How do I review the attached articles I want to have it review them to able to compare with what I have done so far American Economic Association Corporate Income Taxes and the Cost of Capital A Correction Author(s) Franco Modigliani and Merton H Miller Source The American Economic Review, Vol 53, No 3 (Jun , 1963), pp 433 443 Published by American Economic Association Stable URL http www jstor org stable 1809167 Accessed 10 09 2009 09 53 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http www jstor org page info about policies terms jsp JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non commercial use Please contact the publisher regarding any further use of this work Publisher contact information may be obtained at http www jstor org action showPublisher publisherCode aea Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission JSTOR is a not for profit organization founded in 1995 to build trusted digital archives for scholarship We work with the scholarly community to preserve their work and the materials they rely upon, and to build a common research platform that promotes the discovery and use of these resources For more information about JSTOR, please contact support jstor org American Economic Association is collaborating with JSTOR to digitize, preserve and extend access to The American Economic Review http www jstor org COMMUNICATIONS 433 equanimity a writing down of the value of their reserves, or unless one is prepared to forego the possibility of exchange rate adjustment, any major extension of the gold exchange standard is dependent upon the introduction of guarantees It is misleading to suggest that the multiple key currency system is an alternative to a guarantee, as implied by Roosa 6, pp 5 7 and 9 12 IV Coniclusion The most noteworthy conclusion to be drawn from this analysis is that the successful operation of a multiple key currency system would require both exchange guarantees and continuing cooperation between central bankers of a type that would effectively limit their choice as to the form in which they hold their reserves Yet these are two of the conditions whose undesirability has frequently been held to be an obstacle to implementation of the alterlnative proposal to create a world central bank The multiple key currency proposal represents an attempt to avoid the impracticality supposedly associated with a world central bank, but if both proposals in fact depend on the fulfillment of similar conditions, it is difficultto convince oneself that the sacrifice of the additional liquidity that an almost closed system would permit is worth while Unless, of course, the object of the exercise is to reinforce discipline rather than to expand liquidity JOHN WILLIAMSON REFERENCES 1 R Z ALIBER, Foreign Exchange Guarantees and the Dollar Comment, Am Econ Rev , Dec 1962, 52, 1112 16 2 S T 1E3ZAAND G PATTERSON, Foreign Exchange Guaranteesand the Dollar, Am Econ Rev , June 1961, 51, 381 85 3 AND , Foreign Exchange Guaranteesand the Dollar Reply, Amn Econ Rev , Dec 1962, 52, 1117 18 4 F A LUTZ,The Problem of International Equilibrium Amsterdam 1962 5 R NURKSE, International Currency Experience Geneva 1944 6 R V RooSA, Assuring the Free World's Liquidity, Business Review Supplement,Federal Reserve Bank of Philadelphia, Sept 1962 The author is instructor in economics at Princeton University He acknowledges the helpful comments of Fritz Machlup Views expressed are those of the author alone Corporate Income Taxes and the Cost of Capital A Correction The purpose of this communication is to correct an error in our paper The Cost of Capital, Corporation Finance and the Theory of Investment (this Review, June 1958) In our discussion of the effects of the present method of taxing corporations on the valuation of firms, we said (p 272) The deduction of interest in computing taxable corporate profits will prevent the arbitrage process from making the value of all firms in a given class proportional to the expected returns generated by their 434 THE AMERICAN ECONOMICREVIEW physical assets Instead, it can be shown (by the same type of proof used for the original version of Proposition I) that the market values of firms in each class must be proportional in equilibrium to their expected returns net of taxes (that is, to the sum of the interest paid and expectednet stockholderincome) (Italics added ) The statement in italics, unfortunately, is wrong For even though one firm may have an expectedreturn after taxes (our Xr) twice that of another firm in the same risk equivalent class, it will not be the case that the actual return after taxes (our X7) of the first firm will always be twice that of the second, if the two firms have different degrees of leverage ' And since the distribution of returns after taxes of the two firms will not be proportional, there can be no arbitrage process which forces their values to be proportional to their expected after tax returns 2 In fact, it can be shown and this time it really will be shown that arbitrage will make values within any class a function not only of expected after tax returns, but of the tax rate and the degree of leverage This means, among other things, that the tax advantages of debt financing are somewhat greater than we originally suggested and, to this extent, the quantitative difference between the valuations implied by our position and by the traditional view is narrowed It still remains true, however, that under our analysis the tax advantages of debt are the only permanent advantages so that the gulf between the two views in matters of interpretation and policy is as wide as ever I Taxes, Leverage,and the Probability Distribution of After Tax Returns To see how the distribution of after tax earnings is affected by leverage, let us again denote by the random variable X the (long run average) earnings before interest and taxes generated by the currently owned assets of a given firm in some stated risk class, k 3 From our definition of a risk class it follows that X can be expressed in the form XZ, where X is the expected value of X, and the random variable Z X X, having the same value for all firms in class k, is a drawing from a distribution, sav fk(Z) Hence the 1 With some exceptions, which will be noted when they occur, we shall preserve here both the notation and the terminology of the original paper A working knowledge of both on the part of the reader will be presumed 2 Barring, of course, the trivial case of universal linear utility functions Note that in deference to Professor Durand (see his Comment on our paper and our reply, this Review,Sept 1959, 49, 639 69) we here and throughout use quotation marks when referring to arbitrage 3 Thus our X corresponds essentially to the familiar EBIT concept of the finance literature The use of EBIT and related income concepts as the basis of valuation is strictly valid only when the underlying real assets are assumed to have perpetual lives In such a case, of course, EBIT and cash flow are one and the same This was, in effect, the interpretation of X we used in the original paper and we shall retain it here both to preserve continuity and for the considerable simplification it permits in the exposition We should point out, however, that the perpetuity interpretation is much less restrictive than might appear at first glance Beforetax cash flow and EBIT can also safely be equated even where assets have finite lives as soon as these assets attain a steady state age distribution in which annual replacements equal annual depreciation The subject of finite lives of assets will be further discussed in connection with the problem of the cut off rate for investment decisions COMMUNICATIONS 435 random variable XA, measuring the after tax return, can be expressed as (1) Xr (1 r)(X R) where r R (1 r)X rR (1 r)XZ is the marginal corporate income tax rate (assumed equal to the average), and R is the interest bill Since E(XT) XT X (X (1 T)X TR we can in (1) to obtain substitute XfTY'rRfor (1 r)X (2) rR X'( I R)Z rR ) Z Thus, if the tax rate is other than zero, the shape of the distribution of XK will depend not only on the scale of the stream XTand on the distribution of Z, but also on the tax rate and the degree of leverage (one measure of which is R XT) For example, if Var (Z) Var (XY) q2(X)2 f2, (1 we have r implying that for given Xr the variance of after tax returns is smaller, the higher r and the degree of leverage 4 II The Valuation of After Tax Returns Note from equation (1) that, from the investor's point of view, the longrun average stream of after tax returns appears as a sum of two components (1) an uncertain stream (1 r)XZ and (2) a sure stream rR 5 This suggests that the equilibrium market value of the combined stream can be found by capitalizing each component separately More precisely, let p7 be the rate at which the market capitalizes the expected returns net of tax of an unlevered company of size X in class k, i e , pr (1 )X F Vu or Vu (I )X 6 PT 4 It may seem paradoxical at first to say that leverage reduces the variability of outcomes, but remember we are here discussing the variability of total returns, interest plus net profits The variability of stockholder net profits will, of course, be greater in the presence than in the absence of leverage, though relatively less so than in an otherwise comparable world of no taxes The reasons for this will become clearer after the discussion in the next section 5 The