What is Key Rate Duration?
Key Rate Duration (KRD) is a familiar tool used by actuaries and other practitioners to summarize balance sheet value sensitivity across the yield curve. The concept is used in many robust applications. Practitioners utilize a wide range of available methods. This article aims to summarize commonly used KRD methods and explore how their results differ depending on whether spot or forward rates are used as the basis for analysis.
KRD is a structured middle ground along the spectrum of interest rate sensitivity analysis. At one extreme is full cashflow decomposition, where the value sensitivity to interest rates of every projected payment date is evaluated separately. At the other extreme lies traditional duration, which compresses all value sensitivity to interest rates into a single value for an entire portfolio. KRD provides a practical balance, summarizing exposure through sensitivities tied to a defined set of yield curve segments.
To illustrate full cashflow decomposition versus traditional duration and KRD methods in subsequent sections of this article, an example involving deterministic monthly cashflows of $1 for 10 years will be analyzed. These could be contractually deterministic (for example, a term certain annuity), or they could be expected values for more complex instruments such as a mortgage. In the latter case, the analysis ignores any impact of the level or path of interest rates on the amount or timing of cashflows. Illustrating with these deterministic cashflows allows the analysis to focus on applying KRD methods to yield curves without considering interactions with stochastic or other complex valuation methods. (See Figure 1)
Figure 1
Key Rate Duration Block Method on Spot Rates
Ten years of monthly payments of $1. Each payment represents a distinct cashflow date whose sensitivity could be modeled separately (full decomposition) or summarized through key rate durations.
In this example, full cashflow decomposition would evaluate sensitivity for 120 separate payment dates, while traditional duration would compress this to a single measure. KRD provides a middle ground, summarizing exposure in five aggregated quantities. (The block method illustrated in Figure 1 is explained in a later section.)
Calculation Basics: Full Cashflow Decomposition, Traditional Effective Duration and KRD
Before embarking on the various KRD methods, the following reviews the basic math of computing sensitivities to interest rate changes as applied to the range from full cashflow decomposition to KRD.
Full Cashflow Decomposition
Under this method, each individual cashflow is valued as a zero-coupon bond, making spot rates the natural basis for analysis. Utilizing continuously compounded rates, exact calculus derivatives can be computed for each cashflow when performing full decomposition. Consider expected cashflows at times 
with expected cashflow amount at and associated spot rate rt. The present value pt of at discounted to present is:
The sensitivity St of that payment to a change of its spot rate rt is:
The total interest rate sensitivity of all payments, denoted D, is:
Traditional Effective Duration
The exact calculus expression of Equation 2) may be difficult or impossible for some cashflows such as for interest-sensitive or other contingent payments. Traditional effective duration is widely used in these situations because it avoids needing derivatives or continuous compounding. The method approximates the value sensitivity of each payment at at time t with small shocks up and down of size δ to interest rate rt:
Total effective duration sums these approximations for all t and is denoted D̂:
As δ becomes infinitesimally small, the approximation approaches the calculus definition of a derivative:
Key Rate Duration
In the case of KRD, sensitivities are not computed for each cashflow individually. Instead, predefined shocks are applied to the yield curve and the corresponding change in portfolio value is measured. The yield curve, denoted Y(τ), can be expressed in terms of spot rates or forward rates depending on the context. When expressed as spot rates, Y(t)=rt. Portfolio value P(Y) is further defined as:
Note that equation 7) allows for more complex cashflows that are contingent on interest rates or other market quantities. In such cases, at is an expected value and the valuation of P(Y) has been calibrated to satisfy equation 1). Also note that in valuations of P(Y) for such complex cashflow streams, it may be convenient to express Y as forward rates.
The yield curve is represented using M key-rate buckets. For each bucket j = 1, …, M, a KRD sensitivity Kj is defined along with a base shock δ. For each Kj, a function βj (τ) with values ranging from 0 to 1 for every cashflow date τ is specified to scale the shock.
Kj is computed using the following formula:
D̃ is defined as the aggregation or sum of Kj for j = 1, …, M. For certain methods, the weighting function fully covers all cashflows such that the weight sums to 1 for any time τ.
In such cases, the sum of KRD shocks equals a parallel shock of size δ. The following equation is satisfied.
