These are exciting times for Treasury futures. Futures on several maturities of Treasuries should help hedge changes that will occur in a yield curve that is unusually low at present but should rise and change shape with an improving U.S. economy. Now is the time to prepare for that eventuality.
The long end of the yield curve is of particular interest at the moment because the Federal Reserve — having approached the limit of reductions in short-term interest rates — is resorting to purchases of longer-maturity bonds to push more funds into the U.S. economy. Flattening the yield curve will reduce the cost of long-term debt, encouraging investment in fixed-assets, while taking some of the downward pressure off of the dollar by allowing short-term rates to increase. It also may reduce the positive carry currently enjoyed by banks and other institutions with historically low borrowing rates.
Five Treasury futures contracts are included in the following analysis. The short-term portion of the yield curve is represented by two-year, five-year and 10-year T-note futures, while the longer maturities are covered by the 30-year classic and ultra 30-year Treasury bond futures.
The original 30-year bond futures contract has price and yield cash flows that correspond to a 20-year fixed-income security. The ultra 30-year T-bond futures contract is so called because it has a longer maturity, at least 25 years. CME Group created the ultra to better reflect a 30-year maturity. The original bond contract specifications called for 30-year bonds with more than 15 years left to maturity. When the Treasury Department discontinued offering long bonds in 2001, the supply shrank. Because there were none offered for a seven-year period, the remaining supply reflected a shorter maturity. The ultra has cash flow characteristics that match a 30-year bond. Thus, the yield curve is covered by two-, five-, 10-, 20- and 30-year maturities.
Note that Treasury futures, as notional instruments, do not return cash payments in terms of interest and principal. Futures are held and traded primarily to take advantage of price changes before their delivery dates. Long and short Treasury futures contracts require a cash transaction on or around the delivery date because these derivatives are settled by the exchange of Treasury securities that closely match the maturity of the underlying T-note or T-bond.
Three of the Treasury futures maturities are shown on "Yields & rates" (below). The chart shows the eurodollar futures quarterly rates and yield curve, U.S. Treasury yields and T-note futures yields at two-, five- and 10-year maturities. Eurodollar yields form a smooth curve that is slightly higher and parallel to U.S. Treasury yields that are shown at eight maturities from three months to 10 years. Eurodollar yields are based on the geometric means of quarterly rates that extend over a 10-year span.
The three Treasury futures maturities shown on "Yields & rates" are two-, five- and 10-year T-notes. The T-note yields shown are slightly higher than corresponding eurodollar yields. Eurodollar futures — although representing dollar deposits that have more risk than Treasury securities — have the advantage of cash settlement at the delivery date.
Options on T-note and T-bond futures may be analyzed by using the LLP option pricing model available on Excel spreadsheets that can be downloaded from futuresmag.com. For example, an option price curve based on 12 strike prices was computed for the March 2011 five-year T-note futures on Oct. 28, 2010. A portion of the spreadsheet is shown on "Five-year T-note calls" (below).
The analysis results in call option prices predicted by a regression equation, the slope (delta value) of the option price curve at each strike price and upper and lower breakeven prices at expiration, 109 days in the future. Breakeven prices are those that would result in zero gain or loss on a delta-neutral trade between calls and underlying futures.
For the five-year T-note option with 121.00 strike price, the breakeven prices are 123,198 and 119,361. These prices imply a yield spread for the five-year T-note of 1.205% to 1.920%, while the yield on Oct. 28 was 1.665%. The price and yield spread on the measurement date shows the options market’s assessment of yield and price volatility for five-year T-bond futures through the expiration date in March 2011.
"Calls on Treasury futures" (above) shows the option price curves for the five Treasury futures on Oct. 28, 2010. Heights of the curves correspond to the maturities of the underlying note or bond. For example, calls for the 30-year ultra T-bond futures are on the highest curve because longer maturities produce the largest change in price for fixed-income securities with equal risk. As shown on "Breakeven prices & yields" (below), breakeven yield spreads for the five Treasury-based futures increase with larger underlying maturities. Because the five Treasuries have identical risk and equal times to expiration, maturities are the primary reason for different curve heights, breakeven prices and yield spreads.