statement that rR the tax saving per period on the interest payments is a sure stream is subject to two qualifications First, it must be the case that firms can always obtain the tax benefit of their interest deductions either by offsetting them directly against other taxable income in the year incurred or, in the event no such income is available in any given year, by carrying them backward or forward against past or future taxable earnings or, in the extreme case, by merger of the firm with (or its sale to) another firm that can utilize the deduction Second, it must be assumed that the tax rate will remain the same To the extent that neither of these conditions holds exactly then some uncertainty attaches even to the tax savings, though, of course, it is of a different kind and order from that attaching to the stream generated by the assets For simplicity, however, we shall here ignore these possible elements of delay or of uncertainty in the tax saving but it should be kept in mind that this neglect means that the subsequent valuation formulas overstate, if anything, the value of the tax saving for any given Dermanent level of debt 6 Note that here, as in our original paper, we neglect dividend policy and growth in the THE AMERICAN ECONOMICREVIEW 436 and let r be the rate at which the market capitalizes the sure streams generated by debts For simplicity, assume this rate of interest is a constant independent of the size of the debt so that R r or D D 7 R r Then we would expect the value of a levered firm of size X, with a permanent level of debt DL in its capital structure, to be given by (I VL ( (3) TR T)X )DLX VU r Pr In our original paper we asserted instead that, within a risk class, market value would be proportional to expected after tax return XT(cf our original equation 1 1 ), which would imply Xr (4) VL (1 pr r TR T)X p VU TDL pr pr We will now show that if (3) does not hold, investors can secure a more efficient portfolio by switching from relatively overvalued to relatively undervalued firms Suppose first that unlevered firms are overvalued or that VL TDL of the levered firm entitles the holder to the outcome YL (1 (M SL) (m SL) (1 T) (XZ RL) )XZ (1 (m SL) T)RL Consider the following alternative portfolio (1) borrow an amount for which the interest cost will be (Mn SL)(1 r)RL (M SL)(1 r)DL (assuming, of course, that individuals and corporations can borrow at the same rate, r) and (2) invest m plus the amount borrowed, i e , m m(1 r)DL SL M SL (1 T)DL (M SL)1VL' TDL SL in the stock of the unlevered firm The outcome so secured will be (MI SL) (VL UTDL) )XZ (1 Subtracting the interest charges on the borrowed funds leaves an income of Y u (m SL) (L L) (1 T)XZ (M SL)(1 T)RL which will dominate YL if (and only if) VL 1 DL Vu Thus, in equilibrium, both VL T DL Vu and VL T DL I if d1 d dl dl 1 r pT Hence the before tax required rate of return cannot be defined without reference to financial policy In particular, for an investment considered as being financed entirely by new equity capital dD dI O and the required rate of return or marginal cost of equity financing (neglecting flotation costs) would be S Ps 1 r This result is the same as that in the original paper (see equation 32 , p 294) and is applicable to any other sources of financing where the remuneration to the suppliers of capital is not deductible for tax purposes It applies, therefore, to preferred stock (except for certain partially deductible issues of public utilities) and would apply also to retained earnings were it not for the favorable tax treatment of capital gains under the personal income tax For investments considered as being financed entirely by new debt capital dI dD and we find from (7) that (33 c) pD p7 which replaces our original equation (33) in which we had (33) pS 8 pD Ir r 1 r 14 Note that we use the term earnings net of taxes rather than earnings after taxes We feel that to avoid confusion the latter term should be reserved to describe what will actually appear in the firm's accounting statements, namely the net cash flow including the tax savings on the interest (our Xl) Since financing sources cannot in general be allocated to particular investments (see below), the after tax or accounting concept is not useful for capital budgeting purposes, although it can be extremely useful for valuation equations as we saw in the previous section 15Remember that when we speak of the minimum required yield on an investment we are referring in principle only to investments which increase the scale of the firm That is, the new COMMUNICATIONS 441 Thus for borrowed funds (or any other tax deductible source of capital) the marginal cost or before tax required rate of return is simply the market