‘§§i¥§f9§?.i§‘§'t ‘ , K I t :3 44251 322;}? 9“ ' .jww ."'5f: 1&5‘»v.‘x‘<‘r?=1u§'~'v. - v WARRANT VALUATION IN RATIONAL SECURITY MARKETS Thesis for the Degree of Ph. D. M£CHIGAN STATE UNIVERSITY HERBERT IOEL WHNRAUB 1972 This is to certify that the thesisentitled 3.111: V11 t_.;, .,, .. , 1 Sz.:cw;'~t;; AI' ’VJF presented by :3 L I"? P'f‘ t J C‘ 31 ,3. ."P :1 13;: H has been accepted towards fulfillment of the requirements for JKD Jememflgttj‘" Fl‘h. Him Mo: professor DateflWr/f/ /; 72— 0-7639 ‘w amounts av 7" " HUM; E SUNS' F “W BINDEHY INC. V . -:mnv amosns flatulcllfll ABSTRACT WARRANT VALUATION IN RATIONAL SECURITY MARKETS BY Herbert Joel Weinraub Problem The valuation of stock purchase warrants has been studied based on three objectives: First, to deve10p a valuation model that generates a lower boundary value for a given warrant at a given time. Second, to describe and justify a new approach to the incorporation of warrants in the investment decision. Third, to apply the model in a market test to determine the frequency of warrant under- valuation, and the factors that influence the probability of acquiring undervalued warrants. Model develOpment The analysis utilizes the two-parameter model, with the standard deviation and the arithmetic rate of return specified as the parameters. Additionally, the concept of a risk-free rate of interest at which an investors funds can be all or partially employed is incorporated in the model. . Future price expectations can be transformed into Herbert Joel Weinraub a probability analysis that can be used to generate the two parameters for both a common stock (A) and its associated warrant (A'). Since the primary objective is the develop— ment of the warrant premium, it is initially assumed that the warrant sells for its base (non-premium) value. Under the base value assumption, a warrant will lie above and to the right of its associated stock in the two—parameter space. o I / I .. , . / l f A/ M ’ /‘ I / / I / / / ./ / I)" / I I / / / ’ / / / l / / I / / l/-’ ,/ l7 _,__._...-______--_.., ._.p 0 l The model to be develOped is predicated on the assumption that an investment in a warrant should be com- bined with the risk—free rate of interest, and together should be considered as an alternative to an investment in the common stock. If the anticipated risk—return rela— tionship for a stock is deemed acceptable, the alternative "a: .17 In.‘ . u-; n.-' we, «Thai-‘6 5.5.1.. . . t.~|flfl’|’ta3d _ Wit .1. RV. PM .‘fi :\ Herbert Joel Weinraub warrant-i strategy should then be considered with the objec- tive of pr0portioning funds in a manner that achieves a higher u, at no increase in risk. This situation will be possible whenever the warrant occupies a point in the two- parameter Space to the right of the ray iA". The valuation model deve10ped is: P0+D (l + i)n Pw* = the price that places the warrant on the same ray from i as the associated common stock. Pw* = P5 - with, PS the current market price of the common stock. Po the current Option price of the warrant. D = the total dollar dividend per share, over the time horizon of the investment, on the common stock. i = the risk-free rate of return. n = the time horizon of the investment. At Pw* an investor can pr0portion his funds between A' and i in a manner that equates the warrant strategy with the stock, and represents the minimum value a warrant can rationally have at a given time. with any market price less than Pw*, an investor can achieve a higher p with the warrant-i combination, at no increase in risk, relative to the common stock. If an investor's specified degree of acceptable Herbert Joel Weinraub risk is 0A, he will seek that investment on the GAL line furthest to the right. Whenever the market price Of a war— rant is less than its Pw*, a warrant-i combination can be purchased which is to the right of the stock on the GAL line, hence, rational investors will purchase the warrant in lieu of the stock, and this action will tend to force the price of the warrant back to a minimum of Pw*. Market Test Methodology A market test is deve10ped to determine the prac- tical significance of the valuation model. There are 35 randomly generated weeks between the dates Of January 1, 1961 and June 1, 1971. Stock and warrant price Observa— tions are selected from the closing prices of the respec- tive securities on the Friday of each week. The risk-free rate used is the prevailing maximum interest rate on savings accounts Of commercial banks. The investment holding period is the remaining length of a warrant's life. For warrant's Whose Option price changed before maturity, the Option price prevailing on the date examined is used and the war- rant is assumed held until the date the terms changed. In addition, the last Specified Option price is used with the assumption that the warrant is held to eXpiration. A r‘r‘ ( “(I rt [3 Herbert Joel Weinraub warrant is considered undervalued if its market price is below Pw* with either Option price, but counted only once if undervalued under each Option price. Test Results It was found that 4.7 per cent of all stock price Observations had associated undervalued warrants, and 11.3 per cent of all stock prices above their respective Option prices had associated undervaluations. The probability Of undervaluations increased when the remaining maturity Of the warrant was less than two years, and particularly in— creased with less than one year to maturity. There is also an increase in the percentage of undervaluations when the associated stocks have a high Ps/PO ratio. Over two-thirds Of all undervaluations occurred under the above maturity and Ps/PO conditions, although such conditions account for less than 13 per cent of total Observations. The general state of the market (bull or bear) has an effect on the frequency Of undervaluations, with less occurring in bear, more in bull, than would be eXpected if the events were independent. Of all tested dates 63 per cent had at least one undervaluation. The conclusion reached is that warrant undervaluation occurs with suffi— cient frequency to merit investor attention. WARRANT VALUATION IN RATIONAL SECURITY MARKETS BY Herbert Joel Weinraub A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department Of Accounting and Financial Administration 1972 © Copyright by HERBERT JOEL WEINRAUB l 9 7 2 4“ f\ (3 I") #01 71 ACKNOWLEDGMENT S It is impossible to honor the countless people who in some way contributed of themselves and who therefore made my education and this research possible, however the following people deserve my particular and sincerest appreciation. Professor Alan Grunewald, Chairman of my disser- tation committee, who provided the encouragement, guidance, and suggestions that led to the successful culmination Of this research. Professor Alden Olson, whose enthusiasm and valuable suggestions, both as a teacher and member of my dissertation committee, gave me insight into the field of investments. Professor Gardner Jones, Chairman Of the Department of Accounting and Financial Administration, for his encour- agement and meticulous editing of this paper while serving on my committee. Professor Douglas V. Austin, Chairman of the Finance Department, University Of Toledo, for his many valuable contributions toward my professional development. Jo McKenzie, who skillfully typed the final draft of this manuscript. And to my beloved wife Donna, whose faith, sacri- fice, and love made the completion of my graduate work possible, and to whom this volume is dedicated. Of course, I, alone, am responsible for any errors or omissions that may appear on the following pages. ii TABLE OF CONTENTS LIST OF TABLES O O O O O O O O C O O O O O O O 0 LIST OF FIGIIRES O O O O O O O O O O O O O O O 0 Chapter I. II. PURPOSE, TERMINOLOGY, AND REVIEW OF LITERATWE . O O O O O O O O O O 0 Purpose of Study . . . . . . . . . . Review Of Warrant Terminology. . . . Stock Purchase Warrant . . . . . . Associated Common Stock. . . . . . Option Price . . . . . . . . . . . Expiration Date. . . . . . . . . . Dilution Protection. . . . . . . . Theoretical Price. . . . . . . . . Warrant Premium. . . . . . . . . . Leverage . . . . . . . . . . . . . Detachable . . . . . . . . . . . . Marketable Warrant . . . . . . . . Arbitrage. . . . . . . . . . . . . Role of Warrants . . . . . . . . . . Background and Review of Literature. DEVELOPMENT OF THE MODEL . . . . . . . Introduction . . . . . . . . . . . . DevelOpment Of the Basic Model . . . Generalization of the Model. . . . . Implications Of the Basic Model. . . Model Modification Due to Dividend . Model Modification Due to Multi- period Analysis . . . . . . . . . . Summary of Valuation Significance. . iii Page vi H \JUIb.b.b4>uJu:wtvboanapia l9 19 22 33 35 36 37 43 Chapter III. IV. RESEARCH DESIGN AND ANALYSIS OF RES [HITS C C O O O C O O O O O O O O O 0 Market Test Methodology. . . . . . . . . Test Results . . . . . . . . . . . . . . The Risk-Free Rate of Interest . . . . . Relationship Of Pw* tO Other Valuation Concepts. . . . . . . . . . . General Market Test. . . . . . . . . . . The Professional Investor Hypothesis . . Market Test Conclusions. . . . . . . . . SUMMARY, IMPLICATIONS AND QUALIFICATIONS . Model Development. . . . . . . . . . . . Lower Boundary Valuation . . . . . . . . Market Test Methodology. . . . . . . . . Test Results . . . . . . . . . . . . . . Qualifications . . . . . . . . . . . . . Components Of the Model. . . . . . . . . The Need for Additional Research . . . . Contributions Of the Study . . . . . . . BIBLIOGRAPHY. . . . . . . . . . . . . . . . . . . . APPENDIX A-l THE VARYING DEGREES OF UNDERVALUATION, LISTED CERONOLOGICALLY O O O O O C O O O 0 TIME TO MATURITY IN DESCENDING ORDER FOR EACH UNDERVALUED WARRANT. . . . . . . . . UNDERVALUATIONS RELATIVE TO THE NUMBER OF LISTED WARRANTS. . . . . . . . . . . . UNDERVALUATIONS RELATIVE TO THE NUMBER OF WARRANTS WHOSE ASSOCIATED STOCK PRICE WAS GREATER THAN THE WARRANT OPTION PRICE THE RELATIONSHIP OF THE STOCK PRICE TO THE WARRANT OPTION PRICE, PER YEAR, FOR EACH UNDERVALUATION . . . . . . . . . iv page 45 45 47 51 53 58 6O 62 64 64 66 67 68 69 7O 71 72 74 76 77 78 79 80 2-4 LIST OF TABLES Probability