The Mathematics of Financial Modelingand Investment Management

20-Term Structure Page 606 Wednesday, February 4, 2004 1:33 PM

606 The Mathematics of Financial Modeling and Investment Management

Using Spot Rates to the Arbitrage-Free Value of a Bond

Finance theory tells us that the theoretical price of a Treasury security

should be equal to the present value of the cash flow where each cash

flow is discounted at the appropriate theoretical spot rate For example,

if the Treasury spot rates shown in the last column of Exhibit 20.1 are

used to compute the arbitrage-free value of an 8% 10-year Treasury

security, the present value of the cash flow would be found to be

$115.2619. If a 4.8% coupon 10-year Treasury bond is being valued

based on the Treasury spot rates shown in Exhibit 20.1, the arbitrage-

free value is $90.8428.

Suppose that the 8% coupon, 10-year Treasury issue is valued using

the traditional approach based on 6% (i.e., the yield on a 10-year Trea-

sury coupon bond shown in Exhibit 20.1). Discounting all cash flows at

6% would produce a value for the 8% coupon bond of $114.8775. Con-

sider what would happen if the market priced the security at $114.8775.

The value based on the Treasury spot rates is $115.2619. Faced with this

situation, a securities dealer can buy the 8% 10-year issue for $114.8775,

strip off each coupon payment and the maturity value, and sell each cash

flow in the market at the spot rates shown in Exhibit 20.1. By doing so,

the proceeds that will be received by the dealer are $115.2619. This

results in an arbitrage profit of $0.3844 (= $115.2619 – $114.8775).

Securities dealers recognizing this arbitrage opportunity will bid up the

price of the 8% 10-year Treasury issue in order to acquire it and strip it.

Once the price is up to around $115.2619 (the arbitrage-free value), the

arbitrage opportunity is eliminated.

We have just demonstrated how stripping of a Treasury issue will

force the market value to be close to its arbitrage-free value when the

market price is less than the arbitrage-free value. When a Treasury issue’s

market price is greater than the arbitrage-free value, a securities dealer

can capture the arbitrage value by a process referred to as reconstitution.

Basically, the securities dealer can purchase a package of stripped Trea-

sury securities traded in the market so as to create a synthetic Treasury

coupon security that is worth more than the same maturity and the same

coupon Treasury issue. The sale of the resulting synthetic coupon security

that is created will force the price down to its arbitrage-free value.

The Discount Function

A more convenient way of characterizing the term structure of interest rates

is by means of the discount function. The discount function specifies the

present value of a cash flow in the future. It can therefore be interpreted as

the price of a pure risk-free discount bond of a given maturity with a $1

face value. The discount function (Dn) is related to spot rates as follows: