valuing convertible bonds: a new approach · valuation framework is to decompose a convertible bond...
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Valuing Convertible Bonds: A New Approach
John D. Finnerty, PhD, and Mengyi Tu
John D. Finnerty is a Professor of Finance at Fordham University’s Gabelli School of Business and an AcademicAffiliate of AlixPartners LLP. Mengyi Tu is an Associate of AlixPartners LLP.
A recent paper by Finnerty expresses the value of a convertible bond as the value of
the straight bond component plus the value of the option to exchange the bond
component for a specified number of conversion shares and develops a closed-form
convertible bond valuation model. This article illustrates how to apply the model to
value nonredeemable convertible bonds and callable convertible bonds. The article
also compares model and market 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
A convertible bond gives the holder an American
option to convert the bond into common stock by
exchanging it for a specified number of common shares
at any time prior to the bond’s redemption. Often, the firm
has an American call option, which it can use to force
conversion before the bondholders voluntarily convert, if
the conversion option is in-the-money, and the bond-
holders may have one or more European put options,
which they can use to force premature redemption. The
interaction of these options with the firm’s default option
requires a contingent claims valuation model to capture
fully a convertible bond’s complex optionality.
A recent article by Finnerty (2015) models an
investor’s option to exchange the straight bond compo-
nent for the conversion shares and develops a closed-form
convertible bond valuation model. It obtains an explicit
expression for the value of the option to exchange the
straight bond for the conversion shares by applying
Margrabe’s (1978) insight into how to value the option to
exchange one asset for another asset. It then expresses the
value of a convertible bond as the value of the straight
bond component plus the value of the exchange option
component.
This article illustrates how to apply the model to value
nonredeemable convertible bonds and callable convert-
ible bonds. It values callable convertibles by modeling the
firm’s option to force early conversion within a stopping
time framework. The exchange option convertible bond
pricing model is simpler to use than the more mathemat-
ically sophisticated partial differential equation (PDE)
models. The article also reports the results of empirical
tests of the model. The overall average median and mean
pricing errors are�0.18% and 0.21%, respectively, which
are within the average bid-ask spread for the convertible
bond sample during the postcrisis period.
Literature Review
The convertible securities literature reflects two main
strands of research. One set of papers (Lewis, Rogalski,
and Seward 1998; Lewis and Verwijmeren 2011; Nyborg
1996) investigates how the traditional convertible bond
structure—straight bond with the option to convert it into
a fixed number of common shares—has been reengi-
neered.1 These papers analyze the firm’s motivation for
developing innovative structures, including the desire to
mitigate the costs of external financing, such as asset
substitution (Green 1984), financial distress and asym-
metric information (Brown et al. 2011; Nyborg 1995;
Stein 1992), risk uncertainty (Brennan and Schwartz
1988), and over-investment (Mayers 1988); to give
conventional bond investors an equity sweetener (Nyborg
1996); and to manage publicly reported earnings (Lewis
and Verwijmeren 2011). The literature has documented a
rich variety of innovative structures (Bhattacharya 2012;
Lewis and Verwijmeren 2011).
Convertible securities have evolved in response to the
capital market’s growing sophistication and improved
analytical capability. The optionality of convertible
securities is attractive to hedge funds, which accounted
1 The conversion price is usually, but not always, fixed. For example,Hillion and Vermaelen (2004) describe a class of floating priceconvertible securities, which smaller high-risk firms have issued.
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for about 80% of the funds invested in convertible
securities in the United States and possibly an even higher
percentage of European convertibles prior to the recent
financial crisis (Bhattacharya 2012; Horne and Dialynas
2012).2 Hedge funds, in particular, develop arbitrage
strategies that are designed to capitalize on the perceived
mispricing of the convertibles’ embedded options.3 This
article concerns the traditional form of callable convert-
ible bond, which again accounts for the majority of new
issuance as hedge funds have become less of an influence
since the 2008–2009 financial crisis (Bhattacharya 2012).
The second main body of convertible securities
research develops and empirically tests convertible
security pricing models. Contingent claims models for
convertible bond pricing first appeared in the 1970s.
Ingersoll (1977) and Brennan and Schwartz (1977)
developed the first such models in the spirit of the
seminal Black-Scholes-Merton (Black and Scholes 1973;
Merton 1973) contingent claims methodology. They
develop PDE models, which specify a stochastic process
for each factor that drives option value, correlations
between processes, and a set of boundary conditions that
embody the assumed option exercise behavior. Ingersoll
(1977) and Brennan and Schwartz (1977) develop single-
factor structural models that extend Merton’s (1974)
corporate bond valuation model to convertible bonds. The
value of the firm’s assets follows geometric Brownian
motion, and the firm’s equity, convertible securities, and
other debt are contingent claims on the value of its assets.
Debt holders face credit risk because they get fully paid
only if the value of the firm’s assets exceeds what they are
owed.
Ingersoll (1977) notes that analytic solutions are not
readily obtainable for callable convertible bonds because
of their complexity. Brennan and Schwartz (1977)
independently considered the valuation of convertible
bonds within the same framework as Ingersoll (1977) and
obtained many of the same results but under more general
conditions. Brennan and Schwartz (1980) extend the
Brennan and Schwartz (1977) model by assuming that the
short-term riskless rate follows a mean-reverting lognor-
mal stochastic process. In both models, the firm might
default on the convertible bond at maturity, in which case
bondholders receive a fixed fraction of the face value. The
resulting PDE model, which includes four boundary
conditions defining the conversion, call, maturity, and
bankruptcy conditions, must be solved numerically. They
demonstrate that under reasonable assumptions about the
interest-rate process, assuming a nonstochastic riskless
rate would introduce errors of less than 4%. McConnell
and Schwartz (1986) extend the Brennan and Schwartz
(1980) model to value what has proven to be a very
popular form of convertible security, zero-coupon
convertible bonds, which provide for a series of
embedded firm call options and investor put options.
Nyborg (1996) compares PDE models and the simple
single-factor lattice model. In practice, the single-factor
binomial lattice model is one of the most widely used
convertible security valuation models (Bhattacharya
2012; Hull 2012). These models take two important
shortcuts. They assume a constant riskless rate, which
ignores interest rate volatility, and a constant credit
spread, which ignores credit spread volatility, to capture
the default risk and model the convertible bond as a
contingent claim on a single factor, the firm’s stock price.
