Why Key Rate Durations Sum to Approximate Effective Duration

Summary

The document explains why adding key rate durations can approximate a bond’s effective duration under a parallel yield curve shift. Key rate duration measures the sensitivity of present value to one particular curve point, while effective duration measures sensitivity when rates move together. The apparent mismatch between these definitions is resolved by breaking a simultaneous shift into a sequence of single-rate changes.

For two key rates, the change in value from shifting both can be written as two successive differences: first shift one rate while holding the other at its new level, then shift the second. For a small shift, the first difference is close to the partial sensitivity measured at the original curve, and the second corresponds to the other key rate duration. Extending the argument across curve points gives the approximate sum. The explanation relies on small shifts and local approximations; nonlinear effects and finite-shift conventions can make the equality imperfect.

Key ideas

• Key rate duration measures sensitivity to an individual yield curve point.
• Effective duration reflects a parallel shift across the curve.
• A simultaneous shift can be decomposed into sequential changes at each key rate.
• For small shifts, summed key rate sensitivities approximate the parallel-shift sensitivity.

Tags

# Why does the sum of key rate durations approximately equal the effective duration?


# Why does the sum of key rate durations approximately equal the effective duration?












Let's assume the present value of a bond is a function of all key rates i.e. $f\big(KR_1,KR_2,KR_3,...\big)$. Then, $KRD_i = \frac{\partial PV}{\partial KR_i}\big(KR_1,KR_2,...\big)$.

The effective duration assumes a parallel shift to the yield curve which changes all key rates by some amount. However, when all key rates are changing, $KRD_i$ can't be used in the formula for effective duration since it is a partial derivative so why does the formula hold?

## Answer by dm63 (score 2, accepted)

https://quant.stackexchange.com/a/82293

Using only 2 key rates for simplicity : $$KRD= 1/\delta * (f(KR_1+\delta, KR_2+\delta) - f(KR_1, KR_2))$$ but the term in parentheses can be rewritten $$(f(KR_1+\delta,KR_2+\delta) - f(KR_1,KR_2+\delta)) + (f(KR_1,KR_2+\delta) - f(KR_1,KR_2))$$ and you see that the second term is $KRD_2$ and the first term is a very close approximation to $KRD_1$ when $\delta$ is small.

This extends to any number of $KRs$.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.