Question

Suppose we observe the three-year Treasury security rate (₁R₃) to be 4.3 percent, the expected one-year rate next year—E(₂r₁)—to be 4.8 percent, and the expected one-year rate the following year—E(₃r₁)—to be 5.6 percent. If the unbiased expectations theory of the term structure of interest rates holds, what is the one-year Treasury security rate? (Do not round intermediate calculations. Round your answer to 2 decimal places. (e.g., 32.16))

one-year Treasury security rate: _______%

Answer 1

The three-year Treasury security rate (1R3) = 4.3%

The expected one-year rate next year E(2r1) = 4.8 %

The expected one-year rate in third year E(3r1) = 5.6 %

We have to find the one-year Treasury security rate (1R1)

Now according to the formula,

1R3 = [ ( 1 + 1R1)* (1 + E(2r1)* (1 + E(3r1) ] ^(1/3)

or, 1.043 = [ ( 1 + 1R1) * 1.048 * 1.056] ^(1/3)

or, 1.043^3 = [ ( 1 + 1R1) * 1.048 * 1.056]

or, 1.1346 = [ ( 1 + 1R1) * 1.048 * 1.056]

or, ( 1 + 1R1) =  1.1346 / (1.048 * 1.056)

or, ( 1 + 1R1) = 1.025245

or, 1R1 = 0.025245 or 2.5245 % or 2.52 % (answer)