232 Mathematics for Finance
be ln(110, 517. 09 / 100 ,000)∼=10%. Financial intermediaries may simplify your
task by offering a so-called Forward Rate Agreement and perform the above
construction of the loan on your behalf.
Exercise 10.17
Explain how a deposit of $50,000 for six months can be arranged to start
in six months and find the rate ify(0,6) = 6% andy(0,12) = 7%, where
τ= 121.In general, theinitial forward ratef(0,M,N) is an interest rate such thatB(0,N)=B(0,M)e−(N−M)τf(0,M,N),so
f(0,M,N)=−^1
τ(N−M)lnB(0,N)
B(0,M)=−lnB(0,N)−lnB(0,M)
τ(N−M).
Note that this rate is deterministic, since it is worked out using the present bond
prices. It can be conveniently expressed in terms of the initial term structure.
Insert into the above expression the bond prices as determined by the yields,
B(0,N)=e−τNy(0,N)andB(0,M)=e−τMy(0,M),to get
f(0,M,N)=Ny(0,N)−My(0,M)
N−M. (10.5)
Exercise 10.18
Suppose that the following spot rates are provided by central London
banks (LIBOR, the London Interbank Offer Rate, is the rate at which
money can be deposited; LIBID, the London Interbank Bid Rate, is the
rate at which money can be borrowed):Rate LIBOR LIBID
1 month 8.41% 8.59%
2 months 8.44% 8.64%
3 months 9.01% 9.23%
6 months 9.35% 9.54%As a bank manager acting for a customer who wishes to arrange a loan of
$100,000 in a month’s time for a period of 5 months, what rate could you
offer and how would you construct the loan? Suppose that another insti-
tution offers the possibility of making a deposit for 4 months, starting 2