108156.pdf

Solutions 273

Figure S.2 Coupon bond price versus time in Exercise 2.32

2.35By solving the equation

(1 +r)−^1 =0. 89

we find thatr∼= 0 .1236, that is, the effective rate implied by the bond is about

12 .36%. The price of the bond after 75 days will be

B(75/ 365 ,1) =B(0,1) (1 +r)

36575

=0.89 (1 + 0.1236)

36575 ∼

= 0. 9115

and the return will be

K(0, 75 /365) =B(75/^365 B(0,1),1)−B(0,1)=∼^0.^91150. 89 −^0.^89 ∼= 0. 0242 ,

about 2.42%.

2.36The initial price of a six-month unit bond is e−^0.^5 r, whererdenotes the implied

continuous rate. If the bond is to produce a 7% return over six months, then

1 −e−^0.^5 r

e−^0.^5 r =0.^07 ,

which givesr∼= 0 .1353, or 13.53%.

2.37The continuous rate implied by the bond satisfies

e−r=0. 92.

The solution isr∼= 0 .0834. At timetthe bond will be worth 0.92ert. It will

produce a 5% return at a timetsuch that

0 .92ert− 0. 92

0. 92 =0.^05 ,

which givest∼= 0 .5851 years or 213.6days.

2.38At time 0 we buy 1/B(0,1) = erbonds, at time 1 we increase our holdings to

er/B(1,2) = e^2 rbonds, and generally at timenwe purchase e(n+1)rone-year

bonds.