Suppose the bond portfolio is twice as large, $20 million, but that its modified duration is only 4.5 years. Show
that the proper hedge position in T-bond futures is the same as the value just calculated, 100 contracts.
Although the hedge ratio is easy to compute, the hedging problem in practice is more dif-
ficult. We assumed in our example that the yields on the T-bond contract and the bond portfolio
would move perfectly in unison. Although interest rates on various fixed-income instruments
C H A P T E R
2 3
Futures, Swaps, and Risk Management
815
do tend to vary in tandem, there is considerable slippage across sectors of the fixed-income mar-
ket. For example, Figure 23.5 shows that the spread between long-term corporate and 10-year
Treasury bond yields has fluctuated considerably over time. Our hedging strategy would be
fully effective only if the yield spread across the two sectors of the fixed-income market were
constant (or at least perfectly predictable) so that yield changes in both sectors were equal.
This problem highlights the fact that most hedging activity is in fact cross-hedging,
meaning that the hedge vehicle is a different asset than the one to be hedged. To the extent
that there is slippage between prices or yields of the two assets, the hedge will not be
perfect. Cross-hedges can eliminate a large fraction of the total risk of the unprotected
portfolio, but you should be aware that they typically are far from risk-free positions.
6
Interest rate swaps have nothing to do with the Homer-Liebowitz bond swap taxonomy described in Chapter 16.
7
The participants to the swap do not loan each other money. They agree only to exchange a fixed cash flow for a
variable cash flow that depends on the short-term interest rate. This is why the principal is described as
notional.
The notional principal is simply a way to describe the size of the swap agreement. In this example, a 7% fixed
rate is exchanged for the LIBOR rate; the difference between LIBOR and 7% is multiplied by notional principal
to determine the net cash flow.
Swaps are multiperiod extensions of forward contracts. For example, rather than agreeing
to exchange British pounds for U.S. dollars at an agreed-upon forward price at one single
date, a foreign exchange swap would call for an exchange of currencies on several future
dates. The parties might exchange $1.6 million for £1 million in each of the next 5 years.
Similarly, interest rate swaps call for the exchange of a series of cash flows proportional to
a given interest rate for a corresponding series of cash flows proportional to a floating inter-
est rate.
6
One party might exchange a variable cash flow equal to $1 million times a short-
term interest rate for $1 million times a fixed interest rate of 5% for each of the next 7 years.
The swap market is a huge component of the derivatives market, with well over $500
trillion in swap agreements outstanding. We will illustrate how these contracts work by
using a simple interest rate swap as an example.
23.4
Swaps
Consider the manager of a large portfolio that currently includes $100 million par value
of long-term bonds paying an average coupon rate of 7%. The manager believes inter-
est rates are about to rise. As a result, he would like to sell the bonds and replace them
with either short-term or floating-rate issues. However, it would be exceedingly expensive
in terms of transaction costs to replace the portfolio every time the forecast for interest
rates is updated. A cheaper and more flexible approach is to “swap” the $7 million a year
in interest income the portfolio currently generates for an amount of money tied to the
short-term interest rate. That way, if rates do rise, so will the portfolio’s interest income.
A swap dealer might advertise its willingness to exchange, or “swap,” a cash flow
based on the 6-month LIBOR rate for one based on a fixed rate of 7%. (The LIBOR, or
London Interbank Offered Rate, is the interest rate at which banks borrow from each
other in the Eurodollar market. It is the most commonly used short-term interest rate in
the swap market.) The portfolio manager would then enter into a swap agreement with
the dealer to pay 7% on notional principal of $100 million and receive payment of the
LIBOR rate on that amount of notional principal.
7
In other words, the manager swaps a
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