The Mathematics of Financial Modelingand Investment Management

20-Term Structure Page 616 Wednesday, February 4, 2004 1:33 PM

616 The Mathematics of Financial Modeling and Investment Management

for five years; (2) invest in a 6-month instrument and when it matures,

reinvest the proceeds in six-month instruments over the entire 5-year

investment horizon; and (3) invest in a 2-year bond and when it matures,

reinvest the proceeds in a 3-year bond. The risk in the second and third

alternatives is that the return over the 5-year investment horizon is

unknown because rates at which the proceeds can be reinvested until

maturity are unknown.

As noted by John Cox, Jonathan Ingersoll, and Stephen Ross, in

practice, there are at least five variants of the pure expectations theory

that have been put forth in the financial literature.^5

1. Globally equal expected-holding-period return theory


2. Local expectations theory


3. Unbiased expectations theory


4. Return-to-maturity expectations theory


5. Yield-to-maturity theory^6

The globally expected-holding-period return theory asserts that the

expected return for a given holding period is the same regardless of the

maturity of the bonds held. So, for example, an investor who has a

holding period of three years is expected to have the same 5-year return

whether the investor (1) purchased a 1-year bond today and when it

matures reinvests the proceeds in a 4-year bond; (2) purchased a 2-year

bond today and when it matures reinvest the proceeds in a 3-year bond;

or (3) purchased a 10-year bond and sold it at the end of three years.

The globally expected-holding-period return theory is the broadest

interpretation of the pure expectations theory.

The second variant of the pure expectations theory, the local expec-

tations theory, is more restrictive about the relevant holding period for

which the returns are expected to be equal. It is restricted to short-term

holding periods that begin today. An investor with a 6-month holding

period, for example, would have the same expected return if (1) a 6-

month bond is purchased today; (2) a 3-year bond is purchased today;

or (3) a 20-year bond is purchased today.

The unbiased expectations theory asserts that the spot rates that the

market expects in the future are equal to today’s the forward rates.

(^5) John Cox, Jonathan Ingersoll, and Stephen Ross, “A Re-examination of Tradition-
al Hypotheses about the Term Structure of Interest Rates,” Journal of Finance (Sep-
tember 1981), pp. 769–799.
(^6) The labels for the last four variants of the pure expectations theory are those given
by Cox, Ingersoll, and Ross. The first label is given by McEnally and Jordan, “The
Term Structure of Interest Rates,” p. 829.