Pricing securities with multiple risks: A case of exchangeable debt

Journal of Banking & Finance 37 (2013) 1018–1028

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Pricing securities with multiple risks: A case of exchangeable debt Ravi S. Mateti a,⇑, Shantaram P. Hegde b, Tribhuvan Puri c a

Concordia University, 1455 De Maisonneuve Blvd. West, Montreal, Canada H3G 1M8 University of Connecticut, 2100 Hillside Road Unit 1041, Storrs, CT 06269-1041, United States c University of Massachusetts Dartmouth, North Dartmouth, MA 02747-2300, United States b

a r t i c l e

i n f o

Article history: Received 20 July 2011 Accepted 7 November 2012 Available online 20 November 2012 JEL classification: G12 G13 Keywords: Exchangeable bonds Trivariate recombining lattice Risk-neutral setting

a b s t r a c t Building on the work of Das and Sundaram (2007), we develop a widely applicable model to price securities subject to interest rate, equity, and default risks and use it to price exchangeable bonds. The extension features a trivariate recombining lattice instead of the original model’s bivariate recombining lattice. We also show how to estimate some critical non-observable inputs to implement the model by using current market data so that the model’s prices reflect current market information. We test the model on a sample of exchangeable bonds to determine the model’s empirical performance. Besides exchangeable bonds, we can also use the model to price securities such as reverse exchangeable bonds, bonds exchangeable to indexes, and bonds exchangeable to commodities. Ó 2012 Elsevier B.V. All rights reserved.

1. Introduction We develop a widely applicable model to price securities subject to interest rate, equity, and default risks. Our model is an extension of Das and Sundaram’s (2007, henceforth DS) model. The DS model is useful in pricing certain securities subject to the aforementioned risks, but it cannot price many complex securities such as exchangeable bonds, reverse exchangeable bonds, bonds exchangeable to indexes, and bonds exchangeable to commodities. Our motivation for developing the new model is to be able to price these kinds of securities. In this paper, we show how to use our model to price exchangeable bonds. Exchangeable debt is similar to convertible debt, except that the debt is convertible to the stock of a company other than the one issuing the exchangeable debt. The stock to which the debt is convertible will be referred to as the underlying stock. Prior literature examines the valuation effects of exchangeable debt offerings as well the economic motives of these issues. It is of interest to distinguish between two types of exchangeable debt issuers: investment banks and non-financial firms. Investors demand securities that offer preferred packages of current yield, capital appreciation, downside protection, trading liquidity, etc., based on their cash needs and tax situation. But such mixtures are not often available in ⇑ Corresponding author. Tel.: +1 514 848 2424x2379; fax: +1 514 848 4500. E-mail addresses: [email protected] (R.S. Mateti), shegde@business. uconn.edu (S.P. Hegde), [email protected] (T. Puri). 0378-4266/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jbankfin.2012.11.009

conventional stocks and bonds. Investment banks design innovative financial products like exchangeable bonds to fill such a market niche. Exchangeable bonds help investors to manage risks in a different way compared to convertible bonds. Investors in convertible bonds experience two offsetting effects when the issuers of these bonds become riskier. Although the value of the investors’ claim to income from these bonds decreases, the conversion option of the bonds becomes more valuable because of the resulting higher volatility of the issuer’s stock. These two offsetting effects reduce the investment risk (Brennan and Schwartz, 1988). On the other hand, exchangeable bonds offer risk diversification to investors. The investors in these bonds are exposed to the default risk of the issuer and to the equity risk of the underlying stock. In other words, the values of debt and underlying stock of exchangeable bonds depend on the performance of two different firms. Also, from the perspective of investors in exchangeable bonds, such equity-linked securities are generally less risky than the underlying stock, and specific structuring offers higher-than-average current yields at the expense of lower rate of capital appreciation (Arzac, 1997). The non-financial issuers of exchangeable debt can hold a block of the underlying stock for various reasons. The underlying stock may belong to a target in a failed takeover attempt, a subsidiary of the issuer, or an equity carve-out by the issuing firm, and so on. One important reason an issuer chooses to sell exchangeable debt is to divest such a block of stock when having negative information regarding the future prospects of the underlying firm.

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Ghosh et al. (1990) find that the announcement of exchangeable debt offerings has little effect on the stockholders of the issuing firms but a significant negative effect on the stockholders of the underlying firms. Barber (1993) rejects the popular claim of investment banks that exchangeable debt offerings constitute a taxefficient way to divest a large block of stock and notes that the advantage of exchangeable debt offers lies in lower underwriting costs as compared to what will be paid to issue a block of shares in a seasoned equity offering. Danielova et al. (2010) study the factors that influence the choice of a particular block of stocks out of several such holdings to serve as the underlying asset for an exchangeable debt issue. They find that the issuers tend to market-time their selection such that both the operating and the stock market performance of the underlying firm are expected to decline after the issue in contrast to their superior performance before the issue. These findings are consistent with the evidence in Baker and Wurgler (2000) and others who report that firms issue equity (debt) when subsequent returns are expected to be low (high). Danielova and Smart (2012) find significant negative abnormal returns on the underlying stocks on the announcement of mandatory exchangeable debt offerings, consistent with the argument that issuers exploit their private information concerning the overvaluation of the underlying stocks. Many models have been proposed in the past to price convertible bonds but not many to price exchangeable bonds. In this paper, we show how to price exchangeable bonds by extending the DS model and conduct an empirical study using the extended model. We also show how to estimate some critical non-observable inputs to implement the model. The DS model is for pricing securities subject to interest rate, equity, and default risks. It is essentially a bivariate recombining lattice built in a risk-neutral setting. There are two state variables represented on this lattice: (1) the interest rate and (2) the stock price of the issuer of the security that is to be priced. DS represent the dynamics of the interest rate through the dynamics of the forward rates of various maturities. When a forward rate matures at a particular time, it becomes current interest rate for that time. From any node of the lattice, the stock price can take one of three possible values over the next time period, of which one value is zero. In this model, the stock price will be zero when the firm defaults on its debt obligations. The probability of the stock price being zero during the next time period on the lattice is essentially the default risk of the issuer. We make note here that three state variables—the interest rate, the stock price, and the default risk—are captured in just a two-dimensional lattice. This will impart computational efficiency to the model. DS determine the probabilities of the branches from each node of the lattice such that the expected growth rate of the stock is equal to the risk-free rate and is consistent with the recombining nature of the lattice. This results in a bivariate recombining lattice in a risk-neutral setting, which can price many securities that are subject to interest rate, equity, and default risks. It can also price securities that are subject to any two of the above three risks. This adaptability of the model makes it versatile for pricing many different securities. For a detailed description of the model and its implementation, see DS. The DS model in its original form, however, cannot price exchangeable bonds. The price of an exchangeable bond, among other things, depends on its conversion value. The conversion value is the number of shares of the underlying stock that the bond converts to times the price of the underlying stock. Thus, to price exchangeable bonds in the above framework, we must make provision for an additional stock price, the price of the underlying stock, by adding an extra dimension to the lattice. It shall be noted here that we need to consider the stock price of the issuer as well, though it does not determine the conversion value of the bond. This is because the probability of the stock price of the issuer

