Question

Suppose that the current one-year rate (one-year spot rate) and expected one-year T-bill rates over the following three years (i.e., years 2, 3, and 4, respectively) are as follows:

1R1 = 0.5%, E(2r 1) = 1.5%, E(3r1) = 9.9%, E(4r1) = 10.25%

Using the unbiased expectations theory, calculate the current (long-term) rates for one-, two-, three-, and four-year maturity Treasury securities. (Round your answers to 3 decimal places. (e.g., 32.161))

	Current (Long-Term) Rates
One-year	%
Two-year	%
Three-year	%
Four-year	%

Answer 1

According to Pure Expectation theory, short term interest rates are indicator of long term interest rates.

Based on this theory, if you invest in a two year bond, you would generate equal returns as you would have generated if you would have invested in a 1 year bond and then reinvested the proceeds in another 1 year bond.

Now, applying the same concept.

One-year Rate = 0.5% (as given in question)

(1 + 2yr Rate)² = (1 + 1Yr Rate) * (1 + 1 Yr rate 1 Yr from now)

(1 + 2yr Rate)² = (1 + 0.50%) * (1 + 1.5%)

(1 + 2yr Rate)² = (1.0050) * (1.015)

(1 + 2yr Rate)² = 1.020075

1 + 2yr Rate = 1.009988

2 Yr Rate = 0.999%

(1 + 3yr Rate)³ = (1 + 1Yr Rate) * (1 + 1 Yr rate 1 Yr from now) * (1 + 1 Yr rate 2 Yr from now)

(1 + 3yr Rate)³ = (1 + 0.50%) * (1 + 1.5%) * (1 + 9.9%)

(1 + 3yr Rate)³ = (1.0050) * (1.015) * (1.099)

(1 + 3yr Rate)³ = 1.121062

1 + 3yr Rate = 1.038827

3 Yr Rate = 3.883%

(1 + 4yr Rate)⁴ = (1 + 1Yr Rate) * (1 + 1 Yr rate 1 Yr from now) * (1 + 1 Yr rate 2 Yr from now) * * (1 + 1 Yr rate 4 Yr from now)

(1 + 4yr Rate)⁴ = (1 + 0.50%) * (1 + 1.5%) * (1 + 9.9%) * (1 + 10.25%)

(1 + 4yr Rate)⁴ = (1.0050) * (1.015) * (1.099) * (1.1025)

(1 + 4yr Rate)⁴ = 1.235971

1 + 4yr Rate = 1.054392

4 Yr Rate = 5.439%