Business Reviewe Volume 36 Number 3 Ó 2017, American Society of Appraisers

Valuing Convertible Bonds: A New Approach

John D. Finnerty, PhD, and Mengyi Tu

John D. Finnerty is a Professor of at Fordham University’s Gabelli School of Business and an Academic Affiliate of AlixPartners LLP. Mengyi Tu is an Associate of AlixPartners LLP.

A recent paper by Finnerty expresses the value of a convertible as the value of the straight bond component plus the value of the to exchange the bond component for a specified number of conversion shares and develops a closed-form convertible model. This article illustrates how to apply the model to value nonredeemable convertible bonds and callable convertible bonds. The article also compares model and prices for a sample of 148 corporate convertible bonds issued between 2006 and 2010. The average median and mean pricing errors are 0.18% and 0.21%, respectively, which are within the average bid-ask spread for convertible bonds during the postcrisis sample period.

Introduction tests of the model. The overall average median and mean pricing errors are 0.18% and 0.21%, respectively, which A convertible bond gives the holder an American are within the average bid-ask spread for the convertible option to convert the bond into common by bond sample during the postcrisis period. exchanging it for a specified number of common shares at any time prior to the bond’s redemption. Often, the firm Literature Review has an American call option, which it can use to force conversion before the bondholders voluntarily convert, if The convertible securities literature reflects two main the conversion option is in-the-money, and the bond- strands of research. One set of papers (Lewis, Rogalski, holders may have one or more European put options, and Seward 1998; Lewis and Verwijmeren 2011; Nyborg which they can use to force premature redemption. The 1996) investigates how the traditional convertible bond interaction of these options with the firm’s default option structure—straight bond with the option to convert it into requires a contingent claims valuation model to capture a fixed number of common shares—has been reengi- fully a convertible bond’s complex optionality. neered.1 These papers analyze the firm’s motivation for A recent article by Finnerty (2015) models an developing innovative structures, including the desire to investor’s option to exchange the straight bond compo- mitigate the costs of external financing, such as asset nent for the conversion shares and develops a closed-form substitution (Green 1984), financial distress and asym- convertible bond valuation model. It obtains an explicit metric information (Brown et al. 2011; Nyborg 1995; expression for the value of the option to exchange the Stein 1992), risk uncertainty (Brennan and Schwartz straight bond for the conversion shares by applying 1988), and over-investment (Mayers 1988); to give Margrabe’s (1978) insight into how to value the option to conventional bond investors an equity sweetener (Nyborg exchange one asset for another asset. It then expresses the 1996); and to manage publicly reported earnings (Lewis value of a convertible bond as the value of the straight and Verwijmeren 2011). The literature has documented a bond component plus the value of the exchange option rich variety of innovative structures (Bhattacharya 2012; component. Lewis and Verwijmeren 2011). This article illustrates how to apply the model to value Convertible securities have evolved in response to the nonredeemable convertible bonds and callable convert- capital market’s growing sophistication and improved ible bonds. It values callable convertibles by modeling the analytical capability. The optionality of convertible firm’s option to force early conversion within a stopping securities is attractive to hedge funds, which accounted time framework. The exchange option convertible bond pricing model is simpler to use than the more mathemat- 1 The conversion price is usually, but not always, fixed. For example, ically sophisticated partial differential equation (PDE) Hillion and Vermaelen (2004) describe a class of floating price models. The article also reports the results of empirical convertible securities, which smaller high-risk firms have issued.

Business Valuation ReviewTM — Fall 2017 Page 85 ReviewTM for about 80% of the funds invested in convertible default on the convertible bond at , in which case securities in the United States and possibly an even higher bondholders receive a fixed fraction of the face value. The percentage of European convertibles prior to the recent resulting PDE model, which includes four boundary financial crisis (Bhattacharya 2012; Horne and Dialynas conditions defining the conversion, call, maturity, and 2012).2 Hedge funds, in particular, develop bankruptcy conditions, must be solved numerically. They strategies that are designed to capitalize on the perceived demonstrate that under reasonable assumptions about the mispricing of the convertibles’ embedded options.3 This -rate process, assuming a nonstochastic riskless article concerns the traditional form of callable convert- rate would introduce errors of less than 4%. McConnell ible bond, which again accounts for the majority of new and Schwartz (1986) extend the Brennan and Schwartz issuance as hedge funds have become less of an influence (1980) model to value what has proven to be a very since the 2008–2009 financial crisis (Bhattacharya 2012). popular form of convertible , zero- The second main body of convertible securities convertible bonds, which provide for a series of research develops and empirically tests convertible embedded firm call options and investor put options. security pricing models. Contingent claims models for Nyborg (1996) compares PDE models and the simple convertible bond pricing first appeared in the 1970s. single-factor . In practice, the single-factor Ingersoll (1977) and Brennan and Schwartz (1977) binomial lattice model is one of the most widely used developed the first such models in the spirit of the valuation models (Bhattacharya seminal Black-Scholes-Merton (Black and Scholes 1973; 2012; Hull 2012). These models take two important Merton 1973) contingent claims methodology. They shortcuts. They assume a constant riskless rate, which develop PDE models, which specify a stochastic process ignores interest rate , and a constant credit for each factor that drives option value, correlations spread, which ignores credit spread volatility, to capture between processes, and a set of boundary conditions that the default risk and model the convertible bond as a embody the assumed option exercise behavior. Ingersoll contingent claim on a single factor, the firm’s stock price. (1977) and Brennan and Schwartz (1977) develop single- For example, Tsiveriotis and Fernandes (1998) develop a factor structural models that extend Merton’s (1974) lattice model that decomposes convertible bond value valuation model to convertible bonds. The into two components. One applies when the conversion value of the firm’s assets follows geometric Brownian feature is not exercised and the security ends up as debt. motion, and the firm’s equity, convertible securities, and Payments are discounted at the riskless interest rate plus a other debt are contingent claims on the value of its assets. credit spread. The other applies when the conversion Debt holders face credit risk because they get fully paid option is exercised and the bond winds up converted into only if the value of the firm’s assets exceeds what they are . Payments are discounted at the riskless owed. interest rate. Ammann, Kind, and Wilde (2003) test this Ingersoll (1977) notes that analytic solutions are not model on a sample of twenty-one French convertible readily obtainable for callable convertible bonds because bonds and find that it produces values that are on average of their complexity. Brennan and Schwartz (1977) more than 3% higher than the observed market prices and independently considered the valuation of convertible that the overpricing is most severe for out-of-the-money bonds within the same framework as Ingersoll (1977) and convertibles. obtained many of the same results but under more general The lattice approach can handle more than one factor conditions. Brennan and Schwartz (1980) extend the but simplifying assumptions are required to make the Brennan and Schwartz (1977) model by assuming that the model manageable. For example, Hung and Wang (2002) -term riskless rate follows a mean-reverting lognor- describe a two-factor reduced-form model that extends mal stochastic process. In both models, the firm might the Jarrow and Turnbull (1995) model to convertible bonds. The model contains a binomial stock price lattice, 2 The importance of hedge funds in the convertible securities market an uncorrelated binomial interest-rate lattice, exogenous declined following the financial crisis of 2008–2009, but they still represent about 40% of convertible ownership and 50% to 75% of time-varying default probabilities, and an exogenous convertible trading volume in the United States (Bhattacharya 2012). constant recovery rate. Yet, even with these simplifica- 3 capitalizes on any perceived mispricing of the embedded call option by hedging the equity risk so as to realize a tions, the tree is complex because each node emits six supernormal return on the bond component. Equity volatility arbitrage branches. seeks to capitalize on any difference between the implied volatilities for Carayannopoulos and Kalimipalli (2003) extend the a particular stock implicit in the prices of a convertible bond and credit default swaps on the same firm’s bonds. arbitrage seeks trinomial lattice model of Kobayashi, Nakagawa, and to exploit a perceived inconsistency in either the probability of default or Takahashi (2001). They empirically test it on a sample of the expected default recovery implicit in the prices of a convertible bond and other debt of the same firm. See Bhattacharya (2012) and Horne and 434 price observations for twenty-five frequently traded Dialynas (2012) for a more detailed description of these strategies. convertible bonds between January 2001 and September