For example, Tsiveriotis and Fernandes (1998) develop a
lattice model that decomposes convertible bond value
into two components. One applies when the conversion
feature is not exercised and the security ends up as debt.
Payments are discounted at the riskless interest rate plus a
credit spread. The other applies when the conversion
option is exercised and the bond winds up converted into
common stock. Payments are discounted at the riskless
interest rate. Ammann, Kind, and Wilde (2003) test this
model on a sample of twenty-one French convertible
bonds and find that it produces values that are on average
more than 3% higher than the observed market prices and
that the overpricing is most severe for out-of-the-money
convertibles.
The lattice approach can handle more than one factor
but simplifying assumptions are required to make the
model manageable. For example, Hung and Wang (2002)
describe a two-factor reduced-form model that extends
the Jarrow and Turnbull (1995) model to convertible
bonds. The model contains a binomial stock price lattice,
an uncorrelated binomial interest-rate lattice, exogenous
time-varying default probabilities, and an exogenous
constant recovery rate. Yet, even with these simplifica-
tions, the tree is complex because each node emits six
branches.
Carayannopoulos and Kalimipalli (2003) extend the
trinomial lattice model of Kobayashi, Nakagawa, and
Takahashi (2001). They empirically test it on a sample of
434 price observations for twenty-five frequently traded
convertible bonds between January 2001 and September
2 The importance of hedge funds in the convertible securities marketdeclined following the financial crisis of 2008–2009, but they stillrepresent about 40% of convertible ownership and 50% to 75% ofconvertible trading volume in the United States (Bhattacharya 2012).3 Convertible arbitrage capitalizes on any perceived mispricing of theembedded call option by hedging the equity risk so as to realize asupernormal return on the bond component. Equity volatility arbitrageseeks to capitalize on any difference between the implied volatilities fora particular stock implicit in the prices of a convertible bond and creditdefault swaps on the same firm’s bonds. Capital structure arbitrage seeksto exploit a perceived inconsistency in either the probability of default orthe expected default recovery implicit in the prices of a convertible bondand other debt of the same firm. See Bhattacharya (2012) and Horne andDialynas (2012) for a more detailed description of these strategies.
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2002. The median percentage difference between the
model prices and the observed market prices is 5.21%
(overpricing) for convertible bonds with approximately
at-the-money exchange options, between 5.07% and
9.09% (overpricing) when the exchange option is out-
of-the-money, and between�8.54% and �9.94% (under-
pricing) when the exchange option is in-the-money.
This article describes a closed-form exchange option
model for valuing a conventional (nonputable) convertible
bond when the riskless interest rate and the firm’s credit
spread and share price are all stochastic, dividends are paid
at a constant continuous rate, the convertible is callable
according to a prespecified call price schedule, and the
discrete bond coupon rate of interest is reexpressed as an
equivalent constant cash flow yield on the value of the
straight bond component of the convertible bond. The
exchange option model is simpler to apply than lattice
models. We report the results of tests of the model’s
pricing accuracy, which find that the average median and
mean pricing errors are within the average bid-ask spread
for convertible bonds during the sample period.
Exchange Option Valuation Model
The section describes how to value a convertible bond
as a straight bond plus the option to exchange the bond
for the underlying shares. Finnerty (2015) provides a
detailed mathematical derivation of the exchange option
convertible bond pricing model. The essential step in the
valuation framework is to decompose a convertible bond
into a straight bond plus the option to exchange the bond
for the conversion shares. The number of common shares
N into which the bond is convertible (the conversion
ratio) is fixed at the time the bond is issued. The share
price on the valuation date T1 is ST1, and the stock pays
dividends at the continuous yield d. The convertible bond
matures at TM. It remains outstanding until it is converted
or redeemed at T2 � TM. We address how to estimate T2
later in the article.
BðrT1; sT1
; T1; TMÞ (denoted BT1) is the price on the
valuation date T1 of a coupon-bearing bond maturing at
TM . T1 when the short-term riskless rate is rT1and the
short-term credit spread is sT1Define c as the equivalent
continuously compounded average annualized constant
cash flow yield on the value of the bond component. B̂(rt,
st, t, TM) ¼ BT1e�cðT2�T1Þ is the present value of the
forward price of the bond component.
The value of the exchange option is
EðrT1; sT1
;BT1; ST1
; T2;TMÞ ¼
NST1e�dðT2�T1ÞH
lC þ r2CðT2 � T1Þ
rC
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiT2 � T1
p� �
� BT1e�cðT2�T1ÞH
lC
rC
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiT2 � T1
p� �
; ð1Þ
where H(�) is the standard normal cdf:
lC ¼ lnNST1
e�dðT2�T1Þ
BT1e�cðT2�T1Þ
� �� r2
CðT2 � T1Þ2
; ð2Þ
r2CðT2 � T1Þ ¼ Var ln
NST2
BT2
� �� �: ð3Þ
Equation (1) is the value of a European option to
exchange one asset for another (Margrabe 1978). The
price ratio volatility, r2C, can be estimated directly from
stock and bond price time series for the issuing firm.
Finnerty (2015) finds that the common stock price
volatility can be used in place of the price ratio volatility
without any appreciable loss of pricing accuracy. We
adopt that simplification in this article.
The value of the convertible bond is
CðrT1; sT1
;BT1; ST1
; T2; TMÞ ¼ BT1ð1� e�cðT2�T1ÞÞ
þ NST1e�dðT2�T1Þ
3 HlC þ r2
CðT2 � T1ÞrC
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiT2 � T1
p� �
þ BT1
�H � lC
rC
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiT2 � T1
p� �
:
ð4Þ
Equation (4) can be applied very simply when the firm
has a publicly traded bond whose features are similar to
those of the straight bond component or if investors can
determine the market price at which the firm could issue
such a bond. Value the exchange option using equation
(1) and add the market price of the straight bond.
Equation (4) assumes the convertible bond pays accrued
interest to the (forced) conversion date T2. Accrued
interest is usually not paid when bonds are converted,
although there has been an increasing tendency in recent
years to pay accrued interest (Bhattacharya 2012). When
the convertible bond is coupon-bearing and the interest
that has accrued since the last interest payment date must
be forfeited when the bond is converted, subtract the
present value of the interest that is expected to be
forfeited.
