Duration, Convexity, and Key Rate Duration: Measuring Interest Rate Sensitivity

When interest rates move, how much does a portfolio's value change? Duration gives the first-order answer. Convexity refines it. Key rate duration reveals where on the curve the risk lives. Together, these three measures form the toolkit that risk managers use to understand, communicate, and hedge interest rate sensitivity.

The Question

Suppose you manage a $5 billion fixed-income portfolio — a mix of government bonds, corporate securities, and mortgage-backed instruments. The ALCO committee asks: "If rates rise 100 basis points, how much do we lose?"

You need a single number that summarizes the portfolio's sensitivity. Duration provides it. But as you'll see, one number is never enough — and the assumption that rates move uniformly across all maturities (a parallel shift) is rarely what actually happens.

Modified Duration

Modified duration measures the percentage change in a bond's price for a 1% (100bp) change in yield. It is the first derivative of the price-yield function, normalized by price.

A bond with modified duration of 5.0 years will lose approximately 5% of its value when yields rise by 100bp. On a $100 million position, that's a $5 million loss.

Key properties of modified duration:

• Longer maturity → higher duration. A 30-year bond has more cashflows in the distant future, making it more sensitive to rate changes.
• Higher coupon → lower duration. Larger coupons shift more of the cashflow weight toward earlier periods, reducing average sensitivity.
• Higher yield → lower duration. Higher discount rates reduce the present-value weight of distant cashflows.

Where Duration Fails

Duration is a linear approximation of a non-linear relationship. The price-yield curve for any fixed-income instrument is convex — it curves — and duration captures only the slope at one point, not the curvature.

This approximation works well for small rate moves (±25bp or ±50bp). But for larger shocks — the +200bp parallel scenarios used in IRRBB — the error can be material:

Example: 10-year 4% bond, duration = 5.0, convexity = 30. Duration systematically overestimates losses and underestimates gains.

Notice the asymmetry: the actual price falls less than duration predicts when rates rise, and rises more when rates fall. This is the effect of convexity — and it's always in the bondholder's favor for plain vanilla instruments.

Convexity: The Second-Order Correction

Convexity measures the curvature of the price-yield relationship — how much the slope (duration) itself changes as rates move. Adding the convexity term to the Taylor expansion dramatically improves the approximation:

For the +200bp example above: −5.0 × 0.02 + 0.5 × 30 × (0.02)² = −0.10 + 0.006 = −9.4%, much closer to the actual −9.52%.

Convexity matters most when:

• Rate shocks are large (>100bp) — the BCBS scenarios of ±200-300bp make convexity corrections essential.
• Duration is high — long-dated instruments amplify the curvature effect.
• Instruments have embedded options — prepayable mortgages exhibit negative convexity at low rates (price gains are capped by prepayment, but losses are uncapped).

DV01: The Dollar Measure

While duration gives a percentage price change, DV01 (Dollar Value of a Basis Point) gives the absolute dollar change for a 1bp move:

DV01 is the standard measure for hedge sizing. If your portfolio has DV01 of +$500,000 (value increases when rates fall) and you want to neutralize it, you need a swap or short position with DV01 of −$500,000. The arithmetic is straightforward: match the DV01, and the portfolio is immunized against small parallel rate moves.

Key Rate Duration

Modified duration answers "how sensitive is the portfolio to a parallel rate shift?" But rates rarely move in parallel. The front end might rise 300bp while the long end barely moves (a monetary tightening scenario), or the curve might steepen with short rates falling and long rates rising.

Key Rate Duration (KRD) decomposes the total duration into sensitivities at specific points along the curve — typically the same tenors used to build the yield curve (3M, 1Y, 2Y, 5Y, 10Y, 30Y).

The triangular shape ensures that every point on the curve is covered by exactly one perturbation function (they tile the curve without gaps or overlaps), and that the sum of all KRDs recovers the parallel-shift sensitivity.

KRD in Practice

KRD profiles reveal information that total duration hides. Consider two instruments with identical modified duration of 5.0 years:

Both have duration 5.0, but their risk profiles are radically different:

• The bullet bond concentrates almost all its risk at the 10Y tenor — it's a bet on what the 10-year rate does.
• The amortizing loan spreads risk across 1Y through 10Y — it's diversified across the curve, with the largest exposure at 5Y.

Under a steepener scenario (short rates down, long rates up), the bullet bond loses significantly while the amortizing loan's short-end gains partially offset its long-end losses. Same duration, very different outcome.

This is precisely why KRD matters for IRRBB management: the BCBS non-parallel scenarios (short up, short down, steepener, flattener) affect different parts of the curve differently, and KRD is the tool that maps exposure to the curve shape.

For hedging, KRD tells you not just how much to hedge (total DV01) but where — you might need a 2Y swap plus a 10Y swap, in specific proportions, rather than a single 5Y swap. This bucket hedging approach produces a more robust risk reduction than duration-matching alone.

The Bigger Picture

Duration, convexity, and key rate duration form a hierarchy of increasingly granular sensitivity measures:

• Duration answers: "How much does the portfolio move for a parallel shift?" — one number, the first-order approximation.
• Convexity refines: "How much does the duration-only estimate miss for large shocks?" — the second-order correction.
• KRD decomposes: "Where on the curve does the risk live?" — a vector of sensitivities, one per key tenor.

For IRRBB purposes, all three measures should be computed at the individual instrument level, aggregated to the portfolio level, and reported under both the base scenario and each of the six BCBS scenarios. The KRD profile, in particular, is the single most useful input for designing a hedge program that works across non-parallel curve movements — which is where most of the real risk lives.