Convexity: measuring the curve

Why duration is not enough

In the previous module, modified duration turned a yield change into a percentage price change: $\%\Delta PV \approx -\text{ModDur} \times \Delta y$. That rule is a straight line. It treats every basis point as having the same price effect.

The real relationship between a bond's price and its yield-to-maturity is a curve that bows toward the origin. Duration is the slope of the line that just touches (is tangent to) the curve at today's yield. Near today's yield the line and the curve are almost the same, so for small moves duration works well.

Move further away and the two separate. Because the curve sits above the tangent line on both sides, the actual price is higher than the duration estimate whether yields rise or fall. When yields fall, the bond gains more than duration predicts. When yields rise, it loses less.

Convexity is the risk measure that captures this bending. Duration is the first-order (linear) effect of a yield change; convexity is the second-order (non-linear) effect. Used together, they give price estimates much closer to the true prices for big yield moves.

Think of driving on a curved road using only your current heading. For the next few metres the heading is a fine guide. Over a long distance you need to know how sharply the road bends, and that is what convexity adds.

Approximate convexity: three prices and one formula

The quickest way to measure convexity is to reprice the bond. Start from today's full price $PV_0$. Raise the yield by a small amount $\Delta y$ and find $PV_+$; lower it by the same amount and find $PV_-$. These are exactly the prices you already use for approximate modified duration.

Approximate modified duration uses the gap between the two prices: $(PV_- - PV_+)/(2 \times \Delta y \times PV_0)$. Approximate convexity uses how far their sum sits above twice today's price: $(PV_- + PV_+ - 2PV_0)/((\Delta y)^2 \times PV_0)$.

Why does the sum work? If the price-yield line were straight, the gain from a fall in yield and the loss from a rise would cancel, and $PV_- + PV_+$ would equal $2PV_0$. Curvature makes the gain bigger than the loss, so the sum exceeds $2PV_0$. The size of that excess, scaled by $(\Delta y)^2$, is the convexity.

Enter $\Delta y$ as an annual decimal (5 bp = 0.0005). The result is then already an annualized convexity, even for a semiannual bond. Because the method only needs prices, it also works for bonds whose cash flows are uncertain, for example because of embedded options or default risk.

• Use a small shift (1 to 10 bp) and keep many decimals in the prices: the numerator is a tiny number, so rounding the prices to two decimals ruins the answer.
• The denominator squares $\Delta y$. Forgetting the square is the classic error.

The exact method: a duration table with one more column

Convexity can also be built from the cash flows, in the same table you use for Macaulay duration. For each period t you already have the cash flow's present value and its weight $w_t$ (PV of the cash flow ÷ bond price).

Add one column: $t \times (t+1) \times w_t$ divided by $(1+r)^2$, where r is the periodic yield. Multiplying t by t + 1 is what brings in the squared, non-linear term. Summing that column gives convexity measured in periods.

To annualize, divide the sum by the number of periods per year squared: by 1 for an annual-pay bond, by 4 for a semiannual bond. (Duration in periods is divided by periods per year; convexity, being a second-order measure, by its square.)

Both methods give practically the same answer for an option-free bond. A third route is a closed-form formula from calculus, which you will not need to compute.