Note that for a consistent shock size δ, D̃ is exactly equivalent to D̂ (the traditional duration) for some methods. For others, it is only approximately so. Furthermore, for the Sum Methods discussed later, Equations 9) and 10) require modification. The corresponding definitions of D̃ will be defined specifically for those methods.
Tying the Basics Together
It should be apparent that when only one KRD is defined such that β(τ) = 1 for all cashflow dates τ, D̃ = D̂. More generally, for any finite collection of KRD buckets, aggregate KRD D̃ converges to aggregate full cashflow sensitivity D as the shock size δ approaches zero. As the shock size δ → 0, when the KRD partitions are refined to coincide with the timing of cashflows, the KRD sensitivity decomposition resolves into the full cashflow decomposition.
Finally, the exact values of St for some calculations of P(Y) must be estimated numerically via methods like equation 4). Note that full cashflow decomposition can be employed in lieu of KRD for each possible cashflow date.
Yield Curve and Basis for KRD Method Comparison Example
Results for comparison of different KRD methods are based upon treasury rates as of year-end 2024.[1] Missing points on the yield curve were filled in using the Monotone Convex method, which is similar to the U.S. Treasury’s methodology.[2] The comparison focuses on the deterministic case. KRD for stochastic interest rates involves additional considerations.
Continuously compounded interest rates are used throughout the analysis. While the KRD methods illustrated can also be implemented using curtate (discretely compounded) rates, working in continuous space has desirable properties. In particular, parallel shocks applied to spot and forward curves produce equivalent traditional duration measures, and the sum of key rate durations aligns cleanly with traditional duration when computed on a consistent basis. Using curtate rates can create small distortions due to algebraic effects. (See Figure 2)
Figure 2
Treasury Rates (Continuously Compounded) as of 12/31/2024 Monotone Convex Interpolation
Treasury rates as of year-end 2024. Though 10 years of cashflows are illustrated, the yield curve is extended an additional year to demonstrate the string method as shown in Figure 8.
KRD Methods
This section illustrates and compares some common KRD methodologies using the 10-year monthly cashflows previously introduced. The methods differ in how the yield curve is shocked and how sensitivities are apportioned:
- Block Method: Applies flat shocks to all rates within a bucket.
- String Method: Applies tapered shocks for smoother transitions.
- Forward Sum/Backward Sum Methods: Use cumulative application of shocks across buckets and are particularly useful when working with forward rates.
While all methods are indexed to the same set of key-rate locations, they differ in whether those locations are interpreted as interval boundaries (Block and Sum Methods) or reference points around which shocks are constructed (String Method).
Block Method
The Block Method defines each KRD with reference tenors τj at the boundaries of blocks with τ0 equal to zero.
Consider the example shocks below with 5 KRDs. Visual inspection of the chart makes clear that Equation 9) is satisfied with this method; summing all the shocks equals a parallel shift. (See Figure 3)
Figure 3
The Block Method applies level shocks within “blocks” of maturities.
The impact of shocking the fourth KRD block spanning 6 to 8.5 years of spot rates using a shock size of 0.1% is analyzed. The shocked spot yield curve exhibits discontinuities at the endpoints of the KRD block. (See Figure 4)
Figure 4
Block Method on spot rates.
Figure 5 shows the Block Method applied to forward rates. Discontinuities on the shocked forward curve also occur at the endpoints of the 6 to 8.5 years KRD block.
Figure 5
Block Method applied to forward rates.
It may be relevant for some applications to convert between forward and spot rates. When using continuous compounding, converting to forward rates from a shocked yield curve on spot rates results in forward rates equivalent to shocking directly on the forward curve except for large distortions at the end points of the shocked KRD block. Outside of the scale in the graph below (Figure 6), the forward rate at t=6 is 12.2% while the forward rate at t=8.5 is negative 5.2%.
Figure 6
Converting forward rates from spot rates shocked by the Block Method produces distortions.
In contrast, a forward curve shocked by the Block Method converts into a smooth spot yield curve, as demonstrated in Figure 7 below.
Figure 7
Spot rates converted from a forward curve shocked by the Block Method.