Valuing Callable Convertible Bonds
Convertible bonds often have call options, which the
issuer can use to force conversion by calling the bonds for
redemption when the bondholders’ conversion option is
in-the-money. The bondholders’ best strategy in that case
is to convert the bonds into stock because converting
yields greater value than turning the bonds in for cash
redemption.
A firm will try to minimize the cost of the convertible
bond by extracting the maximum option time premium
Business Valuation ReviewTM — Fall 2017 Page 87
Valuing Convertible Bonds: A New Approach
from investors. If the firm could redeem bonds instanta-
neously, then in a frictionless market, it would call the
convertible bonds for redemption as soon as the
conversion value reaches the effective call price (the
stated call price plus accrued interest). The intrinsic value
of the conversion option is zero, and the time premium is
a maximum because the option is at-the-money. Howev-
er, market imperfections and agency costs could make
this strategy impractical. Empirical studies have found
that firms often waited until the conversion price
exceeded the effective redemption price by about 20%
before forcing conversion (Asquith 1995; Asquith and
Mullins 1991). A redemption cushion increases the value
of the convertible bond because it delays the forced
conversion and thereby reduces the investors’ loss of time
premium.
Most convertible bonds issued since 2003 are dividend-
protected. For example, Grundy and Verwijmeren (2012)
find that more than 82% of the convertible bonds issued
between 2003 and 2006 were dividend-protected. Table 1
confirms that this predominance has continued through
2013. The conversion price adjusts downward to reflect
fully each cash dividend payment, preserving the value of
the conversion shares against all but a liquidating
dividend. Consequently, a firm will find it optimal to
force conversion as soon as the conversion value first
reaches the effective call price (zero redemption cushion).
Brennan and Schwartz (1977) and Ingersoll (1977) first
suggested this behavior. Convertible issues that lack
dividend protection may still have a redemption cushion
when the firm calls them.
Upon a dividend event, a dividend-protected convert-
ible bond issue automatically has the conversion ratio
adjust according to the following formula:
CR1 ¼ CR0 3Sd
ðSd � divÞ ; ð5Þ
where CR1 is the conversion ratio in effect after the
payment of a dividend of div per share; CR0 is the
conversion ratio in effect prior to the dividend payment;
and Sd is the cum-dividend stock price. This adjustment is
captured in equations (1), (2), and (4) by setting d ¼ 0.
Finnerty (2015) models forced conversion as a
‘‘stopping time’’ problem. The firm calls the bond to
force conversion the first time its share price reaches the
forced conversion barrier, which determines the forced
conversion date. This date is the ‘‘stopping time’’ because
the bondholder will convert the bond (and stop holding it)
on this date.
The bond indenture specifies a schedule of redemption
prices, Rt , which usually step down in equal annual
amounts. Each redemption price is expressed as the face
amount multiplied by one plus the percentage redemption
premium, e.g., 1,050 when the face amount is $1,000 and
the redemption premium is 5%. The following procedure
can be used to find the expected forced conversion date.
The firm calls the bond to force conversion the first time
the firm’s share price reaches the forced conversion
barrier, which is the forced conversion date, denoted T̂2.
We assume that bondholders will expect the firm to call
the convertible bond to force conversion at T̂2. The forced
conversion barrier is equal to the effective redemption
price RbT2
multiplied by a redemption factor K, which is
equal to one plus the issuer-selected percentage redemp-
tion cushion. Once the issuer selects the percentage
redemption cushion, K, together with the schedule of
optional redemption prices and the stated coupon rate on
the bond uniquely determine the position of the forced
conversion barrier at each future date. In the valuation
model, each point on the forced conversion barrier is
present valued at the risky yield yc on the straight bond
component back to T1 to facilitate a contemporaneous
comparison between the expected redemption price and
the expected conversion value at T1 when the bondhold-
ers are assumed to estimate the expected forced
conversion date T̂2. The risky yield yc is the proper
discount rate because the firm’s ability to make a cash
redemption payment on its debt is subject to the same
default risk as any other cash debt payment the firm
makes.
T̂2 can be found iteratively. Initially set T̂2 equal to the
earliest date the bond can be called. Calculate
NST1e�dðbT2�T1Þ. Choose as the initial optional redemption
value RbT2
and calculate KRbT2
e�ycðbT2�T1Þ. If NST1e�dðbT2�T1Þ
is greater than or equal to KRbT2
e�ycðbT2�T1Þ the initial
optional redemption date would be the assumed forced
conversion date T̂2. If NST1e�dðbT2�T1Þ is less than
KRbT2
e�ycðbT2�T1Þ, forced conversion would be unprofit-
able for the firm. Increment T̂2 by 1 and use the next
period’s optional redemption price as RbT2
. Recalculate
KRbT2
e�ycðbT2�T1Þ a n d c o m p a r e NST1e�dðbT2�T1Þ t o
KRbT2
e�ycðbT2�T1Þ. If NST1e�dðbT2�T1Þ is greater than or equal
to KRbT2
e�ycðbT2�T1Þ, the second period’s earliest optional
redemption date would be the assumed forced conver-
sion date T̂2. Otherwise, increment T̂2 by 1 again and
continue the search process. The search procedure
usually finds T̂2 after at most a few steps. Figure 1
illustrates the search procedure.
For example, suppose K¼ 1.0. Assume NST1¼ 780, d
¼ 0 , KR3e�3yc ¼ 980, KR4e�4yc ¼ 880, a n d
KR5e�5yc ¼ 780. At T̂2 ¼ 3 years, NST1, KR3e�3yc(780
, 980). At T̂2¼ 4 years, NST1,KR4e�4yc (780 , 880). At
T̂2¼ 5 years, NST1¼ KR5e�5yc (780¼ 780). Thus, the best
estimate is T̂2 – T1¼5 years. For most issues, T̂2 – T1 will
Page 88 � 2017, American Society of Appraisers
Business Valuation ReviewTM
not exceed 5.0 years, which also marks the upper end of
the range of acceptable ‘‘breakeven’’ values that outright
buyers of convertible bonds (i.e., investors other than
hedge funds) traditionally used in performing convertible
payback calculations (Ritchie 1997).4
The exchange option model (4) includes the difference
between the value of the option to exchange the bond for
the underlying stock and the value of the firm’s option to
redeem the exchange option within a single calculation.