Page 86 Ó 2017, American Society of Appraisers Valuing Convertible Bonds: A New Approach

2002. The median percentage difference between the where H() is the standard normal cdf: model prices and the observed market prices is 5.21%  NS edðT2T1Þ r2 ðT T Þ (overpricing) for convertible bonds with approximately l ¼ ln T1 C 2 1 ; ð2Þ C cðT2T1Þ at-the-money exchange options, between 5.07% and BT1 e 2 9.09% (overpricing) when the exchange option is out-  of-the-money, and between 8.54% and 9.94% (under- 2 NST2 rCðT2 T1Þ¼Var ln : ð3Þ pricing) when the exchange option is in-the-money. BT2 This article describes a closed-form exchange option Equation (1) is the value of a European option to model for valuing a conventional (nonputable) convertible exchange one asset for another (Margrabe 1978). The bond when the riskless interest rate and the firm’s credit price ratio volatility, r2 , can be estimated directly from spread and price are all stochastic, dividends are paid C at a constant continuous rate, the convertible is callable stock and bond price time series for the issuing firm. according to a prespecified call price schedule, and the Finnerty (2015) finds that the common stock price discrete bond coupon rate of interest is reexpressed as an volatility can be used in place of the price ratio volatility equivalent constant cash flow on the value of the without any appreciable loss of pricing accuracy. We straight bond component of the convertible bond. The adopt that simplification in this article. exchange option model is simpler to apply than lattice The value of the convertible bond is models. We report the results of tests of the model’s cðT2T1Þ CðrT ; sT ; BT ; ST ; T2; TMÞ¼BT ð1 e Þ pricing accuracy, which find that the average median and 1 1 1 1 1 þ NS edðT2T1Þ mean pricing errors are within the average bid-ask spread T1 l þ r2 ðT T Þ for convertible bonds during the sample period. 3 H C pCffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 1 rC T2 T1 Ù Exchange Option Valuation Model ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffilC þ BT1 H p : rC T2 T1 The section describes how to value a convertible bond ð4Þ as a straight bond plus the option to exchange the bond for the underlying shares. Finnerty (2015) provides a Equation (4) can be applied very simply when the firm

detailed mathematical derivation of the exchange option has a publicly traded bond whose features are similar to convertible bond pricing model. The essential step in the those of the straight bond component or if investors can valuation framework is to decompose a convertible bond determine the market price at which the firm could issue into a straight bond plus the option to exchange the bond such a bond. Value the exchange option using equation for the conversion shares. The number of common shares (1) and add the market price of the straight bond. N into which the bond is convertible (the conversion Equation (4) assumes the convertible bond pays accrued ratio) is fixed at the time the bond is issued. The share interest to the (forced) conversion date T2. Accrued price on the valuation date T1 is ST1 , and the stock pays interest is usually not paid when bonds are converted, dividends at the continuous yield d. The convertible bond although there has been an increasing tendency in recent matures at TM. It remains outstanding until it is converted years to pay accrued interest (Bhattacharya 2012). When or redeemed at T2 TM. We address how to estimate T2 the convertible bond is coupon-bearing and the interest later in the article. that has accrued since the last interest payment date must ; ; ; BðrT1 sT1 T1 TMÞ (denoted BT1 ) is the price on the be forfeited when the bond is converted, subtract the valuation date T1 of a coupon-bearing bond maturing at present value of the interest that is expected to be T . T when the short-term riskless rate is r and the M 1 T1 forfeited. short-term credit spread is sT1 Define c as the equivalent continuously compounded average annualized constant Valuing Callable Convertible Bonds ˆ cash flow yield on the value of the bond component. B(rt, cðT2T1Þ st, t, TM) ¼ BT1 e is the present value of the Convertible bonds often have call options, which the forward price of the bond component. issuer can use to force conversion by calling the bonds for The value of the exchange option is redemption when the bondholders’ conversion option is in-the-money. The bondholders’ best strategy in that case Eðr ; s ; B ; S ; T ; T Þ¼ T1 T1 T1 T1 2 M 2 is to convert the bonds into stock because converting l þ r ðT2 T1Þ dðT2T1Þ C Cffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi yields greater value than turning the bonds in for cash NST1 e H p rC T2 T1 redemption. l cðT2T1Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiC A firm will try to minimize the cost of the convertible BT1 e H p ; ð1Þ rC T2 T1 bond by extracting the maximum option time premium

Business Valuation ReviewTM — Fall 2017 Page 87 Business Valuation ReviewTM from investors. If the firm could redeem bonds instanta- premium, e.g., 1,050 when the face amount is $1,000 and neously, then in a frictionless market, it would call the the redemption premium is 5%. The following procedure convertible bonds for redemption as soon as the can be used to find the expected forced conversion date. conversion value reaches the effective call price (the The firm calls the bond to force conversion the first time stated call price plus accrued interest). The intrinsic value the firm’s share price reaches the forced conversion ˆ of the conversion option is zero, and the time premium is barrier, which is the forced conversion date, denoted T2. a maximum because the option is at-the-money. Howev- We assume that bondholders will expect the firm to call ˆ er, market imperfections and agency costs could make the convertible bond to force conversion at T2. The forced this strategy impractical. Empirical studies have found conversion barrier is equal to the effective redemption that firms often waited until the conversion price price R multiplied by a redemption factor K, which is bT exceeded the effective redemption price by about 20% equal to2 one plus the issuer-selected percentage redemp- before forcing conversion (Asquith 1995; Asquith and tion cushion. Once the issuer selects the percentage Mullins 1991). A redemption cushion increases the value redemption cushion, K, together with the schedule of of the convertible bond because it delays the forced optional redemption prices and the stated coupon rate on conversion and thereby reduces the investors’ loss of time the bond uniquely determine the position of the forced premium. conversion barrier at each future date. In the valuation Most convertible bonds issued since 2003 are dividend- model, each point on the forced conversion barrier is protected. For example, Grundy and Verwijmeren (2012) present valued at the risky yield yc on the straight bond find that more than 82% of the convertible bonds issued component back to T1 to facilitate a contemporaneous between 2003 and 2006 were dividend-protected. Table 1 comparison between the expected redemption price and confirms that this predominance has continued through the expected conversion value at T1 when the bondhold- 2013. The conversion price adjusts downward to reflect ers are assumed to estimate the expected forced ˆ fully each cash dividend payment, preserving the value of conversion date T2. The risky yield yc is the proper the conversion shares against all but a liquidating discount rate because the firm’s ability to make a cash dividend. Consequently, a firm will find it optimal to redemption payment on its debt is subject to the same force conversion as soon as the conversion value first default risk as any other cash debt payment the firm reaches the effective call price (zero redemption cushion). makes. ˆ ˆ Brennan and Schwartz (1977) and Ingersoll (1977) first T2 can be found iteratively. Initially set T2 equal to the earliest date the bond can be called. Calculate suggested this behavior. Convertible issues that lack b dividend protection may still have a redemption cushion NS edðT2T1Þ. Choose as the initial optional redemption T1 b b when the firm calls them. ycðT2T1Þ dðT2T1Þ value Rb and calculate KRb e .IfNST1 e T2 T2 b Upon a dividend event, a dividend-protected convert- ycðT2T1Þ is greater than or equal to KRb e the initial ible bond issue automatically has the conversion ratio T2 optional redemption date would be the assumed forced adjust according to the following formula: b ˆ dðT2T1Þ conversion date T2.IfNST1 e is less than d b S KR eycðT2T1Þ, forced conversion would be unprofit- CR ¼ CR 3 ; ð5Þ bT 1 0 d 2 ˆ ðS divÞ able for the firm. Increment T2 by 1 and use the next where CR1 is the conversion ratio in effect after the period’s optional redemption price as Rb . Recalculate b T2 b ycðT2T1Þ dðT2T1Þ payment of a dividend of div per share; CR0 is the KRb e and compare NST1 e to T2 b b conversion ratio in effect prior to the dividend payment; ycðT2T1Þ dðT2T1Þ KRb e .IfNST1 e is greater than or equal d T2 b and S is the cum-dividend stock price. This adjustment is ycðT2T1Þ to KRb e , the second period’s earliest optional captured in equations (1), (2), and (4) by setting d 0. T2 ¼ redemption date would be the assumed forced conver- Finnerty (2015) models forced conversion as a sion date Tˆ . Otherwise, increment Tˆ by 1 again and ‘‘stopping time’’ problem. The firm calls the bond to 2 2 force conversion the first time its share price reaches the continue the search process. The search procedure ˆ forced conversion barrier, which determines the forced usually finds T2 after at most a few steps. Figure 1 conversion date. This date is the ‘‘stopping time’’ because illustrates the search procedure. the bondholder will convert the bond (and stop holding it) For example, suppose K ¼ 1.0. Assume NST1 ¼ 780, d 3y 4y on this date. ¼ 0, KR3e c ¼ 980, KR4e c ¼ 880, a n d 5yc ˆ 3yc The bond specifies a schedule of redemption KR5e ¼ 780. At T2 ¼ 3 years, NST1 , KR3e (780 ˆ 4yc prices, Rt , which usually step down in equal annual , 980). At T2 ¼ 4 years, NST1 ,KR4e (780 , 880). At ˆ 5yc amounts. Each redemption price is expressed as the face T2 ¼ 5 years, NST1¼ KR5e (780 ¼ 780). Thus, the best ˆ ˆ amount multiplied by one plus the percentage redemption estimate is T2 – T1 ¼ 5 years. For most issues, T2 – T1 will

