Matching Key-Rate Exposures

Why Match Exposures at Curve Nodes?

A liability portfolio can be sensitive to several parts of the yield curve at once. A single duration measure summarizes sensitivity to a specified broad rate movement, but it does not show whether the portfolio is especially exposed to, for example, a change near an intermediate maturity. Key-rate matching addresses this by comparing asset and liability sensitivity at selected curve nodes.

A node match is a targeted first-order hedge, not a guarantee that assets and liabilities will respond identically to every possible curve movement. Its usefulness depends on the chosen nodes, the curve-bump definitions, and whether exposures away from those nodes are also controlled.

Definition: Let $r_j$ be the rate at selected curve node $j$. The key-rate duration of a portfolio with value $P$ is $D_j=-(1/P)(\partial P/\partial r_j)$, where the derivative is calculated by changing the rate at node $j$ according to a specified bump shape while holding the other node rates fixed. Its dollar key-rate exposure is $K_j=PD_j$. Use the same nodes and bump shapes for assets and liabilities. A positive $K_j$ means that a small increase in that node rate decreases portfolio value to first order.

Here, rates are expressed as decimals when calculating changes: a rise of 10 basis points is $0.001$. If holdings are measured in units of a chosen market-value amount, exposures must use the same units. For example, an exposure of 2.4 in “$100,000-years” means a dollar key-rate exposure of $2.4\times \$100{,}000$-years.

First-order surplus relation: For small changes in the selected node rates, let $K_{A,j}$ and $K_{L,j}$ be the asset and liability dollar key-rate exposures. If surplus is assets minus liabilities, then $$ \Delta S \approx -\sum_j (K_{A,j}-K_{L,j})\Delta r_j. $$ This approximation assumes fixed cash flows and uses first-order rate sensitivity. It omits convexity and any changes in cash flows or option behavior. For larger shocks, materially nonlinear cash flows, or shocks not represented by the specified node bumps, use appropriate repricing rather than treating this approximation as exact.

Setting Up a Holdings Match

Suppose asset $i$ has market value $V_i$ per unit of holding and key-rate duration $D_{i,j}$ at node $j$. If $x_i$ units are held, its value contribution is $x_iV_i$, and its dollar key-rate exposure contribution is $x_iV_iD_{i,j}$. Matching the liability portfolio at selected nodes therefore means solving

\sum_i x_iV_i=P_L, \qquad \sum_i x_iV_iD_{i,j}=K_{L,j} \quad\text{for each selected node }j,

where $P_L$ is liability present value and $K_{L,j}$ is its dollar key-rate exposure at node $j$. The first equation matches value; the remaining equations match exposures. If the number or structure of available assets does not allow these equations to be satisfied together, an exact match may not exist. Even when a solution exists, nonnegative holdings, liquidity needs, and other constraints may affect whether it is usable.

Fix the exposure convention.
Record the selected nodes, bump shapes, units, and whether exposures refer to a unit holding or the whole portfolio.

Write one equation for value and one for each matched node.
Use the same definitions for assets and liabilities so the equations compare like with like.

Solve and verify.
Check the value equation, every node equation, and any required holding restrictions.

Test what the match leaves exposed.
Assess specified movements between nodes or outside the selected range separately; a match at selected nodes does not determine those responses by itself.

Why Matching Node Exposures Cancels First-Order Risk

The justification follows from the first-order change in each portfolio’s value. Under the fixed-cash-flow, differentiable-value assumption, a small vector of node-rate changes gives

\Delta P_A\approx-\sum_j K_{A,j}\Delta r_j, \qquad \Delta P_L\approx-\sum_j K_{L,j}\Delta r_j.

Subtracting the liability change from the asset change gives

\Delta S=\Delta P_A-\Delta P_L \approx-\sum_j(K_{A,j}-K_{L,j})\Delta r_j.

If $K_{A,j}=K_{L,j}$ at every selected node, each term in this expression is zero. Thus the first-order surplus change is zero for small movements represented as combinations of those node bumps. This argument does not establish that the surplus change is zero for a curve movement outside that representation; second-order effects may also remain even for movements at the selected nodes.

Worked Example: Solving and Checking a Three-Node Match

A hypothetical liability portfolio has a present value of $1,000,000. Use three curve nodes, at 2, 5, and 10 years. Its dollar key-rate exposures are 3.2, 7.4, and 7.6 in units of $100,000-years. Three available assets each have a market value of $100,000 per holding unit. Their key-rate durations, in years at the respective nodes, are: Asset A, $(0.8,0.2,0)$; Asset B, $(0.2,1.4,0.4)$; and Asset C, $(0,0.4,2.0)$. Find nonnegative holdings that match liability value and all three node exposures.

Method. Let $x_A,x_B,x_C$ denote holding units. Since one unit of each asset is worth $100,000, the liability value requires

x_A+x_B+x_C=\frac{\$1{,}000{,}000}{\$100{,}000}=10.

The node exposure equations, in the stated $100,000-years, are

0.8x_A+0.2x_B=3.2,\quad 0.2x_A+1.4x_B+0.4x_C=7.4,\quad 0.4x_B+2.0x_C=7.6.

Use the value equation and the first and third node equations to solve. Multiplying the first node equation by 5 gives $4x_A+x_B=16$, so $x_B=16-4x_A$. The value equation then gives

$$ x_C=10-x_A-x_B=10-x_A-(16-4x_A)=3x_A-6. $$

Substitute into the third node equation:

0.4(16-4x_A)+2(3x_A-6)=7.6, $$ $$ 6.4-1.6x_A+6x_A-12=7.6,\quad 4.4x_A=13.2,\quad x_A=3.

