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An empirical study of a financial signalling model Campbell, Alyce 1987

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AN EMPIRICAL  STUDY OF A FINANCIAL SIGNALLING  MODEL  by  A L Y C E RITA  CAMPBELL  M . B . A . , University of Alberta, A THESIS  1982  S U B M I T T E D I N P A R T I A L F U L F I L M E N T OF  THE REQUIREMENTS FOR THE DEGREE DOCTOR OF  OF  PHILOSOPHY  in T H E F A C U L T Y OF G R A D U A T E S T U D I E S Faculty of Commerce and  Business Administration  We accept this thesis as  conforming  to the required standard  T H E U N I V E R S I T Y OF BRITISH C O L U M B I A July  1987  • A L Y C E RITA C A M P B E L L ,  1987  In presenting this thesis in partial fulfilment of the requirements for an advanced degree at The University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission.  Faculty of Commerce and Business Administration The University of British Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date: July  1987  ABSTRACT Brennan and Kraus(1982,1986) developed a costless signalling model which can explain why managers issue hybrid securities—convertibles(CB's) or bond-warrant  packages(BW's). The model predicts that when the true  standard deviation (a) of the distribution of future to the market, characteristics  firm  value is unknown  the firm's managers will issue a hybrid with specific such that the security's full information value is at a  minimum at the firm's true a. In this fully revealing equilibrium market price is equal to this minimum value. In this study, first the mathematical  properties  of the  hypothesized  bond-valuation model were examined to see if specific functions could have a minimum not at o = 0 or o = °° as required for signalling. The Black-Scholes-Merton model was the valuation model chosen because of ease of use, supporting empirical evidence, and compatibility with the Brennan-Kraus model. Three different variations, developed from Ingersoll(1977a);  Geske( 1977,1979)  Brennan and Schwartz(1977,1978),  and Geske and Johnson(1984);  and  were examined. For all hybrids except  senior CB's, pricing functions with a minimum can be found for plausible input parameters. However, functions with an interior maximum are also plausible. A function with a maximum cannot be used for signalling. Second, bond pricing functions for 105 hybrids were studied. The two main hypotheses were: (1) most hybrids have functions with an interior minimum; (2) market price equals minimum theoretical value. The results do not support the signalling model, although the evidence is ambiguous. For the o range 0.05-0.70, for CB's (BW's) 15(8)  ii  Brennan-Schwartz  functions were everywhere positively sloping, 11(2) had an interior minimum, 22(0) were everywhere negatively sloping, and 35(12) had an interior maximum. Market prices did lie closer to minima than maxima from the Brennan-Schwartz solutions, but the results suggest that the solution as implemented overpriced the CB's. B W ' s were unambiguously overpriced. With consistent overpricing, market prices would naturally lie closer to minima. Average variation in theoretical values was, however, only about 5 percent for CB's and about 10 percent for B W ' s . This, coupled with the shape data, suggests that firms were choosing securities with theoretical values relatively insensitive to a rather than choosing securities to signal a unambiguously.  iii  T A B L E OF CONTENTS Abstract  ii  List of Tables  v  Acknowledgements I. Introduction  vii  and Statement of Research Problem  II. The Signalling and Bond-Valuation Models A . The Brennan-Kraus Model 1. Description of the Model 2. Theoretical Context of the Brennan-Kraus Model B. The Bond-Valuation Model  1 6 6 6 18 24  III. Numerical Methods Used in Calculating Bond Values  30  IV. Theoretical Properties of the Bond-Valuation Models  36  V . Testing the Model With Real Data A . Empirical Hypotheses B. Assumptions Made in Adapting Theoretical Solutions to Real Bonds 44 C. Description of the Sample D. Results 1. Evidence on the Feasible Sigma Range 2. Theoretical Prices Versus Market Prices 3. Properties of the Functions 4. Results of Tests on Sigma from the B & S Solution for A M E X N Y S E firms in the Sample V I . Conclusions and Discussion  43 43 48 49 60 61 66 or 92 97  Bibliography  107  Appendix I. Basic Data on Issues and Issuers  112  Appendix II. Sample Computer Programs 1. Ingersoll Solution, Junior Bond Plus Warrant 2. Geske Solution, Junior Bond Plus Warrant, With Warrant Expiring Earlier Than Bond 3. Geske Solution for Junior Discount Bond Plus Warrant With Senior Discount Bond Maturing Earlier Than Junior Bond 137 4. Brennan-Schwartz Solution, Junior Convertibles 5. Brennan-Schwartz Solution, Bonds Plus Warrants  131 132  iv  134  141 150  List of Tables Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table  1. Solutions to the Bond-Valuation Model Used In This Study 2. Characteristics of the Sample 3. Correlations Between Variables, Whole Sample 4. Correlations Between Variables, Convertibles Only 5. Correlations Between Variables, Bond-Warrants Onry 6. Regressions of C V (Market Price/Market Value of Firm) on Input Variables 7. Frequencies of Curve Shape and Location of Market Price Relative to Theoretical Prices, A l l Solutions 8. Frequencies of Function Shape and Location of Market Price Relative to Theoretical Prices, A l l Solutions (a range 0.05-0.70) 9. Some Characteristics of Sample Functions for Convertibles, Ingersoll Solution ( A L L and N O N E Assumptions) 10. Some Characteristics Of Sample Functions for Bond-Warrants, Ingersoll and Geske Solutions 11. Some Characteristics of Sample Functions, Brennan-Schwartz Solution for a Range 0.05-1.00 12. Characteristics of Convertibles Classified by Shape and Location of Market Price Relative to Theoretical Price Range 13. Characteristics of Convertibles (Subsample with Common Equity Listed on A M E X or N Y S E ) Classified by Shape and Location of Market Price Relative to Theoretical Price Range 14. Characteristics of Bond-Warrants Classified by Shape and Location of Market Price Relative to Theoretical Price Range 15. Regressions of [(Gmax, - M P ) / G m a x „]X100 on Variables Used in Pricing (Brennan-Schwartz Solution) 16. Regressions of [(MP-Gmin )/MP]X100 on Variables Used in Pricing (Brennan-Schwartz Solution) 17. Regressions of [(Gmax - M P ) / G m a x „]X 100 on Variables Used in Pricing Plus S H A P E Variables (Brennan-Schwartz Solution) 18. Regressions of C V (Market Price/Market Value of Firm) on Input Variables Plus S H A P E Variables 19. Statistics on o for Firms on N Y S E or A M E X 20. Frequencies of Curve Shape and Location of Market Price Relative to Theoretical Prices (o Range Defined by Standard Deviation of Stock Returns) 21. Differences Between Market Price and Maximum and Minimum for Issues from Firms with Common Equity Listed on N Y S E or A M E X (a Range Defined by Standard Deviation of Stock Returns) 22. Some Sources and Directions of Error in the Pricing Functions Relative to Market Value A l . Issues Included in this Study A 2 . Market Value (MV), Debt Ratio (D), j, R-C(%), and Exercise Premium (EP) for Issues in the Sample A 3 . New Debt Ratio ( D J , New Capital Ratio (CV), Maturity (MAT), Call Price at Issue (CALL) , Rating (RT), and a of firm (oMP) from Brennan-Schwartz Solution for Issues in Sample 0  39 52 54 55 56 58 63 65 68 69 70 74 77 78  7  81  7 0  7 0  83  7  v  86 90 93 94 95 101 113 116 119  Table A 4 . Data on Theoretical Prices Generated by the Brennan-Schwartz Solution: Location and Value of Maximum and Minimum Values Table A 5 . Data on Theoretical Prices Generated by the Brennan-Schwartz Solution: Theoretical Values at a = 0.05, 0.50, 0.70 Table A 6 . Differences between Maximum Value and Market Price and Minimum Value and Market Price for the Brennan-Scwhartz Solution  vi  122 125 ..128  ACKNOWLEDGEMENTS I am very grateful  to Alan Kraus, Ron Giammarino, Eduardo  Schwartz,  and Michael Brennan for many helpful discussions during the period when I worked on this project.  While completing my doctoral studies I was very  fortunate to receive fellowships from Suncor Ltd., the Leslie G. Wong Memorial Fund, and the Social Sciences and Humanities Research Council of Canada, and I am very grateful for this financial support. M y special thanks go to Duncan and Topaz Campbell.  vii  I. INTRODUCTION AND S T A T E M E N T OF R E S E A R C H P R O B L E M That a firm's financial decisions can perhaps reveal to the market information known only to managers/insiders has been demonstrated in a number of analytical models. Since managers/insiders have relatively unrestricted freedom to choose dividend policy, degree of leverage, and mix of contract types (equity, debt, preferred, loans and so on), and presumably make their choices based on all information, outsiders can consider the choice to be a signal  (perhaps  involuntary and/or noisy) of this private knowledge. Examples of such signalling models are in Leland and Pyle(1977), Ross(1977), Bhattacharya(1979,1980), Downes and Heinkel(1982), and Miller and Rock(1985). As noted by Spence(1976), there are two basic types of signalling mechanisms: mechanisms which are costless in equilibrium such as those based on contingent claims contracts ; and 1  mechanisms which require the signallers to suffer a loss of utility either directly through expenditure of resources or indirectly through suboptimal portfolio holdings. "Cost" is measured as the difference between equilibrium utility levels in the market with asymmetry and in the same market with full information . 2  A contingent claim is an3' claim whose payoff is dependent upon the particular outcome of any event which has several possible outcomes. Thus a wager at the races is a contingent claim. However, the term "contingent claim" has come to mean a security whose payoff is dependent on the value of some other underlying security or asset. The basic examples are call and put options on stocks. A n equilibrium is a price system such that the market clears. A Pareto-optimal equilibrium is a price system such that the market clears, every participant attains the highest possible utility, and at least one participant would be worse off if the price system were changed. Pareto-suboptimal equilibria are equilibria that arise in a market in which there are barriers to free trade such as restrictions imposed by the structure of trading institutions, imperfect information, or features of an asset which prevent it from being privately owned. The equilibria are the best possible given the constraints on free trade but some individuals have a utility level lower than would be attained in the unconstrained market. 2  1  2 Many finance models which yield separating equilibria  3  rely on mechanisms of  the latter type, so the equilibria are costty and therefore  Pareto-suboptimal  compared to corresponding full information equilibria. Direct empirical tests of these models have been few. Most commonly, event studj' methodologj' has been used to look for "information effects",  with the information revealed not tightly  specified". The reason for the relative rarity of direct tests is simple: the information supposedly revealed is not easily parameterized measured  or not easily  or both. In contrast, the signalling model of Brennan and  Kraus(1982,1986) is based on the use of contingent claims, so yields costless equilibria. Also, the model lends itself to testing because the information supposedly being revealed is well-defined, being some or all parameters of the distribution of future firm value. In particular, Brennan and Kraus showed that the issuance of a bond with an attached bond-warrant  package ) 5  stock-like feature (a convertible or a  can signal insider information about o, the  standard  deviation of the distribution. This is the model's third attractive feature:  there  When there is uneven distribution of information about the quality of goods/firms in an economj' there is the possibility that equilibria will develop such that the market clears but the quality of goods/firms is not revealed. This is called a "pooling" equilibrium because the market price is the same for goods/firms of different quality. In such an equilibrium it is not possible to state with certainty the quality of every good/firm uniquely. A n equilibrium in which the market clears and the quality of every good/firm is fully revealed, so that prices reflect differences in quality, is called a "separating" equilibrium. A full information equilibrium is naturally fully separating, but a separating equilibrium can be suboptimal compared to the full information equilibrium, if the signalling mechanism is costly. For example, Dann and Mikkelson(1984) examined the reactions of stock prices to the issuance of convertible debt, to see if financial information was released through restructuring. Other examples are Masulis(1980a,1980b,1983) and Dann(1981). A convertible is a bond which can be converted at the holder's option into some preset amount of some other security. In this study the other security is always common stock. A warrant is a security which gives the holder the right to buy common stock from the firm at some preset price per share. fl  5  3  are so few models which account for the appearance of these complex hybrids. In classical economies, these hybrids give no advantage over straight debt or straight equity, so their appearance is an oddity. In real markets,  some  justifications are that convertible debt is "cheaper" than regular debt and/or that convertibles are a way of selling common stock at a price above the existing market price. Neither explanation makes economic sense. Convertibles are riskier than regular debt, so true cost of convertible debt is greater than cost of straight debt. The notion of selling common stock at a premium presupposes  an  eventual rise in stock price followed by conversion, which is never certain at time of issue. One true advantage of convertibles is that for any firm  the  coupon rate on a convertible will be lower than on a straight bond, so fixed payments  are lower. Further, this capital investment may become "permanent", if  the firm grows in value. These features may be especially attractive to rapidly growing firms. Brennan and Kraus(1982) first demonstated  that if a firm's earnings  are  a convex combination of random variables with joint distribution known to the market but precise combination known only to managers,  the characteristics of  the security chosen to raise capital can costlessly reveal management's knowledge. In fact, in a one-period world with perfect markets a fully revealing, separating equilibrium is generic. When the marginal distributions of the random variables are normal with a common, known mean but different standard deviations, contracts with payoff functions similar to the payoff functions of hybrid securities form a subset of the contracts that can be used to signal the particular combination, which simultaneously signals the standard deviation. This case is formally equivalent to that in which the parameter  of location for any firm's  4 earnings is known and the "family" of possible distributions for that firm's earnings is defined by known upper and lower bounds on the standard deviation. Brennan and Kraus(1986) extended the model to a world with a continuously changing firm value, allowing explicit bond-valuation models derived from contingent claims analysis to be incorporated into the theory. In this version the distribution of firm value is assumed to be defined by a lognormal diffusion process. Such a process is completely described by two parameters: a, expected total (possibly risk-adjusted)  return on the firm; and a, the  standard  deviation of of return on the firm. In this case a is assumed known, so the "family" of possible diffusion processes is defined by known upper and lower bounds on a. This extension is important because these bond-valuation models, particular^ those developed from work by Black and Scholes(1973), have features which make the direct testing of the signalling model feasible: •  Independent evidence suggests that these bond pricing models do, in fact, yield theoretical values close to market prices [for example, see King(1986) and Jones et aZ.(1984, 1985)].  •  The equations are relatively simple, and in principle all the input parameters needed to determine real issue are  the bond pricing function for any particular  observable.  The research problem addressed in this dissertation is the testing of this signalling model, using each of three different solutions to the Black-Scholes-Merton differential equation for contingent claims as the bond valuation model. The investigation of the problem involved two steps. First, some mathematical properties of the three bond-valuation solutions were examined to  5 see if they matched those required by the Brennan-Kraus model. In the second step the properties of the theoretical bond pricing functions for 105 new issues were studied to see if they conformed to the predictions of the signalling model, and market prices were compared to the values predicted by the signalling model. A s well, to see if the a predicted by the bond-valuation model was feasible, for a subsample of firms the a of the firm as determined from the location of market price on the theoretical function was compared to the range of o's defined by the standard deviation of returns on the common stock and this value adjusted  for leverage.  This dissertation has seven chapters. Descriptions of the Brennan-Kraus model and the bond-valuation model are in Chapter II. Chapter III is a discussion of the numerical methods used to generate theoretical bond values. Results of the investigation of the mathematical properties are given in Chapter IV. The data and specifics of the tests for the  105 issues are described in  Chapter V . Empirical results are reported in Chapter V I . The final  chapter  includes a discussion of the results and some suggestions for related future work.  II. T H E S I G N A L L I N G  A. T H E B R E N N A N - K R A U S  1. Description of the The  AND BOND-VALUATION  MODELS  MODEL  Model  model economy is one in which the securities market is complete,  costless (no transactions/bankruptcy costs), and competitive. Taxes are neutral. Managers aim to maximize true current shareholder wealth (that is, managers aim  to maximize the true, full information value of the securities of the existing  shareholders). Investors are all risk-neutral or risk averse utilky maximizers with rational beliefs. With these assumptions and with symmetric information (and homogeneous beliefs), there is no optimal debt/equity ratio for any firm, and changing capital structure is costless. A t time 0, some firms issue new securities to finance new positive-NPV projects and seek funding at least equal to the cost of the project. This cost is known to all investors. There are no restrictions on the type of new security the firm may issue. These firms may or may not have existing projects which are generating cash flows, but if so, the distributions of outcomes of these projects is known with certainty by all investors. The  firms may or may not have outstanding securities other than equity;  however, all securities of any firm, whether existing or new, are assumed to be legally enforcable contracts which specify payouts to the holders as a function of total firm value (hence, of return on the firm's assets), and not as a function of the return on any particular project the firm may be operating. Existing debt issues are assumed to be protected by perfect "me-first" rules.  6  7 Each firm has only one new project available to it and the distribution of returns from this project is fixed. A t time 1, the returns from all the  firm's  projects are revealed, and the payout due to each contract outstanding at that time is fixed. One distributional assumption is made: the distribution function of the post-investment returns on the firm's productive assets is assumed to be characterized by two parameters only, labled a and o. The parameter a determines central tendency (that is, a is mean return for normal distributions, and mean total return for lognormal diffusion processes). The parameter a determines dispersion (that is, a is standard deviation of return for normal distributions and instantaneous  standard deviation of return for diffusion  processes). The market price of each contract at time 0 is a function of the payoff function stipulated in the contract and the post-investment distribution of returns, since payoffs are contingent on firm value at time 1. If I now added the assumption that managers and market participants have the same, correct, information about the distribution functions of the new projects, the result would be a classical economy for which a Pareto-optimal competitive equilibrium could be shown to exist. Suppose, instead, we assume that for some firms only insiders/managers know with certainty both c and a of the distribution of returns at time 1, but investors know only a and upper and lower bounds on a. That is, investors can only define a "family" of distributions of firm value, all with the same a, since they do not know with certainty the value of a . 6  ° N o assumption is made about investors' beliefs about the distribution of firm a's within this range. It is unnecessary because beliefs are revised with the observation of the security choice, as shown by the principal results of the Brennan-Kraus model.  8  The questions addressed by Brennan and Kraus were: under the assumptions made, can a Pareto-optimal, fully revealing (thus fully  separating)  equilibrium exist, and if so, can any conclusions be drawn about how securities are priced? The answer to both questions is yes; however, the proofs require careful interpretation of the assumptions, so these merit further discussion. The key assumption is that managers are assumed to maximize the true intrinsic worth of the securities held by the original shareholders. In a perfect classical economy this is tantamount  to maximizing firm value, so all  positive-NPV projects would be undertaken. Further, in financing decisions this is equivalent to maximizing the difference between price received for any new security and its true intrinsic worth. However, in the classical economy this difference would be zero, since outsiders have the same information as insiders. Is the assumption that managers maximize true full information value reasonable for a market with asymmetric information? As demonstrated  in models  such as those formulated by Akerlof(1970) and Myers and Majluf(1984), when there is asymmetric information and no feasible signalling mechanism, existing shareholders may benefit from the firm not taking all positive N P V projects. The reason is that offers made by outside investors may not reflect true intrinsic worth of the security. If the bid price is too low then the gains from the new project to the shareholders may not outweigh the loss due to underpricing of the security. Further, if existing shareholders have made side bets on the terminal value of the firm, or if there are existing debtholders and the gains from the new project accrue to the debtholders without benefitting the shareholders, may not be unanimous support for undertaking some particular positive-NPV project.  there  9 To eliminate these latter problems, a number of assumptions about market structure and information structure are made. The market is assumed to be complete, and shareholders are assumed to have common prior beliefs about firm value so have no desire to make side-bets. Thus all else equal, the shareholders would unanimous^ support firm value maximization and consequently the acceptance of all positive-NPV projects. To eliminate the possibility of the rejection of positive-NPV projects just because the gains are shifted to existing bondholders, strong bond covenants are assumed and further, bankruptcy is assumed costless (that is, costs of monitoring and enforcement of contracts are assumed away). Finally, in the Brennan-Kraus model managers are free to choose any security type and capital structure. If a consequence of this is that a feasible signalling mechanism exists, then no positve-NPV projects would be omitted because of incorrect pricing. Given these auxiliary assumptions, the assumption of true value maximization on the part of insiders/managers is reasonable. Another barrier to value maximization may arise if the firm has available to it two or more projects. Shareholders have incentive to substitute high-variance for low-variance projects after the new securities have been issued because of their limited liability. This is eliminated by assuming that only one project is available to any  firm . 7  Two other consequences of these assumptions should be noted. No agency problems other than asymmetric information influence the decisions of investors and managers/insiders. Also, there is no optimal capital structure, once the truth A n alternative explanation of convertibles proposed by Green(1984) is that they serve to reduce the incentive of shareholders to shift to riskier projects. The model is mathematically very similar to the Brennan-Kraus model discussed in this dissertation.  is revealed. The Brennan-Kraus economy is thus unrealistic, compared to a real economy, but less so than a classical perfectfy competitive economy. Furthermore the effects of asymmetric information can be examined in detail. This understanding can be the foundation of more complex, more realistic models. Brennan and Kraus provided results on the signalling of both a and a. Only the three relevant to signalling a are presented here. Result I yields what Brennan and Kraus called the "lemons" property of prices of fully revealing (fully separating) equilibria. Result II shows the necessary and sufficient conditions for the existence of a fully revealing equilibrium. Result III specifies the geometric property that payoff functions must have if a is to be signalled. Payoff functions of convertibles and bond-warrant functions exhibit this property. Let z be the vector of parameters that describe the characteristics of the security (for example, for a debt issue, this would include face amount,  payouts  in event of bankruptcy, payments in event of mismanagement of the firm; parameters could be added for timing of payments  and so on); z is an element  of the set of feasible security types labled Z and all investor know the of Z. Let o be the parameter  of dispersion; a is an element of the  S = {CT|L< o<M}, where L and M  other  elements  set  are the lower and upper bounds on the value  of a. Define P(z) as the market price of a security as a function of its characteristics. Let z be the parameter firm with a = & . Note that  vector of the security chosen by the  z is chosen from the set z — {z\zeZ and P(z)=W},  where W is the outlay required to initiate the new project. Let o* be the a the market uses to price the security with characteristics  2. H(z,a) is the  full  information value of a security given its characteristics and the firm's a. That is, H(z,o) is the market price of the security in a classical perfect economy,  11 where z is naturally observable and o for any firm is known with certainty. B y definition, H( z , 6 )= JY(V; z )f(V; d )dV; where Y ( V ; z ) is the payoff function and f(V; &) is the density function (possibly risk- and time- adjusted) over firm value V . Since terminal V depends upon return, H(>) could alternately be written as a function of Y(«) and the density function for the returns to the firm. Since, by assumption, investors have rational beliefs, they use the same function to price the security but must use an estimate a* of &, when they do not have the managers' private knowledge. Brennan and Kraus(1982,1986) then showed: Result I : If a fully revealing (that is, a fully separating) signalling equilibrium exists, then investors can and do correctly price any new security by assuming the worst: that is, investors first find the minimum possible full information value of the security over all feasible a's, then set market price equal to this minimum value; the a of the firm that issued the security is equal to the a* that yields the minimum value. This is equivalent to investors assuming that the security is issued by the firm with the a which makes the security worth the least and being correct. Algebraically speaking: • •  The pricing function is: P(z) = m i n H(z,o) for all feasible z and a; For any particular security with particular characteristics t, P( z ) = H( z ,a*), where a* = arg min^ H ( 2 , a ) and o * = 8 , the true a of the firm choosing 2. In other words, in equilibrium, at o * = & , 8H/9o = 0, and 9 H / 3 a > 0, the standard first- and second-order conditions for a minimum. a  2  2  Result II: A fully revealing equilibrium can exist if and only if for each firm with & there exists some feasible z such that when 2 is chosen by the firm, 9 is the o that minimizes the security's full information value and this minimum value is equal to the outlay required. Algebraically: •  For all firms $ =  {z|2 Z  Result III: Under between the known upper requires that the contract over some range of firm values.  e  and  & = arg m i n  a  H ( z , a ) and H ( z , &) = W } * 0 .  the distributional assumption, for firms with & falling and lower bounds, a fully revealing equilibrium value function as a function of firm value be concave values and convex over some other range of firm  12  The logical basis of result I is as follows. By assumption every manager maximizes the full information value of the equity contracts. This is equivalent to finding  z  such that the function P(z)-H(z, d) is maximized. However, given  equilibrium and rational, competitive behavior on the part of investors, maximum possible is zero, so firms choose  the  z such that P ( z ) = H ( z , & ) . In other  words, even though managers would like to the fool the market by offering securities which have negative N P V at market price, this is not possible given rational behavior and equilibrium. Now suppose there was some feasible z* such that P(z*) was greater i than the minimum feasible full information value. This would imply that for some firm with o — a , P(z*)-H(z*,o ) is greater than zero, which would contradict 0  0  the assumption of rational behavior. Thus such a z* cannot exist, or no  firm  can mimic another firm and profit by i t . 