1019

dropping to zero represents the default risk of the issuer, and this risk surely influences the price of the exchangeable bond. Our extended model incorporates five state variables—the interest rate, the stock price of the issuer of the exchangeable bond, the default risk of the issuer of the exchangeable bond, the price of the underlying stock, and the default risk of the issuer of the underlying stock—in just three dimensions. This makes our model computationally efficient. The remainder of the paper is organized as follows. Section 2 briefly discusses the two main classes of credit risk models and how the DS model draws on ideas from both these classes to incorporate default risk in the model. Section 3 presents our model to price exchangeable bonds. Section 4 explains how to estimate some critical non-observable inputs to implement our model. Section 5 presents the empirical results and a discussion of these results. Section 6 concludes. Appendix A gives a brief description of how to relax the assumptions of our model. 2. Credit risk models Two main classes of credit risk models exist: the structural models and the reduced-form models. Structural models (e.g., Merton, 1974; Black and Cox, 1976) focus on the value of the firm and view equity and debt as contingent claims on this value. The firm defaults when its value hits the default boundary. The probabilities of default are based on the notion of distance-to-default. Implementing this class of models is difficult because the firm value is not observable and involves making restrictive assumptions about the capital structure of the firm. The equity market is mainly used to estimate the parameters of the models belonging to this class. The reduced-form models (e.g., Jarrow and Turnbull, 1995; Madan and Unal, 1995, 2000; Duffie and Singleton, 1999) treat the time of bankruptcy as an exogenous process that does not depend explicitly on the value of the firm, and the conditional probability of default is determined by the default intensity, also called the hazard rate. Therefore, to implement this class of models, we need not estimate the parameters associated with the value of the firm. The credit risk premium is a function of the probability of default and the recovery rate in the event of a default. The DS model draws on ideas from both the approaches to model credit risk. As in the reduced-form approach, the probability of default is represented by a hazard rate process. And, as in the structural approach, the equity value is assumed to drop to zero in the event of a default. However, there is an important difference. A typical reduced-form model mainly uses information in the debt market to estimate the parameters of the hazard rate. However, the present model uses information in both the debt and the equity markets. This is expected to result in an improved estimation of the hazard rate as more information is used and because the equity market is a lot more liquid than the debt market. Jarrow (2001) also recommends estimating the default rate from the information in both of the markets. 3. The model for pricing exchangeable debt An exchangeable bond is like a convertible bond, but the bond is convertible to the stock of a company that is not the issuer of the bond. For example, Morgan Stanley issued a 30-year, 1% coupon bond convertible to 13.1313 shares of Wal-Mart. We now present an extension of the DS model to price exchangeable bonds. We need to extend the original DS model, which is a bivariate recombining lattice, by adding an extra dimension and making it a trivariate recombining lattice. This is because we now also have to consider the price of the underlying stock and the default risk of

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the issuer of the underlying stock. We will, henceforth, identify the issuer of an exchangeable bond as Firm 1, the stock of the issuer of an exchangeable bond as Stock 1, its price as S1, and the conditional probability of default of Firm 1 as k1. Similarly, we will identify the issuer of the underlying stock as Firm 2, the underlying stock as Stock 2, its price as S2, and the conditional probability of default of Firm 2 as k2. We now have a total of five variables— the two stock prices, the interest rate, the default risk of Firm 1, and the default risk of Firm 2. There are two more variables in the extended model than in the original DS model, but only an extra dimension is added to the original model to incorporate these additional variables. Because our extended model incorporates five variables in just three dimensions, it makes the model computationally efficient. We will see shortly that this lattice has 18 branches emanating from each of its nodes. However, all of these branches are not equally important, and we can ignore quite a few of them when writing a program to implement the model. As in the original model, we capture the default risk by the probability of the stock price dropping to zero. The interest rate and S1 determine the probability that Firm 1 will default over the next time period on the lattice, which we take to be a quarter. The interest rate and S2 will determine the probability that Firm 2 will default over the next time period on the lattice. When this default takes place, the conversion value of the bond becomes zero, and the price of the bond behaves like that of a simple risky nonconvertible bond. 3.1. Dynamics of the state variables As in the DS model, we model the interest rate dynamics through the evolution of the forward rates. Using the model of Heath et al. (1990, henceforth HJM model), the discrete-time, risk-neutral dynamics of the forward rates, f(t, T), are given by

pffiffiffi f ðt þ h; TÞ ¼ f ðt; TÞ þ af ðt; TÞh þ rf ðt; TÞX f h

ð1Þ

where f(t, T) is the value at time t of the forward rate that matures at time T, af(t, T) is the drift rate of the forward rate, rf(t, T) is the volatility of the forward rate, Xf is a random variable, with an expected value of zero and a variance of one, that takes one of the values {1, 1}, and h is a fixed length of time, which we take to be a quarter. We note that the drift rate of a forward rate is not independent of its volatility in an arbitrage-free market. See Das and Sundaram (2001) for a recursive relationship between the drift rates and the volatilities. We model the discrete-time, risk-neutral dynamics of S1 by a slight modification of

ln

  pffiffiffi S1 ðt þ hÞ ¼ rðtÞh þ rS1 X S1 h S1 ðtÞ

where r(t) is the short-term risk-free interest rate at time t, X S1 is a random variable that takes one of the values {1, 1, 1}, and rS1 is the volatility of S1. When Firm 1 defaults, the value of X S1 is minus infinity and the stock price drops to zero. The drift rate of the stock price is equal to the risk-free rate in a risk-neutral setting. We, however, need to modify the above dynamics of the stock price to be able to model it on a recombining lattice. Because the drift is not constant and changes with r(t), we eliminate it from the above equation and modify the probabilities of thepffiffibranches at each node h of the lattice such that the mean of X S1 is rðtÞ rS1 . We now do not have a drift term, but still the mean growth rate is r(t). The modified dynamics of S1 are given by

ln

  pffiffiffi S1 ðt þ hÞ ¼ rS1 X S1 h S1 ðtÞ

ð2Þ

Using the same reasoning as above, the dynamics of S2 are given by

ln



 pffiffiffi S2 ðt þ hÞ ¼ rS2 X S2 h S2 ðtÞ

ð3Þ

where rS2 is the volatility of S2 ; X S2 is a random variablepffiffithat takes h one of the values {1, 1, 1}, and the mean of X S2 is rðtÞ rS2 . Given the nature of the dynamics of the three state variables in Eqs. (1)–(3), we will have a recombining lattice. Each node of the lattice represents a particular combination of values of the three state variables—the spot interest rate (or the vector of forward rates) and the two stock prices (S1 and S2). Eighteen branches (2  3  3) emanate from every node of the lattice, and each of these branches corresponds to a particular combination of values of Xf, X S1 , and X S2 as shown in Table 1. B1 is branch 1, B2 is branch 2, and so on. B13 to B18 are the branches relating to the default by Firm 1. The lattice does not extend beyond these branches because the bond ceases to exist once Firm 1 defaults. B3, B6, B9, B12, B12, B15, and B18 are the branches relating to the default by Firm 2. Once this happens, the bond’s conversion value is zero and is effectively a non-convertible bond. However, the lattice does extend beyond these branches because the bond continues to exist even though Firm 2 may have defaulted. 3.2. Risk-neutral probabilities of lattice branches Because we build our lattice in a risk-neutral setting and preserve the observed correlations between the variables, we assign probabilities to the branches from each node of the lattice such that the following five conditions are met. Condition 1. The growth rate of S1 is equal to the risk-free interest rate.