The formula values the forced conversion option (and
prices the convertible bond) using the expected date of
forced conversion, rather than taking the expectation of
the convertible bond price across the different possible
forced conversion dates. The value of the firm’s option to
force early conversion is a convex function of the time to
forced conversion, as illustrated in Figure 2. As a result,
my model tends to understate the value of the forced
conversion option and therefore overstate the convertible
bond’s price. The empirical test results reported later in
this article suggest that any overpricing due to this
convexity effect remaining after calibrating the model to
recent market prices is slight.
Examples
Nonredeemable convertible bond
This section illustrates the application of the exchange
option convertible bond pricing model (4) and tests its
accuracy by comparing the clean model price to the clean
market price for the nonredeemable EMC Corp. (EMC)
1.75% convertible senior notes due December 1, 2011.
These prices are compared on the last trading day of each
month between July 2009 and November 2011. The EMC
notes were issued at par on February 2, 2007, with a
1.75% coupon and a 27.5% conversion premium. They
are convertible into 62.198 shares of EMC common stock
any time before the close of business on the maturity date
making the stated conversion price $16.08 per share. The
true conversion price varies with the value of the bond
component. For example, EMC notes with a 1.75%
coupon and maturing December 1, 2011, would be worth
$1,218.17 on August 31, 2010, which implies a true
conversion price equal to $19.59 per share. The notes
were initially rated BBBþby Standard & Poor’s and were
upgraded to A� on June 16, 2008. The issue had $1,725
million principal amount outstanding, making it relatively
liquid. The issue does not have a sinking fund. EMC
noteholders are dividend-protected, and the exercise of
the conversion option can only be settled in EMC
common stock.
Since the convertible bond is nonredeemable, the bond
and exchange option components are valued assuming
their expected term equals the convertible bond’s
maturity. Table 2 illustrates the step-by-step valuation
process for the EMC 1.75% Convertible Senior Notes as
Figure 1Choosing the forced conversion date T̂2. The figure illustrates an iterative procedure for finding T̂2 and RbT2
.
4 Some convertible security investors perform a payback calculation toassess the relative attractiveness of a direct investment in the firm’scommon stock versus an indirect investment through the purchase of thefirm’s convertible bonds (Bhattacharya 2012). This calculation quantifiesthe trade-off between buying the common stock at a (conversion)premium and recouping this premium over time from the differencebetween the interest on the bonds and any dividends on the underlyingcommon stock. Convertible securities investors who consider thepayback calculation in their convertible securities investment decisionsseem to prefer a payback period that does not exceed the call protectionperiod. Note that exclusive reliance on payback would lead to flawedinvestment decisions for the same reasons it is deficient as a capitalbudgeting decision criterion.
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Valuing Convertible Bonds: A New Approach
Table 1Characteristics of Convertible Bonds Issued between 2003 and 2013
Investment-Grade Notes
Investment-Grade
Total
Noncallable,
Nonputable
Callable,
Nonputable
Putable,
Noncallable
Callable
and Putable
2003
No. of Issues 67 5.97% 49.25% - 44.78%
Total Face Amount ($ 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 � 2017, American Society of Appraisers
Business Valuation ReviewTM
Table 1Extended.
Non-Investment-Grade and Nonrated Notes
Total
Non-Investment-Grade
TotalaNoncallable,
Nonputable
Callable,
Nonputable
Putable,
Noncallable
Callable
and Putable
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
Valuing Convertible Bonds: A New Approach
of August 31, 2010. Panel A provides the assumptions.
We calculate the value of the bond component in Panel B.
If the firm has outstanding a nonconvertible bond of like
maturity and the same terms, or if a similar bond can be
found trading in the capital market, its market price can
be used to value the bond component. If a similar bond
cannot be identified, other market data can be used to
estimate a yield to value the bond component. The
following procedure was employed in the two convertible
bond valuation examples reported in this article. On each
valuation date, linearly interpolate the Treasury constant
maturity5 yields to match the straight bond component’s
time to maturity. Then add the BofA Merrill Lynch U.S.
Corporate Option-Adjusted Spread (OAS)6 for bonds
with the same credit rating as the EMC convertible notes.
The estimated bond value is reported in Panel B.
This simple procedure does not take into account the
idiosyncratic company-specific credit risk of the bond
component. To account for this risk, we also applied an
alternative valuation procedure. We employed a dynamic
calibration technique that uses yield adjustments for the
six previous valuation dates to recalibrate the model
monthly (Finnerty 2015). Specifically, for each calibra-
tion 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
Figure 2The 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
holders are forced to convert at T̂2, 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*.
5 Information concerning the Treasury Constant Maturity Yield isavailable at http://www.federalreserve.gov/releases/h15/current/h15.pdf.
6 The BofA Merrill Lynch OASs are the calculated spreads between acomputed OAS index of all bonds in a given rating category and the spotTreasury curve. An OAS index is constructed using each constituentbond’s OAS weighted by the bond’s market price.
Page 92 � 2017, American Society of Appraisers
Business Valuation ReviewTM
Lynch OAS; and average the six spread adjustments. Add
the average spread adjustment to the valuation date
Treasury yield plus OAS to obtain the discount rate y and
then calculate the value Bt of the straight bond
component:
Bt ¼XJ
j¼1
CFj
1þ y2
� �j ; ð6Þ
where CFj is the interest and principal payment in
semiannual period j and J is the number of periods. We
report the bond valuations obtained using this more
detailed alternative valuation procedure in Panel D of
Table 2.
Next, after valuing the bond component, value the
exchange option using equations (1)–(3). We do this in
Panel C of Table 2. Adjust the current value of the bond
component for the coupon payments expected to be paid
Table 2Illustration of the Valuation Process for the EMC Corp. Nonredeemable 1.75% Convertible Senior Notes Due
December 1, 2011
Panel A. AssumptionsValuation 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 (T2�T1) 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 Bond1 2 3
Implied Volatility
Function
Bloomberg Six-Month
Volatility
Bloomberg Implied
Volatility
Stock Volatility (r) 35.32% 27.72% 30.45%
rz ¼ r(T2�T1)1/2 39.53% 31.03% 34.09%
RatioT1¼ NS(t)/B(t)e�cðT2�T1Þ 1.1503 1.1503 1.1503
lc ¼ ln[RatioT1] � (rz
2/2) 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 Procedure1 2 3
Implied Volatility
Function
Bloomberg Six-Month
Volatility
Bloomberg Implied
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.