8  The formal proof that 3H(«)/3a = 0 and 3 H ( « ) / 3 a 2  2  >  0 is as follows : 9  The firm's problem is assumed to be: • M a x P(z)-H(z,o) When solved it must be true for all a that [9P(z(o))/9z] = [3H(z(a),o/az]; (1) 9 P(z)/9z < 9 H ( z ( o ) , o ) / 9 z . (2) • A fully revealing equilibrium requires P(4) = H ( 4 , a ) . (3) Then, from differentiating (3) [9P(-)/9z][9z/9a] = [9H( • )/3z][3z/9a] + [9H(.)/9a]; =>9H(- )/9a = 0; [from (1)]. From differentiating (3) twice [ 9 P ( ' ) / 9 z ] [ 9 z / 9 a ] + [9P( • ) / 3 z ] [ 3 z / 9 o = [9 H(-)/9z ][9z/9o] + [9H(.)/9z][3 z/3a ] + [ 9 H ( - ) / 3 o ] + 2[9 H(-)/9z3a][9z/3a]; (4) * A d i f f e r e n t proof based on game-theoretic principles is provided by Milgrom and Roberts(1986), who discussed the equilibrium that results when investors use a "sophisticated" strategy in assessing any firm's reporting system for communicating firm value. Investors who are "sophisticated" act as proposed by Brennan and Kraus and assume the worst. This proof was provided by A . Kraus. z  2  2  2  2  2  2  2  9  2  2  2  2  2  2  2  2  2  2  13 (1) * [9 H(-)/3z9a][9z/9a] = {[9 P(-)/9z ] - [9 H(-)/9z9 ]}[9z/9o] •» [9 H(.)/9z9a][9z/9a] < 0; (5) [from (2)]; Combining (4) with (1) =* [ 9 H ( - ) / 9 a ] = - [3 H(-)/3z9c;][9z/30] 2  2  2  2  2  2  2  2  2  =» 9 H ( - ) / 9 a 2  2  2  >  0 [from (5)] • .  The signalling mechanism is costless because shareholders  do not  suffer  any loss of utility due to having to issue a suboptimal security (capital is irrelevant) and there are no increases in transactions/bankruptcy  structure  costs due to  issuing some particular security type. This would not be true in all economies. For  example, suppose for some firm the optimal new security is new equity. In  the Brennan-Kraus world this does not affect the firm's value, but in a world otherwise the same but with interest payments payments  tax-deductible and dividend  not, where firms should have as much debt as possible, the  would have to suffer deadweight costs in adopting a capital structure lower debt/equity  with a  ratio.  The second result follows from the requirements firm  firm  there is no feasible  z  of the first. If for some  such that the sale of the securky yields the  firm  at  least W when correctly priced, the firm will not invest nor will it issue new securities, and there can be no fully revealing equilibrium. On the other hand if for  every firm there exists a feasible security that yields at least W when sold,  every firm can invest and further, firm  by the assumption of value maximization, no  will choose an inappropriate z. The third result follows directly from the first result and the restriction of  the distribution function to a two-parameter  family, and defines more specifically  14  the properties of Y ( V ; z ) . Only a payoff function with the property described can lead to a pricing function with an interior minimum at some o, given the distribution restriction. These results were developed in a one-period world, in which there is no real time difference between issue and payoff. However, as Brennan and Kraus(1986) showed, the results are extendible to a world where securities are continuously traded or firm value is governed by a diffusion process over a finite interval. Assuming that a fully revealing equilibrium exists, all that is required is that at issue the instantaneous first-order and second-order partial derivatives of H with respect to o be zero and greater than zero, respectively. Several points should be noted which may make the basis of this extension clearer. The fact that partial derivatives are being considered does not mean that the o is assumed to be changing. Rather investors are taking V and a to be fixed, but a to be unknown. In their deliberations they are finding the a that best fits observations and their beliefs about managerial behavior;  finding  the minimum is the means of finding the best fit. Secondly, the parameters that are assumed constant should be carefully noted, as these are not completely analogous to the parameters assumed constant in the one-period world with normally distributed firm value. Note that it is a, the risk-adjusted expected total return, that is taken to be known and constant, which means that changes in beliefs about a do not lead to revisions of V . Under what conditions is such an assumption reasonable? In equilibrium in a risk-neutral economy, such as modeled by Black and Scholes(1973), a = r, the riskless rate for all firms. Then V depends only on a, but the value of any security with payoff contingent on the future value of V  depends  both on V and on o. In a risk-averse world the  c is equal to the riskless rate plus an adjustment  for risk such that if the drift  in the stochastic process were set to a, under risk-neutral valuation the value of the firm would be V . Merton(1973a) described an economy where this is the case. Constantinides(1978) used this technique to show how projects could be priced as if in a risk-neutral economy. In application to real economies and real firms with an asymmetric information problem of this type, Results I and III imply that securities with value functions convex in firm value (such as warrants or new equity on a levered firm) will be issued only by firms with the minimum possible a and securities concave in firm value (such as straight debt) will be issued by firms with the maximum possible a. Those firms with non-extreme  a's will offer  convertible bonds or bond-warrant packages with the characteristics that produce a minimum at the firm's true a. What are the specific characteristics z that could be adjusted to yield value functions with minima? The Brennan-Kraus model is very flexible in that the only givens for any firm are: its existing debt, the intrinsic expected value and a of its assets including the new project, and the required investment outlay. The firm is free to choose: 1.  7, the fraction of the firm the warrant- or convertible-bond-holders receive upon exercise;  2.  the exercise price;  3.  the coupon rate on the principal, which determines the future value owing;  4.  the maturities of the bond and warrant components of the security.  These are integral to the complete specification of a basic contract which is a hybrid of debt and equity. The firm is also free to make the contract more  complex by adding features such as a sinking fund provision and a call privilege, and can choose the amount and timing of interim payouts (the coupons). Dividend policy is also a choice variable on which bond value depends indirect^ and some firms may choose to include in the bond contracts restrictions on payouts to stockholders. These results and consequences relative frequencies  can be used to make predictions about the  of security types that will be issued in an economy in which  a number of firms are experiencing this problem. Assume all securities can be classed into one of three types: equity, convertible bonds/bond-warrants  packages,  and straight bonds. There are N firms about to issue new securities, and k N of these have an information asymmetry problem, (that is, k is the fraction of firms with a unknown to the market). The behavior of any individual firm with a known is unpredictable since there is no optimal capital structure.  Such firms  should be indifferent to type of security issued, so collectively they should choose equity, straight debt, and hybrids with equal frequency. For any firm with a unknown, assume that the probabilities of the firm's true a being at the lower bound, at the upper bound, and in between are, respectively, p q=l-p -p . 1  2  Then the expected frequencies  1 ;  p , and 2  of equitj% straight debt, and hybrids  are, respectively, f ={[(l-k)N/3] + p (kN)}/N, f ={[(l-k)N/3] + p (kN)}/N, and 4  1  2  2  f ={[(l-k)N/3] + q(kN)}/N. If k = 0, f =f =f = 1/3. If k = l , f ^ p ^ 3  f =q. 3  x  If 1 >  k >  0, f i  >  2  3  f, only if p  x  > q and f  2  > f  3  f = p „ and a  only if p  2  > q.  However, the former would imply that investors systematically underestimate the firm's true a, while the latter would imply an opposite bias. Such bias is inconsistent with rational beliefs. Note also that if firm o's are continuously distributed over the range of feasible o, p  x  and p  2  go to zero and q goes to 1.  17 In general then f  3  > f  1  and f  > f , if k is not equal to zero. In other  3  2  words, we expect more hybrid issues than either equity issues or straight debt issues in an economy with an asymmetric information problem. Also, of course, if firms with known a were systematically to choose either straight debt or straight equity then the relative frequencies would also be affected. There is, however, no economic reason for such selectivity in the Brennan-Kraus model. If taxes or transactions costs were added, say to make debt preferrable, then signalling would not be costless, in that issuing a hybrid or equity would be Pareto-suboptimal. A fully revealing equilibrium might not even exist, in that for some firms the gains from issuing a suboptimal security might not outweigh the costs. The general conditions for a costly revealing equilibrium can be found in Riley(1979) ° . 1  A numerical example ma} help to clarify this result further. Suppose 7  there are equal numbers of firms with a's of 0.1, 0.2, 0.3, and 0.4 (the only possible a's), and that at time 0, o is unknown to the market for 50 percent of the firms issuing new securities. There are three contract types: warrants/equity, hybrids, and straight debt. Of the firms with a unknown, the model predicts that those with a = 0.1 would issue warrant-like securities; those with a = 0.4, straight- debt; and the remaining firms, hybrids with pricing functions with a minimum at either 0.2 or 0.3. Given that firms with a's known choose all types equally, the following proportions by type would be expected: warrant-like, 29.17%; hybrids with an interior minimum, 41.67%; straight debt, 29.17%. If the range of a were continuous from 0.1 to 0.4, the expected proportions would be  The model might also have to changed fundamentally. Expected value of the firm would also be unknown with certainty, since this would depend on choice of financing.  16.67%, 66.67%, and 16.67%, respectively, since the probability of a firm's a being at either upper or lower bound goes to zero. If there were 100 percent asymmetry the proportions would be 100% hybrids. With 0 percent asymmetry the proportions would be 33.33% warrant-like, 33.33% hybrids, and 33.33% straight debt.  2. Theoretical  The  Context of the Brennan-Kraus M o d e l  Brennan-Kraus model is one of many that have been formulated to  study the effects of asymmetric information and conflicts of interest that arise in real markets. Two broad classes of problems relevant to financial decision-making have been defined: 1.  conflicts of interest between managers and residual claimants that arise when control is separated from ownership;  2.  conflicts of interest between shareholders and bondholders that arise because shareholders have limited liability yet control the firm's operations; within this class four mechanisms for transferring wealth from bondholders to stockholders have been identified: a.  dividend payout beyond what operations can support;  b.  issuance of bonds of the same or higher priority as existing bonds;  c.  substitution of high-risk for low-risk projects;  d.  failure to invest in positive-NPV projects when the benefits would accrue mostly to bondholders.  Most cases can be given one of two lables: 1.  "adverse selection" in which persons on one side of a contract are not able to measure with certainty some payoff-relevant characteristic that is known  19 with certainty by persons on the other side of the contract; 2.  "moral hazard" in which not only is one side of the contract not able to measure with certainty some characteristic, as in adverse selection, but also the informed side is able to change this characteristic.  The Brennan-Kraus model is the result of a study of a particular adverse selection problem within the broad class of problems related to shareholder-bondholder conflict. However, the four mechanisms of wealth transfer noted are situations involving moral hazard, so the Brennan-Kraus study is unlike and somewhat unrelated to many preceding studies. In most cases of adverse selection, the problem is that some insiders (managers/original shareholders) would benefit from being able to reveal their information to the market, but other insiders would benefit from misleading outsiders. If there is no mechanism operating that forces all insiders to be honest, the outsiders cannot distinguish between the truth-tellers and liars and so develop a set of beliefs about the distribution of truth-tellers and liars. Outsiders then set prices according to these beliefs such that they minimize their expected loss of utility due to misrepresentation. As noted by Spence(1976), two types of signalling mechanisms for attaining a fully separating equilibrium are available: 1.  dissipative, in which the truth-tellers suffer some loss of utility in order to distinguish themselves from the liars;thes mechanisms lead to Pareto-suboptimal equilibria;  2.  nondissipative, in which payoffs are made contingent on some observable event, so truth-tellers can costlessly distinguish themselves from liars by their choice of contract terms. A typical model of adverse selection starts with the assumption that for  some payoff-relevant characteristic, there are "good" firms/buyers/sellers and "bad" firms/buyers/sellers. The categorization may be dichotomous or continuous. Since price is higher for "good" firms than for "bad", the "bad" firms benefit from trying to convince the market they are "good". Some rules controlling trading activity are added, then the model economy is checked for the existence of equilibria, and any equilibria are studied to determine whether they are pooling or separating, how close they are to being Pareto-optimal, and whether they can be changed and improved through the addition of some extra  feature.  The classic discussion of this problem is the "market for lemons" paper by Akerlof(1970). Akerlof showed that the cost of dishonest}' is that only "lemons" are traded: buyers bid so low that those with high quality products leave the market. He also noted some extra-market mechanisms that would improve the welfare of all: guarantees, licensing or certification, and reputation. Another classic example is in Rothschild and Stiglitz(1976) who showed that asymmetric information can easily lead to markets for which no competitive equilibria exist. They also explored more fully the notions of pooling and separating equilibria. There are many other examples in which these insights are applied to specific markets. A recent example of a study of adverse selection in bond markets is in Flannery(1986). In his model, "bad" firms are those for which the probability of a favorable outcome from a project is low. He showed that when short- and long-term bonds can be issued, with no transactions  costs  there will be a pooling equilibrium with only short-term bonds issued, but with transactions costs "good" firms will issue short-term bonds while "bad" firms will issue long-term bonds. In both cases, the equilibria are Pareto-suboptimal compared to a full information equilibrium. Some other examples of markets with  21 costly competitive equilibria are in Leland and Pyle(1977), Bhattacharya(1979), Ross(1977), Campbell and Kracaw(1980), Myers and Majluf(1984), and Miller and Rock(1985). The list of nondissipative models is much shorter: some examples are in Bhattacharya(1980) and Heinkel(1982). Haugen and Senbet(1979) discussed in general the use of contingent claims as a costless signalling device. The Brennan-Kraus model, being non-dissipative, is attractive because there is no economic waste. It is unlike many predecessors in several other  aspects.  First, the information being communicated is inherently neither "good" nor "bad" news. There are no "bad" firms which seek to convince the market they are "good" firms. Given that theoretical functions have a minimum, every insider would benefit from misrepresentation, and no insider has incentive to expend resources to force a separating equilibrium. Second, unlike in Akerlof-type models, insiders cannot withdraw from the market, since there is no other source of capital to support the new project. A third difference is that unlike many authors, Brennan and Kraus started by assuming that a fully separating costless equilibrium exists and deduced pricing rules compatible with this assumption and with rational behavior. That hybrids can be a signalling mechanism was an incidental finding that makes the model even more interesting. There are few other well-developed models which explain the appearance of hybrids. Moral hazard problems were assumed away by Brennan and Kraus. However, the existence of moral hazard problems has been suggested as the reason for the complexitj  r  of bond contracts. Conversion features,  sinking fund  provisions, rules of seniority, and restrictions on payouts are examples of features which increase the complexity. Given that at least four mechanisms exist for funnelling wealth from bondholders to shareholders, it is natural that bondholders  22 would demand contracts with provisions designed to protect them. Many  authors,  such as Jensen and Meckling(1976), Myers(1977), Smith and Warner(1979), Mikkelson(1981), Kalay(1982), Ho and Singer(1982), and Green(1984), have looked at the role of various typical provisions in debt contracts for resolving conflict and/or reducing welfare loss. Of these the most relevant to this study is the Green(1984) model which provides an alternate explanation for the appearance of convertibles: perhaps convertibles are used to eliminate the incentive that shareholders  have to  substitute higher variance projects for lower variance projects. Convertibles have this effect because, holding  firm  value constant,  a switch to high risk projects  not only decreases the value of the debt portion of the convertible, but also increases the value of all the equity outstanding, including the equitj' portion of the convertible. Thus the amount of wealth transferred  to shareholders  through  the switch is not as great as with straight debt. If the convertible contract is correctly designed, all losses in value in the debt portion are exactly offset by gains in the equity portion, and the wealth of both shareholders  and new  investors is unchanged. Consequently, shareholders have no incentive to switch projects. In the Green model the argument is developed one step further: the terms of the convertible are such that the value of shareholders' wealth is at maximum when the risk-minimizing mix of high variance and low variance projects is chosen, holding firm value constant. In this case any shift, whether to higher-variance or to lower-variance projects, results in a decrease in value of the shareholders' wealth. Since losses mean gains to the new investors, the value of the convertibles must increase with any shift. Thus, as in the Brennan-Kraus  23 model, the theoretical value function of the convertible is at a minimum at the firm's "true" a. No empirical tests of this model have been reported in financial journals. Green's conditions for elimination of the variance-shifting incentive are stronger than necessary. New investors would be satisfied with a contract that has the same value regardless of the variance of the projects the actually choose  1  1  shareholders  .  Barnea, Haugen and Senbet(1985) extended the Green model by demonstrating that shareholders  can simultaneously resolve the risk incentive  problem and the "perks" problem  1 2  by issuing bonds with both a conversion  feature and a call feature and issuing to the manager stock options. This use of several different options at the same time shows the potential of these models to account for the observed complexity of firms' capital structures. A large body of literature discusses general capital structure  problems,  such as whether or not there is an optimal level of debt or whether or not a change in capital structure is "good" or "bad" news. The most famous of these are, of course, Modigliani and Miller(1958,1963) and Miller(1977). B y assuming perfect capital markets with no taxes, Brennan and Kraus eliminated the issues raised by this stream of research. Thus changes in capital structure are neither "good" news nor "bad" news in the Brennan-Kraus world. This means that the model does not and cannot explain, for example, the negative stock price reaction ~ Other options can also eliminate this moral hazard problem. Barnea, Haugen and Senbet(1980) showed that incorporating a call provision into the bond contract could be effective. The "perks" problem arises when a manager has incentive to consume perquisites such as expensive furniture, expense accounts and so on, because he is only a partial owner of the firm. The classic work on this problem is that of Jensen and Meckling(1976). T  1 2  1  to the issuance of convertibles reported by Dann and Mikkelson(1984). In the Brennan-Kraus world, the reaction in stock price would vary from firm to firm according to the firm's circumstances. A systematic  response would  not be. expected.  B.  T H E B O N D - V A L U A T I O N  The  M O D E L  Brennan-Kraus model is quite general in that it does not depend  upon any particular bond-valuation model, but shows only what properties the pricing functions must exhibit if there is to be a fully revealing equilibrium. The only restriction on any security's characteristics is that jointly  they must lead to  a value function with a unique minimum. Empirical testing requires that the bond-valuation model used by investors and insiders/managers be known. The  model adopted in this study is a well-known model arising from  contingent claims theory, specifically the Black-Scholes-Merton equation. That corporate liabilities are contingent claims or portfolios of options was first noted by  Black and Scholes(1973) and this insight was developed further by  Merton(1974,1977),  and extended by numerous authors. In this framework, every  security of a levered firm is viewed as a contingent claim on the total value of the cash-generating assets of the firm. Stocks can thus be seen simply as options to buy these assets from bondholders. Equivalently, ordinary noncallable risky bonds are equivalent in value to a default-free bond, less a put giving the asset-owners (stockholders)  the right to sell the assets to the bondholders.  According^, every security can be priced using the same general equation which depends on only a few constants: default-free interest rate, current value of the firm,  a of the firm's assets, and timing of payouts. Although the model has not  25 been tested extensively, the results of the few reported studies support the model's validity [King(1986), Jones et aZ.(1984,1985)]. All contingent claims analyses of continuously traded securities are similar. First an author specifies some particular differential equation to be solved. The specification depends upon what is assumed about the securities market and the diffusion process that governs the value of the underlying asset. Then the author states the initial, terminal, and boundary conditions that correspond to the initial and payoff conditions specified by the security contract, and finds (if possible) an explicit solution to the differential equation. Let G(V,T,F) be the value of a bond, where V is the value of the  firm,  T is the maturity date of the bond, and F is the face value of the bond. Also, let r be time remaining to maturity, so 0 < r ^ T . Other variables relevant to pricing are c, the net known payment (such as coupon payments and sinking fund payments) by the firm per unit time to the bond; D, the the known sum of all payments per unit time made by the firm to all securities; r, the constant default-free rate of interest; and a, the standard deviation of returns on the firm's  assets. The three bond-valuation equations adopted in this study are all  specific solutions to the same differential equation: l / 2 o V ( 3 G / 8 V ) + (rV-D)(9G/aV)-OG/3r)-rG+c = 0. 2  2  2  2  This differential equation is that formulated by Black and Scholes(1973), except for the variables representing net payments. These were added by Merton(1974,1977). The formulation is based on a number of assumptions which thus naturally apply to this study: 1.  There are no transactions costs or differential taxes. For corporate liabilities this means no bankruptcy costs, perfect liquidity for all assets, and no  26 differences in taxation on dividends, interest, and capital gains. 2.  The default-free rate of interest is known and constant through time.  3.  Borrowing and lending at the default-free rate are unrestricted.  4.  Short sales, with free use of proceeds, are unrestricted.  5.  Trading is continuous through time.  6.  Over time, firm value changes according to the diffusion process: dV = (aV-D)dr + oVdZ;  where a is the instantaneous expected rate of total return to the firm, D is known net payout per time, a is the intertemporally constant instantaneous standard deviation of return on the firm  1 3  , and  Z is a Gauss-Wiener stochastic process. 7.  All investors have the same beliefs about a, a, and D " . 1  The assumptions about the structure of the market are identical to those made by Brennan and Kraus; the same beliefs about o are arrived at through the attainment of the fully-revealing equilibrium. The diffusion process assumed matches the assumption that the distribution function of firm value can be summarized by two parameters. Brennan and Kraus were silent on the behavior of interest rates. On point about risk aversion versus risk neutrality should be noted. Every contingent claim has a unique price under these assumptions because every The two components for return are capital gains(losses), in this case changes in the market value of a firm's assets, and cash flows received or paid. In general, the market value of most real assets tends to be stable, so variance is primarily due to variance in value of intangible assets (advertizing, good will, research and development) and in cash flows (sales and costs of production). Thus the variance will depend upon the type of assets, the type of products sold, and the type of production methods employed. Recall that the assumption made was that a, the expected risk-adjusted return for a firm, is constant and equal to r, the riskless rate. This allows the expected value of the firm to be taken as a constant and independent of a. 1 4  investor can construct riskless hedges using the contingent claims. Since both risk neutral and risk averse investors will value riskless hedges the same, contingent claims will have the same value in a risk neutral economy as in a risk averse economy [see Cox and Ross(1976) for a full discussion of this point]. The firm  case of a pure-discount ordinary bond issued by a non-dividend paying  with no other debt illustrates the nature of typical boundary conditions. The  payments D and c are equal to zero in this case. The boundary and terminal conditions are: 1.  G(0,r,F) = 0;  2.  G(V,T,F)->Fexp(-rr) as V-»»;  3.  G(V,0,F) = Min(V,F).  The  first  condition says that when the  firm  has no value the debt has no  value. Condition 2 says that as asset values increase, the market value of the . debt approaches the market value of a default-free The  bond with the same  terms.  last condition says that at maturity the bond is worth asset value for  values below F and F for firm values above F . For this case there is an explict solution: G(V,T,F)=Fexp(-r )N(h ) + V [ l - N ( h ) ] ; T  J  1  where N( •) is the standard normal distribution; h = [ln(V/Fexp(-r r))/o/r]+ 1/2 a / r , x  and h = h - a / r . 2  1  Unfortunately, there are no analytical solutions to the equation for complex real-world bonds which pay coupons, can be called, and have other options attached. Such bonds can be evaluated only by using numerical techniques. In this study two of the three solutions examined (labled Ingersoll and Geske) have explicit solutions. The third (labled Brennan-Schwartz) was evaluated using the  28 finite difference technique. Each of these solutions rests on different  assumptions,  with the least realistic being those underlying the Ingersoll solution and the most realistic those underlying the Brennan-Schwartz solution. Ingersoll(1977a) developed a solution for convertibles by showing that the value of a pure-discount convertible bond issued by a firm pajdng no dividends is equal to the sum of the values of a plain risky bond and a warrant (with the appropriate exercise price). In other words, in the pure-discount case, a convertible is just a special case of a bond-warrant  package, with both securities  maturing at the same time. Since each component can be priced with a simple equation, so can the convertible. Similarly, by analogy, the value of a junior convertible maturing at the same time as senior ordinary debt is equal to the value of a plain junior bond plus a  warrant.  Geske(1977, 1979) and Geske and Johnson(1984)  extended  the  Black-Scholes analysis to allow valuation of a security which matures later or earlier than other securities on which its value depends. Geske's solutions are basically multidimensional analogs of the Black-Scholes and Ingersoll solutions, with the dimension of the multivariate normal distribution determined by the number of contingent payouts made prior to the maturity date of the bond. For example, the pricing formula for a plain bond paying L coupons at  sequential  discrete points requires the evaluation of ( L + l ) integrals, with dimensions ranging from one (standard normal) to ( L + l ) (standard  L + l multivariate n o r m a l ) . This 15  complexity means that, in practice, the value of such bonds cannot be directly computed using the Geske theory. However, the theory does generate explicit, For example, for a 20-year bond paying coupons semiannually the solution requires evaluating a 41-dimensional multivariate normal distribution at the specified constants, which depend on the terms of the bond, firm value, and interest rate.  29 relatively simple pricing functions for two-dimensional cases, such as a plain bond maturing later than a simultaneously issued warrant and a senior  pure-discount  bond maturing earlier than a pure-discount junior convertible. The Ingersoll and Geske solutions have the advantage that exact theoretical values for bonds can be calculated directly from the formulae, but have the disadvantage  that the prices are for discount bonds with no special  features, and may not be applicable to real bonds. However, as noted, if the boundary/terminal conditions are adequately  stated, theoretical prices for more  realistic contracts can be generated numerically. These conditions were identified for senior convertibles by Brennan and Schwartz(1977,1982). The problems of evaluating junior issues and bonds with sinking funds were discussed by Jones et a(\(1984,1985). Ho and Singer(1982,1984) also discussed sinking fund provisions. The insights of these authors  1  6  were incorporated into the  specification of the boundary/terminal conditions, making it possible to obtain approximate theoretical values for the hybrid issues examined in this project. These boundary/terminal conditions are described more fully in the next  chapter  in conjunction with a discussion of the numerical method used in this study. This solution method is referred to throughout  as the Brennan-Schwartz solution.  