rate of capitalization for net of tax unlevered streaimsand is thus independent of both the tax rate and the interest rate This required rate is lower than that implied by our original (33), but still considerably higher than that implied by the traditional view (see esp pp 276 77 of our paper) under which the before tax cost of borrowed funds is simply the interest rate, r Having derived the above expressions for the marginal costs of debt and equity financing it may be well to warn readers at this point that these expressions represent at best only the hypothetical extremes insofar as costs are concerned and that neither is directly usable as a cut off criterion for investment planning In particular, care must be taken to avoid falling into the famous Liquigas fallacy of concluding that if a firm intends to float a bond issue in some given year then its cut off rate should be set that year at pD while, if the next issue is to be an equity one, the cut off is ps The point is, of course, that no investment can meaningfully be regarded as 100 per cent equity financed if the firm makes any use of debt capital and most firms do, not only for the tax savings, but for many other reasons having nothing to do with cost in the present static sense (cf our original paper pp 292 93) And no investment can meaningfully be regarded as 100 per cent debt financed when lenders impose strict limitations on the maximum amount a firm can borrow relative to its equity (and when most firms actually plan on normally borrowing less than this external maximum so as to leave themselves with an emergency reserve of unused borrowing power) Since the firm's long run capital structure will thus contain both debt and equity capital, investment planning must recognize that, over the long pull, all of the firm's assets are really financed by a mixture of debt and equity capital even though only one kind of capital may be raised in any particular year More precisely, if L denotes the firm's long run target debt ratio (around which its actual debt ratio will fluctuate as it alternately floats debt issues and retires them with internal or external equity) then the firm can assume, to a first approximation at least, that for any particular investment dD dI L Hence, the relevant marginal cost of capital for investment planning, which we shall here denote by p , is l p TLT T PS D L ps( 1 L ) p DL That is, the appropriate cost of capital for (repetitive) investment decisions over time is, to a first approxiimation,a weigllted average of the costs of debt and equity financinig, the weights being the proportions of each in the target capital structure 16 assets must be in the sarrme class as the old See in this connection, J Hirshleifer, Risk, the Discount Rate and Investment Decisions, Am Econ Rev , May 1961, 51, 112 20 (especially pp 119 20) See also footnote 16 16 From the formulas in the text one can readily derive corresponding expressions for the required net of tax yield, or net of tax cost of capital for any given financing policy Specifi 442 THE AMERICAN ECONOMICREVIEW V Some Concluding Observations Such, then, are the major corrections that must be made to the various formulas and valuation expressions in our earlier paper In general, we can say that the force of these corrections has been to increase somewhat the estimate of the tax advantages of debt financing under our model and consequently to reduce somewhat the quantitative difference between the estimates of the effects of leverage under our model and under the naive traditional view It may be useful to remind readers once again that the existence of a tax advantage for debt financing even the larger advantage of the corrected version does not necessarily mean that corporations should at all times seek to use the maximum possible amount of debt in their capital structures For one thing, other forms of financing, notably retained earnings, may in some circumstances be cheaper still when the tax status of investors under the personal income tax is taken into account More important, there are, as we pointed out, limitations imposed by lenders (see pp 292 93), as well as many other dimensions (and kinds of costs) in realworld problems of financial strategy which are not fully comprehended within the framework of static equilibrium models, either our own or those of the traditional variety These additional considerations, which are typically grouped under the rubric of the need for preserving flexibility, will normally imply the maintenance by the corporation of a substantial reserve of untapped borrowing power The tax advantage of debt may well tend to lower the optimal size of that reserve, but it is hard to believe that advantages of the size