Condition 2. The growth rate of S2 is equal to the risk-free interest rate. Condition 3. The correlation between the interest rate and the return on Stock 1 is equal to the observed correlation, qr;S1 . Condition 4. The correlation between the interest rate and the return on Stock 2 is equal to the observed correlation, qr;S2 . Condition 5. The correlation between the returns on Stock 1 and Stock 2 is equal to the observed correlation, qS1 ;S2 . Conditions 1 and 2 are just like the conditions in the DS model. We have three extra conditions (Conditions 2, 4, and 5) in the extended model compared to the DS model. We take the probabilities assigned to the branches of the lattice in the DS model as our starting point and then use them to derive the probabilities of the branches of the lattice in our model. Six branches emanate from each node of the lattice in the DS model. We reproduce the probabilities of the branches in the original model in Table 2 to facilitate our discussion. Because the DS model is the basis for our extended model, we first examine the rationale for the assignment of probabilities to various branches of the lattice in the DS model before setting out to assign probabilities to various branches of the lattice in the extended model. To distinguish the branches in the original model, as shown in Table 2, from those in the extended model, as shown in Table 1, we use uppercase for the extended model and lowercase for the original model. k1(t) denotes the conditional probability at time t that the issuer of the exchangeable bond will default during the next time period on the lattice. We discuss how to estimate default risk in Section 4.2. The reason for the assignment of the probabilities in Table 2 is as follows. The stochastic variable Xf, which

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R.S. Mateti et al. / Journal of Banking & Finance 37 (2013) 1018–1028 Table 1 Branches of the trivariate lattice.

Xf X S1 X S2

B1

B2

B3

B4

B5

B6

B7

B8

B9

B10

B11

B12

B13

B14

B15

B16

B17

B18

1 1 1

1 1 1

1 1 1

1 1 1

1 1 1

1 1 1

1 1 1

1 1 1

1 1 1

1 1 1

1 1 1

1 1 1

1 1 1

1 1 1

1 1 1

1 1 1

1 1 1

1 1 1

m1 ¼

Table 2 Probabilities of the branches of the bivariate lattice, Das and Sundaram (2007). Branch

Value of Xf

Probability ð1þm1 Þð1k1 ðtÞÞ 4 ð1m1 Þð1k1 ðtÞÞ 4 ð1þm2 Þð1k1 ðtÞÞ 4 ð1m2 Þð1k1 ðtÞÞ 4 k1 ðtÞ 2 k1 ðtÞ 2

1

1

b2

1

1

b3

1

1

b4

1

1

b5

1

1

b6

1

1

appears in the equation representing the discrete-time dynamics of forward rates, has to have an expected value of zero and a variance of one. Given that the possible values of Xf are 1 and 1, the probabilities are prob(Xf = 1) = probðX f ¼ 1Þ ¼ 12. In other words, the probability of any one of the branches b1, b2, or b5 being realized is equal to the probability of any one of the branches b3, b4, or b6 being realized. As we will explain in Section 4.2, the conditional probability of default of the issuer during a particular time period on the lattice is determined by S1 and the interest rate at the beginning of this time period. The nature of the interest rate shock (Xf = 1 or Xf = 1) during this time period does not have any influence on the default during this time period. In other words, an interest rate shock does not affect the probability of default during the time period it occurs. The effect of the interest rate shock will be felt only in the next time period, because the new interest rate is applicable to the period following the occurrence of the shock. Therefore, the probability of default during a particular time period is independent of the nature of interest rate shock during this period. Because branches b5 and b6 comprise the scenario in which Firm 1 defaults, these two branches shall equally share the probability of default. Hence, probðb5Þ ¼ probðb6Þ ¼ k12ðtÞ. We will have interest in the probabilities of the no-default branches b1, b2, b3, and b4 on the condition that no default by Firm 1 occurs. Thus, it is desirable to express the unconditional probabilities of these branches in a form in which the term (1  k1(t)) appears multiplicatively. If we want the conditional probabilities, the term (1  k1(t)) simply drops out. We can check from Table 2 that the sum of conditional probabilities is one. Again, because the expected value of Xf is zero, and the only possible values of Xf are 1 and 1, probðX f ¼ 1=no defaultÞ ¼ probðX f ¼ 1=no defaultÞ ¼ 12. In other words, probððb1 or b2Þ=no defaultÞ ¼ prob ððb3 or b4Þ=no defaultÞ ¼ 12. If a factor m1 is added in the probability of branch b1, then the same must be subtracted in the probability of branch b2 so that probðX f ¼ 1=no defaultÞ ¼ 12. The same reasoning applies to branches b3 and b4, and hence we get the probabilities as shown in Table 2. To determine the values of m1 and m2, we invoke the two of the previously stated conditions, Conditions 1 and 3, which lead to

E

  h pffiffi i S1 ðt þ hÞ ¼ E erS1 X S1 ðtÞ h ¼ erðtÞh S1 ðtÞ

ð4Þ

and

cov½X f ðtÞ; X S1 ðtÞ ¼ qr;S1 Using these two conditions, we solve for m1 and m2 as:

m2 ¼

AB 2

where

Value of X S1

b1

AþB ; 2

ð5Þ

 "  # rðtÞh 1 1k1 ðtÞ A¼  2ða þ bÞ ; 4e ða  bÞ pffiffi pffiffi a ¼ eðrS1 hÞ ; b ¼ eðrS1 hÞ

B¼

2qr;S1 ; ð1  k1 ðtÞÞ

To ensure that the probabilities lie between 0 and 1, we have to impose restrictions on k1(t). From Table 2, we see that probability of branch b1 is non-negative as long as m1 P 1, and the probability of branch b2 is non-negative as long as m1 6 1. Therefore, the probabilities of both branches, b1 and b2, are non-negative as long as 1 6 m1 6 1. This condition also ensures that the probabilities of branches b1 and b2 never exceed 12. By a similar reasoning, we have the condition 1 6 m2 6 1. Using these conditions and the expressions for m1 and m2, we can derive bounds on k1(t) that ensure that we meet the conditions on m1 and m2. See DS for details. We have to extend this original DS model to price exchangeable debt. We need a trivariate lattice, with each node of the lattice representing the values of three state variables—the short-term riskfree interest rate and the two stock prices. As shown earlier, 18 branches emanate from each node of the lattice and we need to assign probabilities to these branches in a risk-neutral setting. We use the probabilities in Table 2 as our starting point and then determine the probabilities of the branches of the extended model. With each of the branches b1–b6, we can associate three possible prices of Stock 2. The stock price, S2, either goes up, goes down, or drops to zero (Firm 2 defaults), just like S1. These scenarios correspond to X S2 ¼ 1; 1; 1, respectively. We have seen that in the DS model branch b1 is for the upward movement of both the interest rate and S1, that is, Xf = 1 and X S1 ¼ 1. Let us denote the unconditional probability of this branch (prob(b1)) by P1 and the conditional probability of this branch given no default by the issuer (prob(b1/no default)) by p1. We use a similar notation for the unconditional and conditional probabilities of the other no-default branches b2, b3, and b4 and the unconditional probabilities of the default branches b5 and b6. Given the probabilities of branches b1–b6, our goal is to divide each of them into three parts corresponding to the three associated movements of X S2 in a risk-neutral setting. We assume that the default of Firm 2 is independent of the branches b1–b6. In other words, the conditional probability of default of Firm 2, k2(t), does not depend on the branches b1–b6. This assumption is not very unrealistic in the context of exchangeable debt, because the stock Firm 1 tries to offload typically does not belong to the same industry as that of Firm 1. Therefore, we can expect the default correlation between Firm 1 and Firm 2 to be low. Given the probability of default of Firm 2, we need six variables to divide the probabilities of the six branches. But, compared to the DS model, as noted above, we have only three additional conditions for introducing the new stock price, S2. With six new variables but only three additional conditions, we have three degrees of freedom. It makes sense to satisfy these conditions given that the Firm 1 does not default. This will reduce the number of variables required to divide the probabilities from six to four because now we do not have to divide the probabilities of the default branches b5 and b6, which still leaves us with one degree of freedom. We can impose any innocuous additional condition on the variables that does not have