Business Valuation ReviewTM — Fall 2017 Page 93
Valuing Convertible Bonds: A New Approach
during its remaining life (the exchange option’s expected
term) to get BT1e�cðT2�T1Þ The value of the underlying
equity is NST1e�dðT2�T1Þwhen cash dividends are paid at
the rate d. The EMC noteholders are dividend-protected,
so the conversion value is unaffected by dividend
payments. Upon a dividend event, a dividend-protected
convertible bond issue automatically has the conversion
ratio adjust according to equation (5). Convertible
noteholders thus receive the value of the cash dividends
EMC will pay between the date of purchase and the date
of conversion.
The price ratio volatility equals the stock price
volatility when the convertible bond is nonredeemable.
Estimate the stock price volatility for each valuation date
by applying the implied volatility function (IVF)
methodology described in Hull and Suo (2002), which
utilizes the volatilities implied by the prices of market-
traded call and put options on a stock to express volatility
as a function of the remaining option term and money-
ness. This IVF is used to infer volatility for the stock
option being valued by extrapolating long-term volatility
from short-term exchange-traded option implied volatil-
ities. It involves fitting a skew function to the implied
volatilities. A typical skew function is of the form7
IVFðx; TÞ ¼ C0x�q þ C1logx*1ffiffiffiTp� �
þ C2 logx½ �2*1
T
� �;
ð7Þ
where x ¼ (strike/forward stock price), T ¼ time to
maturity, and C0, C2, and q are time-dependent
parameters to be estimated for each maturity and C0 is
the at-the-money volatility. The parameters are estimated
by ordinary least squares.
We also performed each valuation using two other
stock price volatility measures obtained from Bloomberg:
the six-month historical stock price volatility and the
implied stock price volatility for a three-month at-the-
money option. Both ignore the exchange option’s
moneyness but have the advantage of being readily
available on a Bloomberg screen.8 We used these
volatility estimates to test whether this simpler approach
could achieve acceptable pricing accuracy.
After valuing the exchange option, we add it to the
value of the bond component to get the model price of the
EMC convertible notes as of August 31, 2010, which we
report in Panel C of Table 2. The pricing error is between
�0.57% and 2.07% depending on the stock price
volatility estimate used. We also valued the exchange
option and the convertible bond in Panel D of Table 2 by
applying the alternative procedure that takes into account
the idiosyncratic company-specific credit risk of the bond
component. It should therefore normally provide a more
accurate valuation, but at the cost of requiring a more
complex calculation. The pricing errors, which are
reported in Panel D, are between 0.03% and 1.08% and
are uniformly lower than the pricing errors for the simple
procedure in Panel C.
We also applied the more detailed valuation procedure
at monthly intervals between July 31, 2009, and
November 30, 2011. Table 3 reports the pricing errors
for the EMC convertible notes when the exchange option
valuation model is recalibrated monthly. The median
pricing error using the IVF methodology is 0.03%, and
the mean pricing error is�0.03%. The two other volatility
estimates lead to slightly larger average pricing errors,
which are still within�1.17%.
Figure 3 illustrates how the clean model price closely
tracks the clean market price between July 31, 2009, and
November 30, 2011. It also breaks down the value of a
EMC note into its bond and exchange option component
values. Figure 4 shows how the value of the EMC
noteholders’ exchange option varies with the implied
volatility and the time to maturity as of July 31, 2009, just
after the financial crisis period ended. The exchange
option is more valuable the higher the implied volatility
and the longer the time to expiration, as option theory
predicts.
Callable convertible bond
This section compares the model price to the actual
market price for the callable (but nonputable) Maxtor
Corporation (STX) 2.375% convertible senior notes due
August 15, 2012. The STX notes were issued at par on
February 1, 2006, with a 2.375% coupon and a 23%
conversion premium. They were initially convertible into
153.1089 shares of STX common stock any time before
the close of business on the maturity date making 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
bond component. For example, STX bonds with a
2.375% coupon and maturing August 15, 2012, would
be worth $1585 on March 31, 2006, which implies a true
conversion price equal to $10.3521 per share. The notes
were initially rated BBþby Standard & Poor’s. The rating
decreased to B on December 12, 2008. The issue had
$326 million principal amount outstanding, which made
7 We obtained this skew function from the convertible bond departmentof a major broker-dealer.8 Bloomberg began reporting an implied volatility surface (IVS) forequity options July 26, 2010. Although this date falls within ourempirical testing period, we did not have a continuous series of IVSvolatilities for the entire period. Finnerty (2015) compares the pricingerrors when using IVF volatilities and Bloomberg IVS volatilities for asubsample of nonredeemable convertible bonds. He finds no statisticallysignificant difference in pricing errors between the valuations based onthe IVF and IVS volatilities.
Page 94 � 2017, American Society of Appraisers
Business Valuation ReviewTM
Figure 3Components 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.
Table 3Pricing 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.
Business Valuation ReviewTM — Fall 2017 Page 95
Valuing Convertible Bonds: A New Approach
it relatively liquid. The issue does not have a sinking
fund. STX noteholders are dividend-protected, and the
exercise of the conversion option can be settled only in
STX common stock.
The STX notes were redeemable beginning on August
20, 2010, at a price equal to 100.68% of the principal
amount plus accrued and unpaid interest to, but not
including, the redemption date. If STX decides to redeem
part or all of the notes, it would have to give the holders at
least 30 days’ but no more than 60 days’ notice. STX
would pay the conversion value unless it fell below the
redemption value. The STX convertible notes are valued
assuming zero redemption cushion, K ¼ 1, and R ¼$1006.80.
We use the stopping time methodology to determine
the notes’ expected forced conversion date T̂2 and
equation (4) to value the notes. Table 4 illustrates the
step-by-step valuation process for the STX 2.375%
convertible senior notes as of March 31, 2010. The
earliest call date is August 20, 2010. Investors expected
STX to force conversion soon after the first call date
(around October 4, 2010). The present value of the
effective redemption price, if forced conversion is
expected in 0.51 years, is $971.88 per bond, which is
less than the dividend-adjusted price of the underlying
STX common stock (d¼ 0), $971.90 per bond. Set T̂2�T1
¼0.51 and RbT2
¼$1006.80 since the STX exchange option
was expected to be in-the-money. Panel A provides the
other assumptions. The value of the straight bond
component using the basic procedure is $902.33, which
is calculated in Panel B. The value of the exchange option
and the value of the convertible bond are calculated in
Panel C using three stock volatility measures. The pricing
errors are between �4.58% and �5.09% and average a
little under 5% underpricing.