contemplated under our model could justify any substantial reduction, let alone their complete elimination Nor do the data cally, let p(L) denote the required net of tax yield for investment financed with a proportion of debt L dD dl (More generally L denotes the proportion financed with tax deductible sources of capital ) Then from (7) we find (8) (L) (1 r) (1 Lr)pT and the various costs can be found by substituting the appropriate value for L In particular, if we substitute in this formula the target leverage ratio, L , we obtain p p(L) (1 TL )pr and p measures the average net of tax cost of capital in the sense described above Although the before tax and the net of tax approaches to the cost of capital provide equally good criteria for investment decisions when assets are assumed to generate perpetual (i e , non depreciating) streams, such is not the case when assets are assumed to have finite lives (even when it is also assumed that the firm's assets are in a steady state age distribution so that our X or EBIT is approximately the same as the net cash flow before taxes) See footnote 3 above In the latter event, the correct method for determining the desirability of an investment would be, in principle, to discount the net of tax stream at the net of tax cost of capital Only under this net of tax approach would it be possible to take into account the deductibility of depreciation (and also to choose the most advantageous depreciation policy for tax purposes) Note that we say that the net of tax approach is correct in principle because, strictly speaking, nothing in our analysis (or anyone else's, for that matter) has yet established that it is indeed legitimate to discount an uncertain stream One can hope that subsequent research will show the analogy to discounting under the certainty case is a valid one but, at the moment, this is still only a hope COMMUNICATIONS 443 indicate that there has in fact been a substantial increase in the use of debt (except relative to preferred stock) by the corporate sector during the recent high tax years 17 As to the differences between our modified model and the traditional one, we feel that they are still large in quantitative terms and still very much worth trying to detect It is not only a matter of the two views having different implications for corporate financial policy (or even for national tax policy) But since the two positions rest on fundamentally different views about investor behavior and the functioning of the capital markets, the results of tests between them may have an important bearing on issues ranging far beyond the immediate one of the effects of leverage on the cost of capital FRANCO MODIGLIANI AND MERTON H MILLER 17 See, e g , Merton H Miller, The Corporate Income Tax and Corporate Financial Policies, in Staff Reports to the Commission on Money and Credit (forthcoming) The authors are, respectively, professor of industrial management, School of Industrial Management, Massachusetts Institute of Technology, and professor of finance, Graduate School of Business, University of Chicago Consumption, Savings and Windfall Gains Comment In her recent article in this Review 3 , Margaret Reid attempted to answer previous articles by Bodkin 1 and Jones 2 challenging the validity of the permanent income hypothesis Bodkin and Jones used income and expenditure data for those consumer units who had received the soldiers' bonus (National Service Life Insurance dividends) during 1950, the year of the urban consumption survey 4 These bonuses were regarded as windfall gains for the purposesof their analyses Professor Reid used data from the same survey, but her windfall gains were represented by other money receipts These are defined as inheritances and occasional large gifts of money from persons outside the family and net receipts from the settlement of fire and accident policies 4, Vol 1, p xxix She assumed that the soldiers' bonus was included, and that it accounted for about one half of other money receipts Here she made an unfortunate mistake in interpreting the data for the main critical purpose of her article The soldiers' bonus is not part of other money receipts (0) but rather a part of disposable money income (Y) It is the main part of an item in the disposable money income category called military pay, allotments, and pensions 4, Vol 11, p xxix This would appear to alter completely the relationship of Professor Reid's main findings to the Bodkin results and to change the windfall interpretation of the 0 variable Surely, fire and accident policy settlements are not windfall income, but rather a (partial) recovery of real assets previously lost Likewise, inheritances are probably best considered as a long anticipated