228 Mathematics for Finance
- the face values ofB: $51, 376 .39,
- the market value ofA, selling at $101.46: $36, 682 .22.
The result is $91, 673 .92. We invest all the money in bondsB, since the
required duration is now 1. (The payoff of these bonds is guaranteed
next year.) We can afford to buy 1, 003 .07 bondsBselling at $91.39.
The terminal value of the investment will be about$100,307.
b) After two years the rate goes up to 16%. We cash the same amount as
above for coupons and zero-coupon bonds, but bondsAare now cheaper,
selling at $83.85, so we have less money in total: $85, 305 .68. However,
the zero-coupon bonds are now cheap as well, selling at $85. 21 ,and we
can afford to buy 1, 001 .07 of them, ending up with$100,107.
- After one year the rate drops to 9%. In a similar way as before, we arrive
at the current value of the investment by adding the coupons ofA, the face
value ofBand the market value of bondsAheld, obtaining $83, 658 .73.
Then we find the weightswA∼= 0 .4013,wB∼= 0 .5987, determining our new
portfolio of 329.56 bondsAand 548.04 bondsB. (We have to sell 165. 50
bondsAand buy 548.04 new bondsB.)
a) After two years the rate goes up to 14%. We cash $3, 295 .55 from the
coupons ofA, which together with the $54, 803 .77 obtained fromBand
the market value of $29, 174 .39 of bondsAgives $87, 273 .72 in total. We
buy 1003.89 new zero-coupon bondsB,ending up with$100,389after
3years.
b) After two years the rate drops to 6%. Our wealth will then be $94, 405 .29,
we can afford to buy 1, 002 .43 bondsB, and the final value of our in-
vestment will be$100,243.
As we can see, we end up with more than $100,000 in each scenario.^1
Exercise 10.15
Design an investment of $20,000 in a portfolio of duration 2 years con-
sisting of two kinds of coupon bonds maturing after 2 years, with annual
coupons, bondAwith $20 coupons and $100 face value, and bondB
with $5 coupons and $500 face value, given that the initial rate is 8%.
How much will this investment be worth after 2 years?(^1) It can be shown that the future value at timetof a bond investment with duration
equal tothas a minimum if the rateyremains unchanged. This means that rate
jumps in a model with yields independent of maturity lead to arbitrage. In an
arbitrage-free model with rate jumps, the yields must therefore depend on maturity.