Another look at yield spreads: the role of liquidity.

Kim, Dong Heon

1. Introduction

The expectations hypothesis (hereafter EH) of the term structure of interest rates states that the long-term rate is determined by the expectation for the short-term rate plus a constant term premium. With rational expectations, one of the EH implications is that the coefficient in a regression of the change in expected future short-term interest rates on the current yield spread between long- and short-term rates is unity. Many empirical studies, such as Shiller, Campbell, and Schoenholtz (1983), Fama (1984), Mankiw and Miron (1986), Fama and Bliss (1987), Hardouvelis (1988), Mishkin (1988), Froot (1989), Simon (1989, 1990), Cook and Hahn (1990), Campbell and Shiller (1991), and Roberds, Runkle, and Whiteman (1996), among others, have shown mixed results, with more evidence against the EH than in its favor. (1) Even though Fama (1984), Hardouvelis (1988), Mishkin (1988), and Simon (1990) have found yield spreads do help predict future rates, the coefficient appears inconsistent with the EH. (2) Thornton (2006) argues that analysis should not be based on the slope coefficient of the test equation only, because even under the alternative hypothesis where the EH does not hold, one can have slopes that are numerically close to the theoretical ones.

Several studies have shown that even if the EH does hold, it would be hard to use it for forecasting due to interest rate smoothing by the Federal Reserve System (Fed). (3) On the other hand, a number of studies have focused on the possibility of a time-varying risk premium and concluded that a time-varying risk premium can help explain the failures of the EH. (4) From a study of time variation in expected excess bond returns, Cochrane and Piazzesi (2005) find that a single factor, which is a single tent-shaped linear combination of forward rates, predicts excess returns on one- to five-year maturity bonds and strengthens the evidence against the EH. Wachter (2006) shows that a consumption-based model in which external habit persistence from Campbell and Cochrane (1999) and the short-term interest rate that makes long bonds risky are the driving forces for time-varying risk premia on real bonds.

However, Evans and Lewis (1994) argue that a time-varying risk premium alone is not sufficient to explain the time-varying term premium observed in the Treasury bill. Dotsey and Otrok (1995) suggest that a deeper understanding of interest rate behavior will be produced by jointly taking into account the behavior of the monetary authority along with a more detailed understanding of what determines term premia. Diebold, Piazzesi, and Rudebusch (2005) state that from a macroeconomic prospective, the short-term interest rate is a policy instrument under the direct control of the central bank, which adjusts the rate to achieve its economic stabilization goals; from a finance prospective, the short rate is a fundamental building block for yields of other maturities, which are just risk-adjusted averages of the expected future short rate. They suggest that a joint macro-finance modeling strategy will provide the most comprehensive understanding of the term structure of interest rates.

Recently, Bansal and Coleman (1996) argued that some assets other than money play a special role in facilitating transactions, which affects the rate of return that they offer. In their model, securities that back checkable deposits provide a transactions service return in addition to their nominal return. Since short-term government bonds facilitate transactions by backing checkable deposits, this results in equilibrium with a lower nominal return for these bonds. Such a view implies that liquidity plays an important role in determining the returns of various securities. So, if liquidity is an important factor determining the returns of financial assets, liquidity may also be important for yield spreads and the term structure of interest rates. But how do investors consider liquidity in allocating their funds between securities of different terms? Since commercial banks are the principal investors and primary dealers in instruments such as federal funds, commercial paper (hereafter CP), and Eurodollar CDs, a study of liquidity demand by commercial banks may provide the key to answering this question. (5,6)

This paper attempts to answer the following question: Can the fact that liquidity plays an important role in explaining how banks determine their allocation of funds explain yield spreads and help provide an explanation for the failure of the EH?

If a bank might, at some point, be unable to turn its assets into ready cash, the bank faces a liquidity risk. Liquidity is a crucial fact of life for banks, and for this reason may have an implication for yield spreads and the term structure of interest rates. In addition, because banks' liquidity can vary as a result of Fed policy, financial market conditions, an individual bank's specific demand for reserves, and so on, banks' liquidity might play an important role in explaining time-varying term premia. Most previous studies, however, have not focused on banks as the main investors in financial markets and, thus, have not considered the role of banks' liquidity.

The paper begins by developing a model of a bank's optimal behavior. In terms of this banking model, the paper follows previous literature such as Cosimano (1987), Cosimano and Van Huyck (1989), Elyasiani, Kopecky, and Van Hoose (1995), Kang (1997), and Hamilton (1998), with the new addition of the cash-in-advance (CIA) constraint of Clower (1967), Lucas (1982), Svensson (1985), Lucas and Stokey (1987), and Bansal and Coleman (1996). In addition, this model incorporates a time-varying risk premium. (7) The paper finds that the CIA constraint plays an important role in determining yield spreads and the term structure of interest rates. The empirical part of the paper shows that the simple EH is not consistent with the empirical evidence, but the EH model allowing for the liquidity premium and the risk premium is a more realistic term structure model. This result implies that the EH might be salvaged when account is taken of the liquidity premium and the risk premium. (8)

The plan of this paper is as follows: Section 2 develops a model that incorporates liquidity into banks' optimal behavior and examines the determination of yield spreads and the term structure of interest rates. Section 3 provides a brief empirical test of the EH of the term structure and obtains empirical results for the theoretical model developed in section 2. A brief summary and concluding remarks are given in section 4.

2. The Model

This section develops a model that incorporates liquidity into banks' optimal behavior. In this model, bank loans are n-period assets and federal funds are one-period assets. There are many banks (N is number of banks) in the banking system. Banks have an infinite horizon. Each period consists of two sessions, the beginning of period t and the end of period t.

