Sizing a Swap Hedge with Present-Value Sensitivity

Why Hedge Notional Depends on Sensitivity

A swap hedge is sized to offset a change in value, not merely to match the portfolio’s market value. A relatively small swap can have a large effect when its value is sensitive to rates; a much larger swap may be needed when each dollar of notional has only a small sensitivity. The key calculation is to compare the portfolio’s dollar change for a stated rate move with the swap’s dollar change for the same move.

The sign convention must be explicit. In this tutorial, PV01 means the signed change in value when the relevant market rates fall by one basis point, where one basis point is $0.0001$ in decimal-rate units. Thus, a position that gains value when rates fall has positive PV01. For a standard fixed-rate swap, a receive-fixed position has positive PV01 and a pay-fixed position has negative PV01, assuming a parallel rate shift and the usual direction of fixed-leg value change.

Definition: A position’s signed PV01 is its approximate dollar value change for a one-basis-point decrease in the specified market rates. A positive PV01 indicates a gain for that rate decrease; a negative PV01 indicates a loss. The PV01 convention must state both the direction of the rate move and whether the number is signed or an unsigned magnitude.

A quoted sensitivity may instead be reported for a one-basis-point increase, or as a positive magnitude without a position sign. Such figures can still be used, but they must first be converted to a common convention. The calculations here use signed PV01 for a one-basis-point rate decrease throughout.

The Signed Notional Equation

Let $P$ denote the portfolio’s signed PV01 after all relevant asset and liability exposures have been combined. Let $p$ be the signed PV01 of a receive-fixed swap per unit of notional, measured using the same rate bump and valuation basis. Define $N$ as a signed notional: $N>0$ means receive fixed, and $N<0$ means pay fixed. Under this convention, the swap’s PV01 is $Np$.

P+Np=0 \qquad\Longrightarrow\qquad N=-\frac{P}{p}.

This equation gives the notional that offsets the portfolio’s first-order value change for the specified rate move. If a swap’s PV01 is quoted per $1 million of receive-fixed notional, the quotient first gives the signed number of millions. Multiply by $1,000,000 to obtain the signed notional in dollars.

Important: With the signed-notional convention above, a negative answer means pay fixed and a positive answer means receive fixed. Do not remove the minus sign before interpreting the hedge direction.

Why the Matching Formula Works

For a small parallel rate change, a position’s value change is approximately proportional to the size of that change. If $\Delta r$ is measured in basis points, then a position with PV01 $P$ has approximate value change $P\Delta r$. A receive-fixed swap with signed notional $N$ and per-million PV01 $p$ has approximate value change $Np\Delta r$, when $N$ is expressed in millions. The combined change is therefore

\Delta V_{\text{combined}} \approx (P+Np)\Delta r.

To make the combined change zero for this rate move, set $P+Np=0$. Solving gives $N=-P/p$. The same result follows by comparing the changes directly for a one-basis-point decrease: the portfolio changes by $P$ dollars and the swap by $Np$ dollars, so the offsetting condition is $P+Np=0$.

The matching is an approximation to a local sensitivity, not a guarantee that the hedge eliminates all value changes. It is most directly applicable when the portfolio and swap sensitivities are measured consistently and the rate move is small. Larger moves, nonparallel changes across maturities, and differences between the portfolio’s and swap’s cash-flow exposures can leave residual risk.

Worked Swap-Hedge Calculations

Worked Example: Match a Portfolio PV01 to a Pay-Fixed Swap

A bond portfolio has a signed PV01 of $+\$64{,}000$ for a one-basis-point fall in rates. A receive-fixed swap has a PV01 of $+\$80$ per $1 million of notional for the same rate move. Find the signed swap notional that offsets the portfolio’s sensitivity, and state the position.

Method. Use the signed-notional equation, keeping the receive-fixed swap’s unit PV01 positive. Since the portfolio gains when rates fall, the offsetting swap must lose for the same move.

First express the notional in millions. Setting the combined PV01 to zero gives

64{,}000+80N_{\text{millions}}=0.

Solving,

N_{\text{millions}}=-\frac{64{,}000}{80}=-800.

The signed notional is $-800$ million dollars, or $-\$800{,}000{,}000$. The negative sign means pay fixed.