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an undue influence on the results and helps us arrive at a solution to the problem. We define the following three conditional probabilities relating to the breakdown of the conditional probability, p1, of branch b1: p11 ¼ probðb1 \ ðX S2 ¼ 1Þ=no default by Firm 1), p12 ¼ probðb1 \ ðX S2 ¼ 1Þ=no default by Firm 1), p13 ¼ probðb1\ ðX S2 ¼ 1Þ=no default by Firm 1). These are the conditional probabilities of the three values of X S2 associated with branch b1. We have similar definitions for the breakdown of the conditional probabilities of the other no-default branches b2–b4. We now derive the probabilities in the risk-neutral setup. Conditions 1 and 3, the common conditions in our model and the DS model, help in determining the conditional probabilities of no-default branches. Along with an additional condition that will be discussed later, Conditions 2, 4, and 5 help in breaking down these conditional probabilities, which we do now. Let us define pffiffi pffiffi c ¼ eðrS2 hÞ and d ¼ eðrS2 hÞ . Condition 2 requires that the growth rate of S2 be equal to the risk-free rate. We have (suppressing the time argument for easy readability)

  S2 ðt þ hÞ E ¼ erðtÞh S2 ðtÞ  

k2 ¼ p11  c þ p1  p11  d 4  

k2 d þ p21  c þ p2  p21  4  

k2 þ p31  c þ p3  p31  d 4  

k2 d þ p41  c þ p4  p41  4

After some manipulation, we get

p11  p21 þ p31  p41 ¼

1 þ p1  p2 þ p3  p4 Þ ðq 2 S1 ;S2

ð8Þ

We have three equations and four unknowns. We need an additional condition to arrive at a solution. We will like to equate two probabilities that are small, and because they are small we hope that this equating will not have a significant influence on the results. It is an empirical fact that the stock prices are much more correlated among themselves than with the interest rates. In Table 2, branches b2 and b4 represent downward movement of S1. The upward movement of S2 associated with these branches is small compared to the other two no-default branches. Therefore, we equate the probabilities p21 and p41. This is just one of the many possible assumptions that can be made. (In Appendix A, we show how to relax this assumption). Making use of the condition p21 = p41, we can express Eqs. (6), (7), and (9) in the matrix notation as

MK ¼ N

ð9Þ

where

2

1

2

1

3

0

p11

1

0

erh dð1k2 Þ cd

1

B qr;S þp1 þp2 p3 p4 C 6 7 B C C 2 M ¼ 4 1 0 1 5; K ¼ @ p21 A; N ¼ B @ A 2 qS1 ;S2 þp1 p2 þp3 p4 1 2 1 p31 2

The solution for the unknowns in matrix K is K = M1N. The completes the determination of probabilities in the risk-neutral setup. 3.3. Conditions required at the nodes of the lattice to take care of the call option of the issuer

After some manipulation, we get

p11 þ p21 þ p31 þ p41 ¼

1 ½erh  dð1  k2 Þ ðc  dÞ

ð6Þ

Condition 4 requires that the correlation between the interest rate and the returns on Stock 2 be equal to the observed correlation, qr;S2 . We have

cov½X f ðtÞ; X S2 ðtÞ ¼

 

k2  1  1 p11  1  1 þ p1  p11  4  

k2  1  1 þ p21  1  1 þ p2  p21  4  

k2  1  1 þ p31  1  1 þ p3  p31  4  

k2 þ p41  1  1 þ p4  p41   1  1 4

¼ qr;S2

1 ðq þ p1 þ p2  p3  p4 Þ 2 r;S2

ð7Þ

Similarly, Condition 5 requires that the correlation between the returns on Stock 1 and Stock 2 be equal to the observed correlation, qS1 ;S2 . We have

 

k2  1  1 cov½X S1 ðtÞ;X S2 ðtÞ ¼ p11  1  1 þ p1  p11  4  

k2  1  1 þ p21  1  1 þ p2  p21  4  

k2 þ p31  1  1 þ p3  p31   1  1 4  

k2  1  1 þ p41  1  1 þ p4  p41  4 ¼ qS1 ;S2

Condition A. If max (present value of expected value of bond at next time step, conversion value of bond) < call price,

value of bond ¼ maxðpresent value of expected value of bond; conversion value of bondÞ Because the investors have the option to convert the bond into common stock, the price of the bond can never be less than the conversion value. However, if the present value at time t of the expected value at time (t + h) is greater than the conversion value, this would be the value of the bond, provided this present value is not greater than the call price. Condition B. If min (present value of expected value of bond, conversion value of bond) > call price,

After some manipulation, we get

p11 þ p21  p31  p41 ¼

To take care of the embedded call option in an exchangeable bond, we impose certain conditions at the nodes of the lattice.

value of bond ¼ conversion value The price of the bond has to be at least as much as the conversion value. If the present value is greater than the conversion value, the firm can call the bond and therefore force conversion (because the conversion value is greater than the call price). Condition C. If present value < call price < conversion value,

value of bond ¼ conversion value This is because the value of the bond has to be at least as much as the conversion value. Condition D. If conversion value < call price < present value,

value of bond ¼ call price

R.S. Mateti et al. / Journal of Banking & Finance 37 (2013) 1018–1028

If the present value is greater than the call price, the firm will call the bonds and force its value to be equal to the call price. This is because the firm, acting in the interests of its shareholders, will not want the price of its debt to be higher than the call price.

