Convexity

Price is a convex, declining function of yield, so a first-order approximation based on duration is a tangent line drawn at the current yield. Convexity is the second term of the Taylor expansion of that price-yield function: it measures how fast the slope itself changes as yield moves. The estimated percentage price change therefore has two pieces. The duration piece is minus modified duration multiplied by the change in yield; the convexity piece is one half of annual convexity multiplied by the square of the change in yield.

Because the yield change is squared, the adjustment is sign-independent for option-free bonds: it pushes the estimate upward whether rates rise or fall. It is also negligible for a one-basis-point shift and material for a shift of one or two percentage points, which is why convexity matters for stress scenarios rather than for daily risk reporting.

A practical estimate uses three prices. Reprice the bond at a yield above and below the current level, sum the two shocked prices, subtract twice the initial price, then divide by the initial price times the squared yield shift.

Convexity rises with the dispersion of cash flows around the bond's duration. Two bonds can share an identical duration and still behave differently in a large rate move: the one whose payments are spread widely across time, rather than concentrated near a single point, has greater curvature. This is the analytical basis for the barbell-versus-bullet comparison, where a portfolio of very short and very long maturities matches the duration of an intermediate bullet but carries higher convexity.

Holding other features constant, convexity increases with maturity, decreases with coupon, and increases as the market yield falls. A long-dated, low-coupon bond trading at a low yield is the most convex structure in the option-free universe, which is why ultra-long sovereign issues produce price moves that a linear estimate cannot capture.

Convexity also scales with price level. Money convexity, obtained by multiplying convexity by the full price of the position, converts the measure into currency terms for portfolio-level risk work.

Once a bond contains an embedded option, the relevant measures are effective duration and effective convexity, computed by shifting the entire benchmark yield curve and revaluing under a model that lets the option be exercised. A callable bond loses upside as yields fall because the issuer will refinance: its price flattens toward the call price, so effective convexity turns negative in that region. Agency mortgage-backed securities show the same pattern through prepayment behaviour, and the resulting negative convexity is central to how mortgage portfolios are hedged. A putable bond works in the opposite direction, adding convexity for the holder.

Markets price this. Because convexity benefits the holder in large moves, more convex structures generally trade at a lower yield than comparable less convex ones, and callable and mortgage paper compensates investors with additional spread. Convexity is therefore not free: an investor buys it by giving up yield, and the trade only pays if realised volatility exceeds what was implicitly paid for.

Expect to combine duration and convexity into a two-term price estimate for a stated yield shift, and to compare that estimate against duration alone. Recognise that the duration-only figure understates the price of the bond after either direction of move for option-free debt, and be ready to state which term dominates for small versus large shocks.

Be fluent with the approximation from three prices, and with the distinction between yield-based measures, appropriate for option-free bonds priced off their own yield to maturity, and effective measures, required whenever cash flows respond to the level of rates. Watch scaling: convexity is quoted on an annual basis, matching the compounding convention of the yield used, and yield changes must be entered in decimal form when applying the one-half convexity term.

Finally, be able to explain convexity as a portfolio construction choice rather than only as an arithmetic correction, including its role in duration-matched immunisation, where higher convexity increases exposure to non-parallel curve shifts.