Check. For a one-basis-point fall, the portfolio gains $\$64{,}000$. The pay-fixed swap has PV01 $-\$80$ per $1 million, so at $800 million its PV01 is $-800\times \$80=-\$64{,}000$. The combined PV01 is $+\$64{,}000-\$64{,}000=\$0$. This avoids the common mistake of choosing receive fixed simply because the portfolio PV01 is positive.

Worked Example: Net Asset and Liability Sensitivities First

A hypothetical fund has asset PV01 of $+\$42{,}000$ and liability PV01 of $+\$17{,}000$, both measured for a one-basis-point fall in rates. The fund wants to hedge the sensitivity of its surplus, defined as asset value minus liability value. A receive-fixed swap has PV01 $+\$80$ per $1 million of notional. Find the signed notional for a first-order hedge.

Method. Calculate the surplus PV01 as asset PV01 minus liability PV01. Then apply the signed-notional equation to the resulting net exposure.

When rates fall one basis point, the assets gain $\$42{,}000$, while the liabilities also gain $\$17{,}000$. Because liabilities are subtracted in the surplus, the surplus change is

P_{\text{surplus}}=42{,}000-17{,}000=25{,}000.

Let $N_{\text{millions}}$ be positive for receive fixed and negative for pay fixed. The hedge equation is

25{,}000+80N_{\text{millions}}=0, \qquad N_{\text{millions}}=-\frac{25{,}000}{80}=-312.5.

The signed notional is $-\$312{,}500{,}000$, so the fund should pay fixed.

Check. A pay-fixed swap with notional $312.5 million has PV01 $-312.5\times \$80=-\$25{,}000$. Added to the surplus PV01 of $+\$25{,}000$, the total is zero. Subtracting the liability sensitivity is essential; adding the asset and liability PV01s would hedge a different quantity.

Worked Example: Estimate PV01 From Bumped Values

A portfolio is valued at $\$8{,}400{,}000$ under the current curve. When all relevant rates are shifted down by one basis point, its value is $\$8{,}431{,}600$; when they are shifted up by one basis point, its value is $\$8{,}368{,}800$. A receive-fixed swap valued at the same current curve has value $\$0$ per $1 million notional. Under the corresponding down and up bumps, its values are $+\$63$ and $-\$61$ per $1 million. Estimate central PV01s and size a swap hedge.

Method. Use a symmetric central bump estimate. For a one-basis-point step, central PV01 is half the difference between the value under the down bump and the value under the up bump. This uses both bumped values rather than relying only on one side.

For the portfolio, the central estimate is

P=\frac{8{,}431{,}600-8{,}368{,}800}{2} =\frac{62{,}800}{2} =\$31{,}400.

The current value is the midpoint reference for the bump exercise; the central estimate can also be checked from the individual changes: the down-bump gain is $\$31{,}600$, and the up-bump loss is $\$31{,}200$, whose average magnitude is $(31{,}600+31{,}200)/2=\$31{,}400$.

For the receive-fixed swap per $1 million, the corresponding central PV01 is

p=\frac{63-(-61)}{2} =\frac{124}{2} =\$62\text{ per \$1 million}.

Thus the signed notional in millions is

N_{\text{millions}}=-\frac{31{,}400}{62} =-506.4516129\ldots

The hedge is approximately $-\$506{,}451{,}613$, or about $-\$506.45$ million to the nearest $\$0.01$ million. The negative sign means pay fixed.

Check. Using the unrounded notional, the swap PV01 is approximately $-506.4516129\times \$62=-\$31{,}400$ per $1 million-based calculation. This offsets the portfolio’s $+\$31{,}400$. Rounding the notional to $-\$506.45$ million leaves a negligible rounding difference. This avoids mixing a one-sided portfolio bump with a central swap bump.

A Reliable Calculation Process

Before dividing to find a notional, organize the information so the two sensitivities refer to the same quantity and rate move. In particular, establish whether the portfolio figure is a liability-inclusive surplus sensitivity or an asset-only sensitivity, and establish whether each swap figure is per dollar, per thousand, or per million of notional.

State the bump and sign convention.
For example, define PV01 as the signed change in value for a one-basis-point fall in the relevant rates.

Combine portfolio exposures.
If hedging a surplus, calculate asset PV01 minus liability PV01. Include existing positions with their signed contributions.

