THE ECONOMICS OF MONEY,BANKING, AND FINANCIAL MARKETS 167

CHAPTER 6

The Risk and Term Structure of Interest Rates

135

Continuing with these calculations, we obtain the general solution for the forward
rate n periods into the future:
i et

(1
n

in
(1

)n
int )n

1

1t

1

(5)

Our discussion indicated that the expectations theory is not entirely satisfactory
because investors must be compensated with liquidity premiums to induce them
to hold longer-term bonds. Hence, we need to modify our analysis, as we did
when discussing the liquidity premium theory, by allowing for these liquidity premiums in estimating predictions of future interest rates.

Recall from the discussion of those theories that because investors prefer to
hold short-term rather than long-term bonds, the n-period interest rate differs from
that indicated by the pure expectations theory by a liquidity premium of *nt. So to
allow for liquidity premiums, we need merely subtract *nt from int in our formula
to derive i et n:
i et

(1
n

in
(1

1t

int

*n 1t )n
*nt )n

1

1

(6)

This measure of i et n is referred to, naturally enough, as the adjusted forward-rate
forecast.
In the case of i et 1, Equation 6 produces the following estimate:
i et

1

=

11 + i2t - *2t22
1 + it

- 1

Using the example from the Liquidity Premium Theory Application on page 128,
at time t the *2t liquidity premium is 0.25%, *1t = 0, the one-year interest rate is 5%,
and the two-year interest rate is 5.75%. Plugging these numbers into our equation
yields the following adjusted forward-rate forecast for one period in the future:
i et

1

=

11 + 0.0575 - 0.002522
1 + 0.05

- 1 = 0.06 = 6%

which is the same as the expected interest rate used in the Application on expectations theory, as it should be.
Our analysis of the term structure thus provides managers of financial institutions with a fairly straightforward procedure for producing interest-rate forecasts.
First they need to estimate *nt, the values of the liquidity premiums for various n.
Then they need merely apply the formula in Equation 6 to derive the market s forecasts of future interest rates.