# Suppose we observe the three-year Treasury security rate (1R3) to be 5.2 percent, the expected one-year rate next year—E(2r1)—to be 6.2 percent

1. Suppose we observe the three-year Treasury security rate (1R3) to be 5.2 percent, the expected one-year rate next year—E(2r1)—to be 6.2 percent, and the expected one-year rate the following year—E(3r1)—to be 6.8 percent. If the unbiased expectations theory of the term structure of interest rates holds, what is the one-year Treasury security rate?

Based on unbiased expectations theory,

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

(1 + 5.2%)3 = (1 + 1 Yr rate) * (1 + 6.2%) * (1 + 6.8%)

1.164253 = (1 + 1 Yr rate) * (1.062) * (1.068)

1.164253 = (1 + 1 Yr rate) * 1.134216

(1 + 1 Yr rate) = 1.0265

1 Yr rate = 0.0265 = 2.65%