4. Estimating non-observable inputs of the model Forward rates, volatilities of forward rates, the two stock prices, volatilities of the stock prices, correlation between the interest rate and stock prices, and credit default swap spreads are the inputs required to implement the model. We also need the values of the parameters of the default intensity function, which we have to estimate, and the actual market prices of exchangeable bonds to compare our model’s prices with. The stock prices, interest rates, credit default swap spreads, and prices of exchangeable bonds are the directly observable data. All other inputs for the model have to be derived from observable variables. All observable data is taken from Bloomberg. 4.1. Forward rate volatilities One of the critical inputs is the volatilities of the forward rates. For a forward rate maturing at time ti, f(t, ti), we want to keep its volatility, rf(t, ti), unchanged from the current time, t0, until it matures at time ti; volatility of f(t, ti) = rf(t, ti) = k (a constant) for t0 6 t 6 ti. Forward rates maturing at different times have different constant volatilities. This aspect of forward rate volatilities is necessary to have a recombing lattice and is one of the distinctive features of the model. For example, consider the forward rate maturing after 2 years. We assume that the volatility of f(t,2) = rf(t,2) = k (a constant) for t0 6 t 6 2. In other words, we want the volatility of a forward rate to depend on the time of maturity rather than the time to maturity of the forward rate. As we move forward in time and the time to maturity of a forward rate gets shorter, the volatility of the forward rate does not change. If we are more inclined to believe that the forward rate volatilities depend on the time to maturity of the forward rate, then we can interpret these time of maturity volatilities as an average of the time to maturity volatilities. If we denote the time to maturity volatilities by vol(ti  t0), then the relationship between the time of maturity volatility, rf(t, ti), and the time to maturity volatilities can be expressed as

rf ðt; ti Þ ¼

1 ðt i  t 0 Þ

Z

ti

volðt k  t 0 Þdt k

ð10Þ

t0

We can estimate the time of maturity forward rate volatilities in several ways. We first present a method to estimate these volatilities implied by the current market data. Later, we show how to estimate these volatilities from historical data. 4.1.1. Estimating normal volatilities from prices of interest rate caplets We can derive the implied forward rate volatilities for use on our lattice from the prices of interest rate caplets. A caplet is a call option on an interest rate. Assume that the face value of the contract is $1, the underlying interest rate is a LIBOR, the caplet rate (strike rate) is R, the tenor of the caplet (length of time for which interest is to be calculated) is s, and the maturity of the caplet is at Ti. If the LIBOR at time Ti for the period (Ti, Ti+1) is L(Ti, Ti+1), then the payoff C at time Ti+1 is given by

C ¼ s max ½LðT i ; T iþ1 Þ  R; 0

ð11Þ

where Ti+1  Ti = s (the tenor of the caplet). The cash flow from the caplet is not paid when the caplet matures at time Ti but after a period equal to the tenor of the caplet, that is, at time Ti+1. A series of caplets constitutes a cap. The market practice to value caps and

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caplets involves the use of Black’s (1976) model, which is essentially a modification of the Black–Scholes model with the drift rate of the price of the underlying asset set equal to zero. In the Black’s model, the forward rates (the underlying variable of caps and caplets) have a log-normal distribution. The market prices of caps and caplets are not quoted in dollars but in terms of Black implied volatilities (just like the implied volatilities derived from stock option prices). We refer to the Black implied volatilities of the forward rates also as log-normal volatilities because the forward rates have a log-normal distribution in the Black’s model. The price of a caplet using the Black’s formula is

Cðt; T i Þ ¼ sPðt; T iþ1 Þ½Lðt; T i ; T iþ1 ÞNðd1 Þ  R  Nðd2 Þ

ð12Þ

where C(t, Ti) is the price at time t of a caplet maturing at time Ti, P(t, Ti+1) is the price at time t of zero-coupon bond maturing at time Ti+1, L(t, Ti, Ti+1) is the forward LIBOR at time t for the period h   Lðt;T i ;T iþ1 Þ (Ti, Ti + 1), R is the caplet rate, d1 ¼ p1ffiffiffiffiffiffiffi ln þ R r T i t p ffiffiffiffiffiffiffiffiffiffiffiffi 2 r ðT i tÞ ; d2 ¼ d1  r T i  t, and r is the volatility of the forward 2 rate. Interestingly, a caplet is equivalent to a certain number of put options on a zero-coupon bond. The payoff from the caplet at time Ti+1 is smax[L(Ti, Ti+1)  R, 0]. At time Ti, the discounted value of this payoff is 1þssR max½LðT i ; T iþ1 Þ  R; 0. The time Ti payoff from the cah i plet can also be expressed as ð1 þ sRÞ max 1þ1sR  1þsLðT1i ;T iþ1 Þ ; 0 . This expression can be interpreted as the payoff from (1 + sR) put options on a zero-coupon bond with a strike price of 1þ1sR and maturing at time Ti. The interpretation becomes clear if we note that 1 is the price at time Ti of a zero-coupon bond with a face 1þsLðT ;T Þ i

iþ1

value of $1 and maturing at time Ti+1. Following Ho and Lee (1986, henceforth the Ho–Lee model), whose model gives a closed-form solution for pricing call and put options on zero-coupon bonds, the price of a put option on a zero-coupon bond is given by

putðt; T i ; T iþ1 Þ ¼ KPðt; T i ÞNðd2 Þ  Pðt; T iþ1 ÞNðd1 Þ

ð13Þ

where put(t, Ti, Ti+1 is the price at time t of a put option that matures at time Ti on a zero-coupon bond maturing at time Ti+1, K is the strike price of the put option, P(t, Ti) is the price at time t of a h i. Pðt;T Þ zero-coupon bond maturing at time T i ; d1 ¼ ln Pðt;Tiþ1i Þ rp þ pffiffiffiffiffiffiffiffiffiffiffiffi rp ; d ¼ d  r ; r ¼ r ðT  T Þ T  t , and r is the constant vol2 1 p p iþ1 i i 2 atility for the spot and forward rates. Finally, we note that the HJM model with constant volatility results in the Ho–Lee model. Because in the DS model the dynamics of the forward rates are modeled using the HJM model, and the volatility of each forward rate is constant until it matures (forward rates maturing at different times have different constant volatilities), it is like fitting a different Ho–Lee model for each forward rate. Thus, we can assume that the forward rate maturing after, say, 1 year is modeled by a Ho–Lee model with a particular volatility, and similarly the forward rate maturing after 2 years is modeled again by a Ho–Lee model but with a different volatility. We now summarize the steps to arrive at the time of maturity volatilities for different forward rates. We refer to the time of maturity volatilities also as normal volatilities because the forward rates are normally distributed in the Ho–Lee model. (i) Suppose that we want the volatility of the forward rate maturing after 1 year. We observe the market price of atthe-money (ATM) caplet maturing after 1 year. This is given in terms of the Black implied volatility, which has to be converted to the dollar price by inputting the Black implied volatility in the Black’s formula. We choose ATM caplets

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because the high trading volume in these caplets is likely to better reveal the market expectations of the forward rate compared to caplets with other caplet rates. (ii) We can interpret the above price of the 1-year caplet as the price of (1 + sRATM) put options with a strike price of 1þs1RATM on a zero-coupon bond. We now use the formula for pricing put options on a zero-coupon bond in the Ho–Lee model. We search for that value of volatility r that best explains the price of the above (1 + sRATM) put options on the zero-coupon bond. The optimum r from the above exercise is chosen as the volatility of the forward rate maturing after 1 year, rf(t, 1). We repeat the above procedure for all other maturities. The volatilities we get are the normal volatilities. We use the above procedure to convert log-normal volatilities to normal volatilities using the prices reported in Bloomberg on January 3, 2006. The results are shown in Table 3. We see that there is a distinct hump in the reported lognormal (Black) volatilities. We fit a logarithmic curve to the plot of normal volatilities to estimate volatilities for maturities for which we may not have information. The resulting logarithmic curve is shown in Fig. 1.