Table for Stock. . . . Probability Table for Warrant. . . Probability Table for Stock (Strategy A) . . . . . . . . . . . Probability Table for Warrant-l Combination (Strategy B) . . . . . Strategy A, The Stock. . . . . . . Strategy B, The Warrant With i . . The Consequences of Undervaluation Stocks Above Their Option Price. . Page 28 29 30 31 42 42 so 57 LIST OF FIGURES Placement Of the Stock and Warrant in the Two-Parameter Space . . . . . . The Effect Of Utility Curves . . . . . The Effect Of the Risk-Free Interest . The Effect Of PrOportioning Funds. . . The Effect Of a Changing Warrant Price Comparative Returns at Varrying Warrant Prices . . . . . . . . . . . . Relationship of Valuation Theories . . The Risk-Return Relationship with Varying Warrant Prices . . . . . . . . Vi Page 23 24 25 26 27 32 54 65 CHAPTER I PURPOSE, TERMINOLOGY, AND REVIEW OF LITERATURE Purpose Of Study The primary Objective of this study is to develOp a model of warrant valuation that establishes a minimum price at which a given warrant at a given time can ration— ally trade. It will be demonstrated that a warrant market price below that determined by the model leads to an in- vestment strategy that is superior to an investment in the warrant's associated common stock. A second Objective is to describe and justify an investment strategy for warrants which represents a new approach to the incorporation Of this vehicle in the investment decision. The final Objec— tive is an application Of the developed model in a market test to determine if irrational pricing leads to substan- tial Opportunities to acquire undervalued warrants, and under what circumstances, if any, are these Opportunities most prevalent. Review of Warrant Terminology Stock Purchase Warrant: A stock warrant is a 1 2 security giving the bearer an Option to purchase a speci— fied number Of common stock shares, at a specified price, on or before a specified date. Marketable warrants trade primarily in the over-the-counter market and on the American Stock Exchange. The New York Stock Exchange has recently allowed listing for a few select warrants. The principle methods by which warrants come into existence are: as part Of the terms of a merger agreement; attached to an issue Of long-term bonds; as part of the lending agreement for intermediate-term funds; as partial compensa- tion for venture capitalists. Associated Common Stock: The associated (related) common stock is that security which must be surrendered by the corporation when the Option of a given warrant is exercised. Option Price: The Option (exercise) price, is the specified sum the warrant holder must pay to the corpora- tion for each share Of common stock received if the warrant is exercised. The Option price may be a constant through- out the length of the warrant's life, or may change periodically at specified dates. Expiration Date: The expiration date of a warrant is the date its terms eXpire. If a warrant remains un— exercised at the close of trading on the expiration date 3 it becomes valueless. Warrant's at issuance can have original expiration dates ranging from a few months to perpetuity. Marketable warrants are normally intermediate— term securities with original life spans of three to ten years. Dilution Protection: Dilution protection is a pro- vision in a warrant's terms designed to protect a warrant holder from loss in the value Of the Option due to stock splits, stock dividends, or additional common stock financing. If a warrant is fully protected against dilu— tion, the number of shares each warrant entitles its bearer to purchase is adjusted whenever there is a change in the amount Of common stock Outstanding. The majority Of warrants are fully protected against dilution, some partially protected, and a few have no protection. Theoretical Price: The theoretical or base price Of a warrant is commonly defined as the current market price Of the associated common stock minus the current Option price Of the warrant, multiplied by the number Of shares the warrant entitles the holder to purchase. Warrant Premium: The premium is the excess over the base (theoretical) price that investors pay for a warrant. Premiums arise due to the willingness of in- vestors to acquire the potentially advantageous leverage 4 that warrants possess, relative to their related common stock. The establishment of the correct premium under a given set Of circumstances has been the focus Of this and past studies in this area. Leverage: The concept Of leverage can be variously defined. As defined here, leverage is the percentage move- ment in the market price of a warrant relative to a one per cent change in the market price of its associated common stock. The leverage is a function of the Option price which is a fixed cost, as Opposed to the variability Of common stock prices. Detachable: A detachable warrant can be removed from the security with which it was originally issued, and traded separately. If a warrant is nondetachable, a trading market for the warrant alone cannot be established. Marketable Warrant: A marketable warrant is one which is detachable. Arbitrage: For the purpose of this study, arbi- trage is defined as the process Of simultaneous purchase and sale of the same or equivalent securities to take advantage of a price discrepancy, when such action will result in immediate profit to the arbitrager. Role Of Warrants The creation and distribution Of warrants arises for a variety of purposes. The primary purpose Of issuance has been as "sweeteners" added to long-term debt to make the issue more salable. For lessor known corporations, or in periods Of relatively tight money, many firms can attract capital at reasonable prices only by including warrants as part Of the security package. In addition, investors normally accept a lower bond yieldwhen warrants are attached, due to the potentially advantageous equity par— ticipation Offered by the security. The effect is to lower the cost Of long—term debt to the corporation, relative to what it would have been with a straight bond issue. Warrants also represent delayed equity financing. When the price of the stock rises above the Option price, the corporation is assured that by the expiration date, at the latest, the warrants will be surrendered together with a specified sum Of cash, and common stock issued in its stead. If a corporation forecasts a need for external equity funds in the future, the firm can arrange the Option terms such that the eXpiration date coincides with the period when the cash inflow is required. In this manner, equity financing in a future period can be arranged as part of a present bond issue, lowering the cost of that 6 issue as well as part Of the flotation costs of the new common stock. A major risk for the corporation is that the price Of the stock may not rise or be maintained above the Option price, and therefore, the warrant will not be exercised and additional funds not received. An additional risk is the stock price rising substantially above the Op- tion price necessitating the sale Of the common well below then current market levels. Warrants may also arise as part Of the terms in a merger agreement. For the issuing corporation the use of warrants in this context represents a deferred payment to the other party. This was particularly true before the effect Of warrants was required to be included in the reported earnings per share on a fully diluted basis. Prior to 1969 the potential increase in common stock was not included when reporting the earnings of the issuing corporation. Warrants have also been used as partial compensation to underwriters and other venture capitalists by a rela- tively new and/or unknown corporation when first going public. 1Opinions Of the Accounting Principles Board, Earnings Per Share, Number 15, May 1969. 7 With the recent issuance Of warrants by AT&T and Tenneco, and with the inclusion Of these securities on the New York Stock Exchange, it seems probable that the use of this security may increase in the future, and in particular, their use by the larger and less speculative corporations. From an investment perspective, the range and availability of warrants is likely to increase, and it seems an appro— priate time to add to the knowledge of prOper warrant valuation. Background and Review Of Literature Giguere.2 Giguere's basic hypothesis is that a mathematical relationship exists between the price of a stock and the price of its associated warrant. He attempts to discover this relationship and determine its predictive accuracy. "In fact, the value of a soundly based, reliable method of evaluating warrants would be to supply us with criteria against which actual market prices can be judged.”3 Giguere states that certain warrant character- istics such as short life to expiration, a variable option 2G. Giguere, "Warrants, A Mathematical Method of Evaluation," The Analysts Journal, NO. 5, 17-25 (November 1958). 3Giguere, p. 17. 8 price, or a thin market, are factors that can distort the mathematical relationship between the securities. TO avoid these problems he selects two perpetual warrants which have constant Option prices. He then plots the prices of the stocks and their warrants and determines their relationship P2 to be, = 33', a parabola with its vertex at the origin. W = the warrant value, P = the price of the stock, a = the Option price. Giguere uses two "ideal" types to determine an equ- ation Of best fit, then applies this equation to other war- rant types tO see if they are under or over priced. There is no theory which explains the reasons for the warrant’s behavior, and two highly specific and uncommon examples are used to formulate a general theory. Morrison.4 Morrison bases his valuation model on a break-even method of analysis. He attempts to determine if, at any given time, a commitment to a warrant is prefer- able to a commitment to its associated common stock. The formulation derived is: A = "“E7— + Z with l-X Y I 4R. J. Morrison, "The Warrants or the Stock", Analysts Journal, Vol. 13, NO. 5 (November 1957), p. 52. 