increase in assets not an increase in transitory income The discovery of this error probably does not affect whatever importance Professor Reid's secondary finding may have the need, in any study of The Pricing of Options and Corporate Liabilities Author(s) Fischer Black and Myron Scholes Source Journal of Political Economy, Vol 81, No 3 (May Jun , 1973), pp 637 654 Published by The University of Chicago Press Stable URL http www jstor org stable 1831029 Accessed 06 06 2015 17 53 UTC Your use of the JSTOR archive indicates your acceptance of the Terms Conditions of Use, available at http www jstor org page info about policies terms jsp JSTOR is a not for profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive We use information technology and tools to increase productivity and facilitate new forms of scholarship For more information about JSTOR, please contact support jstor org The University of Chicago Press is collaborating with JSTOR to digitize, preserve and extend access to Journal of Political Economy http www jstor org This content downloaded from 137 52 76 29 on Sat, 06 Jun 2015 17 53 42 UTC All use subject to JSTOR Terms and Conditions The Pricing of Options and Corporate Liabilities FischerBlack Universityof Chicago MyronScholes MassachusettsInstituteof Technology If optionsare correctlypricedin the market,it shouldnot be possible to make sure profitsby creatingportfoliosof long and shortpositions in options and theirunderlyingstocks Using this principle,a theoreticalvaluationformulafor optionsis derived Since almost all corporateliabilitiescan be viewedas combinationsof options,the formula and the analysisthatled to it are also applicableto corporateliabilities such as commonstock, corporatebonds, and warrants In particular, the formulacan be used to derive the discountthat shouldbe applied to a corporatebond because of the possibilityof default Introduction An option is a securitygiving the right to buy or sell an asset, subject to certain conditions,withina specifiedperiod of time An American option is one that can be exercised at any time up to the date the option expires A European option is one that can be exercised only on a specified future date The price that is paid for the asset when the option is exercisedis called the exercise price or strikingprice The last day on which the option may be exercised is called the expiration date or maturitydate The simplest kind of option is one that gives the right to buy a single share of common stock Throughout most of the paper, we will be discussing this kind of option, which is often referredto as a call option Received for publication November 11, 1970 Final version received May 9, 1972 The inspirationfor this work was provided by Jack L Treynor (1961a, 1961b) We are grateful for extensive comments on earlier drafts by Eugene F Fama, Robert C Merton, and Merton H Miller This work was supported in part by the Ford Foundation 637 This content downloaded from 137 52 76 29 on Sat, 06 Jun 2015 17 53 42 UTC All use subject to JSTOR Terms and Conditions 638 JOURNAL OF POLITICAL ECONOMY In general,it seems clear that the higherthe price of the stock,the greaterthe value of the option When the stock price is much greater thantheexerciseprice,theoptionis almostsureto be exercised The currentvalue of the optionwill thus be approximately equal to the priceof the stockminusthe priceof a pure discountbond that matureson the samedate as theoption,witha facevalue equal to thestriking priceof the option On the otherhand, if the price of the stock is much less than the exerciseprice,the optionis almostsure to expirewithoutbeingexercised, so its value willbe nearzero If the expirationdate of the optionis veryfarin the future,thenthe priceof a bond thatpays the exercisepriceon the maturitydate will be verylow, and the value of the optionwill be approximately equal to the priceof the stock On the otherhand,if the expirationdate is verynear,the value of the optionwill be approximately equal to the stock price minusthe exercise price,or zero,if the stockpriceis less than the exerciseprice Normally, thevalueof an optiondeclinesas its maturity date approaches,if thevalue of thestockdoes not change These generalpropertiesof the relationbetweenthe optionvalue and thestockpriceare oftenillustrated in a diagramlikefigure1 Line A representsthemaximumvalue of theoption,sinceit cannotbe worthmorethan the stock Line B represents the minimumvalue of the option,since its value cannotbe negativeand cannotbe less thanthestockpriceminusthe exerciseprice Lines T1, T2, and T3 representthe value of the optionfor shortermaturities successively Normally,thecurverepresenting the value of an optionwill be concave upward Since it also lies