Question 14.1: Consider five bonds with terms of one to five years, each pa...
Consider five bonds with terms of one to five years, each paying annual interest payments or coupons of 5%, with the next coupons all being due one year from now. The gross redemption yields for these bonds are given below:
| Term (t) | Gross redemption yield (r_t) |
| 1 | 5.20% |
| 2 | 5.25% |
| 3 | 5.35% |
| 4 | 5.35% |
| 5 | 5.60% |
The first value needed is the price of the first bond. If a redemption payment of 100 is assumed, the 5% interest gives a coupon payment of 5 at the same time. This means that given a gross redemption yield of 5.20%, the price of the first bond is:
B\;P_1=\frac{100+5}{1+0.0520}=99.81.
Remembering that spot yields are given in force of interest terms, this
means that the one-year spot rate, s_1, is given by:
99.81=105e^{-s_1}.
Rearranging this gives a value of 5.069% for s_1. Similarly the dirty price for the two-year bond is given by discounting all payments at the gross redemption yield:
B\;P_2=\frac{5}{1+0.0520}+\frac{100+5}{(1+0.0520)^2}=99.54 .
According to the principle of no arbitrage, the value of the coupon in the first year can be found by discounting it at the one-year spot rate. This means that the two-year spot rate must satisfy the following equation:
99.54=5e^{-s_1}+105e^{-s_2}.
Substituting the known value for s_1 and rearranging gives a value for s_2 of 5.116%. This process can be repeated to find s_3, s_4 and s_5, whose values are given in the table below:
| Term (t) | Bond price B\;P_t | Spot interest rate (s_t) |
| 1 | 99.81 | 5.069% |
| 2 | 99.54 | 5.116% |
| 3 | 99.05 | 5.219% |
| 4 | 98.77 | 5.216% |
| 5 | 97.44 | 5.479% |