Comparing STIR Futures Strips with Swap Rates for Convexity

Summary

The document examines how to interpret a swap zero rate when comparing it with rates implied by a strip of short-term interest-rate futures. It cites a textbook example that compounds four quarterly futures-implied forward rates into a cumulative rate, then asks whether the swap-side comparison should use a par swap rate or a compounded version of that rate over the same periods.

The post does not resolve the comparison or give a definitive convexity calculation. It supplies a par-rate expression and raises the possibility that matching compounding conventions and tenor is necessary. Its value is in highlighting that par swap rates and compounded futures-strip rates are different objects; a sound comparison would require consistent cash-flow dates, day-count conventions, discounting, and treatment of futures-versus-forward convexity. Those details are not worked through here.

Key ideas

• A futures strip can be compounded from its implied forward rates to form a cumulative rate.
• The author asks what swap zero rate should be compared with that strip.
• A par swap rate may not be directly comparable to a compounded futures-strip rate.
• The document poses the convention-matching problem but does not provide a solution.

Tags

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


# 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?

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.