Question

The following is a list of prices for zero-coupon bonds with different maturities and par values of $1,000.

Maturity (Years)

Price maturity 1 year = $ 925.15

Price maturity 2 years = 862.57

Price maturity 3 years = 788.66

Price maturity 4 years = 711.00

According to the expectations theory, what is the expected forward rate in the third year?

Answer 1

First, we calculate the 3-year and 4-year spot rates.

Price of zero coupon bond = face value / (1 + yield)^{years to maturity}

$788.66 = $1,000 / (1 + 3-year spot rate)³

3-year spot rate = ($1,000 / $788.66)^1/3 - 1 = 8.2356%

$711.00 = $1,000 / (1 + 4-year spot rate)⁴

4-year spot rate = ($1,000 / $711.00)^1/4 - 1 = 8.9012%

As per expectations theory, investing for 4 years at the 4-year rate should result in the same ending value as investing for 3 years at the 3-year rate, and reinvesting the proceeds after 3 years at the 1-year rate 3 years from now.

Let us say the 1-year rate 3 years from now is R. Then :

(1 + 8.9012%)⁴ = (1 + 8.2356%)³ * (1 + R)

R = ((1 + 8.9012%)⁴ / (1 + 8.2356%)³) - 1

R = 10.9226%

expected forward rate in the third year is 10.9226%