Page 88 Ó 2017, American Society of Appraisers Valuing Convertible Bonds: A New Approach

Figure 1 ˆ ˆ Choosing the forced conversion date T2. The figure illustrates an iterative procedure for finding T2 and Rb . T2 not exceed 5.0 years, which also marks the upper end of Examples the range of acceptable ‘‘breakeven’’ values that outright Nonredeemable convertible bond buyers of convertible bonds (i.e., investors other than hedge funds) traditionally used in performing convertible This section illustrates the application of the exchange 4 payback calculations (Ritchie 1997). option convertible bond pricing model (4) and tests its The exchange option model (4) includes the difference accuracy by comparing the clean model price to the clean between the value of the option to exchange the bond for market price for the nonredeemable EMC Corp. (EMC) the underlying stock and the value of the firm’s option to 1.75% convertible senior notes due December 1, 2011. redeem the exchange option within a single calculation. These prices are compared on the last trading day of each The formula values the forced conversion option (and month between July 2009 and November 2011. The EMC prices the convertible bond) using the expected date of notes were issued at par on February 2, 2007, with a forced conversion, rather than taking the expectation of 1.75% coupon and a 27.5% conversion premium. They the convertible bond price across the different possible are convertible into 62.198 shares of EMC common stock forced conversion dates. The value of the firm’s option to any time before the close of business on the maturity date force early conversion is a convex function of the time to making the stated conversion price $16.08 per share. The forced conversion, as illustrated in Figure 2. As a result, true conversion price varies with the value of the bond my model tends to understate the value of the forced component. For example, EMC notes with a 1.75% conversion option and therefore overstate the convertible coupon and maturing December 1, 2011, would be worth bond’s price. The empirical test results reported later in $1,218.17 on August 31, 2010, which implies a true this article suggest that any overpricing due to this conversion price equal to $19.59 per share. The notes convexity effect remaining after calibrating the model to were initially rated BBBþ by Standard & Poor’s and were recent market prices is slight. upgraded to A on June 16, 2008. The issue had $1,725 million principal amount outstanding, making it relatively 4 Some convertible security investors perform a payback calculation to assess the relative attractiveness of a direct investment in the firm’s liquid. The issue does not have a sinking fund. EMC common stock versus an indirect investment through the purchase of the noteholders are dividend-protected, and the exercise of firm’s convertible bonds (Bhattacharya 2012). This calculation quantifies the conversion option can only be settled in EMC the trade-off between buying the common stock at a (conversion) premium and recouping this premium over time from the difference common stock. between the interest on the bonds and any dividends on the underlying Since the convertible bond is nonredeemable, the bond common stock. Convertible securities investors who consider the payback calculation in their convertible securities investment decisions and exchange option components are valued assuming seem to prefer a payback period that does not exceed the call protection their expected term equals the convertible bond’s period. Note that exclusive reliance on payback would lead to flawed investment decisions for the same reasons it is deficient as a capital maturity. Table 2 illustrates the step-by-step valuation budgeting decision criterion. process for the EMC 1.75% Convertible Senior Notes as

Business Valuation ReviewTM — Fall 2017 Page 89 Business Valuation ReviewTM

Table 1 Characteristics of Convertible Bonds Issued between 2003 and 2013

Investment-Grade Notes

Investment-Grade Noncallable, Callable, Putable, Callable Total Nonputable Nonputable Noncallable and Putable

2003 No. of Issues 67 5.97% 49.25% - 44.78% Total Face Amount ($ mil) 19,256,103,500 5.93% 3.81% - 90.26% No. of Dividend-Protected 3 - - - 100.00% 2004 No. of Issues 94 4.26% 54.26% - 41.49% Total Face Amount ($ mil) 18,671,935,500 1.17% 5.47% - 93.36% No. of Dividend-Protected 13 - - - 100.00% 2005 No. of Issues 51 17.65% 56.86% - 25.49% Total Face Amount ($ mil) 12,760,553,400 2.11% 7.20% - 90.69% No. of Dividend-Protected 10 - - - 100.00% 2006 No. of Issues 65 40.00% 20.00% - 40.00% Total Face Amount ($ mil) 33,633,336,000 54.38% 0.87% - 44.75% No. of Dividend-Protected 31 35.48% - - 64.52% 2007 No. of Issues 40 55.00% 10.00% - 35.00% Total Face Amount ($ mil) 28,027,550,000 35.96% 0.50% - 63.54% No. of Dividend-Protected 20 50.00% - - 50.00% 2008 No. of Issues 20 40.00% 30.00% - 30.00%

Total Face Amount ($ mil) 8,971,492,440 40.31% 1.96% - 57.73% No. of Dividend-Protected 10 50.00% - - 50.00% 2009 No. of Issues 15 93.33% - - 6.67% Total Face Amount ($ mil) 8,565,000,000 95.97% - - 4.03% No. of Dividend-Protected 12 91.67% - - 8.33% 2010 No. of Issues 7 85.71% - - 14.29% Total Face Amount ($ mil) 4,949,403,000 90.01% - - 9.99% No. of Dividend-Protected 7 85.71% - - 14.29% 2011 No. of Issues 15 80.00% - 6.67% 13.33% Total Face Amount ($ mil) 5,307,493,000 80.14% - 10.34% 9.51% No. of Dividend-Protected 14 78.57% - 7.14% 14.29% 2012 No. of Issues 10 100.00% - - - Total Face Amount ($ mil) 4,360,000,000 100.00% - - - No. of Dividend-Protected 9 100.00% - - - 2013 No. of Issues 2 100.00% - - - Total Face Amount ($ mil) 321,602,000 100.00% - - - No. of Dividend-Protected 1 100.00% - - - 2003-2013 No. of Issues 386 30.31% 35.23% 0.26% 34.20% Total Face Amount ($ mil) 144,824,468,840 38.13% 2.27% 0.38% 59.22% No. of Dividend-Protected 130 49.23% - 0.77% 50.00% a Includes high-yield and nonrated convertible bonds.

Page 90 Ó 2017, American Society of Appraisers Valuing Convertible Bonds: A New Approach

Table 1 Extended.

