Another look at yield spreads: the role of liquidity

Southern Economic Journal, April, 2008 by Dong Heon Kim

[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: