Question

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

Bond	Years until Maturity	Yield to Maturity
A	1	8.50	%
B	2	9.50
C	3	10.00
D	4	10.50

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.)

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.)

The current yield curve for default-free zero-coupon bonds is as follows:

Maturity (years)	YTM
1	9.0	%
2	10.0
3	11.0

a. What are the implied one-year forward rates? (Do not round intermediate calculations. Round your answers to 2 decimal places.)

b. Assume that the pure expectations hypothesis of the term structure is correct. If market expectations are accurate, what will the pure yield curve (that is, the yields to maturity on one- and two-year zero-coupon bonds) be next year?

• There will be a shift upwards in next year's curve.

• There will be a shift downwards in next year's curve.

• There will be no change in next year's curve.

c. What will be the yield to maturity on two-year zeros? (Do not round intermediate calculations. Round your answers to 2 decimal places.)

d. If you purchase a two-year zero-coupon bond now, what is the expected total rate of return over the next year? (Hint: Compute the current and expected future prices.) Ignore taxes. (Do not round intermediate calculations. Round your answer to 2 decimal places.)

e. If you purchase a three-year zero-coupon bond now, what is the expected total rate of return over the next year? (Hint: Compute the current and expected future prices.) Ignore taxes. (Do not round intermediate calculations. Round your answer to 2 decimal places.)

Answer 1

1a). If the one-year interest rate, three years from now is i then

(1+y4)^4 = (1+y3)*(1+i)

(1+10.50%)^4 = (1+10%)^3*(1+i)

i = 1.1201 -1 = 12.01%

1b). One year expected interest rates are found out as

r = [(1+yn)^n/(1+yn-1)^n-1] -1

Using this formula, short interest rates are calculated.

Bond	Time	YTM	Short rates
A	1	8.50%
B	2	9.50%	10.51%
C	3	10%	11.01%
D	4	10.50%	12.01%

The yield expectations for next year for zero-coupon bonds of 1-year, 2-year and 3-year maturities will be the geometric means of the short rates, as follows

Maturity	Short rates	YTM	Calculation
1	10.51%	10.51%	equals One-year short rate
2	11.01%	10.76%	((1+11.01%)*(1+10.51))^(1/2) -1
3	12%	11.17%	((1+12%)*(1+11.01%)*(1+10.51))^(1/2)

2a). One-year forward rates will be calculated using the same formula, as for the short rates:

r = [(1+yn)^n/(1+yn-1)^n-1] -1

Maturity	YTM	One-year forward rates	Calculation
1	9%	9.00%
2	10%	11.01%	((1+10%)^2/(1+9%)^1)-1
3	11%	13.03%	((1+11%)^3/(1+10%)^2) -1

2b). If pure expectations hypothesis of term structures is correct then the yield curve next year will shift upwards which indicates that the market expects short-term interest rates in the future to rise.

2c). The YTM on two-year zeros is given as 10% in the table above.

2d & e). Current price of two-year bond = 1,000/(1+10%)^2 = 826.45

Current price of three-year bond = 1,000/(1+11%)^3 = 731.19

Next year, the two-year bond will be a one-year bond and can be valued using the one-year forward rate of 11.01% as:

1,000/(1+11.01%) = 900.83

The three-year bond will be a two-year bond and can be valued using the one-year forward rates of 11.01% and 13.03% as:

1,000/(1+11.01%)*(1+13.03%) = 797.00

Return on the two-year bond = (expected price next year/current price) -1 = (900.83/826.45)-1 = 9.00%

Return on the three-year bond = (expected price next year/current price) -1 = (797/731.19)-1 = 9.00%