The Mathematics of Financial Modelingand Investment Management

20-Term Structure Page 600 Wednesday, February 4, 2004 1:33 PM

600 The Mathematics of Financial Modeling and Investment Management

sons account for this tendency. First, Treasury securities are free of

default risk, and differences in creditworthiness do not affect yield esti-

mates. Second, the Treasury market offers the fewest problems of illi-

quidity or infrequent trading.

Typically in constructing a yield curve using Treasury yields the on-

the-run Treasury issues are used. These are the most recently auctioned

Treasury issues. In the United States, the U.S. Department of the Trea-

sury currently issues 3-month and 6-month Treasury bills and 2-year, 5-

year, and 10-year Treasury notes. Treasury bills are zero-coupon instru-

ments and Treasury notes are coupon-paying instruments. Hence, there

are not many data points from which to construct a Treasury yield

curve, particularly after two years. At one time, the U.S. Treasury issued

30-year securities (referred to as Treasury bonds). However, the Trea-

sury stopped this practice. In constructing a Treasury yield curve, mar-

ket participants use the last issued Treasury bond (which has a maturity

less than 30 years) to estimate the 30-year yield. The 2-year, 5-year, and

10-year Treasury notes and an estimate of the 30-year Treasury bond is

used to construct the Treasury yield curve. On September 5, 2003, Leh-

man Brothers reported the following values for these four yields:

2 year 1.71%

5 year 3.25%

10 year 4.35%

30 year 5.21%

To fill in the yield for the 25 missing whole year maturities (3 year, 4

year, 6 year, 7 year, 8 year, 9 year, 11 year, and so on to the 29-year matu-

rity), the yield for the 25 whole-year maturities are interpolated from the

yield on the surrounding maturities. The simplest interpolation, and the

one most commonly used in practice, is simple linear interpolation.

For example, suppose that we want to fill in the gap for each one

year of maturity. To determine the amount to add to the on-the-run

Treasury yield as we go from the lower maturity to the higher maturity,

the following formula is used:

(yH – yL)/N

where:

yH = yield at higher maturity

yL = yield at lower maturity

N = number of years between two observed maturity points