The Mathematics of Financial Modelingand Investment Management

20-Term Structure Page 603 Wednesday, February 4, 2004 1:33 PM

Term Structure Modeling and Valuation of Bonds and Bond Options 603

month periods from now and has a maturity value of $101.50. Obvi-

ously, in the case of each coupon bond, the value or price of the bond is

equal to the total value of its component zero-coupon instruments.

Valuing a Bond as a Package of Cash Flows

In general, any bond can be viewed as a package of zero-coupon instru-

ments. That is, each zero-coupon instrument in the package has a maturity

equal to its coupon payment date or, in the case of the principal, the matu-

rity date. The value of the bond should equal the value of all the compo-

nent zero-coupon instruments. If this does not hold, it is possible for a

market participant to generate riskless profits by stripping the security and

creating stripped securities. We will demonstrate this later in this chapter.

To determine the value of each zero-coupon instrument, it is neces-

sary to know the yield on a zero-coupon Treasury with that same matu-

rity that we referred to as the spot rate earlier. The spot rate curve is the

graphical depiction of the relationship between the spot rate and its

maturity. Because there are no zero-coupon Treasury debt issues with a

maturity greater than one year issued by the U.S. Department of the

Treasury, it is not possible to construct such a curve solely from obser-

vations of market activity. Rather, it is necessary to derive this curve

from theoretical considerations as applied to the yields of actual Trea-

sury securities. Such a curve is called a theoretical spot rate curve.

Obtaining Spot Rates from the Treasury Yield Curve

We will now explain the process of creating a theoretical spot rate curve

from the yield curve that is based on the observed yields of Treasury

securities. The process involves the following:

1. Select the universe of Treasury securities to be used to construct the
   theoretical spot rates.


2. Obtain the theoretical spot rates using bootstrapping.


3. Create a smooth continuous curve.

We will return to the first and the third tasks later in this chapter. For

now, we want to show how the theoretical spot rates can be obtained

from the interpolated yields on Treasury securities (i.e., the Treasury yield

curve). To simplify the illustration, we will assume that an estimated

Treasury yield curve is as shown in Exhibit 20.1. The 6-month and 1-year

Treasury securities are assumed to be zero-coupon Treasury securities.

The process of extracting the theoretical spot rates from the Trea-

sury yield curve is called bootstrapping. To explain this process, we use

the data for the price, annualized yield (yield to maturity), and maturity