Our -Authors A Model for Valuing Bonds George 0. Wtlltams c'.'.>n their artide: "The evolu­ and Embedded Oi,riotts iS MSiSt~nt pt'6f~ssot of mathe­ tl6fl of bond valuation has pro­ matics and computer science at gressed from discounting all cash Andrew J. Kalotay Manhattan College. He has been a flows at a single rate to discount­ is associate professor of finance frequent consultant to maior fi­ ing each cash flow at its own in the Graduate School of Busi­ nancial institutions on computer­ zero-coupon (spot) rate or, ness of Fordham University. He is related issues ranging from ad­ equivalently, by a series of one­ an authonty in the field of debt vanced workstation technology period forward rates. When a management, having structured and computerized trading sys­ bond has an embedded option transactions for both the issuance tems to simulation techniques. An such as a call or put provision, and retirement sides amounting expert on the theory of random however, consideration must be to over $50 billion. Prior to estab­ walks, he is widely published in given to the volatility of forward lishing his own firm, Dr Kalotay computer science and applied rates. We present a bond valua­ mathematics. His past academic tion model that allows for the was director-researcher in Sal­ positions include a research fac- omon Brothers' bond portfolio discounting of each cash flow at 1.ilty appointment at New York appropriate volatility•dependent, analysis group. Betore coming to University's Courant Institute. Wall Street, he was with the trea­ one-period forward rates This latest technique allows for valua­ sury department of AT&T and FrankJ. Fabozzi with Bell Laboratories. He is a tion of options in a natural, con­ trustee of the Financial Manage­ is editor of the Journal ofPortfo­ sistent manner " ment Association lio Managemem and a consultant to numerous financial institutions and money management firms. He is on the board of directors of 10 NYSE closed-end funds man­ aged by BlackRock Financial Man­ agement and six open-end funds managed by The Guardian. Dr. Fabozzi has edited and authored numerous books on investment management. He recently coau­ thored two books with Professor Franco Modigliani, Capital Mar­ kets· Institutions and Instruments (Prentice Hall) and Mortgage and Mortgage-Backed Securities Mar­ kets (Harvard Business School Press). From 1986 to 1992, he was a full-time professor of finance at MIT's Sloan School of Manage­ ment. A Model for Valuing Bonds and Embedded Options Andrew J. Kalotay, well as stand-alone risk­ bond. The future course of inter­ George 0. Williams and control instruments such as est rates determines when and if the party granted the option is Frank J. Fabozzi swaps, swaptions, caps and likely to alter the cash flows. floors. A second complication is deter­ One can value- a bond by mining the rate at which to dis­ - count the expected cash flows. discounting each of its cash flows at its own zero-cou­ In the good old days, bond valu­ The usual starting point is the pon ("spot'') rate. This pro­ ation was relatively simple. Not yield available on Treasury secu­ only did interest rates exhibit lit­ rities Appropriate spreads must cedure is equivalent to dis­ tle day-to-day volatility, but in the be added to those Treasury yields counting the cash flows at long run they inevitably drifted to reflect additional risks to which a series of one-period for­ up, rather than down. Thus the the investor is exposed. Deter­ ward rates. When a bond ubiquitous call option on long­ mining the appropriate spread is bas one or more embedded term corporate bonds hardly ever not simple, and is beyond the options, however, its cash required the attention of the fi­ scope of this article. nancial manager Those days are flow is uncertain. If a call­ gone. Today, investors face vola­ The ad hoc process for valuing an able bond is called by the tile interest rates, a historically option-free bond (i.e., a bond issuer, for example, its cash with no options) once was to steep yield curve, and complex discount all cash flows at a rate flow will be truncated' bond structures with one or more equal to the yield offered on a embedded options. The frame­ new, full-coupon bond of the work used to value bonds in a To value such a bond, one same maturity. Suppose, for ex­ relatively stable interest rate envi­ must consider the volatility ample, char one needs to value a ronment is inappropriate for val­ of interest rates, as their 10-year, option-free bond. If the uing bonds today. This article sets yield to maturity of an on-the­ volatility will affect the pos­ forth a general model that can be run 10-year bond of comparable sibility of the call option used to value any bond in any credit quality is 8%, then the being exercised One can interest rate environment. do so by constructing a value of the bond under consid­ A Brief History of Bond eration can be taken to be the binomial interest rate tree present value of its cash flows, all that models the random Valuation discounted at 8% evolution offutu,:e interest The value of any bond is the present value of its expected cash According to this approach, the rates. The volatility-depen­ flows This sounds simple Deter­ rate used to discount the cash m dent one-period forward mine the cash flows, and then flows of a 10-year, current-cou- °'O'I rates produced by this tree discount those cash flows at an pon bond would be the same rate ~ can be used to discount the appropriate rate. In practice, it's as that used to discount the cash => cash flows of any bond in not so simple, for two reasons. flow of a 10-year, zero-coupon ~ First, holding aside the possibility bond. Conversely, discounting ~ order to arrive at bond of default, it is not easy to deter­ the cash flows of bonds with dif- ~ value. mine the cash flows for bonds ferent maturities would require ~ with embedded options. Because different discount rates. This ap- Q Given the values of bonds the exercise of options embed­ proach makes little sense because t;; with and without an em­ ded in a bond depends on the it does not consider the cash flow >- <i: bedded option, one can future course of interest rates, the characteristics of the bonds. Con- z obtain the value of the em­ cash flow is a priori uncertain. sider, for example, a portfolio of ::'.; bedded option itself. The The issuer of a callable bond can bonds of similar quality but dif- ~ alter the cash flows to the investor ferent maturities. Imagine two ~ procedure can be used to by calling the bond, while the equal cash flows occurring, say, ~ value multiple or interre­ investor in a puttable bond can five years hence, one coming lated embedded options, as alter the cash flows by putting the from a 30-year bond and the 35 other coming from a 10-year Long a Callable Bond Glossary bond. Why should these two cash flows have different discount = Long an Option-Free Bond ~ Embedded Options: rates, hence different present val­ Options that are part of the ues? + Short a Call Option on the Bond. structure of a bond, as op­ posed to bare options, which Given the drawback of the ad hoc In terms of the value of a callable trade separately from an un­ approach to bond valuation, bond, this means: derlying security. greater recognition has been given to the fact that any bond Value of Callable Bond ~ Forward Rate: The interest rate that will pre­ should be thought of as a package = Value of an Option-Free Bond of cash flows, with each cash flow vail for a specified length of time, starting at some future viewed as a zero-coupon instru­ - Value of a Call Option ment maturing on the date it will date. be received. Thus, rather than on the Bond. using a single discount rate, one ~On-the-Run Yield Curve: But this also means that: The relationship between the should use multiple discount yield-to-maturity and maturity rates, discounting each cash flow for bonds of similar quality at its own rate. Value of an Option-Free Bond trading at par. Value of Callable Bond One difficulty with implementing = ~Option-Free Bond: this approach is that there may + Value of a Call Option A bond that does not have any not exist zero-coupon securities embedded options. from which to derive every dis­ on the Bond. count rate of interest. Even in the .... Spot Rate: absence of zero-coupon securi­ An early procedure to determine The interest rate on a zero­ ties, however, arbirrage argu­ the fairness of a callable bond's coupon instrument. ments can be used to generate market price was to isolate the the theoretical zero-coupon rates implied value of its underlying .... Volatility: an issuer would have to pay were option-free bond by adding an A measure of interest rate un­ it to issue zeros of every maturity. escimate of the embedded call certainty that may loosely be Using these theoretical zero­ option's value to the bond's mar­ regarded as the standard devi­ coupon rates, more popularly re­ ket price.2 The former value ation of interest rates over ferred to as theoretical spot could be estimated by applying one period, normalized by the rates, the theoretical value of a option-pricing theory to interest­ level of interest rates.