The role of the reference rate in an interbank market with imperfect information

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The role of the reference rate in an interbank market with imperfect information Ichiro Muto Bank of Japan, 2-1-1 Nihonbashi-Hongokucho Chuo-ku, Tokyo 103-8660, Japan

A R T I C L E

I N F O

Article history: Received 30 June 2016 Received in revised form 8 March 2017 Accepted 8 March 2017 Available online xxxx JEL classification: E43 E44 G14 Keywords: Interbank market Reference rate Imperfect information Financial stability

A B S T R A C T This study investigates the potential role of the reference rate in an interbank market where individual banks cannot fully identify the nature of underlying shocks affecting their interbank transactions. We find that the reference rate does not always mitigate the market distortion arising from imperfect information. When the number of sample transactions is smaller than a certain threshold, the reference rate magnifies the distortion even if the reference rate is not affected by any reporting noise. The threshold depends on the relative size of aggregate and idiosyncratic shocks. Noise in the reported interest rates, which is potentially increased by banks’ manipulations, distorts individual banks’ inferences about the underlying shocks, and thereby raises the threshold. When noise is highly correlated among multiple sample transactions, perhaps owing to collusive manipulations, it is possible that increasing the number of sample transactions may never mitigate the market distortion. © 2017 Elsevier Inc. All rights reserved.

1. Introduction The roles of the reference rate in interbank markets (henceforth the “interbank reference rate”) have become a major concern for various agents active in financial markets in light of the recent LIBOR manipulation problem. The interbank reference rate represents only a limited number of interest rates reported by panel banks, not interest rates in all transactions throughout the interbank market. However, it provides financial market participants, including panel banks, with precious information concerning the price developments and market conditions of the interbank market. In addition, the interbank reference rate is widely used as a benchmark interest rate in many kinds of financial transactions, such as corporate loans, mortgage loans, and derivative transactions. This means that its developments influence financial transactions engaged in by various entities and can ultimately affect real economic activities. Therefore, understanding the economic functions of the interbank reference rate is important to financial stability as well as monetary policy making. Nevertheless, since the roles of the interbank reference rate have not received much attention until recently, in the aftermath of the LIBOR manipulation problem, academic studies have not sufficiently investigated this topic. There exist at least three issues to be explored. The first is the informational role of the interbank reference rate. It is most fundamentally important to investigate the mechanism through which information provided by the interbank reference rate enhances (or, under some conditions, deteriorates) the efficiency or stability of the interbank market. The second issue is the

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incentive for panel banks to manipulate the market in reporting their individual interest rates. It would certainly be important to investigate why banks have such incentives, how manipulations distort the efficiency or stability of the interbank market, and what kinds of institutional arrangements prevent manipulation. The third issue is the impact of variations of the interbank reference rate, possibly due to manipulations by panel banks, on financial transactions outside the interbank market and on real economic activities. Although these issues are potentially interrelated, to reach clear-cut conclusions, it is better to focus on some specific aspect of the reference rate, rather than to take into account all the issues simultaneously. This study mainly addresses the first issue, that is, the informational (or signaling) role of the reference rate, although it also to some degree investigates the influence of banks’ manipulations. The interbank reference rate should ideally apprise individual banks, which have specific information only on their own transactions, of the aggregate financial conditions that exist throughout the market. By doing so, is should help these banks set more appropriate interest rates for individual interbank transactions, thereby allocating resources more efficiently. We analyze (i) the mechanism by which the interbank reference rate contributes to stabilizing the interbank market, (ii) conditions under which the interbank reference rate might actually serve to destabilize the interbank market, and (iii) the specific properties that might allow a reference rate to help stabilize the interbank market and thereby the effectiveness of monetary policy. We introduce a simple interest rate model in which the individual interbank interest rate is determined by two kinds of components: idiosyncratic shocks and aggregate shocks. This kind of setup with two kinds of shocks is traditional in theoretical studies on the functioning of interbank markets (for example, Allen, Carletti, & Gale, 2009; Allen & Gale, 2000; Freixas, Martin, & Skeie, 2011). However, we additionally assume that banks engaged in individual interbank transactions cannot correctly distinguish the two kinds of shocks. Although this kind of imperfect information environment is similar to the famous island economy modeled in a macroeconomic context (Lucas, 1972, 1973; Phelps, 1970), our study is the first to apply the setup to the interbank market and to examine the role of a reference rate in such an imperfect information environment. It is indeed a realistic setup, because banks usually obtain information on aggregate market conditions by observing the reference rate and thereby learn the reasonable interest rate that should be applied in their own transactions. We choose to use a simple interest rate model, mainly because of the lack of widely accepted canonical models for analyzing the interbank market. Because of its simplicity and generality, our setup is potentially applicable to many theoretical models of interbank markets. In our study, the interbank reference rate is defined as the sample average of interest rates reported by individual banks. The sample size is limited because in reality the number of banks contributing to the panel of interbank reference rates is small– according to Gyntelberg and Wooldridge (2008), between 10 and 20 in many cases (see Table 1). Even in the case of the euro interbank offered rate (Euribor), which has the largest panel of banks, selected from all countries in the Euro area, the panel size is only around 45. Because of such limited sample size, it is possible that individual bank interest rate data, which are largely influenced by idiosyncratic factors such as the bank’s credit risk or liquidity, can significantly influence market transactions in general. Even more seriously, the small sample size potentially allows panel banks to influence the interbank reference rate by manipulating their reports on their individual interest rates. This influence can be magnified if multiple banks simultaneously manipulate their reported interest rates in the same direction. To capture these features, we assume that the interest rates reported by panel banks can deviate from the rates actually used in their transactions through reporting noise, which can be increased by banks’ manipulations and can be correlated among interest rates reported by multiple banks. Since this study

Table 1 Number of contributor banks for the interbank reference rate. Currency

Reference rate

Number of contributor banks

AUD CAD CNY DKK

Libor Libor Shibor Libor Cibor Libor Euribor Hibor Jibor Mibor Libor Tibor Koribor Klibor Libor Phibor Sibor Bibor Libor Sibor

8 12 16 8 12 16 45 20 18 33 16 16 14 11 8 17 13 16 16 15

EUR HKD IDR INR JPY KRW MYR NZD PHP SGD THB USD

Source: Gyntelberg and Wooldridge (2008).