Clearly, the Ingersoll solution is a special case of the Geske solution, which in turn is a special case of the Brennan-Schwartz solution, since all are based on the same differential equation, but incorporate an increasing degree of complexity in the contract terms modeled.  Many authors other than those noted here have contributed to the theory of contingent claims. For fuller bibliographies the reader can consult Brock and Malliaris(1982), Jarrow and Rudd(1983), and Mason and Merton (1985).  III. NUMERICAL METHODS USED IN C A L C U L A T I N G BOND V A L U E S Testing the model empirically required calculating the theoretical bond values as a function of a. For the Ingersoll and Geske solutions these values and the values of first- and second-order  partial derivatives were calculated  directly simply by writing a computer program incorporating the  appropriate  equations. Calculating values for the Geske solution required the evaluation of a bivariate normal integral and a numerical search for a critical firm value. The subroutines  for these were available in the I M S L Library. The programs  are  given in Appendix II. The Brennan-Schwartz solution was approximated using the finite difference method. This method seemed most appropriate  for several reasons. Firstly, the  mathematical properties of the method are extremelj' well  documented  [Noye(1981), Twizell(1984), for example], since the method is extensively used in the physical sciences. Second^, the method has an established "track record" in financial research. Brennan and Schwartz used this technique, demonstrating usefulness  in evaluating senior convertibles and warrants  its  [Brennan and  Schwartz(1977,1982)]. Geske and Shastri(1985) showed that the method is efficient for evaluating simple call and put options, when there are many options to evaluate, in a comparative study of several different  methods.  In the method, the partial derivatives in the differential equations  are  each replaced by a particular discrete approximation, allowing the value of a bond at some time and firm value to be expressed as function of the values of the bond at small time- and value-increments  away. Repeated application of the  approximation at each time- and value-increment from the lower to upper time and value boundaries yields a set of linear equations which, rearranged,  30  and  31 coupled with linear equations representing the boundary/terminal conditions is consistent and has a unique solution. For any function y = f(x), the derivative at any point ( x j j j  is the slope  at that point and may be indicated by the notation dy/dx. This slope can be approximated by dividing [f(x)-f(x )] by [ x - x j , that is by dividing the "rise" by x  the "run" between ( x ^ y j and any other point (x,y). Let [ x - x ] = [ x - x ] = h, 1  where x  2  > x  x  0  2  1  > x . Three approximations involving different pairs of points 0  are commonly used: 1.  dy/dx~[f(x )-f(x )]/h; called the forward difference formula;  2.  dy/dx^UCx^-fCx,,)]/]!; called the backward difference formula;  3.  dy/dx~[f(x )-f(x )]/2h; called the central difference formula.  2  2  1  0  The smaller the size of h, the better the approximation. The error in approximation of (1) and (2) is 0 ( h ) , 1 7  while of (3) it is 0 ( h ) . The choice of 2  the approximations used in solving the differential equation is somewhat  arbitrary,  all things equal. However, the approximation scheme for a particular problem is chosen with several conditions in mind: 1.  ignoring rounding error, the solution from the scheme should converge to the true value as the size of increment approaches zero;  2.  the scheme should be stable; that is, the system should yield solutions regardless of the size of the  3.  reasonable  increment;  the system should not require the fixing of points beyond boundar3' points.  The approximations most suitable for solving the Black-Scholes-Merton equation, given these criteria, are the central difference for the first-order partial derivative " For any two scalar quantities notation r(h) = 0(q(h)) means that independent of h, such that: limit as h->» of |(h)/q(h)|<K.  T  7  depending on h, say r(h) and q(h), there exists some positive constant,  the K,  w i t h respect to firm derivative w i t h as  an  respect to time,  a p p r o x i m a t i o n for the  would lead or  value, V , and the  to problems  at  The central  difference  p a r t i a l derivative w i t h  the  boundary  where  for the  partial  f o r m u l a cannot be  used  respect to time, because  time equals  this  maturity  e x p i r y date. Let  differential  G ( V , T , F ) be the equation  issue; F , the  is the  principal  r e m a i n i n g time  Vmax must  is the  the  and  such  0^i<I,  and  have been  the  firm  suppressed  for c l a r i t y .  noted  at  firm  since  date of  rate; and  security values  p a r t i a l derivative  are  9G/9V  bonds.  Note that V m a x  i s s u i n g the  security being  bond w h e n f i r m where  the  times  The  value  calculated is equal  may  be  and to  7,  and well  evaluated.  value is AV times i ,  arguments of the finite  G function  difference  i and time AT times  G ^ ^ ^ A V )  the  T the  AT such that J AV = Vmax.  the  (j+1)  yields:  2  -G )/Ar. y  substituted  into the  linear s y s t e m  i ° i + i j + i  increments  value A V times  9G/9T.(G  a  default-free  Substitution of the  •  to yield the  the  value of the  a^G/aV^^G^^ + ^ G i ^ i +  are  maturity  A V such t h a t I times  Vmax  •  These  T; r, the  is A r times j , 0 < j < J ,  i J + 1  (or w a r r a n t ,  acquired upon conversion or exercise for convertibles  value of the  time  bond  value; T, the  value for w h i c h  that at  G : : as  approximations  paid at  is equal to 0 for straight  actual  Define  V , firm  increments  m a x i m u m firm  fraction of equity  above  same);  value of the  to maturit}'. Define J discrete  be chosen  warrants,  theoretical  to be  AT = T. Define I discrete  the  r.  b a c k w a r d difference  +  a  composed of equations  * ij + i G  differential equation  +  a  3 i-i,j + i G  =  a  then  the  of the 4 y; G  w  t e r m s are  form: h  e  r  e  rearranged  33  a , x  a , a , and a 2  3  are coefficients that depend on A V , A r , i , j , r, and  4  o  1 8  .  The boundary/terminal conditions ensure continuit}' of value and rational valuation of a security at each time at which an inflow or outflow occurs. For senior callable convertibles issued by a non-dividend paying firm these are: 1.  G ,j  2.  G- • = Min[(iAV),Max(G: :, (iAV)],  =  0  f o r  J  a 1 1  0  ;  T  for all i and j ; 3  -  G  i-(c/AV)J  +  c  y  =  G  ij>  for all i and j ; 4.  G - : = Min(G -,CPj ), for all i and j ; CP;  ;  is call price at j .  ;  The first condition says that if the firm has no value the debt has no value. The second says that the bond is worth the lesser of (1) the value of the firm and (2) the larger of straight debt value or conversion value . The 1 9  third condition stipulates that the value of the bond just before the coupon (or other payment) is paid is worth the same as the value of the bond just  after  the coupon is paid plus cash value of the coupon. The last states that the bond cannot exceed the call price at any time. If the firm pays dividends an additional constraint is that at the time the dividend is paid the bond must be worth the greater of the conversion value (including dividend) and the value of The equations that would result if the central difference with respect to time were used are of the form:  for the  derivative  i i+lJ +l * i,j + l 3 i-lj+l * i - l , j + 2 s i,jThat is, calculation of any point would require a value two time steps ahead. This creates problems at boundary points and adds to the complexity of the linear system. The smaller the increments the greater the number of equations to be solved simultaneously, but the more accurate the solution to the limit of rounding error. This is consistent with the assumption that investors expect firms to call the bond at the theoretically optimal time—when the bond market price equals call price. Ingersoll(1977b) found that firms delay calling their bonds, but no theoretical model has been developed to account for this behavior. a  1 9  G  + a  G  + a  G  + a  G  =  a  G  34  the bond not converted after the dividend has been paid. If the bond is a junior bond with senior debt worth S-j, condition (2) must be modified to: G j = Min[(iAV-S ), MaxCGy^iAV-Sij)]. i  ijj  This says that junior debt is worth the lesser of (1) the value of the firm minus the value of senior debt and (2) the larger of straight debt value and conversion value. The conditions for a bond plus warrant are very similar. At all times the straight bond is worth the minimum of firm value, call price, and current theoretical value. Thus unlike the case of convertibles, the call feature does not limit the value of the option to convert. If firm value does not exceed combined value of all debt then the warrant is worthless. If the firm pays dividends the warrant must be worth the greater of conversion value (including dividend) and the value of the warrant not converted after the dividend has been paid. Note that a coupon-paying bond-warrant package and a convertible on a dividend-pa3'ing firm are not identical as in the case of the discount bonds on non-dividend paying firms, since the exercise price of the warrant is fixed rather than variable, and the warrant holder does not forego future coupon payments. Ideally when the finite difference method is used, the system of equations should be re-solved for smaller and smaller increments until two consecutive solutions converge. This was not feasible given the size of the sample and the cost of computing theoretical values. Instead, the increment sizes were preset at the values at which several typical discount bond test cases converged. However, before theoretical values for a sample bond at all a's were computed at the preset increments, the numerical result was checked for convergence by comparing  for a mid-range a the value at the preset increments to the value using smaller increments. If there was a difference the increment size was again reduced and the process repeated until there was no difference. Values for all o's were then computed using the appropriately-sized increments. Brennan and Schwartz(1977,1982)  and Geske and Shastri(1985) applied  this method to the solution of a single differential equation. In this study, the method was used to solve simultaneously a maximum of three differential equations  (for a bond, a senior bond, and a warrant). This added to the  programming complexity and may have reduced the accuracy of the approximations, since the error in the solution for one equation depends on its own error and the negative of the errors in the solution of the other two equations. There was no way to determine the magnitude of the error, except for simple cases which could also be evaluated directly using the programs for Ingersoll and Geske solutions. For the simple cases the finite difference  method  and the direct calculation method yielded theoretical values which matched within 0.3-0.5 percent for the a range of interest (0.05-1.00 annually). The approximation for more complex cases was expected to be as accurate on theoretical  grounds.  IV. T H E O R E T I C A L PROPERTIES OF T H E BOND-VALUATION MODELS The Brennan and Kraus model requires that bond pricing functions (as a function of a) have a minimum in the interior of the feasible range. The first step then was to study the properties of the bond pricing functions generated by the bond valuation model for a wide range of choice variables (7, exercise premium, coupon rate) and input parameters (firm value, interest rate, debt/firm value ratios). In this phase of the study a function was considered to have an interior minimum if the minimum was not at o — O or o — ~> Two analytical tools are available—mathematical analysis and  brute-force  numerical calculation. The former was applicable only to the Ingersoll and Geske solutions because there is no closed-form solution for the Brennan-Schwartz solution. The latter could, in theory, be applied to all three solutions, but in practice the numerical techniques required to approximate the Brennan-Schwartz solution are extremely costly. Thus calculating specific bond values for a wide range of parameters was feasible only for the Ingersoll and Geske solutions. However, because the Ingersoll and Geske solutions are special cases of the Brennan-Schwartz solution, any property found for the simpler solutions is a property of the more general solution. The versions of Ingersoll, Geske, and Brennan-Schwartz solutions examined are noted in Table 1, along with prototype equations. The equation under the Ingersoll heading is for a junior discount bond. The equations under the Geske heading are for, respective^, a junior bond with an earlier maturing warrant  and a junior bond with earlier maturing senior debt.  The mathematical analysis of the Ingersoll solution revealed that the pricing functions of senior convertibles and junior straight debt can not have an  36  37  interior minimum . In fact, simulation revealed that for many plausible 20  combinations of choice variables and input parameters the specific bond pricing of senior convertibles exhibit a prominent maximum. For senior bond-warrant packages, junior convertibles, and junior bond-warrant packages, in plausible ranges of parameters and choice variables, functions with an interior minimum are possible. The numerical simulations confirmed these results, and also showed that no one shape is dominant. For plausible inputs, functions were found which were everywhere up-rising (warrant-like), or everywhere down-sloping (bond-like) or which had an interior minimum or interior maximum. Mathematical analysis of the two-dimensional versions of the Geske solution yielded little in the way of proofs, due to the complexity of the derivatives. The particular bonds considered were a bond with an earlier expiring warrant, a bond with earlier maturing senior debt, and a senior bond with one coupon paid out prior to maturity . However, the results of the numerical analyses of the 21  bond-warrant case and the earlier expiring senior debt case were entirely consistent with the results for the Ingersoll solution. Functions with a minimum were found, but no one shape was dominant. Not one example of a senior convertible with one earlier coupon yielded a function with an interior minimum;  This is the solution for non-callable pure-discount bonds plus warrants all maturing at the same time, so includes the pure-discount convertible as a special case. The analysis of this case of the Geske solution was aimed at trying to find a result for the one coupon case that could be used in an inductive proof for the L-coupon case. 2  1  38 this is consistent with the idea that the timing of payments on the same security is irrelevant. What generates the minimum is the interaction between two different securities. Setting the time of expiry of the warrant to some time before maturity of the bond caused the value of the package to decrease relative to its value when the maturities were the same. This result was expected, since a fundamental result of option theory is that the value of a warrant declines as time to maturity declines. Another more interesting effect observed was that in manj' ranges of inputs the specific functions had no interior minimum when maturities were the same, but had an interior minimum for an earlier maturing warrant. The results for earlier maturing senior debt were similar. Decreasing the time to the maturity of the senior debt decreased the value of the junior hybrid. Again this was not surprising, because the earlier the senior debt matures, the lower the probability that stock-holders will exercise their call option and pay off the senior debt-holders. Also, as for warrants, some specific pricing functions which had no interior minimum when maturities of junior and senior debt were the same exhibited such a minimum when the senior debt was assumed to mature earlier. The results for a bond with one coupon were opposite. The sooner the coupon was paid the more the bond was worth, clearty in keeping with standard finance theory. Also, as noted, not one specific function with an interior minimum was found. The conclusion is that coupons do not change the basic shape of the pricing function of the underlying bond type, but only raise value at all a values, and perhaps broaden and flatten the function.  Table 1. Solutions to the Bond-Valuation Model Used In This Study Bond type Discount bonds (Ingersoll Solution) G = (F + c)exp(-rT )N(h ) + V[ 1-N(h )] +Bexp(-rT )N(h )-V[ 1-N(j, )]-Bexp(-rT )N(j ) + VN(l )-K exp(-rT )N(l ) 2  2  2  7  1  Senior Junior Senior Junior Junior  2  2  2  1  Minimum  2  2  2  convertible convertible bond plus warrant bond plus warrant bond  no yes yes yes no  Discount bonds (Geske Solution) (1) G = (F+c)exp(-rT )N(h ) + V[ 1-N(h ,)] + Bexp(-rT )N(h )-V[ 1-N(j, )]-Bexp(-rT )N(j ) + V N (k, , h , ,p)-y(F + c + B)exp(-rT )N (k ,h , p) -K exp(-rT )N(k ) 2  2  7  2  2  2  2  2  2  z  2  2  2  2  2  (2) G = (F + c)exp(-rT )N (k ,h ,p) + V[ 1-N (k , h , ,p) ] + B e x p - r T ] N ( k ) - V [ l - N ( j Jl-BexpC-rTJNCJ, ) 2  1  2  2  2  2  2  1  1  4  + V N (k ,1, , p ) - K , exp(-rT )N (k , 1 ,p) 7  a  x  2  2  2  2  Senior bond plus earlier expiring warrant Junior bond plus earlier expiring warrant Senior bond plus warrant with earlier maturing senior bond Junior convertible plus earlier maturing senior bond Senior convertible plus one coupon paid earlier  yes yes yes yes no  Bonds with coupons, call features, sinking funds (Brennan-Schwartz Solution) l / 2 o V (9 2 G/ 9 V ) + (rV-C)OG/SV)-OG/3t)-rG + c = 0 s.t specified boundary/terminal conditions 2  2  2  Senior convertible Junior convertible Junior bond plus earlier expiring warrants (Table 1 continued)  ?? yes yes  40 (Table 1 continued) Variables are: 1. V = firm value; 2. (F + c) = principal of junior bond, c is the coupon; 3. 7 = fraction of firm acquired upon conversion of bond or exercise of warrant; 4. E = stated strike price of warrant; 5. B = principal of senior bond; 6. T = t i m e to maturity of junior bond; 7. T = t i m e to expiry of earlier expiring warrant or time of maturity of earlier maturing bond; 8. r = constant default-free rate; 9. N( •) = univariate normal distribution function; N (•, • ,p) = bivariate normal distribution function, where p = / ( T / T ) ; 10. K = i m p l i e d strike price of warrant, defined as a. K = ( l - 7 ) ( F + c) + 7(F + c + B) b. K = implied strike price of warrant, defined as K = ( 1 - 7 ) E ; c. K = implied strike price of warrant, defined as K = (1- )(F + c) + ( F + c + B)exp[r (T -T,)] 11. G = theoretical price of the bond. 2  1  2  1  2  1  1  2  2  3  3  T  7  2  The constants are: h =[ln(V/(F + c + B)exp(-rT ))]/a /T h = [ln(V/(F+c-rB)exp(-rT ))]/a/T 1  2  2  2  j =[ln(V/Bexp(-rT ))]/oi/T j =[ln(V/Bexp(-rT ))]/o/T a  1  2  2  t  l  2  + 0.5a/T - 0.5a/T  l  2  2  2  2  i;  i ;  l^tln^V/KexpC-rTJJl/oi/Tj l =[ln(7V/Kexp(-rT ))]/a/T 2  + 0.5o/T ; - 0.5a/T ;  + 0.5o/T ; - 0.5o»/T ;  a 2  j =tln(V/Bexp(-rT ))]/a/T j^tmCV/BexpC-rT^j/c/Tx 11  2  2  + 0.5o/T ; - 0.5a/T ; 2  2  2  k =Dn(VA^*exp(-rT ))]/avT + 0 . 5 o / T k = [ln(VAV*exp(-rT ))]/a/T - 0 . 5 a / T V*=the minimum firm value at which the warrant or shareholders would pay the bondholders at T. 1  2  1  1  1  1  i;  i ;  x  would be exercised  The significance of these results for the model are clear. Since functions with an interior minimum can be generated for plausible input parameters, the Brennan and Kraus model cannot be rejected not all convertible/bond-warrant  on theoretical grounds. However,  contracts can be used to signal o. Specifically,  those with functions with an interior maximum would only be issued by firms with no information asymmetry problem. For example, this means that all-equity firms with an as3 mmetric information problem could not issue a convertible to r  convey inside information, but firms that have outstanding debt could do so, because of the difference  in properties of senior and junior convertibles.  Further,  those securities with theoretical functions which are everywhere positively sloped would only be issued by firms with o's at the lower bound, and those with functions everywhere declining by firms with o's at the upper bound. This unexpected variety of security types within the broad class of convertibles/bond-warrants  enriches the market, but makes interpretation  of the  issuance of convertibles more complex. Consider the numerical example again. The theoretical results indicate that in the simple economy there are not three basic securitj' types but rather six: warrants/equity,  warrant-like hybrids, hybrids with  functions with a minimum, hybrids with functions with a maximum, debt-like hybrids, and straight debt. Assume as before that firms with a known to the market choose all types equally. The proportions then would be respectively, 14.58%, 14.58%, 33.33%, 8.33%,  14.58%, and 14.58%. Considering only the  hybrids, the proportions of the four shapes would be 20.58%, 47.06%, 11.76%, and 20.58%. In the continuous case the figures for the six types would be respectively: 8.33%, 8.33%, 58.33%, 8.33%, 8.33%, 8.33%, and for hybrids only would be 10%, 70%, 10%, and 10%. If there were no asymmetry problem, all  four types would be observed equally. If at the other extreme the  assumption  were made that firms with a known to the market never chose hybrids, then the only hybrids that would be expected in the continuous case would hybrids with an interior minimum. In an} arbitrarily chosen economy, the specific 7  proportions expected would depend on the extent of the asymmetry  problem and  on the distribution of firm -a's, but the implications are clear. If asymmetric information is a problem then the majority of hybrids issued should have functions with an interior minimum. The other types are not ruled out by the Brennan and Kraus model, but should be observed with lower  frequency.  V. TESTING T H E M O D E L WITH R E A L D A T A  A. EMPIRICAL HYPOTHESES The model together with the results from the theoretical survey yielded three testable hypotheses: H : Among hybrids, the most frequently occurring curve is a function with an interior minimum in the feasible a range. The alternate hypothesis is that there is no information asymmetry, implying that the observed frequency of each of the four types is the same. 0 1  H : The observed market price is not significantly different from the theoretical minimum in the feasible a range. Again, given the assumptions of the model, if there is no information asymmetry and the market prices the bonds correctly there should be no relationship between market price and theoretical minimum value. 0 2  H : The a predicted by location of the market price on the theoretical curve is the a used to price all the firm's securities. 0 3  The first hypothesis arises from the joint assumptions that the Brennan and Kraus model is correct and that firms with a known that choose hybrids, choose all four equally frequently. Note that the hypotheses are expressed in terms of "feasible a range". In the simulations portion of this study, the properties of the functions over the o range 0 to <= were examined, but it is likely that the market believes that a for most firms lies in a much smaller range, since, in particular, very large standard deviations are inconsistent with reported values. A minimum must thus lie in the market-perceived. feasible range for the model to be supported. In the empirical studies the range was assumed to be no greater than  0.05-1.00  (5 percent to 100 percent per annum). Smaller subranges were also considered. To test the first hypothesis the specific bond pricing functions for 105 new issues were generated  from the Ingersoll, Geske, and Brennan-Schwartz  43  44 solutions and studied. To test the second, the market price at time of issue was compared to both the theoretical maximum and minimum in the feasible range or a subrange. Finally, to test the third hypothesis, the a predicted from location of market price on the theoretical price function was compared to the a estimated from the common stock for those firms whose equity was traded on the NYSE or the AMEX during the 52 weeks following the bond offering. This is only an indirect test, since this determined only whether or not the predicted o was feasible.  B. ASSUMPTIONS M A D E IN ADAPTING T H E O R E T I C A L SOLUTIONS TO R E A L BONDS  Generating bond pricing functions for real bonds from these solutions is not a simple matter. The goal is to retain as much of the simplicity of the solutions as possible, while generating prices that reflect as closely as possible the real prices. The complexity of "other debt" was a major problem. Although firms with simple capital structures were selected, most firms in the sample had outstanding at time of new issue one or more public debt issues and one or more loans, to which various terms and conditions were attached. As Geske(1979) showed, for each unique cashflow date the dimension of the normal distribution used in solving for prices increases by one. Thus for a firm with one discount bond and one discount loan outstanding, the evaluation of a new discount bond plus warrant would require the evaluation of a four-dimensional standard normal function. Because programs for calculating standard normals with dimension greater than two were not available, for the Geske and Ingersoll solutions at  most only two different dates could be accommodated (that is, a junior expiring later than a senior, for example). To approximate Brennan and Schwartz solution values correctly the difference equations for all distinct contingent claims must be solved simultaneously. Thus for a single junior bond plus warrant with a single senior bond outstanding, three equations must be solved jointly. In general, the cost goes up and the error in solution may go up the greater  the number of  equations. It was thus decided that for this solution also only one senior and one junior could be accommodated, for a maximum of three equations if the junior issue included a warrant. The new issue in all cases was treated  as the junior. To obtain a single  senior, all other debt was combined into a composite single issue, under the following assumptions: •  all other debt was assumed to be senior;  •  time to maturity of the composite was assumed to be the face-weighted average of all maturity times;  •  coupon rate on the composite was assumed to be a face-weighted of the various coupon and interest rates;  •  interest was assumed to be paid semi-annually;  •  when the data sources did not provide a coupon/interest rate or maturity, the coupon rate was assumed equal to the current default-free rate and the maturity equal to the maturity of the new issue;  •  options on other debt such as call feature, sinking fund requirement, and right to convert floating rate to fixed rate, were all ignored.  The net effect of these assumptions is probably an understatement  average  of the value  of other debt, since the ignored options have positive value, and an understatement  of the risk of default since the unevenness of major outflows is  smoothed away. There were also some problems that were unique to a particular solution.  46  •  Basic Ingersoll and Geske Solutions: The major problem was that these solutions are for discount bonds, while real bonds are coupon-paying. To account for this, it was assumed that the total future value of the coupons was paid out at maturity, with the annuity factor based on the default-free rate. Thus the "principal" was the sum of the face, plus coupons and accumulated interest. The timing of coupons created a second problem in valuing the warrant portion of convertibles. In both the Ingersoll and Geske solutions, the exercise price of this warrant is the terminal balloon payment. That is, an investor exercises the warrant by giving up principal plus all accumulated interest. With coupon-paying bonds, the investor need give up only the final coupon payment plus principal. In fact, since the option to convert is American the investor may find it optimal to convert at some time prior to maturitj'. The net effect is that in converting coupon-paying bonds to equivalent discount bonds the appropriate exercise price is uncertain, since the number of coupons foregone is part of the cost of exercise. Consequently two extreme endpoints were adopted in determining exercise price. Under the ALL assumption, the holder gives up all the coupons (as in the case of a discount bond), while under the NONE assumption the holder gives up only the terminal cash flow. Note that either solution ignores the inherent risk that interim cashflows (coupons) will not be paid. Both the Ingersoll and Geske solutions are too simple to allow for other options in the new issue contract, such as call features or sinking  47 funds. Dividend payments were ignored for the same  reason . 