9 = break-even point Of stock. warrant Option price. current market price Of warrant. current market price of stock. = total dividends anticipated per share during investor's time horizon. N~<> ll As an example assume: W = $3.00, Y = $6.00, X = $4.00, Z = annual dividend Of 30 cents. $3.00 A = 1 - $4.00/$6.00 + $1.20 = $10.20 Thus, for warrants to be a superior vehicle for the employ— ment Of $1200, the common must increase in price from $6.00 to $10.20. The formula can also be arranged to give the current worth of a warrant with any given estimate Of the future price Of its related stock. Morrison recognizes that the risk relationship of the two securities is not incorporated in his model. In addition, only the mean value Of the two alternative invest- ments are equated, with other possible values ignored. Shelia-5 Shelton intuitively selects six factors he believes may influence the premium investors pay for warrants. By using regression analysis he finds three of the six materially affect the warrant—stock price relation- ship. In order Of importance: the dividend yield Of the 5J. P. Shelton, "Relation of the Price Of a Warrant tO the Price Of its Associated Stock," Financial Analysts Journal, J23:143-51 May; 88—99 July 1967. 10 common; its listing (on the American Exchange or over-the- counter); and the warrant's remaining life span. For ex— ample, the greater the foregone dividend the lower the war- rant's premium, and the longer the life span the larger the premium. More than half a warrant's premium is unexplained by the tested variables, and Shelton attributes this to speculative emotions. His formula is designed to give apprOpriate weight to the factors he found significant. His model is: (.47 - 4.25 yield + .17, if listed) Y ’x longevity in months 3 72 Its interpretation is, initially place the warrant value 47 per cent Of the distance from the tOp to the bottom of the two extremes that arbitrage defines, subtract percentage points if its associated stock pays a dividend, add 17 per cent if the warrant is listed, then multiply the resultant figure by the fourth root Of the life span factor. Sprenkle.6 The Sprenkle study attempts to deter- mine from warrant prices, what Option investors' 6C. M. Sprenkle, "Warrant Prices as Indicators Of Expectations and Preferences", Yale Economic Essays, ll expectations are toward the expected mean and standard deviation of a warrant's associated common stock. A second Objective is to Obtain quantitative measures of investors' risk preferences. He finds that two parameters can be estimated, expected variance and risk preference, and incorporates into his valuation model the assumption that future common stock prices are log-normally distributed. His mathematically determined non—premium warrant value is based upon the assumptions that the expected price Of the stock as well as the distribution Of stock prices above the Option price determines a warrant's basic value. Warrant value equals: X—aifX>a (l) 0 if X < a with a = Option price, and X any expected stock price. If f(X), the distribution of possible stock prices on a target date, is continuous, then; E(Pw) = 3(x - a) f(X) dx (2) a with E(Pw) = the expected warrant value. Equation (2) specifies that the expected price of the stock minus the Option price is equal to the warrant's basic value. With his assumption that future stock prices are 12 log-normally distributed, that is, f(X) is log-normal with variance LnX equal to 02, and mean LnX equal to u, equation (2) takes the form of, Q E(Pw) =IH exp -1/2 [(LnX - u/{lz dx (3)- 3 However, warrant's have a leverage factor compared to their Associated stock. Whatever the (0,“) of the stock, its warrant will have both greater expected return and greater risk. An investor will pay E(Pw) as determined by (3), only if he is neutral toward risk. The price an investor will normally pay for a warrant is equal to the expected value Of that warrant plus a premium determined by what he is willing tO pay for the warrant's leverage. Sprenkle compares actual warrant prices to their E(Pw) to determine what the marginal Option buyer's expec-' tations are toward the future risk Of a common stock, as well as his preferences for that risk. His warrant premium is derived not for the purpose Of finding a rational price for the security, but for its usefulness in indicating Option investors' risk expectations and preferences. Samuelson.7 Samuelson explicitly considers the 7P. A. Samuelson, "Rational Theory of Warrant Pricing", Industrial Management Review, Vol. 6, NO. 2 (Spring, 1965), pp. 13—39. 13 additional worth of an American warrant due to the privilege of being able to convert it at any time on or before matu- rity. His final valuation formula is, (Y-l)Y-l F(x)= YY XY . Y=C_1 With X the current price of the common stock, and C that point where it is no longer advantages to hold the warrant rather than the stock, and therefore, where it sells for its basic (non-premium) value. The formula applies only to the Special case of a perpetual warrant whose associated stock is assumed to have a log-normal distribution of future prices. Y Warrant value Z ,l/’ /'/l ,/// B /, F(x’m Cm // F(X,25) / F(X,4 /‘/ F(X,1) // , | C25 we C4 ——~"""‘ C1 0 ' X common stock price 14 For example, C25 is the point where a warrant with 25 years of remaining life will sell without a premium. He finds that at the maximum, when a common stock sells for four times its Option price, the premium on its associated warrant will have disappeared. The rationale is that arbi- trage will prevent a warrant from selling at a price lower than the locus of points represented by OAB. In addition, as a stock's price increases relative to its Option price, the leverage inherent in the warrant diminishes, and by the time it sells for a multiple of four times its Option, the leverage remaining in the warrant is too small to in— duce investors into paying a premium. The equation Of the line between the origin and the CT intercept is represented by Y, and can be empirically estimated by regressing the log warrant price against the log common price, Y being the regression coefficient. Ayers.8 Ayers attempts to illustrate how warrant prices reflect the risk preferences of investors. He . . 9 . assumes the average investor to be a risk averter. His 8H. F. Ayers, "Risk Aversion in the Warrant Mar— kets," Industrial Management Review, Vol. 5, NO. 1 (Fall, 1963), pp. 45—53. 9Risk aversion is defined as the unwillingness to accept additional risk without additional compensating return. 15 warrant valuation model is, w = ” D [S(t) - E(ty] Pr (S(t)) (1) S E "M where: D a discount factor less than one resulting from risk aversion. W = current expected warrant price. S(t) stock price at time t from now. E(t) Option price at time t from now. Pr(S(t)) = the probability Of s att. Ayers attempts to find the functional form of D. He assumes that future stock prices will have a log—normal distribution. The probability density function for the common stock is, S(t) 2 Sm] _ 1 .n[__]- .1. f Ln [:80 — 2not exp - SO (2) 2021: an unbounded random walk with trend at. His major departure from other studies is the incor- poration Of his assumption of risk averting behavior of in- vestment company managers. He utilizes the Farrar Objec- 10 tive function, where, U = R - A02 with, (3) 10 D. Farrar, "Investment Decision Under Uncer- tainty", Ford Foundation Prize Doctoral Dissertation Series, 1961. 16 C.‘ II expected utility. 5!! ll expected return. A = the coefficient of risk aversion. 02 = expected variance. Equation (3) is a postulated model of an investment manager's decision process. Ayers says that if S (stock price) in equation (2) represents a portfolio rather than a single stock, and if Farrar's model is accurate, investors expect an exponential rate of growth over time, and this rate increases with risk. Therefore, D = e"rt (4), with r = the expected return Of the warrant. Combining equa— tions (1), (2), and (4), he derives his final model, w = “2" [e—rt, was] (5) with We(t) equal to the expected value of the warrant at time (t). The argument Ayers uses to explain the establish- ment of a warrant's market price is predicated on the assumption that investors do not regard any given warrant as an independent investment Opportunity, but the warrant and its common stock together as a single composite Oppor- tunity. He assumes that investors will allocate their funds between the two securities in a manner that maximizes their utility. He claims that investors who are described 17 by the Farrar Objective function dominate the market, so it is their preferences which determine a warrant's market price. ghgg 11 As does Samuelson, Chen incorporates into his model the concept that an American warrant's value is increased due to its being convertible prior to maturity. All things equal, the longer the remaining life Of a war- rant, the greater its theoretical value. In each period the investor must make the choice of either converting the warrant or holding it for one more period. A random walk in log-normal distribution is assumed for future stock prices, and a dynamic programming technique is used to develOp his model. Shelton and Giguere utilize the technique of regres- sion analysis to develop their respective models. They attempt to formulate an equation, which describes the "typical" warrant—stock price relationship that has existed in the past. Each author assumes that if his model matches the warrant market prices that are Observed, he has deve10ped a theory of warrant valuation. At best their methodology leads to a theory on how investors' guide 11A. H. Y. Chen, "A Model of Warrant Pricing in a Dynamic Market", The Journal of Finance, Vol. XXV, NO. 5 (December, 1970), pp. 1041 — 1059. 