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focuses on the informational role of the reference rate, the main issue to be analyzed is how the introduction of the reference rate diminishes or magnifies the efficiency or stability of the interbank market, measured straightforwardly as the deviation of the market-wide interest rate (the average interest rate for all interbank transactions) from the rate in a hypothetical environment with no informational frictions. We also examine how the presence of banks’ reporting noise, which is potentially increased by banks’ manipulations, influences the impact of the reference rate on market distortion. Some previous studies, such as those by Freixas and Holthausen (2004) and Heider, Hoerova, and Holthausen (2009), have constructed theoretical models to explain the mechanisms determining the interest rates in interbank markets. Our study differs in that we pay particular attention to the role of the reference rate in mitigating (or magnifying) the market distortion arising from imperfect information, which has not been sufficiently addressed in previous studies. Our study can be viewed as a complement to the analysis of Duffie, Dworczak, and Zhu (in press), which also focuses on the informational role of benchmarks. They incorporate asymmetric information and search frictions between dealers and buy-side customers in the over-the-counter market, and take into account the endogenous entry–exit problem of customers. They show that under reasonable assumptions, publishing a benchmark enhances welfare because it increases the volume of transactions by encouraging customers’ entry, facilitates more efficient matching between dealers and customers, and reduces search costs.1 In contrast to their searchtheoretic framework, our study focuses on the banks’ inference problem of attempting, under imperfect information, to gauge the price development or market conditions of the entire interbank market. This process is self-referential: the individual bank infers the conditions of the whole interbank market by observing the reference rate, which reflects the average of individual banks’ interest rates, which in turn are determined by the panel banks’ inferences. Our study also contributes to research on the incentive for banks to manipulate interest rates through their reporting. Duffie and Dworczak (2014) investigate how to design a robust mechanism for the interbank reference rate in an environment where private agents have incentive for manipulations.2 Specifically, they examine how the desirable weights of individual interest rates in calculating the reference rate depend on the volume of the sample transactions. In our study, the weight used for calculating the reference rate is equal for all sample transactions, following the actual calculation method. In addition, we examine the impact of collusive manipulations on the market interest rate, whereas Duffie and Dworczak (2014) ignore collusion throughout their study. The remainder of this study is organized as follows. In Section 2, we present our simple interbank interest rate model assuming imperfect information. In Section 3, we provide a benchmark analysis under the assumption that individual banks can observe the aggregate and idiosyncratic components separately. In Section 4, we examine how the results differ when individual banks cannot do this. In Section 5, we investigate how the reference rate mitigates (or magnifies) the market distortion arising from imperfect information. In Section 6, we examine the implications of noise in the reference rate. In Section 7, we discuss the relationship between our analysis and the real-world problem of the reference rate. Finally, Section 8 summarizes our analysis and discusses potential future work in this area, as well as policy implications.

2. Model In general, the interest rate of an individual transaction in the interbank market can be decomposed into the risk-free rate and the risk premium. The risk premium is determined by aggregate factors (which have a common impact on all interbank transactions) and idiosyncratic factors (which influence only individual interbank transactions). Because an individual bank cannot usually observe the interest rates charged in other banks’ transactions, it cannot estimate the market-wide interest rate (the average of interest rates across all interbank transactions). Thus it cannot accurately decompose the fundamental factors (such as liquidity and credit conditions) into aggregate and idiosyncratic factors, especially in the absence of the interbank reference rate. To illustrate this kind of situation, we consider a simple model of interbank interest rates in which each bank cannot fully identify the sources of underlying shocks that influence individual interbank transactions. Suppose that there are an infinite number of transactions (indexed by j = 1, · · · , ∞) in an interbank market.3 The interest rate for the jth transaction is determined by the following two equations: i j = ip + aEj l + (1 − a)Ej ej ,

(1)

y j = l + ej ,

(2)

1 Kobayashi (2012) also employs a model incorporating search frictions to examine whether the calculation of reference rates should be based solely on actual transaction data and whether the use of expert judgment should be allowed to some extent, particularly in times of financial crisis. His analysis leads to a conclusion that an expert judgment could be allowed to some extent in order to reduce liquidity premiums resulting from a temporary surge in uncertainty during crisis periods. 2 Coulter and Shapiro (2014) is another recent study that presents a mechanism to incentivize panel banks to reveal their borrowing costs truthfully. 3 This setup can be interpreted as the situation in which there are a finite number of banks and each bank is engaged in a large number of interbank transactions.

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where i j is the interest rate for the jth transaction and ip is the policy interest rate. y j is the fundamental factor influencing the jth transaction (such as funding liquidity and counterparty credit risk), l is the aggregate component of y j , and e j is the individual component of y j .4 For simplicity, we assume that each shock follows the i.i.d. normal distribution (sl2 > 0 and se2 > 0):   l ∼ N 0, sl2 ,   ej ∼ N 0, se2 . These equations capture a situation in which the banks identify the aggregate shock and the individual shock in the jth transaction and then take account of these shocks in determining their individual interest rate. (1) indicates that the banks in the jth transaction set the interest rate by adjusting the policy interest rate using their estimates of the aggregate component (l) and the individual component (ej ). a and (1 − a) determine the relative influences of the aggregate component and individual component on the interest rate for the jth transaction (0 < a < 1). (2) indicates that the fundamental factor for the jth transaction consists of a shock influencing the entire interbank market (l) and a shock specific to the jth transaction (ej ). Ej is the expectation operator of the banks in the jth transaction. Because the banks in the jth transaction equally observe the fundamental factor y j , the expectations of these banks are identical. An important assumption in this framework is that the aggregate components and individual components of y j have different impacts on the individual interest rate (that is, a = 12 ). There are good reasons for introducing this assumption. First, idiosyncratic shocks of individual transactions can be diversified by engaging in multiple transactions. Allen and Gale (2000) indicate that interbank markets provide optimal liquidity insurance when banks are subject to idiosyncratic liquidity shocks, but may lead to contagion when aggregate liquidity shocks are present. In this situation, banks naturally require higher risk premiums for aggregate factors.5 Second, some possible idiosyncratic factors, such as counterparty credit risk, can differ greatly between a lending bank and a borrowing bank. In this case, optimal contracts might require higher premiums for idiosyncratic factors. Third, it is possible that macroeconomic policy, such as the central bank’s liquidity provision or macro prudential policy, responds differently to aggregate and idiosyncratic shocks. Goodfriend and King (1988) claim that, if the interbank market is efficient, the central bank should respond to aggregate liquidity shocks, but not to idiosyncratic liquidity shocks. Allen et al. (2009) and Freixas et al. (2011) demonstrate that the central bank’s optimal policy responses to aggregate and idiosyncratic liquidity shock can differ. It follows that the risk premiums required for aggregate shocks andfor idiosyncratic shocks can also differ.  It is still arguable which component is more important a < 12 or a > 12 , because both situations can theoretically be assumed. Although empirical evidence on this issue is quite limited, Angelini, Nobili, and Picillo (2011) provide an empirical study on the determinants of individual interbank interest rates around the global financial crisis period using e-MID, a unique Italian database of individual interbank transactions. Their results indicate that before the crisis the characteristics of individual borrowing banks did significantly affect the spreads between interbank reference rates for unsecured and secured deposits. In addition, the large and volatile movements of the spreads after the onset of the financial crisis were mainly driven by aggregate (market-wide) factors, rather than by bank-specific factors. These results can be viewed as consistent with the assumption that a > 12 . However, because the generality of this assumption is yet to be examined, we introduce only an assumption that a = 12 . Since this assumption is quite general, our argument presented below holds in a broad theoretical setup of the interbank market. 3. Benchmark: full information case For benchmarking purposes, we begin by assuming that the banks in the jth transaction can observe the realizations of l and ej separately. Then the banks’ estimates of the aggregate component and the individual component are exactly the same as the true values of l and ej . As a result, the interest rate in the jth transaction is determined as follows: i j = ip + al + (1 − a)ej .