22  Basic B r e n n a n and Schwartz solution: A major conundrum was  •  the  behavior of firms at points of major outflows or inflows of capital (such as at maturity of a senior bond or expiration of a warrant issue). The usual assumption is "perfect antidilution protection", meaning that no new securities except equity can be issued until all existing claims expire or mature. After some pondering, this was adopted in this study. Outflows were thus assumed to be financed hy liquidation of assets, and inflows from exercise of warrants were treated simply as negative dividends. This assumption, while clearly contrarj' to observed practice, eliminated the need for incorporating ad-hoc rules governing capital structure  decisions and  investor behavior in forecasting these decisions. Dividends were assumed to be paid annually with the  amount  determined by assuming that payout ratio (dividend/firm value) rather than dollar payout would remain constant.  Sinking fund payments were  assumed  made only at the end of the year in which they were required, at the lower of sinking fund price or theoretical bond price. This latters reflects the fact that sinking fund requirements  can be met by open market  purchase or a call to bondholders. Certain types of options were ignored because to evaluate them would require simulation of possible realized stock  This assumption had minimal impact because of the low dividend payouts. In the sample, 43 firms paid no dividends in the fiscal year the bond was issued. The average paid out by the other firms was 1.3 percent of firm value. Maximum payout was 6 percent of firm value. Such small variation does not change a function's shape or location of maxima or minima. 2  2  48  price paths or realized interest rate paths (for example, some call provisions include restrictions based on level of interest rates or behavior of stock prices in a period prior to a call).  C. DESCRIPTION OF T H E S A M P L E The 105 issues studied were a sample drawn from all convertible(CB) or bond-warrant(BW) issues offered in 1983 and 1984, as listed in Standard and Poor's Corporation Records. Out of a total of 1039 issues of all types of debt, only 224 (22 percent) were hybrids. Of the 224, 173 were C B offerings and 51 B W offerings. Only 83 CB's and 22 BW's were included in the study because some issues were unsuitable for testing purposes. Offerings were eliminated from the sample if: 1.  market prices were not available;  2.  capital structures of firms were very complex;  3.  offerings were complex, involving complex exchange of securities. The common stocks of 43 of the 83 C B issuers and of 10 of the 22 B W  issuers were traded on the N Y S E and A M E X , so 53 firms were included in the test on o. The sum of the market values of a firm's outstanding s e c u r i t i e s  23  was  used as a proxy for firm value. Market prices of traded debt were collected from Moody's or Standard  and Poor's Bond Guides. Most prices quoted were bid  prices. Book value was used as the "market price" of nontraded debt due to  " ~ A l l short-term debt was included except for net payables, as the latter were considered part of normal operations. This treatment is consistent with, for example, Jones et al. (1984,1985) and Masulis(1980b). For most firms short-term debt constituted between 1 and 5 percent of firm value. 5  3  49  lack of any other suitable proxy. Market prices of stocks and warrants were obtained from the records of the appropriate market (NYSE,AMEX,NASDAQ). Data on firms' capital structures, details of each debt issue, and current annual dividend were obtained from Moody's Manuals or Standard and Poor's Corporations Records. The proxy used for the current default-free rate was the rate on U.S. government intermediate- or long-term securities, as quoted in Standard and Poor's Bond Guide *. 2  An estimate of annual standard deviation for common stock was obtained by annualizing the standard deviation of weekly returns . This was the upper 25  feasible bound on the true o of the firm. The lower feasible bound was the leverage-adjusted standard deviation, calculated by multiplying the standard deviation of the common stock by the ratio of market value of common stock to market value of the firm. This would be the standard deviation of the firm if the debt were riskless.  D. RESULTS  Three types of summary statistics are reported. The first category, labelled "sample characteristics", includes statistics on firm value, debt ratio, 7 (conversion ratio), exercise premium (ratio of the difference between exercise price and current stock price to current stock price), and other characteristics of issues or * There isone Canadian bond in the sample. The bond price was found in the Financial Post and the stock price was taken from the records of the TSE; the interest rate proxy was the rate on Canadian government long-term bonds as 5  quoted in the Bank of Canada Review.  A weekly series was obtained by taking the geometric average of the daily returns as recorded on the CRSP tapes. Since the number of trading days in any calendar week varied, the mean daily return of the actual trading days was calculated, then multiplied by five to yield 52 standard weekly returns. Variance of these returns was then multiplied by 52 to obtain an estimate of the annual variance. 2 5  50 issuers. Also included are correlation matrices and results of regressions of market value of new issue (adjusted  for market value of firm) on the  input parameters. In the second category are. statistics which attempt to summarize  the  properties of the theoretical functions. These include: Basic shape. Four types are distinguished: functions positively sloped at all  •  points in the a range; functions with a minimum in the interior of the range; functions negatively sloped at all points in the range; and with a maximum in the interior of the r a n g e . 2 6  relative to a predefined  functions  Classification was made  a range, so could change if the range  were redefined. Variation i n value. The measure was percentage drop in value from the  •  maximum value in the a range to values at various a's. Percent was used rather than absolute value to allow interfirm comparisons. Slopes. Slope was measured by dividing percentage drop in value from the  •  maximum value to values at various a's by 100 times the  difference  between the a at maximum and a of the other value. When the maximum theoretical value fell at a a used for comparison, the slope was set to zero. The third set of statistics summarize the comparisons between  observed  prices and theoretical prices. These include: In or out of range. For a predefined  •  o range, the function was classed  A few functions had both an interior minimum and an interior maximum in some a ranges. These were classed as functions with an interior maximum since a hybrid issue with such a function could be used for signalling only if the feasible range were suitably narrow and included the a where the function declines to the minimum. 2  6  51 as above, in, or below range if the market price was above, inside, or below the theoretical price range defined by the pricing function. •  Gains to the F i r m . This was calculated as the percentage  difference  between observed market price and the theoretical minimum in a specified a range, indicating the premium the market is above the "signalled" value. •  Losses to the F i r m . This was calculated as the percent difference between theoretical maximum in a specified a range and the observed market price, indicating the loss to the firm if the security's true worth were the maximum value. Again, percent was used to allow interfirm comparisons. Analysis of these gains and losses included both univariate analysis of variance by curve shape and regressions of these variables on input  parameters.  The average values of some characteristics of the sample are presented in Table 2. Market value of the new issue was found to be strongly correlated with market value of firm. The correlations for the whole sample, CB's only, and BW's only, were, respectively, 0.73, 0.80, and 0.93. A l l these values were significant from one-tailed t-tests, assuming bivariate normality. To compensate for this correlation and to allow interfirm comparison of issues, market values of all debt and of the new issue are reported as fractions of market value of the firm. Not shown in Table 2 is the statistic that 9 of 22 B W ' s were not callable at issue and the remaining 13 were callable at  100 plus interest. A t issue 24  CB's were not callable, but only 2 of the remaining 59 were callable at 100 plus interest. Also not shown is the statistic that the average face-weighted average of maturity of other debt was 14.68 for CB's, 13.01 for B W ' s , and 14.33 for the whole sample. The maturities of the warrants were: 0-5 years, 19;  52  Table 2. Characteristics of the Sample Mean  Median  SE  Max  Min  434.72 48.85 0.18 1.66 0.36 0.19 0.20 0.22 0.42 0.30  165.66 30.00 0.16 1.86 0.33 0.18 0.18 0.20 0.41 0.22  93.81 6.19 0.01 0.16 0.02 0.01 0.01 0.02 0.02 0.03  6642.46 400.00 0.70 5.09 0.75 0.60 0.73 0.95 0.88 0.88  31.06 4.00 0.02 -2.25 0.07 0.02 0.02 -0.30 0.00 0.00  Rating!  NR 47(13)  AA-A 7(0)  BBB 10(0)  BB-B 38(9)  CCC 4(0)  Maturity(M)t(Yr)  0-10 19(14)  10-15 16(3)  15-25 70(5)  CB's Mean  SE  BW's Mean  SE  ANC^  484.51 43.83 0.18 2.13 0.32 0.17 0.17 0.21 0.41 0.29  212.44 10.93 0.02 0.26 0.03 0.02 0.02 0.03 0.03 0.03  247.00 65.60 0.21 -0.13 0.51 0.24 0.28 0.24 0.47 0.33  71.91 18.23 0.03 0.31 0.03 0.03 0.03 0.05 0.05 0.06  ** ** ** **  Statistic Firm value(MV)($mill.) Principal($mill.) 7*  R-C(%)* Debt ratio(D)* New debt ratioCDJ* New capital ratio(CV)* Exercise premium(EP)* Book value ratio(B ) Private debt ratio(B ) :|:  1  ;|:  2  By type Statistic MV Principal 7  R-C(%) D D CV EP B, B x  2  *"y"= fraction of firm acquired upon conversion of bond or exercise of •warrant; "R-C(%)"= default-free rate less coupon rate on new bond; "Debt ratio(D)" = market value of all debt/ market value of firm; "New debt ratio(D )"= market value of new bond/market value of firm; "New capital ratio(CV)"= market value of new issue/market value of firm; "Exercise premium(EP)" = exercise price less current stock price/current stock price; " B j / ^ face value of other debt/face value of all debt; " B " = face value of private debt/face value of all debt. tFigures for B W ' s only in brackets n $ * . . i } i t e s C B and B W means are significantly different at 5% level from univariate A N O V A . 1  2  nc  ca  53 6-7 years, 2; 10 years, 1. On average the face-weighted coupon on other debt was 0.95% less that the default-free rate(R) for the whole sample, 1.04% less than R for CB's, and 0.59% less than R for B W ' s . The most noteworthy statistics in Table 2 are the significant differences between B W issues and issuers and CB issues and issuers for several variables. The new issue on average constituted a greater portion of firm value for BW's than for CB's (0.28 versus 0.17). Also, judging by the post-investment debt ratio (D), the bond ratings, and firm values, BW's were issued by smaller, lesser known firms which at time of issue had significantly more debt. The greater  risk  of default perceived for these firms appears to have been translated into B W ' s with shorter terms and higher coupon rates compared to C B ' s . The difference in coupon rates between CB's and BW's is very striking, but without knowing what any firm's straight debt coupon rate would have been, it is difficult to compare the relative benefits of a conversion feature and a warrant. The average value of R-C(%) for straight debt over the sample period was -0.58 (SE = 0.059) for A A bonds, -0.92 (SE = 0.023), and -1.48 (SE = 0.041) for B B B b o n d s . These 27  data suggest that a conversion feature or an attached warrant results in a significant lowering of coupon rates compared to straight debt. Pairwise correlations of the input variables for the whole sample and for convertibles and bond-warrants separately are shown in Tables 3, 4, and 5, along with significance levels assuming bivariate normality. In Table 6 are regression coefficients for regressions of C V (ratio of market value of issue to market value of firm) on input variables. A priori one would expect market  These figures are based on the yields quoted in Standard and Poor's Bond Guide. The yield on long-term Treasury bonds was used as the proxy for the default-free rate. The Bond Guide does not report figures for lower-rated bonds. 2  7  54  Table 3. Correlations Between Variables, Whole Sample MV  D  CV  7  R-C  EP  M  MV  1.00 (1.00)  -0.33 (0.00)  -0.33 (0.00)  -0.36 (0.00)  0.34 (0.00)  -0.04 (0.36)  0.28 (0.00)  0.14 (0.08)  -0.02 (0.44)  D  -0.33 (0.00)  1.00 (1.00)  0.47 (0.00)  0.56 (0.00)  -0.48 (0.00)  0.08 (0.22)  -0.41 (0.00)  0.42 (0.00)  0.44 (0.00)  CV  -0.33 (0.00)  0.47 (0.00)  1.00 (1.00)  0.79 (0.00)  -0.30 (0.00)  -0.09 (0.17)  -0.37 (0.00)  -0.48 (0.00)  -0.37 (0.00)  7  -0.36 (0.00)  0.56 (0.00)  0.79 (0.00)  1.00 (1.00)  -0.16 (0.05)  -0.26 (0.00)  -0.37 (0.00)  -0.23 (0.01)  -0.09 (0.19)  R-C  0.34 (0.00)  -0.48 (0.00)  -0.30 (0.00)  -0.16 (0.05)  1.00 (1.00)  -0.03 (0.38)  0.49 (0.00)  -0.14 (0.07)  -0.10 (0.15)  EP  -0.04 (0.36)  0.08 (0.22)  -0.09 (0.17)  -0.26 (0.00)  -0.03 (0.38)  1.00 (1.00)  0.04 (0.34)  0.07 (0.24)  0.07 (0.24)  M  0.28 (0.00)  -0.41 (0.00)  -0.37 (0.00)  -0.37 (0.00)  0.49 (0.00)  0.04 (0.34)  1.00 (1.00)  -0.05 (0.33)  -0.10 (0.16)  B,  0.14 (0.08)  0.42 (0.00)  -0.48 (0.00)  -0.23 (0.01)  -0.14 (0.07)  -0.07 (0.24)  -0.05 (0.33)  1.00 (1.00)  0.78 (0.00)  B  -0.02 (0.44)  0.44 (0.00)  -0.37 (0.00)  -0.09 (0.19)  -0.10 (0.15)  0.07 (0.24)  -0.10 (0.16)  0.78 (0.00)  1.00 (1.00)  2  B  x  B  2  " M V " = market value of firm; "D"= market value of all debt/MV; " C V " = market value of new issue/MV; "7"= fraction of firm acquired upon conversion of bond or exercise of warrant; "R-C"= default-free rate less coupon-rate on new bond; " E P " = (exercise price less current stock price)/current stock price; "B =book value of other debt/book value of all debt; "B = book value of private debt/book value of all debt. 1  2  Significance from one-tailed t-tests, assuming bivariate normality, in brackets.  55 Table 4. Correlations Between Variables, Convertibles Only MV  D  CV  7  R-C  EP  M  Bx  MV  1.00 (1.00)  -0.35 (0.00)  -0.40 (0.00)  -0.41 (0.00)  0.39 (0.00)  -0.04 (0.36)  0.29 (0.00)  0.17 (0.06)  -0.01 (0.47)  D  -0.35 (0.00)  1.00 (1.00)  0.40 (0.00)  0.68 (0.00)  -0.37 (0.00)  0.08 (0.24)  -0.33 (0.00)  0.44 (0.00)  0.51 (0.00)  CV  -0.40 (0.00)  0.40 (0.00)  1.00 (1.00)  0.90 (0.00)  -0.30 (0.00)  0.00 (0.49)  -0.35 (0.00)  -0.53 (0.00)  -0.36 (0.00)  7  -0.41 (0.00)  0.68 (0.00)  0.90 (0.00)  1.00 (1.00)  -0.36 (0.05)  -0.07 (-0.27)  -0.48 (0.00)  -0.23 (0.02)  -0.06 (0.28)  R-C  0.39 (0.00)  -0.37 (0.00)  -0.30 (0.00)  -0.36 (0.05)  1.00 (1.00)  0.13 (0.12)  0.33 (0.00)  -0.04 (0.36)  -0.08 (0.23)  EP  -0.04 (0.36)  0.08 (0.24)  0.00 (0.49)  -0.07 (0.27)  0.13 (0.12)  1.00 (1.00)  0.14 (0.10)  0.02 (0.45)  0.05 (0.32)  M  0.29 (0.00)  -0.33 (0.00)  -0.35 (0.00)  -0.48 (0.00)  0.33 (0.00)  0.14 (0.10)  1.00 (1.00)  0.04 (0.35)  -0.05 (0.31)  B,  0.17 (0.06)  0.44 (0.00)  -0.53 (0.00)  -0.23 (0.02)  -0.04 (0.36)  0.02 (0.45)  0.04 (0.35)  1.00 (1.00)  0.77 (0.00)  B  -0.01 (0.47)  0.51 (0.00)  -0.36 (0.00)  -0.06 (0.28)  -0.08 (0.23)  0.05 (0.32)  -0.05 (0.31)  0.77 (0.00)  1.00 (1.00)  2  B  2  " M V " = market value of firm; " D " = market value of all debt/MV; " C V " = market value of new issue/MV; ll ll fraction of firm acquired upon conversion of bond 7 - == default-free rate less coupon-rate on new bond; or exercise of warrant; "R-C" " E P " == (exercise: price less current stock price)/current stock price; " B / ^ b o o k value of other debt/book value of all debt; "B ' =book value of private debt/book value of all debt. I:  2  Significance from one-tailed t-tests, assuming bivariate normality, in brackets.  56 Table 5. Correlations Between Variables, Bond-Warrants Only MV  D  CV  7  R-C  EP  M  MV  1.00 (1.00)  -0.15 (0.26)  -0.14 (0.27)  -0.30 (0.09)  0.19 (0.19)  0.00 (0.50)  0.20 (0.18)  -0.07 (0.38)  -0.02 (0.48)  D  -0.15 (0.26)  1.00 (1.00)  0.33 (0.07)  0.32 (0.08)  -0.09 (0.35)  -0.01 (0.49)  0.18 (0.21)  0.28 (0.10)  0.21 (0.17)  CV  -0.14 (0.27)  0.33 (0.07)  1.00 (1.00)  0.68 (0.00)  0.31 (0.08)  -0.32 (0.08)  0.14 (0.27)  -0.75 (0.00)  -0.68 (0.00)  7  -0.30 (0.09)  0.32 (0.08)  0.68 (0.00)  1.00 (1.00)  0.44 (0.02)  -0.59 (0.00)  -0.17 (0.23)  -0.31 (0.08)  -0.18 (0.21)  R-C  0.20 (0.19)  -0.09 (0.35)  0.31 (0.08)  0.44 (0.02)  1.00 (1.00)  -0.28 (0.10)  -0.09 (0.35)  -0.33 (0.07)  -0.06 (0.40)  EP  0.00 (0.50)  -0.01 (0.49)  -0.32 (0.08)  -0.59 (0.00)  -0.28 (0.10)  1.00 (1.00)  -0.01 (0.48)  0.20 (0.18)  0.11 (0.31)  M  0.20 (0.18)  0.18 (0.21)  0.14 (0.27)  -0.17 (0.23)  -0.09 (0.35)  -0.01 (0.48)  1.00 (1.00)  -0.07 (0.38)  -0.12 (0.29)  Bx  -0.07 (0.38)  0.28 (0.10)  -0.75 (0.00)  -0.31 (0.08)  -0.33 (0.07)  0.20 (0.18)  -0.07 (0.38)  1.00 (1.00)  0.85 (0.00)  B  -0.02 (0.47)  0.21 (0.17)  -0.66 (0.00)  -0.18 (0.21)  -0.06 (0.40)  0.11 (0.31)  -0.12 (0.29)  0.85 (0.00)  1.00 (1.00)  2  B,  B  2  " M V "=market value of firm; " D " = market value of all debt/MV; " C V " = market value of new issue/MV; "j"= fraction of firm acquired upon conversion of bond or exercise of warrant; "R-C"= default-free rate less coupon-rate on new bond; " E P " = (exercise price less current stock price)/current stock price; " B " = book value of other debt/book value of all debt; "B "=book value of private debt/book value of all debt. 1  2  Significance from one-tailed t-tests, assuming bivariate normality, in brackets.  57 value (CV) to be positively correlated with fraction of firm acquired upon exercise or conversion (7), negatively correlated with the difference between riskless rate and coupon (R-C(%)), and negatively correlated with exercise premium (EP). In fact, there was essentially no correlation between C V and E P for the whole sample, CB's, and B W ' s , and only a moderate negative correlation between C V and (R-C) for C B ' s , BW's and the whole sample. However, the correlations between C V and 7 were very strong, 0.79, 0.90, and 0.68 for whole sample, CB's, and B W ' s respectively. The variable C V was also negatively correlated with the B  x  variable, which is the ratio of book value of other debt to book value of  all debt. The correlations were -0.48, -0.53, and -0.75 for, respectively, for whole sample, CB's, and BW's. Since this variable is a measure of outstanding other debt, and thus a crude measure of default risk, this negative correlation is reasonable. The positive link between C V and 7 and negative link between C V and other debt were further illustrated in the results of the regressions of C V on the input variables plus categorical variables indicating whether or not a bond was callable at issue and rated at issue. The method used to select predictors from the full model was backward elimination using SPSSX; the criterion for removal was probability of F-to-remove = 0.10. For the whole sample, for CB's, and for BW's, 7 had a significantly positive coefficient. More surprising was the positive sign on the exercise premium variable (EP). The higher the value of this variable, the more out-of-the money the warrant or conversion privilege at issue, so all things equal, securities with higher EP's should be worth less. The R  2  values were quite high, given the relatively few predictor variables. For CB's, 93 percent of the variability of C V was explained by debt ratio (D) 7, E P , maturity  58 Table 6. Regressions of CV (Market Price/Market Value of Firm) on Input Variables Full  CB's + BW's Reduced  CB's Full  Reduced  BW's Full  Reduced  K  0.14 (4.89)  0.12 (7.65)  0.03 (1.55)  0.03 (1.58)  0.20 (0.12)  0.23 (6.42)  MV  9.69* (1.73)  9.00* (1.68)  -3.73* (-1.23)  ...  -2.59* (-0.43)  ...  D  0.33 (6.16)  0.35 (7.17)  -0.12 (-2.44)  -0.09 (-1.93)  0.43 (3.55)  0.46 (6.33)  7  0.40 (5.19)  0.40 (6.28)  0.96 (13.29)  0.94 (13.55)  0.34 (1.84)  0.29 (3.99)  R-C  -0.007 (-1.94)  -0.007 (-2.24)  -0.003 (-1.28)  ....  635.00* (0.06)  ...  EP  0.005 (0.18)  ....  0.050 (2.84)  0.045 (2.68)  0.024 (0.28)  -490.0* (-0.46)  ....  0.0014 (2.12)  0.0013 (2.06)  0.003 (0.84)  0.0032 (0.53)  ...  0.023 (0.69)  -0.001 (-0.20)  ...  -0.007 (-0.15)  ...  M CALL RT  -0.002 (-0.22) -0.005 (-0.51)  —  ....  ...  -  B,  -0.21 (-6.05)  -0.22 (-6.40)  -0.048 (-2.12)  -0.057 (-2.69)  -0.37 (-2.86)  -0.52 (-11.06)  B  -0.093 (-3.18)  -0.092 (-3.24)  -0.036 (-2.20)  -0.034 (-2.09)  -0.11 (-0.95)  ...  2  R-C M  0  0  R R adj 2  2  -0.006 (-1.46)  ....  -0.001 (-0.56)  ....  -0.014 (-0.75)  ...  234.7* (0.32)  ....  502.0* (1.29)  ....  -0.002 (-0.53)  ...  0.85 0.84  0.85 0.85  0.95 0.94  0.94 0.94  0.95 0.91  0.93 0.92  (Table 6 continued)  59  (Table 6 continued) " K " = regression constant; " M V " = market value of firm; "D"= market value of all debt/MV; "y"= fraction of firm acquired upon conversion of bond or exercise of warrant; "R-C"= default-free rate less coupon-rate on new bond; " E P " = (exercise price less current stock price)/current current stock price; " M " = maturity; " C A L L " = dummy variable indicating callable/not callable at issue (not callable = 0); "RT" = dummy variable indicating rated/not rated at issue (rated = 0); " B j ' ^ b o o k value of other debt/book value of all debt; " B " = book value of private debt/book value of all debt; " R - C " = default-free rate minus face-weighted coupon on other debt; " M " = face-weighted average of maturity of other debt. 2  0  0  * Coefficients multiplied by 10 . T-values for coefficients in brackets.  60 of the bond (M), relative face value of other debt (B ), and relative face value 1  of private debt ( B ) , with C V negatively related to D, B j and B . For BW's, 2  2  93 percent of the variability of C V was accounted for by D , 7, and B  1 ;  with  the C V variable negatively related only to Bj_. The negative coefficients on the rating dummy variable, although not significant, indicated that market value was lower for non-rated than for rated bonds. This is not surprising if the rating process provides additional information to the market on a bond's true risk.  1. Evidence on the Feasible Sigma Range Although theoretical functions were generated  for o from 0.05 to 2.00 for  the Ingersoll and Geske solutions, and for a from 0.05 to 1.00 for the Brennan and Schwartz solution it seems highly likely from other evidence that the  market  puts an upper bound on a that is much less than 2.00 and probably less than 1.00 (which is annually 100 percent). Jones et aZ.(1985) reported estimated 0 values for 15 companies. The lowest reported was 0.129 for Cities Service in 1979 and the highest 0.42 for M G M in 1981. Jones et aZ.(1984) reported values for 27 firms using an adjusted equity a. The lowest value was 0.052 for Rapid American in 1981 and the highest 0.663 for N V F in 1980. Dammon et aZ.(1986) reported annual 0 values for equity issues on the N Y S E based on monthly and yearly returns for various time ranges. These can be considered upper bounds because most firms are levered. For 436 firms over the period 1955-1983, the mean standard deviation was 0.32, for a range of 0.124 to 2.39. The firms in this sample were similar to those in the Jones et aZ.(1984) sample in having simple capital structures, but the proportion of nontraded to traded debt was probably higher. Also, for all firms in the Jones et aZ.(1984)  61 sample, the equity was listed on a major exchange, all debt was rated, and the majority (149 of 305) of issues were of investment grade (BBB or higher) [see also Jones et al., 1985]. In contrast, in this sample, the equity of 50 percent of the firms was traded over-the-counter, 45 percent of the issues were not rated, and  68 percent of the rated issues were rated B B or lower. Given these  observations, examining the properties of the functions in the range 0.05 to 0.70 seemed appropriate, although this supposed that none of the firms was an outlier. For comparative purposes some statistics for the range 0.05 to 0.50 were also calculated.  2. Theoretical Prices Versus Market  The  Prices  signalling model is testable only if the bond-valuation model assumed  to hold does indeed generate values that match market valuations. This was tested by comparing market price to the range of theoretical values in both the full a range and in the reduced range 0.05-0.70.. Because inputs were measured with error and simplifying assumptions were adopted to make the solutions empirically useful, it was not expected that market price would be in range for every case. Without measures of the magnitude and direction of the error induced by the assumptions and methodology, a firm prediction of expected success rate was not possible. The somewhat a r b i t r a l assumption made was that a bond-valuation model needs to be successful 80 percent of the time (that is, generate prices in range 80 percent of the time) to be considered as a possibly good generator of market price. This should not, of course, be the only criterion: a model that indicates that market price lies at extreme o's is probably not a good model even if it generates market price in range 100  62  percent of the time. The success rate of a model producing estimates not related to market price can also be determined. For such a model by random chance market prices would fall in range one-third of the time. A biased model would yield values that were either generally too high or generally too low. Table 7 gives the frequencies  for cases of market price above, in, and below range, and  the results of several chi-squared(x ) goodness-of-fit tests, for the full r a n g e . 2  2 8  For CB's for none of the three solution techniques did market prices lie within range 80 percent of the time. However, for B W ' s the less realistic solutions had a very high success rate. For the Brennan and Schwartz solution [CB's(B) and BW's(B)] both null hypotheses  of an 80 percent success rate and equal numbers  in all categories were rejected. To see if the success rate of the simpler solutions was correlated with the success rate of the Brennan-Schwartz solution, Spearman's rank-correlation coefficients were calculated: none were significant. Thus results from the simpler solutions cannot be used even as a guide to predicting results from the more realistic (but much more expensive) Brennan-Schwartz solution. CB's appear to have been overpriced by the Brennan-Schwartz solution, the most realistic model. B W ' s were definitely overpriced; that is, market prices were much less than values indicated by the m o d e l . For testing 2 9  purposes,  For CB's there was essentially no difference between the theoretical values generated by the Ingersoll and Geske solutions, so the results for the Geske solution are not presented here. There are two causes for this similarity. Firstly, for many issues there was very little difference in maturity of the new issue and the composite senior (onty 41 of 83 had different maturities and the average difference for these was about 2 years). This means that the correlation of payoffs at time of issue was very high (measured as the square root of the maturity of the senior to the maturity of the junior). Secondly, actual times were used in calculating equivalent discount debt, further reducing the effect of time differences. The table indicates that for BW's there were 8 cases for which the Brennan and Schwartz solution generated the market price in range. This is somewhat 2  9  63  Table 7. Frequencies of Curve Shape and Location of Market Price Relative to Theoretical Prices, A l l Solutions  STAT IN/OUT 11: above 12: in 13: below X U) 2  X (2) X (3) X (4) 2  2  2  SHAPE F l : up sloped F2: with min. F3: down sloped F4: with max. X (5) X (6) X (7) 2  2  2  CB's I-A  I-N  B-S  BW's I  G  B-S  2 57 24  0 49 33  15 41 27  0 18 4  0 21 0  0 8 14  55.38 35.81 14.76 32.99  na na 3.12* 83.34  12.24 57.26 10.13 46.81  na na 8.90 1.64*  na na na na  na na 1.64 70.32  14 2 0  2 6 43  10 15 16  1 3 10  1 15 2  6 1 0  67  31  42  8  3  15  na 86.48 481.52  57.02 28.56 282.70  30.01 13.32 214.36  9.64 4.46* 62.60  24.52 13.71 0.96*  na 13.73 96.69  "I-A" = Ingersoll, A L L ; " I - N " =Ingersoll, N O N E ; " B - S " =Brennan-Schwartz; "I" = Ingersoll; " G " = Geske X (l)-Test X (2)-Test X (3)-Test X (4)-Test 2  2  2 2  against against against against  equal frequencies frequencies of 10 equal frequencies frequencies of 90  in 3 categories 80 10 in 2 categories (11 + 12), 13 10 in 2 categories (11 + 12), 13  X (5)-Test against- equal frequencies in 4 categories X (6)-Test against equal frequences in 3 categories (F1 + F3), F 2 , F4 X (7)-Test against frequencies of 10 80 10 for categories (F1+F3), F 2 , F4 2 2  2  "*" Indicates x  2  values not significant at the 5 percent level  64 overpricing was not overly troublesome for CB's, since the magnitude was small, but as shown below, the magnitude for BW's was large, with market prices being on average 14.5 percent below the minimum value, on curves for which the average maximum variation was 10 to 15 percent. Various results for B W ' s are tabulated below, but these should be interpreted as only crudely representative of the true pricing functions. The tests most affected by overpricing are those comparing market price to theoretical minimum and maximum, since overpricing would bias the results in favor of the signalling model. Overpricing has less impact on the shape tests because locally (that is, over relatively small changes in input parameters) the bond pricing functions are parallel. As long as mispricing was due to misestimation of a parameter such as market value or default-free rate, rather that to misspecification of the bond-valuation model itself, the shape data will be quite reliable. Results for the reduced a range are given in Table 8. It should be noted that in Table 8 the numbers for I N / O U T for the Ingersoll and Geske solutions refer to recalculated functions which were forced to "hit" the market price in the range 0.05-». That is, for the Ingersoll and Geske solutions when the market price was not generated by the function for measured firm value, the theoretical values were recomputed for firm values down to 0.8 X market value in one-percent steps. The function based on firm value closest to measured firm value that also generated the observed price was taken to be the "correct" pricing function. This is consistent with the expected direction of bias due to (cont'd) misleading in that the value is the sum of bonds plus warrants. In fact, the theoretical prices of bonds and warrants were generated separately but simultaneously, and for not one of these 8 cases did the solution generate the market price for bonds and warrants at the same a. In general, market prices of warrants were located at o's of 0.00-0.10, while market prices of bonds were located o's greater than 0.5. 