18 their actions, not to how warrant's should be theoretically valued. Also, as Shelton admits, his model describes less than half of a warrant's premium fluctuation. Samuelson, Ayers, Sprenkle, and Chen each requires knowledge of the future price action Of common stocks. TO determine the value Of a warrant at time t, the probable price Of its associated stock at t+n must be known, and each assumes that future stock prices have a positive trend and log-normal distribution. CHAPTER II DEVELOPMENT OF THE MODEL Introduction A warrant has value because it is substitutable for the stock, and changes in that value are dependent upon changes in the stock's future market price. The investment characteristics of the two securities are, however, quite different. Compared to its related stock, a warrant normally has a higher expected return in conjunction with a greater potential variance of return, or risk. The greater market price volatility inherent in a warrant is a function Of the fixed Option price required to purchase the associated stock. The pricing of a warrant on any given date is normally a function Of marginal investors' expectations as to the future price performance of the related stock. A premium is established because of the desirability of Obtaining the security's leverage charac— teristics, with the amount Of the premium set so that the greater expected return of the warrant is sufficient to compensate marginal investors for the additional risk 19 20 assumed. The warrant valuation model to be deve10ped in this study differs substantially in concept, methodology, and conclusions from the papers outlined in Chapter One. This study does not attempt to derive a valuation for a given warrant based upon either past warrant-stock price relationships or future stock price predictions. The pri- mary Objective is to develOp a model which generates a rational lower boundary valuation for any given warrant in relation to the current market price Of its associated stock. The model is based upon the precept that a warrant and its associated stock should be considered as two alternative investment strategies, one consisting of the common stock alone, the other of the warrant in combination with a risk-free rate of interest. Considering these strategies as independent Opportunities for investment, and utilizing the two-parameter model, it will be demonstrated that at any given time there is a unique warrant valuation that equates these strategies in terms of their reSpective standard deviations and expected rates Of return. At this unique valuation the alternative strategies will occupy the same point in the two-parameter space, and can therefore be considered as identical assets. It will also be demonstrated that when a warrant's market price is below 21 that generated by the model, a higher return can be attained with the warrant—interest rate strategy than that possible with the associated stock, at no increase in risk. A rational investor will always seek to maximize return sub- ject to a given level of risk, hence the warrant valuation derived will represent a lower boundary below which a mar- ket value cannot be rationally supported. In develOping the basic model the following assump- tions are made: (1) Rational investors. (2) A two security world consisting Of a common stock A, and its related warrant A'. (3) The common stock pays no dividend. (4) One hundred per cent Of investable funds must be employed. (5) A' does not initially sell at a premium. The above assumptions are not critical. Their purpose is to simplify the analysis while develOping the basic model, and each will be later removed, with the attending conse- quences discussed. It is additionally assumed that the warrant when purchased is held to maturity, and when a non-sustainable price discrepancy occurs the adjusting mechanism of arbitrage is a change in the warrant price, not in the stock price. The consequences Of these latter 22 assumptions will also be discussed. The derived warrant value will be labeled Pw*. Development Of the Basic Model The analysis utilizes the two-parameter model, with the standard deviation and the arithmetic rate Of return specified as the parameters. Incorporated in the model is a risk-free rate of interest at which an investor's funds can be all or partially employed. When a range Of potential values and their respective probabilities Of occurrence is established for the common stock A, the expected return (u) and the standard deviation (0) of return can be calculated. Since it has been assumed that the warrant, if purchased, will be held to maturity, the time horizon for the estimates on the stock should coincide with the remaining length of its associated warrant's life. Withlland<5calculated the stock can be plotted in a two-parameter space (Figure 2-1). Assuming the warrant is currently priced at its base value, the Option price can be subtracted from the values esti- mated for the stock, thereby generating the required infor- l mation for calculating the H and 0 for the warrant. When plotted, A' will lie above and to the right of A in the two-parameter space (Figure 2—1). 1Constrained by a lower limit Of zero. 23 Figure 2-1 PLACEMENT OF THE STOCK AND WARRANT IN THE TWO-PARAMETER SPACE There is no indication which of the two securities a given investor will prefer. A' promises a greater ex- pected return, but more risk must be assumed, as measured by 0, to be attained. The determination of which security an investor will favor requires the specification Of his utility preference toward the tradeoff between risk and return. If his utility map is as depicted in Figure 2—2, the investor will chose to buy security A, since the com- bination Of risk and return for that security places him on his highest attainable utility curve. 24 Figure 2-2 THE EFFECT OF UTILITY CURVES An alternative approach is to specify the degree of risk willing to be assumed, then choosing that security which has the greatest u for that risk level. If (3A is specified, the security lying furthest to the right on the horizontal line cAL will be chosen. Since a two security world has been assumed, the common stock A would be chosen over A'. A risk-free investment incorporated into the analysis modified the two-parameter model to that deve10ped 2 by Sharpe. The rays extending out from the risk-free rate 2W. F. Sharpe, Capital Asset Prices: A Theory Of Market Equilibrium Under Conditions of Risk", Journal Of Finance, Vol. XIX, NO. 3 (September, 1964), pp. 425-442. ia: 5811' 317.? 0 pa: iii In “r1 he 25 i are what Sharpe labeled capital market lines, and repre- sent the division Of an investor's funds between i and the respective securities illustrated (Figure 2-3). For ex- ample, if part Of the investable funds are placed in i and part in A, the (u,0) combination the investor will receive falls somewhere on the ray between i and A. If funds are borrowed at rate i, for investment in A, the (u,o) combina- tion will fall on the ray beyond point A. IfcIA is the specified degree Of risk that an investor wishes to assume, he will place 100 per cent Of his investable funds in the common stock, with none invested or borrowed at i. Figure 2-3 THE EFFECT OF THE RISK-FREE INTEREST 0A / L 26 The same reasoning explains the ray i to A' and it is seen that a combination Of i and A' exists that will place the investor at point B on the ray iA' (Figure 2-4). Figure 2-4 THE EFFECT OF PROPORTIONING FUNDS r4 0A / If the correct iA' investment combination is under— taken, the expected return the investor will achieve is greater than what can be achieved by investing in the com- mon stock, and with no attending increase in risk. A rational investor will seek that strategy furthest to the right on the qAL line, thus increasing the demand for portfolio B relative to the demand for the common stock A. Since A' is part of B, the demand for the warrant will in- crease relative to the demand for its related common stock. 27 If Ps is held constant, Pw will increase due to the higher relative demand. The increase in the price of the warrant will move it to the left in the two-parameter space, with the process continuing till the price of the warrant is such that both it and the stock lie on the same ray from i (Figure 2-5). At this point a combination of A' and i can be purchased which equates that investment strategy with the common stock and will result in indifference between investing in the common stock or warrant strategy. Figure 2—5 THE EFFECT OF A CHANGING WARRANT PRICE 0A / ’ ./ (Ath*) / ,/ I A'/ ( mt / // / / / / / ,/ A,B, B/ .631 L /// // 17 u TO aid in illustrating the concepts involved, and in develOping the Pw* model, a hypothetical distribution of common stock prices and other relevant information is 28 set forth below.3 The method Of probability analysis is used, by which an investor at time period t eStimates the values a security may have in period t+n, together with their attendant probabilities. The expected rate of return (a) and the standard deviation (O) Of the return can then be computed. If PO equals $20 the non-premium Pwt will be TABLE 2-1 PROBABILITY TABLE FOR STOCK Assume Pst = $30 Pst+n Gain Per Share ROR(Z) Prob. Expected Gain $45 $15 50.0 .10 1.50 40 10 33.3 .20 2.00 35 5 16.7 .40 2.00 30 0 0.0 .20 0 25 (5) —l6.7 .10 (.50) Pst = The current price of the stock. ROR = The rate of return. Po = The warrant's Option price. Put = The current price of the warrant. uA = E(PiRoRi) = 16.77. 0A = v/fPi(RoRi-uA)2 = 18.26 $10, assuming each warrant entitles the holder to purchase one share Of stock. The probability distribution for the warrant is shown in Table 2-2. 3This is one Of two major distribution classifica- tions that affect the model. The effects of the other distribution will be discussed in a later section. 29 TABLE 2-2 PROBABILITY TABLE FOR WARRANT .Pwt+n Gain Per Warrant ROR(Z) Prob. Expected Gain $25 $15 150.0 .10 1.50 20 10 100.0 .20 2.00 15 5 50.0 .40 2.00 10 0 0.0 .20 0 5 (5) -50.0 .10 (.50) uA' - 50% oA' = 54.77 The Objective is to find a Pw* that equalizes the two alternative investment strategies to the extent that they occupy an identical point in the two—parameter space. That is, to find a warrant price such that thelland o of the warrant-i combination is identical to theliand o of the common stock. With the type of probability distributions hypothesized above, this Objective can be satisfied by equating any two of the possible outcomes. X(LP — Pw*) + [F - (x - Pw*)l (i) T (l) X(Ep - Pw*) + [F - (x - Pw*)] (i) G (2) X = the number of warrants to be purchased. Lp = the lowest possible price the warrants can attain. Ep = the expected price at time t+n per warrant. F = funds to be invested Pw* = the current price of the warrant, consisting of its base price plus premium, that will place the warrant on the same ray from i as the stock. i = the risk-free rate. possible loss in dollars with the stock. expected gain in dollars with the stock. 