(3)

In this study, we measure market distortions arising from informational imperfections by the deviation of the “market interest rate” (denoted by im ), which is defined as the average of all interest rates in the entire interbank market, from its normative level. The normative level of the market interest rate (denoted by im ), which should be realized if all parties have full information, is computed as follows by summing the individual interest rates: im = lim

n→∞

n n 1 j 1 i = ip + al + lim (1 − a)ej = ip + al . n→∞ n n j=1

(4)

j=1

4 In this study, we do not specify the fundamental factors that determine the spread between interbank interest rates and the policy interest rate. The issue of whether liquidity factors or counterparty risk factors were dominant as the determinants of the LIBOR-OIS spread, especially after the onset of the financial crisis, is examined by many authors, such as Michaud and Upper (2008), Taylor and Williams (2009), and Gefang, Koop, and Potter (2011). 5 Another possible situation is proposed by Allen et al. (2009), who consider the situation in which the idiosyncratic shock is not sufficiently large for the banks to fail. Further, it is optimal for the banks to keep enough liquidity to insure themselves against aggregate liquidity shocks.

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Therefore, in the full information case the normative level of the market interest rate is the sum of the policy interest rate and the risk premium for the aggregate component (al). Any upward or downward deviation of the actual market interest rate from the normative level can be viewed as a distortion of the entire interbank market. In the following sections we examine how market-wide distortions arise from informational imperfections and how the interbank reference rate diminishes (or magnifies) the distortions. 4. The case of imperfect information in the absence of the reference rate Next, we assume that the banks in the jth transaction cannot observe the realizations of l and ej separately and observe only the sum of these components: y j . In this situation, the banks need to estimate the aggregate component l in order to determine interest rates for individual transactions. Suppose that the banks know the variances of l and ej : sl2 and se2 . Then standard statistical inference yields the following estimators of l and ej . Ej l = hy j , Ej ej = (1 − h)y j ,

where h ≡

sl2 sl2

+ se2

.

The interest rate in the jth transaction is then determined as follows: i j = ip + ahy j + (1 − a)(1 − h)y j = ip + (ah + (1 − a)(1 − h))(l + ej ).

(5)

By summing the individual interest rates, we compute the market interest rate as follows: im = lim

n→∞

n 1 j i = ip + (ah + (1 − a)(1 − h))l . n

(6)

j=1

m , is computed From Eqs. (4) and (6), the distortion of the market interest rate from its normative level, which is denoted by i as follows: m = im − im = (1 − 2a)(1 − h)l . i

(7)

m can be either positive or negative, the expected size of the distortion is evaluated in terms of its variance: Since i m ) = (1 − 2a)2 (1 − h)2 s 2 . V ar(i l

Because we assume that a = of the interbank reference rate.

1 2,

(8)

m ) is strictly positive. This is the expected size of the market distortion in the absence V ar(i

5. The role of the reference rate Thus far, we have assumed that individual banks cannot observe the interest rates used for other banks’ transactions. As a result, individual banks also cannot observe the market interest rate im . This is quite a realistic assumption, since individual players in a real-world interbank market can rarely observe the average of interest rates across all interbank transactions. Banks usually do observe a sample average for some fraction of interbank transactions, in the form of the interbank reference rate. However, because the reference rate includes only a small number of sample transactions, it does not purely reflect the aggregate market condition, but is disproportionately affected by idiosyncratic factors related to individual sample transactions, such as an individual bank’s credit risk or liquidity. Therefore, it is not obvious whether the existence of the reference rate always mitigates market distortion. Below we examine under what conditions the reference rate mitigates (or magnifies) market distortion. There exists a feedback mechanism in which an individual interest rate is collected as a sample for the reference rate, but the reference rate in turn influences the setting of the individual interest rate. Our static framework investigates the equilibrium of this self-referential system. Here we also assume that the data on interest rates reported by individual banks do not include any reporting noise. Suppose that there exists an institution that organizes an interbank market. The institution collects data on interest rates used in interbank transactions and provides these data to individual banks. Specifically, the institution selects q number of transactions and averages the interest rates used in these transactions: iq ≡

q 1 k i . q

(9)

k=1

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The institution then announces the value of iq to individual banks as the reference rate of interbank interest rates.6 We define yq as the fundamental factor corresponding to iq : yq ≡

q 1 k q y =l+e , q

(10)

k=1

q

q

where e is defined as e ≡

1 q

q  k=1

ek . q

Note that the distribution of e is as follows:   q e ∼ N 0, se2 /q . Here, we assume that individual banks can discover yq by observing the value of iq . They can then infer the value of the aggregate component l.7 q The banks know that yq consists of l and e , although they do not observe these components separately. In addition, the banks know the number of sample transactions (q). Then, statistical inference yields the following estimator of l: Ej l = gyq ,

where g ≡

sl2 sl2

+ se2 /q

.

