Why a Duration Gap Matters
An institution may hold assets to support liabilities whose values also change with interest rates. The relevant quantity is not whether the asset value or liability value moves more on its own, but how their movements combine in surplus, defined here as asset value minus liability value, with any swap value included. If a rate change affects the two sides differently, surplus changes even when no cash flow has yet been received or paid.
A plain-vanilla interest rate swap can change this sensitivity. For example, under the usual valuation convention, a pay-fixed, receive-floating swap tends to gain value when market rates rise; a receive-fixed, pay-floating swap tends to lose value when rates rise. The swap therefore adds its own rate sensitivity to the asset-liability position. The direction of that effect must be checked against the existing surplus exposure.
A basis point is \(0.0001\) in decimal yield. Thus, a 10-basis-point increase means \(\Delta y=10(0.0001)=0.0010\). In the duration approximation, a positive duration means that value decreases when yields rise. This sign applies to ordinary fixed-cash-flow assets and liabilities; the swap’s sign is handled separately using its stated sensitivity or position.
Combining Asset, Liability, and Swap Sensitivities
Since \(S=A-L+W\), the change in surplus is \(\Delta S=\Delta A-\Delta L+\Delta W\). Substituting the duration approximations gives a direct way to interpret the asset-liability gap and the swap contribution.
If the swap’s sensitivity is \(K\) dollars per basis point for a yield increase, then for a parallel move of \(b\) basis points, \(\Delta W\approx Kb\). Equivalently, the asset-liability contribution per basis point is \((-D_AA+D_LL)0.0001\). Both calculations use the same direction convention: a positive number is a surplus gain for a rate increase; a negative number is a surplus loss for a rate increase.
Why the Duration Approximation Works
For a portfolio of fixed cash flows, present value is the sum of each cash flow discounted to the valuation date. Modified duration measures the approximate proportional change in that present value for a small change in yield. In differential form, \(dA/dy\approx-D_AA\), so for a small finite change \(\Delta y\), \(\Delta A\approx-D_AA\Delta y\). Apply the same reasoning to liabilities and subtract their value change from the asset value change.
The swap is added, not subtracted, because \(W\) is already defined as the value of the institution’s swap position. Its sensitivity can be supplied as dollars per basis point, so its estimated contribution is added directly to the asset-liability contribution. This derivation also shows the approximation’s limits: it is local to the current valuation and assumes a small, parallel yield change. It does not establish that every cash flow or every point on the yield curve is hedged.
Worked Examples
Worked Example: Checking Whether a Receive-Fixed Swap Reduces the Gap
An institution has assets worth \(80{,}000{,}000\) dollars with modified duration \(5.1\), and liabilities worth \(72{,}000{,}000\) dollars with modified duration \(6.0\). It holds a receive-fixed, pay-floating swap with a sensitivity of \(-1{,}500\) dollars per basis point for a parallel rate increase. Estimate the change in surplus for a 12.5-basis-point rate increase. Assume a small parallel shift and use the duration approximation.
Method. Calculate the asset-liability surplus sensitivity first, then add the swap sensitivity. This makes the swap’s direction explicit.
For a one-unit yield increase, the asset duration-dollar amount is \(5.1(80{,}000{,}000)=408{,}000{,}000\) dollars, and the liability amount is \(6.0(72{,}000{,}000)=432{,}000{,}000\) dollars. Therefore, the asset-liability surplus change for a one-basis-point rate increase is
The swap contributes \(-1{,}500\) dollars per basis point, so the combined sensitivity is \(2{,}400-1{,}500=900\) dollars per basis point. For a 12.5-basis-point increase, the estimated surplus change is \(900(12.5)=11{,}250\) dollars. The swap reduces the original positive sensitivity from \(2{,}400\) to \(900\) dollars per basis point, but does not eliminate it.
Check. The assets lose approximately \(408{,}000{,}000(0.00125)=510{,}000\) dollars, while the liabilities lose approximately \(432{,}000{,}000(0.00125)=540{,}000\) dollars. Subtracting the liability change gives a \(30{,}000\)-dollar asset-liability surplus gain. The swap loses \(1{,}500(12.5)=18{,}750\) dollars, leaving \(30{,}000-18{,}750=11{,}250\) dollars, as calculated.
Common mistake avoided: A receive-fixed swap has negative, not positive, sensitivity to a rate increase under the convention stated in the problem. Reversing that sign would incorrectly conclude that it increases this positive surplus exposure.
Worked Example: A Pay-Fixed Swap That Reinforces the Existing Exposure
Assets are worth \(30{,}000{,}000\) dollars with modified duration \(3.2\), and liabilities are worth \(26{,}000{,}000\) dollars with modified duration \(3.8\). The institution has a pay-fixed, receive-floating swap whose value increases by \(190\) dollars per basis point when rates rise. Estimate the combined surplus change for a 20-basis-point rate decrease.
Method. Find the rate-increase sensitivity, including the swap, and then apply a negative basis-point change for the rate decrease.
The asset duration-dollar amount is \(3.2(30{,}000{,}000)=96{,}000{,}000\) dollars. The liability amount is \(3.8(26{,}000{,}000)=98{,}800{,}000\) dollars. The asset-liability sensitivity for a one-basis-point increase is
The swap adds \(190\) dollars per basis point for a rate increase. Thus, combined sensitivity is \(280+190=470\) dollars per basis point for an increase. A 20-basis-point decrease is \(-20\) basis points, so the estimated change is \(470(-20)=-9{,}400\) dollars: surplus decreases by approximately \(9{,}400\) dollars.
