Interest Rate Swap Valuation: A Basic Adjustment for Convexity Bias in Eurodollar Futures Quotes

Recently I was tasked with researching and developing a financial model for use in valuing several plain vanilla interest rate swaps held by a large client for the purposes of marking the instruments to market under FASB 815, Derivatives and Hedging. Being the doe-eyed rookie and immersed in study for the CFA examinations, I was the obvious nominee to build and inhabit the weeds of such a model. Finding practical public information on implying forward LIBOR to value derivatives is not easy with large banks relying on proprietary models. After weeks of research, application, and re-application, I produced a forward LIBOR curve within a swap valuation model that I was happy to present to my boss and the client. In this week’s blog, I identify a few pointers I learned along the way that will hopefully provide a foundation to imply forward LIBOR.

Eurodollar Futures contracts set the stage to model forward LIBOR and for that matter, a swap valuation as a whole. These futures provide a strong indication of the future floating rates implicit in LIBOR-based derivative instruments, as long as liquidity is sufficient. The contracts trade on the Chicago Mercantile Exchange with standard contract sizes of $1 million and contracts maturing during the months of March, June, September and December. The contracts settle as an index value equal to 100 minus the then current 3-month LIBOR rate[1]. In other words, Eurodollar Futures prices are driven by the market’s expectation of 3‐month LIBOR upon the settlement of the contract.

Because the floating rate leg of plain vanilla interest rate swap is generally based on 3-month LIBOR, Eurodollar Futures are prime inputs to interest rate swap valuation models. Unfortunately for me, Eurodollar Futures quickly became my friend but were quicker to become my enemy. This is due to a difference in return characteristics between Eurodollar Futures and interest rate swaps.

One way to visualize this difference is to think of an interest rate swap’s return pattern akin to the convexity profile of a fixed rate bond. The receive‐fixed side of an interest rate swap is actually a positively convex position, since for a shift in swap rates up or down of equal magnitude the fixed rate receiver benefits more in a rally (lower rates) than he is hurt in a sell‐off (increasing rates)[2].

On the other hand, the Eurodollar Futures market exhibits a linear pay-off pattern due to the daily mark-to-market requirements on the clearinghouse. Because of this fact, every first-year trader learns that a 1 basis point rate change always translates to a futures price delta of $25, regardless of the level of rates[2]. This payoff pattern is roughly equivalent to the effective duration profile of a bond. The effective duration is a linear estimate of the price change in a fixed rate bond given a change in interest rates. Although accurate for small changes in interest rates, the effective duration measure quickly loses accuracy with larger deltas in interest rates.

The below chart[3] shows the true convexity of a fixed rate bond plotted along with the effective duration estimate of this bond. The focus of this article is on moving from the return profile of a Eurodollar Futures contract (think effective duration estimate, the blue line) to the return profile of the receive-fixed leg of an interest rate swap (think real world bond returns, the green).


In simple economic terms, the discrepancy between the return profiles of interest rate swaps and Eurodollar Futures suggests that swap rates implied by Eurodollar Futures are too high[2]. The Hull‐White term structure calculation is a generally accepted, yet somewhat outdated method to correct for this discrepancy, as shown below:

The convexity-adjusted futures rate, covering the range between t1 and t2 (denominated in years from current date), equals the continuously compounded Eurodollar Futures rate less the above convexity adjustment. This formula incorporates the standard deviation of the change in short-term interest rates (σ) and a mean reversion rate (α)[4]. A typical range of values for the mean reversion rate is 0.001 for negligible effects to 0.1, which could have material effects. For simplicity, a constant default value for mean reversion speed could be assumed. For example, Bloomberg assumes a constant mean reversion rate of 0.03[4]. Although the formula looks complex, it is actually fairly simple with very little inputs. There are really two major takeaways from this formula:
• As the time horizon to Eurodollar settlement increases, the convexity adjustment increases
• Higher interest rate volatility results in larger convexity adjustments

Below is a stylized model of the Hull-White formula which continuously compounds raw Eurodollar quotes and deducts the convexity adjustment to arrive at a convexity adjusted Eurodollar quote. I assumed 1.5{7656ca219931958fe15db644f5e70e9855a8d6dc7ecf752fd1b70e2be3385646} volatility in short-term interest rates and a 0.03 mean reversion rate. Keep in mind that Eurodollar Futures settle on the last trading two London business days before the third Thursday of each contract month. Interest rate swaps generally pay quarterly or monthly, making interpolation necessary to fill in the gaps. It should also be noted that as the Eurodollar contracts extend into the future, the overall adjustment becomes larger.

The total adjustment begins at approximately one basis point and extends to approximately three basis points late in 2015. Admittedly, this convexity adjustment methodology is on the simpler side of the quantitative spectrum. More recent financial engineering methodologies have produced advanced stochastic volatility and Monte Carlo models to address the bias implicit in Eurodollar Futures quotes.

In conclusion, this article only covers the tip of the iceberg that is interest rate swap valuation. Although conversing over Eurodollar convexity bias at the local bar after work is not ideal for the average finance professional, building the proper inputs into any model is crucial to get an accurate result on the back end. As for the doe-eyed analysts out there, watch out for the Eurodollar Futures convexity bias. The adjustment is very important to produce an accurate interest rate swap valuation model, and will definitely help avoid (some) unnecessary grey hairs.

[1] Labuszewski, John. Kamradt, Michael. Gibbs, David. Understanding Eurodollar Futures. Chicago, IL: CME Group, 2013.
[2] Corb, Howard. Interest Rate Swaps and Other Derivatives. New York, NY: Columbia University Press, 2012.
[4] Ron, Uri. A practical Guide to Swap Curve Construction. Ottawa, Ontario: Bank of Canada, 2000.