Put the swap sensitivity on the same basis.
Use the same rate bump, valuation convention, and definition of the exposure. Note whether the quoted sensitivity is per $1 million or another notional amount.

Solve with signed quantities.
Set portfolio PV01 plus swap PV01 equal to zero, and retain the sign of the resulting notional.

Check the direction and units.
Multiply the signed notional by the swap’s unit PV01 and verify that it offsets the portfolio sensitivity in dollars per basis point.

Common Sign and Measurement Traps

Suppose a portfolio’s PV01 is $+\$25{,}000$, while a receive-fixed swap has PV01 $+\$50$ per $1 million. A mistaken calculation might choose a positive notional of $25{,}000/50=500$ million and conclude that receiving fixed is the hedge. But this position has PV01 $+\$25{,}000$, so it adds to the portfolio exposure rather than offsetting it. The correct calculation is

N_{\text{millions}}=-\frac{25{,}000}{50}=-500,

which means pay fixed. The minus sign in the hedge equation is not optional: it ensures the swap’s value change has the opposite sign to the portfolio’s.

Other frequent errors include:

Mixing rate directions. A PV01 for rates rising one basis point has the opposite sign from the PV01 convention used here, for the same local exposure.
Using an unsigned DV01 as though it were signed. A magnitude alone does not identify whether the position gains or loses for the stated rate move.
Forgetting the notional scale. A sensitivity of $\$80$ per $1 million cannot be divided into a portfolio PV01 and then interpreted as dollars without converting from millions.
Hedging the wrong balance-sheet measure. A surplus exposure uses assets minus liabilities, not assets plus liabilities.
Treating the result as an exact hedge for every curve movement. A single PV01 match offsets a local, specified rate shift. It need not offset changes of different sizes or changes that affect maturities differently.

Interpreting the Result

A PV01-matched notional is a local hedge ratio: it balances the estimated dollar response to the chosen one-basis-point rate move. It does not imply that the portfolio and swap have identical cash flows or that their values will move together under every market scenario. The estimate depends on the valuation inputs, the bump method, and the assumed rate movement. For a sufficiently small parallel shift, the sensitivity comparison is useful; for larger or nonparallel shifts, the remaining value change can be material.

The central practical rule is to preserve the signed quantities until the final interpretation. Calculate the exposure being hedged, use a swap PV01 measured on the same basis, solve $P+Np=0$, and then translate the sign of $N$ into receive fixed or pay fixed. This keeps the hedge direction tied to the actual value changes rather than to a memorized direction rule.

Key takeaway: Under the convention that positive signed notional means receive fixed and PV01 measures the value change for a one-basis-point fall in rates, the hedge notional is the negative of portfolio PV01 divided by receive-fixed swap PV01 per unit notional. Verify the resulting dollar sensitivities cancel.

Check Your Understanding

For each problem, use signed PV01 for a one-basis-point fall in rates and signed notional positive for receive fixed. Keep the stated notional units consistent.

A portfolio has PV01 $+\$18{,}000$. A receive-fixed swap has PV01 $+\$45$ per $1 million. Find the signed notional and position. (Answer: $-\$400{,}000{,}000$, pay fixed.)
Assets have PV01 $+\$42{,}000$, and liabilities have PV01 $+\$17{,}000$. A receive-fixed swap has PV01 $+\$80$ per $1 million. Hedge the surplus, defined as assets minus liabilities. (Answer: $-\$312{,}500{,}000$, pay fixed.)
A portfolio’s signed PV01 is $-\$12{,}600$. A receive-fixed swap has PV01 $+\$63$ per $1 million. Find the signed notional. (Answer: $+\$200{,}000{,}000$, receive fixed.)
A portfolio’s value rises by $\$22{,}400$ under a one-basis-point down bump and falls by $\$21{,}600$ under a one-basis-point up bump. A receive-fixed swap’s central PV01 is $+\$55$ per $1 million. Use the central portfolio estimate to find the signed hedge notional. (Answer: $-\$400{,}000{,}000$, pay fixed.)
A desk calculates its residual PV01 as $-\$31{,}000-(-\$51{,}000)-\$6{,}000$. A receive-fixed swap has PV01 $+\$35$ per $1 million. Find the signed notional. (Answer: $-\$400{,}000{,}000$, pay fixed.)

What You'll Learn