Banks' Optimal Behavior Subject to CIA Constraint

I assume that the reserve supply in the banking system does not change unless the Fed changes it. When borrowers cannot repay the loan principal as well as the loan interest rate payment, banks face risks on loans (default risk). I assume that this default risk increases as the quantity of loans or the maturity of loans increases. The model assumes that banks determine optimal balance sheet quantities by taking all interest rates as predetermined. Suppose that a bank j has the following profit function:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [L.sub.j,t] is the quantity of new n-period loans for bank j made in the beginning of period t, [r.sup.L.sub.n,t] is the yield on n-period loans made at time t, [r.sup.D.sub.t] is the yield on federal funds lent or borrowed at time t, [F.sub.j,t] is the federal funds lent or borrowed for bank j at time t, [r.sup.D.sub.t] is the deposit rate, [delta]'s are nonnegative constants, and [[bar.D].sub.j,t] is the level of demand deposits for bank j at time t. I assume that [[bar.D].sub.j,t] is taken to be exogenous. The expressions in parentheses represent the risk on loans each period.

In this case, a bank j has two options at time t - 1. One option is for the bank to hold reserves at the end of period t - 1 in order to lend them over n periods at the beginning of period t. The other option is that the bank rolls over reserves as federal funds for n periods. If bank./" lends to the public over n periods, the bank faces risks on loans and this default risk increases as the quantity of loans or the maturity of loans increases.

Bank j chooses the level of new loans, [L.sub.j,t], and lends this amount to the public at the beginning of period t. The bank will get back the loan at the beginning of period t + n. At the end of period t, it chooses the quantity of federal funds to lend, [F.sub.j,t]. These choices, along with some other exogenous or predetermined factors, determine the level of reserves, [R.sub.j,t], with which the bank will end the period. These other factors are the bank's level of demand deposits, [[bar.D].sub.j,t], with a positive value for [[bar.D].sub.j,t], [[bar.D].sub.j,t-1] i increasing the bank's end-of-period reserve position; the repayment of the federal funds the bank lent the previous period, [F.sub.j,t-1]; and the repayment of the loans the bank made n periods previously, [L.sub.j,t]. The bank's end-of-period reserves thus evolve according to (9)

[R.sub.j,t] = [R.sub.j,t] + [L.sub.j,t-n] + [F.sub.j,t-1] + [[bar.D].sub.j,t] - [[bar.D].j,t-1] - [L.sub.j,t] - [F.sub.j,t]. (1)

In addition, the bank must satisfy thte reserve requirement

[R.sub.j,t] [greater than or equal to][theta][[bar.D].sub.j,t], (2)

where [theta] is the required reserve ratio, (10)

At the beginning of period t, bank j starts with reserve balances given by [R.sub.j,t-1] + [L.sub.j,t-n], the transferred previous reserve balances plus the repayment of the loans made n periods previously. Given the loan rate and the federal funds rate, bank j must choose its loan supply, [L.sub.j,t], before knowing its deposits, [[bar.D].sub.j,t]. This choice is subject to predetermined holdings [R.sub.j,t-1] + [L.sub.j,t-n] of reserve balances. That is, in our model, a borrower from the bank wants cash, and the bank can only extend a loan to such a customer if it has cash (reserves) on hand equal to the amount of the loan. In this case, the bank's need for liquidity (sufficient reserves) can be for transaction purposes as well as the precautionary purpose of meeting depositor withdrawals or the reserve requirement.

I can model the bank's demand for liquidity in the same way that the CIA literature has modeled demand for liquidity by private households. In the conventional CIA formulation, goods must be purchased with cash, and a consumer can only obtain goods if he has cash on hand sufficient to pay for them. In this case, a household's demand for cash is mainly for transaction purposes. In terms of banks, loans are funded with reserves, and the bank can only have loan commitments with sufficient reserve balances. This kind of transaction reserve model can be compatible with Hamilton (1996, 1998), in which banks would want to hold reserves even if there were no reserve requirements for the same reason that members of the public hold cash: Namely, funds are needed to effect transactions. Reserves are useful to banks beyond the purpose of reserve requirements in the sense that reserves provide banks with a transactions service return. In this case, even though banks are subject to the required reserve constraint, the true value of the reserve for liquidity services might not be captured using only the shadow price of the required reserve constraint.

How should one think of the CIA constraint in the light of borrowed reserves? By studying the discount window borrowing by weekly reporting banks disaggregated by Federal Reserve District, Cosimano and Sheehan (1994) show that any single bank seldom visits the discount window and argue that this behavior is consistent either with banks not aggressively managing their discount window borrowings or, more plausibly, with the presence of considerable harassment costs imposed by the discount window officer. Hamilton (1998) states that banks act as if they faced a cost function for borrowing reserves from the Fed, and the cost of borrowing includes nonpecuniary costs in the form of additional regulation, supervision, and inferior credit standing with other banks. From this point of view, banks should not place excessive reliance on the discount window to obtain reserves and fund their loans to the public; thus, they might need to hold excess reserves for liquidity yield purposes. Hence, the CIA constraint that the bank faces might reflect the bank's demand for liquidity.