Then $x_B=16-4(3)=4$ and $x_C=3(3)-6=3$. These holdings are all nonnegative. Check value: $3+4+3=10$, or $1,000,000. Check the exposures:

K_{A,2}=0.8(3)+0.2(4)=3.2, $$ $$ K_{A,5}=0.2(3)+1.4(4)+0.4(3)=0.6+5.6+1.2=7.4, $$ $$ K_{A,10}=0.4(4)+2(3)=1.6+6=7.6.

All three exposures equal the liability targets. The method avoids treating each asset’s duration as though it were its exposure: the duration must be multiplied by the value held.

Node Matches and Movements Between Nodes

A set of key-rate exposures describes sensitivity to the particular bump functions used to define those exposures. It does not, by itself, give a complete description of the response to every possible curve shape. A localized movement between selected nodes may affect asset and liability cash flows differently, even when the exposures at the selected nodes agree exactly.

To assess such a movement, specify the shock and obtain the corresponding sensitivities or reprice the portfolios under that shock. Do not infer an unprovided between-node sensitivity merely by taking an average of the matched node exposures. Interpolation of rates is not automatically interpolation of portfolio sensitivities.

Worked Example: Measuring a Between-Node Residual

Use the holdings from the previous example. Their 2-, 5-, and 10-year exposures match the liabilities. A hypothetical valuation model separately reports dollar sensitivities to a localized 7-year rate bump: the asset holdings have exposure 6.1 and the liabilities have exposure 5.5, both in $100,000-years. The 7-year rate rises by 30 basis points. Estimate the first-order surplus change attributable to this shock.

Method. This is a shock-specific residual calculation. The asset-minus-liability exposure is $6.1-5.5=0.6$ in $100,000-years. Convert the rate change to decimal form: 30 basis points is $0.003$. Using the first-order surplus relation for this specified bump,

\Delta S\approx-(0.6)(0.003)(\$100{,}000) =-\$180.

The approximate surplus change is a loss of $180. The sign is consistent with the asset portfolio having greater exposure: the rate rise decreases asset value more than liability value. The 7-year sensitivities were supplied separately; they cannot be recovered from the three matched node exposures alone.

Residual Exposure Outside the Selected Range

The same limitation applies beyond the outermost selected nodes. Matching exposures at 2, 5, and 10 years does not establish a match for a movement concentrated at, say, 15 years. Long-dated cash flows may respond differently from the selected-node summary. If a relevant outside-range shock is specified, compare the asset and liability responses to that shock directly.

Worked Example: A Shock Beyond the Longest Matched Node

A hypothetical 2-, 5-, and 10-year key-rate match has already been verified. A separate model gives dollar sensitivities to a localized 15-year rate movement of 2.4 for assets and 3.1 for liabilities, in $100,000-years. The 15-year rate falls by 20 basis points. Estimate the first-order surplus change and interpret its sign.

Method. The residual asset-minus-liability exposure is $2.4-3.1=-0.7$ in $100,000-years. The rate change is $-0.002$. Therefore,

\Delta S\approx-(-0.7)(-0.002)(\$100{,}000) =-\$140.

The estimated surplus change is a loss of $140. When the 15-year rate falls, both values increase to first order, but the liability’s larger sensitivity means it increases more. This exposure is outside the selected matching nodes, so the node match provides no basis for setting this residual to zero.

Common Mistakes and Practical Interpretation

Matching durations instead of dollar exposures. A portfolio’s exposure is its value multiplied by its key-rate duration. Compare dollar exposures at each node, not duration figures in isolation.
Mixing bump definitions. Asset and liability key-rate durations must use the same node locations and bump shapes. Otherwise, the numbers do not measure comparable sensitivities.
Forgetting the sign. For a positive exposure, a rate increase produces a first-order value decrease. For surplus, use asset exposure minus liability exposure before applying the negative sign.
Claiming a complete hedge from a node match. A match controls the first-order response to movements represented by the matched bumps. It does not guarantee protection between nodes, outside the range, or against nonlinear effects.
Relying on the approximation for a large shock. The sensitivity formula is first order. For large curve changes or cash flows whose behavior changes with rates, reprice using the relevant model.

Key takeaway: Match asset and liability present values and dollar key-rate exposures using consistent node definitions. Then assess unmatched curve movements separately. A precise match at selected nodes is useful, but its protection is limited to the sensitivities and assumptions actually included.

Check Your Understanding

For each question, use dollar key-rate exposures and express rate changes as decimals.

At two selected nodes, an asset portfolio has exposures $(2.1,4.8)$ and a liability portfolio has exposures $(2.4,4.2)$, in $100,000-years. The node rates change by $(+0.001,-0.002)$. Find the first-order surplus change. (Answer: gain of $150.)
Two assets each have value $100,000 per holding unit and node durations $(0.6,0.2)$ and $(0.2,1.0)$. A liability has value $800,000 and target exposures $(2.8,5.6)$ in $100,000-years. Find nonnegative holdings matching value and both nodes. (Answer: 3 units of the first asset and 5 units of the second.)
A matched portfolio has an asset-minus-liability sensitivity of $0.4$ in $100,000-years to a specified between-node bump. That rate rises by 25 basis points. Estimate the surplus change and state whether the surplus rises or falls. (Answer: decrease of $100.)
At an unselected 15-year shock, asset and liability exposures are 1.8 and 2.3 in $100,000-years, respectively. The 15-year rate falls by 40 basis points. Estimate the first-order surplus change. (Answer: decrease of $200.)

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