Chapter 7
The Liquidity Premium

Unpublished memorandum, 1979

Let the price c07-math-0001 at time u of a discount bond maturing at time v be described by the stochastic differential equation

1 equation

where c07-math-0003 is a Wiener process. As shown in Vasicek (1977) (Chapter 6 of this volume), the mean c07-math-0004 and volatility Οƒ(u, v) of the instantaneous rate of return are related by

2 equation

where c07-math-0006 is the spot rate and c07-math-0007 is the market price of risk. Eq. (1) can be written as

3 equation

Integrate Eq. (3) over u from t to c07-math-0009. We get

4 equation

Now differentiate this equation with respect to v. This produces

5 equation

where

6 equation

is the forward rate, and

7 equation

is the volatility of the forward rate c07-math-0014. The stochastic differential equation corresponding to the integral form (5) is

8 equation

We note that the drift of forward rates is fully determined by their volatilities and the pricing of risk.

The dynamic of the spot rate is described by

9 equation

where c07-math-0017 is the volatility of the spot rate. The drift of the spot rate is equal to the slope of the forward rate curve at the origin, less the market price of risk multiplied by the volatility of the spot rate. This is consistent with equations (21) and (22) in Vasicek (1977).

Put c07-math-0018 in Eq. (5). Then

10 equation

Taking the expectation as of time t yields the equation

11 equation

The liquidity premium (or term premium, as it should be called) c07-math-0021 is given by

12 equation

The liquidity premium in a term structure of interest rates has two components. The first component is driven by the market price of risk. It is equal to the expected integral over the span of the forward rate of the forward rate volatility multiplied by the market price of risk. There is, however, a second component, which is present even if the market price of risk is zero. This component, equal to the negative of the expected aggregate over the forward rate span of the bond price volatility times the forward rate volatility, arises as a result of the nonlinear relationship between prices and rates.

References

  1. Vasicek, Oldrich, A. (1977). β€œAn Equilibrium Characterization of the Term Structure.” Journal of Financial Economics, 5, 177–188.
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