Deriving Swap Hedge Ratios from Duration

Why a Duration-Based Hedge Ratio Works

A swap hedge is sized to offset a position’s exposure to interest-rate changes. Duration gives a first-order estimate of how a position’s value changes when yields move. The hedge ratio follows by matching the signed value changes—not merely comparing the sizes of two positive amounts. That distinction matters especially when the position being hedged is a liability.

The method here assumes a small, parallel change in yields and uses modified durations evaluated at the current valuation date. It is a local approximation: it does not account for convexity, nonparallel curve movements, basis risk, or changes in cash flows. Those effects can leave a residual even when first-order duration exposure is matched.

Definition: Let $V$ be the signed market value of a position: an asset has positive value, and a liability has negative value. If its modified duration is $D$, its first-order change in value for a small yield change $\Delta y$ is $$ \Delta V \approx -D V\,\Delta y. $$ The quantity $-DV$ is the position’s signed dollar sensitivity per unit change in yield. A positive value of $-DV$ means the position’s value increases when yields rise; a negative value means it decreases.

For a portfolio, add the signed sensitivities of its components. If there is one hedged position $H$ and a swap with notional $N$, write the swap’s first-order sensitivity per unit notional as $k_S$. The combined first-order change is $$ \Delta V_{\text{combined}} \approx \left(-D_H V_H + N k_S\right)\Delta y. $$ A duration hedge sets the coefficient of $\Delta y$ equal to zero. Thus, when $k_S\ne 0$, $$ N=\frac{D_H V_H}{k_S}. $$ Here $V_H$ is signed, and $k_S$ includes the direction of the swap position. The resulting notional must be positive; if the calculation would require a negative notional, take the opposite swap position instead.

Deriving the Hedge Condition

The hedge ratio is not a separate rule to memorize: it comes directly from adding the two first-order changes. The derivation also makes the sign convention explicit.

Write the hedged position’s change.
For signed market value $V_H$ and modified duration $D_H$, the change is approximately $-D_HV_H\Delta y$.

Write the swap’s change.
If the swap sensitivity is $k_S$ per unit notional, its change at notional $N$ is approximately $Nk_S\Delta y$.

Set the combined change to zero.
To neutralize the first-order exposure, require $-D_HV_H\Delta y+Nk_S\Delta y=0$. For a nonzero shift, divide by $\Delta y$.

Solve for notional.
The result is $N=D_HV_H/k_S$. Check the sign and verify the sensitivity of both positions is equal in magnitude and opposite in sign.

For a swap, $k_S$ can be obtained from its legs. Consider a payer-fixed position: it receives floating and pays fixed. If the current present values and modified durations of those legs are $V_{\text{float}},D_{\text{float}}$ and $V_{\text{fixed}},D_{\text{fixed}}$, respectively, the swap value is $V_{\text{float}}-V_{\text{fixed}}$. Under the same parallel-shift approximation, its sensitivity per unit notional is $$ k_{\text{payer}}=-D_{\text{float}}V_{\text{float}}+D_{\text{fixed}}V_{\text{fixed}}. $$ A receive-fixed position has the opposite sensitivity. This expression assumes the leg values and durations are stated on a consistent notional basis; if they are specified per one million dollars of notional, the resulting sensitivity is per one million dollars.

Important qualification: A par swap can have zero net market value at inception. Its ordinary modified duration calculated as $-\frac{1}{V}\frac{dV}{dy}$ is then undefined, because $V=0$. Do not multiply its zero net value by a duration to estimate its sensitivity. Use the fixed- and floating-leg sensitivities, or a supplied dollar sensitivity per unit notional. If a nonzero-valued swap position is instead supplied with a meaningful modified duration, its signed sensitivity is $-D_SV_S$.

Worked Applications

Worked Example: Hedging an Asset Portfolio

A hypothetical bond portfolio has market value $ \$12{,}000{,}000 $ and modified duration $5.5$. A payer-fixed swap has, per $ \$1{,}000{,}000 $ of notional, a fixed-leg present value of $ \$980{,}000 $ with modified duration $4.4$, and a floating-leg present value of $ \$980{,}000 $ with modified duration $0.2$. Find the payer-fixed notional that hedges the portfolio’s first-order exposure. Assume a small parallel yield shift, the stated leg durations remain applicable, and convexity is ignored.

Method. Compute the portfolio’s signed dollar sensitivity, then calculate the payer-fixed swap sensitivity from its legs. This avoids treating the swap’s net value as the sensitivity base.

