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 >-
$100 + $5.25 Year 2: z2 = [(1 + f1)(1 + f2)]1t2 - 1, + ( )3 = $102.075 1.0 4 531 ral, consistem manner. Once the Year 3: z3 = [(1 + f1)(1 + f2) value of a bond with embedded This procedure will ascribe a ( 1 + f3)]1/3 - 1. options and the value of its un value of 100 to each of the three derlying option-free bond are securities used to define the on Substituting in the forward one known, the value of the options the-run yield curve. year rates, we can verify this rela themselves can be determined. tionship. Forward Rates Spot Rates and Forward A forward rate is an interest rate Year 1: 3.5% = 3.5%, Rates on a security that begins to pay The relationship among the interest at some time in the fu 112 yields to maturity on securities ture. For example, a one-year rate Year 2. [(1.035)(1.04523)] - 1 one year forward is an interest selling at par with the same credit = 0.0401 = 4 01 %, quality but different maturities is rate on a one-year investment be called the on-the-run yield ginning one year from today; a curve. The yield curve is typically one-year rate two years forward is Year 3: [(1.035)(1.04523) an interest rate on a one-year constructed using the maturities 113 and observed yields of Treasury investment beginning two years (1 05580)] - 1 = 0 04531 securities. As market participants from today. Forward rates, like do not perceive these securities spot rates, can be derived from = 4.531%. to have any default risk, this gov the yield curve, using arbitrage ernment yield curve reflects the arguments. Consider once again Consequently, discounting cash effect of maturity alone on yield. the hypothetical yield curve used flows at arbitrage-free forward However, a theoretical yield to derive the spot rates. Table II one-year rates is equivalent to curve can be constructed for any gives the arbitrage-free forward discounting at the current spot issuer. one-year rates. rates. This can be verified using the 5.25% coupon bond with Spot Rates Note that the forward one-year three years to maturity. The value The yield curve shows the rela rate for Year 1 is the rate on a of the bond found by discounting tionship between the yield on one-year security issued today, at the arbitrage-free forward one full-coupon securities and matu the forward one-year rate for Year year rates is· rity. What we are interested in is 2 is the rate on a one-year security the relationship between the issued one year from today, and $5.25 $5.25 yield on zero-coupon instru so forth In saying that these rates + ments (i.e., spot rates) and matu are arbitrage-free, we mean that (1.035) (1.035)(1.04523) rity. Pure expectations theory and an investor would be indifferent arbitrage arguments can be used between, say, a sequence of three $100 + $5.25 one-year investments at these +------to determine this relationship, (l.03500)(1.04523)(1.05580) called the spot rate curve.5 rates and a three-year investment at the on-the-run three-year rate, = 102.075. Table I gives, for example, the all else being equal. yield curve (based on current on Spot rates and the arbitrage-free This is the same value as found the-run yields) for annual-pay, forward rates are related. Specifi- earlier. full-coupon par bonds, as well as the spot rate for each year Now Interest Rate Volatility consider an option-free bond Table II Foiward Rates Once we allow for embedded with three years remaining to ma One-Year options, consideration must be turity and a coupon rate of 5.25%. Year Forward Rate given to interest rate volatility. The value of this bond is the This can be done by introducing a 1 3500% present value of the cash flows, 2 4523% binomial interest rate tree. This with each cash flow discounted at 3 5580% tree is nothing more than a dis the corresponding spot rate So crete representation of the possi- 37 Figure A Three-Year Binomial Given our assumption that the Interest Rate Tree two one-year rates one year for ward are equally likely, the rela tionship between the two is sim ply:
where e is the base of the natural logarithm 2 71828 . Suppose, for example, that r 1 0 is 4.074% and a is 10% per year. Then:
2 010 r1,u = 4.074%(e · ) = 4.976%
