A brief history of bond valuation

The value of any bond is the present value of its expected cash flows. This sounds simple: Determine the cash flows and then discount those cash flows at an appropriate rate. In practice, it’s not so simple for two reasons. First, holding aside the possibility of default, it is not easy to determine the cash flows for bonds with embedded options. Because the exercise of options embedded in a bond depends on the future course of interest rates, the cash flow is a priori uncertain. The issuer of a callable bond can alter the cash flows to the investor by calling the bond, while the investor in a putable bond can alter the cash flows by putting the bond. The future course of interest rates determines when and if the party granted the option is likely to alter the cash flows.

A second complication is determining the rate at which to discount the expected cash flows. The usual starting point is the yield available on Treasury securities. Appropriate spreads must be added to those Treasury yields to reflect additional risks to which the investor is exposed. Determining the appropriate spread is not simple, and is beyond the scope of this article. The ad hoc process for valuing an option-free bond (i.e., a bond with no options) once was to discount all cash flows at a rate equal to the yield offered on a new full-coupon bond of the same maturity. Suppose, for example, that one needs to value a 10-year option-free bond. If the yield to] maturity of an on-the-run 10-year bond of given credit quality is 8%, then the value of the bond under consideration would be taken to be the present value of its cash flows, all discounted at 8%.

According to this approach, the rate used to discount the cash flows of a 10-year current-coupon bond would be the same rate as that used to discount the cash flow of a 10-year zero-coupon bond. Conversely, discounting the cash flows of bonds with different maturities would require different discount rates. This approach makes little sense because it does not consider the cash flow characteristics of the bonds. Consider, for example, a portfolio of bonds of similar quality but different maturities. Imagine two equal cash flows occurring, say, five years hence, one coming from a 30-year bond and the other coming from a 10-year bond. Why should these two cash flows have different discount rates and hence different present values?

Given the drawback of the ad hoc approach to bond valuation, greater recognition has been given to the fact that any bond should be thought of as a package of cash flows, with each cash flow viewed as a zero-coupon instrument maEagle Tradersg on the date it will be received. Thus, rather than using a single discount rate, one should use multiple discount rates, discounting each cash flow at its own rate.

One difficulty with implementing this approach is that there may not exist zero-coupon securities from which to derive every discount rate of interest. Even in the absence of zero-coupon securities, however, arbitrage arguments can be used to generate the theoretical zero-coupon rate that an issuer would have to pay were it to issue zeros of every maturity. Using these theoretical zero-coupon rates, more popularly referred to as theoretical spot rates, the theoretical value of a bond can be determined. When dealer firms began stripping of full-coupon Treasury securities in August 1982, the actual prices of Treasury securities began moving toward their theoretical values.

Another challenge remains, however—determining the theoretical value of a bond with an embedded option. In the early 1980s, practitioners came to recognize that an option-bearing bond should be viewed as a package of cash flows (i.e., a package of zero-coupon instruments) plus a package of options on those cash flows. For example, a callable bond can be viewed as a package of cash flows plus a package of call options on those cash flows. As such, the position of an investor in a callable bond can be viewed as:

Long a Callable Bond = Long an Option-Free Bond + Short a Call Option on the Bond.

In terms of the value of a callable bond, this means:

Value of Callable Bond = Value of an Option-Free Bond - Value of a Call Option on the Bond.

But this also means that

Value of an Option-Free Bond = Value of Callable Bond + Value of a Call Option on the Bond.

An early procedure to determine the fairness of a callable bond’s market price was to isolate the implied value of its underlying option-free bond by adding an estimate of the embedded call option’s value to the bond's market price. The former value could be estimated by applying option pricing theory as applied to interest rate options.

This insight led to the first generation of valuation models that sought to value a callable bond by estimating the value of the call option. However, estimation of the call option embedded in callable bonds is not that simple. For example, suppose a 20-year bond is not callable for five years after which time it becomes callable at any time on 30-days notice.