(11)

Note that the coefficient g is larger than h when q is greater than unity. The estimator of ej is obtained as follows: Ej ej = y j − Ej l = y j − gyq .

(12)

By substituting Eqs. (11) and (12) into Eq. (1), we determine the interest rate in the jth transaction as follows: i j = ip + agyq + (1 − a)(y j − gyq ).

(13)

From Eqs. (9) and (13), the average interest rate across the sample of q transactions is iq =

q   1 p i + agyq + (1 − a) yk − gyq q k=1

= ip + (ag + (1 − a)(1 − g))yq .

(14)

Therefore, as we assumed above, if the market-organizing institution provides data on iq to individual banks, these banks can obtain the value of yq , via the simple calculation (iq − ip )/(ag + (1 − a)(1 − g)) = yq . From Eqs. (2), (10), and (13), the market interest rate is im = lim

n→∞

n n   1 j 1  p i = lim i + agyq + (1 − a) y j − gyq n→∞ n n j=1

j=1

n    1  p q = lim i + (2a − 1)g l + e + (1 − a) l + ej n→∞ n j=1

p

q

= i + ((2a − 1)g + (1 − a))l + (2a − 1)ge .

(15)

m ) is computed as follows: Therefore, the distortion of the market interest rate from its normative level (i

  m = im − im = (2a − 1) (g − 1)l + geq . i

(16)

6 We assume that every bank not selected in the sample transactions observes the same value of iq . In addition, we assume that a bank selected in the sample transactions observes all sample interest rates except for the bank’s own interest rate. This means that the sample size reflected in the reference rate is q for the former bank and q − 1 for the latter bank. This difference suggests that the estimators of l set by these banks are slightly different. However, since there exist an infinite number of transactions in the interbank market, the impact of this difference on the entire market can be regarded as negligible, and an explicit consideration of this difference does not alter the essence of our argument. 7 For simplicity, we assume that individual banks use only the information of yq . In the Appendix, we confirm that the results are essentially the same even if banks use both yj and yq .

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The variance of the market distortion is     m = (1 − 2a)2 (1 − g)2 s 2 + g2 s 2 /q . V ar i l e

(17)

Therefore, the variance of the market distortion depends on the number of sample transactions (q). If q approaches infinity, V ar(is ) converges to zero: ⎡   2⎣  m 1− lim V ar i = lim (1 − 2a)

q→∞

q→∞

sl2

2

sl2 + se2 /q

 sl2

+

sl2 sl2 + se2 /q

⎤

2 se2

/q⎦ = 0.

(18)

This means that, if the reference rate reflects the interest rates across all interbank transactions, then the market interest rate becomes exactly equal to its normative level. In this case, the reference rate provides perfectly accurate information on the aggregate conditions in the interbank market.   m is calculated as below: But if q = 1, g is equal to hand Var i     m = (1 − 2a)2 (1 − h)2 s 2 + h2 s 2 . V ar i l e

(19)

  m is larger if there is a reference rate than if there is none. This By comparing Eq. (19) with Eq. (8), we find that V ar i means that the presence of the reference rate can magnify market distortion. This phenomenon occurs when the number of sample transactions reported for the reference rate is extremely small, because in that case individual components included in the reference rate do not completely cancel each other out. Since every individual bank observes the same reference rate, the q averaged individual component   included in the reference rate (e ) then distorts each individual interest rate to an equal degree.  m As Eq. (17) shows, V ar i decreases monotonically with q. By comparing Eqs. (8) and (17), we can determine the threshold number of q– the minimum number of sample transactions needed for the reference rate to reduce the variance of market distortion – as is given in the following proposition.   m in the absence of reporting noise, which is given by Eq. (17), is smaller Proposition 1. The variance of the market distortion V ar i than that in the absence of the reference rate, which is given by Eq. (8), if and only if Eq. (20) holds. q>2+

sl2 se2

.

(20)

  m is smaller in Eq. (8) than in Eq. (17) if and only if Proof. Since (1 − 2a)2 > 0, the value of V ar i (1 − g)2 sl2 + g2 se2 /q > (1 − h)2 sl2 . Multiplying by q2 , we can rewrite the above inequality as 

se2 q − sl2 − 2se2



 sl2 q + se2 > 0,

which implies Eq. (20).  Therefore, when the sample is sufficiently large, the reference rate helps to reduce the market distortion arising from informational imperfections, by providing good information on the aggregate shocks that influence the entire interbank market. This helps individual banks to identify more accurately the sources of shocks arising in the interbank market and to set individual interest rates at more appropriate levels. However, Proposition 1 also tells us that the mere presence of a reference rate does not necessarily have this effect. This result is somewhat surprising, because it is usually expected that any increase in public information always improves market efficiency. The result comes from the fact that, when all banks use the same reference rate based on a small sample, idiosyncratic factors broadly affect all banks’ expectations about the aggregate  market.  As Eq. (20) shows, the threshold number of q increases with the proportion of aggregate shock to idiosyncratic shock sl2 /se2 . This is because, when the aggregate shock is relatively large, individual banks expect that the large variations in the reference rate reflect the movements of aggregate factors, and they then use a higher value of g to estimate the aggregate shocks, as is shown in Eq. (11). However, Eq. (17)) indicates that, if g is large, the market distortion is more affected by idiosyncratic shocks, because the reference rate is influenced by idiosyncratic shocks, especially if the sample number q is relatively small. As a result, when sl2 /se2 is large, the minimum number of q Please cite this article as: I. Muto, The role of the reference rate in an interbank market with imperfect information, Global Finance Journal (2017), http://dx.doi.org/10.1016/j.gfj.2017.03.005

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required for the reference rate to mitigate market distortion also becomes large. This means that, when the number of sample transactions is limited, the reference rate does not mitigate the market distortion arising from informational imperfections, even if it is not affected by any reporting noise. 6. Noise in the reference rate In the previous section, we have assumed that the data on interest rates reported by individual banks do not include any reporting noise. However, what if the reference rate does include some noisy components? The recent LIBOR problem suggests that this is a very real possibility, in which case the reference rate might not correctly reflect the interest rates for actual interbank transactions, because panel banks have incentives to manipulate it. In this section, we introduce noise into the model, as an exogenous factor representing deviations in the interest rate reported by sample banks, from the rates used for actual transactions.8 We make an allowance for two different kinds of noise: (i) idiosyncratic noise, which is specific to individual transactions, and (ii) correlated noise, whose impacts are correlated among multiple transactions. The latter noise is introduced to examine the potential impact of collusion in the manipulation of the reference rate. 6.1. The case of idiosyncratic noise As in Section 5, we assume that a market-organizing institution collects the data on interest rates used in individual interbank transactions. Here we also assume that the interest rate reported for the kth transaction (denoted by ˜ık ) includes idiosyncratic noise mk : ˜ık = ik + (1 − a)mk ,

