Key-Rate Exposure and Residual Risk in Swap Hedges

Why a Parallel Hedge Can Leave Curve Risk

A swap hedge can offset a portfolio’s sensitivity to a parallel interest-rate change while leaving substantial exposure to a twist or bend in the yield curve. The reason is that a parallel change moves all relevant rates by the same amount; a nonparallel change does not. Matching one overall duration, or one aggregate sensitivity, checks only the parallel direction.

Key-rate exposures make the remaining risk visible. They describe how the value of a position responds to small changes at specified points on the curve. Comparing the portfolio and swap exposure vectors shows whether the hedge offsets each point’s sensitivity or merely makes their sum equal and opposite.

The key-rate points, curve interpolation method, valuation date, and bump size must be specified consistently for the portfolio and the swap. In this tutorial, exposures are signed dollar changes per basis point, and a curve movement is written as a vector of changes in basis points. Assume the instruments are valued on the same curve and that the shocks are small enough for a first-order estimate. In practice, the underlying exposure may be estimated by revaluing an instrument after a small, localized bump to the curve.

From Key-Rate Exposures to a Value Estimate

Suppose a position has exposures \(E_1,\ldots,E_m\), and the corresponding key rates move by \(\Delta b_1,\ldots,\Delta b_m\) basis points. For a sufficiently small change, its estimated value change is the sum of each exposure multiplied by its associated rate movement.

The approximation is first-order: it leaves out curvature and interactions between rate changes. It is most useful for diagnosing the direction and approximate size of residual exposure, not for claiming an exact value change under a large curve move.

A parallel movement of \(c\) basis points has \(\Delta b_j=c\) at every key rate. Substitution into the estimate gives the aggregate parallel exposure.

Thus, the sum of the signed key-rate exposures is the position’s first-order exposure to a one-basis-point parallel increase. If that sum is zero, the position is first-order neutral to a small parallel move under this model. That condition does not require each individual \(E_j\) to be zero.

Why Matching Overall Duration Is Not Enough

Duration summarizes value sensitivity to an assumed rate movement, commonly a parallel movement. Key-rate exposures refine that summary by distinguishing the parts of the curve at which the sensitivity arises. Two positions can have the same aggregate parallel exposure even though their key-rate exposure vectors differ. They will then respond differently when the curve moves nonparallel.

The relationship can be seen by treating the value \(V\) as a function of the key-rate nodes. For small changes, the first-order change in value is the sum of the partial change in value at each node multiplied by that node’s change. Expressing each partial change per basis point gives the key-rate exposure formula.

For a conventional fixed-income position with value \(V\), modified duration \(D_{\mathrm{mod}}\) summarizes the first-order effect of a parallel change in decimal yield: \(\Delta V\approx -V D_{\mathrm{mod}}\Delta y\). Since one basis point is \(0.0001\) in decimal yield, this corresponds to an aggregate signed exposure of approximately \(-V D_{\mathrm{mod}}(0.0001)\) dollars per basis point. This scalar summary does not specify how that exposure is distributed across key rates. A swap hedge can therefore match the scalar duration exposure without matching the portfolio’s key-rate vector.

This description assumes the selected curve and bump convention are used consistently in valuing both instruments. Real swap portfolios may involve distinct discounting and floating-rate curves, basis risk, or other features. Those require additional modeling; they are not resolved by a key-rate comparison alone.

Worked Examples: Matching and Diagnosing Exposures

Interpreting Residual Risk in Practice

A residual vector should be read together with the movement scenario. A large positive component indicates a potential gain for an increase at that key rate, while a large negative component indicates a potential loss for that increase. The net effect depends on the size and direction of movement at each rate, as well as the limitations of the first-order approximation.

When comparing candidate hedges, a useful sequence is:

• Check that the portfolio and swap exposures use the same key-rate points, valuation basis, and units.
• Compare the sums to assess first-order parallel exposure.
• Subtract or add the swap vector using the actual hedge direction and notional to find the residual vector.
• Apply the specified nonparallel movements component by component, preserving all signs.
• Interpret the result as a local estimate, not a complete valuation under a large shock.

The signed dollar convention used here is especially convenient for combining assets, liabilities, and derivatives: add exposures for positions held and subtract exposures for positions owed or sold. A position’s aggregate exposure can be zero even when its components are material. Accordingly, an effective curve hedge may require attention to the residual vector, not only to a single duration or parallel sensitivity.