- Stochastic Interest Rates 257
annual compounding):
AB
fixed 11 .40% 13 .40%
variable LIBOR + 2% LIBOR + 3%In this case we say that A has comparative advantage over B in the fixed rate,
with B having comparative advantage over A in the variable rate. (Notwith-
standing the fact that the overall credit rating of A is better, as reflected by
the lower interest rates offered.) In these circumstances A should borrow at
the fixed rate, B should borrow at the variable rate, and they can swap their
interest payments.
Consider a principal of $100,000 borrowed for one year and suppose that
LIBOR is 10% and (just for simplicity) remains the same during the first year
of the loan. If A borrows at the variable rate and B at the fixed rate, then
the total interest paid will be $25,400 between them. However, if A borrows
at the fixed rate and B at the variable rate, the interest payments will be only
$24,400 in total. The difference of $1,000 will be available to share between
the two companies if they arrange to swap the rates. (In practice, this amount
would be reduced by a fee charged by the intermediary arranging the deal.)
If LIBOR changes to 9%, say, in the second year of the loan, so will the total
interest payable, but the difference will remain at $1,000.
How should this difference be shared between the two companies? To answer
the question, we assume the term structure of interest rates determined by the
prices of one- and two-year zero-coupon bonds in Example 11.10. In particular,
we can identify LIBOR with the effective short rate implied by the bond prices,
B(0,1)−^1 −1inyearoneandB(1,2)−^1 −1 in year two. These are the same
rates as those implied by the floating coupons in Example 11.10. The fixed
coupons in the same example imply a rate of 10.04%.
Instead of swapping interest payments with B, company A would achieve
the same result by taking a loan of $100,000 at the fixed rate of 11.40% offered
by the bank, buying 1,000 of the fixed-coupon bonds, and writing 1,000 of the
floating-coupon bonds considered in Example 11.10. As a result, company A
will have borrowed $100,000 at the rate 11.40%− 10 .04%+LIBOR = LIBOR+
1 .36%.Compared to the variable rate of LIBOR + 2% offered to company A,
this is a gain of 0.64%. On a $100,000 loan this would mean a gain of $640 in
each year.
By a similar argument, instead of swapping with A, company B could bor-
row $100,000 at the variable rate LIBOR + 3%, buy 1,000 floating-coupon
bonds and write 1,000 fixed-coupon bonds. As a result, B would pay interest
at LIBOR + 3%−LIBOR + 10.04% = 13.04%,a gain of 0.36% as compared to