Option price curve equations are useful for predicting changes in prices several days forward, based on new underlying futures prices with the same set of strike prices. "Three days predicted" (below) shows options on 20-year T-bond futures where the option price-to-strike price is related to the futures price-to-strike price. The curve shows close relationships between the predicted prices over three days and initial prices on Oct. 28.
On Nov. 3 there is some indication of the market prices falling slightly lower than predicted by the fixed equation. This will be an increasing tendency as time to expiration decreases. Variations between current market prices and predicted prices may be used to find temporarily over-valued and under-valued options as prices tend to move back toward the regression curve.
Trial & error
"T-note futures yield & duration" (below) shows one section of an Excel spreadsheet that calculates the yield on a T-note or T-bond given the listed price. The duration for the futures contract is computed automatically on the spreadsheet at the same time it is calculating the yield from a given price.
The two-year T-note has a price listed as 109% of $200,000 par, plus 27.25 times 1/32nd of 1% of par. In decimal form, the price is $218,852. Interest of $6,000 is paid semiannually, with $200,000 par value received at maturity. The yield that corresponds to a $218,852 price must be computed by trial and error, trying out different yields until the computed total present value is approximately equal to the listed price. The trial-and-error process may be done manually, which is how this table was created, or accomplished by a computer program that gradually centers in on the required discount yield. Yields and prices also may be looked up on tables of price-to-yield and yield-to-price available online at CME Group.
CME Group publishes a Treasury price index online — the Dow Jones Chicago Board of Trade (CBOT) Treasury Price Index — based on five-year, 10-year and 20-year maturities. The "T-note futures yield & duration" spreadsheet includes a section that produces data similar to the Dow Jones CBOT index, and shows how the index is computed.
Because duration is an important element in the index calculation, it may be well to describe duration in more detail. Duration is the weighted average of time to maturity of any asset. The weights are equal to the present value of cash flows in each time period divided by the asset’s total present value.
Duration is shown computed on "T-note futures yield & duration" (above), where the fourth column shows the calculation of weights for each time period and column five multiplies the weight and the time period number. The weighted average time to maturity, or duration, of the two-year note is computed as 3.8372 six-month periods or 1.9186 years. Duration changes constantly with new data for market yields and periodic cash flows; however, it is always equal to or less than calendar time to maturity.
Duration is a critical factor in hedging individual financial instruments and entire portfolios. In theory, a portfolio that has a given weighted duration that includes its total holdings may be hedged by a long or short position in a single Treasury futures contract with the same computed duration.
It is important to realize that prices on fixed-income securities change in relation to their durations rather than simply time to maturity. "Treasury price index" (below), calculated on Oct. 28, 2010, includes modified durations (where duration is divided by (1+ i), with i equal to the computed yield). For example, the weighted average maturity of the 30-year ultra T-bond is 15.6497 years. The time pattern of cash flows determines duration, resulting in the $100,000 par value received on the 30-year bond shrinking in its impact on price because of the longer time to maturity.
If we were to begin a Treasury price index similar to the Dow Jones CBOT Treasury Index, using just the five-, 10- and 20-year maturities as shown on "Treasury price index," the three prices would be weighted. The index weights are calculated by dividing the modified duration of the 30-year T-bond by the modified duration of each of the other maturities. In this way, the pricing effects of different maturities are equalized and the index is made comparable over time.
The importance of the Treasuries in hedging, speculating and forecasting interest rates and yields of all maturities hardly can be overstated. Their futures and options will be of increasing usefulness as the Federal Reserve uses financial markets in its efforts to accelerate economic recovery.
Paul Cretien is an investment analyst and financial case writer. His e-mail is PaulDCretien@aol.com.