However, the strict CIA requirement ignores real-life institutions such as credit cards available to consumers or within-day overdraft privileges available to banks on their accounts with the Fed. Even so, the requirement of needing actual cash on hand for certain transactions seems to capture the key idea of what is meant by liquidity and has proven to be a useful framework for thinking about liquidity demand by private households. I propose that it may also be fruitful for seeing how the need for liquidity may impact the rates of return on assets of different maturities held by banks. Thus, I propose that bank j faces a CIA or liquidity constraint such as the following:

[L.sub.j,t] [less than or equal to][R.sub.j,t-1] + [L.sub.j,t-n], (3)

where [R.sub.j,t-1] is the reserve balance transferred from the previous period to the start of period t for the bank j. In this case, I assume that the repayment of outstanding loans contributes to the bank's liquidity balances at the beginning of period t. Thus, the bank enters period t with predetermined holdings of reserves, as the liquidity balances and the bank's new loans must obey the CIA constraint. (11)

The Equilibrium

Bank j faces the choice of new loans at the beginning of period t and the choice of federal funds at the end of period t. The state variables that are relevant for the bank's decision of the quantity of federal funds to lend at the end of period t are [L.sub.j,t], [L.sub.j,t-1], ..., [L.sub.j,t-n], [F.sub.j,t-1], [R.sub.j,t-1], and [[bar.D].sub.j,t]. Consider the following value function formulation of the decision at the end of period t:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

subject to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [U.sub.j,t](.) denotes the lifetime value of bank j's optimal program as of the second session of period t, whereas [V.sub.j,t+1](L.sub.j,t],[L.sub.j,t-1], ..., [L.sub.j,t- n+1],[F.sub.j,t],[R.sub.j,t],[[bar.D].sub.j,t]) is the value as of the first session of period t + 1, and [beta] denotes the discount factor.

The bank's problem is to maximize the value function [V.sub.j,t](.) subject to the CIA constraint (Eqn. 3) at the first session of period t and the value function [U.sub.j,t](.) subject to the balance-sheet constraint (Eqn. 1) and required reserve constraint (Eqn. 2) at the second session of period t. In other words, bank j chooses federal funds to maximize the value function [U.sub.j,t](.) subject to the bank's balance-sheet constraint and the required reserve constraint at the end of period t. The first-order conditions are as follows:

[r.sup.F.sub.t] + [beta][E.sub.t][partial derivative][V.sub.j,t+1]/[partial derivative][F.sub.j,t] = [beta][E.sub.t][partial derivative][V.sub.j,t+1]/[partial derivative][R.sub.j,t] + [[lambda].sub.j,t], (4)

[R.sub.j,t][greater than or equal to] [theta] [[bar.D].sub.j,t], with equality if [[lambda].sub.j,t] > 0, (5)

where [[lambda].sub.j,t] is the Lagrange multiplier of the required reserve constraint and represents the implicit price of this constraint. Note that the balance-sheet constraint can be substituted for [R.sub.j,t] in the value function and the required reserve constraint, so we do not have a Lagrange multiplier for the balance-sheet constraint.

At the beginning of period t, bank j starts with reserve balances given by [R.sub.j,t-1] + [L.sub.j,t-n]. Given the loan rate and the federal funds rate, bank j must choose its loan supply, [L.sub.j,t], before knowing the deposits, [[bar.D].sub.j,t]. The state variables for the decision at the beginning of period t are [L.sub.j,t- 1], [L.sub.j,t-2], ..., [L.sub.j,t-n], [F.sub.j,t-1], [R.sub.j,t-1], and [[bar.D].sub.j,t-1]. The value function of the beginning of period at time t is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

subject to

[L.sub.j,t] [less than or equal to] [R.sub.j,t-1] + [L.sub.j,t-n].

Maximizing the value function [V.sub.j,t](.) subject to the CIA constraint at the beginning of period t, I have the following first-order conditions:

[r.sup.L.sub.n,t] + [partial derivative][U.sub.j,t]/[partial derivative][L.sub.j,t] - ([[delta].sub.0] + [delta][L.sub.j,t]) = [[eta].sub.j,t], (6)

[L.sub.j,t-n] + [R.sub.j,t-1][greater than or equal to][L.sub.j,t], with equality if [[eta].sub.j,t] > 0, (7)

where [[eta].sub.j,t] is the Lagrange multiplier of the CIA constraint and represents the shadow price of this constraint. Using the envelope conditions from the value functions at the beginning of period t and at the end of period t, I have:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

since [beta][E.sub.t][[eta].sub.j,t+1] + [[lambda].sub.j,t] = [r.sup.F.sub.t], ..., [[beta].sup.n-1][E.sub.t][[eta].sub.j,t+n-1] + [[beta].sup.n-2][E.sub.t] [[lambda].sub.j,t+n-2] = [[beta].sup.n-2][E.sub.t][r.sup.F.sub.t+n-2] and [[beta].sup.n-1][E.sub.t][[lambda].sub.j,t+n-1] = [[beta].sup.n-1][E.sub.t] [r.sup.F.sub.t+n-1] - [[beta].sup.n][E.sub.t][[eta].sub.j,t+n]. Using the above equations and the first-order conditions, it is straightforward to characterize the equilibrium as satisfying the following equations: (12)

[r.sup.F.sub.t] = [beta][E.sub.t][[eta].sub.j,t+1] + [[lambda].sub.j,t], (8)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (9)

Equation 9 can be rewritten as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (10)

Equation 10 states that the n-period loan rate is the weighted average of the current federal funds rate and expected future federal funds rate over n period, plus the cost resulting from the risk (or transaction costs) on loans and the cost of loss of the liquidity benefit for bank j. The term in parentheses on the right-hand side is the risk premium, and the second bracket of the right-hand side is the liquidity premium. The liquidity premium depends on the difference between the current shadow price of the CIA constraint and the expected shadow price of the CIA constraint at time t + n. If the bank j expects the liquidity benefit of the loan lent at time t to be higher at the beginning of time t + n when the bank gets back the loan lent at the beginning of time t, the bank does not require as much compensation for liquidity loss. On the other hand, if the bank expects the liquidity benefit of the loan lent to be lower at the beginning of time t + n than today, the bank will require more compensation for liquidity loss. Bank j chooses [F.sub.j,t] (and hence a value for [[lambda].sub.j,t] and [E.sub.j,t] and [[eta].sub.j,t+1]) so as to satisfy Equation 8.

