and late consumers.

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Our assumption is less realistic than the one made by Diamond (1997) because some households nowadays participate directly in financial markets. But it has the advantage to greatly simplify the model. Here, we sacrifice realism for tractability. 3.3. Financial intermediaries Following Allen and Gale (2004), we distinguish between two types of FIs, namely GFIs and banks. GFIs issue complete contracts, i.e., contracts that are contingent on the states of nature H and L. On the contrary, banks issue incomplete contracts, i.e., demand deposits. These contracts promise returns that are not contingent on the state of nature. The incompleteness of contracts is a technological restriction that is simply assumed. It is not derived from first principles. Another interpretation is that FIs issue non-contingent contracts when there is an information asymmetry between depositors and their banks on the realized state of nature H or L, and depositors cannot verify the realized state (because contracts cannot be contingent on variables that are not verifiable). Since the information on agents’ type is asymmetric, Arrow– Debreu securities contingent on agents’ types cannot exist at T = 0. However, a GFI can optimally share the risk of turning out to be type 1. We assume that there is free entry in the intermediation business. Hence, the GFI solves:

max k

ðr 1 Þ1a ðr2H Þ1a ðr 2L Þ1a þ ð1  kÞPðHÞ þ ð1  kÞPðLÞ ; 1a 1a 1a

ð5Þ

s.t.:

y þ zB þ p2 ðHÞnB ðHÞ þ p2 ðLÞnB ðLÞ ¼ 1;

ð6Þ

kr 1 ¼ y; ð1  kÞr 2H ¼ RðHÞzB þ nB ðHÞ;

ð7Þ ð8Þ

ð1  kÞr 2L ¼ RðLÞzB þ nB ðLÞ; 1a

PðHÞ

1a

ð9Þ 1a

ðr 2H Þ ðr 2L Þ ðr 1 Þ þ PðLÞ P ; 1a 1a 1a

ð10Þ

where r1 is the consumption of a type 1 agent at T = 1, r2L is the consumption of a type 2 agent at T = 2 in state L, and r2H is the consumption of a type 2 agent at T = 2 in state H. Eq. (6) is the budget constraint at T = 0, where y is the proportion of endowments invested in the short asset, zB is the proportion invested in the long asset, and nB(s) is the quantity of Arrow– Debreu securities contingent on s = H, L that are bought by the intermediary. Eq. (7) is the budget constraint at T = 1. It says that the GFI meets withdrawals kr1 by early consumers with its investments y in the short asset. Eqs. (8) and (9) are the budget constraints at T = 2 in states s = H, L. They say that the GFI meets withdrawals (1  k)r2S with the return R(s)zB on the long asset and the quantity nB(s) of the contingent good that has been bought at T = 0. Eq. (10) is the incentive-compatibility constraint. The lefthand side is the expected utility of a type 2 agent who does not withdraw at T = 1, while the right-hand side is his expected utility when he withdraws at T = 1. This constraint ensures that type 2 agents will not withdraw at T = 1. The bank solves the same problem, except that it faces the supplementary constraint r2H = r2L. That is, it can only issue a non-contingent contract. 4. The Arrow–Debreu equilibrium Financial markets open at T = 0. On these markets, the FI and the capitalist can buy or sell the two Arrow–Debreu securities contingent on H and L. Financial markets are in equilibrium when two conditions are satisfied.

First, it must be true that:

kr 1 ¼ y;

ð11Þ

ð1  kÞNr2H þ cK ðHÞ ¼ NRðHÞð1 þ K  yÞ;

ð12Þ

ð1  kÞNr2L þ cK ðLÞ ¼ NRðLÞð1 þ K  yÞ;

ð13Þ

cK ðHÞ P 0;

ð14Þ

cK ðLÞ P 0;

