Another look at yield spreads: the role of liquidity.

• Author: 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...