below the 45 line, A, we can see that the $40 A $3 TI c$20 oa $10 0l o ' 48 Add ' r 0 T $10 $20 1 $30 $40 Stock Price (ExercisePrice $20) FiG 1 The relationbetweenoptionvalue and stock price This content downloaded from 137 52 76 29 on Sat, 06 Jun 2015 17 53 42 UTC All use subject to JSTOR Terms and Conditions OPTIONS 639 AND LIABILITIES option will be more volatile than the stock A given percentage change in the stock price,holding maturityconstant,will result in a largerpercentage change in the option value The relative volatility of the option is not constant,however It depends on both the stock price and maturity Most of the previous work on the valuation of options has been expressed in termsof warrants For example, Sprenkle (1961), Ayres (1963), Boness (1964), Samuelson (1965), Baumol, Malkiel, and Quandt (1966), and Chen (1970) all produced valuation formulas of the same general form Their formulas,however, were not complete, since they all involved one or more arbitraryparameters For example Sprinkle's formulaforthe value of an option can be written as follows kxN(b1) lnkx c by 1 v2(t 2 t) vv (t t) lnkx c b2 k cN(b2) 2 v2(t tAv(t t) t) In this expression,x is the stock price, c is the exercise price, t is the maturitydate, t is the currentdate, v2 is the variance rate of the returnon the stock,' In is the natural logarithm,and N(b) is the cumulative normal density function But k and k are unknown parameters Sprenkle (1961) definesk as the ratio of the expected value of the stock price at the time the warrantmatures to the currentstock price, and k as a discount factor that depends on the risk of the 'stock He tries to estimate the values of k and k empirically,but finds that he is unable to do so More typically, Samuelson (1965) has unknown parameters x and (, where x is the rate of expected returnon the stock, and ( is the rate of expected returnon the warrant or the discount rate to be applied to the warrant 2He assumes that the distributionof possible values of the stock when the warrant matures is log normal and takes the expected value of this distribution,cuttingit offat the exercise price He then discounts this expected value to the present at the rate ( Unfortunately,there seems to be no model of the pricingof securitiesunder conditions of capital market 1 The variance rate of the return on a securityis the limit, as the size of the intervalof measurementgoes to zero, of the variance of the returnover that interval divided by the length of the interval 2The rate of expected returnon a securityis the limit,as the size of the interval of measurementgoes to zero, of the expected return over that interval divided by the lengthof the interval This content downloaded from 137 52 76 29 on Sat, 06 Jun 2015 17 53 42 UTC All use subject to JSTOR Terms and Conditions 640 JOURNAL OF POLITICAL ECONOMY equilibrium thatwouldmakethisan appropriate procedurefordetermining thevalueof a warrant In a subsequentpaper,Samuelsonand Merton(1969) recognizethe fact thatdiscounting the expectedvalue of the distribution of possiblevalues of the warrantwhenit is exercisedis not an appropriateprocedure They advancethe theoryby treatingthe optionpriceas a functionof the stock price They also recognizethat the discountratesare determined in part by the requirement thatinvestorsbe willingto hold all of the outstanding amountsof both the stockand the option But theydo not make use of thefactthatinvestors mustholdotherassetsas well,so thattheriskof an optionor stockthat affectsits discountrate is only that part of the risk thatcannotbe diversified away Their finalformuladependson the shape of theutilityfunctionthattheyassumeforthe typicalinvestor One of theconceptsthatwe use in developingour modelis expressedby Thorp and Kassouf (1967) They obtain an empiricalvaluationformula forwarrants by fitting a curveto actual warrantprices Then theyuse this formulato calculatetheratioof sharesof stockto optionsneededto create a hedgedpositionby goinglong in one securityand shortin the other Whattheyfailto pursueis thefactthatin equilibrium, theexpectedreturn on such a hedgedpositionmustbe equal to the returnon a risklessasset What we show below is that this equilibriumconditioncan be used to derivea theoretical valuationformula The Valuation Formula In derivingour formulaforthevalue of an optionin termsof the priceof the stock,we will assume ideal conditions in the marketforthe stock and fortheoption a) The short term interestrateis knownand is constantthroughtime b) The stockprice followsa randomwalk in continuoustime with a variancerateproportional to the square of the stockprice Thus the distribution of possiblestock pricesat the end of any finiteintervalis lognormal The variancerateof the returnon