Key Rate Duration For Nonparallel Yield Curves

Why a Single Duration Can Miss Curve Risk

A single duration summarizes how value responds to a parallel change in interest rates. But yield curves do not always move in parallel: short rates may rise while long rates fall, or one maturity may change more than the others. A portfolio can have modest sensitivity to a parallel shift and still be exposed to these changes in curve shape.

Key rate duration measures sensitivity to a change associated with one selected maturity of the yield curve. The result is not a universal property independent of the valuation setup: it depends on how the curve is bumped around that maturity, the size of the bump, and how cash flows are repriced. Those choices should be stated whenever key rate durations are reported.

Definition: A key rate duration at maturity $T_i$ is the approximate percentage decrease in an instrument’s value for a one-unit increase in the yield-curve rate associated with $T_i$, when that rate is bumped according to a specified localized curve-shift rule and other key rates are held fixed. For a symmetric bump of size $\Delta y$, a common estimate is

\operatorname{KRD}_i = \frac{P_i^{-}-P_i^{+}}{2P_0\Delta y}.

Here $P_0$ is the base value, $P_i^{+}$ is the value after an upward bump, and $P_i^{-}$ is the value after a downward bump. The bump $\Delta y$ is expressed as a decimal rate change, not as a number of basis points.

Define the Bump Before Calculating

A yield curve is specified at a set of maturities, or nodes. To measure sensitivity at one node, the curve is shifted locally rather than moving every node by the same amount. One common convention uses a triangular shift: the selected node changes by the full bump, neighboring nodes remain fixed, and the shift between nodes is interpolated. Other conventions are possible. Results from different conventions may differ, especially for cash flows between nodes, so comparisons require consistent bump definitions.

For an instrument with value $P_0$, an upward shift in the relevant yield generally lowers value, so $P_i^{+}<P_0$. A downward shift generally raises value, so $P_i^{-}>P_0$. The symmetric formula uses both repricings and is a central-difference estimate of the local slope. It is usually more accurate than using just one side when the price-yield relationship is curved, but it remains an approximation.

Worked Example: Calculate a Five-Year Key Rate Duration

A hypothetical fixed-income portfolio has a base value of $1,250,000. The valuation system applies a localized, symmetric five-year curve bump of 5 basis points, holding the other key rates fixed. The repriced values are $1,244,375 after the upward bump and $1,255,625 after the downward bump. Calculate the five-year key rate duration.

Method. Use the symmetric central-difference formula. Convert 5 basis points to a decimal: $\Delta y=5/10{,}000=0.0005$.

The difference between the downward- and upward-bumped values is $P_5^{-}-P_5^{+}=\$1{,}255{,}625-\$1{,}244{,}375=\$11{,}250$. The denominator is $2P_0\Delta y=2(\$1{,}250{,}000)(0.0005)=\$1{,}250$.

\operatorname{KRD}_5 = \frac{11{,}250}{1{,}250} = 9.0000.

The portfolio’s five-year key rate duration is 9.0000. Locally, a 1% increase in the five-year key rate, under this bump convention, corresponds to an approximate 9% decrease in value. The input prices are also symmetric about the base value: each differs from $1,250,000 by $5,625. This is a useful check on the supplied repricings.

Common mistake avoided. Using 5 rather than 0.0005 for the bump would treat five basis points as a 500% rate change and make the duration 10,000 times too small.

Why the Symmetric Formula Works

Let $P(y)$ denote value as a function of the selected key rate, with the rest of the bump convention fixed. For a small change $\Delta y$, a central difference estimates the derivative:

\frac{dP}{dy} \approx \frac{P(y+\Delta y)-P(y-\Delta y)}{2\Delta y}.

Duration is defined with a negative sign because value typically falls when yields rise. Dividing the negative price derivative by the base value gives the approximate percentage sensitivity per unit rate change:

-\frac{1}{P_0}\frac{dP}{dy} \approx -\frac{P_i^{+}-P_i^{-}}{2P_0\Delta y} = \frac{P_i^{-}-P_i^{+}}{2P_0\Delta y}.

This also explains why the two repricings appear in the order $P_i^{-}-P_i^{+}$. If the upward bump lowers value and the downward bump raises it, the numerator is positive, as a conventional duration should be. The estimate becomes less reliable if the bump is so large that the local slope changes substantially across the interval, or if the instrument’s value changes discontinuously.

Using Key Rate Durations for a Nonparallel Move

Suppose a portfolio has key rate durations at several maturities and the corresponding key rates change by different amounts. A first-order estimate adds the effects of the individual curve changes:

\frac{\Delta P}{P_0} \approx -\sum_i \operatorname{KRD}_i\,\Delta y_i, \qquad \Delta P \approx -P_0\sum_i \operatorname{KRD}_i\,\Delta y_i.

Each $\Delta y_i$ is a decimal change in the associated key rate. A positive estimated $\Delta P$ is a gain; a negative estimate is a loss. This approximation treats the key rate sensitivities as locally constant and omits higher-order effects such as convexity. It is most useful for modest changes around the base curve.

Worked Example: Estimate Value Change Under a Twisted Curve

A hypothetical portfolio is worth $20,000,000. Its key rate durations are 1.5 at two years, 3.2 at five years, and 4.1 at ten years. Under a specified curve-shift convention, the two-year rate rises by 20 basis points, the five-year rate falls by 10 basis points, and the ten-year rate rises by 5 basis points. Estimate the portfolio’s value change using key rate durations.

