**F****uture changes**

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.

**Bond duration**

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.