Key-Rate Duration for Liability Portfolios

Why Measure Sensitivity at Individual Curve Nodes?

A liability portfolio can have cash flows at several maturities, and a change in one part of the yield curve need not resemble a parallel shift of the whole curve. A single duration measure summarizes sensitivity to a broad rate movement; key-rate duration separates that sensitivity by selected curve maturities. It helps describe which curve regions matter most to the present value of a portfolio.

The calculation is a controlled experiment: change one selected yield-curve node, keep the other nodes fixed, revalue the cash flows using the resulting curve, and compare the new value with the original. The same procedure applies to an asset portfolio or to the positive present value of a liability portfolio. The interpretation depends on which value is being measured.

Definition: The key-rate duration at node $k$, denoted here by $KRD_k$, is the negative proportional derivative of a portfolio's present value $P$ with respect to a change in the yield at that node, with the other curve nodes held fixed: $$ KRD_k=-\frac{1}{P}\frac{\partial P}{\partial z_k}. $$ Here $z_k$ is the effective annual spot rate at node $k$, expressed as a decimal. Thus a rate change of $0.0001$ is one basis point. A positive key-rate duration means that a small increase at that node decreases present value, other nodes held fixed.

Specify the Curve and the Bump

A key-rate duration requires a curve model. For the calculations in this tutorial, assume a set of effective annual spot rates at specified maturity nodes. Between adjacent nodes, interpolate the spot rate linearly in maturity. A node perturbation changes that node's rate by $h$, leaves the other node rates fixed, and changes intermediate rates according to the same interpolation rule. Cash flows are certain and are discounted at the effective annual spot rate for their payment maturity. There are no taxes, transaction costs, or credit-spread changes in the calculation.

For a cash flow $C_t$ due at time $t$, its present value under this model is $C_t(1+z(t))^{-t}$. A liability's present value is calculated as the positive present value of its promised outflows. This convention makes the key-rate duration of a usual fixed positive cash flow positive: increasing its discount rate reduces its present value.

In practice, the derivative in the definition is often estimated by bumping the node both up and down by a small amount. The central-difference estimate is:

KRD_k \approx \frac{P_{k,-}-P_{k,+}}{2hP_0},

where $P_0$ is the unperturbed present value, $P_{k,+}$ is the present value after increasing node $k$ by $h$, and $P_{k,-}$ is the value after decreasing that node by $h$. This symmetric calculation estimates the local derivative while reducing the one-sided approximation error. The answer is a sensitivity per unit change in the annual rate; it is conventionally reported in years. It is not itself the dollar change for a particular shock.

Worked Example: Bumping a Liability's Payment-Maturity Node

A hypothetical liability pays $108.16$ dollars at time 2 years. The 2-year effective annual spot rate is $4\%$, and this maturity is a curve node. Estimate its 2-year key-rate duration using a symmetric 1-basis-point bump. Show the original and bumped present values, and check the result against the derivative formula.

Method. Revalue the payment with the node at $4.01\%$ and $3.99\%$. Since $h=0.0001$, the central-difference denominator is $2hP_0$. The common mistake to avoid is treating 1 basis point as $0.01$ in the rate formula; in decimal form it is $0.0001$.

The original present value is:

P_0=\frac{108.16}{(1.04)^2} =\frac{108.16}{1.0816} =100.00.

With the rate increased to $4.01\%$, the present value is:

P_{+}=\frac{108.16}{(1.0401)^2} =100\left(\frac{1.04}{1.0401}\right)^2 \approx 99.9808.

With the rate decreased to $3.99\%$, it is:

P_{-}=\frac{108.16}{(1.0399)^2} =100\left(\frac{1.04}{1.0399}\right)^2 \approx 100.0192.

Therefore, using the displayed rounded present values:

KRD_2\approx\frac{100.0192-99.9808}{2(0.0001)(100.00)} =\frac{0.0384}{0.0200} =1.9200.

The exact local derivative result is $2/1.04\approx1.9231$, so the central-difference estimate is close. The small difference is due to rounding the bumped values and the fact that a finite bump estimates, rather than equals, the derivative. Both calculations indicate that a 1% increase at this node would produce approximately a $1.92\%$ decrease in value, if the local sensitivity remained representative over that change.

Why the Formula Works

The direct derivative gives a useful check on bump-and-reprice calculations and shows how cash-flow timing and interpolation determine a node's sensitivity. Let $w_k(t)$ be the fraction of a change at node $k$ passed through to the interpolated spot rate at maturity $t$. At node $k$, $w_k(t)=1$; at another node held fixed, it is zero. With piecewise-linear interpolation, an intermediate maturity has the corresponding linear interpolation weight.