(21)

and that this noise follows the i.i.d. normal distribution:   mk ∼ N 0, sm2 . The institution provides the average of the reported (not necessarily actual) interest rates: ˜ıq ≡

q q 1 k 1 k q ˜ı = i + (1 − a)m , q q k=1

(22)

k=1

q

where m is defined as q

m ≡

q 1 k m . q

(23)

k=1

q

The distribution of m is as follows:   q m ∼ N 0, sm2 /q . ˜ q as the fundamental factor corresponding to ˜ıq : We define y ˜ q ≡ l + eq + m q . y

(24)

˜ q by observing the value of ˜ıq . The banks know that y ˜ q consists of l, eq , We assume that individual banks can come to know y q and m , but they do not observe each component separately. Then, statistical inference yields the following estimator of l: ˜ q, Ej l = dy

where d ≡

sl2 sl2

+

se2

/q + sm2 /q

.

(25)

The estimator of ej is obtained as follows ˜ q. E j e j = y j − E j l = y j − dy

(26)

8 For simplicity, we assume that the noise is unbiased. This assumption is valid because there may be gains in one of the numerous derivative products of reference rates, which could go in any direction.

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By substituting Eqs. (25) and (26) into Eq. (1), we obtain the interest rate in the jth transaction as follows: ˜ q + (1 − a)(y j − dy ˜ q ). ij = ip + ady

(27)

From Eqs. (23), (24), and (27), the average of the reported interest rates is ˜ıq =

q 1 k ˜ q + (1 − a)(yk − dy ˜ q )) + (1 − a)mq (i + ady q k=1

˜ q. = ip + (1 − a + (2a − 1)d)y

(28)

Therefore, as we assumed above, if the market-organizing institution provides ˜ıq to individual banks, the banks can obtain ˜ q via the simple calculation (îq − ip )/(1 − a + (2a − 1)d) = y ˜ q. the value of y From Eqs. (2), (24), and (27), the market interest rate is im = lim

n→∞

n n 1 j 1 p ˜ q + (1 − a)(yj − dy ˜ q )) i = lim (i + ady n→∞ n n j=1

j=1

q

q

= ip + (1 − a + (2a − 1)d)l + (2a − 1)d(e + m ).

(29)

m ) is Therefore, the distortion of the market interest rate from its normative level (i m = im − im = (1 − 2a)(1 − d)l + (2a − 1)d(eq + mq ). i

(30)

The variance of the market distortion is     m = (1 − 2a)2 (1 − d)2 s 2 + (1 − 2a)2 d2 s 2 + s 2 /q. V ar i l e m

(31)

Therefore,   m in the presence of idiosyncratic noise, which is Proposition 2. For any given sm2 > 0 , the variance of the market distortion Var i given by Eq. (31), has the following properties:   m decreases monotonically with q, and (i) V ar i   m = 0 for q → ∞. (ii) V ar i This proposition means that even if idiosyncratic noise is included in the reported interest rates, the market distortion can be reduced (ultimately, to zero) by basing the reference rate on a larger number of sample transactions. A natural question is under what condition the variance of the market distortion is larger than it is in the absence of the reference rate. By comparing Eqs. (8) and (31), we obtain the following proposition.   m in the presence of idiosyncratic noise, which is given by Eq. (31), is Proposition 3. The variance of the market distortion V ar i smaller than that in the absence of the reference rate, which is given by Eq. (8), if and only if Eq. (32) holds. q>2+

sl2 se2

 +

2se2 + sl2 se4

sm2 .

(32)

  m is smaller in Eq. (8) than in Eq. (31) if and only if Proof. Since (1 − 2a)2 > 0, the value of V ar i   (1 − d)2 sl2 + d2 se2 + sm2 /q > (1 − h)2 sl2 . Multiplying by q2 , we rewrite the above inequality as 

     se4 q − sl2 + 2se2 se2 + sm2 sl2 q + se2 + sm2 > 0,

which implies Eq. (32).  Please cite this article as: I. Muto, The role of the reference rate in an interbank market with imperfect information, Global Finance Journal (2017), http://dx.doi.org/10.1016/j.gfj.2017.03.005

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Therefore the threshold number of transactions q increases monotonically with the variance of noise sm2 . This means that the noise in the interest rates reported by banks enlarges the variance of the market distortion, and thereby increases the number of sample transactions required to attain any benefit from using the reference rate. The result indicates that the reference rate should ideally be based on actual transactions, since the threshold value of q is smallest if there is no noise in reported interest rates. 6.2. The case of correlated noise The above exercise indicates that if noise in the reported interest rates is idiosyncratic to each sample transaction, the variance of the market distortion can be mitigated (or even eliminated) by increasing the number of such transactions. However, a salient feature of the LIBOR manipulation problem is that large banks engaged in collusive manipulation. According to McConnell (2013), “Colluding with competitors to share information and to manipulate markets is obviously illegal, yet it was common practice in the LIBOR markets.” Collusive manipulation means that the reported interest rates are simultaneously manipulated in the same direction. To consider its implications, we need to introduce a setup where noise in the interest rates reported by banks is correlated among multiple transactions. Suppose that the interest rate reported for the kth transaction ˜ık includes the weighted average of idiosyncratic noise mk and common noise n:   ˜ık = ik + (1 − a) (1 − b)mk + bn ,

(33)

Common noise n has the same influence on all transactions, and it follows the i.i.d. normal distribution:   n ∼ N 0, sn2 . The parameter b represents the correlation among noise in sample transactions (1 ≥ b ≥ 0). If b is zero, all noise is idiosyncratic to sample transactions. If b is unity, all sample transactions have exactly the same noise. The institution provides the average of the reported (not necessarily actual) interest rates: ˜ıq ≡

q q   1 k 1 k q ˜ı = i + (1 − a) (1 − b)m + bn . q q k=1

(34)

k=1

˜ q as the fundamental factor corresponding to ˜ıq : We define y ˜ q ≡ l + eq + (1 − b)mq + bn. y

(35)

˜ q by observing the value of ˜ıq . The banks know that y ˜ q consists We assume that individual banks can acquire the knowledge of y q q of l, e , m , and n, but they do not observe each component separately. Then, statistical inference yields the following estimator of l: ˜ q, Ej l = w y

where w ≡

sl2 sl2 + se2 /q + (1 − b)2 sm2 /q + b2 sn2

.

