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Hedging Interest Rate Risk
Like equity managers, fixed-income managers also sometimes desire to hedge market risk,
in this case resulting from movements in the entire structure of interest rates. Consider, for
example, these problems:
1. A fixed-income manager holds a bond portfolio on which considerable capital gains
have been earned. She foresees an increase in interest rates but is reluctant to sell
her portfolio and replace it with a lower-duration mix of bonds because such rebal-
ancing would result in large trading costs as well as realization of capital gains for
tax purposes. Still, she would like to hedge her exposure to interest rate increases.
2. A corporation plans to issue bonds to the public. It believes that now is a good time
to act, but it cannot issue the bonds for another 3 months because of the lags inher-
ent in SEC registration. It would like to hedge the uncertainty surrounding the yield
at which it eventually will be able to sell the bonds.
3. A pension fund will receive a large cash inflow next month that it plans to invest in
long-term bonds. It is concerned that interest rates may fall by the time it can make the
investment and would like to lock in the yield currently available on long-term issues.
In each of these cases, the investment manager wishes to hedge interest rate uncertainty.
To illustrate the procedures that might be followed, we will focus on the first example,
and suppose that the portfolio manager has a $10 million bond portfolio with a modified
duration of 9 years.
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If, as feared, market interest rates increase and the bond portfolio’s
yield also rises, say, by 10 basis points (.10%), the fund will suffer a capital loss. Recall
from Chapter 16 that the capital loss in percentage terms will be the product of modified
duration, D *, and the change in the portfolio yield. Therefore, the loss will be
D* 3 Dy 5 9 3 .10% 5 .90%
or $90,000. This establishes that the sensitivity of the value of the unprotected portfolio
to changes in market yields is $9,000 per 1 basis point change in the yield. Market prac-
titioners call this ratio the price value of a basis point, or PVBP. The PVBP represents
the sensitivity of the dollar value of the portfolio to changes in interest rates. Here, we’ve
shown that
PVBP
5
Change in portfolio value
Predicted change in yield
5
$90,000
10 basis points
5 $9,000 per basis point
One way to hedge this risk is to take an offsetting position in an interest rate futures
contract, for example, the Treasury bond contract. The bond nominally calls for delivery
of $100,000 par value T-bonds with 6% coupons and 20-year maturity. In practice, the
contract delivery terms are fairly complicated because many bonds with different coupon
rates and maturities may be substituted to settle the contract. However, we will assume that
the bond to be delivered already is known and has a modified duration of 10 years. Finally,
suppose that the futures price currently is $90 per $100 par value. Because the contract
requires delivery of $100,000 par value of bonds, the contract multiplier is $1,000.
23.3
Interest Rate Futures
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Recall that modified duration, D *, is related to duration, D, by the formula D * 5 D /(1 1 y ), where y is the bond’s
yield to maturity. If the bond pays coupons semiannually, then y should be measured as a semiannual yield.
For simplicity, we will assume annual coupon payments, and treat y as the effective annual yield to maturity.
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P A R T V I
Options, Futures, and Other Derivatives
Given these data, we can calculate the PVBP for the futures contract. If the yield on the
delivery bond increases by 10 basis points, the bond value will fall by D * 3 .1% 5 10 3
.1%
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1%. The futures price also will decline 1%, from 90 to 89.10.
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Because the
contract multiplier is $1,000, the gain on each short contract will be $1,000 3 .90 5 $900.
Therefore, the PVBP for one futures contract is $900/10-basis-point change, or $90 for
a change in yield of 1 basis point.
Now we can easily calculate the hedge ratio as follows:
H
5
PVBP of portfolio
PVBP of hedge vehicle
5
$9,000
$90 per contract
5 100 contracts
Therefore, 100 T-bond futures contracts will offset the portfolio’s exposure to interest rate
fluctuations.
Notice that this is another example of a market-neutral strategy. In Example 23.5, which
illustrated an equity-hedging strategy, stock-index futures were used to drive a portfolio
beta to zero. In this application, we used a T-bond contract to drive the interest rate expo-
sure of a bond position to zero. The hedged fixed-income position has a duration (or a
PVBP) of zero. The source of risk differs, but the hedging strategy is essentially the same.
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This assumes the futures price will be exactly proportional to the bond price, which ought to be nearly true.
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