For one cash flow at time $t$, the present value is $C_t(1+z(t))^{-t}$. Differentiating with respect to node $z_k$, while holding the other nodes fixed, gives:

\frac{\partial}{\partial z_k} \left[C_t(1+z(t))^{-t}\right] =-tC_t(1+z(t))^{-t-1}w_k(t) =-\frac{t}{1+z(t)}PV_t\,w_k(t).

For a portfolio, add the derivatives for its cash flows and divide by the total present value according to the definition. The resulting local sensitivity is:

KRD_k =\frac{\displaystyle\sum_t \frac{t}{1+z(t)}PV_t\,w_k(t)} {\displaystyle\sum_t PV_t}.

This derivation assumes the effective annual spot-rate discounting and the stated interpolation rule. If all cash flows occur exactly at node $k$, each has weight 1 for that node, and the expression reduces to a present-value-weighted average of $t/(1+z_k)$. Cash flows between nodes contribute to the duration of each relevant node according to their interpolation weights; assigning each cash flow wholly to its nearest node would not follow this curve model.

Worked Example: Two Liability Payments at Separate Nodes

A hypothetical liability portfolio has a present value of $100$ dollars from a payment at year 2 and $200$ dollars from a payment at year 5. The 2-year and 5-year effective annual spot rates are $4\%$ and $6\%$, respectively, and both payment maturities are curve nodes. Estimate the key-rate duration at each node using a symmetric 1-basis-point bump. The stated present values are the unperturbed values.

Method. Bump only one node at a time and reprice only the cash flow whose maturity is at that node. For the 2-year payment, the promised amount that gives an unperturbed present value of $100$ is $100(1.04)^2=108.16$. For the 5-year payment, the corresponding amount is $200(1.06)^5\approx267.6451$. The other curve node remains unchanged in each test.

For the 2-year node, the bumped present values are:

P_{2,+}=100\left(\frac{1.04}{1.0401}\right)^2\approx99.9808, \qquad P_{2,-}=100\left(\frac{1.04}{1.0399}\right)^2\approx100.0192.

The total unperturbed portfolio value is $P_0=100+200=300$. The year-5 payment does not change when only the 2-year node is perturbed, so it cancels from the difference. Thus:

KRD_2\approx \frac{(200+100.0192)-(200+99.9808)} {2(0.0001)(300)} =\frac{0.0384}{0.0600} =0.6400.

For the 5-year node, the bumped present values of that payment are:

P_{5,+}=200\left(\frac{1.06}{1.0601}\right)^5\approx199.9057, \qquad P_{5,-}=200\left(\frac{1.06}{1.0599}\right)^5\approx200.0944.

The year-2 payment is unchanged during this second bump. Therefore:

KRD_5\approx \frac{(100+200.0944)-(100+199.9057)} {2(0.0001)(300)} =\frac{0.1887}{0.0600} \approx3.1450.

The year-5 node has the larger proportional sensitivity for this portfolio because the later payment has a greater rate sensitivity and represents two-thirds of its unperturbed value. The check from the derivative formula gives $KRD_2=(100/300)(2/1.04)\approx0.6410$ and $KRD_5=(200/300)(5/1.06)\approx3.1447$, close to the bump estimates. Small differences arise from rounded values and finite bumps.

Intermediate Maturities and Curve Interpolation

A payment between two curve nodes can respond to a perturbation at either node. Suppose the nodes are at years 1, 3, and 5, with linear interpolation in maturity. A year-2 cash flow is halfway between the first two nodes: a change at year 3 passes through at weight $1/2$, while a change at year 5 has weight zero. A year-4 cash flow is halfway between the year-3 and year-5 nodes, so a year-3 change also passes through at weight $1/2$.

This is why a key-rate calculation is not generally the same as grouping cash flows by nearest maturity. The weights come from the specified curve construction. A different interpolation method, or a different rule for how the curve behaves outside its first and last nodes, can produce different key-rate sensitivities. If a curve model uses different weights, use those weights consistently in both the bump and the derivative calculation.

Worked Example: A Middle-Node Bump with Interpolated Rates

A hypothetical liability portfolio has payments at years 2 and 4. Under a flat effective annual spot curve of $5\%$, each payment has an unperturbed present value of $100$ dollars. The curve nodes are at years 1, 3, and 5, and zero rates are linearly interpolated by maturity. Estimate the key-rate duration at year 3 using a symmetric 1-basis-point bump, and explain which payments respond.

