# From continuous compounding to simple compounding - convexity adjustment

I have derived the convexity adjustment expression for futures rates using the Ho-Lee model, to arrive at the following: $$ForwardRate = FuturesRate - \frac{1}{2}\sigma^2T_1T_2$$ where $T_1$ refers to the time when the forward rate starts, $T_2$ when it finishes and $\sigma$ refers to the volatility of the short rate process.

I have derived the above expression in continuous time assuming continuous compounding, but my futures rate is a simply compounded rate. Is the following conversion to simple compounding correct? $$\left(1 + ForwardRate\times(T_2-T_1)\right)^{(T_2-T_1)} = \left(1 + FuturesRate\times(T_2-T_1)\right)^{(T_2-T_1)} - \frac{1}{2}\sigma^2T_1T_2$$

I am under the impression I'm terribly wrong!

Since the simple interest $r_{s}$ and the continuous compounded interest $r_{c}$ are connected by $$(1 + r_{s} \cdot (T_{2}-T_{1})) = e^{r_{c} \cdot (T_{2}-T_{1})}$$ it follows for the continuous compounded interest: $$r_{c} = \frac{1}{T_{2}-T_{1}} \cdot \ln{(1+r_{s} \cdot (T_{1}-T{2}))}$$ your convexity formula becomes than:
$$\frac{1}{T_{2}-T_{1}} \cdot \ln{(1 + ForwardRate \cdot (T_{2}-T_{1}))} = \frac{1}{T_{2}-T_{1}} \cdot \ln{(1+FutureRate \cdot (T_{2}-T_{1}))} - \frac{1}{2}\sigma^{2}T_{1}T{2}$$ This is your formula with $ForwardRate$ und $FutureRate$ expressed as simple interest.
• Thanks for clearing my confusion. Shouldnt the last formula be: $$\frac{1}{T_2-T_1}\ln(1 + Fwd(T_2-T_1)) = \frac{1}{T_2-T_1}\ln(1 + Fut(T_2-T_1)) - \frac{1}{2}\sigma^2T_1T_2$$ – Alfie Feb 12 '17 at 12:45