Method. Convert each change to a decimal and apply the first-order weighted sum: $\Delta y_2=0.0020$, $\Delta y_5=-0.0010$, and $\Delta y_{10}=0.0005$.

The three duration-weighted changes are $1.5(0.0020)=0.0030$, $3.2(-0.0010)=-0.0032$, and $4.1(0.0005)=0.00205$. Their sum is $0.0030-0.0032+0.00205=0.00185$.

\Delta P \approx -(\$20{,}000{,}000)(0.00185) = -\$37{,}000.

The estimated value change is a loss of $37,000. The separate contributions are a loss of $60,000 from the two-year rise, a gain of $64,000 from the five-year fall, and a loss of $41,000 from the ten-year rise. They sum to $-\$60{,}000+\$64{,}000-\$41{,}000=-\$37{,}000$, confirming the result.

Common mistake avoided. A fall in the five-year rate has a positive value effect for a position with positive key rate duration. Treating every curve change as a rise would reverse that contribution and misstate the net estimate.

Interpreting a Key Rate Profile

The collection of key rate durations shows where a portfolio is sensitive along the curve. A relatively large five-year value, for example, indicates more exposure to the specified five-year curve movement than a smaller value at another key rate, when comparing on the same base-value and bump convention. A negative key rate duration is possible for instruments whose value responds atypically to a rate change; it should prompt a review of the instrument, valuation method, and bump definition rather than being automatically changed to a positive number.

For a purely parallel shift, the sum of key rate durations is often used as an approximation to parallel duration. This relationship depends on the bump construction: the localized shift functions need to combine to represent the intended parallel move over relevant maturities. If they do not, the sum need not equal a separately measured parallel duration. A practical check is to compare the summed key rate estimate with a direct parallel bump and investigate material differences.

Key rate duration does not itself identify the probability of a curve move or predict what rates will do. It measures conditional sensitivity under a stated valuation setup. It is also not a substitute for repricing when a large or unusual scenario makes the linear approximation unreliable. In those cases, apply the full curve scenario to the valuation model and compare the resulting value with the sensitivity estimate.

Key takeaway: A key rate duration is meaningful only together with its maturity, bump size, and curve-shift convention. Use decimal rate changes, preserve the sign of each nonparallel move, and treat the weighted-sum estimate as a local approximation.

Worked Example: Check a Key Rate Duration for a Zero-Coupon Bond

A hypothetical two-year zero-coupon bond pays $1,000 at maturity. Its two-year spot rate is 5.00% with annual compounding. For this example, only the two-year key rate is bumped; the other curve nodes are held fixed. Calculate its two-year key rate duration using symmetric one-basis-point bumps.

Method. Under annual compounding, the base value of a single payment $C$ at time $t$ is $C/(1+y)^t$. This bond is valued using its specified spot rate, so $P_0=\$1{,}000/(1.05)^2=\$907.029478$, or $907.03 to the nearest cent.

The bump is $\Delta y=0.0001$. The upward-bumped rate is $0.0501$, and the downward-bumped rate is $0.0499$. Thus $P^{+}=\$1{,}000/(1.0501)^2\approx \$906.86$, and $P^{-}=\$1{,}000/(1.0499)^2\approx \$907.20$. Using the unrounded repriced values in the duration calculation, the price difference is approximately $0.3455 and the denominator is $2(\$907.029478)(0.0001)=\$0.1814059$.

\operatorname{KRD}_2 \approx \frac{0.3455}{0.1814059} \approx 1.9048.

The two-year key rate duration is 1.9048, reported to four decimal places. The upward-bumped price is lower than the downward-bumped price, so the sign is consistent with positive rate sensitivity. The calculation uses unrounded prices; rounding both prices to cents before subtracting would lose precision relative to the small one-basis-point bump.

Common mistake avoided. Do not discount the two-year cash flow at a different maturity’s rate simply because another curve node is being discussed. Here the selected two-year spot rate determines the discount factor for the two-year payment.

Common Exam Traps

Mixing basis points and decimals. Ten basis points is $0.0010$, not $0.10$ or 10.
Using the wrong sign. With the stated formula, the downward-bumped value comes first. Reversing the subtraction would produce a negative result for a conventional positive-duration position.
Assuming every key rate bump is identical. Localized shifts depend on the bump convention and interpolation between curve nodes. State the convention before comparing results.
Treating an approximate value change as an exact repricing. The weighted sum is a first-order estimate. Larger shifts, pronounced curvature, or embedded options can make full repricing important.
Adding key rate durations without checking the curve-shift design. Their sum approximates parallel duration only when the collection of localized shifts represents a parallel move in a compatible way.

Check Your Understanding

For each problem, use the stated bump convention and retain the sign of each rate change.

A hypothetical $5,000,000 portfolio has values of $4,997,500 and $5,002,500 after an upward and downward five-basis-point localized bump, respectively. Calculate the key rate duration. (Answer: 1.0000)
A $8,000,000 position has a three-year key rate duration of 3.0. If the three-year rate rises by 12 basis points, estimate the value change. (Answer: −$28,800)
A hypothetical $10,000,000 portfolio has key rate durations of 2.0 at two years and 4.0 at seven years. The two-year rate falls by 5 basis points while the seven-year rate rises by 8 basis points. Estimate the combined value change. (Answer: −$22,000)
A $2,000,000 position is repriced at $1,996,800 after an upward ten-basis-point bump and $2,003,200 after a downward ten-basis-point bump. Calculate its key rate duration and verify the sign using the repriced values. (Answer: 1.6000; positive because the downward-bumped value exceeds the upward-bumped value.)

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