Question

Suppose that the prices of zero-coupon bonds with various maturities are given in the following table. The face value of each bond is $1,000. 2 Maturity Price (Years) 976.20 858.39 01:00:49 3 788.92 4 721.80 670.48 a. Calculate the forward rate of interest for each year. (Round your answers to 2 decimal places.) Maturity (years) Forward Rate % % % % 3 4 5 b. How could you construct a 1-year forward loan beginning in year 3? (Round your Rate of synthetic loan answer to 2 decimal places.) Face value Rate of synthetic loan

Answer 1

a).

Formula		[(1,000/Pn)^(1/n)] -1	[(1+yn)^n/(1+yn-1)^n-1] -1
Maturity (years) n	Price (P)	Yield (y)	Forward rate (f)
1	976.20	2.44%	-
2	858.39	7.93%	13.72%
3	788.92	8.22%	8.81%
4	721.80	8.49%	9.30%
5	670.48	8.32%	7.65%

b). 1-year forward loan beginning in year 3:

Short sell a 3-year zero bond at 788.92. The number of 4-year zeros you'll need to buy to offset this, will be

3-year zero price/4-year zero price = 788.92/721.80 = 1.09299

Face value of the loan = par value of zero*number of zeros bought = 1,000*1.09299 = 1,092.99

Rate of synthetic loan = (cash inflow in year 4/cash outflow in year 3) -1 = (1,092.99/1,000) -1 = 9.30%

c). 1-year forward loan beginning in year 4:

Again, short sell a 4-year zero bond at 721.80. Buy 5-year zeros.

Number of 5-year zeros you can buy = 721.80/670.48 = 1.07654

Face value of loan = par value of zero*number of zeros bought = 1,000*1.07654 = 1,076.54

Rate of synthetic loan = (cash inflow in year 5/cash outflow in year 4) -1 = (1,076.54/1,000) -1 = 7.65%