Question

Consider the following $1,000 par value zero-coupon bonds:

Bond	Years Until Maturity	Yield to Maturity
A	1	7.75%
B	2	8.75
C	3	9.25
D	4	9.75

a. According to the expectations hypothesis, what is the market’s expectation of the one-year interest rate three years from now? (Do not round intermediate calculations. Round your answer to 2 decimal places.)

Interest rate

?%

b. What are the expected values of next year’s yields on bonds with maturities of (a) 1 year; (b) 2 years; (c) 3 years? (Do not round intermediate calculations. Round your answer to 2 decimal places.

Maturity (years)	YTM
1	?%
2	?%
3	?%

Answer 1

(a) The yield to maturity of the zero-coupon bonds of each maturity correspond with the spot rate for that particular maturity

1-Year Spot Rate = r1 = 7.75 %, 2-Year Spot Rate = r2 = 8.75 %, 3-Year Spot Rate = r3 = 9.25 % and 4-Year Spot Rate = r4 = 9.75 %

Let the 1-Year Spot Rate 3-Years from now be 3f1

Therefore, as per pure expectations theory = (1+r3)^(3) x (1+3f1) = (1+r4)^(4)

(1.0925)^(3) x (1+3f1) = (1.0975)^(4)

3f1 = [(1.0975)^(4) / (1.0925)^(3)] - 1 = 0.11264 or 11.264 % ~ 11.26%

(b) The yield on a zero-coupon bond of 1-year maturity, 1 - year from now is the 1-year forward rate 1-year from now. Similarly, yield of a two-year bond, 1-year from now is the 2-year forward rate 1-year from now and so on.

(i) 1-year forward rate 1-year later = 1f1 = [(1+r2)^(2)/(1+r1)] - 1 = =[(1.0875)^(2)/(1.0775)] - 1 = 0.09759 or 9.759% ~ 9.76 %

(ii) 2-year forward rate 1-year later = 1f2 = [(1+r3)^(3)/(1+r1)]^(1/2) - 1 = [(1.0925)^(3)/(1.0775)]^(1/2) - 1 = 0.10008 or 10.008 % ~ 10.01 %

(iii) 3-year forward rate 1-year later = 1f3 = [(1+r4)^(4) / (1+r1)]^(1/3)-1 = [(1.0975)^(4)/(1.0775)]^(1/3) - 1 = 0.10425 or 10.425 % ~ 10.42 %