08 II II 30 Using the information in Tables 2-1 and 2-2, and assuming i = 5 per cent, and F = $3000: x(5 - Pw*) + [3000 - x . Pw*)] (.05) = -500 (1) X(15 - Pw*) + [3000 - (X - Pw*)] (.05) = 500 (2) Solving, X = 100 Pw* = $10.95 The premium = Pw* - Pw = $10.95 - $10.00 = $0.95 The distribution for the rate Of return on the common stock remains unchanged (Table 2—3). However, the distribution Of the rate of return for the warrant in Table 2-4 was based upon a beginning value Of $10 (no premium), and the investment at that time did not involve the (A', i) combination. The recalculated distributions for each strategy are shown below. TABLE 2-3 PROBABILITY TABLE FOR STOCK (STRATEGY A) Pst+n RORA Prob. $ profit A on $3,000 $45 50.0 .10 $1,500 40 33.3 .20 1,000 35 16. 7 . 40 500 _ a 30 0.0 .20 o “A ' 16'7‘ 25 ~16.7 .10 -500 0A = 18.26 31 TABLE 2-4 PROBABILITY TABLE FOR WARRANT —1 COMBINATION (STRATEGY B) Pwt+n RORA, i Prob. $ profit B on $3,000 9 $25 50.0 .10 $1,500 20 33.3 .20 1,000 15 16.7 .40 500 . _ . 10 0.0 .20 o “A i ‘ 16°74 5 -16.7 .10 -500 0A' = 18.26 It is apparent that when the warrant sells for a premium of 95 cents, the two alternative investment strat- egies, strategy A consisting Of 100 shares Of the common stock, and strategy B, consisting of 100 warrants plus the balance Of F in i, are identical and will occupy the same point in the two—parameter space. Each strategy has identi- cal rates Of return and standard deviations. Below is a sample calculation used to determine the probability distribution for strategy B. X = 100 Pw* = $10.95 Total investment in Warrants = $1095 Investment in i = F — $1095 = $1905 If Pw increases to $25, a profit of $25 - $10.95 = $14.05 on each warrant will be realized, or a total warrant profit of $1405. In addition, the profit from i will be $1905(.05) 32 = $95. Therefore, the total (A',i) profit will be $1500. As the above demonstrates, when the warrant sells for a 95 cent premium, and is purchased in prOper combina- tion with i, the two alternative investment strategies facing an investor will be identical in terms of their respective expected profitability and risk. As a result, since the cost of strategy B is a function pf Pw*, and since a Pw* Of $10.95 is the unique price that equates the two alternatives, a market price greater than $10.95 makes the common stock investment relatively more attractive, and a market price less than $10.95 makes the warrant in- vestment, in combination with i, relatively more attractive. Figure 2-6 COMPARATIVE RETURNS AT VARYING WARRANT PRICES I ,' '/ '9 VI) AIII£ A," // I , / I / / / / B" A B , 01y) I. /:-’ /T L ,/ I / / / / I / x /’ / //, 33 At any Pw* less than $10.95 a combination of (A'i) can be purchased, such as portfolio B, that will have a greater u than the common, at no increase in risk (Figure 2-6). At any Pw* greater than $10.95, the (A',i) combination, such as B' will have less 0 for a given risk than the common. One strategy will always be favored compared to the other except when the warrant sells for Pw*. The action of arbitragers will insure a shifting of funds in and out of the warrant strategy depending upon whether the market price of the warrant is over or under Pw*. This action will tend to maintain the warrant's price at Pw*. Generalization of the Model Previously, two simultaneous equations were used to solve for Pw*. Generalizing the equations, XLP - XPW* + iF - iXPW* = T (1) XEP - XPw* + iF - IXPw* = G (2) Solving for X, XLP - XEP = T - G X(LP - EP) = T - G T - G X = LP - EP (3) Using (1) to solve for Pw*, Pw*(-X - ix) = T - iF - XLP m S a I a I P '11 I XLP iY' (4) 34 Substituting for X T - iF - (T -_§)LP LP - EP Pw* = (5) —————-‘T+G (1+1) LP - EP Pw* = EP( - T + 1F) + LP(G - 1F) (6) (G - T (l + i) T and G have been expressed in terms Of the common stock, representing the dollar loss and gain. EP and LP are ex- pressed in terms of the warrant. Converting EP and LP into common stock equivalent terms, and restating T and G, EPW = EPs - PO LPw = LPs - PO T = (P5 - PO)X. X =‘E— , therefore T = F( -l + LPS ) P5 P8 G = (EPs - Ps)X, therefore G = F( 1 -.LB ) Ps Substituting these terms into quation (6), (7) (-l + £2) (-EP + Po) + (53 - l)(Lp - po) + i(EP — LP) Pw* = Ps 43" ‘ Lil (1 + 1) Ps Equation (7) reduces to, PW* __. PS 4' iPS - PO (8) l + i Recalculating the numerical example using (8): Pw* = $30 + .05(§30) - $20 = $11.50 = $10.95 1 + .05 1.50 . c . 35 Implications Of the Basic Model (1) Pw* represents the unique warrant price which equates the strategy Of investing in the warrant-i combina— tion, with that of investing in the common stock. The model is deve10ped from the perspective of the common stock in— vestor who first determines whether a given stock at a given market price is an acceptable vehicle for the employ- ment Of his funds. If it is, he should then compare the market price Of the associated warrant (if one exists) with its calculated Pw*, and purchase the warrant—i combina- tion if Pw* is greater than the market price. (2) If a warrant can be purchased in the market for less than its Pw*, an investor is guaranteed Of a rate of return on the warrant—i combination which will at all times be greater than the rate Of return on the common stock, re— gardless Of the future performance Of that stock. If the stock increases in value over time, the warrant strategy will increase to a greater extent. If the stock declines in value, the warrant strategy will decline to a lessor extent. The greater the difference between the market price Of the warrant and its Pw*, the greater the differ— ence in the performance Of the two strategies. (3) Pw* is an equilibrium price. In a rational market a movement away from Pw* will set up an arbitrage 36 process that will tend to reestablish the price. (4) It is evident that the model for Pw* (formula 8) is independent of the prOSpective probability distribu- tion Of rate of return for the common stock. Probability distributions were used to develOp the model, but once the formula for Pw* was generalized, probability distributions became incidental for establishing a warrant's price. ‘. hilt-”t '1' - .— _’ l ‘ The last point does not belie the importance Of estimating the future price performance Of common stocks. 1 The efficient utilization Of the model is dependent upon these estimates. The underlying position Of this paper is that a warrant should be considered as an alternative to its related common stock, after that stock has been decided upon as an acceptable vehicle for investment. However, it should be reiterated that if a warrant can be purchased for less than its indicated Pw*, then regardless of the future price performance of the common stock, the (A',i) combination will have superior performance. Model Modification Due to Dividend It has been assumed that the common stock pays no dividend. In fact, the dividend which the stock is ex- pected to pay is an important consideration when valuing its related warrant. Warrants do not convey the rights of 37 ownership, they merely represent Options to buy such rights. As such, dividends never accrue to warrant holders. From the perspective of the warrant investor, each dividend paid by the corporation represents lost income relative to the common stock investor. If the common in the example paid a dividend, in addition to having the hypothesized appreciation, the two strategies would no longer be equal at Pw*. The dollar profit for the warrant strategy would be less than that for the stock strategy by the total amount Of the dividends paid over the time horizon, on the number Of shares held. Putting the dividend consideration into the general- ized model, Ps + iPs - PO - D l + i Pw* :- (9) with, D = the total dollar dividend to be paid per share, over the time horizon. Model Modification Due to Multiperiod Analysis The time horizOn for the investment has been im— plicitly assumed a single unit period regardless Of the remaining length of the warrant's life. In correcting for this simplification it must be recognized that a warrant can have n years of life remaining, and the i in equation 38 (9) must be adjusted to so reflect. Equation (9) becomes, Ps + [(1+i)n - 1] Ps — PO - D *= Pw (1 + i)“ (10) reducing, Pw* = PS (1 + i)“ - PO - D (11) (l + i)n and, Po + D * = P3 - ———————- PW (l + i)n (12) Equation (12) represents the completed model. PO is the future sum that must be paid upon exercise Of the warrant. Relative to the owner of the common, it represents a negative cash flow to be incurred in the future if the stock is acquired. D represents the total dollar dividend which the stock will pay over the period, and is also a future negative cash flow to the warrant holder, relative to the stockholder. Equation (12) stipulates that the current value of the warrant is equal to the current value Of the common, minus the future negative cash flows the warrant holder will incur, discounted to present value. The develOpment of Pw* as an arbitrage supported valuation has been based upon the supposition that a war- rant investment is combined with an investment in the risk- free rate Of interest. The Objective was to find the Pw* that would equate the warrant strategy with an investment ,. J 1' L-fl—n—hp Wu ‘- ..—_—._—.——L-‘ 5 — ‘ lu- 39 already deemed acceptable. If, through market imperfec- tions, the market price of the warrant is less than Pw*, the warrant strategy will be purchased in lieu Of its asso- ciated stock, thereby assuring greater relative market per- formance than the common can provide. It must be recognized that many investors may have a risk-return preference that precludes them from consider- i ing the employment of any portion Of their funds in the relatively low yielding i. These investors will consider the warrant alone (rather than in combination with i) as a possible investment vehicle. They may be willing to accept the higher risk inherent in a warrant, in order to attain the higher anticipated return. Since investors vary as to their individual risk-return preferences, it is possible for a market price higher than Pw* to be rationally sup- ported. Given a favorable attitude toward the future price performance Of a given stock, the lower the demand for an increased return per incremental increase in risk, the higher the price an investor will be willing to pay for the warrant.4 With the above consideration, Pw* can no longer be considered as the valuation for a given warrant at a given 4Subject to an upper boundary, the price Of the stock. 