In the second year, the one-year rate has three possible values, which we denote as follows: = one-year rate two years forward, assum ing rates fall in the first and second years, r2,uu = one-year rate two Today 1 Year 2Years 3 Years years forward, assum ing rates rise in the first and second years; and ble evolution over time of the by r0 . What we have assumed in = one-year rate two one-period rate based on some creating this tree is that the one years forward, assum year rate can take on two possible assumption about interest rate ing rates either rise in volatility. The construction of this values one year from today. In the first year and fall in tree is illustrated below. what follows, we also assume that these two future rate environ the second year or fall Binomial Interest Rate Tree ments are equally likely. For con in the first year and Figure A shows an example of a venience, we refer to the higher rise in the second binomial interest rate tree. In this of the two rates in the next period year. tree, the nodes are spaced one as resulting from a "rise" in rates year apart in time (left to right). and the lower as resulting from a The relationship between r 2 uD Each node N is labeled with one "fall" in rates (although we stress and the other two forward rates is or more subscripts indicating the that the rate has not necessarily as follows. path the one-year rate followed in fallen or risen relative to its last value). The tree we construct will m reaching that node. The one-year O'I be a discrete representation of a O'I rate may move to one of two values in the following year: U lognormal distribution over time and LJ.Jz of future one-year rates with a ::::, represents the higher of these two values and D represents the certain volatility. ~ lower. It is imponam m nme chat ~ <( these rates are not necessarily We use the following notation to For example, if r DD is 4.53% z 2 a:: higher or lower than the one-year describe the tree in the first year. and, once again, a is 10%, then ::::, rate in the preceding year. Thus Let: ~ 2 010 "' N00 simply means that the one r200=4.S3%(e • ) = 5.532% ~ year rate followed the upper path a = assumed volatility of
In calculating the present value of expected future cash flows, the appropriate discount rate is the one-year rate at the node where Today 1 Year 2Years 3 Years we seek the value. Now, there are •r equals forward one-year rate if rate falls. two future values to discount 1 the value if the one-year rate rises over the coming year and the value if it falls. We assume both the notation by letting rr denote year foiward is 4% times 10%, outcomes are equally likely, so the one-year rate t years foiward which equals 0.4%, or 40 basis we can simply average the two if interest rates always decline, points. However, if the current future values. Assuming that the since all the other rates t years one-year rate is 12%, the standard one-year rate is r.. at the node foiward will depend on that rate. deviation of the one-year rate one being valued, and letting Figure B shows the interest rate year foiward would be 12% times tree using this simplified nota 10%, or 120 basis points V = the bond's value at the tion. node in question, Determining the Value at a Vu = the bond's value in one Before we go on to show how to Node year if the one-year rate use this binomial interest rate To find the value of the bond at a rises,
tree to value bonds, let's focus on node, first calculate the bond's V0 = the bond's value m one rn two issues. First, what does the value at the two nodes to the right year if the one-year rate a, volatility parameter u in the ex of the n6de of interest For exam a, 2 falls and pression e rr represent? Second, ple, in Figure B, to determine the C = the coupon payment, UJz how do we find the value of the bond's value at node Nu, values at :::i bond at each node? nodes Nuu and NuD must be de the cash flow at a node is either. ~ termined. (Hold aside for now ~ <( Volatility and Standard how we get these two values, as z er: Deviation we will see, the process involves Vu+ C :::i The standard deviation of the starting from the last year in the ~ one-year rate one year foiward is tree and work,ng backward to get if the one-year rate rises or ~ equal to r u 7 The standard devi the final solution, so these two >- 0 V=98.582 • C=4.00 /Nu rl,U =5.496% V= 99.567 ~ N• C=O r =3.500% NUD C:4.00 0 ·E] / V=99.522 C=4.00 ND ~- r l,D = 4.500% ~-~ NDD~ for the tree, r0 , is simply the cur Step 2: Determine the corre rent one-year rate, 3.5%. This rate sponding value for the one-year was chosen because it will value a rate one year forward if rates rise. one-year bond with a 3.5% cou As explained earlier, this rate if the one-year rate rises and 2 pon at 100. equals r1e u. Because r1 is 4.5%, the forward one-year rate if rates 2 01 There are two possible one-year rise is 5.496% (= 4.5%e · °). This value is reported in Figure C 1 + r. rates one year forward-one if rates fall and one if rates rise. at node Nu. What we want to find are the two if the one-year rate falls. The Step 3: Compute the bond's value of the bond at the node is forward rates consistent with the value in each of the two interest then found as follows: volatility assumption that result in a value of 100 for a 4% full rate states one year from now, coupon, two-year bond While using the following steps. ~ V = ~ [Vu + C + Vu + cl· one can construct a simple alge a. Determine the bond's value ~ 2 1 + r. 1 + r. braic expression for these iwo two years from now. Since => rates, the formulation becomes we are using a two-year ~ Constructing a Binomial increasingly complex beyond the bond, the bond's value is its ~ Interest Rate Tree first year. Furthermore, because maturity value of $100 plus z V= 99.071 • C=4.00 /Nv T l,U = 4.976% V= 1.00 ~ C=O N• NUD C=4.00 r =3.500% 0 / ·8 V= 99.929 C=4.00 ND ~- r .D 4.074% 1 = ~-~ NDD~ c. Calculate the bond's present Step 5: Compare the value ob Step 1: Select a value of 4.074% value at node N0 . The ap tained m Step 4 with the target for r 1 . propriate discount rate is value of $100. If the two values the forward one-year rate as are the same, then the r 1 used in Step 2: The corresponding value suming rates fall, or 4.5%. this trial is the one we seek. It is for the forward one-year rate The present value is $99 522 the forward one-year rate to be if rates rise is 4.976% 2 01 (= $104/1.045) and is the used in the binomial interest rate (= 4.074%e · °). value of V0 . tree for a decline in rates, as well as the rate to determine the rate Step 3: The bond's value one Step 4: Add the coupon to the year from now is determined cash flows Vu and V computed to be used if rates rise. If, how 0 ever, the value found in Step 4 is from the bond's value two years in Step 3 to get the total cash flow from now-$104, just as in the at Nu and N • respectively. Then not equal to the target value of 0 first trial, the bond's present val- calculate the average of the $100, our assumed value is not present values of rhese two cash consistent with the volatility as ue at node Nu-Vu, or $99.071 (= $104/1.04976); and the bond's flows, using the assumed value of sumption and the yield curve. In present value at node N0 -V0 or LU r , 3.5%, for r*. this case, repeat the five steps 0 $99.929 (= $104/1.04074). s with a different value for r 1 . 0 ~ 1 [Vu + C V+ ~ V=- + Cl Step 4: Add the coupon to both :::r If we use 4.5% for r 1 , a value of • V= 100 ~[§]NDDD C=4.50 + $99.929 + $4.000] year from now of 4.074% and or the forward one-year rate. We 4.976%. The desired value of r2 is should expect to find this result, 1.035 4.53%. Figure E shows the com as our bond is option-free. This pleted binomial interest rate tree. clearly demonstrates that, for an = ($99 586 + $100 414)/2 Using this binomial interest rate option-free bond, the tree-based tree, we can value a one-year, valuation model is consistent with = $100.000 two-year or three-year bond, the standard valuation model. Step 5: Since the average present whether option-free or not. value is equal to the target value Valuing a Callable Bond Using the Binomial Tree The binomial interest rate tree ~ of $100, r 1 is 0.04074 or 4.074%. °' To illustrate how to use the bino can also be applied to callable u.., mial interest rate tree, consider bonds. The valuation process pro z We're not done. Suppose we warn ::::, to "grow" the tree for one more an option-free 5.25% bond with ceeds in the same fashion as in ~ year-that is, we need to deter two years remaining to maturity. the case of an option-free bond, ~ . Assume that the issuer's on-the with one exception: When the call ~ mine r2 We will use a three-year, <( z on-the-run 4.5% coupon bond to run yield curve is the one given in option can be exercised by the a:: ::::, get r2 . The same five steps are Table I so that the appropriate issuer, the bond's value at a node ~ used in an iterative process to binomial interest rate tree is the must be changed to reflect the V, tii find the one-year rates two years one in Figure E. Figure F shows lesser of its value if it is not called ~ forward. Our objective now is to the various·values obtained in the (i.e., the value obtained by apply <( z find the value of r 2 that will pro discounting process. It produces ing the recursive valuation for <( duce a value of $100 for the 4.5% a bond value of $102.075. mula described above) or the call on-the-run bond and that will be price zV <( consistent with (1) a volatility as It is important to note that this z u::: sumption of 10%, (2) a current value is identical to the bond Consider a 5.25% bond with forward one-year rate of 3.5% and value found earlier by discount three years remaining to maturity 42 (3) the two forward rates one ing at either the zero-coupon rate that has an embedded call option Figure F Valuing an Option-Free Bond with Three Years to Maturity and a Coupon of 5.25% ·§=100 ;,uu C=5.25 ------, • V=98.588 N C=5.25 uu ;I( ',==6.151% "· ~ • V= 99.461 Nu C=5.25 r1,u=4.976% Jf ,------,JfN UUD L--~·=·~~~-J V= 102.075 V= 99.732 • C=O N • C=5.25 N UD 2,UD r0 = 3.500% r = 5.532% /I ~ ~ V= 101.333 C =5.25 ND• r l,D = 4.074% ;u:I ;=~~ I .....------, ~ • V = 100.689 N DD C=5.25 r 2,DD = 4.530% ~-~ NDDD~ exercisable in years one and two age of interrelated embedded $102.075 and the value of the at $100. Figure G shows the value options, such as those found in callable bond is $101.432, so the at each node of the binomial in sinking fund bonds.9 value of the call option 1s $0.643. terest rate tree. The discounting process is identical to that shown Determining the Call Option Transaction Costs in Figure F, except that at two Value When an issuer calls a bond, the nodes, ND and NDD, the values From our earlier discussion of debr is generally refinanced. The from the recursive valuation for the relationships between the issuer will, in general, incur trans mula ($101.002 at ND and value of a callable bond, the value action costs in the refinancing. $100.689 at NDD) exceed the call of a noncallable bond and the er, These costs may be injected into O'I price ($100); they have therefore value of the call option, we know the valuation model by scaling up O'I been struck out and replaced that: the call price appropriately It LU with $100. The value for chis call 3 would be imprudent for an issuer ~ able bond is $101432 Value of Call Option to exercise a call option unless E Note that the technique we have the net-present-value savings real- ;{ Value of Option-Free Bond JUSt illustrated allows the direct = ized by calling the bond repre- ~ sented a substantial fraction of the ~ valuation of a bond with an em - Value of Callable Bond forfeited option value (i.e., unless V, bedded option. While others have t;:; implemented interest rate pro the intrinsic value of the call op ~ <( cesses and valuation models on But we have just seen how to tion dominated its incremental z <( discrete tree structures, they have determine the values of both a time value). Indeed, the option ..J <( generally valued options directly, noncallable bond and a callable exercise rule embodied in the 0 z not as an integral part of the bond. The difference between the valuation model presented here <( 8 z underlying security. The distinc two values is the value of the call assumes that an option will be C: tion becomes even more impor option. In our example, the value exercised only if its incremental tant when confronted wnh a pack- of the noncallable bond 1s time value is zero. 43 Figure G Valuing a Callable Bond with Three Years to Maturity and a Coupon of 5.25%, Callable in Years One and Two at 100 ·[§]=100 :;uuu C=5.25 ..----- • V=~8.388 Nuu C=5.25 V=99.461 ~ ',uu••.1m, ~, ;::'.;'.: I Nu• C=S.25 Jf r1.u = 4.976% ..------, ----..... ,;,, V= 101.432 ~ V:99.732 • C=O N • C:5.25 N UD r0 = 3.500% r2,UD = 5.532% ~ V=100(191.~,1' ~ • C = 5.25 • 8V = 100 N NUDD C = 5.25 D r1.D = 4.074% ~ V = 100 (199.689) .l'f NDD• C=S.25 r 2,DD = 4.530% ------'~8• V= 100 NDDD C=S.25 By including transaction costs in put option exercisable annually puttable bond is $102.075, so the every option-exercise decision, in years one and two at $100 value of the put option is une obtains an adjusted option A'isume that the appropriate bino -$0.448. The negative sign indi value that does not penalize re mial interest rate tree tor this ~ates tht:! issuer has sold the op funding. One must consider, issuer is the one in Figure F. tion or, equivalently, the investor however, what will happen at the Figure H shows the binomial in has purchased the option. final maturity. Debt is rarely ex terest rate tree with the bond tinguished at maturity; it is gener values altered at two nodes CNuu The same general framework can ally refinanced, again with trans and N00) because the bond val be used to value a bond with action expenses. Failure to ues at these two nodes are less multiple or interrelated embed ~ include these costs at maturity than $100, the value at which the ded options. The bond values at °' will penalize the decision to ex bond can be put. The value of this each node are simply altered to zLi.I ercise the call option on a bond puttable bond is $102.523. reflect the exercise of any of the ::, close to maturity options. ~ As the value of a nonputtable ~ Other Embedded Options bond can be expressed as the OAS, Effective Duration and value