Method. The year-2 and year-4 rates each move by half of the year-3 node bump. The promised payments are $100(1.05)^2=110.25$ dollars at year 2 and $100(1.05)^4=121.550625$ dollars at year 4. The base total present value is $200$ dollars.

Under a positive 1-basis-point bump at year 3, each affected spot rate rises by $0.5$ basis point, or $0.00005$. The repriced total is:

P_{3,+} =100\left(\frac{1.05}{1.05005}\right)^2 +100\left(\frac{1.05}{1.05005}\right)^4 \approx199.9714.

Under the negative bump, each affected rate falls by $0.00005$:

P_{3,-} =100\left(\frac{1.05}{1.04995}\right)^2 +100\left(\frac{1.05}{1.04995}\right)^4 \approx200.0286.

The central-difference estimate is:

KRD_3\approx \frac{200.0286-199.9714}{2(0.0001)(200)} =\frac{0.0572}{0.0400} \approx1.4300.

The derivative formula provides an independent check. Each cash flow has weight $1/2$ at the year-3 node, so:

KRD_3 =\frac{100\left(\frac{2}{1.05}\right)\left(\frac12\right) +100\left(\frac{4}{1.05}\right)\left(\frac12\right)} {200} =\frac{300/1.05}{200} \approx1.4286.

The bump estimate is close to the derivative result. The key point is that both payments contribute to year-3 sensitivity even though neither is due at year 3. The common mistake of assigning both payments to their nearest nodes would miss this response.

Interpreting Dollar Sensitivity and Avoiding Traps

For a small change $\Delta z_k$ at one node, the first-order dollar change in present value is:

\Delta P\approx-P_0KRD_k\Delta z_k.

For a positive-valued liability portfolio, this describes the change in the present value of its promised outflows. For an asset portfolio, it describes the change in asset value. The formula is a local approximation: larger shocks can produce a material difference from a full revaluation because present value is not a linear function of rates. If several nodes change, first-order changes can be added across nodes, but that approximation does not capture nonlinear effects.

For example, if a liability portfolio has value $500{,}000$ dollars and its key-rate duration at one node is $1.8$, a 10-basis-point increase at that node gives the approximation:

\Delta P\approx-(500{,}000)(1.8)(0.001)=-900\text{ dollars}.

The liability's present value is estimated to fall by $900$ dollars. The sign is negative because an increase in the discount rate reduces present value. The same positive key-rate duration convention applies to asset values: their estimated change is also negative for an isolated upward node bump.

A common trap is to forget either the sign or the units. With the central-difference convention, the numerator is $P_{k,-}-P_{k,+}$, not the reverse. Also, the rate change must be entered as a decimal: 10 basis points is $0.001$, not $0.10$. A second trap is to perturb every node at once and call the result a key-rate duration; that instead measures sensitivity to the particular multi-node shock used.

Key takeaway: A key-rate duration is a proportional, local sensitivity to one specified curve node under a specified curve and interpolation model. Bump one node at a time, reprice all cash flows affected by that node, and use the rate change in decimal form when translating duration into an approximate dollar change.

Check Your Understanding

Use the stated spot-rate and interpolation assumptions. Unless a question says otherwise, treat the requested sensitivity as a local derivative rather than a finite-bump estimate.

A single liability cash flow has present value $150$ dollars at year 3, where the effective annual spot rate is $5\%$ and year 3 is a curve node. Find its key-rate duration at year 3. (Answer: $3/1.05\approx2.8571$ years)
A liability portfolio has value $200$ dollars: $80$ dollars of present value at a 2-year node with rate $4\%$, and $120$ dollars at a separate 4-year node with rate $5\%$. Find the key-rate durations at the 2-year and 4-year nodes. (Answer: $0.7692$ years at year 2 and $2.2857$ years at year 4)
Nodes are at years 1, 3, and 5, with rates interpolated linearly. Two liability payments have present values $90$ dollars at year 2 and $110$ dollars at year 4 under a flat $4\%$ curve. Find the key-rate duration at the year-3 node. (Answer: $\big[90(2/1.04)(0.5)+110(4/1.04)(0.5)\big]/200\approx1.4904$ years)
An asset portfolio is worth $300{,}000$ dollars and has key-rate duration $2.4$ at one node. Estimate the dollar change in asset value for an isolated 5-basis-point increase at that node. (Answer: approximately $-360$ dollars)
A liability portfolio has value $800{,}000$ dollars and key-rate duration $1.25$ at a node. Estimate the change in its present value for an isolated 12-basis-point decrease at that node. (Answer: approximately $+1{,}200$ dollars)

What You'll Learn