Implication for the Aggregate Banking

To consider the implication for aggregate banking (market equilibrium), I sum Equation 1 and take an average for Equations 8-10. This yields

[r.sup.F.sub.t] = [beta][E.sub.t][[eta].sub.t+1] + [[lambda].sub.t], (11)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (12)

[R.sub.t] = [R.sub.t-1] + [L.sub.t-n] + [F.sub.t-1] + [[bar.D].sub.t] - [[bar.D].sub.t-1] - [L.sub.t] - [F.sub.t], (13)

where [[lambda].sub.t] = 1/N [[summation].sup.N.sub.j] = 1 [[lambda].sub.j,t], [E.sub.t][[eta].sub.t+1] = 1/N [[summation].sup.N.sub.j] = 1 [E.sub.t][[eta].sub.j,t+1], [R.sub.t-i] = 1/N [[summation].sup.N.sub.j = 1] [R.sub.j,t- i], for i = 0, 1,

[L.sub.t-n] = 1/N [[summation].sup.N.sub.j=1][L.sub.j,t-n], [F.sub.t-1] = 1/N [[summation].sup.N.sub.j=1][F.sub.j,t- 1], [[bar.D].sub.t-1] = 1/N [[summation].sup.N.sub.j=1][[bar.D].sub.j,t-i], for i = 0, 1, [L.sub.t] = 1/N [[summation].sup.N.sub.j=1][L.sub.j,t], and [F.sub.t] = 1/N [[summation].sup.N.sub.j=1][F.sub.j,t].

In equilibrium, [F.sub.t] must be zero, and the exogenous supply of reserves and demand for loans will determine [[lambda].sub.t] and [E.sub.t][[eta].sub.t+1], which, together with Equation 11, determine [r.sup.F.sub.t]. The interest rate adjusts to clear the market. Equation 11 bears an analogy with Svensson's (1985) paper. It states that the current federal funds rate is the sum of the discounted expected value of next period's shadow price of the CIA constraint and the shadow price of the required reserve constraint at time t. Even if the required reserve constraint is non-binding and the shadow price of it is zero, the existence of a binding liquidity constraint would warrant a positive federal funds rate. So reserves are held against federal funds for the future liquidity services they provide, and the value of these liquidity services is the value of relaxing the future liquidity constraint. Equation 12 is an Euler equation and an optimizing condition between the loan market and the federal funds market. This equation implies that the marginal benefit of increasing the volume of loans is equal to the marginal benefit of lending federal funds. Equation 12 can be rewritten as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (14)

Equation 14 states that the n-period loan rate is the weighted average of the current federal funds rate and the expected future federal funds rate over n period plus the cost resulting from the risk (or transaction costs) on loans and the cost of loss of the liquidity benefit. Multiplying Equation 12 by [beta], taking the expectation at time t - 1, and rearranging I get:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (15)

Equation 15 is the term structure model that incorporates a time-varying liquidity premium and a risk premium; given information at time t - 1. One point to make is that if the CIA constraint is binding, monetary policy draining or injecting reserve balances can have an impact not only on the federal funds rate and the loan rate, but also on the term premium and, thus, monetary policy can affect the term structure of interest rates.

Term Structure Implication

Equation 15 determines the term structure of interest rates between the n-period loan rate and the one-period federal funds rate. This term structure model incorporates time-varying liquidity and risk premia. Since banks' liquidity can vary over time, the liquidity difference shows up on the term structure model as the liquidity premium. Hence, banks' liquidity plays an important role in explaining the time-varying term premium and thus can help to explain the widespread rejection of the EH. Consequently, a term structure model that incorporates banks' liquidity demand into banks' optimal behavior might provide an alternative to the simple EH.

For clarity, setting [beta] = 1, Equation 15 can be rewritten as follows:

[r.sup.L.sub.n,t] = 1/n [E.sub.t][n-1.summation over (i=0)][r.sup.F.sub.t+i] + ([[delta].sub.0] + [[delta].sub.1][L.sub.t]) + 1/n [[[eta].sub.t] - [E.sub.t][[eta].sub.t+n]]. (16)

To explore whether this model can quantitatively match the features of the EH under the maintained hypothesis concerning the liquidity premium and risk premium, I need quantitative magnitudes of the shadow prices. Recall from Equation 11 that, in equilibrium, the current federal funds rate is the expected value of next period's discounted shadow price of the CIA constraint, assuming the required reserve constraint is not binding. (13) Thus, we can take conditional expectations of both sides of Equation 16 based on information available at t - 1, as in Equation 15, to use the previous federal funds rate as the expected value of the discounted shadow price of the CIA constraint. Similarly, I can assume [E.sub.t-1] [r.sup.F.sub.t+n-1] = [E.sub.t-1][[eta].sub.t+n]. Then, Equation 16 can be rewritten

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (17)

Further assuming rational expectations, let [v.sub.t+i], i = 0, 1, 2, ..., n - 1, [e.sub.t], and [[xi].sub.t] denote the subsequent forecast errors orthogonal to information available at time t - 1: (14)

[v.sub.t+i] = [r.sup.F.sub.t+i] - [E.sub.t-1][r.sup.F.sub.t+i], i = 0, 1, 2, ..., n - 1, (18)

[e.sub.t] = [r.sup.L.sub.n,t] - [E.sub.t-1][r.sup.L.sub.n,t], (19)

[[xi].sub.t] = [L.sub.t] - [E.sub.t-1][L.sub.t]. (20)

I also assume that [e.sub.t], [[xi].sub.t] and [v.sub.t+j] for all j's are serially uncorrelated and mutually independent. Substituting Equations 18 20 into Equation 17 and rearranging it, I get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (21)

Equation 21 provides plausible parameter values for the risk premium and liquidity premium under the above assumptions. The second term on the right-hand side of Equation 21, [[delta].sub.0] + [[delta].sub.1][L.sub.t], and the third term, 1/n (r.sup.F.sub.t-1] - [r.sup.F.sub.t+n-1]), capture the risk premium and the liquidity premium, respectively. Subtracting [r.sup.F.sub.t] from both sides of Equation 21 and rearranging results in

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (22)

The model 22 implies that the simple EH does not hold because of the liquidity premium and the risk premium. I can estimate this model and examine if the EH might be a more realistic model when the liquidity premium and the risk premium play a role.