Panel D of Table 4 values the STX 2.375% convertible
senior notes using the more detailed alternative proce-
dure, which takes into account the idiosyncratic compa-
ny-specific credit risk of the bond component more
accurately by recalibrating the model monthly. As
Figure 4Value 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.
Page 96 � 2017, American Society of Appraisers
Business Valuation ReviewTM
expected, the more detailed procedure achieves greater
accuracy with pricing errors between �2.16% and
�3.43% and averaging less than 3% underpricing.
Table 5 reports the pricing errors for the STX 2.375%
convertible senior notes when they are valued on the last
trading day of each month between July 2009 and August
2010 and the model is recalibrated monthly. The median
pricing error using the IVF methodology is�0.84%, and
the mean pricing error is�1.22%. The two other volatility
estimates lead to somewhat larger average pricing errors.
Figure 5 compares the clean market price to the clean
model price for the STX convertible notes under two
assumptions: the firm forces conversion optimally and the
firm never forces conversion. The model price assuming
optimal forced conversion closely tracks the market price,
but the model price assuming there will be no forced
Table 4Illustration of the Valuation Process for the Maxtor Corp. Callable 2.375% Convertible Senior Notes Due August 15,
2012
Panel A. AssumptionsValuation 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 ComponentTime 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 Bond1 2 3
Implied Volatility
Function
Bloomberg Six-Month
Volatility
Bloomberg Implied
Volatility
Stock Volatility (r) 48.01% 45.75% 47.82%
rz ¼ r(T2�T1)1/2 34.32% 32.71% 34.18%
RatioT1¼ NS(t)/B(t)e�cðT2�T1Þ 1.0902 1.0902 1.0902
lc ¼ ln[RatioT1]�(rz
2/2) 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 Procedure1 2 3
Implied Volatility
Function
Bloomberg Six-Month
Volatility
Bloomberg Implied
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.
Business Valuation ReviewTM — Fall 2017 Page 97
Valuing Convertible Bonds: A New Approach
Table 5Pricing 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 5Model 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 � 2017, American Society of Appraisers
Business Valuation ReviewTM
conversion significantly overstates the market price for
almost the entire period, which confirms that forced
conversion risk is priced by the market.
Finnerty (2015) describes how the valuation model (4)
can be further extended to value convertible bonds that
are both callable and putable.
Empirical Tests
Finnerty (2015) tested the accuracy of the exchange
option convertible bond pricing model (4) using market
prices for a sample of 148 convertible bond issues.
Month-end TRACE bond prices were obtained from
Bloomberg for the period January 31, 2006, through
January 31, 2014.9 Clean model prices were compared to
observed clean market prices monthly for three subsam-
ples of bonds issued between January 2, 2006, and
December 31, 2010, with a principal amount of at least
$100 million: eighty-seven nonredeemable (noncallable
and nonputable), six callable (but nonputable), and fifty-
five callable and putable convertible bonds.
The model’s accuracy was tested based on month-end
prices between January 2006 and January 2014. Since
this period includes the financial crisis of 2008–2009, the
testing was performed for the full period and for three
subperiods: precrisis (January 2006 through December
2007), crisis period (January 2008 through June 2009),
and postcrisis (July 2009 through January 2014). The
model’s pricing accuracy was also investigated during the
short selling ban period (September 2008 through
February 2009). The short selling restrictions, including
the outright ban on short selling the SEC imposed
between September 19, 2008, and October 8, 2008,
severely disrupted the convertibles market. The SEC’s
short selling restrictions prohibited the short selling that
hedge funds use to hedge their convertible bond
investments making convertible bond investing riskier
and driving some hedge funds from the market. As a
result, the model would be expected to overprice
convertible securities during the short sale ban period.
To value each convertible bond, Finnerty (2015)
valued the straight bond component using the more
detailed valuation procedure described earlier in the
article, applied equations (1)–(3) to value the exchange
option, and summed the two component values. So long
as the convertible is dividend-protected, a zero redemp-
tion cushion was assumed. Finnerty (2015) also assumed
optimal option exercise by firms and investors. Common
stock volatility was used in place of the price ratio
volatility in equations (1)–(4). Stock price volatility was
estimated for each valuation date by applying the IVF
methodology described in Hull and Suo (2002), which
utilizes the volatilities implied by the prices of market-
traded call and put options on a stock to express volatility
as a function of the remaining option term and money-
ness. Each convertible issue was also valued using two
other stock price volatility measures that were obtained
from Bloomberg: the six-month historical stock price
volatility and the implied stock price volatility for a three-
month at-the-money option. Using these volatility
estimates tests whether this simpler approach could
achieve acceptable pricing accuracy.
Table 6 reports the median and mean pricing errors for
the nonredeemable convertible bond subsample in Panel
A, the callable convertible bond subsample in Panel B,
and the callable and putable convertible bond subsample
in Panel C. The simple arithmetic average for each
statistic is reported. The clean model price is calculated
for each convertible outstanding at each month-end. The
pricing error is the difference between the clean model
price and the clean market price divided by the clean
market price.
For the full period and IVF volatility, the average
median (mean) pricing errors are�0.04% (0.22%) for the
nonredeemable convertible bonds; �0.50% (�0.16%) for
callable convertible bonds; �0.38% (0.24%) for callable
and putable convertible bonds; and �0.18% (0.21%)
overall. The Bloomberg six-month and at-the-money
implied volatilities lead to similar average pricing errors.
All the average median and mean pricing errors are within
60.63% during the full period, although they are
somewhat smaller for the nonredeemable convertibles
than for the redeemable convertibles.
The average median and average mean pricing errors
for the nonredeemable convertible bonds are smallest
during the postcrisis period and largest during the short
selling ban period for all three volatility estimation
methods. They are also largest during the short selling
ban period for all three volatility estimation methods for
callable convertibles and callable and putable convert-
ibles. No clear pattern of dominance is evident in the
precrisis, crisis, and postcrisis subperiod pricing errors for
callable convertibles or callable and putable convertibles.