of. a puttable bond minus 2 The bond valuation framework Effective Convexity ~ pn;sented here can be used to the value of a put option on that Our model determines the theo Q analyze securities such as putta bond: retical value of a bond. For exam ~ ble bonds, options on interest ple, if the observed market price Value of Put Option ~ rate swaps, caps and floors on of the three-year, 5.25% callable ~ floating-rate notes, and the op = Value of Option-Free Bond bond is $101 and the theoretical _, tional accelerated redemption value is $101.432, the bond is ~ granted to an issuer in fulfilling - Value of Puttable Bond. underpriced by $0.432. Bond ~ sinking fund requirements. Let's market participants, however, ~ consider a puttable bond. Sup- In our example, the value of the prefer to think not in dollar pose that a 5.25% bond with three puttable bond is $102.523 and the terms, but in terms of a yield 44 years remaining to maturity has a value of the corresponding non- spread: A "cheap" bond trades at Figure H Valuing a Puttable Bond with Three Years to Maturity and a Coupon of 5.25%, Puttable in Years One and 1\vo at 100 .------.~I ;:;~ I • V= 100 (98.58~ Nuu C=S.25 .If r2,UU = 6. 757% ,------, • V= 100.261 ~~ Nu C=5.25 Jf r i.u = 4.976% NUUD~ .------~ ,------,/ V= 102.523 V = 100 (99.732) • C=O N • C=5.25 N UD r0 = 3.500% r 2,UD = 5.532% .------,;1f ~ V= 101.461 • C=5.25 ND '1.D = 4.074% .....-----,;1 ;:;~ I ~. V = 100.689 N C=S.25 DD r 2,DD = 4.530% ~§• V= 100 NDDD C:5.25 a higher spread and a "rich" bond that a bond fairly priced relative points. Any measure of convexity at a lower spread relative to some to an issuer's yield curve will have should also be derived from basis. an OAS of zero whether the bond changes in the yield curve, rather is option-free or has embedded than the yield to maturity. The market convention has been options. Effective convexity recognizes the to think of a yield spread as the dependence of cash flows on in difference between rhe yield m Investors also want to know the terest rates. maturity on a particular bond and sensitivity of a bond's price to the yield on a Treasury bond or changes in interest rates. Unwar If V(O) is the value of a bond, other benchmark security of com ranted emphasis has been placed including accrued interest, V(d) parable maturity. This is inappro on modified duration, which is a is its value when the tree is priate. however, because yield to measure of the sensitivity of a shifted up by d basis points and maturity is just a conversion of a bond's price, not to changes in V(-A) is its value when the tree dollar price imo a yield, given the interest rates, but to changes in is shifted down by a similar ~ contractual cash flows, and may the bond's yield. It has the further amount Then· :::, have little to do with the underly shortcoming of ignoring any de ~ ing yield curve that determines pendence of cash flows on inter ~ Effective Duration <( price. A step in the right direction est rate levels, hence is unsuited z cc:: is to determine a discounting for bonds with embedded op V( - A) -V(d) :::, spread over the issuer's spot or tions. The correct duration mea =------Q V, forward rate curve. In terms of sure--called effective duration 2V(O)A tii our binomial interest rate tree. quantifies prfce sensitivity to :::; <( one seeks the constant spread small changes in interest rates z and <( that, when added to all the for while simultaneously allowing for -' <( ward rates on the tree, makes the changing cash flows. In terms of 0 Effective Convexity z theoretical value equal to the our binomial interest rate tree, <( z market price. This quantity is price response to changing inter V( - A) - 2V(O) + V(A) u::: called the option-adjusted spread est rates is found by shifting the =------(OAS). It is "option-adjusted" in tree up and down by a few basis 45 After-Tax Valuation cess is carried out so as to mini Contingent Claims," Journal of Fi Certain market participants, par mize the issuer's after-tax cost of nance 41 (1986), pp 1011-29, and ticularly corporations, must value servicing the debt. In this case, R Dattatreya and F J Fabozzi, "A Simplified Model for Valuing Debt bonds on an after-tax basis. The what's good for the goose is also Options, " Journal of Ponfolio Man standard approach is to discount good for the gander The reason agement 15 (1989), pp 64-73 after-tax cash flows at after-tax is that this is a three-player game, 4 For an explanation of these options rates. One begins by multiplying with the third player being the and their valuation using the model rhe entire on-the-run yield curve taxman. presented in this article, see