ð15Þ

with r2H = r2L when the FI is a bank. In Eq. (11), the left-hand side is the consumption of type 1 agents at T = 1. The right-hand side represents the resources available at T = 1. Eq. (11) says that the consumptions at T = 1 are financed by the investments in the short asset. In Eq. (12), the left-hand side represents the consumptions at T = 2 when the realized state of nature is H. Since the FI pays r2H to type 2 agents and (1  k)N agents are type 2, the consumption of type 2 agents is (1  k)Nr2H. The consumption of the capitalist is cK(H). The righthand side represents the resources available at T = 2 in the state of nature H. They are provided by the investments in the long asset which mature at T = 2. Since the long asset yields R(H) in state of nature H, the resources available at T = 2 in this state are NR(H)(1 + K  y). By the same logic, equilibrium in the state of nature L is given by Eq. (13). Also, Eqs. (14) and (15) say that in equilibrium, the consumption of the capitalist cannot be negative. The second condition is that arbitrage opportunities cannot exist in equilibrium. Therefore:

p2 ðHÞRðHÞ þ p2 ðLÞRðLÞ ¼ 1:

ð16Þ

By investing 1 at T = 0 in the long asset, an individual obtains R(H) in state of nature H or R(L) in state of nature L. Obtaining this returns profile by investing in Arrow–Debreu securities costs p2(H)R(H) + p2(L)R(L). These two costs must be equal. If this is not the case, there is an arbitrage opportunity, and thus financial markets are not in equilibrium. When the incentive-compatibility constraint is not binding, the equilibrium price p2(H) is given by the implicit function:

n

RðLÞð1 þ KÞ  cKNðLÞ h

o

h

ð1kÞ PðLÞRðLÞ k 1p2 ðHÞRðHÞ

 RðHÞð1 þ KÞ þ

i1=a

ð1kÞ PðHÞ k p2 ðHÞ

i1=a

 þ RðHÞ

þ RðLÞ

cK ðHÞ ¼ 0: N

ð17Þ

When the incentive-compatibility constraint is binding, the equilibrium price p2(H) is given by the implicit function:

n

RðLÞð1 þ KÞ  cKNðLÞ

on

ð1kÞ k

ð1kÞ ½PðHÞ ð1aÞ=a k

g

 RðHÞð1 þ KÞ þ

o

g1=a ½PðHÞgð1aÞ=a þ PðLÞ1=ða1Þ þ RðHÞ

cK ðHÞ ¼ 0; N

þ PðLÞ1=ða1Þ þ RðLÞ ð18Þ

2 ðHÞRðHÞÞ where g  PðHÞð1p . p ðHÞPðLÞRðLÞ 2

Since the implicit functions (17) and (18) cannot be transformed into explicit functions, we use numerical simulations to study the model. 5. Numerical simulations The risk-free rate r on a zero-coupon bond that is bought at T = 0 and yields 1 at T = 2 is given by p2(H) + p2(L) = 1/(1 + r). We simulate the model with P(H) = P(L) = 0.5, R(H) = 1.3, R(L) = 1, and K = 0.1. The main findings appear when we compare Figs. 2 and 3. In Fig. 2 (Fig. 3), when a increases, the risk-free rate decreases (increases). Consequently, bond prices increase (decrease).

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Fig. 2. General financial intermediaries: the effect of the depositors’risk aversion on the risk-free long rate.

Fig. 4. General financial intermediaries: the effect of the capitalists’risk aversion on the risk-free long rate.

Fig. 3. Banks: the effect of the depositors’risk aversion on the risk-free long rate.

Fig. 5. Banks: the effect of the capitalits’risk aversion on the risk-free long rate.

This because consumers face two kinds of risks. Firstly, there is the risk of turning out to be a type 1 agent. Secondly, there is the risk of turning out to be a type 2 agent, because type 2 agents bear the aggregate risk. Hence, when consumers are more risk-averse, the GFI provides more insurance against these two risks. Liquidity insurance requires cross-subsidization between early and late consumers. That is, the GFI increases r1 and decreases the expected return P(H)r2H + P(L)r2L at T = 2. Simultaneously, the GFI provides more insurance against the aggregate risk by reducing the difference between r2H and r2L. If the GFI can invest in the risky asset and a riskless long asset, markets for hedging the aggregate risk are complete because these two assets are linearly independent. When a increases, the GFI invests more in the short asset in order to increase r1. The GFI increases the liquidity of its portfolio in order to satisfy the higher demand for insurance against liquidity shocks. It also invests less in the risky asset and more in the riskless long asset in order to decrease r2H  r2L. Also, the price of the Arrow–Debreu security contingent on H decreases, whereas the price of the Arrow–Debreu security contingent on L increases. When the GFI increases its supply of liquidity, the prices of contingent claims that pay off when the return on the risky long asset is high (low) decrease (increase). But we see in Fig. 3 that when FIs are banks, an increase in a causes an increase of r, i.e., a fall in bond prices. This because a bank is constrained to issuing non-contingent contracts. Hence, the bank is obliged to completely insure its depositors against the aggregate risk. As a result, when a increases, the bank provides more insurance against liquidity shocks by increasing crosssubsidization between type 1 and type 2 agents. The bank increases r1 and decreases r2. If a bank has access to the risky and risk-free long assets, it will never invest in the risky asset. The economic interpretation is as follows. Since the bank is constrained to completely insuring its depositors against aggregate shocks, it provides consumption at T = 2 only by investing in the riskless long asset. Hence, when a increases, the bank increases