2  9  65 Table 8. Frequencies of Function Shape and Location of Market Price Relative to Theoretical Prices, A l l Solutions (a range 0.05-0.70) STAT IN/OUT 11: above 12: in 13: below X (D X (2) X (3) X (4) 2  2 2  2  SHAPE F l : up sloped F2: with min. F3: down sloped F4: with max. X (5) X (6) X (7) 2  2 2  CB's I-A  I-N  B-S  BW's I  G  B-S  1 78 4  0 36 45  16 38 29  0 16 6  0 19 2  0 4 18  135.54 10.45 66.78 2.39*  na na 190.14  8.84 70.92 7.53 57.36  na na 4.54 7.29  na na 13.76 0.01*  na na 8.91 126.08  15 2 0  2 6 42  15 11 22  1 3 10  1 9 8  8 2 0  66  31  35  8  3  12  na 84.68 473.18  55.54 26.73 221.80  16.04 15.13 231.36  9.64 4.46* 62.60  8.52 3.43* 26.68  na 6.91 72.77  1.25*  "I-A" = Ingersoll, A L L ; " I - N " = Ingersoll, N O N E ; " B - S " = Brennan-Schwartz; "I" = Ingersoll; " G " = Geske. X (l)-Test X (2)-Test X (3)-Test (4)-Test 2  2  2  2  against against against against  equal frequencies frequencies of 10 equal frequencies frequencies of 90  in 3 categories 80 10 in 2 categories (11 + 12), 13 10 in 2 categories (11 + 12), 13  X (5)-Test against equal frequencies in 4 categories X (6)-Test against equal frequencies in 3 categories, (F1+F3), F 2 , F 4 X (7)-Test against frequencies of 10 80 10 for categories (F1 + F3), F 2 , F4 2 2  2  "*" Indicates values not significant at the 5 percent level.  66  mismeasurement  of senior debt and firm value, and does not change the shape,  slope, or variation of a curve, since the curves, holding other parameters constant, are essentially parallel. However, this forced the solutions to generate "hits" only. Such recalculation was deemed inappropriate solution given the relatively realistic assumptions  for the Brennan and Schwartz  made, so Brennan-Schwartz  curves were taken "as is". Thus not all I N / O U T figures in Table 8 are directly comparable to those in Table 7. The frequencies  for the S H A P E variable for all  three solutions are directly comparable. For both CB's and B W ' s the number of cases in range was fewer for the reduced o range than for the full range. This is not a surprising result, given that maxima or minima in the full range could lie above 0.70. For CB's, for the Ingersoll ( A L L ) solution, the restriction had little impact (at the full range all solutions were in range, since these were recalculated values as explained). However, for Ingersoll (NONE) the restriction resulted in more than half being outside the range. Restricting the range changed the results for the Brennan-Schwartz solution very little. This was also true for BW's for all three solutions. Noteworthy for the Brennan-Schwartz solution, in particular, is that a success rate of 80 percent was strongly These results further  rejected.  supports the conclusion that the solution as implemented  overpriced these securities.  3. Properties of the Functions  Tables 7 and 8 also give the frequencies  of curve shape for the various  solutions over the a ranges studied as well as the results for some x Two  2  tests.  null hypotheses were tested: whether the frequency of occurrence was  the  67  same for all shapes and whether curves with minima (category F2) were the most common. For the full range, both null hypotheses were soundly rejected, except for B W ' s priced according to the Geske solution. For this case the most frequent shape was one with an interior minimum. The most interesting point is that for the Brennan-Schwartz solution for both CB's and BW's curves with an interior maximum were most common, not least common as expected. The results for the restricted range (Table 8) were little different from the results for the full range. Except for the case of B W ' s evaluated with the Geske solution, functions with an interior minimum were not the most  frequent  and in fact were the least frequently occurring. The null hypotheses of equal numbers in all categories and of F2 curves being the most frequent were both strongly rejected. For the Brennan and Schwartz solution, for both CB's and BW's, functions with an interior maximum ( S H A P E F4) were the most frequently occurring, a result inconsistent with signalling o as a motive for issuing hybrids. In Tables 9, 10, and 11 are some statistics which summarize the properties of the sample functions for all three solutions. The a of the market price (MP) was taken to be the a the market price on the theoretical curve  3 0  .  In Table 9 are reported results for CB's as priced by the Ingersoll solution. Both the percent differences between maximum theoretical value (Gmax) and values at various a's and the slopes reflect the fact that a curve with an interior maximum was the most common. Average maximum variation was about  ™ I n general, the o of M P was recorded as the midpoint of the 0.05-long interval whose endpoints corresponded to the a for a value lower than M P and the a for a value higher than M P . If M P was generated at more than one a because the theoretical curve had an interior minimum or maximum, the lower a was taken as the firm's a. When the market price did not lie on the function, a of the value on the curve closest to market price was taken to be the a of the firm.  68 Table 9. Some Characteristics of Sample Functions for Convertibles, Ingersoll Solution ( A L L and N O N E Assumptions). Mean  Median  SE  Max  Min  C B ' s , Ingersoll(ALL) G m a x „ - G » / G m a x „ (%) Gmax o u"Go 5 /Gmax „ (%) Slope Gmax o o -G /Gmax (%) Slope Gmax o o -G /Gmax (%) Slope G m a x - M P / G m a x (%) Gmax -MP/Gmax (%) Gmax -MP/Gmax (%) M P - G m i n /MP(%) MP-Gmin /MP(%) MP-Gmin /MP(%) a of M P  15.07* 21.58* 0.71* 8.47* -0.23* 12.30* -0.25* 15.12* 14.36* 13.29* 9.24* 7.62* 7.49* 0.21  17.73 20.00 0.76 6.83 -0.30 13.10 -0.31 13.25 13.22 12.55 7.30 6.41 6.23 0.13  1.05 1.33 0.04 0.78 0.02 0.60 0.02 1.12 1.02 0.91 2.06 0.86 0.86 0.03  29.18 98.41 1.86 54.73 0.37 29.87 0.23 58.27 53.51 45.90 97.57 31.46 31.46 1.25  0.00 7.98 0.04 0.07 -0.64 2.00 -0.56 0.00 -1.95 -5.51 0.00 -7.75 -7.75 0.05  CB's, lngersoll(NONE) Gmax -Gco/Gmax (%) Gmax o o -G o s /Gmax (%) Slope Gmax o o " o /Gmax (%) Slope Gmax o o -G o /Gmax (%) Slope Gmax -MP/Gmax (%) G m a x , -MP/Gmax (%) Gmax -MP/Gmax (%) M P - G m i n /MP(%) MP-Gmin /MP(%) MP-Gmin /MP(%) a of M P  30.81* 2.97* 0.02 18.17* -0.40* 25.36* -0.41* 28.45* 28.03* 27.68* 7.93* -1.14* -11.33* 0.84  35.02 0.00 0.00 18.56 -0.44 27.29 -0.46 28.65 28.41 28.46 4.65 -0.43 -12.32 0.78  1.26 1.45 0.01 0.69 0.02 0.93 0.02 0.75 0.79 0.82 1.35 1.14 0.97 0.03  45.04 98.41 0.51 53.30 0.36 50.36 0.08 55.13 50.66 43.74 97.57 41.13 21.94 1.25  0.00 0.00 0.00 0.45 -0.62 3.43 -0.88 9.32 8.94 3.75 -5.19 -24.58 -33.61 0.05  Statistic  2  0  2  2  0  2 0  2  5 0  2 0 0  2  7 0  2 0 0  2 0 0  2 0 0  7 0  7 0  5 0  5 0  2 0 0 70 50  200  2 0 0  2  2 0 0  G  2  2 0 0  5  2  7  2 0 0  2 0 0  2 0 0  0  7 0  5 0  s 0  2 0 0 70  50  G = theoretical value at 0 = 0.05; G = theoretical value at 0=0.50; G = theoretical value at o = 0.70. Gmax(Gmin) = maximum (minimum) theoretical value in range 0 = 0.05-2.00; Gmax(Gmin) =maximum(minimum) theoretical value in range o = 0.05-0.70; Gmax(Gmin) =maximum (minimum) theoretical value in range o = 0.05-0.50. M P = market price of issue. 0 5  5 0  7 0  2 0 0  7 0  5 0  Slopes are for values in the line preceding. "* "Indicates that the mean value is significantly different from zero at the 5 percent level from a one-tailed t-test.  69 Table 10. Some Characteristics of Sample Functions for Bond-Warrants, Ingersoll and Geske Solutions Mean  Median  SE  Max  Min  B W ' s , (Ingersoll) G m a x -G°°/Gmax (%) Gmax o o -G o /Gmax (%) Slope G m a x - G / G m a x „(%) Slope Gmax o o -G o /Gmax (%) Slope Gmax -MP/Gmax (%) Gmax -MP/Gmax, (%) Gmax - M P / G m a x „(%) M P - G m i n /MP(%) MP-Gmin /MP(%) M P - G m i n /MP(%) a of M P  39.79* 7.69* 0.07* 30.44* -0.62* 35.75* -0.53* 22.12* 20.21* 18.54* 32.15* 50.96* 30.97* 0.49  38.03 0.00 0.00 30.73 -0.66 34.98 -0.54 24.66 19.14 19.10 25.09 29.09 22.37 0.38  5.50 4.21 0.03 3.34 0.46 4.14 0.08 2.26 1.96 2.16 5.75 14.04 11.23 0.08  83.48 78 0.40 61.20 0.22 73.74 0.16 42.90 36.14 36.14 81.82 214.80 213.73 01.25  0.00 0.00 0.00 2.95 -1.36 5.68 -1.13 6.41 6.41 1.48 0.05 15.72 23.77 0.08  B W ' s , (Geske) G m a x -G=°/Gmax (%) G m a x o - G / G m a x (%) Slope G m a x o - G / G m a x (%) Slope Gmax o o -G /Gmax (%) Slope Gmax -MP/Gmax (%) Gmax -MP/Gmax (%) G m a x -MP/Gmax (%) M P - G m i n /MP(%) MP-Gmin /MP(%) MP-Gmin /MP(%) a of M P  35.13* 7.81* 0.06* 38.87* -0.73* 40.11* -0.54* 15.26* 11.54* 9.13* 39.67* 89.67* 74.87* 0.36  35.34 0.00 0.00 38.64 -0.73 40.87 -0.63 14.64 12.70 11.92 37.47 56.62 45.65 0.33  5.95 4.17 0.02 4.70 0.13 4.93 0.10 2.14 1.72 2.57 5.67 19.14 17.71 0.06  82.97 79.1 0.41 78.05 0.30 77.34 0.26 33.56 23.51 23.51 81.82 273.88 268.78 1.25  0.00 0.00 0.00 6.35 -1.73 . 8.65 -1.19 1.26 0.37 -21.77 2.33 -11.53 -17.11 0.08  Statistic  2 0 0  2 0  2  0  5  2 0 0  2 0 0  5 0  2  20  2 0 0  5  2 0 0  2 0 0  7 0  0  5 0  5  2 0 0 70  5 0  2 0 0  2 0 0  2  0  0 5  20 0  2  0  5 0  20 0  2  7 0  2 0 0  2 0 0  2 0 0  7 0  7 0  5 0  5 0  2 0 0 70  50  G =theoretical value at o = 0.05; G = theoretical value at a = 0.50; G = theoretical value at a = 0.70. Gmax(Gmin) = maximum (minimum) theoretical value in range o = 0.05-2.00; Gmax(Gmin) = maximum (minimum) theoretical value in range o = 0.05-0.70; Gmax(Gmin) = maximum (minimum) theoretical value in range o=0.05-0.50. M P = m a r k e t price of issue. 0 5  5 0  7 0  2 0 0  7 0  5 0  Slopes are for values in preceding line. "* "Indicates that mean is significantly different from zero at the 5 percent level from a one-tailed t-test.  70  Table  11. Some Characteristics of Sample Functions, Brennan-Schwartz for the a Range 0.05-1.00  Statistic CB's Gmaxj - G ,,/Gmaxj (%) Gmax j o o -G /Gmax (%) Slope Gmax - G / G m a x (%) Slope Gmax o o -G /Gmax „ (%) Slope G m a x ! - M P / G m a x (%) Gmax -MP/Gmax (%) Gmax -MP/Gmax (%) MP-Gmin /MP(%) MP-Gmin /MP(%) MP-Gmin /MP(%) o of M P M  1  0  00  0 5  1 0 1  x 0 0  5 0  x  y 00  7 0  x 0  0 0  7 0  7 0  5 0  5 0  1 0  0  0  10  0  x  0  5 0  x 0 0  7  x 0 0  1 0 0  x 0 0  7 0  7 0  5 0  5 0  1 0 0  70 50  Max  Min  4.77* 3.86* 0.06* 3.00* -0.01 4.01* -0.04*  4.52 0.18 0.01 2.06 -0.02 3.30  0.48 0.67 0.01 0.41 0.01 0.38 0.01 1.07  16.77 33.02 0.35 22.35 0.62 14.41  0.00 0.00 0.00 0.00 -0.27 0.00  -0.05 3.70  0.98 0.92  0.31  15.11* 8.79* 0.17* 7.91* -0.11 10.39* -0.20* 28.96* 27.90* 27.72* -14.50* -14.98* -18.29* 0.48  13.16 6.48 0.16 6.27 -0.01 7.14 -0.21 28.17 25.47 23.64 -9.09 -12.59 -14.09 0.56  3.21  50  t c  SE  3.15 3.15 3.20 0.28 0.58 0.28  1 0 0  x  Median  5.96* 5.51* 1.26 0.62 -0.88  10  BW's Gmax; - G IGmax „(%) G m a x o - G s/Gmax; „(%) Slope Gmax o o -G /Gmax (%) Slope Gmax i o o -G o /Gmax (%) Slope Gmax -MP/Gmax (%) Gmax -MP/Gmax (%) Gmax -MP/Gmax (%) MP-Gmin /MP(%) MP-Gmin /MP(%) MP-Gmin /MP(%) o of M P  Mean  6.43*  1 00  Solution  0.99 0.83 0.73 0.03  1.58 0.03 1.38 0.09 1.81 0.08 2.69 2.61 2.64 7.18 3.73 3.00 0.09  0.48  -0.25  41.26 32.17 25.11 18.58 19.11 15.91 1.00  -13.93 -13.93 -13.93 -23.69 -18.77 -18.17  47.32 30.40 0.50 19.42 0.43 27.53 0.61 55.00 55.00 55.00 37.97 17.12 5.86 1.00  0.00 0.00 0.00 0.46 -1.43 0.15 -0.88 6.58 6.58 6.58 -106.94 -51.68 -51.68 0.05  0.05  G„ = theoretical value at o = 0.05; G = theoretical value at o = 0.50; G , = theoretical value at o = 0.70. G m a x ( G m i n ) =maximum (minimum) theoretical value in range a = 0.05-1.00; Gmax(Gmin) = maximum (minimum) theoretical value in range o — 0.05-0.70; Gmax(Gmin) = maximum (minimum) theoretical value in range 0 = 0.05-0.50. MP=market price of issue 5  s 0  0  10 0  7 0  5 0  Slopes are for values in the line  preceding.  "* "Indicates that the mean value is significantly different percent level from a one-tailed t-test.  from zero at the 5  71 22 percent for the A L L assumption and about 30 percent for the  NONE  assumption, so average curves were not very flat. The market price was significantly above the minimum value for the A L L assumption in both full and restricted ranges, and was significantly below the minimum in the restricted range for the N O N E assumption. That the Ingersoll solution with the assumption is a poor bond evaluation model is demonstrated  NONE  by the average o  for market price—0.84, a value outside the most likely feasible range. The results for BW's for the simpler solutions are given in Table 10. The figures for both solutions reflect the mixture of slope types. Most noticible is the large average variation in theoretical values (about 36 percent for the Ingersoll solution and 40 percent for the Geske solution). Also striking is how far  market  prices would be from minimum values if these solutions were correct. Table 11 contains the results for the Brennan-Schwartz solution for both CB's and B W ' s . These statistics differ markedly from the statistics for the simpler solutions. The average percent differences between Gmax and values at various a's reflect, as did the results for the simpler solutions, the fact that the most frequently observed shape was a curve with a maximum. The differences were much less, showing that the more realistic functions were much flatter functions generated by the simpler models. This is consistent with the  than  results  from the simulations that were reported earlier. This relative flatness comes from a more realistic approximation of the true effects of coupon payments and call provisions, both of which serve to broaden and flatten curves. Call provisions effectively reduce the maximum value a bond can attain, thus reducing the range of prices over the a range. For CB's this call provision, as noted, restricts both the value of the debt portion and the value of the warrant portion. This is not  the case for bond-warrants, so more variation could be expected for the latter type. For C B ' s , average maximum variation was only about 5 percent over the a range 0.05-1.00, while for BW's average variation was about 15 percent. Market price of CB's was on average 6.43 percent below maximum theoretical value and 1.26 percent above minimum theoretical value, but the latter value was not significantly different from zero. Corresponding figures for BW's are 28.96 percent below maximum and 14.5 percent below minimum, both significantly different from zero at the 5 percent level. In the range a = 0.05-0.5 market price of CB's was about 5.51 percent below the maximum value in the range and 0.88 percent below the minimum. The corresponding values for BW's are 27.72 percent below maximum and 18.29 percent below the minimum. For CB's these results compare very favorably to earlier results. Jones et aZ.(1984,1985) and King(1986) both found that theoretical prices from Brennan and Schwartz type solutions (based on point estimates of o) deviated systematically from market prices. Jones et oJ.(1984) found that overall in their sample of 305 straight bonds, the predicted price was about 4.5 percent higher than market price, while for the subsample of noninvestment grade bonds the predicted price was about 10 percent higher than the market p r i c e . King(1986) 3 1  reported results for 103 CB's (also based on point estimates of a), and found on average overpricing of 4.5 percent for one date and 3.0 percent for the second date. For the subsample in which bid prices were compared to predicted prices, average overpricing was 9.2 percent and 8.4 percent at the respective dates. No studies of bond-warrants have been reported so it is hard to evaluate these results. Certainly the poor agreement  "  between market price and theoretical  These figures are from Jones et aZ.(1984), Table 3.  73 price is disappointing compared to the results for the convertibles. This problem needs to be researched  further.  To see if S H A P E and I N / O U T for the Brennan and Schwartz solution were related to firm or issue characteristics, group means were compared using univariate A N O V A .  It should be noted that the B W sample is too small for the  A N O V A to be more than a qualitative guide to how firms/issues differed. Unfortunately C B and B W samples were too small to allow comparison of group means with the sample classified by both S H A P E and IN/OUT. The numbers in each category for CB's were: Fl-Il,  0; F1-I2, 3; F1-I3, 12;  F2-I1, 2; F2-I2, 6; F2-I3, 3; F3-I1, 5; F3-I2, 12; F3-I3, 5; F4-I1, 9; F4-I2, 17; F4-I3, 9. The  x  value in the test for independence of S H A P E and I N / O U T was 17.27  2  (significance was 0.008), so the hypothesis that the variables were independent was rejected. Note, however, that the expected frequency of 4 of 12 cells was less that 5, so the x Kendall's r - C  32  2  values must be interpreted with caution. However,  for this table was only -0.29 (significance is 0.0017) which  indicates only a moderate negative association between S H A P E and IN/OUT. It is puzzling that for S H A P E S F 2 , F 3 , and F4 there was a symmetry, with about one-half in range and the other half split above and below range, but for S H A P E F l , 80 percent of the sample was below range.  " ~^ 3  Kendall's r-C is a nonparametric measure of association and is defined as: r-C = [2m(P-Q)]/N (m-l); where P = number of concordant pairs; Q = number of discordant pairs; N = sample size squared; and m = minimum of number of rows and number of columns in contingencj table. 2  2  7  74 Table 12. Characteristics of Convertibles Classified by Shape and Location of Market Price Relative to Theoretical Price Range STAT  Fl  F2  F3  F4  11  12  13  FREQ  15  11  22  35  16  38  29  Gl*t  18.35 (2.02)  4.98 (2.28)  1.40 (1.07)  3.82 (1.20)  -4.10 (0.86)  3.09 (0.50)  15.27 (1.40)  G2*t  -7.87 (1.84)  1.03 (1.43)  2.81 (1.14)  2.74 (1.23)  8.69 (1.07)  3.60 (0.41)  -7.75 (0.97)  MVt  219.65 (63.46)  277.17 (95.23)  338.93 (77.97)  754.60 (264.91)  1453.10 (536.38)  299.87 (51.78)  191.96 (37.49)  D*t  0.34 (0.04)  0.50 (0.03)  0.29 (0.02)  0.28 (0.03)  0.24 (0.03)  0.32 (0.02)  0.38 (0.03)  CV  0.15 (0.01)  0.16 (0.02)  0.19 (0.02)  0.18 (0.02)  0.14 (0.02)  0.18 (0.01)  0.18 (0.02)  Ti  0.18 (0.02)  0.21 (0.03)  0.18 (0.02)  0.16 (0.02)  0.13 (0.02)  0.18 (0.01)  0.20 (0.02)  R-C(%)  2.28 (0.27)  1.42 (0.47)  2.03 (0.33)  2.35 (0.20)  2.72 (0.31)  2.13 (0.21)  1.80 (0.25)  EP*  0.09 (0.02)  0.24 (0.05)  0.16 (0.03)  0.29 (0.03)  0.19 (0.04)  0.24 (0.04)  0.18 (0.03)  M*t  17.47 (1.38)  21.36 (1.19)  20.36 (1.11)  21.86 (0.58)  23.44 (0.88)  20.60 (0.66)  19.03 (0.94)  B *  0.50 (0.05)  0.65 (0.04)  0.36 (0.04)  0.32 (0.05)  0.41 (0.06)  0.35 (0.04)  0.47 (0.04)  B *  0.41 (0.07)  0.53 (0.08)  0.20 (0.04)  0.21 (0.04)  0.22 (0.05)  0.28 (0.04)  0.33 (0.05)  0.46 (0.19)  0.75 (0.34)  1.08 (0.34)  1.35 (0.25)  1.34 (0.31)  1.05 (0.25)  0.86 (0.23)  a.  x  2  R-C  0  " G l " = G m a x - M P / G m a x ( % ) ; "G2" = M P - G m i n /MP(%). i d i t e s variable for which group means for S H A P E are significantly different at the 5 percent level. "t" Indicates variable for which group means for IN/OUT are significantly different at the 5 percent level. Standard errors are in brackets 70  M * I I  n  ca  70  7  0  Two other multiple classifications of CB's and B W ' s that would be of interest in A N O V A if a sufficient sample were available are S H A P E by callable/not callable at issue and IN/OUT by callable/not callable at issue. A call feature would serve to flatten a curve and decrease overall value, so S H A P E  at  issue could be related to structure of the call provision. Deferrment of the call option would result in steeper curves, all things e q u a l . The numbers in each 3 3  category of call timing by S H A P E and I N / O U T for CB's were: Fl-not callable, 12; Fl-callable, 3; F2-not callable, 1; F2-callable, 10; F3-not callable, 1; F3-callable, 21; F4-not callable, 10; F4-callable, 25; 11- not callable, 0; Il-callable, 16; 12- not callable, 7; I2-callable, 31; 13- not callable, 17; I3-callable, 12. There seems to have been a link between  a bond being non-callable and having  the F l S H A P E (up-rising) and also between being non-callable at issue and being overpriced by the model. The x  2  test for independence for S H A P E by callable  yielded a value of 27.50 (significance = 0.000, with 2 of 8 cells having expected frequency  less than 5). The value for I N / O U T by callable was  (significance = 0.000, with 1 of 6 cells with expected frequency  20.99 less that 5).  In both cases, then, the independence of the two pairs of variables was  rejected.  Kendall's r-C for S H A P E by callable was 0.22 (significance = 0.02) and for I N / O U T by callable was -0.48 (significance = 0.00). The former indicates only moderate positive association, while the latter indicates relatively strong  3  3  A l l bonds not callable at issue became callable at some later date.  negative  76  association. Of course, with such a small sample these results must be interpreted cautiously. Results from the A N O V A ' s are in Tables 12, 13, and 14. In the C B sample, market prices were on average significantly below the minimum value in the range for securities with S H A P E F l functions but were on average significantly above the minimum for S H A P E F3 and S H A P E F4 functions. This difference in behavior was also observed in the subsample of firms with equity traded on the N Y S E or A M E X , and in the B W sample, where the  percent  below minimum was much greater for F l curves than for F 4 curves. This odd result could be explained plausibly in several ways: (1) the  effect  was a genuine market effect and attributable to some extra risk not accounted for in the bond-valuation model that the market perceived for firms that chose to issue Fl-shaped securities; (2) the model overpriced the warrant component more than it overpriced the bond component or incorrectly accounted for a delayed call feature;  (3) assuming that these were issued by low-variance firms as  suggested  by the Brennan and Kraus model, the model overpriced lower variance securities more than higher variance securities. Limited support for the latter comes from Jones et aZ.(1984) who found that overall average overpricing was greater for low-variance than for high-variance f i r m s *. . The debt ratio and book-value ratios 3  for F l firms were relatively large, but not so large as for F 2 firms. There was also an interesting trend in the coupon rates on other debt, with highest rates for F l issues and lowest for F4 issues, although the group means were found to be not significantly different. These characteristics, coupled with the lower exercise  "This i s i n contrast to Jones et o.Z.(1985) who found that the securities of low-variance firms were underpriced and the securities of high-variance firms were overpriced. 3  77 Table 13. Characteristics of Convertibles (Subsample with Common Equity Listed on A M E X or N Y S E ) Classified by Shape and Location of Market Price Relative to Theoretical Price Range STAT  Fl  F2  F3  F4  11  12  13  4  6  14  19  12  20  11  Gl*t  19.32 (4.79)  7.09 (3.17)  0.66 (1.32)  2.23 (1.69)  -3.95 (1.09)  2.98 (0.69)  14.46 (2.62)  G2*t  -6.53 (2.92)  -0.55 (2.22)  2.85 (1.48)  3.91 (1.68)  8.92 (1.42)  2.59 (0.42)  -6.74 (1.57)  MVt  192.42 (22.13)  353.71 (173.89)  414.27 (114.06)  1133.33 (471.33)  1778.89 (693.26)  337.03 (83.11)  194.34 (44.06)  D*t  0.39 (0.13)  0.52 (0.06)  0.29 (0.30)  0.27 (0.04)  0.28 (0.04)  0.28 (0.03)  0.47 (0.07)  CV  0.16 (0.02)  0.15 (0.03)  0.20 (0.03)  0.17 (0.02)  0.15 (0.03)  0.18 (0.02)  0.19 (0.03)  t  0.21 (0.05)  0.20 (0.05)  0.20 (0.03)  0.16 (0.02)  0.14 (0.32)  0.17 (0.02)  0.25 (0.03)  R-C(%)  1.92 (0.44)  1.35 (0.57)  2.34 (0.39)  2.62 (0.25)  2.76 (0.30)  2.39 (0.29)  1.58 (0.39)  EP*  0.07 (0.07)  0.20 (0.05)  0.13 (0.03)  0.26 (0.04)  0.20 (0.06)  0.19 (0.04)  0.18 (0.04)  M*t  16.25 (1.25)  21.67 (1.67)  20.21 (1.44)  22.11 (0.79)  23.33 (1.12)  20.65 (0.92)  18.64 (1.36)  B,*  0.51 (0.12)  0.72 (0.04)  0.34 (0.23)  0.32 (0.06)  0.44 (0.07)  0.32 (0.06)  0.49 (0.07)  B *  0.47 (0.13)  0.59 (0.09)  0.19 (0.06)  0.21 (0.06)  0.24 (0.08)  0.25 (0.02)  0.39 (0.09)  0.48 (0.22)  1.18 (0.58)  1.21 (0.55)  1.74 (0.36)  1.32 (0.34)  1.39 (0.42)  1.39 (0.49)  FREQ  7  2  R-C  0  " G l " = G m a x - M P / G m a x (%); "G2" = M P - G m i n /MP(%). "*" Indicates variables for which group means for S H A P E are significantly different at the 5 percent level. "t" Indicates variables for which group means for IN/OUT are significantly different at the 5 percent level. Standard errors are in brackets. 7 0  7 0  7 0  78 Table 14. Characteristics of Bond-Warrants Classified by Shape and Location of Market Price Relative to Theoretical Price Range Fl  F2  F3  F4  11  12  13  8  2  0  12  0  4  18  Gl*t  35.22 (3.65)  33.19 (13.51)  na (na)  22.89 (2.97)  na (na)  14.17 (3.06)  30.95 (2.63)  G2*t  -25.96 (5.12)  -16.87 (12.04)  na (na)  -7.34 (4.76)  na (na)  10.33 (1.27)  -20.60 (1.23)  147.75 (20.62)  702.81 (629.36)  na (na)  237.19 (90.33)  na (na)  417.88 (270.28)  209.00 (66.89)  D  0.45 (0.05)  0.48 (0.16)  na (na)  0.55 (0.04)  na (na)  0.59 (0.03)  0.49 (0.04)  CV  0.24 (0.03)  0.19 (0.08)  na (na)  0.33 (0.05)  na (na)  0.25 (0.03)  0.29 (0.04)  7  0.23 (0.04)  0.11 (0.06)  na (na)  0.21 (0.05)  na (na)  0.12 (0.01)  0.23 (0.04)  R-C(%)  -0.40 (0.47)  0.74 (0.67)  na (na)  -0.09 (0.47)  na (na)  0.17 (0.44)  -0.20 (0.37)  EP  0.19 (0.06)  0.03 (0.09)  na (na)  0.31 (0.08)  na (na)  0.48 (0.16)  0.19 (0.05)  Ml  11.88 (1.91)  10.00 (0.00)  na (na)  13.83 (1.47)  na (na)  16.25 (2.40)  12.00 (1.14)  B»  0.51 (0.07)  0.49 (0.31)  na (na)  0.44 (0.07)  na (na)  0.57 (0.03)  0.45 (0.06)  B  0.40 (0.08)  0.49 (0.31)  na (na)  0.26 (0.74)  na (na)  0.44 (0.06)  0.31 (0.06)  0.66 (0.28)  1.15 (1.15)  na (na)  0.45 (0.25)  na (na)  0.63 (0.18)  0.58 (0.28)  STAT FREQ  MV  2  R-C  0  " G l " = G m a x - M P / G m a x , ( % ) ; "G2" = M P - G m i n /MP(%). "* "Indicates variable for which group means for S H A P E are significantly different at the 5 percent level " t " Indicates variable for which group means for IN/OUT are significantly different at the 5 percent level. Standard errors are in brackets. 7 0  7  7 0  79 premia, shorter maturities and delayed call seem to be consistent with the extra risk explanation in that the investor wanted more protection for his/her investment in the firm  3  5  . Lower exercise premium and shorter maturities also  seemed to be a characteristic of BW's with F l S H A P E , although the sample is too small for firm conclusions to be drawn. Firms with CB's with prices above range (and thus underpriced by the bond model) had a higher market value and lower debt ratio than those with prices below range. Further, although not significantly different at the 5 percent level, the group means for coupon on other debt indicate a lower coupon rate on other debt for securities underpriced than for securities overpriced. These characteristics suggest lower risk of default, which could explain the relatively higher prices. The issue characteristics that varied across the I N / O U T  categories  were y and maturity, with those above range having lower y and longer term. These results can be compared, albeit roughly, to those of Jones et al.( 1984) who  ^ Thatcher(1986) suggested that "two-tiered" call provisions (callable from issue, except for refinancing at a lower rate, for some years, then callable for any purpose thereafter) could be used to resolve informational asymmetry about expected cashflows or to resolve the risk incentive problem described earlier, and concluded that firms with high debt ratios, high default risk, and currently outstanding debt with long maturities would use two-tiered call provisions. Robbins and Schatzberg(1986) showed in a very limited model that risk-averse agents with compensation contingent upon realized value of a firm would prefer to issue callable bonds if the prospects of the firm were good. "Bad news" firms would issue equity. In this sample almost all F l securities had what Thatcher labelled as "standard" call provisions, meaning not callable for some period then callable thereafter. F2 and F3 securities were almost all callable at issue. The average debt ratio was highest for F2 securities, lowest for F3 and F4 securities. The ratio of book value of other debt to book value of all debt was highest for F l and F2 securities. Group means of face-weighted average maturity of other debt for each of the four shapes were 15.12, 17.71,12.19, and 15.10 which were not significantly different according to A N O V A . Thus average maturity of existing debt was shorter than the maturity of the new issue. These results do not support Thatcher's model. If "not callable" equals "bad news" then the price of the new securities could reflect this and be lower than bond valuation model predicts.  80 found overpricing to be greater firms,  for junior bonds  for higher-leveraged compared to lower-leveraged  compared to senior bonds,  and for high-coupon  versus  low-coupon bonds. W i t h i n a category type, they found overpricing lower for investment grade this sample, we  than  for non-investment  should observe that below range  and higher coupon rate than the However, the  above  range  Jones et al. results. w i t h the  grade. I f Jones  above range  firms had  firms,  The lower 7 for above range  willing to hold more  It m i g h t also be the  less likely to default,  so requires  price, regressions  relationship between  of the difference between  market  dependent)  and T a b l e 16 ( M P - G m i n „/MP(%) as  are  them w a s  variables m a y  ratios based  on book  predicted and  observed  price and theoretical m a x i m u m was regression  15 ( G m a x - M P / G m a x ( % )  dependent).  7 0  B o t h the  7 0  high in absolute  value. Regressions containing both  from  m a y have been too low because  coefficients were artificially  C B ' s alone, the  these  the u s u a l problem of  t-values  as  7 variable  1  thus have suffered  variances of the  are  ratio v a r i a b l e ( D ) were included, even though the correlation  multicollinearity—namely, that the  For  shown i n T a b l e 7  and the new debt between  2  the  equity  backward e l i m i n a t i o n , w i t h probability of F-to-remove set to 0.10. The along w i t h the R  with  market  or theoretical m i n i m u m on input variables were r u n . T h e method used  coefficients  case.  consistent  t h a t the  less of an  categories.  light on the  is indeed the  case  value were not significantly To shed further  a higher debt ratio,  managers/owners  of the  the  apply to  is less consistent  incentive to contribute capital. The group means different across  results  securities m i g h t be  agency theory i n that the  of the f i r m .  