Bonds, duration, and convexity

~9 min read

🎯 Interview frequent

Why it matters

A bond is the simplest interesting present-value machine: a schedule of coupons plus principal, discounted on a curve. Duration and convexity are how you talk about interest-rate risk without re-pricing the whole book every time the curve twitches. Rates desks, risk, and almost every “fixed income 101” interview loop live here.

Core concepts

Price is discounted cash flows. For a coupon bond with cash flows \(C_{t_i}\) and discount factors \(DF(t_i)\): \(P = \sum_i C_{t_i}\, DF(t_i)\). If you use a single yield \(y\) (YTM), you are fitting one number that makes that sum equal the market price — YTM is an internal rate of return, not a curve.
Price and yield move inversely. If required yields rise, every cash flow is worth less today, so the bond cheapens. Khan Academy’s “bond prices vs interest rates” video is the intuition; the formula is just TVM (see TVM / discounting).
Macaulay duration is the PV-weighted average time of the cash flows: \(D_{\text{Mac}} = \sum_i t_i \cdot w_i\) where \(w_i = C_{t_i} DF(t_i) / P\). A zero’s Macaulay duration equals its maturity; a coupon bond’s is strictly less.
Modified duration is the first-order sensitivity: \(\frac{1}{P}\frac{dP}{dy} = -D_{\text{mod}}\). For a 1bp (0.01%) parallel yield rise, \(\Delta P / P \approx -D_{\text{mod}} \times 0.0001\). This is the number risk systems mean by “duration.”
Convexity is the second derivative: \(\frac{1}{P}\frac{d^2P}{dy^2}\). For vanilla (option-free) bonds, convexity is positive: the price-yield curve bends up. Duration underestimates the gain when yields fall and overestimates the loss when yields rise.
The Taylor approximation interviewers want: \(\frac{\Delta P}{P} \approx -D_{\text{mod}}\,\Delta y + \tfrac12 \,\text{Conv}\,(\Delta y)^2\). Use duration for small moves; add convexity when the move is large or when comparing two bonds with similar duration.
What duration does not capture. It assumes a parallel yield shift of a single YTM (or a parallel curve shift, in DV01 language). Real curves twist and butterfly. Option-embedded bonds (callables, MBS) can have negative convexity when the embedded option kicks in.
DV01 / PV01 is the dollar value of 1bp: \(DV01 \approx D_{\text{mod}} \times P \times 0.0001\). Portfolio risk is often quoted in DV01, not “duration years.”

Mental model

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

  small dy: duration (the tangent) is enough
  large dy: add +1/2 convexity (dy)^2  — you sit *above* the tangent

A barbell (cash in short + long zeros) has more convexity than a bullet (cash in the middle) at the same duration — that’s why convexity has value when volatility of yields is real.

Interview questions

Why does a bond’s price fall when yields rise? Give the one-line TVM answer, then the duration answer. Answer: TVM: higher \(y\) shrinks every discount factor, so PV falls. Duration: \(\Delta P \approx -D_{\text{mod}} P \Delta y\), so a positive \(\Delta y\) produces a negative \(\Delta P\). Duration is the local linearization of that TVM fact.
Two bonds, same duration, same yield. Why might you still prefer one? Answer: Convexity (and cash-flow timing / spread / liquidity). Higher convexity is better for the long: you gain more if yields move a lot in either direction. Also credit, optionality, and whether duration was computed off YTM vs. a full curve.
A 2-year 6% annual coupon bond, par, yield 6%. Rough Macaulay duration — bigger or smaller than 2? Why? Answer: Smaller than 2. Some PV arrives as the year-1 coupon, which pulls the PV-weighted average time below maturity. Only a zero has \(D_{\text{Mac}} = T\).
You hedge a long 10y bond with a short 2y note, matching DV01. What risk remains? Answer: Curve shape risk (2s10s steepener/flattener), convexity mismatch (the 10y has more convexity), and any spread/credit basis. DV01 matching only kills a parallel 1bp shift.
Callable bond: why can effective convexity go negative? Answer: As yields fall, the issuer is more likely to call, so price stops rising toward the call price. The price-yield curve flattens or bends down on the low-yield side — that’s negative convexity, and duration can drop as yields drop (unlike a vanilla bond).

Watch

Introduction to bonds — Khan Academy. What a bond is: par, coupon, and why a company issues one.
Relationship between bond prices and interest rates — Khan Academy. Why prices and yields move inversely, with a simple coupon example.
Bond convexity — Bionic Turtle. Convexity as the second moment of the cash-flow times, and why the duration tangent sits below the curve.