cross-subsidization between early and late consumers. That is, it increases r1 and decreases r2. Consequently, the bank invests more in the short asset and less in the risk-free long asset. Since the demand of the riskless long asset falls, its price falls and the risk-free long interest rate rises. Our simulations also show the effect on the risk-free long rate of shocks on the capitalist’s risk aversion coefficient r. Whatever the kind of FI, when r increases bond prices increase (see Figs. 4 and 5). This because when r increases, the risk-bearing capacity of the economy decreases. Hence, the capitalist must be more remunerated to insure consumers against the aggregate risk. As a result, p2(L) increases and p2(H) decreases. But since arbitrage opportunities cannot exist in equilibrium, i.e., p2(H)R(H) + p2(L)R(L) = 1, and R(H) > R(L), in absolute value the rise of p2(L) is higher than the fall of p2(H). Consequently, the interest rate falls and bond prices increase. Finally, our model allows to analyze the impact of a liquidity shock. 9 When k increases, i.e., when liquidity preference increases, the interest rate increases (see Figs. 6 and 7). This because when k increases, the number of type 2 agents decreases. Hence, the bank demands less Arrow–Debreu securities contingent on L. As a result, when liquidity preference increases, the price p2(L) decreases and the price p2(H) increases. The net effect is a fall of p2(H) + p2(L), i.e., an increase of r. We summarize the results of the comparative statics in the following table: GFI Bank

a increases

r increases

r falls r increases

r falls r falls

k increases r increases r increases

9 Ding et al. (2009) analyze how liquidity shocks impact the performance of hedge funds. Griffiths et al. (2010) analyze how the bid-ask spreads on 1-month Treasury bills is affected by liquidity shocks.

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the interim period. In Allen and Gale (2005), a capital market opens at the interim period, and this generates cash-in-the-marketpricing and negative bubbles. It seems potentially interesting to investigate how our analysis is impacted if a market opens both at the first period and at the second period. Acknowledgements I thank Ike Mathur (the Editor) and an anonymous referee for very helpful comments and suggestions. The financial support of the Fondation du Risque (Chaire Groupama) and the ANR (project RISK) is gratefully acknowledged. Fig. 6. General financial intermediaries: the effect of liquidity shocks on the riskfree long rate.

Fig. 7. Banks: the effect of liquidity shocks on the risk-free long rate.

Overall, the model generates a short-term volatility of the real interest rate. It is also able to explain why, contrary to the conventional wisdom, the real interest rate may increase during a financial crisis, as it did in the fall of 2008. In particular, in this model a liquidity crisis can be characterized by a positive shock on k. If this shock is sufficiently large, r increases. Moreover, if banks dominate the financial intermediation sector, a higher risk aversion of depositors will tend to increase r. 6. Conclusion This paper constructs a general equilibrium model of the interaction between FIs and financial markets that generates a volatility of the real interest rate due to shocks on risk aversion and liquidity preference. The model has several limitations. Firstly, our economy is a real one. Moreover, we investigate the effect of shocks on the demand for liquidity, not on the supply of liquidity. Introducing money into our framework could provide insights on the relationship between monetary policy and the real interest rate. Secondly, we impose exogenously that banks issue incomplete contracts. Providing micro-foundations for this restriction would greatly enrich the model. Thirdly, in our model no financial market opens at

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