4.1.2. Estimating forward rate volatilities from Treasury constant maturity rates Next, we show how to estimate forward rate volatilities from historical data. We need data on Treasury constant maturity rates (TCMR, henceforth), and they are freely available on the Web site of Federal Reserve Bank of St. Louis. The Federal Reserve Bank of St. Louis calculates TCMRs for 1-month, 3-month, 6-month, 1-year, 2-year, 3-year, 5-year, 7-year, 10-year, 20-year, and 30-year periods every business day. The purpose of TCMR series is to estimate the yield on, say, a 2-year note on a given day, even if there is no Treasury issue with 2 years to maturity trading on that day. The estimate is made by interpolating yields from on-the-run Treasury issues. The most recent Treasury issue for a particular time to maturity is the on-the-run issue for that maturity. It rolls to offthe-run after the next issue for that maturity, which becomes onthe-run. On the day of issue of, say, a 2-year note, it is easy to determine the Treasury rate for a 2-year maturity from the price of a 2-year note. However, a day later, if we want the Treasury rate for a 2-year maturity, we do not have any Treasuries that have 2 years remaining to maturity from which we can determine the 2-year rate. The 2-year note issued a day ago now has 1 day less to maturity. Cubic splines are used to smooth between on-therun issues to determine TCMRs for the key maturities mentioned earlier. Because on-the-run issues trade close to par and TCMRs are estimated from these issues, we can treat TCMRs as par coupon bond yields. Because we have chosen each time period to be a quarter on our lattice, we need forward rate volatilities for periods spaced out at quarterly intervals—3 months, 6 months, 9 months, 1 year, 1 year and 3 months, and so on – to implement the model. Suppose that we want to estimate the forward rate volatilities until 3 years. To do this, we first need TCMRs spaced out at quarterly intervals until 3 years. Some TCMRs for these periods are readily available, and others have to be estimated by interpolating from the readily available ones. We can either use cubic splines or simple linear interpolation. Table 4 shows the reported TCMRs on January 3, 2006 for periods 3 months, 6 months, 1 year, 2 years, and

Fig. 1. Forward rate volatilities.

3 years. We use simple linear interpolation of readily available TCMRs to estimate the TCMRs for other periods shown in the table. Next, we estimate the zero-coupon bond prices from TCMRs. Again, we note that TCMRs can be interpreted as par coupon bond yields because these rates are estimated from on-the-run Treasury issues, which trade close to par value. We can easily estimate the price of the zero-coupon maturing after 3 months, P(0, 0.25), from 1 the 3-month constant maturity rate: Pð0; 0:25Þ ¼ 1þ0:250:0416 ¼ $0:989707. Similarly, the price of zero-coupon bond maturing after 1 6 months is Pð0; 0:50Þ ¼ 1þ0:500:0440 ¼ $0:978474. The price of zero1 coupon bond maturing after 9 months is Pð0; 0:75Þ ¼ 1þ0:750:0439 ¼ $0:968125. The calculation of the above zero-coupon prices is straightforward. This is because these short-maturity Treasury bonds do not pay any interest. Now, for the calculation of the 1-year zero-coupon price, we have to take the coupon payments into account. The 1-year bond is expected to pay interest after 6 months and when it matures after 1 year. Assuming that the face value of the bond is $1, an amount equal to half of the Treasury constant maturity rate (this is assumed to be the coupon rate) is paid when interest is due. For the 1-year bond, this amount is $0:0438 ¼ $0:0219. The 2 1-year bond, therefore, makes a payment of $0.0219 after 6 months and $0.0219 plus $1 (principal repayment) after 1 year. The present value of these payments has to be $1 because par coupon bonds trade at par: 1 = P(0, 0.50)  0.0219 + P(0, 1)  (0.0219 + 1). Solving for P(0, 1), we get the price of zero-coupon bond maturing after 1 year to be P(0, 1) = $0.9576. Using the same procedure, we can solve for the price of zero-coupon bond maturing after 1 year and 3 months, P(0, 1.25). We note that this bond pays interest after 6 months and 1 year, assuming interest is paid every 6 months. After the payment of the last interest at the end of 1 year, we still have 3 months left for the maturity of this bond. It is assumed that interest accrues for these last 3 months before maturity and is paid along with the face value on maturity. With this assumption, we have: 1 = P(0, 0.50)  0.02185 + P(0, 1)  0.02185 + P(0, 1.25)  (0.010925 + 1). Substituting the values of P(0, 0.50) and P(0, 1) that we determined earlier and solving for P(0, 1.25) gives us $0.94735. Using this bootstrapping procedure, we can derive the zero-coupon bonds for other maturities. After deriving zero-coupon bond prices, we need to estimate the continuously compounded forward rates implicit in these zero-coupon bond prices. The continuously compounded 3-month forward rate (annualized), f(0, 0.25), can

Table 3 Lognormal and normal volatilities.

Log-normal (Black) volatility Normal volatility

1 year

2 years

3 years

4 years

5 years

6 years

7 years

8 years

0.1189 0.00566

0.1664 0.00777

0.1887 0.00889

0.1997 0.00945

0.2037 0.00962

0.2073 0.00978

0.2079 0.00987

0.2076 0.00991

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R.S. Mateti et al. / Journal of Banking & Finance 37 (2013) 1018–1028 Table 4 Observed and interpolated Treasury constant maturity rates, TCMR. 3 month

6 month

9 month

1 year

1 year 3 month

1 year 6 month

TCMR

4.16%

4.40%

4.39%

4.38%

4.37%

4.36%

TCMR

1 year 9 month 4.35%

2 year 4.34%

2 year 3 month 4.33%

2 year 6 month 4.32%

2 year 9 month 4.31%

3 year 4.30%

be found using P(0, 0.25) and P(0, 0.50) and is equal to ln½½Pð0;0:25Þ  Pð0;0:50Þ 4 ¼ 0:045659. We can find forward rates for other maturities using a similar procedure. We derive the forward rates for various maturities every day in the sample period. Then, we create a time series of daily changes in forward rates for each maturity and calculate the standard deviations of these time series. Multiplying standard pffiffiffiffiffiffiffiffiffi deviations by 260 (assuming 260 trading days in a year) will give us the annualized volatility of the forward rates. Finally, after estimating the volatilities of forward rates for various maturities, we need to make one final adjustment. We recall that the volatilities in the DS model depend on the time of maturity, rather than the time to maturity, of the forward rates. We can interpret the lattice forward rate volatilities as the average volatilities for different maturities. For example, the lattice volatility of the 1-year forward rate will be the average of the volatilities of forward rates with maturities up to 1 year. Because we are estimating volatilities at quarterly intervals, the 1-year forward rate volatility (on the lattice) will be the average of the 3-month, 6month, 9-month, and 1-year volatilities calculated above. 4.2. Estimating the parameters of the default intensity function We show now how to estimate the parameters of the default intensity function. We use the same default intensity function as in DS:

nðtÞ ¼ e½a0 þa1 rðtÞa2 ln SðtÞþa3 ðtt0 Þ

ð14Þ

where n(t) is the default intensity at time t, a0, a1, a2, and a3 are yet to be determined parameters, r(t) is the interest rate at time t, and S(t) is the stock price at time t of the firm whose probability of default we are trying to estimate. Unlike in the structural approach in which the firm defaults when the value of the firm hits the default boundary, in the reduced-form approach the time to bankruptcy is treated as an exogenous process that does not depend explicitly on the value of the firm. In the above function, the default intensity depends on three variables—interest rate, stock price, and time. The default intensity decreases when the risk-free interest rate increases. This is because the risk neutral drift rate of the stock is higher and therefore the value of the firm moves away from the default boundary at a faster risk-neutral rate. The default intensity also decreases also when the stock price increases. This is because the value of the firm is further away from the default boundary when the stock price is higher. The effect of time on the intensity depends on the current level of the intensity. If the intensity is currently very high but the firm manages to survive, then the intensity is expected to decrease in the future. In this case, the default intensity decreases with time. The converse holds if the current level of intensity is low. Given the intensity of default, the probability of default is given by

kðtÞ ¼ 1  eðnðtÞhÞ

ð15Þ

where k(t) is the probability at time t of a default occurring during the next time period on the lattice, and other variables have the same meaning as defined before.