No clear pattern of dominance is evident in the full period
average pricing errors across the three volatility estima-
tion methods and the three samples of convertibles.
9 The sample of convertible bonds has the following characteristics: (a)issued in the United States between January 2, 2006, and December 31,2010; (b) initial issue size of $100 million or greater; (c) convertible intothe bond issuer’s common stock; (d) TRACE prices available at leastweekly between the issue date and the earliest of conversion,redemption, or January 31, 2014; (e) rated by Moody’s InvestorsService or Standard & Poor’s throughout the sample period; and (f) fixedcoupon rate. The sample end date was chosen so as to have at least threeyears of pricing data for each bond to test the model. Characteristics (b)and (d) are designed to exclude relatively illiquid issues; (c) excludesbonds exchangeable either for another firm’s common stock or for abasket of stocks; (e) is needed to value the straight bond component; and(f) is designed to exclude bonds with contingent or floating interest rates.
Business Valuation ReviewTM — Fall 2017 Page 99
Valuing Convertible Bonds: A New Approach
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52
71
33
25
54
71
53
25
53
71
53
25
To
tal
Nu
mb
ero
fO
bse
rvat
ion
s4
55
46
20
11
42
27
92
40
84
66
25
95
12
74
27
93
46
74
60
55
91
12
64
27
51
46
0
Av
g.
Min
.D
iffe
ren
ce(%
)(8
.15
)(1
.67
)(9
.99
)(5
.59
)(6
.39
)(1
0.6
8)
(2.4
5)
(10
.22
)(8
.81
)(3
.47
)(9
.77
)(2
.03
)(1
2.2
8)
(7.0
9)
(7.3
4)
Av
g.
25
thP
erce
nti
le(%
)(0
.65
)0
.05
(0.8
7)
(0.4
4)
(0.3
2)
(0.9
2)
0.0
7(0
.74
)(0
.75
)0
.39
(0.8
3)
0.0
0(1
.11
)(0
.57
)(0
.18
)
Av
g.
Med
ian
Dif
fere
nce
(%)
(0.0
4)
1.3
7(0
.61
)(0
.25
)2
.14
(0.4
8)
2.5
51
.29
(0.5
4)
7.7
5(0
.37
)1
.15
(1.0
3)
(0.2
8)
5.0
6
Av
g.
75
thP
erce
nti
le(%
)0
.86
3.7
2(0
.25
)0
.21
5.7
00
.06
6.4
34
.13
(0.3
9)
18
.30
0.2
93
.21
(0.9
3)
0.2
11
2.4
1
Av
g.
Max
.D
iffe
ren
ce(%
)1
1.3
04
.19
15
.61
5.8
61
2.4
61
6.3
57
.41
21
.57
7.2
41
9.4
11
6.2
13
.68
22
.99
7.7
42
0.9
7
Av
g.
Mea
nD
iffe
ren
ce(%
)0
.22
1.2
30
.45
(0.1
8)
2.5
8(0
.11
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.57
2.9
5(0
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.83
(0.0
1)
1.0
21
.07
(0.2
4)
5.9
1
Pa
nel
B.
Ca
lla
ble
(bu
tN
on
-Pu
tab
le)
Co
nv
erti
ble
Bo
nd
sA
vg
.N
o.
of
Ob
serv
atio
ns
per
Bo
nd
52
18
16
18
46
22
61
71
86
57
21
17
18
6
To
tal
No
.o
fO
bse
rvat
ion
s3
12
10
89
41
10
26
37
21
58
10
41
10
34
34
21
28
10
41
10
34
Av
g.
Min
.D
iffe
ren
ce(%
)(8
.10
)(7
.49
)(1
1.1
2)
(9.5
6)
(2.4
8)
(9.4
8)
(7.1
0)
(7.0
0)
(11
.22
)(4
.70
)(9
.11
)(9
.26
)(1
2.9
3)
(9.4
0)
(5.1
0)
Av
g.
25
thP
erce
nti
le(%
)(3
.02
)(3
.04
)(3
.00
)(3
.23
)(1
.02
)(2
.54
)(2
.78
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.72
)(4
.08
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.94
)(3
.12
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.39
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.58
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Av
g.
Med
ian
Dif
fere
nce
(%)
(0.5
0)
(0.5
6)
(0.3
5)
(0.8
2)
2.9
00
.44
(0.1
0)
1.8
2(2
.30
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.33
(0.6
3)
(1.1
2)
1.0
1(0
.83
)0
.18
Av
g.
75
thP
erce
nti
le(%
)2
.88
2.4
93
.27
2.4
08
.12
4.3
23
.24
6.4
80
.24
5.5
82
.95
1.8
95
.45
2.4
64
.17
Av
g.
Max
.D
iffe
ren
ce(%
)1
1.9
68
.16
13
.97
11
.97
13
.32
12
.32
8.9
81
3.1
61
2.8
81
1.2
31
5.2
41
1.2
51
5.5
31
2.8
51
5.2
7
Av
g.
Mea
nD
iffe
ren
ce(%
)(0
.16
)(0
.30
)0
.32
(0.7
0)
4.8
00
.44
0.1
62
.47
(1.9
7)
2.9
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.31
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(1.1
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3.2
8
Pa
nel
C.
Ca
lla
ble
an
dP
uta
ble
Co
nv
erti
ble
Bo
nd
sA
vg
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o.
of
Ob
serv
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ns
per
Bo
nd
49
51
43
05
50
61
43
05
49
51
43
05
To
tal
No
.o
fO
bse
rvat
ion
s2
68
62
84
74
31
65
92
57
27
74
32
37
87
16
64
28
02
70
82
86
75
41
66
82
63
Av
g.
Min
.D
iffe
ren
ce(%
)(1
5.3
6)
(1.3
1)
(15
.41
)(1
0.4
2)
(6.2
0)
(15
.10
)(2
.72
)(1
2.5
9)
(12
.40
)(4
.00
)(1
4.4
3)
(1.3
6)
(13
.79
)(1
0.3
3)
(5.9
8)
Av
g.
25
thP
erce
nti
le(%
)(2
.82
)(0
.84
)(5
.47
)(2
.74
)(0
.44
)(3
.15
)(1
.36
)(2
.50
)(3
.65
)0
.44
(2.5
7)
(0.6
2)
(4.4
9)
(2.5
7)
(1.2
0)
Av
g.
