4.11.3 Rates and Key-Rate Duration

Basic Prerequisites: 01 · From Zero and Calculus & Optimization.

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1. Intuition & Practical Objective

Interest-rate risk is the largest sensitivity exposure in the financial system, and it is the one where the choice of risk factor matters most. A bond does not trade off “the interest rate” - there is no such thing. There is a curve: a rate for every maturity. The job here is to replace one useless scalar with a ladder of exposures, then show that the ladder is not optional.

Three levels of resolution, in increasing honesty:

1. DV01 (dollar value of a basis point). The money lost if the whole curve moves up by one basis point: $\text{DV01}=-\partial V/\partial y\times 10^{-4}$. One number. Useful, and blind to shape.
2. Modified duration and convexity. DV01 restated as a relative measure ($D_{\text{mod}}=\text{DV01}/V\times 10^4$) plus the second-order term that makes duration wrong for large moves.
3. Key-rate durations (KRDs). The sensitivity to each curve node separately: $KR{D}_i=-\partial V/\partial y_i\times 10^{-4}$. This is the risk-factor map for rates, and it is what a swap desk actually hedges, node by node, with a portfolio of instruments chosen to reproduce the ladder.

The essential property. A parallel shift of the curve is one scenario; the curve moving in every other way is infinitely many. Duration hedges the first and only the first. The KRD ladder is the minimal representation that lets you hedge the others.

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2. Mathematical Ground Truth & Derivations

Price, duration, convexity. For a set of cash flows $C_i$ at times $t_i$ discounted at zero rates $y_i$,

$$V=\sum _i{C}_i(1+y_i{)}^{-t_i},D_{\text{mod}}=-\frac{1}{V}\frac{\partial V}{\partial y}=\frac{1}{V}\sum _i{t}_i{C}_i(1+y_i{)}^{-t_i-1},\mathcal{C}=\frac{1}{V}\frac{{\partial }^{2}V}{\partial y^2}.$$

For a parallel shift $\Delta y$, the second-order expansion is

$$\Delta V\approx -\underset{\text{DV01}\times \Delta y\text{ in basis points}}{\underbrace{D_{\text{mod}}V\Delta y}}+\underset{\text{convexity correction}}{\underbrace{\frac{1}{2}\mathcal{C}V(\Delta y{)}^2}}.$$

Convexity is always favourable to a long bond position: the price rises more than duration predicts on a rally and falls less than duration predicts on a selloff. That asymmetry is a real, priced asset - and it is exactly the second-order sensitivity term this folder keeps returning to.

Key-rate durations (the canonical definitions). Let the curve be parameterised by node rates $y_1,\dots ,y_n$ (zero rates, or par swap rates - the choice is a convention, but fixed before the ladder is computed). Then

$$KR{D}_i=-\frac{\partial V}{\partial y_i}\times 10^{-4}\approx -\frac{V(y_i+h)-V(y_i-h)}{2}(\text{per }1\text{bp},h=10^{-4}),\sum _{i}KR{D}_i={\text{DV01}}_{\text{parallel}}.$$

The summation identity is the fundamental consistency check of any rate-risk system: the key-rate ladder must sum to the parallel DV01. When it does not, the curve interpolation or the bump convention is inconsistent with the parallel-shift convention.

Portfolio aggregation. Because the ladder is a vector, books aggregate node by node:

$$KR{D}_i^{\text{book}}=\sum _{\text{desks}}KR{D}_i^{\text{desk}}.$$

Two desks can each have large DV01 and cancel exactly in the ladder - the classic “matched book” - while their convexities add rather than cancel. That mismatch is the origin of the “convexity book” and of the P&L that appears when the curve does not move in parallel (see also 03 · Scenario Construction for twist/bowing scenarios).

Other rate risk factors, for completeness: swap-spread and basis sensitivities (spread of a curve to a benchmark), inflation sensitivities (real vs nominal nodes), and FX sensitivity for a foreign-currency curve. Each is another node in the exposure vector.

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3. Computational Implementation - DV01, convexity, and the KRD ladder of a two-bond book

We price two coupon bonds off a flat $4\%$ zero curve, compute modified duration/DV01/convexity, build the full key-rate ladder, verify the summation identity, and then show how far duration alone gets you on a real parallel shift. Stdlib only.

Three things to read off.

1. The ladder is the risk. The book’s DV01 is $0.130916$, but it is not spread evenly: the 5-year bond’s final coupon/principal node carries $0.043467$ ($33\%$ of the total) and the 10-year’s carries $0.068206$ ($52\%$). The intermediate nodes carry almost nothing ($\approx 0.003$ each). A parallel-shift hedge is therefore a poor hedge against almost any actual curve move - the classic pain trade of a coupon-bond book.
2. The summation identity holds to machine precision ($\text{diff}=1.42\times 10^{-14}$). This is the check that a risk system’s curve parameterisation and its bump are mutually consistent.
3. Convexity is a real correction, not trivia. On a $+100$bp parallel move, duration alone predicts $-13.09$ against an actual $-12.56$ - a $0.53 error on a $212.56 book ($+4.2\%$ of the loss). At $+200$bp the error is $2.05 (8.5%$). The sign is favourable (long convexity cushions a selloff) and it is symmetric - the same convexity term helps on a rally, which is exactly what makes convexity a priced asset.

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4. Failure Modes & First-Principles Breakdowns

1. Hedging the parallel shift is not hedging the curve. With a $33\%/52\%$ concentration at the 5y and 10y nodes, a parallel hedge leaves the book massively exposed to a steepening or flattening. You must hedge the ladder, or accept a curve-shape position explicitly.
2. Duration is a first-order number; the loss is not. Duration alone overstates a $+100$bp loss by $4.2\%$ here and $8.5\%$ at $+200$bp, and the miss grows roughly quadratically in the shock. A duration-only risk system systematically mis-states large rate moves - and it is the large moves that matter.
3. The choice of nodes is a modelling decision that changes the answer. Zero-rate nodes, par-swap nodes, and cash-flow buckets give different ladders that all “sum to DV01”. A ladder is only comparable to another ladder if both use the same curve and bump convention.
4. DV01 is linear, the book is not. Two desks with offsetting DV01 still have additive convexity - the matched book earns or loses on curve moves beyond parallel, and the risk of that lives in the second-order term, invisible to every DV01 report.
5. Interpolation risk. Key-rate ladders are built on an interpolated curve; linear interpolation of zero rates and log-linear interpolation of discount factors produce different node sensitivities and, in the limit of coarse nodes, materially different hedges.

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5. References

• Hull, John C.: Options, Futures, and Other Derivatives (11th ed.)
• Hull, John C.: Risk Management and Financial Institutions
• Alexander, Carol: Market Risk Analysis, Vol. III (2008)
• J.P. Morgan / RiskMetrics: Technical Document, 4th ed. (1996)
• BCBS: Minimum Capital Requirements for Market Risk (2019, FRTB, d457)

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6. Connected Graph Bridges

• Back: 01 · From Zero · 02 · Delta, Gamma, Vega
• Continue: 04 · Factor Exposures · Index Hub
• Sibling: Interest-Rate & Term-Structure Models · Scenario Construction (twist & bowing shocks)
• Regulatory application: Basel & Regulation - FRTB sensitivities-based method