9.32 Convexity

Convexity Formula (for coupon bond):

[latex]Convexity (In Periods)=\frac{(1)(2)(PV-CF_{1})+(2)(3)(PV-CF_{2})+...+(t)(t+1)(PV-CF_{n})}{(1+y/p)^{2}(Price)}[/latex]

Definition:

Convexity estimates the difference between the actual price and the price estimate obtained by using duration. Convexity is a second order estimate of the rate of change in the price/yield curve, i.e., the rate of change in the dollar duration.

Example:

C = 0.08; N = 5; P = 2; YTM = 0.10

Note:

(p² + p) = p (p + 1). The former formula is used in finance literature; the latter is simpler to understand. With the latter formulation, in the first row, we have 1 × 2. In the second, we note 2 × 3, then, 3 × 4, and so on. This speaks more clearly to the curvature of the line (in this writer’s opinion). The “p²” in the former implies curvature – which was not present in the Duration formula.

Convexity (as stated consistent with “periods”) =

[latex]\frac{7,966.405}{(1.05)^{2}(92.2783)}=78.304 (periods)[/latex]

Convexity (consistent with years) =

[latex]\frac{78.304}{(2)^{2}}= 19.576[/latex]

While the convexity number calculated means nothing by itself, it can be used in comparison. “Periods” does not mean 78 periods. The greater the convexity (number), the greater the curvature of the price-yield curve. Duration and Convexity are positively related.

If the bond noted above bore a Zero-coupon, its convexity would be greater, i.e., “curvier”:

[(10)(11)(61.39)] / [61.39 x 1.05² ]) / 2² = 24.94 (compared with 19.576 above)