Non-Investment-Grade and Nonrated Notes

Non-Investment-Grade Noncallable, Callable, Putable, Callable Totala Nonputable Nonputable Noncallable and Putable Total

153 34.64% 26.14% - 39.22% 220 31,273,734,700 27.32% 19.29% - 53.39% 50,529,838,200 19 21.05% 10.53% - 68.42% 22

207 15.94% 14.01% 1.45% 68.60% 301 40,013,103,500 13.45% 7.82% 0.97% 77.76% 58,685,039,000 49 16.33% 4.08% 4.08% 75.51% 62

117 25.64% 16.24% 0.85% 57.26% 168 24,049,904,000 22.66% 12.68% 0.34% 64.32% 36,810,457,400 41 17.07% 9.76% - 73.17% 51

111 40.54% 14.41% 1.80% 43.24% 176 28,175,484,400 28.00% 9.02% 0.85% 62.13% 61,808,820,400 63 31.75% 11.11% 3.17% 53.97% 94

163 52.15% 5.52% 0.61% 41.72% 203 46,241,591,000 55.82% 2.32% 0.04% 41.82% 74,269,141,000 94 55.32% 3.19% - 41.49% 114

82 60.98% 12.20% 2.44% 24.39% 102

17,647,155,150 65.22% 5.12% 1.93% 27.73% 26,618,647,590 44 56.82% 4.55% - 38.64% 54

98 71.43% 10.20% 2.04% 16.33% 113 22,225,530,740 78.11% 7.53% 2.02% 12.33% 30,790,530,740 82 75.61% 8.54% 2.44% 13.41% 94

50 64.00% 10.00% 4.00% 22.00% 57 12,839,856,000 70.89% 10.82% 2.06% 16.23% 17,789,259,000 46 65.22% 8.70% 4.35% 21.74% 53

69 78.26% 4.35% 1.45% 15.94% 84 12,476,360,000 75.12% 4.12% 0.80% 19.96% 17,783,853,000 58 77.59% 5.17% 1.72% 15.52% 72

62 77.42% 4.84% - 17.74% 72 15,475,871,500 72.18% 3.53% - 24.29% 19,835,871,500 59 77.97% 5.08% - 16.95% 68

52 69.23% 17.31% 1.92% 11.54% 54 12,909,916,000 76.08% 7.94% 0.02% 15.96% 13,231,518,000 39 84.62% 5.13% - 10.26% 40

1,164 46.05% 13.14% 1.29% 39.52% 1,550 263,328,506,990 46.11% 8.31% 0.72% 44.87% 408,152,975,830 594 55.89% 6.57% 1.52% 36.03% 724

Business Valuation ReviewTM — Fall 2017 Page 91 Business Valuation ReviewTM

Figure 2 The value of a convertible bond and the firm’s option to force conversion. The voluntary conversion date that maximizes the value of the convertible bond is T2*, assuming the firm cannot force conversion prior to that date. Convertible security ˆ holders are forced to convert at T2, which results in a loss of value equal to the difference between the convertible security’s value assuming optimal voluntary conversion and its value with forced early conversion. The value of the firm’s option to force conversion, which is measured by the investors’ loss of value resulting from forced conversion, declines as the exchange option seasons and reaches zero at T2*. of August 31, 2010. Panel A provides the assumptions. Corporate Option-Adjusted Spread (OAS)6 for bonds We calculate the value of the bond component in Panel B. with the same credit rating as the EMC convertible notes. If the firm has outstanding a nonconvertible bond of like The estimated bond value is reported in Panel B. maturity and the same terms, or if a similar bond can be This simple procedure does not take into account the found trading in the capital market, its market price can idiosyncratic company-specific credit risk of the bond be used to value the bond component. If a similar bond component. To account for this risk, we also applied an cannot be identified, other market data can be used to alternative valuation procedure. We employed a dynamic estimate a yield to value the bond component. The calibration technique that uses yield adjustments for the following procedure was employed in the two convertible six previous valuation dates to recalibrate the model bond valuation examples reported in this article. On each monthly (Finnerty 2015). Specifically, for each calibra- valuation date, linearly interpolate the Treasury constant 5 tion date, solve for the yield on the bond component that maturity yields to match the straight bond component’s equates the clean model price to the clean market price; time to maturity. Then add the BofA Merrill Lynch U.S. calculate the difference between the computed yield and the sum of the Treasury yield plus the BofA Merrill

6 The BofA Merrill Lynch OASs are the calculated spreads between a computed OAS index of all bonds in a given rating category and the spot 5 Information concerning the Treasury Constant Maturity Yield is Treasury curve. An OAS index is constructed using each constituent available at http://www.federalreserve.gov/releases/h15/current/h15.pdf. bond’s OAS weighted by the bond’s market price.

Page 92 Ó 2017, American Society of Appraisers Valuing Convertible Bonds: A New Approach

Table 2 Illustration of the Valuation Process for the EMC Corp. Nonredeemable 1.75% Convertible Senior Notes Due December 1, 2011

Panel A. Assumptions Valuation Date (T1) August 31, 2010 Coupon rate 1.7500%

Share Price at T1 (ST1 ) $18.02 Price at issue (BT1) $1,000.00 Maturity December 1, 2011 Conversion ratio (N) 62.1980 Dividend-Protected Yes Expected conversion (T2) December 1, 2011 Continuous Dividend Yield (d) 0.0% Credit rating A Panel B. Value of the Bond Component Time to Expected Conversion (T2T1) 1.25 years Interpolated Treasury Constant Maturity Yield 0.31% Option-Adjusted Spread (OAS), A-rated 1.77% Bond Cash Flow Yield (c) 2.08% Value of Straight Bond Component $995.99 Panel C. Value of the Exchange Option and the Convertible Bond 12 3 Implied Volatility Bloomberg Six-Month Bloomberg Implied Function Volatility Volatility Stock Volatility (r) 35.32% 27.72% 30.45% 1/2 rz ¼ r(T2T1) 39.53% 31.03% 34.09% cðT2T1Þ RatioT1 ¼ NS(t)/B(t)e 1.1503 1.1503 1.1503 l r 2 c ¼ ln[RatioT1] ( z /2) 0.0619 0.0919 0.0819 Value of Exchange Option* $247.43 $215.22 $226.68 Value of Convertible Bond† $1,243.43 $1,211.22 $1,222.68 Reported Market Price $1,218.17 $1,218.17 $1,218.17 Pricing Error‡ 2.07% 0.57% 0.37%

Panel D. Value of the Convertible Bond under the Alternative Procedure 12 3 Implied Volatility Bloomberg Six-Month Bloomberg Implied Function Volatility Volatility Average Spread Adjustment§ 3.43% 1.44% 0.49% Bond Yield after Calibration (y) 5.50% 0.64% 2.57% Value of Straight Bond Component $955.21 $1,013.81 $989.97 Value of Exchange Option* $276.13 $204.67 $230.21 Value of Convertible Bond§ $1,231.34 $1,218.48 $1,220.18 Pricing Error‡ 1.08% 0.03% 0.16% * Using equations (1)–(3). † Equals value of straight bond component plus value of exchange option. ‡ Pricing error is calculated as the difference between the clean model price and the reported clean market price divided by the clean market price. § For each date, solve for the yield on the bond component that equates the clean model price to the clean market price; calculate the difference between the computed yield and the sum of the Treasury yield plus the BofA Merrill Lynch OAS; and average the spread adjustments for the six previous valuation dates.