views these f i r m s as  which  had longer maturities w h i c h  " p e r k s " model from  et al.  inflated.  significant explanatory variables for predicting  the  81 Table 15. Regressions of [(Gmax, - M P ) / G m a x ]X100 on Variables Used In Pricing (Brennan-Schwartz Solution) 0  K  Full  CB's + BW's Reduced  CB's Full  Reduced  BW's Full  Reduced  36.21 (5.46)  33.18 (6.60)  17.27 (3.22)  16.79 (6.92)  44.09 (1.30)  24.64 (3.53)  -0.002 (-2.71)  -0.19* (-2.69)  -0.006 (-0.44)  -6.18* (-0.56)  MV  7 0  ....  ...  D  25.48 (1.81)  19.57 (2.94)  -21.53 (-1.59)  -15.40 (-1.88)  10.0 (0.18)  Dx  -30.42 (-1.30)  -24.80 (-2.55)  -117.61 (-3.74)  -119.53 (-4.94)  -104.64 (-1.17)  -70.59 (-3.82)  7  -6.01 (-0.34)  133.52 (3.76)  129.40 (4.59)  55.17 (1.16)  48.75 (2.70)  R-C  -1.42 (-1.83)  EP  -7.12 (-1.18)  M  -0.89 (-3.98)  CALL  -8.79 (-3.98)  RT  -0.51 (-0.24)  Bx  -0.45 (-0.48)  B  -3.59 (-0.54)  —  -0.087 (-0.11) 0.44 (0.29)  2  R-C M  0  0  R R adj 2  2  0.54 0.49  (Table 15 continued)  —  -1.58 (-2.30)  0.025 (0.04)  —  0.27 (0.094)  —  ...  1.69 (0.35)  ....  5.00 (0.23)  ...  -0.89 (-4.47)  -0.13 (-0.67)  ....  -0.50 (-0.60)  ...  -8.17 (-3.97)  -10.00 (-6.34)  -1.60 (-0.19)  ...  -5.02 (-0.41)  ...  ....  ....  —  0.049 (0.04) 6.99 (1.72)  -10.32 (-7.11) — ....  -5.03 (-1.13)  —  ....  -1.96 (-0.37)  ....  ....  0.094 (0.91)  0.53 0.50  0.62 0.56  -34.55 (-0.66)  ...  13.88 (0.48)  ...  -1.36 (-0.27)  ...  —  1.24 (1.51)  0.80 (2.05)  0.60 0.57  0.56 -0.16  0.48 0.39  82 (Table 15 continued) " K " = regression constant; " M V " = market value of firm; " D " = market value of all debt/MV; "0/'= market value of new bond/MV; "7" = fraction of firm acquired upon conversion of bond or exercise of warrant; "R-C(%)"= default-free rate less coupon-rate on new bond; " E P " = (exercise price-current stock price)/current stock price; " M " = maturity; " C A L L " = dummy variable indicating callable/not callable at issue (O = not callable); "RT" = dummy variable indicating rated/not rated at issue (0 = rated); "B^' = ?ace value of other debt/face value of all debt; " B " = face value of private debt/face value of all debt; " R - C " = default-free rate minus face-weighted average of coupon rates on other debt(%); " M " = face-weighted average of maturity dates of other debt. 2  0  a  * Coefficients multiplied by 10 . T-values for coefficients in brackets.  83 Table 16. Regressions of [(MP-Gmin )/MP]X100 on Variables Used In Pricing (Brennan-Schwartz Solution). 70  Full K  -34.48 (-3.47)  MV  -1.45* (-0.08)  D  -20.31 (-0.94)  Dx  85.73 (2.34)  7  CB's  +  BW's  Reduced -35.20 (-5.30) ....  CB's  Full  -10.66 (-2.09) 0.001 (1.73)  Reduced -7.99 (-3.73) ...  BW's  Full  -49.70 (-0.61) 0.012 (0.36)  Reduced -33.18 (-3.06) ...  18.35 (1.42)  16.03 (2.05)  -16.35 (-0.12)  69.23 (3.32)  93.37 (3.09)  103.39 (4.47)  253.49 (1.17)  129.47 (2.86)  -47.63 (-1.78)  -56.58 (-2.74)  -108.21 (-3.18)  -119.93 (-4.37)  -166.56 (-1.45)  -102.95 (-2.43)  R-C  1.84 (1.56)  2.28 (2.30)  -0.15 (-0.27)  EP  5.03 (0.56)  M  1.01 (3.01)  1.00 (3.26)  CALL  8.08 (2.42)  7.39 (2.40)  RT  -0.87 (-0.28)  Bx  1.05 (0.076)  Bx  10.25 (1.05)  R-C  0  M„ R* R adj J  -0.16 (-0.095) -0.15 (-0.63) 0.38 0.28  (Table 16 continued)  ....  ....  ....  4.35 (0.93) 0.17 (0.95) 8.62 (5.66) -0.44 (-0.32) -7.41 (-1.24)  ....  — ....  8.90 (6.34) — ....  -1.24 (-0.18) -24.78 (-0.48)  —  0.51 (0.25) 1.76 (0.09) 7.27 (0.25)  ...  61.95 (0.49)  ...  6.06 (1.41)  ...  -2.32 (-0.033)  ...  —  -0.12 (-0.23)  ...  6.06 (0.49)  ...  —  -0.096 (-0.97)  -2.57 (-1.20)  ...  ...  —  0.37 0.34  0.56 0.47  0.53 0.48  0.50 -0.31  0.33 0.26  84 (Table  16  continued)  " K " = regression constant; " M V " = m a r k e t value of firm; " D " = m a r k e t value of all d e b t / M V ; "0/' = m a r k e t value of new b o n d / M V ; "y"= fraction of firm acquired upon conversion of bond or exercise of w a r r a n t ; " R - C " = default-free rate less coupon-rate on new bond; " E P " = (exercise price less current stock price)/current stock price; " M " = m a t u r i t y ; " C A L L " = d u m m y variable indicating callable/not callable at issue (not callable = 0); " R T " = d u m m y variable indicating rated/not rated at issue (rated = 0); " B " = face value of other debt/face value of a l l debt; " B " = face value of private debt/face value of a l l debt; " R - C " = default-free rate m i n u s face-weighted average of coupons on other debt(%); " M " = face-weighted average m a t u r i t y of other debt. x  2  0  0  * Coefficients  multiplied by  T-values for coefficients in  10 . brackets  85 difference between market price and maximum theoretical value were debt ratio, new debt ratio, market value of firm, 7, and call feature.  That is, all things  equal, the percent difference between theoretical maximum and market price was greater for bonds not callable at issue. The difference decreased  as firm value,  debt ratio, and new debt ratio increased but increased as 7 increased. These variables explained 60 percent of the variabilttty. The tolerances for the 7 and D  x  variables were acceptable, despite their strong correlation. It would have been  desirable to have re-estimated this regression using some method to adjust for possible multicollinearity. However, the most applicable methods require more information—either more data or incorporation of restrictions based on prior knowledge. Neither was available. The variables significant in explaining the percent difference between market price and theoretical minimum were the same as in the previously described regression, except for the absence of the value of the firm variable(MV), but with opposite sign. R  2  market  was, as noted, 0.53.  For B W ' s alone, the most significant variables explaining the  difference  between market price and theoretical maximum were new debt ratio (negative association), 7 (positive association), and composite maturity of other debt (positive association). These three variables explained 48 percent of the variability. Thirty-two percent of the variability in the percent difference between  market  price and minimum theoretical value was explained by 7 and new debt ratio, but signs on coefficients were opposite. The caution given above, about multicollinearity, applies in the interpretation of the significance of these coefficients. The same regressions were run but with the addition of dummy variables for S H A P E . The final results for percent difference between theoretical maximum  86  Table  17.  Regressions of f ( G m a x - M P ) / G m a x ] X 1 0 0 on V a r i a b l e s U s e d P r i c i n g Plus S H A P E V a r i a b l e s (Brennan-Schwartz Solution) 7 0  K MV  Full  C B ' s + BW's Reduced  33.55 (5.66)  31.65 (7.07)  -442.3* (-0.41)  D  25.52 (1.97)  —  15.31 (2.57)  7 0  CB's  In  Full  Reduced  BW's Full  Reduced  15.46 (3.20)  16.71 (8.69)  46.99 (1.50)  47.00 (8.89)  -0.002 (-2.80)  -15.85* (-2.49)  -0.018 (-1.10)  ... ...  -22.73 (-1.85)  ....  19.51 (0.37)  Dx  -15.42 (-0.69)  ....  -102.42 (-3.49)  -76.79 (-4.08)  -42.64 (-0.41)  7  -6.55 (-0.41)  ....  131.71 (4.00)  80.76 (4.64)  3.45 (0.043)  R-C  -1.19 (-1.16)  -1.49 (-2.41)  -0.39 (-0.71)  ....  EP  0.69 (-0.12)  ....  8.82 (1.75)  ....  -1.01 (-0.039)  ...  M  -0.65 (-3.21)  -0.58 (-3.14)  0.004 (0.026)  ....  0.86 (0.79)  ...  CALL  -4.91 (-2.25)  -5.21 (-2.61)  -5.12 (-2.88)  RT  0.079 (0.042)  —  -0.16 (-0.13)  2.61 (0.31)  —  10.84 (1.85)  -6.54 (-1.12)  —  -6.33 (-1.58)  0.49 (0.67)  0.20 (0.40)  ....  —  0.038 (0.41)  ....  Bx B  2  R-C M  0  -0.017 (-0.13)  0  ....  -6.41 (-3.97) ... ....  ...  0.035 (0.012)  -19.12 (-1.44) -1.59 (-0.12)  ... ...  —  -13.55 (-2.77) —  11.49 (0.21)  ...  -23.87 (-0.70)  ...  -3.04 (-0.52)  —  0.20 (0.21)  —  -12.45 (-3.70)  -11.84 (-3.67)  -10.97 (-3.73)  -9.91 (-4.08)  TS3  -14.87 (-4.96)  -14.93 (-5.48)  -11.00 (-4.29)  -9.21 (-4.07)  TS4  -11.21 (-4.01)  -11.31 (-5.08)  -7.79 (-2.92)  -6.62 (-3.21)  -23.90 (-4.49)  -20.36 (-4.22)  0.65 0.63 0.59 0.61 17 continued)  0.71 0.64  0.67 0.65  0.72 0.01  0.49 0.43  R R adj 2  2  (Table  14.34 (0.68)  ...  TS2  ...  ...  87 (Table  17  continued)  " K " = regression constant; " M V " = m a r k e t value of firm; " D " = m a r k e t value of a l l d e b t / M V ; " " = fraction of f i r m acquired upon conversion of bond or exercise of w a r r a n t ; " R - C " = default-free rate less coupon-rate on new bond; " E P " = (exercise price less current stock price)/current current stock price; " M " = m a t u r i t y ; " C A L L " = d u m m y variable indicating callable/not callable at issue (not callable = 0); " R T " = d u m m y variable indicating rated/not rated at issue (rated = 0); " B / ^ b o o k value of other debt/book value of a l l debt; " B " = b o o k value of private debt/book value of all debt; " R - C " = default-free rate minus face-weighted average of coupon rates on other debt; " M " = face-weighted average of maturities of other debt; " T S 2 " = d u m m y variable indicating S H A P E F 2 / n o t F 2 at issue (F2 = l); " T S 3 " = d u m m y variable indicating S H A P E F3/not F 3 at issue (F3 = l); " T S 4 " = d u m m y variable indicating S H A P E F 4 / n o t F 4 at issue (F4 = l ) . 7  2  0  0  * Coefficients multiplied by T-values for coefficients  in  10 . brackets.  88  and market price are given in Table 17. For CB's the R  2  was 0.67, slightly  greater than 0.60 for the final model without the dummy variables. However the models were non-nested: the debt ratio and market variables were not included in the reduced model with the dummy variables. Thus they were not directly comparable. The base case was "not callable-at-issue, F l " securities and the results show that there were significant differences between S H A P E F l securities and all three other shaped securities, with the average difference Gmax  7 0  between  and market price being significantly greater for F l securities. Also the  results indicate that the average difference was greater for securities not callable at issue. This is consistent with the results from the contingency tables noted previously. When the regressions were repeated  using a different base case no  significant differences were found between F 2 and F3 curves but significant differences were found between F2 and F4 curves, and F3 and F 4 curves. For the whole sample the repeated  regressions with different base cases revealed no  significant differences between F2 and F 3 , F 2 and F4, and F 3 and F 4 curves. The results for the B W ' s are quite different from the results without the dummy variables. Without the dummy variables, y and new debt ratio ( D J were significant, but with dummy variables, the C A L L variable and the F 4 dummy variable were significant. In the latter case these results indicate that the percent difference between theoretical maximum and market price was significantly greater for not-callable-at-issue F l securities than for callable F 4 securities. It is interesting that this result agrees with the results for CB's, despite the small sample size and the overpricing. The results for the percent difference between market price and theoretical  89  minimum were similar in that the dummy variables proved to have significant coefficients. •  The final result for CB's was:  G2 = -7.37 + 62.7D -68.8-y + 7.06CALL + 5.99TS2 +4.10TS3 + 4.53TS4 1  (-3.72X3.17X-2.30) (4.12)  (2.30)  (1.69)  (2.08)  R = 0.52; R adj = 0.48. 2  2  The final result for BW's was: •  G2 = -67.78 + 34.42CALL+45.06F4 (-5.62) (3.08)  (4.08)  R =0.48; R adj = 0.42. 2  2  The addition/substitution of SHAPE variables did not result in more of the variability of this variable being explained for CB's but did improve the prediction for the BW's. To summarize, average overpricing was greater for convertible bonds not callable at issue, F l securities, and securities with higher 7's. These may not be unrelated because all things equal the higher the 7, the more valuable the warrant component relative to the bond component, and the longer the call is delayed the more steep the theoretical curves. To see if market price was related to SHAPE, regressions were run with CV (market price/market value of the firm) as the dependent and the input variables plus SHAPE dummy variables as the explanatory variables. The results are presented in Table 18. The base case was " F l , not-callable-at-issue" securities. For CB's the inclusion of SHAPE dummy variables added little to the explanatory power of the equation. The results show that market price was lower for F l curves than for F3 and F4 curves. Regressions repeated with a different base case revealed that there was a difference between F2 and F3  90 Table 18. Regressions of CV (Market Price/Market Value of Firm) on Input Variables plus SHAPE Variables. Full  CB's + BW's Reduced  CB's Full  Reduced  BW's Full  Reduced  K  0.14 (4.59)  0.12 (7.65)  0.026 (1.31)  0.011 (0.65)  0.15 (1.15)  0.16 (3.88)  MV  7.74* (1.32)  9.00* (1.68)  -5.26* (-1.69)  -7.98* (-3.02)  -5.92* (-0.085)  ...  D  0.33 (5.85)  0.35 (7.16)  -0.12 (-2.35)  -0.15 (-4.24)  0.27 (1.34)  0.36 (4.49)  7  0.40 (5.18)  0.40 (6.28)  0.95 (13.12)  1.02 (19.84)  0.58 (1.83)  0.39 (5.01)  R-C  -0.007 (-1.85)  -0.007 (-2.25)  -0.002 (-0.99)  ....  -15.42 (0.012)  ...  EP  -0.009 (-0.29)  ....  0.024 (1.15)  0.068 (0.62)  ...  M  -586.27* (-0.53)  CALL  0.002 (0.18)  — ....  —  0.0013 (1.79)  0.0014 (2.10)  -701.9 (-0.15)  ...  -0.002 (-0.26)  ...  0.068 (1.28)  0.053 (2.14)  -0.023 (-0.42)  ...  -0.39 (-2.50)  -0.45 (-8.65) ...  RT  -0.008 (-0.74)  B,  -0.20 (-5.41)  -0.22 (-6.40)  -0.037 (-1.53)  B  -0.086 (-2.88)  -0.092 (-3.24)  -0.032 (-1.97)  -0.039 (-2.58)  -0.017 (-0.11  -0.006 (-1.67)  ....  -0.003 (-1.28)  ....  -0.013 (-0.54)  ...  108.03* (0.15)  ....  395.0* (1.01)  ....  0.001 (0.26)  ...  0.004 (0.043)  ...  ...  2  R-C M  0  0  TS2  -0.016 (-0.87)  TS3  0.003 (0.19)  TS4  0.014 (0.92)  R R adj 2  2  0.86 0.84  (Table 18 continued)  —  — ....  —  0.85 0.85  -0.002 (-0.43)  0.007 (0.53)  ....  —  —  0.016 (1.57)  0.013 (1.84)  ...  0.024 (2.20)  0.027 (4.09)  0.067 (1.13)  0.061 (2.31)  0.95 0.94  0.94 0.94  0.96 0.88  0.95 0.93  91 (Table 18 continued) " K " = regression constant; " M V " = market value of firm; " D " = market value of all debt/MV; "y" = fraction of firm acquired upon conversion of bond or exercise of warrant; "R-C"= default-free rate less coupon-rate on new bond; " E P " = (exercise price less current stock price)/current current stock price; " M " = maturity; " C A L L " = dummy variable indicating callable/not callable at issue (not callable = 0); "RT" = dummy variable indicating rated/not rated at issue (rated = 0); " B " = face value of other debt/face value of all debt; " B " = face value of private debt/face value of all debt; "R-C„" = default-free rate minus face-weighted average of coupon rates on other debt; " M " = face-weighted average of maturities of other debt; "TS2"=dummy variable indicating S H A P E F2/not F 2 at issue ( F 2 = l ) ; "TS3"=dummy variable indicating S H A P E F3/not F3 at issue (F3 = l ) ; "TS4"=dummy variable indicating S H A P E F4/not F4 at issue ( F 4 = l ) . x  2  0  * Coefficients multiplied by 10 . T-values for coefficients in brackets.  curves and F2 and F4 curves, with values higher for F3 and F4 curves. There was also a difference between F3 and F4 curves. For BW's adding the SHAPE dummy variables resulted in an equation with about 2 percent more explanatory power than the equation with these were omitted. The results indicate that BW prices were lower for F l curves than for F4 curves and for not-callable-at issue securities. For some reason for both CB's and BW's F l securities were less valued by the market, all things equal. The results, unfortunately, do not provide any  strong clues as to why this should be.  4. Results of Tests on Sigma from the B&S Solution for A M E X or NYSE firms in the Sample As noted, the equity of 53 firms in the sample was traded on the NYSE or AMEX for 52 weeks following issue of the CB or BW. An estimate of the annual standard deviation of the equity was obtained from the CRSP Daily Return File, as outlined previously. Table 19 reports the statistics for the estimates of standard deviation for this subsample, figures from Dammon et a/.(1985), and statistics for the theoretical a of the market price. For only 6 of 43 firms issuing CB's was the a of the market price within the range bounded by the standard deviation of the stock and the leverage-adjusted standard deviation. However, the numbers above and below were very similar(18 and  19). For none of the BW firms was a within the range and 8 out of 10  were above the range. Frequencies of curve shape and whether market price was in or out of range are given in Table 20. The hypotheses that there were equal numbers of all shapes and that SHAPE F2 was most common were strongly rejected. The  93  Table 19. Statistics on a for Firms on NYSE or AMEX STAT  Mean  Median  SEt  Max  Min  0.40 0.27 0.35 o.ost -0.04  0.40 0.25 0.35 0.03 -0.08  0.02 0.02 0.04 0.04 0.05  0.78 0.60 1.00 0.84 0.75  0.14 0.10 0.05 -0.47 -0.57  0.40 0.20 0.68 0.48t 0.28t  0.37 0.16 0.73 0.58 0.36  0.03 0.03 0.12 0.12 0.11  0.60 0.35 1.00 0.69 0.63  0.32 0.11 0.05 -0.17 -0.28  0.35  0.31  0.20  3.27  0.10  0.32  0.29  0.17  2.38  0.12  CB's(B)  oS o-L oMP aMP-aL aMP-oS BW's(B)  aS oL oMP 0MP-0L  oMP-aS 1963-77* oS 1955-1983* *  aS  oS = standard deviation of common stock; aL = aS X (MVequity/MVfirm); aMP=o of MP from B&S solution. JFor Dammon et al. results, standard deviation, not standard error, shown. *Results from Dammon et al. for a sample of 736 firms; **Results from Dammon et al. for a sample of 436 firms. "t "Indicates that mean is significantly different from zero at 5 percent level from one-tailed t-test  94 Table 20. Frequencies of Curve Shape and Location of Market Price Relative to Theoretical Prices (a Range Defined by Standard Deviation of Stock Returns) STAT INI OUT 11 12 13  X (D 2  X (2) X (3) X (4) 2 2 2  SHAPE Fl F2 F3 F4 X (5) X (6) X (7) 2 2 2  X X X X  2 2 2 2  (l)-Test (2)-Test (3)-Test (4)-Test  against against against against  equal frequencies frequencies of 10 equal frequencies frequencies of 90  CB's  BW's  17 4 22  0 0 10  12.05 137.23 0.023* 80.95  na na na na  14 2 23 4  5 0 0 5  26.30 53.91 279.21  na na na  in 3 categories 80 10 in 2 categories (11+12), 13 10 in 2 categories (11 + 12), 13  X (5)-Test against equal frequencies in 4 "categories X (6)-Test against equal frequencies in 3 categories (F1+F3), F 2 , F4 X (7)-Test against frequencies of 10 80 10 for categories (F1 + F3), F 2 , F4 2  2 2  "*" Indicates values not significant at the 5 percent level  95  Table 21. Differences Between Market Price and Maximum and Minimum for Issues from Firms with Common Equity Listed on N Y S E or A M E X , (o Range Defined by Standard Deviation of Stock Returns) STAT  Mean  Median  SE  Max  Min  2.80* -1.14  1.08 -0.10  1.11 1.28  21.42 14.27  -14.58 -27.58  0.07* -0.11* -0.07* 0.13*  0.00 -0.03 0.00 0.03  0.02 0.03 0.02 0.04  0.75 0.00 0.00 1.63  0.00 -0.96 -0.78 0.00  23.98* -28.01*  22.01 20.03  2.86 6.61  42.97 -9.23  14.11 -76.11  0.16* -0.28* -0.11* 0.13*  0.13 -0.14 -0.00 0.03  0.05 0.11 0.05 0.07  0.39 0.00 0.00 0.69  0.00 -0.00 -0.42 0.00  CB's Gmax-MP/Gmax(%) MP-Gmin/MP(%) Slopes 51 52 53 54  BW's Gmax-MP/Gmax(%) MP-Gmin/MP(%) Slopes 51 52 53 54  51 = Slope, Gmax to value at leverage-adjusted estimate of a; 52 = Slope, Gmax to value at estimated a of stock; 53 = Slope, Gmin to value at leverage-adjusted estimate of a; 54 = Slope, Gmin to value at a of stock; "* "Indicates that the mean is significantly different from zero at the 5 percent level from a one-tailed t-test.  hypothesis that the model generated market prices 80 percent of the time was also strongly rejected, but the hypothesis that there were equal numbers below range and either in or above range could not be rejected. This latter is consistent with the previous evidence of some overpricing. Comparisons of market price to maximum and minimum in the range bounded by the standard deviation of returns on equity are given in Table 21. For CB's the market price was on average  1.14 percent below the minimum  value in the range and 2.8 percent below the maximum, compared to 5.51 and 0.88 for the whole sample, for the range a = 0.05-0.5. For B W ' s , the market prices were on average 23.98 percent below the maximum value and 28.01 percent below the minimum value, compared to 27.72 percent and 18.29 percent respectively for the whole sample in the range o = 0.05-0.5. These results are consistent with the earlier finding that the curves were relatively flat in the relevant a range. Within these reduced ranges the curves were on average extremely flat.  VI. CONCLUSIONS A N D DISCUSSION  The match between the predictions of the model (plus the auxiliary assumptions) and the data drawn from the economy is relatively poor. The Brennan and Kraus model can be used to explain the appearance of only some hybrids and these not unambiguously. The conclusion is that the model, while it certainly cannot be rejected, is not the dominant factor explaining choice of security type in real economies. Firstly, consider the numbers and the shape data. Only 22 percent of new bonds were hybrids. Out of all new issues including new equity only about 10 percent were CB's or BW's. If information asymmetry were the only problem influencing choice of security type, hybrids should be the most frequently occurring, not least frequently occurring, security type and curves with a minimum should be the most frequently occurring, not least frequently occurring. Both these hypotheses are rejected, which is here taken as evidence against the explanatory power of the Brennan-Kraus model. Note that the shape data depends strongly on the bond-valuation model being a correct indicator of shape, despite the simplifications made. The contingent claims model seems, however, fairly robust, so the conclusion that curves with a minimum occur least frequently is relative^ robust. Specific hypotheses about relative frequency do depend on assumptions about the extent of the information problem and the behavior of firms with no asymmetry problem. However, the general conclusion that the Brennan and Kraus model is not a dominant factor is quite robust. Consider the effect of making alternative assumptions about "other" firm behavior. One possibility is to assume that firms with o known tend for reasons outside the model tend not to choose 97  98 hybrids when simpler securities are available . The endpoint case is that these 36  firms never choose hybrids. This could explain the low frequency of hybrids overall, implying asymmetric information is an uncommon problem. However, curves with a maximum could never be chosen if the Brennan and Kraus model were the only influencing factor. The shape data strongly reject this case. A more moderate version consistent with low frequency and the shape data is that firms with a known occasionally but infrequently choose hybrids. In this case any curves not with a maximum could (but not conclusivelj' do!) indicate signalling, but the model is not rejected because of the appearance of curves with a maximum. For the CB's 48 of 83 cases or about 60% could indicate signalling. If this is representative of all hybrid issues, then about 6 percent of all new issues could be explained by the model. (The percentage would be a little higher if signalling firms also sometimes issued straight debt or straight equity, which is not ruled out by the model.) However, if firms with no information problem occasionally choose hybrids then they could just as easily occasionally choose hybrids of any shape. Therefore we could also reasonably conclude on the grounds of parsimony that none of the hybrids was issued in response to an asymmetric information problem. Overall, then, regardless of what is assumed about the other firms, these data suggest that the Brennan and Kraus model taken alone does not describe very well the process that leads to the issuance of hybrids instead of straight debt or equity. These considerations do not rule out the possibility that there is a costly equilibrium in which hybrids are issued to reveal o. For example, if taxes or We can reject immediately as untenable either that firms with a known always choose hybrids or that these firms often choose hybrids. The relative frequency of hybrids is simply too low.  differential transactions costs were added and were designed such that straight debt was the preferred choice of firms with no asymmetry, then hybrids would only be issued by firms for which the benefit from having the security correctly priced would outweigh the lost value due to having a suboptimal capital structure. In such an economy the bond pricing model would potentially be inapplicable, depending as it does on perfect markets. Without knowing the details of a particular costly equilibrium it is not possible to say whether the observed frequencies are actually consistent with such a model. It is not even possible to say that curves with a maximum are ruled out although by Riley's(1979) criteria, such curves would appear to be  unsatisfactory.  Secondly, consider the relative flatness of the curves. This suggests that firms were choosing securities which would yield the firm about the same amount of dollars no matter what a estimate the market used. This, coupled with the shape data, suggests that firms either were choosing securities which were known to be insensitive to misestimation of a or were deliberately choosing securities which tend to obscure rather that reveal the firm's true a. The result most consistent with the model is that market prices lay close to the minimum value as predicted, although as Tables 12, 13, and 14 showed, the degree of closeness differed by curve shape. This would be a spurious result if, as indicated by the x  2  tests, securities were on average overpriced.  What might cause such overpricing? The specific functions studied are likely inaccurate due to the simplifying assumptions made, measurement  error,  and the omission of some other options included in bond contracts, notably the interest rate restrictions on call privileges, the redemption feature on warrants, some options associated with the sinking fund feature, and all options on other  100 debt. A determination of the amount and significance  of deviation between real  and theoretical prices attributable to each assumption and each source of measurement  error was beyond the scope of this investigation.  earlier, other researchers have also found evidence tend to deviate  systematically  However, as noted  that solution-generated  values  from market prices. Jones et aZ(1984,1985) in a  regression analysis of percent error (ratio of predicted price less actual price to actual price) for a sample of 305 significantly negatively  straight bonds found that error  was  related to variance, maturity, and financial leverage,  and  signficantly positively related to current yield and bond rating (with A's= l,...,C's = 9). It is interesting to note that Jones et aZ.(1984)  and King(1986)  found the opposite effect for variance, but the same direction of bias for time-to-maturity. From this study, the only relatively firm conclusion is that error was  greater for bonds which were not callable at issue and greater for  with value functions  securities  that were upward-sloping functions of o. Within the CB's  there was no strong evidence that coupon rate was related to under- or over-pricing. The A N O V A showed that overpricing seemed negatively firm value and maturity, and positively  related to debt ratio and 7.  related to Some  potential effects on theoretical prices for these biases are summarized in Table  22.  In general the bonds in my sample were low-rated juniors for which bid prices were often reported and for which the value of other debt may have underestimated.  Also by Jones et al. standard of 0.31,  more than half of the  firms in the sample were highly levered. The bond-warrants had significantly shorter terms and significantly higher coupons than the convertibles. If the bond-valuation model as expressed  in the Brennan-Schwartz solution does  been  101 Table 22. Some Sources and Directions of Error in the Pricing Functions to Market Value Problem  Effect on Theoretical  In Models i n General* Overpricing of low-o bonds! Underpricing of high-a bonds  Curve should be twisted  Underpricing of low-G bonds Overpricing of high-C bonds  CB's too low, BW's too high  Overpricing of juniors  Values too high  Overpricing low-rated bonds  Values too high  Constant o,R** Tax assumptions *  ???? ????  :l=  In A p p l i c a t i o n to Sample I and G Solutions Timing of coupons ignored Call feature ignored Dividends ignored B&S Solution Complexity of options ignored Options in other debt ignored Perfect antidilution protection Dividend policy fixed Use of numerical approximations In data "Other debt" assumptions Capital structures measured with error Use of bid not sales price  Relative  Values  Values too low Values too high Values too high Values too high Values too high Values too high Values too high Values inaccurate. direction of bias unknown Values too high Values inaccurate, direction unbiased? Values too high  *As noted in King(1986) and Jones et al. (1984,1985) tJones et a/.(1984) and King(1986) reported opposite effects; the effect noted here by Jones et aZ.(1985); attributed by authors to errors in point estimates of o. **Some underpricing/overpricing effects are due to these assumptions, but it is not known whether these also affect the properties of the functions in more subtle ways.  102 significantly overprice junior, low-rated, high coupon, short-term bonds issued by highly levered firms, significant overpricing of bond-warrants would be  expected,  and this is indeed the case in this sample. The convertibles were also generally junior, lower-rated bonds but average  rating was  higher than for bond-warrants  and average coupon-rate lower, and on average debt ratio of the issuing firm was lower. This suggests that the overpricing of convertibles should be less than for bond-warrants, which is also the case. These directions of bias are supported somewhat  by the results given in Tables 12,  13, and 14.  