How to Calculate Unbiased Expectations Theory

Investing may be an art, but even the greatest artists have a strong understanding of the science at the foundation of their craft. In the world of bonds, yields, and interest rates, the Unbiased Expectations Theory is one element of the science that investors should know. Let's explore the theory, understand what it teaches us, and where it falls short in practice.

What is the Unbiased Expectations Theory? Unbiased Expectations Theory states that current long-term interest rates contain an implicit prediction of future short term interest rates. More specifically, the theory posits that an investor should earn the same amount of interest from an investment in a single two-year bond today as they would with two consecutive investments in one-year bonds.

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The two one-year bonds would each have a lower interest rate individually compared to the two year bond, however because of compounding interest, Unbiased Expectations Theory predicts that the net outcome would be equal. If we assume the theory to be true, we can use it to make practical predictions about the future of bond yields for our own investing.

Putting the theory into practice Let's assume that the current bond market offers a two-year bond with an interest rate of 10% and a one-year bond at 9%. With this information, we can use the Unbiased Expectations Theory to predict what the one-year bond's interest rate will be next year.

To do the calculation, first add 1 to the two-year bond's interest rate, which in this case gives us 1.1 (or 110%).

Next, we take this result and square it: 1.1 squared gives us 1.21.

The next step is to divide this number by the current year's one-year interest rate plus one. In this example, that means 1.21 divided by 1.09 (9% + 1 = 1.09), which yields 1.11.

The final step is to subtract 1 from that last calculation, giving us the predicted one-year interest rate for next year, of 11.1% in this example.

This means that for an investor to earn an equivalent return to today's two-year bond, she would have to invest in a one-year bond today at 9% and hope that next year's one-year bond yield increased to 11.1%.

Preferred Habitat Theory: Taking bond predictions to the next level In the real world though, Unbiased Expectation Theory doesn't work all that well. When the yield curve is stable, interest rates have a tendency to stay the same and long term rates tend to include a premium over the sum of sequential short term investments.

Preferred Habitat Theory expands on Unbiased Expectations Theory to explain this reality. According to this theory, investors will invest in a shorter term bond over a longer term bond because the shorter term bond carries less interest rate risk. Interest rates over fairly predictable over a short term time span. Over the longer term though, they can vary wildly. For bond investors, that uncertainty is risky. So, unless the long term bond offers an extra incentive to compensate for the increased risk, investors will stick with more predictable, shorter term investments.

That extra compensation, or "risk premium," is why longer term bonds tend to pay a higher yield than shorter term alternatives.

By including the bonds' maturity alongside yield, the Preferred Habitat Theory is able to better predict how bond yields will behave in real world markets. Without considering maturity, the Unbiased Expectations Theory comes up short.

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