# If I know the Price, DV01, and Duration of a Fixed Income instrument, is their approximation for the Convexity?

As the title says, I am looking to see if there is a good approximation for the convexity of a Fixed Income instrument. Say I know all the parameters of the instrument, can the Convexity be written as a function of these?

• No, unless you have 2 durations at 2 different rates
– NN2
Apr 3 at 12:05
• Assuming you know the price $P(y)$ and duration $D(y)$ for a given yield $y$, do you have data on Duration $D(\Delta y)$ (or Price $P(\Delta y)$) of the FI instrument for small parallel shifts of the yield curve? If so, then it's possible to estimate the convexity of the bond. Otherwise, the answer is generally no. Apr 3 at 12:10
• @Sharad Yes I do! Apr 3 at 12:42
• "Pricing and Trading Interest Rate Derivatives" has an appendix section entitled Estimating the gamma of IRS. The formulae are simple yet quite accurate. For an IRS it relies on knowing the delta of the swap and its start date and maturity date. I expect similar formulas exist for bonds. The analogy for a bond would be to take its risk/bp divide by 10,000 and multiply by the number of years to maturity plus 1. E.g a 30y bond in 1mm notional with a risk of 2100 p/bp has an estimated convexity of 6.51 units/bp. Give it a go an compare with real values.
– Attack68
Apr 3 at 14:11

## 1 Answer

Let $$P$$ represent the current price and let $$P(-\Delta y)$$ and $$P(+\Delta y)$$ represent the projected prices if the yield curve is shifted in parallel by the amounts $$-\Delta y$$ and $$+\Delta y$$ respectively (we need to exercise care with respect to the assumptions under which these projected prices are obtained, for example one typically assumes that spreads remain constant). Then:

\begin{align*} \mbox{Convexity} &= \frac{1}{P} \frac{d^2P}{dP^2} \\ &\approx \frac{1}{P} \frac{\left(\frac{P(-\Delta y) - P}{\Delta y} - \frac{P - P(+\Delta y)}{\Delta y}\right)}{\Delta y} \\ &= \frac{1}{P} \frac{P(+\Delta y)+P(-\Delta y) - 2P}{(\Delta y)^2} \end{align*}

You can recycle a version of the argument if you have information on the Durations $$D(-\Delta y)$$ and $$D(+\Delta y)$$ for small parallel shifts of the yield curve.

• This makes sense! In practice, how big of a change in yield should be used? Apr 3 at 14:41
• It'll be somewhat dependent on what kind of FI instrument you're looking at and what your hedging practices are so maybe this is something you can decide empirically. In other words, calculate the convexity for various different bonds assuming 1bp, 10bp, 15bp and 25bp shifts and see how stable the estimate is. You may also want to simulate your hedging PnL using the various assumptions. For mortgage-backed securities, I typically see somewhere between 15-25bps. Apr 3 at 15:39