Solutions 299
timet(note that the bond price grows by a factor of eyt),Dt=eytP^1 (y)(
(τn 1 −t)C 1 e−y(τn^1 −t)+···+(τnN−t)(CN+F)e−y(τnN−t))=P^1 (y)(
(τn 1 −t)C 1 e−τn^1 y+···+(τnN−t)(CN+F)e−τnNy)=D 0 −t,
since the weightsC 1 e−τn^1 y/P(y),C 2 e−τn^2 y/P(y),...,(CN+F)e−τnNy/P(y)
adduptoone.
10.14Denote the annual payments byC 1 ,C 2 and the face value byF,sothat
P(y)=C 1 e−y+(C 2 +F)e−^2 y,D(y)=C^1 e−y+2(C 2 +F)e− 2 y
P(y).
Compute the derivative ofD(y) to see that it is negative:
d
dyD(y)=−C 1 (C 2 +F)e−^3 y
P(y)^2 <^0.10.15We first find the prices and durations of the bonds:PA(y)∼= 120 .72,PB(y)∼=
434 .95,DA(y)∼= 1 .8471,DB(y)∼= 1 .9894. The weightswA∼=− 7 .46%,wB∼=
107 .46% give duration 2, which means that we have to buy 49.41 bondsBand
issue 12.35 bondsA. After one year we shall receive $247.05 from the coupons
ofBand will have to pay the same amount for the coupons ofA.Our final
amount will be $23, 470. 22 ,exactly equal to the future value of $20,000 at 8%,
independently of any rate changes.
10.16If the term structure is to be flat, then the yieldy(0,6) = 8.16% applies to
any other maturity, which givesB(0,3) = 0.9798 dollars andB(0,9) = 0. 9406
dollars.
10.17Issue and sell 500 bonds maturing in 6 months with $100 face value, obtaining
$48, 522 .28. Use this sum to buy 520.4054 one-year bonds. After 6 months
settle the bonds issued by paying $50,000. After one year cash the face value
of the bonds purchased. The resulting rate is 8%.
10.18You need to deposit 100,000e−^8 .41%/^12 ∼= 99 , 301 .62 dollars for one month,
which will grow to the desired level of $100,000, and borrow the same amount
for 6 months at 9.54%. Your customer will receive $100,000 after 1 month
and will have to pay 99, 301 .62e^9 .54%/^2 ∼= 104 , 153 .09 dollars after 6 months,
which implies a forward rate of 9.77%.(The rate can be obtained directly from
(10.5).)
The rate for a 4-month loan starting in 2 months is
f(0, 2 ,6) =^6 ×^9 .35%− 42 ×^8 .64%∼= 10 .09%,so a deposit at 10.23% would give an arbitrage opportunity.
10.19To see that the forward ratesf(n, N) may be negative, let us analyse the case
withn= 0 for simplicity. Then
f(0,N)=(N+1)y(0,N+1)−Ny(0,N)