Table 1 below shows the resulting value sensitivities for the 10-year monthly cashflows applying the Block Method compared to the full cashflow decomposition (Continuous) and traditional duration (Parallel) calculations. Because spot rate and forward rate shocks represent different perturbations of the yield curve, the allocation of sensitivity among individual KRD buckets is not generally expected to coincide, even when aggregate sensitivity is similar.
Table 1
Block Method total sensitivities for 10 years of $1 cashflows per month. The continuous method applies a calculus derivative to each cashflow. The other methods use a discrete shock size of 0.1%.
Figures 4 through 7 illustrate that spot rate and forward rate shocks produce different term structures. Consequently, the value sensitivities for individual KRD buckets are not directly comparable across the two bases. The value sensitivity by block on a spot rate basis can be compared to the continuous method. The total KRD of D̃ can likewise be compared to the traditional duration D̂ and to the sum of value sensitivities D using the continuous method.
For the small shock size of 0.1%, the deviations between methods are minor. When only spot rates are of interest, applying the Block Method to spot rates may have intuitive appeal since the total sensitivity is exactly equivalent to traditional (parallel shift) duration for the same shock size. When working with forward rates, the method may seem intuitive, but its total sensitivity is not equivalent to traditional parallel shift duration. Keep in mind that the finite shock size is a numerical approximation of the calculus derivative computation of D.
String Method
The String Method is named for its analogy of pulling on a string at a reference tenor while being anchored to adjacent tenors. This method is defined by using a reference tenor τ̄j for each KRD interval, rather than the interval endpoints τj used in the Block and Sum Methods. The notation reflects that τ̄j denotes the time at which the string shock attains its maximum, rather than an interval boundary. The first and last KRD intervals are defined so that the total shocks sum to 1 for any possible cashflow date.
The notation of Equation 12) is compact but possibly confusing. The key is that, except for within the first and last KRD intervals, any time t not equal to a reference tenor τ̄j is covered by overlapping KRD βj(t) functions. For example, consider t = 1.25 and the String KRD intervals illustrated in Figure 8 below. This point is covered by β1(t = 1.25) = 0.875 and β2(t = 1.25) = 0.125. As for all t, the sum of β1(t = 1.25) and β2(t = 1.25) is 1.
Figure 8
The String Method is named to allude to pulling a string at a key reference tenor anchored to the adjacent key reference tenors.
As for the Block Method, the analysis examines the second-to-last KRD. Whereas the Block Method applied a uniform shock from 6 to 8.5 years, the String Method can be thought in terms of a reference tenor at seven years. Note from Figure 9 that the resulting shock is not symmetric around seven, grading upwards for two years from below the reference tenor and downwards for the following three years above it.
Figure 9
String Method on spot rates.
Applying the same String Method directly to forward rates produces the shocked forward curve shown in Figure 10.
Figure 10
String Method on forward rates.
Converting from the spot yield curve shocked by the String Method to forward rates results in discontinuities at the reference tenors for the last three KRD periods. However, the resulting jumps are materially less than those under the Block Method. Nevertheless, the pattern reflects the artificial nature of the shock relative to realistic term-structures. (See Figure 11)
Figure 11
Forward rates must be alternately increased and reduced to yield the string pattern to spot rates.
Like the Block Method, shocking a forward curve via the String Method produces a smooth spot curve. (See Figure 12)
Figure 12
Spot rates converted from a forward curve shocked by the String Method.
Table 2 below compares the total sensitivities of the String Method on the 10-year illustrative monthly cashflows to the full cashflow decomposition (Continuous) and traditional duration (Parallel) methods.
Table 2
String Method total sensitivities for 10 years of $1 cashflows per month. The continuous method applies a calculus derivative to each cashflow. The other methods use a discrete shock size of 0.1%.
The String Method does not truly lend itself to comparison with the full cashflow decomposition by KRD bucket. The continuous spot sensitivities correspond to the same buckets as the Block Method. With the small shock size of 0.1%, the total of all sensitivities is relatively close. However, the String Method produces sensitivities such that their sum D̃ does not align exactly with D̂. This can be attributed to the method trying to condense all 120 individual payments into amounts at the reference tenors.
