Current interest rates on bonds of different maturities look like this:

Question

Maturity Rate
1-year 2.50%
2-year 2.75%
3-year 3.00%
If the expectations hypothesis is true, what interest rate would you expect a 1-year bond to pay, 2 years from now?

What will be an ideal response?

Answer 1

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 2-year bond and then when that bond matures, the investor buys a brand new 1-year bond. It is the rate on this second bond that the question is asking about....what is the rate on a 1-year bond, 2 years from now? The expectations hypothesis says that the investor who buys the 2-year bond today and then buys another 1-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 expected return on the 1-year bond, two years from now. The investor who buys the 2-year bond today and then the 1-year bond after that will earn the following return:

(1.0275)^2 × (1 + E(r)) - 1

Or we could say that $1 invested in this way will grow to $1(1.0275)^2(1 + E(r)) 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

(1.0275)^2(1 + (E(r)) = 1.092727

Solve for E(r) and you obtain 3.50%.

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 2-year bond and earns 5.5% over 2 years (2.75% per year for 2 years). How much does Investor B need to earn in the 3rd year to get the same return that Investor A earned? The answer is 3.5% because Investor B earns 2.75% + 2.75% + 3.5% = 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.