Convexity is an issue that is confronted when planning hedged trades between eurodollar futures and futures contracts based on fixed-income securities, such as interest rate swaps and Treasury notes. A price curve for bonds, T-note futures and interest rate swap futures is convex in relationship with market yields. This means prices for these instruments rise more rapidly as yields fall than the price declines as yields rise — thus, a steeper curve when yields are low and a flatter curve for larger yields. Although the price curve of eurodollar futures is not convex, the convexity of other interest rate contracts indirectly determines eurodollar futures prices.
While the prices of fixed-interest securities change in varying amounts per basis point depending on current level of rates and yields, eurodollar futures prices change at $25 per basis point (1/4 year times $1,000,000 underlying notional principal times 0.01%). As yields fall, we expect the bond, T-note or swap price curve to increase up and away from eurodollar futures prices.
Conversely, when market yields increase, the eurodollar futures price will continue to drop at a constant dollar rate while the bond price drops at an increasingly slower rate, widening the distance between the two price curves.
Modified durations
“Convexity” (below) shows that as the market yield increases or decreases, deviations of the 10-year swap futures price from the eurodollar price straight line gradually expand. Although the differences appear small on the chart, the underlying security is a $100,000 interest rate swap in which dollar differences between the eurodollar and swap price curves are minor in relation to the swap futures price.
A eurodollar futures price line can be drawn tangent to the convex price curve of a fixed-interest financial instrument. The process begins by calculating the duration of the fixed income security — for example, 10-year swap futures. The price on April 11, 116-004 ($116,125), corresponds to an annual yield to maturity of 2.1952% for the 4% coupon rate with nominal interest on $100,000 paid semiannually. Duration is the weighted average time to maturity for the swap futures contract, when the weight of each semiannual time period is the present value of that period’s cash flow as a proportion of the contract’s $116,125 present value.
The duration, 8.4823 years, is converted to modified duration by dividing by (1 + the current yield) or 1.021952. Thus, modified duration equals 8.3001. An approximate price change for any yield distance from the original $116,125 is computed by multiplying minus modified duration, times the change in yield, times the original price. For example, an increase of 50 basis points produces an approximate price change for the swap futures contract of minus $4,819.50; and the same dollar amount will occur on the plus side when the yield decreases by 50 basis points.
The reference to “approximate” price changes is because the modified duration dollar changes are taking place along the straight line tangent to the convex price curve — thus, they are approximate changes for the 10-year swap futures but are accurate for changes along the tangent line.
The eurodollar futures contract progresses in each direction from the initial $116,125 at a constant rate of $96.39 per basis point of yield change (or $4,819.5 for a 50-basis-point change). To equalize the eurodollar futures and swap futures slopes at this beginning yield and price, 3.8556 eurodollar futures are used for each swap futures.
A view of the swap futures and eurodollar futures prices at various yields is shown on “Price differences” (below). The yields, futures prices for 10-year interest rate swap and three-month eurodollars, and price differences illustrate the price advantage of swap futures over eurodollar futures at various yields that range between plus and minus 200 basis points from the current yield.
Price adjustment
Because of the convexity advantage held by a fixed-income security over eurodollar futures used in a hedge, an adjustment is made by the market — increasing the yield and decreasing the price of eurodollar futures at each maturity relative to comparative T-note, U.S. Treasury securities and interest rate swap futures.
One way to look at the necessary adjustment in yield is to measure the change in yield required to make the interest rate swap price equal to the eurodollar price at each market yield. “Yield differences” (below) shows the yield changes necessary to move the swap futures price down to equal the eurodollar futures price at selected market yields. The largest adjustments are 22.6 and 17.3 basis points at the plus and minus 200 basis point variations from the current market yield.
A less restricted definition of convexity is to use the term in reference to any nonlinear price — yield relationships. For example, eurodollar futures are marked to market daily. These may be hedged against futures on fixed-income securities that have no intermediate cash flows and thus have an advantage over eurodollar futures in terms of margin requirements. Increasing rates result in higher margin, while decreasing rates mean investing excess margin at lower rates.
Convexity variation
An additional type of convexity is caused by the optionality of interest-rate contracts. During recent years when mortgage and bond rates fell, profiting from large capital gains on these fixed-interest contracts failed to materialize when bond issuers called their issues to finance at lower rates, and when mortgage borrowers repaid their loans and opted for refinancing.
For swap-eurodollar convexity, “Price differences” and “Yield differences” are graphic representations of convexity adjustments. These curves tend to add drama to the convexity that appeared almost as a straight line on “Convexity.” The same calculations could be carried further up and down in market yields, with ever-expanding differences in price as yields increase or decline.
Differences between eurodollar yields and yields on U.S. Treasury securities are shown on “Yields and rates” (below). On April 11, Treasury yields increased from near zero, at 0.14%, to 2.04% at the 10-year maturity. At the same time, the quarterly rates for 10 years of eurodollar futures extended from 0.51% to 2.34%, and the quarterly eurodollar yields derived from geometric mean rates at each maturity approximately were parallel with U.S. Treasury yields and slightly higher in part because of the convexity adjustment discussed earlier.
On April 11, the 10-year yields for U.S. Treasury securities, June 2012 interest rate swap futures and eurodollar futures were 2.04%, 2.20% and 2.34%, respectively. The spreads between yields at the 10-year maturity approximate those that are suggested by “Yield differences.”
Pricing consequences
As shown, price and yield calculations on the 10-year swap futures are computed based on the $100,000 notional principal with an annual coupon rate of 4% with interest paid semiannually. Given a listed or hypothetical price, the yield may be calculated by finding the yield resulting in a matching price or discount rate. CME Group publishes a table of price and yield changes for T-note and swap futures online.
The pattern of eurodollar rates over 40 quarters changes shape in response to shifting yield curves of U.S. Treasury securities and proxies, such as interest rate swaps and T-note futures. In this process, eurodollar futures create their own form of convexity with changes in the curve of quarterly rates-to-yields.
“Rates-to-yields” (above) shows that on three dates in years 2007, 2009 and 2012 the ratio is smaller when yields are high and rise to top levels when yields are low. The following table includes eurodollar yields at quarters 1, 20 and 40 for maturities of 0, 5 and 10 years in addition to the U.S. Treasury yield at those dates.
|
Quarter |
2/1/07 |
1/30/09 |
4/11/12 |
|
1 |
5.37% |
1.25% |
0.50% |
|
20 |
5.18% |
2.58% |
1.29% |
|
40 |
5.35% |
2.64% |
2.34% |
|
Treasury |
|||
|
20 |
4.84% |
1.88% |
0.88% |
Because of convexity adjustments and shifts in the pattern of rates-to-yields, future changes in market yields will produce significant changes in eurodollar rates, yields and prices. By being aware of this relationship, traders can be prepared better for future shifts in market values.
Paul Cretien is an investment analyst and financial case writer. His e-mail is PaulDCretien@aol.com.