Forward Sum Method
The Forward Sum Method starts with the shock used for the first KRD period as in the Block Method but then adds shocks to successive KRD blocks without removing shocks to previous KRD blocks. As illustrated in Figure 13 below, this method does not satisfy Equation 9) term-by-term, reflecting the way the shocks are constructed.
Figure 13
The Forward Sum Method is a level shock from t=0 to the reference tenor of the KRD.
Without adjustment to the individual KRD sensitivities, D̃ = KM. It is common, given K0 = 0, to compute modified KRD sensitivities, K′j:
K′j is comparable to Kj for the previous methods discussed (Block and String) when computed on the same basis (spot or forward). This adjustment is applied at the level of the underlying shocks, so that although Equation 9) is not satisfied, its intended parallel-shift interpretation is preserved. The aggregate KRD, D̃, is then:
The Forward Sum Method is again illustrated using the second-to-last KRD interval. Like the Block Method with the same boundary reference tenors 6 and 8.5 years, the shocked spot and forward curves have a discontinuity at the 8.5-year tenor. However, there is no discontinuity at the six-year tenor like the Block Method produces. (See Figure 14)
Figure 14
Shocked spot rates by the Forward Sum Method.
Applying the Forward Sum Method directly to forward rates produces the analogous pattern shown in Figure 15, with the discontinuity occurring at the ending tenor of the KRD interval.
Figure 15
Shocked forward rates by the Forward Sum Method.
Like the Block Method, the Forward Sum Method introduces a large distortion at the later reference tenor when converting from shocked spot to forward rates. Not shown on the scale in the chart below (Figure 16), the forward rate immediately after 8.5 years is negative 5.2%.
Figure 16
When spot rates are shocked by the Forward Sum Method, converting to forward rates introduces a large distortion.
Figure 17 below shows that once again, the spot rates derived from shocking forward rates form a smooth curve.
Figure 17
Spot rates converted from a forward curve shocked by the Forward Sum Method.
Table 3 below compares the sensitivities for the 10-year illustrative monthly cashflows using the Forward Sum Method to the full cashflow decomposition (Continuous) and traditional duration (Parallel) methods.
Table 3
Forward Sum Method total sensitivities for 10 years of $1 cashflows per month. The continuous method applies a calculus derivative to each cashflow. The other methods use a discrete shock size of 0.1%.
This method produces an aggregate KRD exactly equal to traditional duration when computed using the same shock size and basis, whether using spot or forward rates. By contrast, the Block Method does not preserve this equivalence when applied to forward rates. This illustrates that the primary motivation for the use of the Sum Methods relates to forward-rate applications.
Backward Sum Method
The Backward Sum Method is like the Forward Sum Method but, as the name implies, the method first applies shocks starting with the last KRD block and then adds shocks sequentially to earlier KRD blocks without removing the shocks to the later KRD blocks. (See Figure 18)
Figure 18
The Backward Sum Method starts with the last KRD and successively adds level shocks backward to earlier segments.
Without adjustment to the individual KRD sensitivities, D̃ = K1. Given K(M+1) = 0, modified KRD sensitivities K′j are computed as follows:
K′j is comparable to Kj for the methods already summarized when computed on the same basis (spot or forward). As with the Forward Sum Method, this adjustment is applied at the level of the underlying shocks, so that the intended parallel-shift interpretation is preserved. The aggregate KRD, D̃, is as for Equation 15) then:
The Backward Sum Method produces discontinuities for spot and forward rates. The tenors at which the discontinuities occur vary by method. The Block Method discontinuities are at both the beginning and ending tenors. The Forward Sum Method results in a discontinuity at the ending tenor. Figure 19 shows that the Backward Sum Method discontinuity is at the beginning tenor.
Figure 19
Spot rates shocked by the Backward Sum Method.
The same pattern occurs when the Backward Sum Method is applied directly to forward rates, as shown in Figure 20, with the discontinuity occurring at the beginning tenor of the KRD interval.
Figure 20
Forward rates shocked by the Backward Sum Method.
As shown in Figure 21 below, the forward rates derived from shocking spot rates display a large distortion at the beginning of the KRD block. Outside of the scale of this chart, the forward rate is 12.2% just after six years.
Figure 21
Forward rates have a large distortion when derived from spot rates shocked by the Backward Sum Method.