3. Estimation of the Term Structure Model

The Data

The weekly data set we use runs from February 1, 1984, to December 27, 2000, giving 882 observations, (15) Interest rates are taken from Federal Reserve Statistical Release H. 15 provided by the Federal Reserve Board of Governors. I also take quantities of loans from item H.8 (assets and liabilities of all commercial banks in the United States) of the Federal Reserve Statistical Release. (16) Since this series refers to outstanding loans at each period, I use the change in this series as a proxy for the volume of new loans extended. We consider federal funds rates as a short-term rate and one-month and three-month CP rates as long-term rates. All interest rates are averages of seven calendar days ending on Wednesday and annualized using a 360-day year. (17) Here, the CP rate is viewed as a substitute for the rate on banks' loans to financial and industrial companies.

Figure 1 shows movements of the federal funds rate and the one-month and three-month CP rates during this sample. One interesting feature is that the federal funds rate fluctuated around the CP rates before the 1990 U.S. recession, but after the recession the CP rates were higher than the federal funds rate. Sample period averages of the federal funds rate, one-month, and three-month CP rates are 6.23%, 6.28%, and 6.29%, respectively, and thus the one-month and three-month CP rates are higher on average by five and six basis points, respectively, than the federal funds rate. However, the standard deviation of the federal funds rate is 1.93, higher than those of the one-month and three-month CP rates, 1.84 and 1.82, respectively, which implies that the volatility of the federal funds rate is somewhat higher than those of the CP rates.

[FIGURE 1 OMITTED]

A Test of the Expectations Hypothesis

I start from a test of the simple EH, which implies that the long-term rate is a weighted average of the current short-term rate and expected future short-term rates and that the current spread between the long-term rate and short-term rate predicts the change in future short-term rates. That is,

[r.sup.L.sub.n,t] = 1/n [E.sub.t][n-1.summation over (i=0)][r.sup.F.sub.t+i], (23)

where [r.sup.F.sub.n,t] and [r.sup.F.sub.t] are the n-period CP rate (our substitute for the n-period loan rate) and one-period federal funds rate, respectively. Assuming rational expectations, one can rearrange Equation 23 to yield the following relationship as the term structure regression for empirical investigation:

Model I: [[delta].sub.0] = [[delta].sub.1] = 0, [[eta].sub.t] = [E.sub.t][[eta].sub.t+n] 1/n [n-1.summation over (i=0)][r.sup.F.sub.t+i] - [r.sup.F.sub.t] = [alpha] + [phi] ([r.sup.L.sub.n,t] - [r.sup.F.sub.t]) + [[epsilon].sub.t], (24)

where [[epsilon].sub.t] = 1/n [n-1.summation over (i=0)][r.sup.F.sub.t+i] - 1/n [E.sub.t][n-1.summation over (i=0)] [r.sup.F.sub.t+i] and should be uncorrelated with any variable known at time t. Here, n corresponds to 4 or 12 weeks for one- and three-month commercial paper, respectively. Equation 24 can be estimated by OLS with autocorrelation-heteroskedasticity consistent errors. According to the simple expectations hypothesis, [alpha] = 0, and [phi] = 1. This test can be nested within our term structure model (Equation 22) by imposing the restrictions [[delta].sub.0] = [[delta].sub.1] = 0, and [[eta].sub.t] = [E.sub.t][[eta].sub.t+n].

Table 1 shows the results for estimation of Equation 24. The coefficient on the spread is significantly less than unity and different from zero at conventional significance levels. In addition, the estimated coefficients on the constant are significantly less than zero. These results are very similar to those of previous empirical studies. (18)

Test of the Expectations Hypothesis with Liquidity Premium and Risk Premium

Our model developed in section 2 implies that the simple EH does not hold because of the liquidity premium and the risk premium. Since banks' optimal behavior is subject to a CIA constraint, banks' liquidity causes the shadow price of the CIA constraint to play an important role in yield spreads and the term structure of interest rates. Thus, the model suggests that I need to incorporate a liquidity premium and a risk premium into the simple EH. The model (Equation 22) forms the basis of the tests of the term structure on which to focus. Subtracting 1/n [r.sup.F.sub.t+n-1] from both sides of Equation 22 to avoid including the ex-post future interest rate in the equation as a regressor, I get:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (25)

Then, Equation 25 involves running the regression:

Model II: [[delta].sub.0][not equal to]0, [[delta].sub.1][not equal to]0, [E.sub.t-1][[eta].sub.t] = [r.sup.F.sub.t-1], [E.sub.t-1][[eta].sub.t+n] = [E.sub.t-1][r.sup.F.sub.1+n-1]