(36)

The estimator of ej is obtained as follows ˜ q. Ej e j = y j − Ej l = y j − w y

(37)

By substituting Eqs. (36) and (37) into Eq. (1), we determine the interest rate in the jth transaction as follows: ˜ q + (1 − a)(y j − w y ˜ q ). ij = ip + aw y

(38)

From Eqs. (34), (35), and (38), the average of the reported interest rates is ˜ıq =

q 1 k ˜ q + (1 − a)(yk − w y ˜ q )) + (1 − a)((1 − b)mq + bn) (i + aw y q k=1

˜ q. = ip + (1 − a + (2a − 1)w)y

(39)

Therefore, as we assumed above, if the market-organizing institution provides ˜ıq to individual banks, the banks can obtain ˜ q via the simple calculation (îq − ip )/(1 − a + (2a − 1)w) = y ˜ q. the value of y Please cite this article as: I. Muto, The role of the reference rate in an interbank market with imperfect information, Global Finance Journal (2017), http://dx.doi.org/10.1016/j.gfj.2017.03.005

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From Eqs. (2), (35), and (38), the market interest rate is im = lim

n→∞

n n 1 j 1 p ˜ q + (1 − a)(y j − w y ˜ q )) i = lim (i + aw y n→∞ n n j=1

j=1

q

q

= ip + (1 − a + (2a − 1)w)l + (2a − 1)w(e + (1 − b) m + bn).

(40)

m ) is Therefore, the distortion of the market interest rate from its normative level (i m = im − im = (1 − 2a)(1 − w)l + (2a − 1)w(eq + (1 − b)mq + bn). i

(41)

The variance of the market distortion is m ) = (1 − 2a)2 (1 − w)2 s 2 + (1 − 2a)2 w 2 V ar(i l

   se2 + (1 − b)2 sm2 /q + b2 sn2 .

(42)

m ) and q, we obtain the following proposition. Therefore, as for the relationship between V ar(i m ) in the presence of correlated noise, Proposition 4. For any given sn2 > 0 and b > 0, the variance of the market distortion V ar(i which is given by Eq. (42), has the following properties: m ) decreases monotonically with q, and (i) V ar(i  (ii) V ar(im ) is strictly positive for q → ∞.

Therefore, in contrast to the case of idiosyncratic noise, the variance of market distortion in the presence of correlated noise (as given by b > 0) is not completely eliminated even if the number of sample transactions is increased to infinity. This is a distinctive feature of correlated noise. Next we consider under what conditions the reference rate reduces (or magnifies) the variance of the market distortion. Given the number of sample transactions q, we obtain the following proposition. m ) in the presence of correlated noise, which is given Proposition 5. For any given q > 0, the variance of the market distortion Var(i by Eq. (42), has the following properties: m ) increases with b if 1 ≥ b > (i) V ar(i m ) decreases with b if (ii) V ar(i

sm2 /q sm2 /q+sn2

sm2 /q , sm2 /q+sn2

> b ≥ 0, and

m ) takes the minimum value with respect to b if b = (iii) V ar(i

sm2 /q . sm2 /q+sn2

Proof. Define T ≡ se2 /q + (1 − b)2 sm2 /q + b2 sn2 . Then, Eq. (42) can be rewritten as follows: m ) = (1 − 2a)2 V ar(i

 1−

sl2

−1 2 −2   2 2 4 2 sl + s l s l + T T . sl + T

By differentiating with b, we obtain 

∂ V ar(im )/∂ b = 2(1 − 2a)2 sl4 sl2 + T

−2     b sm2 /q + sn2 − sm2 /q ,

which implies (i), (ii), and (iii).  m ) depends on the relative size of s 2 /q and s 2 . When Proposition 5 is illustrated in Fig. 1. The maximum value of V ar(i m n 2 2  m /q > sn , V ar(i ) takes the maximum value when b = 0. When sm /q < sn2 , V ar(im ) takes the maximum value when b = 1. m ) is larger or smaller than the value in the absence of any noise, which is given by Eq. (19), depends on q and b. Whether V ar(i

sm2

One natural question is whether increasing the number of the sample transactions always mitigate the market distortion or m ) takes the not. To examine this issue, suppose that the number of sample transactions goes to infinity (q → ∞). Then, V ar(i following value:  m ) = (1 − 2a)2 V ar(i

b2 sn2 sl2 + b2 sn2

2

 sl2 + (1 − 2a)2

sl2 sl2 + b2 sn2

2 b2 sn2 .

(43)

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  m . Note: The figure corresponds to the case of s 2 /q > s 2 . Fig. 1. Relationship between b and V ar i m n

m ) = 0. When b = 1, V ar(i m ) = (1 − 2a)2 s 2 , which is the maximum value of V ar(i m ). This indicates that when b = 0, V ar(i n This situation is illustrated in Fig. 2. A comparison between Eqs. (8) and (43) leads to the following proposition.

  m has the following properties. Proposition 6. When q → ∞, the variance of the market distortion Var i

(i) When

 

se4 sl2

(

sl2 +se2

)

2

> sn2 , V ar im in the presence of correlated noise, which is given by Eq. (43), is smaller than it is in the

absence of the reference rate, which is given by Eq. (8) for any value of 1 ≥ b ≥ 0.

  m when q goes to infinity. Fig. 2. Relationship between b and V ar i

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(ii) When

se4 sl2

(

sl2 +se2

)

2

13

  m in the presence of correlated noise, which is given by Eq. (43), is smaller than it is in the ≤ sn2 , V ar i

absence of the reference rate, which is given by Eq. (8) if and only if b is smaller than the threshold value b∗ , which is given as follows:  b∗ = 

se2 sl2 +se2

 1−

(iii) When

(

se4 sl2 2 2 sl +se2

)



sl sn

se2 sl2 +se2

2 .