Med
ian
Dif
fere
nce
(%)
(0.3
8)
0.6
4(0
.15
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.58
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.87
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1)
0.8
40
.95
(0.9
4)
8.7
7(0
.27
)0
.64
0.0
2(0
.60
)7
.34
Av
g.
75
thP
erce
nti
le(%
)2
.36
3.0
05
.60
1.4
81
7.7
52
.89
3.2
06
.81
1.4
11
8.6
62
.16
2.7
85
.18
1.5
01
5.4
1
Av
g.
Max
.D
iffe
ren
ce(%
)2
6.9
05
.79
26
.26
10
.23
27
.06
26
.57
6.4
22
7.9
78
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27
.49
24
.99
5.8
52
4.0
49
.35
24
.88
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g.
Mea
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iffe
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8)
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11
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41
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No
te.
Th
ista
ble
rep
ort
sth
em
ean
and
med
ian
per
cen
tag
ep
rici
ng
erro
rsfo
rth
eex
chan
ge
op
tio
nco
nv
erti
ble
bo
nd
val
uat
ion
mo
del
for
87
no
nre
dee
mab
leco
nv
erti
ble
bo
nd
sin
Pan
elA
,si
xca
llab
le(b
ut
no
tp
uta
ble
)co
nv
erti
ble
bo
nd
sin
Pan
elB
,an
dfi
fty
-fiv
eca
llab
lean
dp
uta
ble
con
ver
tible
bo
nd
sin
Pan
elC
,in
each
case
acro
ssth
efu
llsa
mp
lep
erio
das
wel
las
fou
rsu
bp
erio
ds.
Fu
lld
eno
tes
the
full
sam
ple
per
iod
,w
hic
hex
ten
ds
fro
mth
eis
sue
dat
eto
the
earl
ier
of
the
bo
nd
’sm
atu
rity
and
Jan
uar
y3
1,
20
14
.P
re-
den
ote
sth
e
pre
cris
isp
erio
den
din
go
nD
ecem
ber
31
,2
00
7;
Du
rin
gd
eno
tes
the
fin
anci
alcr
isis
per
iod
,w
hic
hex
ten
ds
fro
mJa
nu
ary
2,
20
08
,th
rou
gh
Jun
e3
0,
20
09
;P
ost
-d
eno
tes
the
po
stcr
isis
per
iod
,w
hic
hb
egin
sJu
ly1
,2
00
9;
and
Ban
den
ote
sth
esh
ort
sell
ing
ban
per
iod
,w
hic
hex
ten
ds
fro
mS
epte
mb
er1
,2
00
8,to
Feb
ruar
y2
8,2
00
9.P
erce
nta
ge
pri
cin
ger
ror
isca
lcu
late
das
the
dif
fere
nce
bet
wee
nth
ecl
ean
mo
del
pri
cean
dth
ecl
ean
mar
ket
pri
ced
ivid
edb
yth
ecl
ean
mar
ket
pri
ce.T
hre
em
eth
od
olo
gie
sar
eu
sed
toca
lcu
late
the
vo
lati
lity
of
the
exch
ang
eo
pti
on
:(1
)Im
pli
edV
ola
tili
tyF
un
ctio
n,
(2)
Blo
om
ber
gsi
x-m
on
thh
isto
rica
lst
ock
pri
cev
ola
tili
ty,
and
(3)
Blo
om
ber
gth
ree-
mo
nth
at-t
he-
mo
ney
imp
lied
vo
lati
lity
.T
he
firs
tro
wo
fea
chp
anel
rep
ort
sth
eav
erag
en
um
ber
of
ob
serv
atio
ns
per
bo
nd
(ro
un
ded
)ac
ross
all
the
mo
nth
sfo
rw
hic
hth
ere
are
avai
lab
lem
ark
etp
rice
s.T
he
tota
l
nu
mb
ero
fm
on
thly
ob
serv
atio
ns
iseq
ual
toth
eav
erag
en
um
ber
of
ob
serv
atio
ns
per
bo
nd
mu
ltip
lied
by
the
sam
ple
size
.S
ixst
atis
tics
are
calc
ula
ted
for
the
seri
eso
fm
on
thly
pri
ces
for
each
bo
nd
:th
em
inim
um
,2
5th
per
cen
tile
,m
edia
n,
75
thp
erce
nti
le,
max
imu
m,
and
mea
no
fth
em
on
thly
per
cen
tag
eer
rors
.T
he
tab
lere
po
rts
the
aver
age
val
ue
for
each
stat
isti
cfo
rth
eei
gh
ty-s
even
no
nre
dee
mab
leb
on
ds,
the
six
call
able
bo
nd
s,an
dth
efi
fty
-fiv
eca
llab
lean
dp
uta
ble
bo
nd
sfo
rth
efu
llte
stp
erio
dan
dfo
rea
chsu
bp
erio
dan
dfo
rea
ch
of
the
thre
ev
ola
tili
tym
eth
od
olo
gie
s.N
um
ber
wit
hin
par
enth
eses
ind
icat
esa
neg
ativ
ev
alu
e.
Page 100 � 2017, American Society of Appraisers
Business Valuation ReviewTM
Two conclusions emerge from this analysis. The model
is more accurate in valuing nonredeemable and callable
(but not putable) convertibles than in valuing callable and
putable convertibles. Second, although the IVF estimate
generally results in more accurate valuations, using the
readily available Bloomberg implied volatility in place of
the IVF estimate results in little loss of accuracy.
Conclusion
This article describes a new closed-form contingent-
claims model for valuing a convertible bond. The model
values convertibles as the sum of the value of the straight
bond component and the value of the option to exchange
the straight bond for the underlying conversion shares.
The article provides explicit formulas for the value of the
exchange option and the value of the convertible bond. It
also compares market and model prices for a sample of
148 convertible bonds. The average median and mean
pricing errors are�0.18% and 0.21%, respectively, which
are within the average bid-ask spread for convertibles
during the postcrisis period.
The closed-form exchange option pricing model is easy
to use. Calculate the value of the straight bond and add
the value of the exchange option. The model can be
extended to value putable convertibles. Moreover, there is
little loss of pricing accuracy when the firm’s stock price
volatility is used in place of the price ratio volatility. This
finding is consistent with the general valuation practice of
assuming that the interest rate is constant and the firm’s
share price is the only source of volatility.
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Business Valuation ReviewTM