From the July 01, 2012 issue of Futures Magazine • Subscribe!

Convexity analysis in fixed income

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.

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