Jones et aL(1984,1985) showed that the assumptions interest rates,  of  nonstochastic  constant variance, and neutral taxes can account for some of the  overpricing or underpricing depending on the specifics  of the group of  issues being considered. The assumption of perfect liquidity may also be a cause of misspricing in this sample. Many of the warrant issues and smaller bond issues may be thinly traded, exposing holders to a liquidity risk not captured adequately in the Black-Scholes-Merton model. The capital structure assumption of perfect antidilution protection also cannot be ignored as a possible on the value of securities  source of error. The assumption has most impact  which mature considerably later than securities  cause a large inflow or outflow  which  of capital. Most warrants expire five or more  years before the partner bond matures, and firms could issue new debt on the strength of this new capital. Short-term notes would be economically senior to the outstanding debt. Outflows to service any new debt would reduce the of the bond, existing debt and existing equity, which in present value  values  terms  would reduce the value of the warrant. The situation for earlier maturing senior  103  issues is completely analogous—the firm could choose to issue new debt rather than liquidate assets . 3 7  The evidence seems to be against accepting the Brennan-Kraus model as a good model of reality; however, the behavioral notions that underlie signalling theory are still applicable. Akerlof(1970) and others since have shown that when buyers cannot distinguish quality differences they will offer a price based on the average expected quality. If sellers of superior quality goods can withdraw then the average quality goes down until the "average" equals the worst. In the Brennan-Kraus world, sellers cannot withdraw but can signal quality differences because they can choose from a virtually unlimited set of distinct securities. If in a real economy this set were restricted, thus preventing signalling, a reasonable conjecture is that a value-maximizing firm would choose the next best alternative and offer a securitj' for which theoretical price varied little over the a range, thus minimizing to insiders/current owners the cost of outsiders' misspecification of a. Investors would accept such securities because they would be obtaining a securitj' worth not less that the price paid and possibly more. The results from the Brennan-Schwartz solution seem in keeping with this model of the world, although it could also be argued that the shape of the value function is irrelevant and these results are due to coincidence. The assumptions made by Brennan and Kraus provide some clues as to why in reality the choice set might be restricted. Firstly, all agency-type  Jones ei al.discussed the effect of this assumption as it related to the option to call straight bonds and showed that failure to include the option to refund called debt could lead to overpricing of juniors and longer term debt. 3 7  104  problems except asymmetric information about a have been assumed  away.  Further, the Brennan-Kraus world is perfect so there are no bankruptcy costs, no transactions costs, and no differential taxation, and capital structure policy is irrelevant. Real firms may experience other agency/signalling problems, such as market uncertainty about the value of the new project, the managerial incentive problem, and the incentive shareholders have to shift from low-variance to high-variance projects. As Barnea et oi.(1985) showed, complex securities multiple options could be used to resolve  with  several of these problems at the  same  time. Clearly, the choice set of hybrids which would satisfactorily minimize total agency cost would be restricted and the resultant characteristics would likely appear to be suboptimal for any problem considered in isolation. For example, Flannery(1986) demonstrated in a very simple model that under some circumstances, term to maturity can signal inside information about the value of a project. Barnea et ci.(1985) outlined how callable debt could be used to resolve the risk incentive problem. Restricting the fraction of shares  a manager/owner  can sell is known to be a solution to the "excessive perks" problem. For each choice variable it is possible to propose at least one agency/signalling equilibrium which results in a restriction on that choice variable. Adding bankruptcy and differential taxation imposes further constraints. Together these limit the choice of coupon payout and face, the former tending to keep these low relative to firm value, the latter tending to keep these high relative to firm value. If some or all of these correspond to real world processes, securities consistent alternatives.  any real firm's set of issuable  with value maximization may contain only a very few  The data themselves suggest this might be the case. For example,  105 19 of 22 warrants  had the same maturity—5 years, and maturities of most  bonds are a multiple of 5 years. As another example most firms chose 7's in the range 0.10 to 0.25. As a final example, firms issuing bond-warrants  appear  to have been restricted to shorter terms and higher coupon rates, compared to firms issuing convertibles. Without knowing the exact nature of any firm's true constrained optimization problem, any signalling model cannot be totally dismissed, even if the evidence is not supportive, as in this study. There are several promising lines of research  arising from this preliminary  work. Extending the Brennan-Kraus model to a world with more than one agency problem per firm could yield further  insights into the use of hybrids.  Extending the model to a multiperiod world would also be useful if it is assumed that timing of information flow into the market plays a role in determining some of features of security  contracts.  It would also be useful to develop the Green(1984) model to the point that it could be empirically tested and comparisons made between the  correctness  of its predictions and the predictions of this signalling model. A t first glance, these results seem compatible the role of convertibles proposed by Green. Investors  want only to ensure that they will not suffer from project switching,  and a contract with theoretical value insensitive to a would satisfy this demand. 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(1976): Informational aspects of market structure; Quarterly Journal of Economics, Vol. 90, pp. 591-597. Thatcher, J.S.(1985): The choice of call provision terms: evidence of the existence of agency costs of debt; Journal of Finance, Vol. 40, pp. ,549-561. Twizell, E.H.(1984): Computational Methods for Partial Differential Equations; Ellis Horwood Limited and Halsted Press, Chichester, England; 276 pages.  I  APPENDIX I. BASIC D A T A ON ISSUES AND ISSUERS. This appendix includes a list of the issues included in the sample and tables showing various data about the issues or issuers.  112  113 Table A l . Issues Included in this Study Issuer  Date (D/M/YR)  Face ($mil.)  Coupon (%)  Type  1. 2. 3. 4. 5. 6. 7. 8. 9. 10.  The Advest Group, Inc. Anixter Bros., Inc. Bearings, Inc. Calny Corp. ChemLawn Corp. Collins Foods Intl., Inc. Comdisco Corp. Communications Indus., Inc. Computer Consoles, Inc. Diagnostic/Retrieval Sys., Inc.  1/3/83 1/1/83 1/2/84 1/11/84 1/9/83 15/12/83 1/5/83 15/10/84 15/2/83 1/8/83  27.50 42.00 44.75 20.00 15.00 30.00 250.00 57.50 80.00 25.00  9.00 8.25 8.50 12.00 10.00 12.00 8.00 9.00 7.75 8.50  CB CB CB CB CB BW CB CB CB CB  11. 12. 13. 14. 15. 16. 17. 18. 19. 20.  Digital Equipment Corp. Digital Switch Corp. Energy Factors, Inc. Equitec Financial Gp., Inc. Geothermal Resources, Inc. Grant Industries, Inc. Hechinger Co. HPSC, Inc. Home Depot, Inc. Kellwood Co.  1/9/84 15/5/83 1/10/83 1/9/84 15/11/84 1/6/83 15/9/84 15/1/84 1/7/84 15/10/84  400.00 110.00 29.96 20.00 15.00 12.50 86.25 20.00 86.25 30.00  8.00 8.00 10.00 10.00 13.00 8.50 8.50 10.00 8.50 9.00  CB CB CB CB BW CB CB BW CB CB  21. 22. 23. 24. 25. 26. 27. 28. 29.  M G M / U A Entertainment Co. Kulicke & Soffa Indus., Inc. M S A Realty Corp. Major Realty Corp. Minstar, Inc. New Plan Realty Trust Pioneer-Std. Electron., Inc. Ply-Gem Industries, Inc. Po Folks, Inc.  15/4/83 1/3/83 1/4/84 1/9/83 1/2/84 1/2/83 1/8/83 15/10/84 5/7/84  400.00 35.00 75.00 12.00 100.00 30.00 22.00 37.00 18.50  10.00 8.00 9.25 10.75 12.50 9.75 9.00 10.00 11.50  BW CB BW CB BW CB CB CB CB  30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40.  Preway, Inc. The Price Co. Property Capital Trust Rehab Hospital Ser v. Corp. Rockaway Corp. Ltd. Scientific Leasing Inc. Sparkman Energy Corp. Spendthrift Farms, Inc. Team, Inc. Texscan Corp. Timeplex, Inc.  15/8/84 31/8/84 15/5/83 1/7/84 1/10/83 15/6/83 28/6/83 21/3/84 9/5/84 15/6/83 1/10/83  57.50 75.00 40.00 15.00 15.00 25.00 14.85 30.00 5.00 40.00 35.00  13.88 8.00 9.75 10.00 9.00 8.25 10.25 12.50 13.00 8.50 7.75  BW CB CB CB CB CB CB BW CB CB CB  114 41. 42. 43. 44. 45. 46. 47. 48. 49. 50.  Transtechnology Corp. Triangle Industries Triton Energj^ Corp. U R S Corp. U S Playing Card Corp. Walker Telecomm. Corp. Wang Laboratories, Inc. Wang Laboratories, Inc. Wedtech Corp. Winners Corp.  15/9/83 1/9/83 15/11/84 15/3/83 15/12/83 30/5/84 15/5/84 1/6/83 15/6/84 19/5/83  16.0 90.00 40.00 25.00 30.00 5.68 100.00 165.00 40.00 25.00  9.00 11.50 13.50 8.75 9.88 11.00 9.00 7.75 13.00 8.25  CB BW BW CB BW CB CB CB CB CB  51. 52. 53. 54. 55. 56. 57. 58. 59. 60.  Mobile Communications Corp. Mobile Communications Corp. Instrument Sj'stems, Inc. Fourth Financial Corp. F P A Corp. Computervision Corp. Collins Industries, Inc. Action Industries, Inc. Alaska Airlines, Inc. BankAmerica Realty Invest.  15/4/85 15/10/84 15/4/84 1/4/83 15/8/83 1/12/84 1/11/84 1/4/83 1/1/84 1/6/83  25.00 30.00 28.75 22.00 30.00 110.00 13.80 17.50 35.00 50.00  9.00 11.00 11.00 8.50 12.63 8.00 10.50 9.00 9.00 9.50  CB CB CB CB BW CB BW CB CB CB  61. 62. 63. 64. 65. 66. 67. 68. 69. 70.  Aviation Group, Inc. Atlantic Research Corp. Cessna Aircraft Corp. Comtech Inc. Consul Restaurant Corp. Inter-Regional F i n . Grp. Isaly Co. Ltd. Easco Corp. Financial News Network Inc. The Cannon Grp., Inc.  15/6/83 1/5/83 1/7/83 1/2/83 1/11/83 15/4/83 15/6/83 15/12/84 16/1/84 15/3/84  25.00 30.00 100.00 10.00 25.00 27.50 5.00 25.00 10.93 32.00  8.00 8.00 8.00 13.00 10.00 10.00 10.00 10.00 9.00 9.00  CB CB CB BW CB CB CB CB CB CB  71. 72. 73. 74. 75. 76. 77. 78. 79. 80.  The Cannon Grp., Inc. Grolier Inc. Naugles, Inc. Merry Land & Invest. L t d . Intl. Banknote Co., Inc. Kinney System, Inc. Maclean Hunter Ltd. Leggett and Piatt, Inc. Midlantic Banks Inc. Olin Corp.  1/11/84 1/11/83 30/4/84 1/7/83 15/9/83 15/5/83 27/4/84 15/6/83 15/8/84 1/6/83  70.00 82.00 30.00 6.60 40.00 22.50 55.00 20.00 50.00 100.00  12.38 13.00 13.75 9.00 10.00 12.00 8.25 8.13 10.50 8.75  BW BW BW CB BW BW CB CB CB CB  81. 82. 83. 84. 85.  Macmillan, Inc. Nat. Convenience Stores Inc. Nat. Health Corp. M E D I Q Inc. Chi-Chi's, Inc.  1/1/83 1/12/83 15/2/84 15/12/83 11/10/84  50.00 50.00 21.00 30.00 50.00  8.75 9.00 8.75 9.50 9.00  CB CB CB CB CB  115 86. 87. 88. 89. 90.  Aeronca, Inc. Leisure and Technology, Inc. Merrill Bankshares Co. N A F C O Financial Grp. Inc. Harnischfeger Corp.  91. 92. 93. 94. 95. 96. 97. 98. 99. 100.  Greatwest Hospitals, Inc. First Union R E Equity VICORP Restaurants, Inc. Wachovia Corp. U.S. Trust Corp. Pennsylvania REIT. Quaker State Oil Refin. Corp. Storer Communications, Inc. State Street Boston Corp. Symbol Technologies, Inc.  101. 102. 103. 104. 105.  First Amarillo BankCorp. Intl. Lease Fin. Corp. Grumman Corp. Jacobsen Stores Inc. Knoll Intl. Inc.  1/2/83 15/9/83 1/5/83 1/9/83 15/4/84  10.00 27.50 8.00 23.00 100.00  12.50 12.00 9.00 9.00 12.00  CB CB CB CB BW  1/7/83 1/9/84 15/3/84 1/11/84 15/10/83 1/7/83 15/3/83 15/5/83 1/5/83 1/4/83  35.00 38.50 50.00 100.00 16.50 35.00 50.00 230.00 50.00 4.40  8.25 10.25 13.25 8.75 10.00 9.75 8.88 10.00 7.75 12.75  CB CBV CB CB CB CB CB BW CB CB  1/10/83 30/8/83 15/8/84 1/4/83 15/8/83  15.00 26.00 50.00 10.00 30.00  10.00 9.25 9.25 10.00 8.25  CB CB CB BW CB  116 Table A2. Market Value (MV), Debt Ratio (D), y, R-C(%), and Exercise Premium (EP) for Issues in Sample ID*  MV($mil)t  7t  R-C(%)t  EPt  Debt Ratio t  1. 2. 3. 4. 5. 6. 7. 8. 9. 10.  140.21 448.29 196.31 62.93 332.03 213.89 1150.74 289.05 298.79 81.63  0.18 0.08 0.21 0.26 0.04 0.10 0.20 0.16 0.25 0.23  1.60 2.18 3.32 -0.63 1.86 -0.22 2.68 2.59 3.17 3.28  -0.10 0.31 0.31 0.40 0.32 0.10 0.12 0.32 0.11 0.45  0.26 0.15 0.36 0.37 0.15 0.31 0.24 0.21 0.29 0.32  11. 12. 13. 14. 15. 16. 17. 18. 19. 20.  6642.46 1037.68 51.98 99.52 129.36 45.24 462.31 116.93 676.07 190.70  0.06 0.08 0.57 0.17 0.13 0.26 0.17 0.34 0.11 0.34  4.72 2.86 1.21 2.33 -1.48 2.36 3.80 1.57 5.09 3.16  0.11 0.38 0.20 0.47 0.38 0.14 0.20 0.38 0.16 0.19  0.11 0.11 0.63 0.38 0.57 0.28 0.24 0.62 0.15 0.75  21. 22. 23. 24. 25. 26. 27. 28. 29. 30.  1332.18 165.12 122.85 58.35 228.60 164.79 119.01 99.10 99.79 131.72  0.05 0.18 0.69 0.19 0.34 0.22 0.23 0.40 0.15 0.33  1.41 1.56 3.34 11 13 13 78 30 09 16  0.12 0.26 -0.31 0.33 0.25 -0.08 0.57 0.16 0.63 0.07  0.32 0.25 0.61 0.32 0.50 0.30 0.48 0.51 0.41 0.73  31. 32. 33. 34. 35. 36. 37. 38. 39. 40.  1044.96 185.44 103.28 74.59 131.00 74.79 141.86 32.45 193.03 181.59  0.06 0.20 0.17 0.19 0.22 0.19 0.05 0.25 0.17 0.16  4.72 1.09 2.65 2.20 2.95 0.95 -0.13 0.0 2.29 3.50  0.25 0.16 -0.02 0.21 0.26 0.23 0.40 0.36 0.35 0.31  0.07 0.26 0.27 0.28 0.46 0.24 0.25 0.66 0.23 0.20  117 41. 42. 43. 44. 45. 46. 47. 48. 49. 50.  104.20 121.43 168.43 73.77 101.72 31.06 4473.54 5806.34 161.40 121.07  0.13 0.32 0.13 0.31 0.30 0.15 0.02 0.02 0.28 0.25  2.60 0.36 -2.25 2.85 1.67 2.50 4.00 3.11 0.75 2.15  0.25 0.16 0.49 0.18 0.06 0.78 0.16 0.28 0.30 0.10  0.15 0.67 0.36 0.41 0.25 0.31 0.12 0.09 0.50 0.38  51. 52. 53. 54. 55. 56. 57. 58. 59. 60.  199.48 137.73 94.46 165.88 162.63 1234.18 44.59 158.71 255.49 259.76  0.14 0.21 0.35 0.15 0.29 0.08 0.13 0.12 0.15 0.17  0.33 1.16 1.59 2.10 -1.22 3.49 0.87 1.60 1.78 1.36  -0.01 0.29 0.11 -0.05 -0.04 0.21 0.95 -0.02 0.19 0.26  0.35 0.47 0.43 0.13 0.66 0.14 0.62 0.16 0.36 0.29  61. 62. 63. 64. 65. 66. 67. 68. 69. 70.  107.63 213.07 658.29 62.74 110.62 473.63 56.13 215.64 57.46 167.60  0.20 0.14 0.14 0.16 0.22 0.14 0.12 0.16 0.21 0.23  1.86 1.86 3.20 -1.12 1.60 0.18 1.02 1.46 2.36 2.76  0.22 0.12 0.19 0.22 0.17 0.09 0.11 0.05 0.04 0.23  0.24 0.21 0.21 0.38 0.32 0.73 0.33 0.46 0.33 0.49  71. 72. 73. 74. 75. 76. 77. 78. 79. 80.  182.19 172.19 120.10 36.78 186.93 113.29 927.82 392.70 524.68 1165.99  0.02 0.14 0.11 0.29 0.24 0.28 0.08 0.05 0.10 0.10  -1.01 -2.03 -1.10 2.20 1.59 -1.76 5.06 2.67 2.16 2.11  0.48 0.37 0.31 0.14 0.19 0.11 0.14 0.06 0.16 0.19  0.50 0.66 0.63 0.55 0.40 0.41 0.32 0.15 0.23 0.37  81. 82. 83. 84. 85.  284.69 420.00 120.26 162.65 445.39  0.14 0.11 0.26 0.20 0.10  2.17 3.26 2.98 1.99 3.60  0.21 0.69 0.11 0.19 0.25  0.25 0.45 0.55 0.37 0.17  -  118 86. 87.  -0.02 0.21 0.11  0.41  0.25 0.24  0.29 0.50  2.95 2.22 -1.26 3.33 1.57 1.35 1.79 0.24  0.17 0.12 0.44 0.20 0.02 0.08 0.40  0.48 0.45 0.58 0.19 0.23 0.46 0.19 0.60  0.10 0.12  2.49 -1.79  0.15 -0.01  0.33 0.17  0.13 0.13 0.05 0.16 0.12  1.21 2.61 3.43 0.06 2.93  0.31 0.12  0.22 0.57 0.22 0.64 0.21  -1.57 -0.64  32.20 105.66 78.03 79.97 293.33  0.36 0.40 0.10 0.24 0.13  1.18 2.21 0.66  95. 96. 97. 98.  165.66 471.53 274.87 1097.46 200.07 108.23 521.64 1213.53  0.26 0.09 0.23 0.09 0.08 0.37 0.10 0.11  99. 100.  516.68 35.30  101. 102. 103. 104.  97.14 375.35 1116.35 73.44  105.  183.73  88. 89. 90. 91. 92. 93. 94.  *ID  0.19  0.23 -0.07 0.45  0.72 0.21  is the number assigned to the issue in Table A l . 7 = fraction of firm acquired upon exercise or t M V = m a r k e t value of firm; conversion; R-C=default-free rate less coupon rate in percent; E P = (exercise price minus stock price)/stock price; Debt ratio = m a r k e t value of all debt/market value of firm.  119 Table A3. New Debt Ratio (D ), New Capital Ratio (CV), Maturity (MAT), Call Price at Issue (CALL), Rating (RT), and a of firm (aMP) from Brennan-Schwartz Solution for Issues in Sample x  ID*  cvt  MATt  CALLt  RTt  aMP1  1. 2. 3. 4. 5. 6. 7. 8. 9. 10.  0.23 0.09 0.22 0.32 0.05 0.12 0.22 0.20 0.28 0.30  0.23 0.09 0.22 0.32 0.05 0.13 0.22 0.20 0.28 0.30  25 20 25 15 25 25 20 25 15 15  107.80 not 108.50 107.50 109.00 100.00 107.00 109.00 105.80 not  NR BB + BBBNR NR BB + BB + BBNR B-  0.05 0.05 1.00 0.20 0.38 0.05 0.43 0.38 0.30 0.15  11. 12. 13. 14. 15. 16. 17. 18. 19. 20.  0.06 0.10 0.59 0.18 0.11 0.28 0.19 0.13 0.13 0.15  0.06 0.10 0.59 0.18 0.12 0.28 0.19 0.18 0.13 0.15  25 20 20 20 7 20 25 10 25 15  108.00 107.20 not 110.00 not 107.60 108.50 not 107.90 not  AA B NR NR NR NR BBB NR BBNR  0.35 0.05 0.05 0.48 1.00 0.63 0.55 0.05 0.20 0.05  21. 22. 23. 24. 25. 26. 27. 28. 29. 30.  0.25 0.22 0.59 0.20 0.31 0.21 0.18 0.38 0.16 0.37  0.26 0.22 0.73 0.20 0.36 0.21 0.18 0.38 0.16 0.40  10 25 9 20 12 15 15 15 20 15  100.00 not not 109.80 100.00 104.00 not 100.00 not not  B + NR NR CCC NR B + BBNR NR  0.60 0.23 0.73 0.55 0.05 0.13 0.35 0.05 0.05 1.00  31. 32. 33. 34. 35. 36. 37. 38. 39. 40.  0.07 0.21 0.16 0.20 0.18 0.20 0.18 0.15 0.21 0.18  0.07 0.21 0.16 0.20 0.18 0.20 0.19 0.15 0.21 0.18  25 25 15 20 20 20 10 20 20 25  108.00 108.80 100.00 108.10 107.40 110.00 not 113.00 107.70 107.20  NR BBB + NR BNR NR NR NR B B-  0.50 0.13 0.05 0.50 0.57 0.77 0.80 0.45 0.27 0.68  120  41. 42. 43. 44. 45. 46. 47. 48. 49. 50.  0.15 0.60 0.21 0.32 0.20 0.17 0.02 0.03 0.24 0.21  0.15 0.62 0.23 0.32 0.30 0.17 0.02 0.03 0.24 0.21  20 20 10 25 10 15 25 25 20 20  108.10 100.00 not 107.00 100.00 not 107.20 109.00 not 106.60  B BBNR NR CCC NR BBB BB-  0.52 0.77 1.00 0.85 0.05 0.05 0.40 0.40 0.07 0.05  51. 52. 53. 54. 55. 56. 57. 58. 59. 60.  0.14 0.22 0.31 0.13 0.17 0.09 0.27 0.11 0.14 0.19  0.14 0.22 0.31 0.13 0.20 0.09 0.28 0.11 0.14 0.19  20 20 10 20 10 25 10 15 20 25  111.00 109.00 109.60 not 100.00 108.00 not not not 108.50  BBCCC NR B+ NR NR CCC BB + A-  0.05 0.32 0.90 0.05 0.05 0.35 0.52 0.05 0.05 0.30  61. 62. 63. 64. 65. 66. 67. 68. 69. 70.  0.22 0.15 0.16 0.16 0.24 0.06 0.09 0.13 0.20 0.18  0.22 0.15 0.16 0.21 0.24 0.06 0.09 0.13 0.20 0.18  20 25 25 8 20 20 15 25 7 25  107.20 106.40 not 100.00 109.00 108.90 not 110.00 not 108.10  NR NR BBB NR NR NR NR NR NR' B-  0.43 0.05 0.20 0.05 0.32 0.25 0.05 0.68 0.05 0.55  71. 72. 73. 74. 75. 76. 77. 78. 79. 80.  0.23 0.41 0.25 0.19 0.15 0.17 0.06 0.05 0.10 0.09  0.28 0.42 0.26 0.19 0.21 0.25 0.06 0.05 0.10 0.09  10 20 15 10 10 10 20 18 25 25  100.00 100.00 100.00 not not 100.00 not 106.50 not 107.90  BB NR NR NR NR NR BBBA BBB +  0.05 1.00 0.18 0.18 1.00 0.05 0.05 0.40 0.05 0.85  81. 82. 83. 84.  0.18 0.12 0.17 0.18  0.18 0.12 0.17 0.18  25 25 15 20  108.80 108.10 not 108.50  BBB B BNR  0.20 0.20 0.05 0.43  121 85. 86. 87. 88. 89. 90.  0.12 0.34 0.27 0.10 0.29 0.26  0.12 0.34 0.27 0.10 0.29 0.28  25 10 15 25 20 20  108.10 109.00 112.50 not 108.10 100.00  B + NR NR NR B + B +  0.05 0.05 0.30 0.05 0.38 0.68  91. 92. 93. 94. 95. 96. 97. 98. 99. 100.  0.21 0.08 0.18 0.09 0.09 0.33 0.10 0.15 0.10 0.12  0.21 0.08 0.18 0.09 0.09 0.33 0.10 0.16 0.10 0.12  20 25 25 25 25 20 25 20 25 10  108.30 110.30 112.20 108.80 not not 107.10 not 106.70 110.00  BABAA ABBBBB + BBA NR  0.23 0.38 0.52 0.50 0.05 0.05 0.05 0.65 1.00 1.00  101. 102. 103. 104. 105.  0.15 0.07 0.05 0.10 0.15  0.15 0.07 0.05 0.11 0.15  25 20 25 10 20  not 109.30 109.30 101.50 not  BB BBBBB + NR NR  0.05 0.52 0.48 0.25 0.18  *ID is the number assigned to issue in Table A l . t D = market value of new debt/market value of firm; C V = market value of new issue/market value of firm; MAT=maturity of debt issue; C A L L = call price at issue (not=not callable); RT = rating (NR=not rated); a M P = o of firm indicated by location of market price of issue on theoretical curve. x  122 Table A4. Data on Theoretical Prices Generated by the Brennan-Schwartz Solution: Location and Value of Maximum and Minimum Values ID*  aMAXt  aMINt  Gmax  1. 2. 3. 4. 5. 6. 7. 8. 9. 10.  0.05 0.55 0.05 0.20 0.18 1.00 0.05 0.30 0.30 0.45  1.00 . 0.05 1.00 1.00 1.00 0.05 1.00 1.00 1.00 0.05  28.72 48.24 45.26 20.00 15.20 44.62 255.92 57.45 81.24 25.82  26.64 42.01 42.68 17.53 13.45 38.25 240.56 51.38 77.01 23.83  11. 12. 13. 14. 15. 16. 17. 18. 19. 20.  0.35 0.35 0.40 0.05 0.38 0.05 0.25 0.80 0.05 1.00  1.00 1.00 0.05 1.00 1.00 1.00 1.00 0.05 1.00 0.05  400.80 105.75 35.21 19.92 26.50 13.02 88.10 46.20 85.24 49.54  382.44 91.38 32.76 16.58 20.77 12.35 81.52 42.96 80.83 33.18  21. 22. 23. 24. 25. 26. 27. 28. 29. 30.  0.85 0.50 0.40 0.15 1.00 1.00 0.05 0.05 0.32 0.30  0.60 0.05 0.80 1.00 0.05 0.05 0.50 0.80 0.05 1.00  439.47 36.88 103.06 12.55 130.88 39.62 21.98 36.74 18.77 77.89  369.30 35.14 79.89 10.87 124.43 34.28 20.82 34.69 17.83 64.76  31. 32. 33. 34. 35. 36. 37. 38. 39. 40.  0.50 0.35 1.00 0.05 0.05 0.05 0.55 0.80 0.27 0.30  1.00 1.00 0.05 1.00 0.60 1.00 1.00 0.45 1.00 1.00  72.52 41.12 20.02 15.40 25.24 15.82 34.09 5.65 39.70 34.04  66.30 37.90 18.05 14.17 23.21 14.67 24.89 5.10 35.11 30.65  1 0 0  t  Gmin,,  123 41. 42. 43. 44. 45. 46. 47. 48. 49. 50.  0.30 0.20 0.35 0.27 1.00 0.55 0.40 0.40 0.05 1.00  1.00 1.00 1.00 1.00 0.05 0.05 1.00 1.00 0.70 0.05  15.88 96.09 51.15 25.46 49.75 6.27 101.76 149.58 26.09 49.67  14.56 62.28 40.40 23.44 43.73 6.20 96.72 144.02 25.42 44.35  51. 52. 53. 54. 55. 56. 57. 58. 59. 60.  0.85 0.10 0.05 1.00 0.95 0.35 0.20 0.70 1.00 0.30  0.10 0.95 0.90 0.05 0.05 1.00 1.00 0.05 0.05 1.00  26.06 31.17 31.16 27.75 53.42 110.94 14.75 22.77 41.47 48.50  26.03 25.72 30.32 25.28 37.18 102.78 7.77 20.05 37.26 45.50  61. 62. 63. 64. 65. 66. 67. 68. 69. 70.  0.30 0.05 0.90 1.00 0.05 0.30 1.00 1.00 0.70 1.00  1.00 1.00 0.05 0.05 1.00 0.25 0.05 0.40 0.05 0.55  24.99 30.49 113.95 16.99 26.23 29.95 6.90 27.50 15.24 32.89  23.00 28.84 101.37 13.70 24.39 29.90 5.79 25.99 13.70 31.05  71. 72. 73. 74. 75. 76. 77. 78. 79. 80.  1.00 0.65 0.30 1.00 0.40 1.00 0.80 0.05 0.50 1.00  0.05 1.00 0.90 0.05 1.00 0.05 0.05 1.00 0.05 0.45  90.03 90.29 34.04 7.46 58.87 43.43 74.23 20.87 61.73 105.50  75.20 60.73 21.57 6.61 52.51 35.46 69.39 20.32 56.45 102.67  81. 82. 83. 84. 85.  0.20 0.20 1.00 0.05 0.05  1.00 1.00 0.05 0.70 1.00  49.47 48.15 30.01 30.95 49.93  42.80 41.85 23.40 28.67 46.20  86. 87. 88. 89. 90.  0.05 1.00 1.00 0.25 0.25  0.70 0.30 0.05 1.00 1.00  10.90 38.84 9.52 23.28 95.94  10.73 31.05 8.18 21.04 68.61  91. 92. 93. 94. 95. 96. 97. 98. 99. 100.  1.00 0.05 0.05 0.05 0.70 0.40 0.05 0.25 0.05 0.05  0.60 0.08 0.55 1.00 0.05 0.05 0.85 1.00 1.00 1.00  36.05 39.65 55.30 102.79 19.84 46.27 52.55 250.58 52.03 4.40  34.36 36.75 49.25 97.20 17.96 43.20 51.13 177.05 50.66 4.22  101. 102. 103. 104. 105.  0.50 0.55 0.05 0.05 0.50  0.05 0.40 1.00 0.25 0.05  16.49 27.31 51.20 15.01 30.42  15.38 25.83 48.40 11.26 27.70  *ID is the number assigned to the issue in Table A l . t a M A X = a of theoretical maximum; a M I N = o of theoretical minimum; GmaxOmin), =theoretical maximum (minimum) in range 0.05-1.00. 0 0  125 . Table A5. Data on Theoretical Prices Generated by the Brennan-Schwartz Solution: Theoretical Values at o = 0.05, 0.50, 0.70 ID*  G„ t  Gsol  1. 2. 3. 4. 5. 6. 7. 8. 9. 10.  28.72 42.01 45.26 20.00' 15.19 38.25 255.92 57.28 81.24 23.84  28.23 48.16 44.90 19.16 14.59 41.93 251.58 56.23 80.17 25.79  27.47 47.16 43.89 18.23 13.90 44.21 250.05 53.96 78.64 25.22  32.72 41.58 42.51 20.00 15.00 28.26 253.75 57.21 84.80 24.50  11. 12. 13. 14. 15. 16. 17. 18. 19. 20.  400.25 104.31 32.76 19.92 26.50 13.02 88.00 42.96 85.24 33.18  398.41 104.71 35.12 18.24 21.78 12.77 86.03 45.24 84.30 38.47  393.84 100.08 34.84 17.05 21.32 12.56 83.44 46.13 82.70 42.90  424.00 104.31 30.56 18.00 15.07 12.63 85.39 20.76 84.96 29.10  21. 22. 23. 24. 25. 26. 27. 28. 29. 30.  421.37 35.14 95.52 12.55 124.43 34.28 21.98 36.74 17.83 73.62  398.53 36.88 102.47 11.84 128.90 38.31 20.82 35.42 18.56 75.45  437.80 36.43 94.68 11.29 125.38 39.10 21.20 34.70 18.15 73.10  351.61 35.88 89.40 11.64 83.20 34.50 21.34 37.19 15.54 52.50  31. 32. 33. 34. 35. 36. 37. 38. 39. 40.  72.09 38.46 18.05 15.40 25.24 15.82 33.22 5.50 39.58 33.92  72.52 40.69 19.60 15.04 23.41 15.43 33.35 5.13 38.58 33.32  70.10 39.10 19.95 14.63 23.37 14.99 30.43 5.40 36.82 31.96  74.63 39.60 16.65 15.00 23.25 14.85 26.79 4.85 39.70 32.20  5  G  7 0  t  MPt  126 41. 42. 43. 44. 45. 46. 47. 48. 49. 50.  15.82 94.57 48.16 25.45 43.73 5.85 101.02 149.31 26.09 44.35  15.70 78.50 50.09 24.67 46.62 6.26 101.76 149.36 25.59 48.92  15.19 75.61 44.95 24.07 48.61 6.21 101.69 144.64 25.42 49.17  15.68 75.60 39.01 23.75 30.92 5.34 109.50 154.50 26.00 39.20  51. 52. 53. 54. 55. 56. 57. 58. 59. 60.  26.03 31.13 31.16 25.28 37.18 110.24 14.44 20.05 37.25 46.38  26.04 27.95 30.75 27.06 43.11 110.29 12.59 22.70 40.28 47.50  26.06 26.73 30.48 27.69 45.28 107.25 10.69 22.77 40.72 46.45  28.13 29.78 29.04 22.32 32.96 114.40 12.52 18.02 35.63 49.38  61. 62. 63. 64. 65. 66. 67. 68. 69. 70.  24.97 30.49 101.37 13.70 26.23 29.94 5.79 26.45 13.70 32.89  24.55 29.82 113.35 15.18 25.39 29.95 6.61 26.27 14.93 31.20  23.83 29.29 113.67 16.25 24.63 29.95 6.73 27.03 15.24 31.33  24.75 32.10 106.00 13.00 26.06 28.46 4.95 27.00 11.53 30.88  71. 72. 73. 74. 75. 76. 77. 78. 79. 80.  75.20 82.03 30.03 6.61 55.37 35.46 68.39 20.87 56.45 104.76  85.48 88.90 31.82 7.04 58.60 39.94 73.39 20.68 61.73 102.83  86.37 86.32 27.38 7.15 56.95 41.51 73.18 20.42 61.56 103.07  50.93 73.13 31.80 6.93 38.60 28.81 56.10 20.80 51.38 105.00  81. 82. 83. 84. 85.  49.30 48.15 23.40 30.95 49.93  46.62 44.24 26.57 29.44 48.30  44.17 42.15 27.97 28.67 47.23  50.00 49.19 20.58 29.70 51.00  127 86. 87. 88. 89. 90.  10.90 31.56 8.18 23.25 91.58  10.83 32.44 9.24 22.71 89.95  10.73 35.14 9.38 21.91 78.05  10.90 28.19 8.00 23.11 83.00  91. 92. 93. 94. 95. 96. 97. 98. 99. 100.  36.01 39.65 55.30 102.79 17.96 43.20 52.55 225.35 52.03 4.40  34.50 37.51 49.63 102.02 19.73 46.16 52.20 201.91 51.53 4.40  34.74 36.84 49.45 100.30 19.84 45.79 51.55 193.38 51.02 4.30  35.70 38.40 49.60 102.00 16.58 35.35 54.00 196.69 49.75 4.05  101. 102. 103. 104. 105.  15.38 26.56 51.20 15.01 27.70  16.49 26.47 49.80 13.08 30.42  16.31 27.31 49.64 12.24 29.99  15.00 26.88 50.00 8.00 28.50  *ID is the number assigned to the issue in Table A l . t G = v a l u e at a = 0.05; G = v a l u e at a = 0.50; G = v a l u e at a = 0.70; MP = market price of issue. 0 5  5 0  7 0  128 Table A6. Differences between Maximum Value and Market Price and Minimum Value and Market Price, for the Brennan-Schwartz Solution ID*  Git  G2t  G3t  G4f  1. 2. 3. 4. 5. 6. 7. 8. 9. 10.  -13.93 13.81 6.08 0.0 1.32 36.08 0.85 0.42 -4.38 5.11  16.05 -1.03 -3.25 8.85 7.33 -35.35 1.46 5.68 7.26 2.73  -13.93 13.66 6.08 0.0 1.32 32.60 0.85 0.42 -4.38 5.11  13.72 -1.03 -5.62 4.20 2.73 -35.35 0.86 1.71 5.46 2.73  11. 12. 13. 14. 15. 16. 17. 18. 19. 20.  -5.79 1.36 13.21 9.64 43.13 3.00 3.08 55.00 0.33 32.17  7.11 4.06 -7.20 5.28 -41.47 0.55 2.28 -106.94 2.66 -14.02  -5.79 1.36 13.21 9.64 43.13 3.00 3.08 54.11 0.33 24.36  6.04 0.0 -7.20 -1.33 -44.53 -1.11 -0.75 -106.94 0.78 -14.02  21. 22. 23. 24. 25. 26. 27. 28. 29. 30.  19.69 2.71 13.25 7.25 35.45 11.76 2.91 -1.22 17.21 32.60  -5.03 2.06 -5.91 3.01 -49.56 0.64 2.44 6.70 -14.74 -39.24  16.56 2.71 13.25 7.25 35.45 9.95 2.91 -1.22 17.21 32.60  -13.34 2.06 -6.85 -1.72 -49.56 0.64 2.44 4.76 -14.74 -40.23  31. 32. 33. 34. 35. 36. 37. 38. 39. 40.  -2.91 3.70 16.54 2.60 7.88 6.13 21.41 11.82 0.0 5.41  6.07 2.88 -8.41 2.47 0.17 -0.94 -13.59 -5.15 7.25 0.75  -2.91 3.70 15.05 2.60 7.88 6.13 19.67 11.82 0.0 5.41  3.40 2.88 -8.41 -0.27 -0.69 -3.91 -24.00 -5.15 2.82 -3.48  129 41. 42. 43. 44. 45. 46. 47. 48. 49. 50.  1.26 21.32 23.73 6.72 36.39 14.83 -7.61 -3.29 0.34 20.28  3.13 -0.01 -15.23 -1.35 -41.43 -9.55 7.74 6.38 2.23 -13.14  1.26 21.32 23.73 6.72 33.68 14.70 -7.61 -3.29 0.34 19.87  -0.13 -3.84 -23.46 -3.87 -41.43 -9.55 7.74 3.36 1.58 -13.14  51. 52. 53. 54. 55. 56. 57. 58. 59. 60.  -7.94 4.46 6.80 19.39 27.21 -3.12 15.12 20.86 12.50 -1.81  7.47 10.24 -4.96 -13.26 •12.80 6.25 14.62 -11.27 -4.55 6.08  -8.03 4.46 6.80 17.52 23.54 -3.12 15.12 20.62 11.54 -1.81  7.47 6.15 -5.89 -13.26 -12.80 3.64 -0.56 -11.27 -4.55 6.08  61. 62. 63. 64. 65. 66. 67. 68. 69. 70.  0.96 -5.28 6.75 20.00 0.65 4.97 26.45 0.11 24.34 6.11  3.72 8.75 4.37 -5.38 5.49 -5.06 -16.97 3.74 -18.82 -0.55  0.96 -5.28 6.48 14.36 0.65 4.97 25.11 -2.08 22.77 6.11  0.81 7.10 4.37 -5.38 2.57 -5.06 -16.97 3.74 -18.82 -1.04  71. 72. 73. 74. 75. 76. 77. 78. 79. 80.  41.03 19.01 6.58 3.08 34.43 30.60 23.56 0.34 16.77 -0.23  -47.65 -12.17 13.90 4.62 -43.45 -23.08 -21.91 1.83 -9.8.7 2.22  40.42 17.74 6.58 1.56 34.43 27.87 23.56 0.34 16.77 -0.23  -47.65 -12.17 5.57 4.62 -43.45 -23.08 -21.91 0.58 -9.87 2.22  81. 82. 83. 84. 85.  -1.07 -2.16 26.42 4.04 -2.14  11.66 14.31 •13.70 3.47 7.39  -1.07 -2.16 22.54 4.04 -2.14  6.76 10.06 -13.70 0.88 5.29  130 86. 87. 88. 89. 90.  0.0 19.78 14.71 0.73 13.49  1.56 -10.15 -2.25 5.19 5.96  0.0 13.10 13.42 0.73 13.49  0.64 -10.15 -2.25 1.73 -8.37  91. 92. 93. 94. 95. 96. 97. 98. 99. 100.  0.86 3.15 10.31 0.77 16.43 23.60 -2.76 21.51 4.38 7.95  3.75 4.30 0.71 1.67 -8.32 -22.21 4.54 1.68 -2.55 -6.17  0.86 3.15 10.31 0.77 15.97 23.60 -2.76 21.51 4.38 7.95  3.36 4.30 -0.06 -0.02 -8.32 -22.21 3.33 -2.65 -3.58 -8.64  101. 102. 103. 104. 105.  9.04 1.57 2.34 46.70 6.31  -2.53 3.91 0.72 -40.75 2.81  9.04 -1.20 2.34 46.70 6.31  -2.53 3.91 0.40 -40.75 2.81  *ID is the number assigned to the issue in Table A l . t G 1 = (Gmax -MP)X 100/Gmax ; G2 = (MP-Gmin, )X 100/MP; G3 = ( G m a x - M P ) X 1 0 0 / G m a x ; G4 = ( M P - G m i n ) X 100/MP; where Gmax(Gmin) = maximum (minimum) theoretical value in a range 0.05-0.70; Gmax(Gmin) =maximum (minimum) theoretical value in o range 0.05-0.50; M P = market price of new issue. 7 0  5O  7 0  5 0  7 0  0  5O  s 0  APPENDIX II. S A M P L E C O M P U T E R PROGRAMS This appendix contains samples of programs used to generate theoretical values for the Ingersoll, Geske, and Brennan-Schwartz solutions to the Black-Scholes-Merton differential equation. A l l programs are written in F O R T R A N IV with  WATFIV.  131  132  1. Ingersoll Solution, Junior Bond Plus Warrant This program was used to calculate the value of a junior discount bond plus warrant, with junior bond, senior discount bond, and warrant all maturing or expiring at the same time. Since a junior convertible is a special case of a junior bond plus warrant, this program can also be used for junior convertibles. For junior convertibles, the exercise price (EX) is set to the face value of the junior bond. This program could also be used for discount convertibles and discount bonds plus warrants when there is no other debt, simply by setting the face of senior debt to zero and modifying any statement that has face of senior debt in the denominator. The variables are: • V--firm value; • R--default-free interest rate; • F l - f a c e value of senior debt; • F2-face value of junior debt; • EX~stated exercise price of warrant; • GAM—fraction of firm gained upon exercise of warrant; • T-maturity date of junior bond; • S--standard deviation per period; • ZOP-calculated value of warrant; • BD-calculated value of junior bond; • B N - s u m of ZOP and B D  DIMENSION S(40), ZOP(40),BD(40),BN(40) REAL H1,H2,K1,K2,L1,L2,NH1,NH2,NK1,NK2,NL1,NL2 C The input parameters are entered in this set of statements.  