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Answer: The expectations hypothesis says that the equilibrium in the bond market is such that
investors earn the same expected return over a given time horizon whether they buy a single
long-term bond or a sequence of short-term bonds. In this example, suppose an investor buys a 3-
year bond. Their total return will be (1.03)3 – 1 = 9.2727%. Or to say this another way, $1
invested in the 3-year bond will grow to $1(1.03)3 = $1.092727 over 3 years.
Suppose instead that an investor buys a 1-year bond and then when that bond matures, the
investor buys a brand new 2-year bond. It is the rate on this second bond that the question is
asking about….what is the rate on a 2-year bond, 1 year from now? The expectations hypothesis
says that the investor who buys the 1-year bond today and then buys another 2-year bond should
earn the same return as an investor who buys the 3-year bond today. Both investors are investing
over a 3-year horizon, and they should earn the same return no matter what type of bonds they
hold over that horizon. So let E(r) be the annual expected return on the 2-year bond, two years
from now. The investor who buys the 1-year bond today and then the 2-year bond after that will
earn the following return:
(1.025) × (1+E(r))2 – 1
Or we could say that $1 invested in this way will grow to $1(1.025)(1+E(r))2 over the 3 years.
But as already stated, this has to equal what the other investor earns, so equating the returns from
the two strategies we have
(1.025)(1+(E(r))2 = 1.092727
Solve for E(r):
(1+E(r))2 = 1.0927/1.025
1 + E(r) = (1.092727/1.025)(0.5)
E(r) = 0.0325 = 3.25%
Here’s an intuitive way to get this answer.
For simplicity, ignore compounding for a moment. “Investor A” buys the 3-year bond paying 3%
and earns 9% over 3 years. “Investor B” buys the 1-year bond and earns 2.5%. How much does
Investor B need to earn in the next 2 years to get the same return that Investor A earned? The
answer is 3.25% because Investor B earns 2.5% + 3.25% + 3.25% = 9%, the same as Investor A.
Because we have ignored compounding, the math here should be considered as an approximate
solution, but as you can see, the answer is nearly the same (i.e., the same to two decimal places)
as the answer we obtained by doing math that did not ignore compounding.
Diff: 3
Topic: Term Structure of Interest Rates
Learning Obj.: LG 1
Learning Outcome: F-05
AACSB: Analytical Thinking