(44)

  m in the presence of correlated noise, which is given by Eq. (43), is larger than it is in the absence ≤ sn2 , Var i

of the reference rate, which is given by Eq. (8) if and only if b is larger than the threshold value b∗ , which is given by Eq. (44).   m in the presence of the reference rate with correlated noise is smaller than the value in the Proof. The maximum value of V ar i absence of the reference rate when (1 − 2a)2 sn2 < (1 − 2a)2 (1 − h)2 sl2 , which leads to (i). In addition, Eqs. (8) and (43) are equalized when  (1 − h)2 sl2 =

b2 sn2

2

sl2 + b2 sn2

 sl2 +

sl2 sl2 + b2 sn2

2 b2 sn2 .

This equality can be rewritten as 

1 − (1 − h)2 sl2 − (1 − h)2 sn2 b2

  sn2 b2 + sl2 = 0,

which implies (ii) and (iii).  (i) means that if the amount of noise is small, the reference rate always mitigates market distortion, even if the noise is correlated. (ii) and (iii) indicate that, if the noise is larger, whether or not the reference rate mitigates the market distortion depends on the correlation among the noise components in the reported interest rates. The most important implication in the above proposition is (iii), which means that if the variance of common noise is large and/or the correlation among the noise components in the reported interest rates is high, then the reference rate does not contribute to reducing the variance of the market distortion even if the sample is quite large. This result suggests that collusive manipulation seriously undermines the role of the reference rate. This negative effect can never be eliminated simply by changing the sampling or calculation method of the reference rate. 7. Discussion We have introduced a simple interbank interest rate model with unobserved components to investigate the potential role of the reference rate in stabilizing or destabilizing the market-wide interest rate. Because of its simplicity and generality, our model is potentially applicable to different theoretical setups of the interbank market. However, readers may wonder whether our theoretical setup is relevant to the real-world problems of the interbank reference rate and whether our argument is quantitatively important. The following are some remarks on these issues. First, we have assumed that the reference rate basically reflects the interest rates used for actual transactions, although they might be affected by reporting noise. In reality, a reference rate, such as LIBOR, is based on surveys, and not necessarily on the interest rates used for actual transactions. However, even if some reported interest rates are purely hypothetical, we consider the essence of our argument applicable. The survey asks banks to give their best estimates of the representative interbank interest rate. Any deviations of reported interest rates from that estimate can be regarded as noise in the reference rate. Once the reference rate is affected by the noise, it distorts the market, because banks use the reference rate to determine the interbank interest rate used for their actual transactions. Second, although in our model the reference rate is determined simultaneously with individual interbank interest rates, so that the self-referential system is in equilibrium, it is certainly possible to extend our setup to incorporate dynamic interactions. For example, suppose that aggregate shocks follow an autoregressive model in which banks estimate the past values of aggregate shocks by using the past values of the reference rate, and then compute the current value of aggregate shocks by extrapolating from the autoregressive model. In this situation, the noise included in the past values of the reference rate affects Please cite this article as: I. Muto, The role of the reference rate in an interbank market with imperfect information, Global Finance Journal (2017), http://dx.doi.org/10.1016/j.gfj.2017.03.005

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banks’ estimates of the current value of aggregate shocks, and as a result, individual interest rates reported by banks are affected by the current and past values of the reporting noise. This situation is essentially the same as the one in our static model. Third, although the present study is purely theoretical, Eisl, Jankowitsch, and Subrahmanyam (2014) offer a quantitative analysis of the potential impact of manipulation on the reference rate. Using the actual submission data of individual interest rates, they calculate the effects on the reference rate of single or multiple banks manipulating their submissions. They report that the potential impact of manipulation by a single bank, calculated by taking the observed submissions as given, reaches 1.16 basis points (bp) for 3 M Australian Dollar (AUD) LIBOR, 0.48 bp for 3 M US Dollar (USD) LIBOR, and 0.17 bp for 3 M EURIBOR. They further indicate that the collusion of multiple banks magnifies the potential impact to 3.50 bp for 3 M AUD LIBOR, 1.61 bp for 3 M USD LIBOR, and 0.53 bp for 3 M EURIBOR. These are sizable impacts — even without taking account of the feedback effect from the reference rate to the individual interbank interest rates, which plays a central role in our theoretical analysis and which should further increase the effect of manipulation. The possibility of LIBOR manipulation during the global financial crisis has also been empirically examined by Gyntelberg and Wooldridge (2008), Abrantes-Metz, Kraten, Metz, and Seow (2012), and Kuo, Skeie, and Vickery (2012). Fourth, although we have not discussed recent changes in monetary policies, our framework is capable of taking these changes into account. After the global financial crisis, central banks in advanced economies largely shifted their policy instruments from short-term interest rates to quantitative measures, such as the monetary base or the quantity of government bond purchases. However, even under quantitative monetary easing regimes, many central banks virtually control short-term interest rates by adjusting the interest rates on excess reserves. In this environment, the range of variations in overnight interest rates is virtually controlled by central banks, and these rates are observable by individual banks — just as, in our setup, the reference rate is observable. However, quantitative easing enhances market-wide liquidity in the interbank market, which means that it becomes a major component of the aggregate factor determining interest rates in that market. In our framework, this situation is interpreted as meaning that aggregate factors play a more important role than idiosyncratic factors — so that, under Proposition 1, the sample size required for the reference rate to reduce market distortion is relatively large. 8. Conclusion In this study, we have investigated the potential role of the reference rate in an interbank market where individual banks cannot fully identify the nature of underlying shocks affecting their interbank transactions. We have found that the reference rate does not always mitigate the market distortion that arises when individual banks cannot fully identify the nature of underlying shocks affecting their interbank transactions. When the number of sample transactions is smaller than a certain threshold, the reference rate magnifies the distortion, even if the reference rate is not affected by any reporting noise. The threshold depends on the relative size of aggregate and idiosyncratic shocks. Noise in the reported interest rates, which can be increased by banks’ manipulations, distorts individual banks’ inferences about the underlying shocks, and thereby raises the threshold. When noise is highly correlated among multiple sample transactions, perhaps owing to collusive manipulations, it is possible that increasing the number of sample transactions may never mitigate the market distortion. To clarify the implications of the reference rate’s role as an information source, we have used a simple interest rate model with imperfect information, which is sufficiently general and potentially applicable to many theoretical models of interbank markets. One important extension for future research is to introduce banks’ incentive for manipulating their interest rates. Our study has assumed that such manipulations arise exogenously, by introducing stochastic noise in the reported interest rates. Although this treatment helps us draw clear-cut implications about the information-providing role of the reference rate, researches should examine how these implications might change with the introduction of an endogenous incentive for manipulation. For example, it might be the case that banks’ incentive to manipulate is large when panel size is quite small and each bank has strong potential to influence the reference rate. A future study might examine the quantitative implications of such a relationship. From a broader perspective, the impact of interbank market stability (or instability) on the aggregate economy, through the linkage between the interbank markets and other financial markets, should be examined using dynamic general equilibrium (DSGE) models, as Woodford (2010) has suggested.9 Furthermore, our analysis does not consider potential problems arising from the “offshore” nature of LIBOR. Some studies briefly explain why offshore rates are used as reference rates in many developed countries; however, future research should more deeply analyze the potential pitfalls.10 Our analysis gives some rationale for some recent proposals to reform the LIBOR mechanism. First of all, our study supports basing the reference rate on transactions, rather than quotes, since such a change reduces market distortions due to imperfect information, and thereby increases the reference rate’s stabilizing effect. Second, Wheatley (2012) recommended ensuring sufficiently large panel sizes for LIBOR, because (i) large panels imply that individual submissions have a limited impact on the published benchmark, weakening banks’ incentive to manipulate and (ii) an increase in the number of contributors could raise the overall representativeness of the LIBOR benchmark. The report also recommended ceasing the compilation and publication