The portfolio is an asset, so its signed value is $+\$12{,}000{,}000$. Its sensitivity per unit yield change is $$ -D_HV_H=-(5.5)(\$12{,}000{,}000)=-\$66{,}000{,}000. $$ Thus, if yields rise, the portfolio loses value to first order.

For each $ \$1{,}000{,}000 $ of payer-fixed notional, the swap sensitivity is $$ k_{\text{payer}}=-(0.2)(\$980{,}000)+(4.4)(\$980{,}000) =-\$196{,}000+\$4{,}312{,}000 =\$4{,}116{,}000. $$ This sensitivity is positive: the payer-fixed swap gains value to first order when yields rise. The required notional is $$ N=\frac{\$66{,}000{,}000}{\$4{,}116{,}000\text{ per } \$1{,}000{,}000} =16.0349854\text{ million dollars}. $$ Rounded to the nearest cent, the payer-fixed notional is $ \$16{,}034{,}985.42 $.

Check. The swap’s total sensitivity is $16.0349854(\$4{,}116{,}000)=\$66{,}000{,}000$ per unit yield, opposite the portfolio’s $-\$66{,}000{,}000$. For a $0.001$ yield increase, the portfolio changes by approximately $-\$66{,}000$, while the swap changes by approximately $+\$66{,}000$. The first-order changes cancel.

Common mistake avoided: Choosing a receiver-fixed swap because “rates rising hurt bonds” would reinforce the portfolio’s loss. The hedge direction must be chosen from the signed sensitivities.

Worked Example: Hedging a Liability Position

A hypothetical liability has a positive present-value amount of $ \$8{,}000{,}000 $ and modified duration $7$. The hedger holds the liability position, so its signed market value is negative. A payer-fixed swap has a sensitivity of $+\$3{,}500{,}000$ per unit yield change for each $ \$1{,}000{,}000 $ of notional. Find the swap direction and notional needed to hedge the liability position. Assume a small parallel shift, and assume a receive-fixed position has the opposite sensitivity to the payer-fixed position.

Method. First assign the liability its signed value. Then equate the sensitivity of the swap to the opposite of the sensitivity of that signed liability position.

The signed liability position has $V_H=-\$8{,}000{,}000$. Its sensitivity is $$ -D_HV_H=-(7)(-\$8{,}000{,}000)=+\$56{,}000{,}000. $$ The positive sensitivity means the signed value of the liability position increases when yields rise. The swap must therefore have negative sensitivity. Since the payer-fixed swap has positive sensitivity, use a receive-fixed swap with sensitivity $-\$3{,}500{,}000$ per unit yield per $ \$1{,}000{,}000 $ of notional.

The notional needed is $$ N=\frac{\$56{,}000{,}000}{\$3{,}500{,}000\text{ per } \$1{,}000{,}000} =16\text{ million dollars}. $$ The hedge is a receive-fixed swap with notional $ \$16{,}000{,}000 $.

Check. The liability position’s sensitivity is $+\$56{,}000{,}000$; the receive-fixed swap’s sensitivity is $16(-\$3{,}500{,}000)=-\$56{,}000{,}000$, per unit yield change. For a $0.002$ increase in yields, their first-order changes are $+\$112{,}000$ and $-\$112{,}000$.

Common mistake avoided: Treating the liability’s positive present-value amount as if it were a positive asset gives sensitivity $-\$56{,}000{,}000$ and would incorrectly select a payer-fixed swap. That calculation describes the change in the positive amount used to report the liability, not the change in the signed value of the liability position being hedged.

Worked Example: Hedging a Net Asset-Liability Position

A hypothetical balance-sheet position consists of an asset worth $ \$9{,}000{,}000 $ with modified duration $4$, and a liability with positive present-value amount $ \$2{,}000{,}000 $ and modified duration $6$. A payer-fixed swap has, per $ \$1{,}000{,}000 $ of notional, fixed-leg present value $ \$1{,}050{,}000 $ and modified duration $5$, and floating-leg present value $ \$1{,}050{,}000 $ and modified duration $0.2$. Find the hedge direction and notional. Assume a small parallel yield shift, the stated durations apply, and convexity is ignored.

Method. Add the signed dollar sensitivities of the asset and liability before sizing the swap. Hedging the asset alone would fail to account for the liability’s offsetting exposure.