1/n [n-2.summation over (i=0)][r.sup.F.sub.t+i] - [r.sup.F.sub.t] = [alpha] + [phi]([r.sup.L.sub.n,t] - [r.sup.F.sub.t] + [[gamma].sub.1][r.sup.F.sub.t-1] + [[gamma].sub.2][L.sub.t] + [[epsilon].sub.t], (26)

where [L.sub.t] is the quantity of new loans at time t, and [[epsilon].sub.t] = 1/n [[summation].sup.n-2.sub.i=0][v.sub.t] + i + [[delta].sub.1][[xi].sub.t] - [e.sub.t]. Regression 26 differs from all tests of EH in the existing literature, where the regressand is not 1/n [[summation].sup.n-2.sub.i=0][r.sup.F.sub.t], but 1/n [[summation].sup.n-1.sub.i=0] [r.sup.F.sub.t+i] - [r.sup.F.sub.t]. Equation 26 cannot be estimated by OLS because [[epsilon].sub.t] is correlated with the regressors [r.sup.L.sub.n,t] and [r.sup.F] Rational expectations require [[epsilon].sub.t] to be uncorrelated

with anything known to banks at time t - 1, but [r.sup.L.sub.n,t] and [r.sup.F.sub.t] are not in the date t - 1 information set. However, Equation 26 can be estimated by instrumental variables using valid instruments. I consider 2SLS with constant, lagged federal funds rates, and lagged quantities of new loans as instruments. I employ Hansen's (1982) method in order to check the overidentifying restrictions and, thus, test these conjectures about the correct set of instruments. Hansen's test statistic has an asymptotic [chi square] distribution with r - k degrees of freedom if the model is correctly specified, where r is the number of instruments and k is the number of estimated coefficients. According to Model II, [alpha] = [[delta].sub.0], [phi] = 1, [[gamma].sub.1] = [-n.sup.-1], and [[gamma].sub.2] = -[[delta].sub.1]. Table 2 shows the results. In both cases, the estimated coefficients on the spread are close to unity. Indeed, the t-statistic for testing the null hypothesis that the coefficient of the spread is unity is -0.564 for the one-month CP rate and -0.053 for the three-month CP rate, respectively, and thus in both cases implies that the estimated coefficients are not statistically significantly different from unity at the 5% level, in contrast with the estimated coefficients on the spread in Model I. In addition, following Hansen's (1982) method, [[chi square].sub.1] for the one-month CP rate and [[chi square].sub.2] for the three-month CP rate are 1.288 and 5.816, respectively, so the null hypothesis that Model II is correctly specified is accepted at the 5% level. These results imply that these instruments are valid. (19)

All the estimated coefficients have the signs predicted by the theoretical model developed in section 2, and all estimated coefficients are statistically significant at the conventional level except the coefficients on [L.sub.t]. In particular, the estimated coefficients on the liquidity premium are close to the values that the theoretical model implies (-0.25 for the one-month CP rate and -0.084 for the three-month CP rate) and statistically significant, indicating that liquidity plays an important role in explaining the term premium. However, among two components reflecting the risk premium, the constant is statistically significant, whereas the estimated coefficients on Lt are not significantly different from zero.

Another Application: Euro-Dollar Rates

Since I did not have a direct measure of the loan rate, I used the CP rates as a proxy for the bank loan rate instead. However, as pointed out in Kashyap, Stein, and Wilcox (1993), bank loans are special, and the commercial paper might be an imperfect substitute. In addition, Stigum (1990) and Cook and LaRoche (1993) state that historically the CP market has been remarkably free of default risk in contrast to bank loans. From this point of view, the CP rate might not be a good choice of proxy. To investigate this issue, I consider the Euro-dollar (hereafter ED) rate as another proxy. Even though the ED is a liability and the ED rate is a deposit rate, it is subject to default risk and can fluctuate in accordance with banks' liquidity demand. The one-month and three-month ED rates are taken from Statistical Release provided by the Federal Reserve Board of Governors for the sample.

Figure 2 plots movements in these ED rates and the federal funds rate. The ED rates show very similar movements to the CD rates displayed in Figure 1. Table 3 shows estimation results for Model I and Model II.

[FIGURE 2 OMITTED]

The estimated coefficients on the spread are close to unity, and the t-statistic for testing the null hypothesis that the coefficient of the spread is unity, is - 1.264 for the one-month ED rate and -0.267 for the three-month ED rate, respectively, and thus I cannot reject the null hypothesis at the 5% level in both cases. In the case of Model II, Hansen's test statistics are 0.7 for the one-month ED rate and 10.29 for the three-month ED rate, respectively, and I do not reject the null, indicating that the model is correctly specified. The estimated coefficients on the liquidity premium are statistically significant and quite close to the values expected by the theoretical model; although, the estimated coefficients on the risk premium components are not significantly different from zero in contrast to the case of the CP rates. Overall, the results are qualitatively similar to those of the CP rates.

4. Conclusion

This paper has focused on commercial banks as the main investors in financial markets, and banks' liquidity as an important component that determines yield spreads over securities with different terms. To this end, I have developed a term structure model that incorporates liquidity demands by commercial banks into a model for banks' optimal behavior. The paper has shown that the shadow price of the CIA constraint plays an important role in determining the yield spread. Moreover, the empirical study has shown that the simple EH is not consistent with the empirical evidence, but when we incorporate the liquidity premium and the risk premium resulting from transaction activities into the term structure of interest rates, the EH model describes yield spreads and the term structure more realistically. This result implies that the EH might be salvaged under the maintained hypothesis concerning the liquidity premium and the risk premium.

The results of the paper have an important implication. As most households' transaction activities are subject to their liquidity conditions, so are banks' transaction activities. When banks allocate their funds into financial securities of different maturities, they incorporate information about liquidity as well as risk into their portfolio management decisions. Investors know that long-term assets are relatively less liquid than short-term assets, and the difference in liquidity among these financial assets is incorporated into their returns. This might be one reason why previous studies have not produced a consensus about the empirical failure of the simple EH. I feel that the story presented here provides a useful alternative to the simple EH.