40 time. It is a lower boundary valuation. If the market price of the warrant falls below Pw* the demand for the warrant strategy must increase if the situation is recog- nized by rational investors. Demand must increase at this undervalued price because superior performance of the war- rant strategy is guaranteed. Therefore, rational investors in the market for the common, will switch their purchase to the warrant strategy, forcing the warrant price back to Pw*. In constructing the model it was also assumed that if the warrant is purchased it will be held to its expira- tion date. As a warrant approaches maturity its premium declines. Immediately before expiration a premium will not be paid for a warrant and it will trade at base value. When develOping the model the expected values for the war- rant strategy incorporated this assumption. The warrant value for t+n did not include an allowance for a premium. The assumption is valid if the warrant is held to eXpira- tion. However, allowance must be made for the possibility of selling the warrant prior to its maturity date. If this occurred the subsequent purchaser of the security will pay a premium, making the rate Of return on the warrant strategy greater than under the base value assumption. Hence a purchaser of the warrant strategy, at any given time, will be willing to pay more than Pw* for the warrant ' ,7...” 41 if he contemplates its sale before expiration. How much more than Pw* the warrant is worth to a current investor is a function of the remaining lifeSpan of the warrant at the time Of its subsequent sale, as well as the number Of subse- quent sales the warrant will be subject to before expiring. Therefore, based upon the above, a market price greater than Pw* can be rationally supported. However Pw* still represents a lower boundary valuation for the security. In the numerical illustration presented earlier, V:‘L‘.~‘,-' A . hypothetical probability distributions for the future price action Of a common stock and its related warrant were set forth.5 This was one of two types of distributions that could have been hypothesized. The delineation Of the distribution classifications is a function of the number Of zero warrant values that are assumed possible. When one or less zero warrant values are forecast, use Of the model generates a Pw* which equates all characteristics of the two alternative strategies. As indicated in the dollar profit columns, Pw* equates more than the u and O of the strategies, it results in an equality Of all possible Out— comes. It makes the strategies identical not only in their occupation Of a point in the two-parameter Space, but 5Tables 2-1 and 2-2. 42 identical in every reSpect. When the range Of the probability distribution is such that more than one zero warrant value is possible, the meaning of Pw* is modified. To illustrate, additional values, below that which hypothesized, are added to the example.6 TABLE 2-5 TABLE 2-6 STRATEGY A, STRATEGY B, THE STOCK THE WARRANT WITH 1 Pst+n $ Profit Pwt+n $ Profit $45 $1,500 $25 $1,500 40 1,000 20 1,000 35 500 15 500 30 0 10 0 25 (500) 5 (500) 20 (1,000) 0 (1,000) 15 (1,500) 0 (1,000) 10 (2,000) 0 (1,000) 5 (2,500) 0 (1,000) The upper part of the profit distribution for the warrant strategy is identical to that Of the stock strategy, but the part below the first possible zero value for the warrant is muted compared to the possible performance of the stock. A warrant's value can never fall below zero, 6Probabilities are omitted as they are not important to the illustration Of the concept involved. 0 . a. Alum-JAE" I I V5223}? ' ’ . .I 43 and this will occur when the stock still has a positive value and the possibility of further decline. As a result, the warrant strategy has a built in limited loss relative to the stock strategy. To the extent that the limited loss aspect of the warrant strategy gives additional value to an investor, a market price higher than Pw* can be rationally supported. The extent Of the higher price will be a function Of the relative market power Of the investors who, at any given time, are forecasting multiple zero distributions, and the value they place on the limited loss aspect Of the strategy. The Pw* remains the price that investors who do not fore- cast such a distribution will be willing to pay, and again represents a lower boundary valuation. Summary Of Valuation Significance In the basic analysis the hypothetical distribution of future stock prices was assumed to be Of the type that led to a maximum Of one possible zero value for the related warrant. Under these circumstances the valuation model generates a Pw* which results in a warrant-i strategy identical in every respect to an investment in the associ- ated stock. They are, in fact, identical assets, and as such, a cost differential in their purchase cannot exist 44 in a rational market. However, other factors may be present that lead to a modification Of the interpretation Of Pw* from the only valuation to a lower boundary valuation. Individual risk-return preferences, the Opportunity Of selling a warrant prior to its eXpiration, and the possi- bility of muted distributions, all combine to rationally ta. '3 support a higher warrant price than its Pw*. How much i I higher is dependent On the force Of these factors at any 5 given time, but no factor can rationally support a warrant E. market price less than Pw*. CHAPTER III RESEARCH DESIGN AND ANALYSIS OF RESULTS As deve10ped in Chapter II, the Pw* model generates . n'. I“; a warrant price which is a lower boundary valuation for a given warrant at a given time. Factors not explicitly in- corporated in the model may be dominant, which justify a i rational investor paying a higher price for a warrant than its indicated Pw*. But if the assumptions upon which the model is based are accepted, a market price lower than Pw* always guarantees better investment performance with the warrant strategy than that possible with the warrant's associated stock. Market Test Methodology A market test is developed to determine the prac- tical significance Of the valuation model. The Objective Of the test is twofold: to determine the frequency of undervalued warrant's; and to determine the factors, if any, that provide an environment that leads to a warrant being undervalued. The test period is from January 1, 1961 to June 1. 45 46 1971, with 35 randomly generated weeks within that period. The specific dates selected are the Friday Of each randomly generated week. Each warrant that was outstanding and traded on the American or New York Stock Exchange, on each selected date, is included in the sample.1 The information required for utilization of the model consists Of the mar- ket price of the common stock, the market price of the stock's associated warrant, the Option price Of the war- rant, the dividend Of the common stock, the length of the warrant's remaining life, and the risk-free rate Of interest. The market prices of each stock and warrant are taken from the closing price Of each security on the Fri— day of each week selected. The Option price Of each war- rant is the prevailing Option price On the respective dates. If the Option price on the date examined to the maturity Of the warrant is a constant, it is assumed that the warrant will be held to maturity. If the contract terms stipulate an increase in the Option price before the warrant's expiration, the prevailing Option price is used with the assumption that the warrant will be held to lWarrants whose terms associated value with a multiplicity Of securities, and warrants with perpetual lives were omitted from the sample. 47 the date the Option price increases, and in addition, the final Option price stipulated is used with the assumption that the warrant will be held to maturity. The total dollar dividend the holder of each stock will receive during the holding period is computed by assuming that the prevailing dividend will remain constant during the time span of the warrant's life. The rate selected as a risk-free rate of interest is the maximum allowable interest rate payable on savings deposits Of 12 months or more by commercial banks which are members Of the Federal Reserve System. Test Results During the period covered by the study, there were a total of 816 price Observations for warrants, and an equal number of associated stock price Observations. During the test period there were 38 instances of under— valuation, as defined by the model, or 4.7 per cent of all Observations. A warrant whose Option price changed over time was counted as undervalued if its calculated Pw* was greater than its market price under any of its multiple of Option prices. If Pw* was greater than the current market price under more than one Option price, it was considered as only one undervaluation. For example, on June 23, 1967 Realty Equities warrant had a current Option price of $5.71 48 which in time increased to $6.34. It was determined as undervalued regardless of Which Option price was used in the model, but was counted as one undervaluation. It is felt that counting this and similar securities as multiple undervaluations would overstate the number of Opportunities for investing in warrants of this nature. Because the model cannot be used unless the market price of the stock is greater than the warrant's option price, this class of price observations is segregated from all others. Of the 816 observations, 336 (41.2 per cent) have a P3 1 P0. The 38 undervaluations constitute 11.3 per cent of all warrants whose associated stock price is greater than the