The asset sensitivity is $$ -(4)(\$9{,}000{,}000)=-\$36{,}000{,}000. $$ The liability’s signed value is $-\$2{,}000{,}000$, so its sensitivity is $$ -(6)(-\$2{,}000{,}000)=+\$12{,}000{,}000. $$ The combined sensitivity is $-\$36{,}000{,}000+\$12{,}000{,}000=-\$24{,}000{,}000$ per unit yield change.

The payer-fixed swap sensitivity per $ \$1{,}000{,}000 $ of notional is $$ -(0.2)(\$1{,}050{,}000)+(5)(\$1{,}050{,}000) =-\$210{,}000+\$5{,}250{,}000 =\$5{,}040{,}000. $$ Because the net position has negative sensitivity, use a payer-fixed swap. Its required notional is $$ N=\frac{\$24{,}000{,}000}{\$5{,}040{,}000\text{ per } \$1{,}000{,}000} =4.7619048\text{ million dollars}. $$ Rounded to the nearest cent, the payer-fixed notional is $ \$4{,}761{,}904.76 $.

Check. At that notional, the swap sensitivity is $4.7619048(\$5{,}040{,}000)=+\$24{,}000{,}000$ per unit yield, offsetting the net position’s $-\$24{,}000{,}000$. For a $0.001$ yield decrease, the net position gains approximately $ \$24{,}000 $, and the swap loses approximately $ \$24{,}000 $.

Common mistake avoided: Applying the hedge ratio to the asset alone would target a sensitivity of $-\$36{,}000{,}000$, overstating the hedge because the liability already contributes $+\$12{,}000{,}000$ of sensitivity.

Interpreting the Result and Its Limits

A duration hedge neutralizes only the first-order response to the yield change specified by the model. For a finite rate move, convexity can make the actual change differ from the duration estimate. Also, a single duration number describes exposure to a parallel shift; it does not ensure that a swap and a portfolio respond equally to changes concentrated at different maturities.

Before accepting a calculated hedge ratio, check three things: the signed value of each position, the sensitivity and direction of the swap, and the units used for notional. If swap sensitivity is quoted per $ \$1{,}000{,}000 $, the computed ratio is in millions of dollars, not dollars. The hedge should then be checked by applying a small positive yield change and confirming that the two first-order value changes have opposite signs and equal magnitudes.

Key takeaway: Use signed market values in $\Delta V\approx-DV\Delta y$, add sensitivities across the positions being hedged, and choose the swap direction that gives the opposite sensitivity. A par swap’s zero net value is not a usable duration base; calculate its sensitivity from its legs or use a supplied sensitivity per unit notional.

Check Your Understanding

For each problem, use signed position values and assume a small parallel yield shift unless stated otherwise.

A hypothetical asset portfolio is worth $ \$5{,}000{,}000 $ and has modified duration $4$. A payer-fixed swap has sensitivity $+\$2{,}000{,}000$ per unit yield change for each $ \$1{,}000{,}000 $ of notional. Find the payer-fixed hedge notional. (Answer: $ \$10{,}000{,}000 $)
A liability position has signed value $-\$3{,}000{,}000$ and modified duration $5$. A receiver-fixed swap has sensitivity $-\$1{,}250{,}000$ per unit yield change per $ \$1{,}000{,}000 $ of notional. Find the hedge notional. (Answer: $ \$12{,}000{,}000 $ receiver-fixed)
A position contains an asset worth $ \$7{,}000{,}000 $ with duration $3$, and a liability with signed value $-\$2{,}000{,}000$ and duration $4$. A payer-fixed swap has sensitivity $+\$2{,}600{,}000$ per unit yield per $ \$1{,}000{,}000 $ of notional. Find the hedge direction and notional. (Answer: $ \$5{,}000{,}000 $ payer-fixed)
A liability position has signed value $-\$4{,}000{,}000$ and duration $5$. Per $ \$1{,}000{,}000 $ notional, a payer-fixed swap has fixed-leg PV $ \$800{,}000 $ and duration $6$, and floating-leg PV $ \$800{,}000 $ and duration $0.25$. Find the hedge direction and notional. (Answer: $ \$4{,}347{,}826.09 $ receive-fixed)
An asset worth $ \$10{,}000{,}000 $ has duration $2.5$; a liability position has signed value $-\$1{,}000{,}000$ and duration $3$. A receive-fixed swap has sensitivity $-\$2{,}200{,}000$ per unit yield per $ \$1{,}000{,}000 $ of notional. Find the notional that hedges the combined position. (Answer: $ \$10{,}000{,}000 $ receive-fixed)

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