Duration and Convexity

Question

I am searching to estimate the evolution of my portfolio duration following a yield increase/decrease. Can i use the convexity? I mean IR delta x (- convexity) = Duration delta

Is it correct?

Thanks a lot !

Answer 1

Yes, you can use convexity although the formula you have is not quite correct. Think of the portfolio as a single bond with price $P(y)$, where $y$ is the yield of the portfolio (we're making the assumption that the duration hedging of the portfolio is based on a single risk variable, the yield to maturity of the portfolio). Then, we have the usual definitions for modified duration $D$ and convexity $C$: $$ D = -\frac{1}{P}\frac{dP}{dy} $$ $$ C = \frac{1}{P}\frac{d^2P}{dy^2} $$ We can rewrite the expression for $C$ in terms of $D$: $$ \begin{align} C &= \frac{1}{P}\frac{d}{dy} \left[ \frac{dP}{dy} \right] \\ &= \frac{1}{P}\frac{d}{dy} \left[ -PD \right] \\ &= D^2 - \frac{dD}{dy} \end{align} $$ This suggests that for a given change in yield $\Delta y$, we can approximate the change in duration, $\Delta D$, by: $$ \Delta D \approx (D^2 - C)\Delta y $$

Example. Consider a default-free bond with a face of 100, a coupon of 6%, a yield of 5% and a term of 10 years. Assume annual compounding. Then, we can directly calculate $D = 7.52$ and $C = 72.17$. If yields increase by 25bps, then direct calculation shows the new duration $D' = 7.48$. On the other hand, using our formula above gives: $$ \Delta D \approx (7.52^2 - 72.17)*(0.25/100) = -0.04 $$