Duration and convexity

~9 min read

🎯 Interview frequent

Why it matters

Duration and convexity are how a credit or rates desk talks about interest-rate risk without re-pricing an entire book every time the curve twitches by a basis point. This is the most-tested quantitative topic in fixed-income interviews across desks — you are expected to derive the intuition on a whiteboard, not just cite the formula.

Core concepts

Macaulay duration is the present-value-weighted average time of a bond’s cash flows: \(D_{\text{Mac}} = \sum_i t_i \cdot w_i\), where \(w_i = \frac{C_{t_i} DF(t_i)}{P}\). A zero-coupon bond’s Macaulay duration equals its maturity exactly; any coupon-paying bond’s duration is strictly less, because some value arrives before maturity.
Modified duration converts that time measure into a price sensitivity: \(D_{\text{mod}} = -\frac{1}{P}\frac{dP}{dy}\). For a small parallel yield change \(\Delta y\): \(\frac{\Delta P}{P} \approx -D_{\text{mod}} \times \Delta y\). This is the number risk systems and traders mean when they say “duration.”
DV01 / PV01 is the dollar value of a 1bp move: \(\text{DV01} \approx D_{\text{mod}} \times P \times 0.0001\). Portfolio and hedge sizing is almost always done in DV01 terms, not “duration years,” because DV01 aggregates additively across positions of different sizes.
Convexity is the curvature term — the second derivative of price with respect to yield: \(\text{Conv} = \frac{1}{P}\frac{d^2P}{dy^2}\). For plain-vanilla (option-free) bonds, convexity is always positive: the price/yield curve bows upward, so duration understates the price gain when yields fall and overstates the price loss when yields rise.
The two-term Taylor expansion is what interviewers actually want on the whiteboard: \[ \frac{\Delta P}{P} \approx -D_{\text{mod}} \, \Delta y + \frac{1}{2}\, \text{Conv} \, (\Delta y)^2 \] Use duration alone for small moves; add the convexity term for large moves or when comparing two bonds with similar duration but different cash-flow dispersion.
Negative convexity is the credit-desk trap. Vanilla bonds are always positively convex, but callable bonds and MBS can go negatively convex: as yields fall, the issuer becomes more likely to call, capping upside and flattening (or inverting) the price/yield curve on the low-yield side. Effective duration can shrink as yields drop — the opposite of a bullet bond.
What parallel-shift duration misses. Both duration and convexity, as defined above, assume a single parallel move in one YTM (equivalently, in a flat curve). Real curves twist, steepen, and flatten. Key-rate duration decomposes sensitivity by curve segment — the tool a rates desk reaches for when “duration” alone isn’t precise enough. For credit bonds specifically, spread duration (sensitivity to the credit spread holding the risk-free curve fixed) is a distinct, and often larger, risk than rate duration — see Rate risk vs. credit risk.

Mental model

  Price
    |        actual price-yield curve  (convex, bows up)
    |              *  *
    |            *      *
    |          *          *
    |        *   tangent = -duration
    |      *  /
    |    *   /
    |  *    /
    | *    /
    +------------------------ yield
         y0

  small dy: duration (the tangent line) is enough
  large dy: add +1/2 convexity (dy)^2 — actual price sits ABOVE the tangent

  Callable bond, low-yield region: curve flattens / bends DOWN
  as the call becomes likely to be exercised — negative convexity

A barbell (cash concentrated at short and long maturities) has more convexity than a bullet (cash concentrated in the middle) at the same duration — which is exactly why convexity carries positive value whenever yield volatility is priced.

Interview questions

Derive, in one line, why a bond’s price falls when yields rise. Answer: Price is the sum of cash flows discounted at yield \(y\); raising \(y\) shrinks every discount factor \(1/(1+y)^t\), so the sum falls. Duration is just the first-order (linearized) size of that effect: \(\Delta P \approx -D_{\text{mod}} P \Delta y\).
Two bonds have identical duration and yield today. Why might you still prefer one? Answer: Convexity (and credit, optionality, liquidity). Higher convexity is better for a long position — you gain more when yields fall a lot and lose less when yields rise a lot than the duration-only estimate would suggest, holding duration fixed.
A 5-year, 6% annual coupon bond priced at par. Is its Macaulay duration bigger or smaller than 5? Why? Answer: Smaller than 5. Coupons paid before year 5 pull the PV-weighted average time of cash flows below the final maturity. Only a zero-coupon bond has \(D_{\text{Mac}} = T\) exactly.
You hedge a long 10-year corporate bond with a short 2-year Treasury note, matched on DV01. What risk is left on the book? Answer: Curve-shape risk (a 2s10s steepener or flattener hurts you even at matched DV01), convexity mismatch (the 10-year has materially more convexity), and — specific to a corporate — all of the credit spread risk, since you only hedged rate risk, not the issuer’s default/spread exposure.
Why can effective duration on a callable bond fall as yields fall — the opposite of a plain bond? Answer: As yields drop, the issuer is increasingly likely to call and refinance cheaper, so price appreciation stalls near the call price. The price/yield curve flattens (or bends down) on the low-yield side, which is negative convexity, and the effective (option-adjusted) duration shrinks exactly when a vanilla bond’s would be largest.

Watch

Relationship between bond prices and interest rates — Khan Academy. The inverse price/yield relationship with a simple coupon example — the foundation duration formalizes.
Bond convexity — Bionic Turtle. Convexity as the second moment of cash-flow timing, and why the actual curve sits above the duration tangent.