A J by (1 - Tax Rate) Using the Kalotay and G O Williams, "The resulting after-tax yield curve, The Challenge of Valuation and Management of one can determine either a set of Bonds with Sinking Fund Provi Implementation sions," Fmanc1al Analysts JournaL zero-coupon rates or a set of for To transform the basic interest ward one-period rates, which can March/April 1992 rate tree into a practical tool re 5 The method used is called "boot then be used to discount after-tax quires several refinements. For strapping " For an illustration of cash flows. As in the case of pre one thing, the spacing of the node how this is done, see Chapter 1 O in tax valuation, however, this dis lines in the tree must be much Fabozzi, Bond Markets, Analysis and counting methodology does not finer, particularly if American op Strategies, op cit take the volatility of interest rates tions are to be valued. However 6 Note that this tree is said to be re mto account, hence is unable to combman~ so that N uo is equivalent the fine spacing required to valu~ capture the value of any embed to N ou in the second year, and m ded options. short-dated securities becomes the third year N uuo is equivalent to computationally inefficient if one both N uou and Nouu We have sim Using a binomial tree for after-tax seeks to value, say, 30-year bonds. ply selected one label for a node While one can introduce time rather than cluuering up the exhibit valuation would seem to be sim with unnecessary information ple. There is a significant catch, dependent node spacing, caution is required; it is easy to distort the 7 This can be seen by noting that e2a however. If an after-tax binomial = 1 + 2u Then the standard devia ir:iterest rate tree is constructed by term structure of volatility. Other tion offorward one-period rates is simply multiplying each one practical difficulties include the management of cash flows that 2 period rate on a pretax tree by (1 re " - r r + 2ur - r - Tax Rate). the after-tax ex fall between two node lines. ----=ur pected cost of an option-free 2 2 bond is dependem on volatility. If 8 See. for example. Black, Derman and Toy, "A One-Factor Model of one constructs an interest rate Footnotes - Interest Rares and its Application to tree based on an after-tax yield 1 For an illustration of coupon strip ping and why the theoretical pnce of Treasury Bond Options, " op cit Like curve-so that the after-tax dis a Treasury bond will be equal to the ours, their model is a onefactor (the counted present value of the af present value of the cash flows dis short-term interest rate) process rep ter-tax cash flows of a bond is counted at the theoretical spot rates, resented on a binomial tree How independent of volatility-the im see Chapter 10 in F J Fabozzi, Bond ever, theirs takes as inputs a zero plied underlying pretax interest Markets, Analysis and Strategies (En coupon yield curve and its volatility rate process is different from that glewood Cliffs, NJ Prentzce Hall, term structure The stochastic differ ential equations underlying the two obtained directly from the pretax 1993) interest rate processes are also differ yield curve. 2 See, for example, M. L Dunetz and J Mahoney, "Using Duration and Con ent 9 See Kalotay and Williams, "The Valu There is another interesting im vexity in the Analysis of Callable Bonds, " Financial Analysts JournaL ation and Management of Bonds plication of the after-tax valuation with Sinking Fund Provisions, " op MayJune 1988, and F J Fabozzi, of option-bearing bonds. Con Fixed Income Mathematics (Chicago Clt sider for a moment the pretax Probus Publishing, 1988) valuation of a callable bond. The 3 See F Black, E Derman and W Toy, valuation process described here "A One-Factor Model of Interest is essentially an optimization pro Rates and its Application to Treasury cedure designed to produce the Bond Options," Financial Analysts lowest possible value for an in Journal, January/February 1990, J VI Hull and A White, "Valuing Denva t;:; vestor or, equivalently, the lowest tive Securities Using the Explicit Fi ~ possible expected cost of servic <( nite Difference Method," Journal of z ing the outstanding debt for the <( Financial and Quantitative Analysis -' issuer. It follows, then, that any <( 25 (1990), pp 87-100, J Hull and 0 other set of option-exercise deci z A White, "Pncing Interest-rate Denv <( sions can only increase the value ative Securities, " Review of Fmanetal z u:: of the bond to the investor (who, Studies 3 (1990), pp 573-92, T S Y we assume, is nm a raxpayer). But Ho and S B Lee, "Term Structure 46 suppose that the valuation pro- Movements and Pncing Interest Rate