This paper is based on chapter 1 of the author's University of California San Diego PhD dissertation. Deep discussions with James D. Hamilton are gratefully acknowledged. For helpful comments and suggestions, the author thanks Kent Kimbrough (coeditor), two anonymous referees, Keith Blackburn, Wouter Den Haan, John Duca, Marjorie Flavin, Andreas Gottschling, Takeo Hoshi, Garett Jones, Alex Kane, Chang-Jin Kim, Soyoung Kim, Jong-Wha Lee, Denise Osborn, Marianne Sensier, Kwan Ho Shin, and seminar participants at Korea University, University of California San Diego, University of Manchester, University of Nottingham, University of Southampton, Bank of Korea, Korean Econometric Society Macroworkshop, RES Conference 2001, Western Economic Association Meeting 2001, and KAEA/KEA Conference 2002. Financial support from Korea University Grant, the Institute of Economic Research at Korea University, University of Manchester, and the Royal Economic Society is gratefully acknowledged.

Received March 2006; accepted November 2006.

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(1) Rudebusch (1995) refers to the 'U-shaped' pattern of the predictability of the yield curve. Roberds and Whiteman (1999) state the existence of a "predictability smile" in the term structure of interest rates: Spreads between long maturity rates and short rates predict subsequent movements in interest rates provided the long horizon is three months or less or if the long horizon is two years or more, but not for intermediate maturities.

(2) For recent EH tests based on VAR specification, see Bekaert and Hodrick (2001), Thornton (2004), Bataa, Kim, and Osborn (2006), Bekaert, Wei, and Xing (2007), and Sarno, Thornton, and Valente (2007).

(3) See Mankiw and Miron (1986), McCallum (1994), and Rudebusch (1995) for the relation between the EH and Federal Reserve behavior.

(4) For a time-varying risk premium, see Engle, Lilien, and Robins (1987), Simon (1989, 1990), Friedman and Kuttner (1992), and Lee (1995), among others.

(5) Stigum (1990) states that "in the money market, in particular, banks are players of such major importance that any serious discussion of the various markets that comprise the money market must be prefaced with a careful look at banking" (p. 117). Cook and LaRoche (1993) also emphasize that commercial banks play an important role in the money market. Kashyap, Rajan, and Stein (2002) argue that a bank that offers liquidity on demand must invest in certain costly "overhead" in order to carry out its job effectively and that since banks often lend via commitments, their lending and deposit-taking may be two manifestations of one primitive function the provision of liquidity on demand. Their model shows that there is a synergy between the two activities to the extent that both require banks to hold large balances of liquidity assets.

(6) Detailed information on the amount of Treasuries of alternate maturity held by banks is not available on the Flow of Funds. However, commercial banks' holdings of securities account for 23% of bank assets as of January 2006 (www.federalreserve.gov/releases/h8/current). The volume of transactions of government securities by commercial banks was about 20% in the money market in the 1980s (Stigum 1990).

(7) Since the paper draws attention to the role of liquidity in determining the yield structure, it might be interesting to consider incorporation of liquidity into a dynamic general equilibrium model. For example, this idea can be extended to the situation where long-term government bonds are less liquid than short-term government bonds, and thus the liquidity difference might affect the yield structure of these assets.

(8) Since most of the empirical term structure literature concentrates on the term structure for government securities, our term structure model might be limited for general application. Even so, the key idea of what is meant by liquidity seems to provide a significant implication for the term structure.

(9) We assume that interest income less default loss is paid out as stockholder dividends, and so these terms do not affect the level of reserves. In practice, federal funds interest is paid by banks a week after the loan, and term loan interest is paid much later. Abstracting from the effects of interest payments on reserves seems a useful simplification, which is unlikely to matter for the results presented here.

(10) The analysis of Anderson and Rasche (2001) suggests that the willingness of bank regulators to permit use of deposit sweeping software has made statutory reserve requirements a "voluntary constraint" for most banks, and thus the economic burden of statutory reserve requirements is zero.

(11) Chami and Cosimano (2001) point out that capital requirements have effectively replaced reserve requirements as the main constraint on the behavior of banks and explore the implications of risk-based capital requirements for monetary policy. From this point of view, an anonymous referee raised an issue about the possibility that the result of the paper would change as a result of the phase out of reserve requirements in terms of Chami and Cosimano (2001). However, the CIA constraint (or liquidity constraint) is neither reserve requirement nor capital requirement in the spirit of replacement of liquidity needs, and thus this issue might not erode qualitatively the role of the CIA constraint in the model presented here.

(12) More details for the derivation of Equation 8-10 are provided in the appendix of Kim (2003).

(13) Frost (1971) shows that banks hold excess reserves because the cost associated with constantly adjusting reserve positions is greater than the interest earned on short-term securities, and the profitability of holding excess reserves when interest rates are very low makes the banks' demand for excess reserves kinked at a low rate of interest. In practice, excess reserves from the data for depository institution that are taken from Statistical Release provided by the Federal Reserve Board of Governors are always positive during the sample period of our empirical study. Therefore, our assumption is consistent with the U.S. data over this sample period.

(14) The anonymous referee raised an issue about the possibility of correlation between shocks in Equations 18-20. Indeed, we might consider that the disturbance to the federal funds market affects immediately the demand for the loan and thus the loan market. However, since we are not focusing on the effect of structural shocks on the term structure, we assume that these structural shocks are mutually independent.

(15) Because the Fed changed from lagged reserve accounting to contemporaneous reserve accounting in February 1984, we use only data after February 1984.

(16) These quantities of loans are loans and leases in bank credit by weekly reporting banks. These quantities are slightly different depending on all commercial banks, domestic commercial banks, and large commercial banks. However, the estimation results were quantitatively and qualitatively similar.

