The yield on 1-year Treasury securities is 6%, 2-year securities yield 6.2%, 3-year securities yield 6.3%, and 4-year securities yield 6.5%. There is no maturity risk premium. Using expectations theory and geometric averages, forecast the yields on the following securities: A 2-year security, 1 year from now?

Question

The yield on 1-year Treasury securities is 6%, 2-year securities yield 6.2%, 3-year securities yield 6.3%, and 4-year securities yield 6.5%. There is no maturity risk premium. Using expectations theory and geometric averages, forecast the yields on the following securities:  A 2-year security, 1 year from now?

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Theory Explanation: The Expectations Theory suggests that the yield of a longer-term bond is an average of the expected short-term interest rates that people anticipate over the maturity period of the longer-term bond. Since there is no maturity risk premium, we can use the given yields to calculate the expected yield on a 2-year security, 1 year from now.Calculate Expected Yield: First, we need to find the expected yield on a 1-year security, 1 year from now. We can do this by using the yields given for the 2-year and 1-year securities. According to the Expectations Theory, the yield on the 2-year security should be the geometric average of the yield on the 1-year security now and the expected yield on the 1-year security next year.Calculate EY1: The formula for the geometric average of two numbers, A and B, is the square root of their product: √(A*B). We can rearrange the formula to solve for the expected yield on the 1-year security next year (which we'll call EY1). The current yield on the 1-year security is 6% (or 0.06 as a decimal), and the yield on the 2-year security is 6.2% (or 0.062 as a decimal). The formula becomes: EY1 = (0.062^2) / 0.06.Forecast Yield FY2: Now we calculate EY1 using the formula: EY1 = (0.062^2) / 0.06 = (0.003844) / 0.06 ≈ 0.06407 or 6.407%.Calculate FY2: Next, we need to forecast the yield on a 2-year security, 1 year from now. This will be the geometric average of the expected yield on the 1-year security next year (which we just calculated as 6.407%) and the yield on the 3-year security (since it will be a 2-year security in one year), which is 6.3% (or 0.063 as a decimal).Using the geometric average formula again, we calculate the forecasted yield on the 2-year security, 1 year from now: FY2 = √(EY1 * Y3) = √(0.06407 * 0.063).Performing the calculation, we get: FY2 = √(0.06407 * 0.063) ≈ √(0.00403641) ≈ 0.0635 or 6.35%.

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