20-Term Structure Page 617 Wednesday, February 4, 2004 1:33 PM
Term Structure Modeling and Valuation of Bonds and Bond Options 617Thus, the forward rates are viewed as the market’s consensus of future
interest rates. The return-to-maturity theory asserts that the return that
can be realized if a zero-coupon bond is held to maturity is the same
return expected by following a strategy of buying shorter term maturity
bonds and reinvesting them until the maturity of the zero-coupon bond.
For example, if an investor purchases a 5-year zero-coupon bond, then
the known return from holding that bond to maturity is the same as the
expected return from buying a 6-month bond today and reinvesting the
proceeds when it matures in another six-month bond and then continu-
ing to reinvest in six-month instruments until the end of the fifth year.
The yield-to-maturity theory asserts the same as in the return-to-matu-
rity theory except that this variant of the pure expectations theory is in
terms of periodic returns.
As Cox, Ingersoll, and Ross have demonstrated, these interpreta-
tions are not exact equivalents nor are they consistent with each other,
in large part because they offer different treatments of the two risks
associated with realizing a return (i.e., interest rate risk and reinvest-
ment risk). Furthermore, Cox, Ingersoll, and Ross showed that only one
of the five variants of the pure expectations theory is consistent with
equilibrium: the local expectations theory.Liquidity Theory
We have explained that the drawback of the pure expectations theory is
that it does not consider the risks associated with investing in bonds.
Nonetheless, there is indeed risk in holding a long-term bond for one
period, and that risk increases with the bond’s maturity because matu-
rity and price volatility are directly related. Given this uncertainty, and
the reasonable consideration that investors typically do not like uncer-
tainty, some economists and financial analysts have suggested a different
theory. This theory states that investors will hold longer-term maturities
if they are offered a long-term rate higher than the average of expected
future rates by a risk premium that is positively related to the term to
maturity. Put differently, the forward rates should reflect both interest
rate expectations and a “liquidity” premium (really a risk premium),
and the premium should be higher for longer maturities.
According to this theory, which is called the liquidity theory of the
term structure, the implied forward rates will not be an unbiased esti-
mate of the market’s expectations of future interest rates because they
embody a liquidity premium. Thus, an upward-sloping yield curve may
reflect expectations that future interest rates either (1) will rise, or (2)
will be flat or even fall, but with a liquidity premium increasing fast
enough with maturity so as to produce an upward-sloping yield curve.