(17) The original CP rates are business-daily averages of offering rates on CP placed by several leading dealers for firms whose bond rating is AA or the equivalent. After we multiply Friday's interest rate by three and use the value of the previous business day for holidays, we construct weekly averages of seven-day series.

(18) Rudebusch (1995) provides an excellent survey of previous empirical results.

(19) When we included lagged CP rate as an instrument on the estimation of Equation 26, we rejected the null hypothesis of Hansen's test, which implies that lagged CP rates are not valid instruments. In addition, we estimated the equation using more lagged federal funds rates and lagged quantities of new loans as instruments. In some cases, we rejected the null hypothesis that the model is correctly specified, but overall the estimated results were similar.

Dong Heon Kim, Department of Economics, Korea University, 5-1 Anam-dong, Seongbuk-Gu, Seoul, 136-701, South Korea; E-mail [email protected], University of Manchester, UK.

Table 1. The Expectation Hypothesis Test without
Liquidity Premium and Risk Premium

 1/n [n - 1.summation over (i = 0)] [r.sup.F.sub.t] =
 [alpha] + [phi] ([r.sup.L.sub.n,t] - [r.sup.F.sub.t])
 + [[epsilon].sub.t]

Maturity [??] [??] [R.sup.2]

One-month CP -0.020 ** (0.009) 0.297 *** (0.083) 0.169
Three-month CP -0.052 * (0.028) 0.488 *** (0.082) 0.218

* The numbers in parentheses are Newey and West's (1987)
autocorrelation-heteroskedasticity consistent standard errors
corrected with four lags for one-month CP rate and 12 lags for
the three-month CP rate. ***, **, and * denote statistical
significance at the 1%, 5%, and 10% level in a two-tailed test,
respectively.

Table 2. The Expectations Hypothesis Test with Liquidity
Premium and Risk Premium: Two-Stage Least Squares Estimation

 1/n [n-2.summation over (i = 0)] [r.sup.F.sub.t+i] -
 [r.sup.F.sub.t] = [alpha] + [phi]([r.sup.L.sub.n, t] -
 [r.sup.F.sub.t]) + [[gamma].sub.1][r.sup.F.sub.t-1] +
 [[gamma].sub.2][L.sub.t] + [[epsilon].sub.t]

Maturity [??] [??]

One-month CP -0.308 *** (0.069) 0.903 *** (0.172)
Three-month CP -0.385 *** (0.129) 0.989 *** (0.208)

 1/n [n-2.summation over (i = 0)] [r.sup.F.sub.t+i] -
 [r.sup.F.sub.t] = [alpha] + [phi]([r.sup.L.sub.n, t] -
 [r.sup.F.sub.t]) + [[gamma].sub.1][r.sup.F.sub.t-1] +
 [[gamma].sub.2][L.sub.t] + [[epsilon].sub.t]

Maturity [[??].sub.1] [[??].sub.2]

One-month CP -0.206 *** (0.010) -0.005 (0.011)
Three-month CP -0.047 ** (0.021) 0.027 (0.019)

 1/n [n-2.summation over (i = 0)] [r.sup.F.sub.t+i] -
 [r.sup.F.sub.t] = [alpha] + [phi]([r.sup.L.sub.n, t] -
 [r.sup.F.sub.t]) + [[gamma].sub.1][r.sup.F.sub.t-1] +
 [[gamma].sub.2][L.sub.t] + [[epsilon].sub.t]

Maturity Instrument

One-month CP constant, [r.sup.F.sub.t-2], ..., [r.sup.F.sub.t-4],
 [L.sub.t - 1]
Three-month CP constant, [r.sup.F.sub.t-1], ..., [r.sup.F.sub.t-4],
 [L.sub.t - 2]

The numbers in parentheses are Newey and West's (1987)
autocorrelation-heteroskedasticity consistent standard
errors corrected with four lags
for the one-month CP rate and 12 lags for the three-month CP
rate. *** and ** denote statistical significance at the 1%
and 5% level in a two-tailed test of the null hypothesis that
the coefficient of the spread is zero, respectively. In addition,
the t-statistic for testing the null hypothesis that the coefficient
of the spread is unity is -0.564 for one-month CP rate and -0.053
for three-month CP rate, respectively.

Table 3. The Expectations Hypothesis Test with Liquidity
Premium and Risk Premium: Federal Funds Rate and the ED Rates

Model Maturity [??] [??]

M. I One-month -0.033 *** (0.009) 0.334 *** (0.094)
 Three-month -0.107 *** (0.024) 0.457 *** (0.094)

M. II One-month -0.080 (0.050) 0.813 *** (0.148)
 Three-month -0.023 (0.085) 0.972 *** (0.105)

Model Maturity [[??].sub.1] [[??].sub.2]

M. I One-month
 Three-month

M. II One-month -0.250 *** (0.006) 0.004 (0.011)
 Three-month -0.114 *** (0.015) 0.005 (0.006)

Model Maturity Instrument

M. I One-month
 Three-month

M. II One-month constant, [r.sup.F.sub.t-2],
 ..., [r.sup.F.sub.t-4], [L.sub.t-1]
 Three-month constant, [r.sup.F.sub.t-2],
 ..., [r.sup.F.sub.t-12], [L.sub.t-2]

The numbers in parentheses are Newey and West's (1987) autocorrelation-
heteroskedasticity consistent standard errors corrected with four lags
for the one-month ED rate and 12 lags for the three-month ED rate. ***
denotes statistical significance at the 1% level in a two-tailed test.
M.I and M.II denote the Model I in Equation 24 and the Model II in
Equation 26, respectively.