Converting in the other direction, from shocked forward rates to spot rates, again produces a smooth curve. (See Figure 22 below)
Figure 22
Spot rates derived from shocking forward rates by the Backward Sum Method.
Table 4 below compares the sensitivities for the 10-year illustrative monthly cashflows for the Backward Sum Method to the full cashflow decomposition (Continuous) and traditional duration (Parallel) methods.
Table 4
Backward Sum Method total sensitivities for 10 years of $1 cashflows per month. The continuous method applies a calculus derivative to each cashflow. The other methods use a discrete shock size of 0.1%.
Like the Forward Sum Method, this approach produces an aggregate KRD exactly equal to traditional duration (Parallel) when using the same shock size. It has the additional feature that the final KRD coincides with the corresponding Block Method KRD values, and that individual KRDs are not dependent on earlier KRD blocks.
Discussion and Observations
This article reviewed four common methods for calculating key rate durations—Block, String, Forward Sum, and Backward Sum—and examined how they behave under both spot and forward rate frameworks. While each method aims to summarize interest rate sensitivity across a segmented yield curve, the results can differ, especially when converting between spot and forward rates.
A key takeaway is that the sum of KRDs equals traditional duration (parallel shocks) for the Sum Methods and for the Block Method when applied to spot rates. The Block Method applied to forward rates and the String Method (on both spot and forward rates) introduce deviations relative to traditional duration. Table 5 summarizes these results.
Table 5
Do KRDs sum to traditional duration for a given method and basis combination?
Traditional duration has an intuitive appeal as a reference for KRD methodology; however, full cash-flow decomposition offers the most precise information on the value and sensitivity of a portfolio to interest rates. In aggregating full cashflow sensitivities into KRD buckets, KRD methods require finite shock sizes δ. Aggregate KRD converges to aggregate full cashflow decomposition as δ→0 while KRD sensitivity decomposition resolves into full cashflow decomposition as the KRD partitions are refined to match the cashflow timing.
Another recurring theme is that conversions between spot and forward representations can introduce distortions, especially when starting with a shocked spot curve and converting to forward rates. The reverse (shocking forwards and converting to spots) tends to produce smoother results, suggesting that for applications requiring both representations, it may be preferable to construct shocks directly to forward rates.
In terms of practical usage:
- Block Method on spot rates is intuitive and aligns closely with traditional duration measures, making it well-suited to deterministic cashflow modeling.
- String Method introduces smoother transitions across segments, potentially better representing realistic term structure movements, though it requires more care in interpretation. Though it does not exactly align with traditional duration, it can be intuitively understood as redistributing sensitivities toward the specified reference tenors when shocking spot rates.
- Sum Methods, particularly in forward space, may be attractive in stochastic or dynamic models where forward rates are more directly tied to economic interpretation or model calibration. These methods may be preferred over the Block Method when matching traditional duration is desired.
Ultimately, the method chosen for calculating KRDs should consider the structure of the instrument or portfolio being analyzed, the modeling context, and the desired interpretability of results. There may be no universally best approach, but understanding the differences between methods equips practitioners to make more informed choices.
This article is provided for informational and educational purposes only. Neither the Society of Actuaries nor the respective authors’ employers make any endorsement, representation or guarantee with regard to any content, and disclaim any liability in connection with the use or misuse of any information provided herein. This article should not be construed as professional or financial advice. Statements of fact and opinions expressed herein are those of the individual authors and are not necessarily those of the Society of Actuaries or the respective authors’ employers.
Mark Evans, FSA, MAAA, FLMI/M, works as actuarial consultant for Investors Heritage in Frankfort, Ky. He can be contacted at Mark@appliedstochastic.com.
Joe Stoutenburg, FSA, MAAA, works as an independent consultant. He can be contacted at stoutej@gmail.com.
Endnotes
[1] U.S. Department of the Treasury, “Daily Treasury Par Yield Curve Rates,” Daily Treasury Rates, https://home.treasury.gov/resource-center/data-chart-center/interest-rates/TextView?type=daily_treasury_yield_curve.
[2] U.S. Department of the Treasury, “Treasury Yield Curve Methodology,” February 18, 2025, https://home.treasury.gov/policy-issues/financing-the-government/interest-rate-statistics/treasury-yield-curve-methodology.