Check. For a rate decrease, asset value rises by approximately \(96{,}000{,}000(0.0020)=192{,}000\) dollars and liability value rises by \(98{,}800{,}000(0.0020)=197{,}600\) dollars. The asset-liability surplus change is \(192{,}000-197{,}600=-5{,}600\) dollars. The swap loses \(190(20)=3{,}800\) dollars when rates fall. The combined change is \(-5{,}600-3{,}800=-9{,}400\) dollars.
Common mistake avoided: The liabilities are more rate-sensitive in dollars than the assets: \(98.8\) million dollars versus \(96\) million dollars of duration-dollar exposure. Because liabilities are subtracted, this makes the asset-liability surplus sensitivity positive for a rate increase. The pay-fixed swap reinforces that sensitivity; it does not hedge it.
Worked Example: Quantifying the Residual After a Partial Hedge
An institution has assets worth \(45{,}000{,}000\) dollars with modified duration \(4.4\) and liabilities worth \(40{,}000{,}000\) dollars with modified duration \(5.0\). A receive-fixed swap loses \(125\) dollars per basis point when rates rise. Estimate the combined surplus change for an 18-basis-point decrease. State what the duration estimate does and does not establish.
Method. Convert the asset-liability duration exposure to dollars per basis point, combine it with the swap sensitivity, and apply the signed rate change.
The asset duration-dollar amount is \(45{,}000{,}000(4.4)=198{,}000{,}000\) dollars. The liability amount is \(40{,}000{,}000(5.0)=200{,}000{,}000\) dollars. Before the swap, the surplus sensitivity is
The swap sensitivity is \(-125\) dollars per basis point for an increase, so the combined sensitivity is \(200-125=75\) dollars per basis point for an increase. An 18-basis-point decrease is \(-18\) basis points; therefore, the estimated combined change is \(75(-18)=-1{,}350\) dollars.
Check. For the rate decrease, assets gain approximately \(198{,}000{,}000(0.0018)=356{,}400\) dollars and liabilities gain \(200{,}000{,}000(0.0018)=360{,}000\) dollars. The asset-liability surplus therefore falls by \(3{,}600\) dollars. The receive-fixed swap gains \(125(18)=2{,}250\) dollars as rates fall. The net change is \(-3{,}600+2{,}250=-1{,}350\) dollars.
The swap has reduced the first-order sensitivity from \(200\) to \(75\) dollars per basis point for an increase, but the calculation does not show that the asset, liability, and swap cash flows occur on matching dates. Nor does it show that the hedge works equally well if rates at different maturities move by different amounts.
Common mistake avoided: Applying the swap’s stated sensitivity for a rate increase unchanged to a rate decrease would give the wrong direction. The signed move is \(-18\) basis points, so the swap’s value change is \((-125)(-18)=+2{,}250\) dollars.
What a Reduced Duration Gap Does Not Hedge
A close match in first-order surplus sensitivity is useful, but it is not the same as matching all future cash flows. The asset and liability payments may occur at different dates, and a standard swap has its own accrual periods and payment dates. Even when their approximate value changes offset for a particular small parallel movement, intermediate cash needs or differences in payment timing may remain.
The calculation also assumes a parallel yield shift: the same change in yield is applied across the maturities relevant to the assets, liabilities, and swap. If short- and long-term rates move by different amounts, the single duration estimate may not capture the resulting changes accurately. In addition, duration is a local approximation; for larger rate changes, the relationship between yield and value may not be linear.
For an exam calculation, keep the scope of the result precise: a duration-based swap hedge can reduce estimated surplus sensitivity for the specified rate movement and valuation assumptions. It does not prove that the position is risk-free, that cash flows match, or that every possible curve movement is offset.
Check Your Understanding
For each numerical question, use the small, parallel-shift duration approximation. Treat a positive swap sensitivity as a gain per basis point when rates rise.
- Assets are worth \(25{,}000{,}000\) dollars with duration \(4.0\); liabilities are worth \(23{,}000{,}000\) dollars with duration \(4.5\). There is no swap. Estimate the surplus change for an 8-basis-point increase. (Answer: surplus increases by approximately \(2{,}800\) dollars.)
- Assets are worth \(60{,}000{,}000\) dollars with duration \(3.5\); liabilities are worth \(55{,}000{,}000\) dollars with duration \(4.0\). A receive-fixed swap has sensitivity \(-650\) dollars per basis point for a rate increase. Estimate the surplus change for a 20-basis-point decrease. (Answer: surplus decreases by approximately \(7{,}000\) dollars.)
- Assets are worth \(18{,}000{,}000\) dollars with duration \(2.8\); liabilities are worth \(16{,}000{,}000\) dollars with duration \(3.3\). A pay-fixed swap has sensitivity \(+90\) dollars per basis point for a rate increase. Estimate the surplus change for a 7-basis-point increase. (Answer: surplus increases by approximately \(2{,}310\) dollars.)
- Assets are worth \(100{,}000{,}000\) dollars with duration \(6.2\); liabilities are worth \(94{,}000{,}000\) dollars with duration \(6.5\). A receive-fixed swap has sensitivity \(-250\) dollars per basis point for a rate increase. Estimate the surplus change for a 10-basis-point decrease. (Answer: surplus increases by approximately \(11{,}500\) dollars.)
- An institution’s asset and liability durations imply nearly zero surplus sensitivity for a small parallel shift after a swap is included. Name two risks that this result alone does not establish to be eliminated. (Answer: cash-flow timing mismatch and exposure to nonparallel yield-curve changes are not established to be eliminated.)