Lynch OAS; and average the six spread adjustments. Add where CFj is the interest and principal payment in the average spread adjustment to the valuation date semiannual period j and J is the number of periods. We Treasury yield plus OAS to obtain the discount rate y and report the bond valuations obtained using this more then calculate the value Bt of the straight bond detailed alternative valuation procedure in Panel D of component: Table 2. Next, after valuing the bond component, value the XJ CF exchange option using equations (1)–(3). We do this in B ¼ ÀÁj ; ð6Þ t y j Panel C of Table 2. Adjust the current value of the bond j¼1 1 þ 2 component for the coupon payments expected to be paid

Business Valuation ReviewTM — Fall 2017 Page 93 Business Valuation ReviewTM during its remaining life (the exchange option’s expected EMC convertible notes as of August 31, 2010, which we cðT2T1Þ term) to get BT1 e The value of the underlying report in Panel C of Table 2. The pricing error is between dðT2T1Þ equity is NST1 e when cash dividends are paid at 0.57% and 2.07% depending on the stock price the rate d. The EMC noteholders are dividend-protected, volatility estimate used. We also valued the exchange so the conversion value is unaffected by dividend option and the convertible bond in Panel D of Table 2 by payments. Upon a dividend event, a dividend-protected applying the alternative procedure that takes into account convertible bond issue automatically has the conversion the idiosyncratic company-specific credit risk of the bond ratio adjust according to equation (5). Convertible component. It should therefore normally provide a more noteholders thus receive the value of the cash dividends accurate valuation, but at the cost of requiring a more EMC will pay between the date of purchase and the date complex calculation. The pricing errors, which are of conversion. reported in Panel D, are between 0.03% and 1.08% and The price ratio volatility equals the stock price are uniformly lower than the pricing errors for the simple volatility when the convertible bond is nonredeemable. procedure in Panel C. Estimate the stock price volatility for each valuation date We also applied the more detailed valuation procedure by applying the implied volatility function (IVF) at monthly intervals between July 31, 2009, and methodology described in Hull and Suo (2002), which November 30, 2011. Table 3 reports the pricing errors utilizes the volatilities implied by the prices of market- for the EMC convertible notes when the exchange option traded call and put options on a stock to express volatility valuation model is recalibrated monthly. The median as a function of the remaining option term and money- pricing error using the IVF methodology is 0.03%, and ness. This IVF is used to infer volatility for the stock the mean pricing error is 0.03%. The two other volatility option being valued by extrapolating long-term volatility estimates lead to slightly larger average pricing errors, from short-term exchange-traded option implied volatil- which are still within 1.17%. ities. It involves fitting a skew function to the implied Figure 3 illustrates how the clean model price closely volatilities. A typical skew function is of the form7 tracks the clean market price between July 31, 2009, and   November 30, 2011. It also breaks down the value of a q 1 2 1 EMC note into its bond and exchange option component IVFðx; TÞ¼C0x þ C1logx* pffiffiffi þ C2½logx * ; T T values. Figure 4 shows how the value of the EMC ð7Þ noteholders’ exchange option varies with the implied volatility and the time to maturity as of July 31, 2009, just where x ¼ (strike/forward stock price), T ¼ time to after the financial crisis period ended. The exchange maturity, and C0, C2,andq are time-dependent option is more valuable the higher the implied volatility parameters to be estimated for each maturity and C0 is and the longer the time to expiration, as option theory the at-the-money volatility. The parameters are estimated predicts. by ordinary least squares. We also performed each valuation using two other Callable convertible bond stock price volatility measures obtained from Bloomberg: the six-month historical stock price volatility and the This section compares the model price to the actual implied stock price volatility for a three-month at-the- market price for the callable (but nonputable) Maxtor money option. Both ignore the exchange option’s Corporation (STX) 2.375% convertible senior notes due moneyness but have the advantage of being readily August 15, 2012. The STX notes were issued at par on available on a Bloomberg screen.8 We used these February 1, 2006, with a 2.375% coupon and a 23% volatility estimates to test whether this simpler approach conversion premium. They were initially convertible into could achieve acceptable pricing accuracy. 153.1089 shares of STX common stock any time before After valuing the exchange option, we add it to the the close of business on the maturity date making the value of the bond component to get the model price of the stated conversion price $6.5313 per share. The conver- sion ratio was 60.2074 shares as of the valuation date. The true conversion price varies with the value of the 7 We obtained this skew function from the convertible bond department of a major broker-dealer. bond component. For example, STX bonds with a 8 Bloomberg began reporting an implied volatility surface (IVS) for 2.375% coupon and maturing August 15, 2012, would equity options July 26, 2010. Although this date falls within our be worth $1585 on March 31, 2006, which implies a true empirical testing period, we did not have a continuous series of IVS volatilities for the entire period. Finnerty (2015) compares the pricing conversion price equal to $10.3521 per share. The notes errors when using IVF volatilities and Bloomberg IVS volatilities for a were initially rated BBþ by Standard & Poor’s. The rating subsample of nonredeemable convertible bonds. He finds no statistically significant difference in pricing errors between the valuations based on decreased to B on December 12, 2008. The issue had the IVF and IVS volatilities. $326 million principal amount outstanding, which made

Page 94 Ó 2017, American Society of Appraisers Valuing Convertible Bonds: A New Approach

Table 3 Pricing Errors for the EMC Corp. Nonredeemable 1.75% Convertible Senior Notes Due December 1, 2011*

Implied Volatility Function Bloomberg Six-Month Volatility Bloomberg Implied Volatility

Number of Observations 28 28 28 Maximum Difference (%) 2.30 1.28 3.18 Minimum Difference (%) 2.21 4.11 3.28 Mean Difference (%) 0.03 1.17 0.75 Mean Absolute Difference (%) 0.58 1.30 1.14 Median Difference (%) 0.03 0.80 0.81 Median Absolute Difference (%) 0.42 0.92 0.85 Standard Deviation (%) 0.84 1.34 1.27 Note. This table uses the exchange option convertible bond pricing model to value the nonredeemable EMC Corp. 1.75% convertible senior notes due December 1, 2011, between July 2009 and November 2011. The model price before calibration is estimated assuming that the bond component of the convertible note has a yield consistent with the Treasury constant maturity yield term structure plus the BofA Merrill Lynch U.S. corporate OAS for bonds with commensurate credit rating on each valuation date. The calibration technique uses a moving estimation window with yields for the preceding six months as the calibration period. Three methodologies are used to calculate the volatility of the exchange option: (1) Implied Volatility Function; (2) Bloomberg six-month historical stock price volatility; and (3) Bloomberg three-month at-the-money implied volatility. * Pricing error is calculated as the difference between the clean model price and the reported clean market price divided by the clean market price.

Figure 3 Components of the value of the nonredeemable EMC Corp. 1.75% convertible senior notes due December 1, 2011. This figure plots the clean market price and the clean model price and plots the value of the straight bond component and the value of the exchange option at monthly intervals between July 2009 and November 2011.

Business Valuation ReviewTM — Fall 2017 Page 95 Business Valuation ReviewTM

Figure 4 Value of bondholders’ exchange option as a function of the implied volatility and the time to expiration of the option. The nonredeemable EMC Corp. 1.75% convertible senior notes due December 1, 2011, are valued and the sensitivities are calculated as of July 31, 2009. it relatively liquid. The issue does not have a sinking STX to force conversion soon after the first call date fund. STX noteholders are dividend-protected, and the (around October 4, 2010). The present value of the exercise of the conversion option can be settled only in effective redemption price, if forced conversion is STX common stock. expected in 0.51 years, is $971.88 per bond, which is The STX notes were redeemable beginning on August less than the dividend-adjusted price of the underlying ˆ 20, 2010, at a price equal to 100.68% of the principal STX common stock (d ¼ 0), $971.90 per bond. Set T2T1 amount plus accrued and unpaid interest to, but not ¼ 0.51 and R ¼ $1006.80 since the STX exchange option bT including, the redemption date. If STX decides to redeem was expected2 to be in-the-money. Panel A provides the part or all of the notes, it would have to give the holders at other assumptions. The value of the straight bond least 30 days’ but no more than 60 days’ notice. STX component using the basic procedure is $902.33, which would pay the conversion value unless it fell below the is calculated in Panel B. The value of the exchange option redemption value. The STX convertible notes are valued and the value of the convertible bond are calculated in assuming zero redemption cushion, K ¼ 1, and R ¼ Panel C using three stock volatility measures. The pricing $1006.80. errors are between 4.58% and 5.09% and average a We use the stopping time methodology to determine little under 5% underpricing. ˆ the notes’ expected forced conversion date T2 and Panel D of Table 4 values the STX 2.375% convertible equation (4) to value the notes. Table 4 illustrates the senior notes using the more detailed alternative proce- step-by-step valuation process for the STX 2.375% dure, which takes into account the idiosyncratic compa- convertible senior notes as of March 31, 2010. The ny-specific credit risk of the bond component more earliest call date is August 20, 2010. Investors expected accurately by recalibrating the model monthly. As

Page 96 Ó 2017, American Society of Appraisers Valuing Convertible Bonds: A New Approach