The task is to estimate the parameters a0, a1, a2, and a3. We make use of the credit default spreads observed in the market to estimate these parameters. The DS model shows how to determine the credit default spreads from the model if the parameters a0, a1, a2, and a3 are known. Now the problem is reversed—given the observed spreads we need to determine the values of the parameters that best explain the observed spreads. We have four parameters to estimate, and if we have credit default spreads for four different maturities, we can get a good estimate. We undertake a large unconstrained minimization exercise for this. The objective function is the sum of squared deviations between the observed spreads and the spreads given by the model. It is a large non-linear function, and the number of terms in it depends on the number of nodes on the lattice. We need a sophisticated algorithm for this problem. We find that MATLABs Optimization Toolbox is useful for solving this problem. Of the many functions available in this toolbox, we find that the fminsearch function best handles this problem. The fminsearch function uses the simplex search method, which is a direct search method that does not use numerical or analytical gradients. In our problem, we have four unknowns, a0, a1, a2, and a3. The stated algorithm creates a simplex in a fourdimensional space that is characterized by five distinct vectors as its vertices. At each iteration, a new point in or near the current simplex is generated. The value of the function at this new point is compared with the functions values at the old vertices of the simplex, and one of them is replaced with the new point, which results in a new simplex. This procedure is repeated until an acceptable solution is found or the number of iterations reaches a predetermined limit. 5. Empirical results on exchangeable bonds We test our model on a sample of exchangeable bonds. We price these bonds on January 3, 2006, but there is no special reason for choosing this day. If we cannot find the market price of an exchangeable bond on Bloomberg on this day, we try to price it on the day closest to the chosen day when the price is available, provided this day is not too distant from the chosen day. Because of the difficulty in obtaining market prices on specific dates as these bonds are not traded frequently, we have only a small sample of 22 exchangeable bonds. Also, it so turns out that the issuers of exchangeable bonds in our sample are all financial institutions. Though this is not by design, one can view our empirical results as a test of our model using a sample of exchangeable bonds issued by financial institutions. Our results are shown in Table 5. As we can see, our model gives reasonably accurate prices as the mean absolute percentage deviation between the market prices and the model prices is just 2.47%. However, there is one thing that stands out as we compare the market prices with the model prices. For most of the bonds (16 out of 22), the model prices are greater than the market prices. This result needs an explanation. There have been many studies on how the market prices of convertible bonds compare with model prices. It is well documented in the literature that convertible bonds are underpriced in the market relative to their model prices. However, there have not been many similar studies on exchangeable bonds probably because of the greater importance of the convertible bond market

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Table 5 Market prices versus model prices of exchangeable bonds. Company

Stock convertible to

Date

Market price

Model price

Abs. dev (%)

Bear Stearns Bank of America Goldman Sachs Goldman Sachs Goldman Sachs Goldman Sachs Salomon Holdings Morgan Stanley Morgan Stanley Morgan Stanley Morgan Stanley Morgan Stanley Wachovia Wachovia Merrill Lynch Merrill Lynch Merrill Lynch Lehman Bros. Lehman Bros. Lehman Bros. Lehman Bros. Lehman Bros.

Fifth Third Bancorp NASDAQ-100 WYE BJ Services Whirlpool Cendant Corp. Pfizer 3M Wal-Mart CISCO General Electric CA Corning Johnson & Johnson Coca-Cola McDonalds Berkshire Hathaway Deere Amgen Bristol Meyers Microsoft Cendant Corp.

1/18/06 3/10/06 11/28/05 1/18/06 2/09/06 1/25/06 1/25/06 1/03/06 1/05/06 1/19/06 1/31/06 4/28/06 2/21/06 2/21/06 1/04/06 1/11/06 1/05/06 5/03/06 4/28/06 4/28/06 5/02/06 12/02/05

843.3 1032 972.7 1040.88 960.8 924.5 945 871.25 858.75 961.33 896.25 937.5 1005.35 1006.7 885.0 970.0 952.1 1221.5 877.5 880.0 1001.5 875.0

883.16 1046.6 972.9 1017.3 989.67 929.85 950.77 916.96 850.44 984.08 904.47 967.0 953.64 976.65 902.16 990.57 928.6 1229.7 926.07 925.37 982.0 884.74

4.7 1.4 0 2.3 3.0 0.6 0.6 5.2 1.0 2.4 1.0 3.1 5.1 3.0 1.9 2.1 2.5 0.7 5.5 5.2 1.9 1.1

Mean absolute deviation (%)

and also because pricing exchangeable bonds is more complex. One study that is relevant in explaining our results is Ammann et al. (2003). The authors investigate the pricing of convertible bonds in the French convertible bond market and also include exchangeable bonds in their study. They choose to investigate this market because it is the largest convertible bond market in Europe, these bonds are exchange traded, and accurate daily prices are easily available. They find that exchangeable bonds are underpriced in the market relative to the model prices on average by 3.65%, whereas convertible bonds are underpriced on average by 3.04%. The authors note that despite their rather different risk-return characteristics, standard convertible bonds and exchangeable bonds are priced similarly in the market relative to the model prices. We think that this study is important in explaining our results even though these findings pertain to the French convertible and exchangeable bond market and not the US market. Our empirical results are consistent with these findings. Another study we refer to is Carayannopoulos and Kalimipalli (2003). The authors study how the model prices of convertible bonds compare with the market prices in the US market. Though this study is not about exchangeable bonds, we think that this study is relevant in light of the observation in Ammann et al. (2003) that the market prices of exchangeable bonds relative to the model prices are similar to the market prices of convertible bonds relative to the model prices. Carayannopoulos and Kalimipalli’s (2003) main finding is that convertible bonds with the conversion option out-of-the-money are generally underpriced in the market relative to the model prices, and that convertible bonds with the conversion option in-the-money are overpriced in the market relative to the model prices. The reason for the overpricing in the market relative to the model, as they note, is that bond call options are generally not exercised optimally by issuers and call bonds only when the bond prices are significantly above the call prices. Thus, a theoretical model that assumes an optimal call policy will underestimate the embedded conversion option. This result will be more pronounced when the conversion option is deeper in-the-money. Our results are not inconsistent with these findings. Besides the above empirical findings, we also examine how liquidity risk can explain our results. Prior literature has well documented the importance of liquidity risk in determining bond

