Five Hundred Contracts Is Right, and Face Value Is Not the Reason

Cutting a hundred million of a thirty-year bond down to fifty takes five hundred futures, and the usual arithmetic of fifty million over a hundred thousand lands there only because the contract's duration per dollar of face happens to match the bond's. What a hedge matches is dollars per basis point: 56,288.92 against 112.5778. Hold a thirty-year zero instead and the same job needs 1,234 contracts, while five hundred three-month contracts would cover 22.2 per cent of it.

You are long a hundred million dollars of face in a thirty-year bond and you want the interest rate exposure cut to fifty million. Bond futures come on a hundred thousand dollars of face each. Short five hundred of them.

Five hundred is the right number. The arithmetic almost everybody offers for it, $50\text{m} / 100\text{k}$ , is arithmetic on the wrong quantity, and it happens to land on the right answer here because of a hypothesis nobody states. Change the bond and the same division gives 1,234.

The shortcut, and the hypothesis inside it

Face value is a label printed on a contract. It is not what hurts you when rates move. What hurts you is the change in the market value of the position, and for a small yield move that is the modified duration times the value:

\frac{\Delta V}{V} \;\approx\; -D_{\text{mod}}\,\Delta y \qquad\Longrightarrow\qquad \frac{\Delta V}{\Delta y} \;=\; -D_{\text{mod}}\,V

(1)

The bond in question is an eight per cent thirty-year par bond, and its modified duration is derived rather than assumed: $D_{\text{mod}} = a_{30}(0.08) = 11.2578$ , the annuity factor, which is what a par bond's modified duration always equals. So the fifty million of exposure you want gone is worth

50{,}000{,}000 \times 11.2578 \times 10^{-4} \;=\; \$56{,}288.92 \ \text{per basis point}

(2)

and one contract, on a hundred thousand of the same bond, is worth $\$112.5778$ per basis point. Divide:

N \;=\; \frac{56{,}288.92}{112.5778} \;=\; 500.000

(3)

Same answer, and now it is possible to see why the face-value version worked. Write the general hedge ratio and let the two durations be equal:

N \;=\; \frac{D_{P}\,V_{P}}{D_{F}\,V_{F}} \;\xrightarrow[\;D_{P} = D_{F}\;]{}\; \frac{V_{P}}{V_{F}} \;=\; \frac{50{,}000{,}000}{100{,}000}

(4)

The duration cancels. That cancellation is the whole justification for the shortcut, and it is conditional on the contract's sensitivity per dollar of face matching the bond's.

What a hedge matches

A hedge is sized so that the two legs move by the same number of dollars for the same shift in the curve. The natural unit is dollars per basis point, sometimes written as the price value of a basis point. Matching notionals matches dollars per basis point only when the two instruments have the same duration per dollar of face, which is a coincidence rather than a rule.

Forty-five times, and the wrong end of the curve

The tempting follow-up is that any interest rate contract will do, since rates are rates. Take a three-month contract on a million of notional. Its sensitivity is fixed by its tenor:

1{,}000{,}000 \times \tfrac{1}{4} \times 10^{-4} \;=\; \$25 \ \text{per basis point}

(5)

Per dollar of face, the thirty-year bond moves $11.2578 / 0.25 = 45.03$ times as much for the same shift in yield. So five hundred three-month contracts remove $500 \times 25 = 12{,}500$ dollars per basis point out of the 56,289 you were aiming at, which is 22.2 per cent of the job. Covering the whole exposure would take 2,252 of them.

Fig. 1 — Matching the contract count does not match the risk. The short bar is the same five hundred contracts bought at the wrong end of the curve.

Even at 2,252 the position would be wrong in a way the arithmetic cannot fix, because it hedges the three-month rate against a thirty-year exposure. Parallel shifts would net out and a curve twist, which is the common case, would leave the book fully exposed at the long end while showing a hedge on the risk report.

Repriced rather than differentiated

Equation (1) is a first derivative, so a hedge built on it is only locally right. The way to find out how local is to reprice rather than to differentiate. Shifting the curve by minus a hundred, minus twenty-five, plus twenty-five and plus a hundred basis points and revaluing both legs exactly, the hedged book keeps 0.5000 of the unhedged move in every case, to four decimal places.

Fig. 2 — Exact revaluation at four shifts, not a duration estimate. The ratio is 0.5000 in all four because both legs are the same instrument.

That exactness is not evidence that duration hedging is exact. It is a consequence of the setup: both legs are the same bond, so the hedged book is literally a fifty-million-dollar long position and there is nothing for convexity to disagree about. The moment the delivered bond differs from the one you hold, the halving is only first-order true and the residual grows with the square of the yield move.

The same position in a zero needs 1,234

Here is the counterexample that gives the caveat teeth. Hold a hundred million dollars of market value in a thirty-year zero-coupon bond instead, and want half of it hedged as before. A zero's Macaulay duration is its maturity, so its modified duration is

D_{\text{mod}} \;=\; \frac{30}{1.08} \;=\; 27.7778

(6)

which is 2.4674 times the par bond's. Equation (4) scales the count by exactly that ratio:

N \;=\; 500 \times \frac{27.7778}{11.2578} \;=\; 1{,}233.7

(7)

Selling 500 contracts against that book would take out about forty per cent of what you meant to take out and report a completed hedge. So the face-value shortcut is not a rule with an exception. It is one point where two durations happened to coincide, and equation (4) is the rule.

When there is no duration to use

Duration assumes the instrument you hold and the instrument you trade respond to the same yield. For a corporate bond that assumption fails at the first credit event, because the spread moves independently of the Treasury curve and the contract tracks only the curve. Duration ratios then flatter the hedge.

The usual substitute is empirical: regress the position's daily returns on the contract's, and use the slope as the hedge ratio. That is honest about what it is doing and it inherits the usual problems. The slope is a sample statistic with a standard error, it drifts as credit conditions change, and it was measured in whatever regime the sample covered rather than the one you are about to hedge.

Two more things sit between equation (4) and a real trade. The deliverable is chosen by the short, so the contract tracks the cheapest bond in the basket rather than a fixed one, and its effective duration is that bond's divided by its conversion factor. And the delivery timing is optional, which means the hedger is short an option nobody priced. Neither changes the arithmetic above, and both change the number you would actually put in the market.

Sources and further reading

The sensitivity in equations (1) and (6) Bond duration
The general form of equation (4) Hedge (finance)
The unit everything here is quoted in Basis point
The contracts being shorted United States Treasury security
The regression slope of the last section Beta (finance)

The duration was derived from the bond's own cash flows rather than looked up, the hedge was confirmed by exact revaluation at four yield shifts rather than by differentiating, and the contract count was reached twice by routes that share no step: fifty million over a hundred thousand, and 56,288.92 over 112.5778.