R=? V=? Fl = ? F2 = ? EX = ? GAM=? T=? E = EX * (1.0-GAM) + GAM • (F1 + F2) C This loop changes the standard deviation.  DO 50 K= 1,10,1 AK = FLOAT(K) S(K) = 0.05+ AK* 0.05  133 C The group of statements sets the constants used in calculating univariate normals. AO = ALOG(V/(Fl •EXP(-R*T))) A l = ALOG(V/((Fl +F2)*EXP(-R*T))) A2 = ALOG(V • GAM/(E • EXP(-R*T))) S1 = S(K)*SQRT(T) L l = (A0/Sl) + 0.5*Sl L2 = (A0/Sl)-0.5*S1 H l = (Al/Sl) + 0.5*Sl H2 = (A1/Sl)-0.5*S1 K1 = (A2/S1) + 0.5*S1 K2 = (A2/S1)-0.5*S1 TWO = SQRT(2.00) C In this section univariate normals are calculated. NL1 = (ERF(Ll/TWO) +1.00)/2.00 NL2= (ERF(L2/TWO) + 1.00V2.00 NH1 = (ERF(Hl/TWO) +1.00)/2.00 NH2 = (ERF(H2/TWO) +1.00)/2.00 NK1 = (ERF(Kl/TWO) +1.00)/2.00 NK2 = (ERF(K2/TWO) +1.00)/2.00 C In this section bond and warrant values are calculated. ZOP(K) = (G A M • V) • NK1 -(E * EXP(-R* T)) • NK2 BD(K) = V*NL1-(F1*EXP(-R T))-V NH1 + ((FH-F2) EXP(-R T))*NH2 BN(K) = ZOP(K) + BD(K) 50 CONTINUE !S  !::  !!I  !!!  C This section causes the theoretical values to be printed. WRITE(7,101) V,F1,F2,GAM,R,T WRITE(7,102)(BN(K),K= 1,10) 101 FORMAT('0',1X,'V = ',F10.5,'F1 = ',F10.5,'F2 = ',F10.5, 'GAM = ',F10.5,'R= ',F10.5,'T= ',F10.5) 102 FORMATU0F12.5) STOP END  134  2. Geske Solution, Junior Bond Plus Warrant, With Warrant Expiring Earlier Than Bond This program was used to calculate the value of a junior discount bond plus warrant, with the warrant expiring earlier than the senior and junior discount bonds. The program calls tow subroutines from the IMSL library: M D B N O R which calculates bivariate normals, and Z X M I N which finds the value of X that minimizes a stated function. This latter was used to find the minimum firm value at which the warrant would be exercised.  • • • • • • • • • •  The variables are: V—firm value; VJ--critical firm value at expiry date of warrant, below which the warrant would not be exercised; R--default-free interest rate; Fl--face value of senior debt; F2-face value of junior debt; EX--stated exercise price of warrant; GAM-fraction of firm obtained upon exercise of warrant; Tl-maturity date of junior bond; T2--expiry date of warrant (T2 is less than TI); S = standard deviation per period; ZOP—calculated value of warrant; BD—calculated value of junior bond;  •  BN--sum of Z O P and BD  • •  DIMENSION S(40), ZOP(40),BD(40),BN(40) R E A L H1,H2,K1,K2,L1,L2,NH1,NH2,NK1,NK2,NL1,NL2 R E A L LL1,LL2, NLL1,NLL2 COMMON BJ,SS,E,GAM C The input parameters are entered in the following statements. R=? V=? Fl = ? F2 = ? EX=? GAM = ? Tl =? T2 = ? C This loop changes the standard deviation. DO 50 K= 1,10,1 A K = FLOAT(K) S(K) = 0.05 + AK*0.05  135 C The group of statements sets the parameters for finding the critical firm value at T 2 . T3 = T1-T2 B J = (Fl + F2)*(EXP(-R*T3)) SS = S(K)*SQRT(T3) E = EX*(1.0-GAM) XIN = ((E/GAM) + BJ) C A L L VBAR(XIN,VJ) C The parameters for calculating univariate and bivariate normals are calculated in this section. CORR=SQRT(T2/Tl) AO = ALOG( V/(F 1 • EXP(-R* T1))) A l = ALOG(V/((Fl + F2)*EXP(-R*T1))) A2 = ALOG(V/(V J EXP(-R*T2))) 51 = S(K)*SQRT(T1) 52 = S(K)*SQRT(T2) L l = (AO/SI)+0.5 SI L2 = (A0/S1)-0.5*S1 LL1 = (A1/S1) + 0.5*S1 LL2 = (A1/S1)-0.5*S1 K1 = (A2/S2)+0.5*S2 K2 = (A2/S2)-0.5 S2 TWO = SQRT(2.00) !;!  1S  C In this section univariate normals are calculated. NL1 = (ERF(LIZTWO) +1.00)/2.00 NL2 = (ERF(L2/TWO) +1.00)/2.00 NLL1 = (ERF(LL 1/TWO) +1.00)/2.00 NLL2 = (ERF(LL2/TWO) +1.00)/2.00 NK1 = (ERF(Kl/TWO) + 1.00)/2.00 NK2 = (ERF(K2/TWO) +1.00)/2.00 C In this section bivariate normals are calculated. C A L L MDBNOR(LLl,Kl,CORR,ZHl,IER) C A L L MDBNOR(LL2,K2,CORR,ZH2,IER) C In this section bond and warrant values are calculated. ZOP(K) = (GAM * V) * ZH 1-(E * EXP(-R* T2)) • NK2-(GAM * (Fl + F2)) • ZH2 BD(K) = V*NL1-(F1*EXP(-R T1)).V*NLL1+((F1 + F2)*EXP(.R T1)) NLL2 BN(K) = ZOP(K) + BD(K) 50 CONTINUE 5!  !!!  :!!  C This section causes theoretical values to be printed. WRITE(7,101)V,F1,F2,GAM,R,T1,T2 WRITE(7,102)(BN(K),K= 1,10) 101 FORMAT('0\1X,'V = ',F10.5,'F1 = ',F10.5,'F2 = \F10.5, 'GAM= ',F10.5,'R= ',F10.5,T1 = ',F10.5,'T2 = ',F10.5) 102 FORMAT(10F12.5) END C In this subroutine the critical value of the firm at T2 is determined. SUBROUTINE VBAR(Z1,Z2) EXTERNAL FUNCT INTEGER N,NSIG,MAXFN,IOPT R E A L H(1),G(1),W(3),F,X(1) N=l NSIG=5 MAXFN=100 IOPT=3 X(1) = Z1 C A L L ZXMIN(FUNCT,N,NSIG,MAXFN,IOPT,X,H,G,F,W,IER) Z2 = X(1) RETURN C In this subroutine the function to be minimized is specified. The function minimized is the value of the warrant at T2. SUBROUTINE FUNCT(N,X,F) INTEGER N COMMON BB,ST,ES,GAMA TWO = SQRT(2.00) A10 = (X/BB) IF(A10)1,1,2 1 A11 = 0.0 GO T O 3 2 A l l = ALOG(X/B) 3 AK1 = (A11/ST)+0.5*ST AK2 = (A11/ST)-0.5*ST ZA1 = (ERF(AK 1/TWO) +1.00)/2.00 ZA2 = (ERF(AK2/TWO) +1.00)/2.00 WF = (X*ZA1-BB*ZA2)*GAMA F=ABS(WF-ES) RETURN END STOP END  137  3. Geske Solution for Junior Discount Bond Plus Warrant With Senior Discount Bond Maturing Earlier Than Junior Bond  This program was used to calculate the value of a junior discount bond plus warrant with the senior discount bond maturing earlier than the junior discount bond plus warrant. This program could also be used to calculate the value of a junior discount convertible b}' setting the exercise price (EX) to the face value of the junior bond. The program calls two subroutines from the IMSL library: MDBNOR, which calculates bivariate normals and ZXMIN which is used to find the value of X which minimizes a stated function. This is used to find the smallest value of the firm at which shareholders would choose to exercise their option and pay out to senior debtholders the principal owing.  • • • • • • • • • • • • •  The variables are: V-firm value; VJ--critical firm value at maturity date of senior bond, below which shareholders would not pay off debtholders; R-default-free interest rate; Fl-face value of senior debt; F2--face value of junior debt; EX—exercise price; GAM—fraction of firm obtained upon exercise of warrant; Tl-maturity date of junior bond; T2-expiry date of senior (T2 is less than TI); S-standard deviation per period; ZOP-calculated value of warrant; BD-calculated value of junior bond; BN-sum of ZOP and BD.  DIMENSION S(40), ZOP(40),BD(40),BN(40),B0(40) R E A L H1,H2,K1,K2,L1,L2,NH1,NH2,NK1,NK2,NL1,NL2 COMMON BJ,SS,E,GAM  C The input parameters are entered in the following statements. R=? V=? Fl =? F2 = ? EX = ? GAM = ? Tl =? T2 = ?  C This loop changes the standard deviation. DO 50 K = 1,10,1 A K = FLOAT(K) S(K) = 0.05 + AK*0.05  C The group of statements sets the parameters for finding the critical value T2.  T3 = T1-T2 B J = F2*(EXP(-R*T3)) SS = S(K)*SQRT(T3) E = F1 IF(K.EQ.l) V J = F2*EXP(-R*T3) VV = V J XIN = VV-0.02*VV C A L L VBAR(XIN,VJ) C This group of statements sets parameters for finding the normal values. CORR=SQRT(T2/Tl) EX = (1.0-GAM) F2 + G A M * (F2 + F1 * EXP(-R * T3)) A0 = ALOG(V/(Fl) 'EXP(-R*T2))) A l = ALOG(V/(F2)*EXP(-R*Tl))) A2 = ALOG(V/(VJ*EXP(.R*T2))) A3 = ALOG((GAM*V)/EX EXP(-R*Tl))> S i = S(K)*SQRT(Tl) S2 = S(K)*SQRT(T2) D1 = (-A0/S2)-0.5*S2 D2 = (A0/S2)-0.5*S2 H1 = (-A1/S1)-0.5*S1 H2 = (A1/S1)-0.5*S1 H3=(Al/Sl) + 0.5*Sl K1 = (-A2/S2)-0.5*S2 K2 = (A2/S2)-0.5 S2 K3 = (A2/S2) + 0.5*S2 L1 = (A3/S1) + 0.5*S1 L2 = (A3/S1)-0.5*S1 CORN = (-CORR) TWO = SQRT(2.00) !!!  !!  :  !!!  C In this section the univariate normals are calculated. NH1 = (ERF(Hl/TWO) +1.00)/2.00 NH2 = (ERF(H2/TWO) +1.00)/2.00 NL1 = (ERF(LIZTWO) +1.00)/2.00 NL2 = (ERF(L2/TWO) +1.00)/2.00 ND 1 = (ERF(D 1/TWO) +1.00)/2.00 ND2 = (ERF(D2/TWO) +1.00)/2.00 NK 1 = (ERF(K 1/TWO) +1.00)/2.00 NK2 = (ERF(K2/TWO) +1.00)/2.00 NK3 = (ERF(K3/TWO) +1.00)/2.00 C In this section the bivariate normals are calculated. C A L L MDBNOR(K2,H2,CORR,ZHl,IER) C A L L MDBNOR(K3,Hl,CORN,ZH2,IER)  139 CALL CALL  MDBNOR(K3,Ll,CORR,ZH3,IER) MDBNOR(K2,L2,CORR,ZH4,IER)  C In this section bond and warrant values are calculated. ZOP(K) = (G A M * V) * ZH3-(EX • EXP(-R • T1)) • ZH4 BO(K) = -F1 • EXP(-R " T2) • ND2-V * ND 1 + F1 • EXP(R * T2) • NK2 + V • NK1 BD(K) = V * ZH2 + (F2 * EXP(-R*T1)) * ZH1 + BO(K) BN(K) = ZOP(K) + BD(K) 50 CONTINUE C This section causes the theoretical values to be printed. WRITE(7,101) V , F 1.F2.G AM,R,T 1 ,T2 WRITE(7,102)(BN(K),K= 1,10) 101 FORMAT('0',1X,'V = ',F10.5,'F1 = \F10.5,'F2 = \F10.5, 'GAM = ',F10.5,'R = ',F10.5,'T1 = ',F10.5,'T2 = \F10.5) 102 FORMAT(10F12.5) END C In this subroutine the critical firm value at T2 is determined. SUBROUTINE VBAR(Z1,Z2) E X T E R N A L FUNCT INTEGER N,NSIG,MAXFN,IOPT R E A L H(1),G(1),W(3),F,X(1) N=l NSIG = 5 MAXFN=100 IOPT=3 X(1) = Z1 C A L L ZXMIN(FUNCT,N,NSIG,MAXFN,IOPT,X,H,G,F,W,IER) Z2 = X(1) RETURN C In this subroutine the function to be minimized is specified. In this program the function being minimized is stock value at T2. SUBROUTINE FUNCT(N,X,F) INTEGER N COMMON BB,ST,ES,GAMA TWO = SQRT(2.00) A10 = (X/BB) IF(A10)1,1,2 1 A11 = 0.0 GO TO 3 2 A l l = ALOG(X/BB) 3 AK1 = (A11/ST)+0.5*ST AK2 = (All/ST)-0.5*ST ZA1 = (ERF( AK1/TWO) +1.00)/2.00  ZA2 = (ERF(AK2/TWO) +1.00)/2.00 WF= (X*ZA1-BB*ZA2) F = ABS(WF-ES) RETURN END STOP END  141 4. Brennan-Schwartz Solution, Junior Convertibles  This program calculates the value of a junior convertible using the finite difference method to approximate the differential equation. The • • • • • • • • • • • • • • • • • • • • • •  variables are: Bl—Theoretical value of junior bond; B2-Theoretical value of senior bond; VO-Market value of firm ; RCl-Coupon rate on junior bond; RC2-Coupon rate on senior bond; DP--Dividend payments to shareholders; SNK-Sinking fund payment; GAM~Fraction of firm acquired upon conversion; NCP-Number of different call prices; CP(NCP)-Call prices for junior bond; CPA(l) is call price at maturity; CPA(NCP) is call price at issue; Fl-Face amount, junior bond; FS-Face amount, senior bond; VAR-Variance rate for return on firm's assets; RF-Default-free interest rate; TNl-Number of months to maturity of junior; NT3-TN1 less number of months to maturity of senior; NCSPAC-Spacing in months of coupon payments; NDSPAC-Spacing in months of dividend payments; NSK—Number of months prior to maturity of last sinking fund payment; NSK2-Number of months prior to maturity of first sinking fund payment; NK—Number of time steps; XK--Number of firm value steps.  DIMENSION B 1(600,30),CI(370),D(370),CP(370),CPA(32) /BB(600),CC(600),FF(600),MC(25), /BBS(600),CCS(600),FFS(600),CS(370),B2(600,30),SK(370) WRITE(6,888) 888 FORMAT(3X,'THESE RESULTS A R E F O R ?') 304 FORMAT(2X,F12.5,3X,F12.5,3X,2I6) 201 FORMAT(2X,10F11.4)  C The input parameters are entered in this set of statements. V0 = ? RF=? Fl =? FS = ? RC1 = ? RC2 = ? CIP = RC1*F1 CIS = RC2*FS NCSPAC=?  142  DP = ? NDSPAC=? SNK = ? NSK = ? NSK2=? NSKl=NSK2+2 GAM = ? TN1 = ? NT1 = IFIX(TN1) NT3 = ? MS = NT1-NT3+1 NT=NT1+1 NK = ? XK = TN 1/FLO AT(NK) Q1 = 0.5*RF*XK Q4=1.0 + Q l + Q l NCP = ? C This set of statements is used to set call prices. CPA(1) = ? CPA(2) = ? DO 82 J = l , ? , l A J = FLOAT(J) CPA(J + 2) = ? 82 CONTINUE CPA(NCP) = ? C In this set of statements the number of months each call price is applicable is entered. NCPP = NCP + 1 MC(2) = ? MC(3) = ? MC(4) = ? MC(5) = ? MC(NCPP) = ? C In this loop the call price at each time point is determined. MC(1)=1 CP(1) = CPA(1) DO 3 J = 2,NCPP MC( J) = MC( J) + MC( J-1) N = MC(J-1)+1 M = MC(J) DO 4 K = N,M CP(K) = CPA(J-1) 4 CONTINUE 3 CONTINUE  143  WRITE(6,304)CP(1),CP(NT),NT1,M NNN = 0 C Initial conditions are set in this loop. DO 5 J=1,NT D(J) = 0.0 CI(J) = 0.0 CS(J) = 0.0 SK(J) = 0.0 IF(J.EQ.NT)GO TO 5 C The interest payment adjustment  is calculated in this set of statements.  C AD J = CIP * (FLO AT((NCSPAC-NNN))/FLO AT(NCSPAC)) CP(J)=CP(J) + C A D J NNN = NNN +1 IF(NNN.GE.NCSPAC)NNN=0 C The bond values at V = 0.0 are set in these statements. IF(J.GT.30) GO TO 5 B1(1,J) = 0.0 B2(1,J) = 0.0 5 CONTINUE C The timing and amount of the cashflows is determined in this set of statements. DO 6 J=1,NT1,NCSPAC 6 CI(J) = CIP DO 66 J = NT3,NT1,NCSPAC CS(J) = CIS 66 CONTINUE IF(DP.EQ.O.OOO) GO T O 11 DO 7 J = 2,NT,NDSPAC 7 D(J) = DP 11 CONTINUE IF(SNK.EQ.0.00)GO TO 15 DO 14 J = NSK,NSK1,12 14 SK(J) = SNK + SNK*RC1 15 CONTINUE C The number and size of firm value increments is set in these statements. F C = F1 + CI(1) FCS = FS + CS(NT3) F M A X = 10.0*((FC/GAM) + FCS) V M A X = AMAX1(V0,FMAX) NX = ?  N X H = IFIX(VM AX/NX) X H = FLO AT(NXH) NSS = IFIX(VMAX/XH) NS = NSS + 1 NS1 = NSS NS2 = NSS-1 NSL = IFIX(0.9 • VO/XH) NSH = IFIX(1.1 * VO/XH) NHH = IFIX(3.0 * VO/XH) C The size of firm value adjustments due to outflows are set in thse statements. NCI = IFIX(CIP/XH) NCIS = IFIX(CIS/XH) ND = IFIX(DP/XH) NB = IFIX(FCS/XH) C The number of time increments is set in this statement. NTZ = IFIX(1.0/XK) + 1 V M = NS2*XH C The value of firm standard deviation is set in this loop. DO 98 L L = 1,20,1 A L L = FLO AT(LL) SIG1 = 0.05* A L L SIG = SIG1/SQRT(12.0000) Q3 = 0.5*((SlG)t2)*XK BB(1) = Q4 + Q3 + Q3 CC(1) = -Q3-Q1 BBS(1) = BB(1) CCS(1) = CC(1) C Boundary conditions at maturity are checked in these statements. DO 9 K=1,NS M = K-1 V = M*XH B2(K,1) = AMIN1(FCS,V) IF(NT3.GT. 1)B2(K, 1) = 0.000 V V = AMAX1((V-B2(K,1)),0.000) GV = G A M * V V B 1(K, 1) = A M A X K G V , AMIN 1(VV,FC)) 9 CONTINUE IF(LL.GT.l)GO TO 99 WRITE(6,201)(B2(IH, 1),IH = NSL,NSH) WRITE(6,201)(B 1(IJ, 1),IJ = NSL,NSH) 9 CONTINUE  145 C This loop causes recalculation of values at each time point. DO 100 J=1,NT1 C J = CI(J) C S J = CS(J) D J = D(J) S K J = SK(J) C P J = CP(J) DO 142 K=1,NS M = K-1 V =M XH :  C Values are calculated at maturity of senior bond in this set of statements. IF(J.EQ.1.0R.J.NE.NT3) GO T O 117 KB = MAX0((K-NB),1) BSJ=FCS B J = B1(KB,1) VS = V-BSJ-BJ IF(VS.LE.0.00)B2(K, 1) = AMIN1(V,FCS) IF(VS.GT.0.00)B2(K, 1) = FCS V V = AMAX1((V-B2(K,1)),0.000) GV = G A M * V V IF(VS.LE.0.00)B 1(K, 1) = V V IF(VS.GT.0.00)B1(K,1) = AMAX1(GV,BJ) 117 CONTINUE C Values at times of coupon payments on senior bond are calculated in this set of statements. IF(J.EQ.l.OR.CSJ.EQ.0.0.OR.J.EQ.NT3) KS = MAXO((K-NCIS), 1) KSS = MAXO((KS-NCIS), 1) BSJ = B2(KS,1) + C S J B J = B1(KS,1) VS = V-BSJ-BJ B2(K, 1) = AMIN 1(BS J , V) IF( J.LT.NT3)B2(K, 1) = 0.00 V V = AMAX 1((V-B2(K, 1)),0.000) GV = G A M V V IF(VS.LE.0.00)B 1(K, 1) = V V IF(VS.GT.0.00)B1(K,1) = AMAX1(GV,BJ) 107 CONTINUE  GO T O 107  !!  C Values at times of coupon payments of junior bond are calculated in this set of statements. IF(J.EQ.l.OR.CJ.EQ.0.0) K K = MAX0((K-NCI),1) B S J J = B2(K,1)  GO T O 103  146 B S J = B2(KK,1) B J = B1(KK,1) + C J VS = V-BSJ-BJ IF(VS.GT.0.00)B2(K, 1) = BSJ IF(VS.LE.0.00)B2(K, 1) = AMIN 1(BS J J , V) IF(J.LT.NT3)B2(K, 1) = 0.00 V V = AMAX1((V-B2(K,1)),0.00) B 1(K, 1) = AMIN 1 (B J , VV) 103 CONTINUE C Values at times of sinking fund payments are calculated in this set of statements. IF(SKJ.EQ.O.O) GO T O 113 GS = SNK/F1 S L J = AMIN1(SKJ,(GS*B1(K,1))) NG = IFLX(SLJ/XH) K T = MAX0((K-NG),1) B S J J = B2(K,1) B S J = B2(KT,1) B J = B1(KT,1) + S L J VS = V-BSJ-BJ IF(VS.GT.0.000)B2(K,1) = BSJ IF(VS.LE.0.000)B2(K,1) = AMIN1(BSJJ,V) IF( J.LT.NT3)B2(K, 1) = 0.000 V V = AMAX1((V-B2(K, 1)),0.000) B1 (K, 1) = AMIN 1(B J , V V) 113 CONTINUE C Values at times of dividend payments are calculated in this set of statements. IF(DJ.EQ.O.O) G O TO 102 DJJ=(DJ/V0)*V NY = IFIX(DJJ/XH) K L = MAX0((K-NY),1) BL = B1(KL,1) BLL=B2(KL,1) B2(K,1) = B L L IF( J.LT.NT3)B2(K, 1) = 0.000 VS = AMAX1((V-B2(K, 1)),0.000) GV=GAM*VS B1(K,1) = AMAX1(BL,GV) 102 CONTINUE C Value of junior bond is compared to call price in this set of statements. IF( J.LT.NT3)B2(K, 1) = 0.000 VS = V-B2(K,1) VSS = AMAX1(VS,0.000) GV = GAM*VSS  147  B E = B1(K,1) B E E = A M A X 1(BE,GV) B 1(K, 1) = AMIN 1(BEE,CP J) 142 CONTINUE NSJ = NS1 CPP = CPJ  C This is loop for the time increments. DO 20 I = 2,NTZ 12 = 1-1 IF(CPJ.GT.VM)GO TO 203 CPZ = CP J-FLOAT(I2) * (CIP/(FLOAT((NTZ-1) •NCSPAC))) DO 44 K=1,NS M = K-1 V = M*XH VS = GAM*(V-B2(K,12)) IF(VS.LE.CPZ)NSJ = K IF(VS.GT.CPZ)GO TO 237 44 CONTINUE 237 C P P = C P Z 203 CONTINUE  C In this set of statements the constants for the approximation are calculated. FF(1) = B1(2,I2) FFS(1) = B2(2,I2) DO 28 K=1,NS1 M = K-1 L =K+1 Y K = FLO AT(K) Q3K = Q3 * Y K * YK A K = Q1*YK-Q3K IF(K.EQ.1)AK = 0.0 IF(K.EQ.NSl) A K = -1.0 AKS=AK BK = Q4 + Q3K + Q3K IF(K.EQ.NSl) BK=1.0 BKS=BK CK = -Q1*YK-Q3K CKS=CK FK = B1(L,I2) FKS = B2(L,I2) IF(K.EQ.NS1) F K = G A M * X H IF(K.EQ.NS1) FKS = 0.0001*XH IF(NSJ.LT.NS1.AND.K.EQ.NSJ) AK = 0.0 IF(NSJ.LT.NS 1.AND.K.EQ.NSJ) FK = C P P  148  C In this section the constants are transformed for use in solving the linear system of difference equations. IF(K.EQ.l.)GO TO 33 IF(ABS(AKS).LT.0.0001) GO TO 34 BBSM = BBS(M)/AKS BBS(K) = BBSM*BKS-CCS(M) CCS(K) = BBSM*CKS FFS(K) = BBSM *FKS-FFS(M) IF(ABS(BBS(K)).LT.10000.00)GO TO 33 BBS(K) = 0.0001 *BBS(K) CCS(K) = 0.0001 *CCS(K) FFS(K) = 0.0001*FFS(K) GO TO 33 34 BBS(K) = BKS CCS(K) = CKS FFS(K) = FKS 33 CONTINUE IF(K.EQ.l) GO TO 23 IF(NSJ.LT.NSl.AND.K.GT.NSJ)GO TO 23 IF(ABS(AK).LT.0.0001) GO TO 24 BBM = BB(M)/AK BB(K) = BBM*BK-CC(M) CC(K) = BBM*CK FF(K) = BBM*FK-FF(M) IF(ABS(BB(K)).LT. 10000.0) GO TO 23 BB(K) = 0.0001 *BB(K) CC(K) = 0.0001 CC(K) FF(K) = 0.0001 *FF(K) GO TO 23 24 BB(K) = BK CC(K) = C K FF(K) = F K 23 CONTINUE 28 CONTINUE 51  C The linear system is solved in this set of statements. B1(NS1 +1,1) = FF(NS1)/BB(NS1) B2(NS1 + 1,I) = FFS(NS1)/BBS(NS1) DO 25 KX=1,NS2 K = NS1 + 1-KX M = K-1 L =K +1 BX = (FF(M)-CC(M)*B1(L,I))/BB(M) BXS = (FFS(M)-CCS(M)*B2(L,I))/BBS(M) B2(K,I) = AMIN1(BXS,B2(L,I)) IF(NSJ.LT.NSl.AND.K.GT.NSJ)GO TO 29 B1(K,I) = AMIN1(BX,B1(L,I)) GO TO 25  29 B1(K,I) = C P P 25 CONTINUE 20 CONTINUE DO 122 K=1,NS B1(K,1) = B1(K,NTZ) 122 B2(K,1) = B2(K,NTZ) 100 CONTINUE C This section causes the theoretical prices to be printed. WRITE(6,200)SIG,RF,CIP,NCSPAC,GAM,F1,XH,XK,NSS,NT1,V0, /FS,MS 200 FORMAT(/2X,'SIG = ',F12.5,2X,'RF=',F12.5,2X,'C = ',F5.2,2X, / 'C I=',I4,2X,'GAM = ',F6.3,2X,'F1 = ',F10.5, / 2X,'XH = ',F10.5,2X,'XK = \F6.4,2X,'NS = ',I4,2X, / 2X,'NT1 = ',I4,2X,'V0 = ',F12.5,2X,'FS = ',F12.5, / 2X,'MS = ',I4) WRITE(6,201) (B1(JN,1),JN= 1,NSH),B1(1,1),B1(NS1,1) WRITE(6,201) (B2(IG,1),IG= l,NSH),B2(l,l),B2(NSl,l) 98 CONTINUE STOP END  150  5. Brennan-Schwartz Solution, Bonds Plus Warrants  This program calculates approximately the theoretical value of a junior bond plus warrant using the finite difference method.  • • • • • • • • • • • • • • • • • • • • • • •  The variables are: Bl-Theoretical value of junior bond; B2-Theoretical value of senior bond; V0--Market value of firm ; RCl-Coupon rate on junior bond; RC2-Coupon rate on senior bond; DP-Dividend payments to shareholders; SNK-Sinking fund payment; GAM-Fraction of firm acquired upon conversion; NCP-Number of different call prices; CP(NCP)--Call prices for junior bond; CPA(l) is call price at maturity; CPA(NCP) is call price at issue; Fl-Face amount, junior bond; FS-Face amount, senior bond; VAR-Variance rate for return on firm's assets; RF-Default-free interest rate; TNl-Number of months to maturity of junior; NT3--TN1 less number of months to maturity of senior; NT4-TN1 less number of months to expiry of warrant; NCSPAC-Spacing in months of coupon payments; NDSPAC-Spacing in months of dividend payments; NSK-Number of months prior to maturity of last sinking fund payment; NSK2-Number of months prior to maturity of first sinking fund payment; NK-Number of time steps; XK—Number of firm value steps;  DIMENSION B 1(600,30),CI(312),D(312),CP(312),CPA(20), /BB(600),CC(600),FF(600),MC(21),W(600,30),BW(600,30), /BBS(600),CCS(600),FFS(600),CS(312),B2(600,30),SK(312), /BBW(600),CCW(600),FFW(600) WRITE(6,888) 888 FORMAT(3X,'THESE RESULTS A R E F O R ') 304 FORMAT(2X,F16.5,3X,F12.5,3X,2I6) 201 FORMAT(2X,10F11.4)  C The input parameters are entered in this set of statements. V0 = ? EX = ? RF=? Fl =? F2 = ? RC1 = ?  RC2 = ? CIP = R C r F l CIS = RC2*F2 NCSPAC=? DP = ? NDSPAC=? SNK = ? NSK = ? NSK2=? NSKl=NSK2+2 GAM = ? TN1 = ? NT1 = IFIX(TN1) NT3 = ? NT4 = ? MS = NT1-NT3 + 1 MW = NT1-NT4 + 1 NT=NT1+1 NK = ? XK = TN 1/FLO AT(NK) Ql = 0.5*RF*XK Q4=1.0 + Q1+Q1 NCP = ? :  C The call prices can be entered with this set of statements. CPA(1) = ? CPA(2) = ? DO 82 J=l,? A J = FLOAT(J) CPA(J + 2) = ? 82 CONTINUE CPA(6) = 25000.00 C The months between call price changes are entered through this set of statements. NCPP=NCP+1 MC(2) = ? MC(3) = ? MC(4) = ? MC(NCPP) = ? C The call prices at each time are set through these statements. MC(1)=1 CP(1) = CPA(1) DO 3 J = 2,NCPP MC( J) = MC( J) + MC( J-l) N = MC(J-1) + 1  M = MC(J) DO 4 K = N,M CP(K) = CPA(J-1) 4 CONTINUE 3 CONTINUE WRITE(6,304)CP(1),CP(NT),NT1,M NNN = 0 C This set of statements sets initial conditions. DO 5 J=1,NT D(J) = 0.0 CI(J) = 0.0 CS(J) = 0.0 SK(J) = 0.0 IF(J.EQ.NT)GO TO 5 C The interest adjustment to price is calculated in this set of statements. CAD J = CIP*(FLOAT((NCSPAC-NNN))/FLOAT(NCSPAC)) CP(J) = CP(J) + C A D J 92 NNN = NNN +1 IF(NNN.GE.NCSPAC)NNN = 0 C The values at V = 0.0 are determined with this set of statements. IF(J.GT.20) GO TO 5 B1(1,J) = 0.000 B2(1,J) = 0.000 BW(1,J) = 0.000 W(1,J) = 0.000 5 CONTINUE C The amount and timing of cashflows is set in this group of statements. DO 6 J=1,NT1,NCSPAC 6 CI(J) = CIP DO 66 J = NT3,NT1,NCSPAC CS(J)=CIS 66 CONTINUE IF(DP.EQ.O.OOO) GO TO 11 DO 7 J = 2,NT,NDSPAC 7 D(J) = DP 11 CONTINUE IF(SNK.EQ.0.000)GO TO 15 DO 14 J=NSK,NSK1,12 14 SK(J) = SNK + SNK*RC1 15 CONTINUE  153 C The number and size of the firm value increments is determined with this set of statements FC = F1 + CI(1) FCS = F2 + CS(NT3) FMAX = 10.0 * ((EX/GAM) + FC + FCS-EX) V M A X = AMAX1 (V0,FMAX) NX = ? NXH = MAX0(IFIX(VMAX/NX), 1) XH = FLO AT(NXH) NSS = IFIX(VMAX/XH) NS=NSS+1 NS1 = NSS NS2 = NSS-1 NSL = IFIX(0.9'V0/XH) NSH = IFIX( 1.1 * VO/XH) NHH = IFIX(2.0* VO/XH) NCI = IFIX(CIP/XH) C This adjustments to firm value from outflows/inflows are calculated in this set of statements. NCIS = IFIX(CIS/XH) ND = IFIX(DP/XH) NB = IFIX(FCS/XH) NW = IFIX(EX/XH) NTZ = IFIX(1.0/XK) + 1 VM=NS2*XH C The value of firm standard deviation is set with this loop. DO 98 L L = 1,20,1 A L L = FLO AT(LL) SIG1 = 0.05* A L L SIG = SIG1/SQRT(12.0000) Q3 = 0.5*((SIG)t2)*XK BB(1) = Q4 + Q3 + Q3 CC(1) = -Q3-Q1 BBS(1) = BB(1) CCS(1)=CC(1) BBW(1) = BB(1) CCW(1) = CC(1) C The values at time of expiry of the junior bond are calculated in this set of statements. DO 9 K=1,NS M = K-1 V = M*XH B2(K, 1) = AMIN 1(FCS, V)  154 IF(NT3.GT. 1)B2(K, 1) = 0.000 V V = AMAX1((V-B2(K,1)),0.000) B 1(K, 1) = AMIN 1(VV,FC) IF(NT4.GT.l)GO TO 88 VS = V-B 1(K, 1)-B2(K, 1) + EX GV=GAM*VS W(K, 1) = AMAX1((GV-EX),0.000) GO TO 79 88 W(K,1) = 0.000 79 BW(K,1) = B1(K,1) + W(K,1) 9 CONTINUE IF(LL.GT.l)GO TO 99 WRITE(6,201)(B2( JX, 1), JX = NSL,NSH),B2(1,1),B2(NS 1,1) WRITE(6,201)(BW(IJ, 1),IJ = NSL,NSH),BW(1,1),B W(NS 1,1) WRITE(6,201)(B1(IX,1),IX = NSL,NSH),B1(1,D,B1(NS1,1) 9 CONTINUE  C This loop causes values at each time point to be calculated. DO 100 J=1,NT1 C J = CI(J) C S J = CS(J) D J = D(J) S K J = SK(J) C P J = CP(J) DO 113 K=1,NS M = K-1 V = M*XH  C Values at maturity of the senior bond are calculated in this set of statements. IF(NT3.EQ.l)GO TO 117 IF(J.EQ.1.0R.J.NE.NT3) GO TO 117 NB = IFIX(FCS/XH) K B = MAX0((K-NB),1) BSJ=FCS VDIF=V-FCS B J = B1(KB,1) IF(B J.LE.0.00.AND.VDIF.GT.0.00)B J = VDIF WJ = W(KB,1) VS = V-BSJ-BJ-WJ IF(VS.LE.0.00)B2(K,1) = AMIN1(V,FCS) IF(VS.GT.0.00)B2(K, 1) = FCS IF(B2(K,1).GT.V)B2(K,1)=V VDIF=V-B2(K,1) V V = A M A X 1 (VDIF,0.000) IF(BJ.EQ.0.0. AND.VDIF.GT.0.00)B J = VDIF IF(VS.GT.0.00)B1(K, 1)=B J IF(VS.LE.0.00)B 1(K, 1) = V V VX = V-B1(K,1)-B2(K,1)+EX  155 GV = (GAM*VX)-EX IF(VS.GT.0.00)W(K,1) = AMAX1(WJ,GV) IF(VS.LE.0.00.OR. J.LT.NT4)W(K, 1) = 0.00 117 CONTINUE  C Values at time of payment of coupons on senior bond are calculated in this set of statements. IF(J.EQ.l.OR.CSJ.EQ.0.0.OR.J.LE.NT3) GO TO 107 NCIS = IFIX(CIS/XH) KS = MAXO((K-NCIS), 1) BSJ = B2(KS,1) + C S J B J = B1(KS,1) WJ = W(KS,1) VS = V-BSJ-BJ-WJ IF(VS.LE.0.000)B2(K,1) = AMIN1(BSJ,V) IF(VS.GT.0.000)B2(K,1) = BSJ V V = AMAX1((V-B2(K,1)),0.000) IF(VS.LE.0.0000)B1(K,1) = AMIN1(VV,BJ) IF(VS.GT.0.000)B 1(K, 1) = B J V L = V - B 1(K, 1)-B2(K, 1) + EX GV = G A M * V L - E X IF(VS.GT.0.00)W(K,1) = AMAX1(WJ,GV) IF(VS.LE.0.00.OR. J.LT.NT4)W(K, 1) = 0.000 107 CONTINUE  C Values at time of payment of coupons on junior bond are calculated in this set of statements. IF(J.EQ.l.OR.CJ.EQ.0.0) GO TO 103 KK = MAX0((K-NCT), 1) BSJJ = B2(K,1) BSJ = B2(KK,1) BM = B1(KK,1) + C J WJ = W(KK,1) VS = V-BSJ-BM-WJ IF(VS.GT.0.00)B2(K,1) = BSJ IF(VS.LE.0.00)B2(K, 1) = AMIN 1(BS J J , V) IF(J.LT.NT3)B2(K,1) = 0.000 V V = AMAX1((V-B2(K,1)),0.000) IF(VS.GT.0.00)B 1(K, 1) = B M IF(VS.LE.0.00)B 1(K, 1) = AMIN 1(VV,BM) V C = V-B 1(K, 1)-B2(K, 1)+EX GV = (GAM*VC)-EX IF(VS.GT.0.00)W(K,1) = AMAX1(WJ,GV) IF(VS.LE.0.00.OR. J.LT.NT4)W(K, 1) = 0.00 103 CONTINUE  156  C Values at the time of sinking fund payments are calculated in this group of statments. IF(SKJ.EQ.O.O) GO TO 113 GS = SNK/F1 S L J = AMIN1(SKJ,(GS*B1(K,1))) NG = IFIX(SLJ/XH) KT=MAX0((K-NG),1) B S J J = B2(K,1) BSJ = B2(KT,1) B J = B1(KT,1) + S L J WJ = W(KT,1) VS = V-BSJ-BJ-WJ IF(VS.GT.0.000)B2(K, 1) = BS J IF(VS.LE.0.000)B2(K, 1) = AMIN 1(BS J J , V) IF( J.LT.NT3)B2(K, 1) = 0.000 V V = AMAX1((V-B2(K,1)),0.000) IF(VS.LE.0.000)B 1(K, 1)=AMIN 1(VV,B J) IF(VS.GT.0.000)B 1(K, 1) = B J VB = V-B 1(K, 1)-B2(K, 1)+EX IF(VS.GT.0.00)W(K, 1) = A M A X 1( W J,(G A M * VB-EX)) IF(VS.LE.0.00.OR. J.LT.NT4)W(K, 1) = 0.00 113 CONTINUE DO 118 K=1,NS M = K-1 V = M*XH  C Values at the time of expiry of the warrant are calculated in this set of statements. IF(J.EQ.1.0R.J.NE.NT4) GO TO 118 KP=K+NW IF(KP.GE.NS)KP = NS BBJ = B1(KP,1) BSSJ = B2(KP,1) BSJ = B2(K,1) B J = B1(K,1) VS = V-BB J-BSS J + E X GV=GAM*VS AX = GV-EX W(K, 1) = AMAX1(AX,0.00) IF(AX.LE.0.00)B2(K, 1) = BS J IF(AX.LE.0.00)B 1 (K, 1) = B J IF(AX.GT.0.00)B1(K, 1) = BB J IF(AX.GT.0.00)B2(K, 1) = BSSJ 118 CONTINUE IF(DJ.EQ.O.O) GO TO 291 DO 227 K=1,NS M = K-1 V = M*XH  157  C Values at time of dividend payments are calculated in this set of statements. D J J = (DJ/VO)*V NY = IFIX(DJJ/XH) KL=MAX0((K-NY),1) B1(K,1) = B1(KL,1) B2(K,1) = B2(KL,1) WJ = W(KL,1) 227 CONTINUE DO 172 K=1,NS M = K-1 V = M'XH KJ=K+NW IF(KJ.GT.NS)KJ = NS VS = V-B 1(K J , 1)-B2(KJ, 1) + EX WJ = W(K,1) AV=GAM*VS IF((AV-EX).GT.WJ)GO TO 170 W(K,1) = AMAX1(WJ,0.000) GO TO 243 170 W(K,1) = AV-EX B2(K,1) = B2(KJ,1) B1(K,1) = B1(KJ,1) 243 CONTINUE IF(J.LT.NT3)B2(K,1) = 0.000 171 IF(J.LT.NT4)W(K,1) = 0.000 172 CONTINUE 291 CONTINUE  C Bond value and call price are compared in this set of statements. DO 142 K=1,NS M = K-1 V = M*XH B E = B1(K,1) WE = W(K,1) B1(K,1) = AMIN1(BE,CPJ) IF( J.LT.NT3)B2(K, 1)=0.000 IF( J.LT.NT4)W(K, 1) = 0.000 VSS = V-B 1(K, 1)-B2(K, 1) GVX = GAM*VSS G V = G A M • (VSS + EX)-EX IF(W(K, 1).GT.GVX)W(K, 1) = AMAX1(GV,0.00) IF(W(K,1).LT.0.00)W(K,1) = 0.00 142 CONTINUE NSJ=NS1 NSW=NS1 CPP=CPJ  158  C Values at the time increments are calculated in this loop. DO 20 I=2,NTZ 12 = 1-1 IF(CPJ.GT.VM)GO TO 203 CPZ = CPJ-FLOAT(I2)*(CIP/(FLOAT((NTZ-l)*NCSPAC))) 203 CONTINUE DO 44 K=1,NS M=K-1 V = M*XH VS = B1(K,I2) VW = W(K,I2) IF(CPJ.GT.VM)GO TO 204 IF(VS.LE.CPZ)NS J = K 204 CONTINUE IF(VW.LE.CPW)NSW = K 44 CONTINUE 237 CPP = CPZ C Constants for the difference equations are calculated in this set of statements. FF(1) = B1(2,I2) FFS(1) = B2(2,I2) FFW(1) = W(2,I2) DO 28 K=1,NS1 M = K-1 L =K+1 YK = FLOAT(K) Q3K = Q3*YK*YK A K = Q1*YK-Q3K IF(K.EQ.1)AK=0.0 IF(K.EQ.NSl) A K = -1.0 AKS = AK AWS = A K BK = Q4 + Q3K + Q3K IF(K.EQ.NSl) BK=1.0 BKS = BK BWS = BK CK = -Ql*YK-Q3K CKS=CK CWS = CK FK = Bl(L,I2) FKS = B2(L,I2) FWS = W(L,I2) IF(K.EQ.NSl) FK = 0.0001*XH IF(K.EQ.NS1) F K S = 0.0001*XH IF(K.EQ.NS1) FWS = G A M ' X H IF(NSJ.LT.NS1.AND.K.EQ.NSJ) A K = 0.0 IF(NSJ.LT.NS1.AND.K.EQ.NSJ) F K = CPP  159 IF(NSW.LT.NS 1.AND.K.EQ.NSW) AWS = 0.0 IF(NSW.LT.NS 1.AND.K.EQ.NSW) FWS = CPW  C The constants in the transformed difference equations are calculated in this section. IF(K.EQ.l)GO TO 33 IF(ABS(AKS).LT.0.0001) GO TO 34 BBSM = BBS(M)/AKS BBS(K) = BBSM*BKS-CCS(M) CCS(K) = BBSM*CKS FFS(K) = BBSM • FKS-FFS(M) IF(ABS(BBS(K)).LT.10000.00)GO TO 33 BBS(K) = 0.0001 *BBS(K) CCS(K) = 0.0001 • CCS(K) FFS(K) = 0.0001 *FFS(K) GO TO 33 34 BBS(K) = BKS CCS(K) = CKS FFS(K) = FKS 33 CONTINUE IF(K.EQ.l)GO TO 53 IF(NSW.LT.NSl.AND.K.GT.NSW)GO T O 53 IF(ABS(AWS).LT.0.0001) GO TO 54 BBWM = BBW(M)/AWS BBW(K) = BBWM*BWS-CCW(M) CC W(K) = BBWM* C WS FFW(K) = BBWM*FWS-FFW(M) IF(ABS(BBW(K)).LT.10000.00)GO TO 53 BB W(K) = 0.0001 • BB W(K) CCW(K) = 0.0001 *CC W(K) FFW(K) = 0.0001 *FFW(K) GO TO 53 54 BBW(K) = BWS CCW(K) = CWS FFW(K) = FWS 53 CONTINUE IF(K.EQ.l) GO T O 23 IF(NSJ.LT.NSl.AND.K.GT.NSJ)GO TO 23 IF(ABS(AK).LT.0.0001) GO TO 24 BBM = BB(M)/AK BB(K) = BBM*BK-CC(M) CC(K) = BBM*CK FF(K) = BBM*FK-FF(M) IF(ABS(BB(K)).LT. 10000.0) GO T O 23 BB(K) = 0.0001*BB(K) CC(K) = 0.0001 *CC(K) FF(K)=0.0001*FF(K) GO TO 23 24 BB(K) = BK  CC(K) = CK FF(K) = F K 23 CONTINUE 28 CONTINUE  C The bond values are calculated in this set of equations. B1(NS1 +1,1) = FF(NS1)/BB(NS1) B2(NS1 + 1,I) = FFS(NS1)/BBS(NS1) W(NS 1 +1,1) = FFW(NS 1)/BBW(NS 1) DO 25 KX=1,NS2 K = NS1 + 1-KX M = K-1 L=K+1 BX = (FF(M)-CC(M) *B 1(L,I))/BB(M) BXS = (FFS(M)-CCS(M) * B2(L,I))/BBS(M) WX = (FFW(M)-CCW(M)*W(L,I))/BBW(M) B2(K,I) = AMIN1(BXS,B2(L,I)) IF(NSW.LT.NSl.AND.K.GT.NSW)GO TO 31 W(K,I) = AMIN 1 (WX, W(L,I)) GO TO 37 31 W(K,I) = CPW 37 CONTINUE IF( J.LT.NT4)W(K,I) = 0.00 IF( J.LT.NT3)B2(K,I) = 0.00 IF(NSJ.LT.NSl.AND.K.GT.NSJ)GO TO 29 B1(K,I) = AMIN1(BX,B1(L,I)) GO TO 241 29 B1(K,I) = C P P 241 CONTINUE 25 CONTINUE 20 CONTINUE DO 122 K=1,NS B1(K,1) = B1(K,NTZ) B2(K,1) = B2(K,NTZ) 122 W(K,1) = W(K,NTZ) 100 CONTINUE DO 923 K=1,NS BW(K, 1) = B 1(K, 1) + W(K, 1) 923 CONTINUE  C This section causes the results to be printed. WRITE(6,200)SIG,RF,CIP,NCSPAC,GAM,F1,XH,XK,NSS,NT1,V0,F2, /MS,MW,EX 200 FORMAT(/2X,'SIG = ',F12.5,2X,'RF = ',F12.5,2X,'C = ',F5.2,2X, / 'C I = ',I4,2X,'GAM = ',F6.3,2X,'F1 = ',F10.5, / 2X,'XH = ',F10.5,2X,'XK = ',F6.4,2X,'NS = ',14, / 2X,'NT1 = ',I4,2X,'V0 = ',F12.5,2X,'FS = ',F12.5, / 2X,'MS = ',I4,2X,'MW = ',I4,2X,'EX = ',F10.5) WRITE(6,201) (BW(JL,1),JL= l,NHH),BW(l,l),BW(NSl,l) WRITE(6,201) (B1(JN,1),JN= l,NHH),Bl(l,l),Bl(NSl,l) WRITE(6,201) (W(JY,1),JY= 1,NHH),W(1,1),W(NS1,1) WRITE(6,201) (B2(IG,1),IG= 1,NSH),B2(1,1),B2(NS1,1) 98 CONTINUE STOP END  

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