Chapter 9 VALUING BONDS WITH EMBEDDED OPTIONS LEARNING OUTCOME STATEMENTS After reading this chapter you should be able to: • explain the importance of the benchmark interest rates in interpreting spread measures. • explain the binomial interest rate modeling strategy. • explain the backward induction valuation methodology within the binomial interest rate tree framework. • compute the value of a callable bond from an interest rate tree given the call schedule and the rule for calling a bond. • explain how the value of an embedded option is determined. • explain the relationship among the values of a bond with an embedded option, the corresponding option-free bond, and the embedded option. • explain the effect of volatility on the arbitrage-free value of a bond with an embedded option. • calculate an option-adjusted spread using the binomial model and interpret an option-adjusted spread with respect to the benchmark interest rates. • calculate effective duration and effective convexity using the binomial model. • compute the value of a putable bond using the binomial model. • describe the basic features of a convertible bond. • compute and explain the meaning of the following for a convertible bond: conversion value, straight value, market conversion price, market conversion premium per share, market conversion premium ratio, premium payback period, and premium over straight value. • discuss the components of a convertible bond’s value that must be included in an option-based valuation approach. • compare the risk/return characteristics of a convertible bond’s value to the risk/return characteristics of the under- lying common stock. 293 294 Valuing Bonds with Embedded Options SECTION I: INTRODUCTION The presence of an embedded option in a bond structure makes the valuation of such bonds complicated. In this chapter, we present a model to value bonds that have one or more embedded options and where the value of the embedded options depends on future interest rates. Examples of such embedded options are call and put provisions and caps (i.e., maximum interest rate) in floating-rate securities. While there are several models that have been pro- posed to value bonds with embedded options, our focus will be on models that provide an “arbitrage-free value” for a security. At the end of this chapter, we will discuss the valuation of convertible bonds. The complexity here is that these bonds are typically callable and may be putable. Thus, the valuation of convertible bonds must take into account not only embedded options that depend on future interest rates (i.e., the call and the put options) but also the future price movement of the common stock (i.e., the call option on the common stock). In order to understand how to value a bond with an embedded option, there are several fundamental concepts that must be reviewed. We will do this in Sections II, III, IV, and V. In Section II, the key elements involved in develop- ing a bond valuation model are explained. In Section III, an overview of the bond valuation process is provided. Since the valuation of bonds requires benchmark interest rates, the various benchmarks are described in Section IV. In this section we also explain how to interpret spread measures relative to a particular benchmark. In Section V, the valuation of an option-free bond is reviewed using a numerical illustration. We first introduced the concepts described in this sec- tion at Level I (Chapter 5). The bond used in the illustration in this section to show how to value an option-free bond is then used in the remainder of the chapter to show how to value that bond if there is one or more embedded options. SECTION II: ELEMENTS OF A BOND VALUATION MODEL The valuation process begins with determining benchmark interest rates. As will be explained later in this section, there are three potential markets where benchmark interest rates can be obtained: • the Treasury market • a sector of the bond market • the market for the issuer’s securities An arbitrage-free value for an option-free bond is obtained by first generating the spot rates (or forward rates). When used to discount cash flows, the spot rates are the rates that would produce a model value equal to the observed market price for each on-the-run security in the benchmark. For example, if the Treasury market is the benchmark, an arbitrage-free model would produce a value for each on-the-run Treasury issue that is equal to its observed market price. In the Treasury market, the on-the-run issues are the most recently auctioned issues. (Note that all such securities issued by the U.S. Department of the Treasury are option free.) If the market used to establish the benchmark is a sector of the bond market or the market for the issuer’s securities, the on-the-run issues are esti- mates of what the market price would be if newly issued option-free securities with different maturities are sold. In deriving the interest rates that should be used to value a bond with an embedded option, the same princi- ple must be maintained. No matter how complex the valuation model, when each on-the-run issue for a benchmark security is valued using the model, the value produced should be equal to the on-the-run issue’s market price. The on-the-run issues for a given benchmark are assumed to be fairly priced.1 The first complication in building a model to value bonds with embedded options is that the future cash flows will depend on what happens to interest rates in the future. This means that future interest rates must be considered. 1 Market participants also refer to this characteristic of a model as one that “calibrates to the market.” Chapter 9 295 This is incorporated into a valuation model by considering how interest rates can change based on some assumed inter- est rate volatility. In the previous chapter, we explained what interest rate volatility is and how it is estimated. Given the assumed interest rate volatility, an interest rate “tree” representing possible future interest rates consistent with the vol- atility assumption can be constructed. It is from the interest rate tree that two important elements in the valuation pro- cess are obtained. First, the interest rates on the tree are used to generate the cash flows taking into account the embedded option. Second, the interest rates on the tree are used to compute the present value of the cash flows. For a given interest rate volatility, there are several interest rate models that have been used in practice to construct an interest rate tree. An interest rate model is a probabilistic description of how interest rates can change over the life of the bond. An interest rate model does this by making an assumption about the relationship between the level of short-term interest rates and the interest rate volatility as measured by the standard deviation. A discus- sion of the various interest rate models that have been suggested in the finance literature and that are used by practi- tioners in developing valuation models is beyond the scope of this chapter.2 What is important to understand is that the interest rate models commonly used are based on how short-term interest rates can evolve (i.e., change) over time. Consequently, these interest rate models are referred to as one-factor models, where “factor” means only one interest rate is being modeled over time. More complex models would consider how more than one interest rate changes over time. For example, an interest rate model can specify how the short-term interest rate and the long- term interest rate can change over time. Such a model is called a two-factor model. Given an interest rate model and an interest rate volatility assumption, it can be assumed that interest rates can realize one of two possible rates in the next period. A valuation model that makes this assumption in creating an interest rate tree is called a binomial model. There are valuation models that assume that interest rates can take on three possi- ble rates in the next period and these models are called trinomial models. There are even more complex models that assume in creating an interest rate tree that more than three possible rates in the next period can be realized. These models that assume discrete change in interest rates are referred to as “discrete-time option pricing models.” It makes sense that option valuation technology is employed to value a bond with an embedded option because the valuation requires an estimate of what the value of the embedded option is worth. However, a discussion of the underlying theory of discrete-time pricing models in general and the binomial model in particular are beyond the scope of this chapter.3 As we will see later in this chapter, when a discrete-time option pricing model is portrayed in graph form, it shows the different paths that interest rates can take. The graphical presentation looks like a lattice.4 Hence, dis- crete-time option pricing models are sometimes referred to as “lattice models.” Since the pattern of the interest rate paths also look like the branches of a tree, the graphical presentation is referred to as an interest rate tree. Regardless of the assumption about how many possible rates can be realized in the next period, the interest rate tree generated must produce a value for the securities in the benchmark that is equal to their observed market price — that is, it must produce an arbitrage-free value. Consequently, if the Treasury market is used for the bench- mark interest rates, the interest rate tree generated must produce a value for each on-the-run Treasury issue that is equal to its observed market price.