220 Mathematics for Finance
If the lifetime of our investment exceeds one year, we will be facing the
problem of reinvesting coupons. In the following example we assume that the
coupons are used to purchase the same bond.
Example 10.3
We begin as in Example 10.2, but our intention is to terminate the investment
after 3 years. After one year we reinvest the coupons obtained in the same, now
a 3-year, coupon bond. Consider the following scenarios after one year:
- The rate remains the same for the period of our investment,y(0) =y(1) =
y(2) =y(3) = 12%. The bond price is $93.48, so for the $108.96 received
from coupons we can buy 1.17 additional bonds, increasing the number of
bonds held to 12.06. We can monitor the value of our investment by simply
multiplying the number of bonds held by the current bond price. We repeat
this in the following year. After three years we cash the coupons and sell
the bonds, the final value of the investment being $1, 433 .33. This number
will be used as a benchmark for other scenarios. Observe that
1 , 433. 33 ∼= 1 ,000e^3 ×12%,the same as the value after 3 years of $1,000 invested on zero-coupon bonds.
The building blocks of our investment are summarised in the table below.Year 0 1 2 3
Rate 12% 12% 12% 12%
PV of coupon 1 $8. 87 $10. 00
PV of coupon 2 $7. 87 $8. 87 $10. 00
PV of coupon 3 $6. 98 $7. 87 $8. 87 $10. 00
PV of coupon 4 $6. 19 $6. 98 $7. 87 $8. 87
PV of face value $61. 88 $69. 77 $78. 66 $88. 69
Bond price $91. 78 $93. 48 $95. 40 $97. 56
Cashed coupons $108. 96 $120. 60 $133. 26
Additional bonds 1. 17 1. 26
Number of bonds 10. 90 12. 06 13. 33
Value of investment $1, 000. 00 $1, 127. 50 $1, 271. 25 $1, 433. 33- Suppose that the rate goes down by 2% after one year and then remains
at the new level. The drop of the rate results in an increase of all bond
prices. The number of additional bonds that can be bought for the coupons
is lower than in scenario 1. Nevertheless, the final value of the investment is