Table 4 Illustration of the Valuation Process for the Maxtor Corp. Callable 2.375% Convertible Senior Notes Due August 15, 2012

Panel A. Assumptions Valuation Date (T1) March 31, 2010 Coupon rate 2.3750%

Share Price at T1 (ST1) $16.14 Price at issue (BT1) $1,000.00 Maturity August 15, 2012 Conversion ratio (N) 60.2074 Dividend-Protected Yes Expected conversion (T2) October 4, 2010 Continuous Dividend Yield (d) 0.0% Credit rating B Panel B. Value of the Bond Component Time to Expected Conversion (T2 T1) 0.51 years Interpolated Treasury Constant Maturity Yield 1.24% Option-Adjusted Spread (OAS), B-rated 5.67% Bond Cash Flow Yield (c) 6.91% Value of Straight Bond Component $902.33 Panel C. Value of the Exchange Option and the Convertible Bond 12 3 Implied Volatility Bloomberg Six-Month Bloomberg Implied Function Volatility Volatility Stock Volatility (r) 48.01% 45.75% 47.82% 1/2 rz ¼ r(T2T1) 34.32% 32.71% 34.18% cðT2T1Þ RatioT1 ¼ NS(t)/B(t)e 1.0902 1.0902 1.0902 l r 2 c ¼ ln[RatioT1]( z /2) 0.0274 0.0328 0.0279 Value of Exchange Option* $171.09 $165.37 $170.60 Value of Convertible Bond† $1,073.42 $1,067.70 $1,072.93 Reported Market Price $1,125.00 $1,125.00 $1,125.00 Pricing Error‡ 4.58% 5.09% 4.63%

Panel D. Value of the Convertible Bond under the Alternative Procedure 12 3 Implied Volatility Bloomberg Six-Month Bloomberg Implied Function Volatility Volatility Average Spread Adjustment§ 1.55% 1.84% 2.61% Bond Yield after Calibration (y) 5.36% 5.07% 4.30% Value of Straight Bond Component $934.29 $940.51 $957.03 Value of Exchange Option* $154.94 $145.96 $143.69 Value of Convertible Bond† $1,089.23 $1,086.46 $1,100.73 Pricing Error‡ 3.18% 3.43% 2.16% * Using equations (1)–(3). † Equals value of straight bond component plus value of exchange option. ‡ Pricing error is calculated as the difference between the clean model price and the reported clean market price divided by the clean market price. § For each date, solve for the yield on the bond component that equates the clean model price to the clean market price; calculate the difference between the computed yield and the sum of the Treasury yield plus the BofA Merrill Lynch OAS; and average the spread adjustments for the six previous valuation dates.

expected, the more detailed procedure achieves greater the mean pricing error is 1.22%. The two other volatility accuracy with pricing errors between 2.16% and estimates lead to somewhat larger average pricing errors. 3.43% and averaging less than 3% underpricing. Figure 5 compares the clean market price to the clean Table 5 reports the pricing errors for the STX 2.375% model price for the STX convertible notes under two convertible senior notes when they are valued on the last assumptions: the firm forces conversion optimally and the trading day of each month between July 2009 and August firm never forces conversion. The model price assuming 2010 and the model is recalibrated monthly. The median optimal forced conversion closely tracks the market price, pricing error using the IVF methodology is 0.84%, and but the model price assuming there will be no forced

Business Valuation ReviewTM — Fall 2017 Page 97 Business Valuation ReviewTM

Table 5 Pricing Errors for the Maxtor Corp. Callable 2.375% Convertible Senior Notes Due August 15, 2012*

Implied Volatility Function Bloomberg Six-Month Volatility Bloomberg Implied Volatility

Number of Observations 13 13 13 Maximum Difference (%) 9.22 12.89 12.74 Minimum Difference (%) 11.46 15.39 12.03 Mean Difference (%) 1.22 3.54 1.68 Mean Absolute Difference (%) 3.94 6.91 5.47 Median Difference (%) 0.84 3.42 2.16 Median Absolute Difference (%) 3.18 7.71 4.83 Standard Deviation (%) 5.22 7.63 6.71 Note. This table uses the exchange option convertible bond pricing model to value the callable Maxtor Corp. 2.375% convertible senior notes due August 15, 2012, between July 2009 and August 2010. (TRACE stopped providing market price data for this bond after August 2010.) The model price before calibration is estimated assuming that the bond component of the convertible note has a yield consistent with the Treasury constant maturity yield term structure plus the BofA Merrill Lynch U.S. corporate OAS for bonds with commensurate credit rating on each valuation date. The calibration technique uses a moving estimation window with yields for the preceding six months as the calibration period. Three methodologies are used to calculate the volatility of the exchange option: (1) Implied Volatility Function, (2) Bloomberg six-month historical stock price volatility, and (3) Bloomberg three-month at-the-money implied volatility. * Pricing error is calculated as the difference between the clean model price and the reported clean market price divided by the clean market price.

Figure 5 Model price versus market price of the callable and nonputable Maxtor Corp. 2.375% convertible senior notes due August 15, 2012. The figure plots the clean model price and the clean market price at monthly intervals between July 2009 and August 2010. The expected time to forced conversion is determined using the stopping time model. The figure also plots the clean model price of the convertible note assuming no forced conversion.

Page 98 Ó 2017, American Society of Appraisers Valuing Convertible Bonds: A New Approach conversion significantly overstates the market price for article, applied equations (1)–(3) to value the exchange almost the entire period, which confirms that forced option, and summed the two component values. So long conversion risk is priced by the market. as the convertible is dividend-protected, a zero redemp- Finnerty (2015) describes how the valuation model (4) tion cushion was assumed. Finnerty (2015) also assumed can be further extended to value convertible bonds that optimal option exercise by firms and investors. Common are both callable and putable. stock volatility was used in place of the price ratio volatility in equations (1)–(4). Stock price volatility was Empirical Tests estimated for each valuation date by applying the IVF methodology described in Hull and Suo (2002), which Finnerty (2015) tested the accuracy of the exchange utilizes the volatilities implied by the prices of market- option convertible bond pricing model (4) using market traded call and put options on a stock to express volatility prices for a sample of 148 convertible bond issues. as a function of the remaining option term and money- Month-end TRACE bond prices were obtained from ness. Each convertible issue was also valued using two Bloomberg for the period January 31, 2006, through other stock price volatility measures that were obtained January 31, 2014.9 Clean model prices were compared to from Bloomberg: the six-month historical stock price observed clean market prices monthly for three subsam- volatility and the implied stock price volatility for a three- ples of bonds issued between January 2, 2006, and month at-the-money option. Using these volatility December 31, 2010, with a principal amount of at least estimates tests whether this simpler approach could $100 million: eighty-seven nonredeemable (noncallable achieve acceptable pricing accuracy. and nonputable), six callable (but nonputable), and fifty- Table 6 reports the median and mean pricing errors for five callable and putable convertible bonds. the nonredeemable convertible bond subsample in Panel The model’s accuracy was tested based on month-end A, the callable convertible bond subsample in Panel B, prices between January 2006 and January 2014. Since and the callable and putable convertible bond subsample this period includes the financial crisis of 2008–2009, the in Panel C. The simple arithmetic average for each testing was performed for the full period and for three statistic is reported. The clean model price is calculated subperiods: precrisis (January 2006 through December for each convertible outstanding at each month-end. The 2007), crisis period (January 2008 through June 2009), pricing error is the difference between the clean model and postcrisis (July 2009 through January 2014). The price and the clean market price divided by the clean model’s pricing accuracy was also investigated during the market price. short selling ban period (September 2008 through For the full period and IVF volatility, the average February 2009). The short selling restrictions, including median (mean) pricing errors are 0.04% (0.22%) for the the outright ban on short selling the SEC imposed nonredeemable convertible bonds; 0.50% ( 0.16%) for between September 19, 2008, and October 8, 2008, callable convertible bonds; 0.38% (0.24%) for callable severely disrupted the convertibles market. The SEC’s short selling restrictions prohibited the short selling that and putable convertible bonds; and 0.18% (0.21%) hedge funds use to hedge their convertible bond overall. The Bloomberg six-month and at-the-money investments making convertible bond investing riskier implied volatilities lead to similar average pricing errors. and driving some hedge funds from the market. As a All the average median and mean pricing errors are within result, the model would be expected to overprice 60.63% during the full period, although they are convertible securities during the short sale ban period. somewhat smaller for the nonredeemable convertibles To value each convertible bond, Finnerty (2015) than for the redeemable convertibles. valued the straight bond component using the more The average median and average mean pricing errors detailed valuation procedure described earlier in the for the nonredeemable convertible bonds are smallest during the postcrisis period and largest during the short selling ban period for all three volatility estimation 9 The sample of convertible bonds has the following characteristics: (a) issued in the United States between January 2, 2006, and December 31, methods. They are also largest during the short selling 2010; (b) initial issue size of $100 million or greater; (c) convertible into ban period for all three volatility estimation methods for the bond issuer’s common stock; (d) TRACE prices available at least weekly between the issue date and the earliest of conversion, callable convertibles and callable and putable convert- redemption, or January 31, 2014; (e) rated by Moody’s Investors ibles. No clear pattern of dominance is evident in the Service or Standard & Poor’s throughout the sample period; and (f) fixed precrisis, crisis, and postcrisis subperiod pricing errors for coupon rate. The sample end date was chosen so as to have at least three years of pricing data for each bond to test the model. Characteristics (b) callable convertibles or callable and putable convertibles. and (d) are designed to exclude relatively illiquid issues; (c) excludes No clear pattern of dominance is evident in the full period bonds exchangeable either for another firm’s common stock or for a basket of ; (e) is needed to value the straight bond component; and average pricing errors across the three volatility estima- (f) is designed to exclude bonds with contingent or floating interest rates. tion methods and the three samples of convertibles.

