Question

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

Bond	Years to Maturity	YTM(%)
A	1	6.0	%
B	2	7.0
C	3	7.5
D	4	8.0

According to the expectations hypothesis, what is the market’s expectation of the yield curve one year from now? Specifically, what are the expected values of next year’s yields on bonds with maturities of (a) one year? (b) two years? (c) three years?

Bond	YTM	YTM (%)
B	1
C	2
D	3

Answer 1

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

a]

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

(1 + 7%)2 = (1 + 6%) * (1 + R)

R = (1.072 / 1.06) - 1

R = 8.01%

1-year rate 1 year from now is 8.01%

b]

Let us say the 2-year rate 1 year from now is R. Then :

(1 + 7.5%)3 = (1 + 6%) * (1 + R)2

R = (1.0753 / 1.06)1/2 - 1

R = 8.26%

2-year rate 1 year from now is 8.26%

c]

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

(1 + 8%)4 = (1 + 6%) * (1 + R)3

R = (1.084 / 1.06)1/3 - 1

R = 8.68%

3-year rate 1 year from now is 8.68%