# How do I calculate implied convexity from futures vs swaps?

From STIR Futures - Trading Euribor and Eurodollar futures by Stephen Aikin, convexity is determined by comparing the zero rate on a swap with an equivalent set of futures. For example, using futures, the calculation is shown as (top of Page 39):

(1+2.0%×0.25)×(1+2.5%×0.25)×(1+2.2%×0.25)×(1+2.3%×0.25)=2.167%

where 2.0%, 2.5%, 2.2%, 2.3% are the Implied Forward Rates from the Z1,H2,M2,U2 futures respectively.

Aikin suggests comparing this with the zero rate on a swap. What does the zero rate on a swap mean exactly? Initially, I thought it referred to the Par Rate given by:

$$S = \frac{\sum_{j=1}^{n_{2}} r_{j}d_{j}v_{j}}{\sum_{i=1}^{n_{1}} d_{i}v_{i}}$$

But I'm unsure if this should be directly comparable to the futures strip. As I write this question, I'm considering whether the comparable rate from the swaps market should be calculated as (1+S%×0.25)^4?

• When the author says "zero rates" I believe he means rates from a zero-coupon-style yield curve estimated from swaps. Essentially you use the implied prices of zero coupon bonds maturing at $t_1$ and $t_2$ get the swap market's estimate of the forward rate between $t_1$ and $t_2$. There is an extensive discussion of Discount Factors and Zero rates in the prior pages. Jan 1 at 17:18