Business Valuation ReviewTM — Fall 2017 Page 99

Table 6 uiesVlainReview Valuation Business ae100 Page Tests of the Exchange Option Convertible Bond Pricing Model’s Average Pricing Errors

Implied Volatility Function Bloomberg Six-Month Volatility Bloomberg Implied Volatility

Full Pre- During Post- Ban Full Pre- During Post- Ban Full Pre- During Post- Ban

Panel A. Nonredeemable Convertible Bonds Avg. No. of Observations per Bond 52 7 13 32 5 54 7 15 32 5 53 7 15 32 5 Total Number of Observations 4554 620 1142 2792 408 4662 595 1274 2793 467 4605 591 1264 2751 460 Avg. Min. Difference (%) (8.15) (1.67) (9.99) (5.59) (6.39) (10.68) (2.45) (10.22) (8.81) (3.47) (9.77) (2.03) (12.28) (7.09) (7.34) Avg. 25th Percentile (%) (0.65) 0.05 (0.87) (0.44) (0.32) (0.92) 0.07 (0.74) (0.75) 0.39 (0.83) 0.00 (1.11) (0.57) (0.18)

Avg. Median Difference (%) (0.04) 1.37 (0.61) (0.25) 2.14 (0.48) 2.55 1.29 (0.54) 7.75 (0.37) 1.15 (1.03) (0.28) 5.06 TM Avg. 75th Percentile (%) 0.86 3.72 (0.25) 0.21 5.70 0.06 6.43 4.13 (0.39) 18.30 0.29 3.21 (0.93) 0.21 12.41 Avg. Max. Difference (%) 11.30 4.19 15.61 5.86 12.46 16.35 7.41 21.57 7.24 19.41 16.21 3.68 22.99 7.74 20.97 Avg. Mean Difference (%) 0.22 1.23 0.45 (0.18) 2.58 (0.11) 2.57 2.95 (0.77) 7.83 (0.01) 1.02 1.07 (0.24) 5.91 Panel B. Callable (but Non-Putable) Convertible Bonds Avg. No. of Observations per Bond 52 18 16 18 4 62 26 17 18 6 57 21 17 18 6 Total No. of Observations 312 108 94 110 26 372 158 104 110 34 342 128 104 110 34 Avg. Min. Difference (%) (8.10) (7.49) (11.12) (9.56) (2.48) (9.48) (7.10) (7.00) (11.22) (4.70) (9.11) (9.26) (12.93) (9.40) (5.10) Avg. 25th Percentile (%) (3.02) (3.04) (3.00) (3.23) (1.02) (2.54) (2.78) (1.72) (4.08) (1.94) (3.12) (3.39) (2.30) (3.23) (2.58) Avg. Median Difference (%) (0.50) (0.56) (0.35) (0.82) 2.90 0.44 (0.10) 1.82 (2.30) 1.33 (0.63) (1.12) 1.01 (0.83) 0.18 Avg. 75th Percentile (%) 2.88 2.49 3.27 2.40 8.12 4.32 3.24 6.48 0.24 5.58 2.95 1.89 5.45 2.46 4.17 Avg. Max. Difference (%) 11.96 8.16 13.97 11.97 13.32 12.32 8.98 13.16 12.88 11.23 15.24 11.25 15.53 12.85 15.27 Avg. Mean Difference (%) (0.16) (0.30) 0.32 (0.70) 4.80 0.44 0.16 2.47 (1.97) 2.98 (0.31) (0.59) 0.85 (1.11) 3.28 Panel C. Callable and Putable Convertible Bonds Avg. No. of Observations per Bond 49 5 14 30 5 50 6 14 30 5 49 5 14 30 5 Total No. of Observations 2686 284 743 1659 257 2774 323 787 1664 280 2708 286 754 1668 263

Ó Avg. Min. Difference (%) (15.36) (1.31) (15.41) (10.42) (6.20) (15.10) (2.72) (12.59) (12.40) (4.00) (14.43) (1.36) (13.79) (10.33) (5.98)

07 mrcnSceyo Appraisers of Society American 2017, Avg. 25th Percentile (%) (2.82) (0.84) (5.47) (2.74) (0.44) (3.15) (1.36) (2.50) (3.65) 0.44 (2.57) (0.62) (4.49) (2.57) (1.20) Avg. Median Difference (%) (0.38) 0.64 (0.15) (0.58) 7.87 (0.31) 0.84 0.95 (0.94) 8.77 (0.27) 0.64 0.02 (0.60) 7.34 Avg. 75th Percentile (%) 2.36 3.00 5.60 1.48 17.75 2.89 3.20 6.81 1.41 18.66 2.16 2.78 5.18 1.50 15.41 Avg. Max. Difference (%) 26.90 5.79 26.26 10.23 27.06 26.57 6.42 27.97 8.36 27.49 24.99 5.85 24.04 9.35 24.88 Avg. Mean Difference (%) 0.24 1.44 1.38 (0.48) 9.29 0.36 1.14 3.30 (1.18) 11.34 0.24 1.50 1.41 (0.50) 8.30 Note. This table reports the mean and median percentage pricing errors for the exchange option convertible bond valuation model for 87 nonredeemable convertible bonds in Panel A, six callable (but not putable) convertible bonds in Panel B, and fifty-five callable and putable convertible bonds in Panel C, in each case across the full sample period as well as four subperiods. Full denotes the full sample period, which extends from the issue date to the earlier of the bond’s maturity and January 31, 2014. Pre- denotes the precrisis period ending on December 31, 2007; During denotes the financial crisis period, which extends from January 2, 2008, through June 30, 2009; Post- denotes the postcrisis period, which begins July 1, 2009; and Ban denotes the short selling ban period, which extends from September 1, 2008, to February 28, 2009. Percentage pricing error is calculated as the difference between the clean model price and the clean market price divided by the clean market price. Three methodologies are used to calculate the volatility of the exchange option: (1) Implied Volatility Function, (2) Bloomberg six-month historical stock price volatility, and (3) Bloomberg three-month at-the-money implied volatility. The first row of each panel reports the average number of observations per bond (rounded) across all the months for which there are available market prices. The total number of monthly observations is equal to the average number of observations per bond multiplied by the sample size. Six statistics are calculated for the series of monthly prices for each bond: the minimum, 25th percentile, median, 75th percentile, maximum, and mean of the monthly percentage errors. The table reports the average value for each statistic for the eighty-seven nonredeemable bonds, the six callable bonds, and the fifty-five callable and putable bonds for the full test period and for each subperiod and for each of the three volatility methodologies. Number within parentheses indicates a negative value. Valuing Convertible Bonds: A New Approach