Option price. A total of 85 different warrants comprise the 816 observations. Of these, 48 separate warrants comprise the 336 observations when the price Of the associated stock is greater than the Option price, and 12 different warrants accounted for the 38 undervaluations. Hence, 14.1 per cent (12/85) of all warrants are undervalued at least once during the period, and 25 per cent (12/48) Of all warrants whose associated stock price is greater than Po are under— valued at least once. The degree of undervaluation ranged from one cent to several dollars, (see Appendixzx). The presentation in Table 3-1 helps to illustrate 49 the meaning and consequences of undervaluation. On January 31, 1969 the closing price of AMK common stock was $45.13. The closing price of the associated warrant was $18.00. The valuation model generated a Pw* of $19.04, hence, by definition, the warrant was undervalued. In accordance with the concept of undervaluation used here, a combination of the warrant, risk-free rate of interest should have been purchased in lieu of the common stock. It can be reason— ably assumed that the purchaser of AMK stock was Optimistic about the future price performance of that security, but aware of the risk inherent with any investment. In any event, a greater return on investment (or a smaller loss) could have been guaranteed, regardless Of any future price movements, had the stock been foregone and the warrant-i combination purchased instead. Column A illustrates the dollar return on each strategy if the price of the stock is unchanged from the current price on the warrant's date Of expiration. Column B shows the respective returns if the price of the stock is higher on the date Of expiration, column C if the stock price is lower. Column D is a special case showing the effect if the stock price falls below the Option price. In the first two cases the profit on the warrant-i strategy is greater than what can be realized on the stock. In the 50 Nsm.esewM ~o~.emwm 45.nomflw aa.msam AHHO + Aosv Ammosv uguoua uaz “Nae ek.mmafiw e~.mmasm ea.mmaam «a.mmaaw saq. x ARV po>fiooou uwmumucfi HmuOH Aaav Aoowamv Aommamo one” Ammamv Ase I Amv muamuuaa ao Ammoav aauoum Aoav cm can” ommam macaw «at. at em>amomu uaaoa< Amo ow om.mw on.m~» no.0Hm Aoa . mac aoaaauaaxm am «can» has maamm mamas msamm msamm Aoomam I mamewO a as usaoa< ARV coma” cows” coma” coma” Amuamuums cogs “saunas so “moo Ace 824mm<3 Ameksmm NMSNAO anus» sham Ase + Ame Ammoso “smote umz Ame aka» okmm cam» ohm» maaaaa>aa Hates ASS AmHONWO Amanmv Assam om AHO I Ame Ammoav uamoum any coma” coca» cocoa mane» mama um em>aoomu uaaoa< a~o msmem name” name» mane» Amauaam ooav sooum mo umoo AHO macaw zazzoo oo.mmm n as oo.oew a ma oo.ocm I ma mH.mew n as costs xuoum onammez= so mmozmaommzco ems Hum mgm 2. In addition, 26.8 per cent of all such stocks had an associated undervalued warrant, compared to 4.7 per cent for all observations. W “‘-“.“‘—— 1...... ‘— r7, CHAPTER IV SUMMARY, IMPLICATIONS AND QUALIFICATIONS Model Development Warrants are considered as alternative investments to their related common stocks. After a given stock has been analyzed and judged an acceptable vehicle for invest— ment, its associated warrant (if one exists) would then be analyzed with the objective of determining if the market price of said warrant were undervalued. If found under- valued, the warrant, in combination with a risk-free invest- ment, would guarantee a higher return on the funds that would have been committed with a direct purchase of the stock. Hence, warrant undervaluation affords the oppor- tunity of improving upon a risk-return relationship which has already been found acceptable. A warrant is defined as undervalued if its current market price places it on a ray from i to the right of a ray from i to its related stock (Figure 4-1). As illus— trated, an investor can pr0portion funds between i and A' (the warrant) and achieve a higher u with no increase in 64 65 Figure 4-1 THE RISK-RETURN RELATIONSHIP WITH VARYING WARRANT PRICES risk relative to the associated stock. In the valuation model, PO + D * = — . PW PS (1 + 1) n Pw* represents the market price of a warrant that will place it on the same ray from i as its associated stock (point A"). At Pw* the pr0portioning of funds between A' and i will place the alternative investment strategies on an identical point in the two-parameter Space. At this point the warrant strategy offers no relative advantage compared to a direct investment in the common stock. A 66 market price higher than Pw* moves the warrant to a ray left of iA (A"'). In sum, Pw* represents the unique value which equates an investment in the warrant-i strategy to an investment in the warrant's related stock. A market price less than Pw* allows the Opportunity to pr0portioning funds so as to achieve a higher p with no increase in risk. :I Lower Boundary Valuation A market price higher than Pw* can be rationally justified on several grounds. If an investor's risk-return w- --. «imam.-. .. preference is such that it precludes the allocation of funds in i, the warrant will not be looked upon as an alternative to its related stock, but as an independent Opportunity. An investor will pay more than Pw* so long as the projected incremental return, relative to the stock, is great enough to compensate for the increased risk inherent in a warrant. In addition, the value of a warrant will decline to zero while its related stock still has a positive value. Since the return on the funds invested in i is independent of the stock price, the warrant-i strategy has limited loss potential relative to a direct investment in the stock. To the extent an investor places a premium on this limited loss, a higher price than Pw* can be justified. Lastly, the Pw* model implies the holding of a warrant to its date 67 of maturity, hence at its sale or conversion the premium will have dissipated. If a warrant is sold before maturity the subsequent buyer will pay a premium, generating addi- tional cash inflow to the warrant holder not explicitly in- corporated in the model. To the extent an investor foresees this possibility, he may be willing to pay more than Pw* for the security. ‘3»... 4 Though the above assumptions may justify a higher price than Pw*, a lower price will always allow the stock i- investor to improve his potential return, at no increase in risk, by purchasing the warrant—i combination in lieu Of the common. Therefore, Pw* is a lower boundary valuation for the warrant, below which, no market price is rationally justifiable. Market Test Methodology A market test is deve10ped to determine the prac- tical significance of the valuation model. There are 35 randomly generated weeks between the dates of January 1, 1961, and June 1, 1971. Stock and warrant price observa- tions are selected from the closing prices of the respec- tive securities on the Friday of each week. The risk-free rate used is the prevailing maximum interest rate on savings accounts of commercial banks. The investment holding period 68 is the remaining length of a warrant's life. For warrant's whose Option price changes before maturity, the option price prevailing on the date examined is used, and the warrant is assumed held until the date the terms change. In addi- tion, the last specified Option price is used with the assumption that the warrant is held to expiration. A war- rant is considered undervalued if its market price is below Pw* with either Option price, but counted only once if undervalued under each option price. Test Results It was found that 4.7 per cent of all stock price Observations have associated undervalued warrants, and 11.3 per cent of all stock prices above their respective option prices have associated undervaluations. The probability of undervaluations increases when the remaining maturity of the warrant is less than two years, and particularly with less than one year to maturity. There is also an in— crease in the percentage of undervaluations when the asso— ciated stocks have a high Ps/Po ratio. Over two-thirds of all undervaluations occurred under the above maturity and Ps/PO conditions, although such conditions accounted for less than 13 per cent of total Observations. The general state of the market (bull or bear) has 3? glam; '- ‘v§“—__H a . ‘ . 1. a ‘ ' ' 69 an effect on the frequency of undervaluations, with less occurring in bear, more in bull, than would be expected if the events were independent. Of all tested dates, 63 per cent have at least one undervaluation. The conclusion reached is that warrant undervaluations occur with suffi- cient frequency to merit investor attention. Qualifications Undervalued warrants as defined and determined do not imply attractive investments. A warrant is only under- “'1'“ .- -~e~w~fl .-_ _.-..