9 Sudo (2012) explores the roles played by the interbank reference rate in business cycle fluctuations using a DSGE model with credit frictions, which is developed and estimated by Muto, Sudo, and Yoneyama (2016). 10 Gyntelberg and Wooldridge (2008) present some potential reasons why the offshore rate is preferred as the reference rate, including (i) fewer regulatory distortions, (ii) greater liquidity (for example, in the London and Singapore markets), and (iii) greater diversity of participants in offshore markets (making them less vulnerable to the actions of a few large institutions).

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of LIBOR for those currencies and tenors for which there are insufficient trade data to corroborate submissions. These recommendations are perfectly consistent with our main results, which indicate that large transaction samples reduce market distortions, and that a reference rate based on an insufficient number of sample transactions distorts the entire market. Third, our analysis also reveals that in the face of collusive manipulations, increasing the number of sample transactions does not mitigate the market distortion. This means that we need regulatory or supervisory measures to prevent collusion, or some fundamental reforms in the process of setting the reference rate (such as basing it solely on actual transaction data). In this direction, the Bank for International Settlements (2013) listed disciplines for improving the LIBOR mechanism to be shared among various institutions including (i) incorporating more information on the interest rates used for actual transactions, (ii) strengthening the governance of the rate-setting process, and (iii) improving the transparency of the rate-setting process. As these measures should reduce the variance and correlations of the noise of the individual interest rates in sample transactions, we strongly endorse these disciplines. Acknowledgments An earlier version of this paper, entitled “A simple interest rate model with unobserved components: The role of the interbank reference rate,” was written as a contribution to the report of “Towards better reference rate practices: A central bank perspective,” released on March 2013 by a working group established by the Bank for International Settlements Economic Consultative Committee (ECC) and chaired by Hiroshi Nakaso. The author is grateful to Koichiro Kamada, Kosuke Aoki, Teruyoshi Kobayashi, Takeshi Kimura, Hibiki Ichiue, Kenji Nishizaki, Takashi Nagahata, Nao Sudo, and Takemasa Oda for their advice and comments. The author is also indebted to Editor Ali M. Fatemi and two anonymous referees for helpful comments. The views expressed in this paper are those of the author and do not necessarily reflect the official views of the Bank of Japan. Appendix A. When banks use y j and yq to infer the aggregate component In Section 5, we assumed that individual banks use the information obtainable from the reference rate (yq ) only to infer the aggregate component (l). Here we assume that banks in the j th transaction additionally use the information on the fundamental shock in their individual transaction (y j ). This means that the banks’ estimators of l and ej take the following form: Ej l = g1 y j + g2 yq .

(A1)

Ej e j = y j − Ej l .

(A2)

Then, from Eqs. (2), (10), and (A1), the parameters of g1 and g2 are computed as follows: g1 =

g2 =

sl2 (1 + q)sl2 + se2 qsl2 (1 + q)sl2 + se2

,

(A3)

.

(A4)

By substituting Eqs. (A1) and (A2) into Eq. (1), we determine the interest rate in the jth transaction as follows: ij = ip + (ag1 + (1 − a)(1 − g1 ))y j + (1 − 2a)g2 yq .

(A5)

The average interest rate across this sample of transactions is iq =

q  1 p i + (ag1 + (1 − a)(1 − g1 ))yk + (1 − 2a)g2 yq q k=1

= ip + (a(g1 + g2 ) + (1 − a)(1 − g1 − g2 ))yq .

(A6)

Therefore, if the market-organizing institution provides iq to individual banks, the banks can obtain the value of yq , from the simple calculation (iq − ip )/(a(g1 + g2 ) + (1 − a)(1 − g1 − g2 )) = yq . From Eqs. (2), (10), and (A5), the market interest rate is im = lim

n→∞

n n  1 j 1  p i = lim i + (ag1 + (1 − a)(1 − g1 )) y j + (1 − 2a)g2 yq n→∞ n n j=1

j=1

q

= ip + ((1 − g1 − g2 ) + a(2(g1 + g2 ) − 1)) l + (1 − 2a)g2 e .

(A7)

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  m is Therefore, the distortion of the market interest rate from its normative level i m = im − im = (1 − 2a)(1 − g − g )l + (1 − 2a)g eq . i 1 2 2

(A8)

The variance of the market distortion is     m = (1 − 2a)2 (1 − g − g )2 s 2 + g2 s 2 /q . V ar i 1 2 l 2 e

(A9)

    m decreases monotonically with q . In the limiting case in which q → ∞, V ar i m converges to This indicates that V ar i    m zero. By comparing Eqs. (8) and (A9), we find that V ar i is smaller in the presence of the reference rate than it is in the absence of the reference rate if and only if the following condition holds: q>

sl4 − se4 se2 sl2

.

(A10)

Therefore, in order to mitigate the variance of the market distortion, q needs to be larger than the threshold value. These results are essentially the same as in Section 5.

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