2.47

prices. There is enough evidence to show that illiquidity of bonds results in their prices being lower. For example, Longstaff et al. (2005) use the information in credit default swaps to find the composition of corporate bond spreads. They find that for bonds of all credit ratings, the default risk is the most important component of the yield spreads. This is true even for bonds of the highest rated investment-grade firms. They also find evidence of a significant non-default component in yield spreads. The non-default component of yield spreads is time varying and strongly related to bond-specific measures of illiquidity and overall liquidity of fixed income markets. Chen et al. (2007) use a large sample of 4000 bonds, both investment and speculative grade, to arrive at similar conclusions. They find that liquidity risk is important in determining both the level and changes in corporate bond yield spreads. These two studies provide an explanation of why the determinants of credit risk alone cannot explain the level and changes in corporate bond yield spreads. Assuming that the entire yield spread represents default risk, as was commonly assumed in the past, is incorrect. Also, trying to infer default probabilities from yield spreads is not advisable as the liquidity risk component of yield spreads is not directly related to default risk. While on the topic of liquidity risk, a study by Batta et al. (2010), though not directly related to the issue we are trying to address, tries to determine whether excess returns from convertible bond arbitrage is entirely due to pricing inefficiencies or is partly due to exposure to some systematic risk factors. Their tests show that when liquidity is excluded as a risk factor, a good portion of abnormal returns from convertible arbitrage strategies seems to be driven by underpricing of convertible bonds and overpricing of underlying equity. The authors then explain why liquidity risk is magnified in a convertible arbitrage strategy and show that when liquidity is included as a risk factor, the abnormal returns diminish significantly. Going by all this evidence, liquidity is a priced factor in the bond market, and illiquidity of bonds should lower their market prices. We expect the liquidity of securities like convertible and exchangeable bonds to be lower than that of regular bonds because of various reasons. Some of the possible reasons, besides the ones that apply to regular bonds, are difficulties in valuing these products, complex bond covenants, and asymmetric information between the issuer and the investor. Because of these, liquidity problems

R.S. Mateti et al. / Journal of Banking & Finance 37 (2013) 1018–1028

can be more pronounced for convertible and exchangeable bonds than for regular bonds. In sum, our model for pricing exchangeable bonds takes into account three important risks affecting their prices, namely, interest rate, equity, and credit risks. However, it ignores a relatively less important yet potentially significant risk, the liquidity risk. Ignoring liquidity risk is likely to result in our model prices being greater than what they should be. Also, as pointed out earlier, the optimal call policy as assumed in the model is not often implemented in practice by issuers of convertible or exchangeable bonds. Assuming so in the model will undervalue the embedded conversion option and underestimate the value of the exchangeable bond. This has the opposite effect of ignoring liquidity risk in the model. From our empirical results, it seems that the effect of ignoring liquidity risk in our model outweighs the assumption of optimal call policy by the issuer. This provides an explanation of why for most of the exchangeable bonds in our sample, the market prices are lower than the model prices.

6. Concluding remarks In this paper, we show how to price exchangeable bonds by extending the Das and Sundaram’s (2007) model. Our extended model is a trivariate recombining lattice built in a risk-neutral setting. It represents the stock price of the issuer, the price of the underlying stock, and the risk-free interest rate. It also represents the default risk of the issuer of the exchangeable bond and the default risk of the issuer of the underlying stock by the probabilities of the corresponding stock prices dropping to zero. Representing five processes—the two stock prices, the risk-free interest rate, and the two default intensity processes—in just three dimensions is one of the distinctive features of our model. This feature makes the model computationally efficient. Because we build our model is a risk-neutral setting, we assign probabilities to the 18 branches from each node of the lattice such that the drift rate of the two stock prices is equal to the risk-free rate and the drift of the forward rates satisfies the HJM condition. We also make sure that we preserve the observed correlations between the variables. Besides exchangeable bonds, we can also use the model to price securities like reverse exchangeable bonds, bonds exchangeable to indexes, and bonds exchangeable to commodities. We also show how to estimate some critical non-observable inputs of the model. We show two methods to estimate the volatility of the forward rates. The first method makes use of the current market prices of interest rate caplets to estimate the forward rate volatilities. The second method shows how to estimate forward rate volatilities from historical data on Treasury constant maturity rates. We prefer the first method because the data is current and the information contained in it is forward looking. We also show how to estimate the parameters of the default intensity function. From the default intensity function, we can know the probability of default, which reflects credit risk. The market information used to estimate these parameters are the credit default swap spreads. The CDS market is highly liquid, and there is no better source of credit risk information than this market. The model makes efficient use of this credit risk information and incorporates it in the parameters of the default intensity function. Our model does not take into account liquidity risk. Though this risk is relatively less important than other risks in the model, it can still be significant. This is a limitation of our model. Ignoring liquidity risk results in the model prices being higher than what they should be. The extent to which they are higher depends on the severity of liquidity risk. However, this overestimation in prices can be partially offset by the undervaluation of the conversion option resulting from the assumption of optimal call policy by the is-

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suer. Issuers of convertible and exchangeable bonds do not frequently exercise the embedded call option optimally. On the whole, we believe that these issues do not have a significant negative influence on the performance of our model as our empirical results bear out. We apply the extended model to a sample of 22 exchangeable bonds, and the results are encouraging. The mean absolute percentage deviation between the actual market prices and the model prices is just 2.47%.

Appendix A A.1. Relaxing the assumptions of the extended model In our model, we assume that the correlation of default between the issuer of the exchangeable bond and the issuer the underlying stock is zero. We also assume that the probabilities p21 and p41 are equal. In Appendix A, we explain briefly how to relax these two assumptions. Before considering relaxing the assumption of zero default correlation between the two firms, it is advisable to keep the following issues in mind. We think the assumption of independent defaults is reasonable because the two firms are usually from different industries. We also note that the default correlation between firms decreases as the length of time period considered for simultaneous defaults decreases (Lucas, 1995). Because the time period on our lattice is just a quarter of a year, this further contributes to the default correlation between the two firms being very low. Despite this, if we have to introduce a non-zero default correlation between the firms in our model, we run into the problem of accurately estimating a very small non-zero default correlation. Also, it requires making some assumptions to estimate the default correlation. In effect, we will have a reasonable assumption of zero default correlation being replaced by a different set of assumptions relating to the estimation of the default correlation. Therefore, in most cases we do not think introducing a non-zero default correlation is necessarily a good thing given that it is expected to be very small to begin with. However, in the few scenarios where the reader feels that it is better to estimate the default correlation and use it in pricing of exchangeable bonds, Lucas (1995) will be helpful in estimating the default correlation and revising the conditional joint probabilities of no-default branches and X S2 ¼ 1ðp13 ; p23 ; p33 , and p43). Once we have these probabilities, we can find the conditional joint probabilities of the no-default branches and the other values of X S2 as explained earlier. In extending the DS model to arrive at our model to price exchangeable bonds, we have values of 4 additional variables to determine but only 3 additional conditions. Because we have more variables than conditions, the variables cannot be determined uniquely. The assumption we make of equating the probabilities p21 and p41 shows how we deal with this problem. We see from above why this is a reasonable assumption. However, there are other ways to get around this problem. To get an extra condition we are looking for, one way is to divide the correlation of returns between the two stocks into two parts: the correlation of returns when the interest rate is up, and the correlation of returns when the interest rate is down. Besides giving us the extra condition we need, we think such a division of the correlation makes economic sense too. Interest rate is a systematic factor and its movement may influence the correlation of returns between the two stocks. The difference between the revised model we have after relaxing the two assumptions and the model we presented earlier only relates to the probabilities assigned to various branches of the lattice. The structure of the lattice and the conditions used at the nodes of the lattice during backward induction remain the same.

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