~- .3. valued relative to the current market price of its asso- ciated stock. If such stock is overvalued and/or a price decline is imminent, a loss will likely be realized whether the stock is purchased directly or indirectly through the warrant strategy. If an investor is not Optimistic con— cerning the future price action of the stock, the warrant strategy should not be undertaken even if the warrant is classified as undervalued. The valuation model is not designed for use independent of an investment decision on common stock. A decision on a given stock should first be deve10ped, and if found acceptable, the model should then be applied to its associated warrant, with that warrant being purchased in combination with i, if found undervalued. The model can also be used to scan all available 7O warrants. If any are classified as undervalued, commensu- rate analysis of the associated stocks can be undertaken to determine if one or more are acceptable. In this manner, analysis can be confined to situations presenting under- valued opportunities, but not all such Opportunities present profitable investment possibilities. Components of the Model There are five variables required for utilization of the model. The P3 is the current market price of the stock, and requires no estimation. The Po and n are part of the warrant's contractual terms and also known with certainty at any given time. The rate used for i is subject to estimation and individual determination. The rate on savings deposits in commercial banks is used for this paper. It is felt to most closely approximate the abstract concept of risk—free interest. Such deposits are insured against loss, and the rate is relatively stable. Dependent upon the time horizon of the investment, and the concept of what constitutes risk for an investor, other rates may reasonably be used for i. For example, if a warrant has less than one year to maturity, the rate on Treasury bills or the savings rate may be used, whichever is higher. If a warrant has ten 71 years to maturity, an adjustment to i may be desired to reflect the possibility of lower future interest rates. Of the information required, D presents the greatest uncertainty, the greatest need for estimation, and the greatest degree of subjectivity. But again, the warrant should be considered, the model used, only after the stock has been analyzed. If fundamental (value) analysis is em- ployed, prospective dividends will have already been esti- mated when determining the desirability of the stock, and since the warrant represents an alternative, the same esti- mates can be employed in the model. The models developed by Samuelson, Chen, Ayers, and Sprenkle, all require esti— mates Of future stock prices. Dividends are far more stable than either stock prices or earnings,1 hence their estimates should tend to be more accurate and dependable. The Need for Additional Research Warrants are only one type of Option available in the security markets, and the model deve10ped may have applicability to other Option contracts. There are several areas that may provide fruitful research. John Lintner, "Distribution of Incomes of Corpo— rations Among Dividends, Retained Earnings, and Taxes", American Economic Review, XLIV (May, 1956), pp. 97-113. 72 1. A study Of investors who frequently purchase stock Options, to determine if the concept of profession- alism has a bearing on valuation. 2. A study of put and call Option contracts, utilizing the deve10ped model. Call Options, in particular, are identical to warrants with short maturities, and their shorter time horizon should reduce if not eliminate the un— certainty Of dividend and interest rate estimations. 3. The examination of premium determination on convertible bonds and preferred stocks, incorporating the interest and dividend yield on the respective securities. Contributions of the Study This research should provide a new perspective to the interrelationships between the investment character- istics of warrants and their related common stocks. It presents an investment strategy, that of combining a war— rant with a risk-free investment, that has not been previously explored, plus the justification of such a strategy, which together should result in more efficient allocation of an investor's funds. Secondly, this paper should result in a supple— mentary definition of what constitutes a warrant's lower boundary valuation. This additional definition will lead 73 to a more rational pricing policy and a narrower trading range for warrants as a whole. Third, the model deve10ped in Chapter II has more practical applicability than those requiring a tenuous assumption of future stock price movements. From an in- vestor's view, the benefit of a warrant valuation model should be its usefulness as an aid in the formulation of a rational investment decision. To the extent that in- vestors consider estimates of future dividends more reli- able than those of future stock prices, the model presented here will be a more accurate tool in the investment deci— sion process. BIBLIOGRAPHY Ayers, H. F. "Risk Aversion in the Warrant Markets," Industrial Management Review, Vol. 5, No. l Bierman, Jr., Harold. "The Valuation of Stock Options," Journal of Financial and Quantitative Analysis, Vol. II, No. 3 (September, 1967), 327-334. Chen, A. H. Y. "A Model of Warrant Pricing in a Dynamic Market," Journal of Finance, Vol. XXV, No. 5 (December, 1970), 1041-1059. Cootner, Paul H. "Random Vs. Systematic Changes," Industrial Management Review, Vol. 3, No. 2 (Spring, 1962), 24—45. . The Random Character of Stock Market Prices. Cambridge, Mass.: The M.I.T. Press, 1964. Giguere, G. "Warrants, a Mathematical Method of Evalu- ation," The Analysts Journal, No. 5 (November, 1958), 17-25. Kurnow, E., Glasset, G. and Ottman, F. Statistics for Business Decisions, Homewood, Illinois: Richard D. Irwin, Inc., 1959. Lintner, John. "Security Prices, Risk, and Maximal Gains from Diversification," Journal of Finance, Vol. XX, NO. 4 (December, 1965), 587-615. "Distribution of Incomes of Corporations Among Dividends, Retained Earnings, and Taxes," American Economic Review, Vol. XLIV (May, 1956), 97-113. Markowitz, Harry. "Portfolio Selection," Journal of Finance, Vol. VII, NO. 1 (March, 1952), 77-91. Morrison, R. J. "The Warrants or the Stock," Analysts Journal, Vol. 13, NO. 5 (November, 1957), 52—57. Opinions of the Accounting Principles Board. Earnings Per Share, NO. 15, May, 1969. 74 75 Samuelson, Paul A. "Rational Theory of Warrant Pricing," Industrial Management Review, Vol. 6, No. 2 (Spring, 1965), 13-39. Sharpe, W. F. "Capital Asset Prices: a Theory Of Market Equilibrium Under Conditions of Uncertainty," Journal of Finance, Vol. XIX, No. 3 (September, 1964), 425-442. Shelton, J. P. "Relation of the Price of a Warrant to the Price of its Associated Stock," Financial Analysts Journal, Vol. 23 (May, 1967), 143-151 (July, 1967), 88-99. Sprenkle, C. M. "Warrant Prices as Indicators of Expecta- tions and Preferences," Yale Economic Essays, APPENDIX A DEGREE AND TIME TO MATURITY OF UNDERVALUED WARRANTS 76 TABLE A—l THE VARYING DEGREES OF UNDERVALUATION, LISTED CHRONOLOGICALLY Warrant Model Market Price,7) Date Warrant Market Price Price Model Price ‘° 11/15/63 TWA 12.88 13.79 93.4 1/29/65 TWA 33.00 36.56 90.3 4/16/65 TWA 35.38 38.60 91.7 Realty Equities 2.00 2.29 87.3 8/13/65 TWA 29.13 31.07 93.8 Realty Equities 1.88 2.14 87.9 11/19/65 TWA 35.88 40.07 89.5 Realty Equities 5.63 7.18 78.4 3/17/67 TWA 66.25 66.31 99.9 Realty Equities 5.63 6.44 87.4 Braniff Airlines 41.50 48.09 86.3 Universal American 1.50 1.66 90.4 4/7/67 TWA 56.75 56.81 99.9 Realty Equities 6.13 7.55 81.2 Braniff Airlines 35.38 39.66 89.2 5/5/67 Realty Equities 7.63 9.13 83.6 6/23/67 Realty Equities 6.50 9.13 71.2 1/26/68 Realty Equities 15.00 15.80 94.9 Frontier Airlines 8.88 10.77 82.5 Uris Buildings 23.50 23.82 98.7 3/15/68 Frontier Airlines 8.00 9.61 83.2 8/30/68 Realty Equities 27.38 30.40 90.1 Martin Marietta 21.75 21.88 99.4 11/8/68 Realty Equities 31.00 31.66 97.9 1/31/69 AMK 18.00 19.04 94.5 Frontier Airlines 8.75 16.37 53.5 11/21/69 Daylin 14.50 23.37 62.0 12/5/69 Daylin 14.25 22.36 63.7 1/1/71 Kaufman and Broad 22.62 23.93 94.5 1/22/71 " " " 26.00 28.02 92.8 2/5/71 " " " 25.37 27.40 92.6 2/26/71 " " " 28.50 30.37 93.8 5/7/71 Daylin 9.62 13.82 69.6 Kaufman and Broad 40.00 41.32 96.8 Williams 29.75 30.38 97.9 5/14/71 Daylin 9.25 12.94 71.5 Kaufman and Broad 37.75 39.80 94.8 Lerner Stores 31.12 31.58 98.5 77 TABLE Ar2 TIME TO MATURITY IN DESCENDING ORDER FOR EACH UNDERVALUED WARRANT Years, Days Warrant 19,279 19,238 19,120 19,106 19,34 18,351 18,30 17,318 17,311 11,109 10,16 9,29 8,307 8,229 8,110 8,12 7,95 6,278 6,238 6,74 L‘CDOUIU! 98 Braniff Airlines Braniff ” Daylin Daylin Frontier Airlines Frontier " Frontier Daylin Daylin Lerner Stores TWA AMK TWA TWA TWA TWA Uris Building TWA TWA . Realty Equities Realty " Realty " Williams Realty Equities Realty " Realty " Realty " Kaufman and Broad Kaufman " " Kaufman Kaufman Kaufman Kaufman Realty Equities Realty " Realty " Martin Marietta Universal American APPENDIX B UNDERVALUED WARRANTS RELATIVE TO CHARACTERISTICS OF THEIR ASSOCIATED COMMON STOCKS 78 TABLE B-l UNDERVALUATIONS RELATIVE TO THE NUMBER OF LISTED WARRANTS Number of Undervalued (%) Date Eligible Warrants* Undervaluations Eligible 5/14/71 61 3 4.92 5/7/71 60 3 5.00 2/26/71 56 l 1.79 2/5/71 52 1 1.92 1/22/71 56 l 1.76 1/1/71 54 1 1.85 12/1/70 33 0 0 10/12/70 30 0 0 5/11/70 31 0 0 4/13/70 30 0 0 12/5/69 43 1 2.33 11/21/69 42 l 2.38 1/31/69 23 2 8.70 11/8/69 18 l 5.56 8/30/68 16 2 12.50 3/15/68 14 1 7.14 1/26/68 14 3 21.43 6/23/67 11 1 9.09 5/5/67 11 l 9.09 4/7/67 11 3 27.27 3/17/67 12 4 33.33 8/22/66 11 0 0 7/22/66 11 0 0 3/11/66 10 0 0 11/19/65 10 2 20.00 8/13/65 10 2 20.00 4/16/65 10 2 20.00 1/29/65 11 l 9.09 11/15/63 11 l 9.09 9/27/63 11 0 0 6/7/63 11 0 0 1/4/63 12 0 0 6/8/61 7 0 0 4/14/61 7 0 0 3/9/61 __6_ ._Q 0 Total or Average 816 38 4.66% *Listed on the American or New York Stock Exchange. 79 TABLE B-Z UNDERVALUATIONS RELATIVE TO THE NUMBER OF WARRANTS WHOSE ASSOCIATED STOCK PRICE WAS GREATER THAN THE WARRANT OPTION PRICE Number of Stocks Ps > Po (7) Undervalued (%) Date With P8 > Po Eligible ° Ps > Po 5/14/71 29 47.54 10.34 5/7/71 27 45.00 11.11 2/26/71 23 41.07 4.35 2/5/71 21 40.38 4.76 1/22/71 18 32.14 5.56 1/1/71 11 20.37 8.33 12/1/70 6 18.18 0 10/12/70 6 20.00 0 5/1/70 2 6.45 0 4/13/70 6 20.00 0 12/5.69 14 32.56 6.67 11/21/69 16 38.10 5.88 1/31/69 19 82.61 10.53 11/8/68 14 77.78 7.14 8/30/68 9 56.25 20.00 3/15/68 11 78.57 9.09 1/26/68 12 85.71 25.00 6/23/67 8 72.73 12.50 5/5/67 6 54.55 16.67 4/7/67 7 63.64 42.86 3/17/67 8 66.67 50.00 8/22/66 6 54.55 0 7/22/66 7 63.64 0 3/11/66 5 50.00 0 11/19/65 4 40.00 50.00 8/13/65 4 40.00 50.00 4/16/65 4 40.00 50.00 1/29/65 4 36.36 25.00 11/15/63 4 36.36 20.00 9/27/63 5 45.45 0 6/7/63 3 27.27 0 1/4/63 3 25.00 0 6/8/61 5 71.43 0 4/14/61 5 71.43 0 3/9/61 __;4 66.67 0 Total or Average 336 41.18% 11.31% 80 TABLE B-3 THE RELATIONSHIP OF THE STOCK PRICE TO THE WARRANT OPTION PRICE, PER YEAR, FOR EACH UNDERVALUATION Number of Date Undervaluations Ps/Po 5/14/71 3 1.128, 2.746, 3.083 5/7/71 3 1.167, 2.815, 2.338 2/26/71 1 2.301 2/5/71 1 2.163 1/22/71 1 2.192 1/1/71 1 2.001 12/5/69 1 1.628 11/21/69 1 1.672 1/31/69 2 1.308, 1.172 11/8/68 1 5.191 8/30/68 2 1.479, 5.025 3/15/68 1 1.125 1/26/68 3 1.192, 3.095, 2.929 6/23/67 1 2.277 5/5/67 1 2.277 4/7/67 3 1.012, 2.058, 3.591 3/17/67 4 1.127, 1.905, 4.023, 1.119 11/19/65 2 2.065, 2.551 8/13/65 2 1.323, 2.136 4/16/65 2 1.345, 2.472 1/29/65 1 2.375 11/15/63 1 1.301