Two conclusions emerge from this analysis. The model for Call and Conversion.’’ Journal of Finance 32 is more accurate in valuing nonredeemable and callable (December):1699–1715. (but not putable) convertibles than in valuing callable and Brennan, Michael J., and Eduardo S. Schwartz. 1980. putable convertibles. Second, although the IVF estimate ‘‘Analyzing Convertible Bonds.’’ Journal of Financial generally results in more accurate valuations, using the and Quantitative Analysis 15 (November):907–929. readily available Bloomberg implied volatility in place of Brennan, Michael J., and Eduardo S. Schwartz. 1988. the IVF estimate results in little loss of accuracy. ‘‘The Case for Convertibles.’’ Journal of Applied 1 (Summer):55–64. Conclusion Brown, Stephen J., Bruce D. Grundy, Craig M. Lewis, This article describes a new closed-form contingent- and Patrick Verwijmeren. 2012. ‘‘Convertibles and claims model for valuing a convertible bond. The model Hedge Funds as Distributors of Equity Exposure.’’ values convertibles as the sum of the value of the straight Review of Financial Studies 25 (October):3077–3112. bond component and the value of the option to exchange Carayannopoulos, Peter, and Madhu Kalimipalli. 2003. the straight bond for the underlying conversion shares. ‘‘Convertible Bond Prices and Inherent Biases.’’ The article provides explicit formulas for the value of the Journal of 13 (December):64–73. exchange option and the value of the convertible bond. It EMC Corporation. 2007. Prospectus for 1.75% Convert- also compares market and model prices for a sample of ible Senior Notes due December 1, 2011. February 2. 148 convertible bonds. The average median and mean Finnerty, John D. 2015. ‘‘Valuing Convertible Bonds and pricing errors are 0.18% and 0.21%, respectively, which the Option to Exchange Bonds for Stock.’’ Journal of are within the average bid-ask spread for convertibles Corporate Finance 31 (April):91–115. during the postcrisis period. The closed-form exchange option pricing model is easy Green, Richard. 1984. ‘‘Investment Incentives, Debt, and to use. Calculate the value of the straight bond and add Warrants.’’ Journal of 13 the value of the exchange option. The model can be (March):115–136. extended to value putable convertibles. Moreover, there is Grundy, Bruce D., and Patrick Verwijmeren. 2012. little loss of pricing accuracy when the firm’s stock price ‘‘Dividend-Protected Convertible Bonds and the Dis- volatility is used in place of the price ratio volatility. This appearance of Call Delay.’’ University of Melbourne finding is consistent with the general valuation practice of working paper, June 23. assuming that the interest rate is constant and the firm’s Hillion, Pierre, and Theo Vermaelen. 2004. ‘‘Death Spiral share price is the only source of volatility. Convertibles.’’ Journal of Financial Economics 71 (February):381–415. References Horne, Jonathan L., and Chris P. Dialynas. 2012. Ammann, Manuel, Axel Kind, and Christian Wilde. ‘‘Convertible Securities and Their Investment Applica- 2003. ‘‘Are Convertible Bonds Underpriced? An tion.’’ In The Handbook of Fixed Income Securities. 8th Analysis of the French Market.’’ Journal of Banking ed. Ed. Frank J. Fabozzi. New York: McGraw-Hill. & Finance 27 (April):635–653. Hull, John C. 2012. Options, Futures, and Other Asquith, Paul. 1995. ‘‘Convertible Bonds Are Not Called Derivatives. 8th ed. Upper Saddle River, NJ: Prentice Late.’’ Journal of Finance 50 (September):1275–1289. Hall. Asquith, Paul, and David W. Mullins, Jr. 1991. Hull, John, and Wulin Suo. 2002. ‘‘A Methodology for ‘‘Convertible Debt: Corporate Call Policy and Volun- Assessing and Its Application to the tary Conversion.’’ Journal of Finance 46 (Septem- Implied Volatility Function Model.’’ Journal of ber):1273–1290. Financial and Quantitative Analysis 37 (June):297– Bhattacharya, Mihir. 2012. ‘‘Convertible Securities: Their 318. Structures, Valuation, and Trading.’’ In The Handbook Hung, Mao-Wei, and Jr-Yan Wang. 2002. ‘‘Pricing of Fixed Income Securities. 8th ed. Ed. Frank J. Convertible Bonds Subject to Default Risk.’’ Journal Fabozzi. New York: McGraw-Hill. of Derivatives 10 (Winter):75–87. Black, Fischer, and Myron Scholes. 1973. ‘‘The Pricing Ingersoll, Jonathan E., Jr. 1977. ‘‘A Contingent-Claims of Options and Corporate Liabilities.’’ Journal of Valuation of Convertible Securities.’’ Journal of Political Economy 81 (May-June):637–659. Financial Economics 4 (May):289–321. Brennan, Michael J., and Eduardo S. Schwartz. 1977. Jarrow, Robert A., and Stuart M. Turnbull. 1995. ‘‘The ‘‘Convertible Bonds: Valuation and Optimal Strategies Pricing and Hedging of Options on Financial Securities

Business Valuation ReviewTM — Fall 2017 Page 101 Business Valuation ReviewTM

Subject to Credit Risk.’’ Journal of Finance 50 McConnell, John J., and Eduardo S. Schwartz. 1986. (March):53–85. ‘‘LYON Taming.’’ Journal of Finance 41 (July):561– Kobayashi, Takao, Naruhisa Nakagawa, and Akihiko 576. Takahashi. 2001. ‘‘Pricing Convertible Bonds with Merton, Robert C. 1973. ‘‘Theory of Rational Option Default Risk.’’ Journal of Fixed Income 11 (Decem- Pricing.’’ Bell Journal of Economics and Management ber):20–29. Science 4 (Spring):141–183. Lewis, Craig M., Richard J. Rogalski, and James K. Merton, Robert C. 1974. ‘‘On the Pricing of Corporate Seward. 1998. ‘‘Agency Problems, Information Asym- Debt: The Risk Structure of Interest Rates.’’ Journal of metries, and Convertible Debt Security Design.’’ Finance 29 (May):449–470. Journal of Financial Intermediation 7 (January):32–59. Nyborg, Kjell G. 1995. ‘‘Convertible Debt as Delayed Equity: Forced versus Voluntary Conversion and the Lewis, Craig M., and Patrick Verwijmeren. 2011. Information Role of Call Policy.’’ Journal of Financial ‘‘Convertible Security Design and Contract Innova- Intermediation 4 (October):358–395. tion.’’ Journal of Corporate Finance 17 (Septem- Nyborg, Kjell G. 1996. ‘‘The Use and Pricing of ber):809–831. Convertible Bonds.’’ Applied 3 Margrabe, William. 1978. ‘‘The Value of an Option to (September):167–190. Exchange One Asset for Another.’’ Journal of Finance Ritchie, John C., Jr. 1997. ‘‘Convertible Securities.’’ In 33 (March):177–186. The Handbook of Fixed Income Securities. 5th ed. Ed. Maxtor Corporation. 2006. Prospectus for 2.375% Frank J. Fabozzi. Chicago: Irwin. Convertible Senior Notes due August 15, 2012. Stein, Jeremy C. 1992. ‘‘Convertible Bonds as Backdoor February 1. Equity Financing.’’ Journal of Financial Economics 32 Mayers, David. 1988. ‘‘Why Firms Issue Convertible (August):3–22. Bonds: The Matching of Financial and Real Investment Tsiveriotis, Kostas, and Chris Fernandes. 1998. ‘‘Valuing Options.’’ Journal of Financial Economics 47 (Janu- Convertible Bonds with Credit Risk.’’ Journal of Fixed ary):83–102. Income 8 (September):95–102.

Page 102 Ó 2017, American Society of Appraisers