the stockis constant c) The stockpays no dividendsor otherdistributions d) The option is European, that is, it can only be exercisedat maturity e) There are no transactioncosts in buyingor sellingthe stockor the option f) It is possibleto borrowany fractionof the price of a securityto buy it or to hold it, at the short term interestrate g) There are no penaltiesto shortselling A sellerwho does not own a securitywillsimplyacceptthepriceof thesecurityfroma buyer,and will agree to settle with the buyer on some futuredate by payinghim an amountequal to thepriceof the securityon thatdate This content downloaded from 137 52 76 29 on Sat, 06 Jun 2015 17 53 42 UTC All use subject to JSTOR Terms and Conditions OPTIONS AND LIABILITIES 64I Under these assumptions,the value of the option will depend only on the price of the stock and time and on variables that are taken to be known constants Thus, it is possible to create a hedged position, consistingof a long position in the stock and a short position in the option, whose value will not depend on the price of the stock, but will depend only on time and the values of known constants XWriting w(x, t) for the value of the option as a functionof the stock price x and time t, the number of options that must be sold short against one share of stock long is 1 wI(x,t) (1) In expression (1), the subscript refersto the partial derivative of w(x,t) with respect to its firstargument To see that the value of such a hedged position does not depend on the price of the stock, note that the ratio of the change in the option value to the change in the stock price, when the change in the stock price is small, is w1(x,t) To a first approximation, if the stock price changes by an amount Ax, the option price will change by an amount w1(xt) Ax, and the number of options given by expression (1) will change by an amount Ax Thus, the change in the value of a long position in the stock will be approximately offsetby the change in value of a short position in 11w options As the variables x and t change, the number of options to be sold short to create a hedged position with one share of stock changes If the hedge is maintained continuously,then the approximationsmentionedabove become exact, and the return on the hedged position is completely independent of the change in the value of the stock In fact, the returnon the hedged position becomes certain 3 To illustrate the formationof the hedged position, let us refer to the solid line (T ) in figure1 and assume that the price of the stock starts at $15 00, so that the value of the option starts at $5 00 Assume also that the slope of the line at that point is 1112 This means that the hedged position is created by buying one share of stock and selling two options short One share of stock costs $15 00, and the sale of two options brings in $10 00, so the equity in this position is $5 00 If the hedged position is not changed as the price of the stock changes, then there is some uncertaintyin the value of the equity at the end of a finiteinterval Suppose that two options go from$10 00 to $15 75 when the stock goes from$15 00 to $20 00, and that they go from $10 00 to $5 75 when the stock goes from $15 00 to $10 00 Thus, the equity goes from $5 00 to $4 25 when the stock changes by $5 00 in either direction This is a $ 75 decline in the equity for a $5 00 change in the stock in either direction 4 3 This was pointed out to us by Robert Merton 4 These figuresare purely for illustrativepurposes They correspondroughlyto the way figure1 was drawn, but not to an option on any actual security This content downloaded from 137 52 76 29 on Sat, 06 Jun 2015 17 53 42 UTC All use subject to JSTOR Terms and Conditions 642 JOURNAL OF POLITICAL ECONOMY In addition,the curve shifts (say fromT2 to T3 in fig 1) as the maturityof the optionschanges The resultingdecline in value of the optionsmeansan increasein theequityin thehedgedpositionand tendsto offsetthepossiblelossesdue to a largechangein thestockprice Note that the declinein the equityvalue due to a large changein the stockpriceis small The ratio of the declinein the equity value to the magnitudeof the changein the stockpricebecomessmalleras the magnitudeof thechangein thestockpricebecomessmaller Note also that the directionof the changein the equityvalue is independentof the directionof the changein the stockprice This meansthat underour assumptionthat the stock price followsa continuousrandom walk and that the returnhas a constantvariancerate, the covariance betweenthe returnon theequityand the returnon the stockwill be zero If the stockpriceand the value of the marketportfolio followa joint continuousrandomwalk withconstantcovariancerate,it means that the covariancebetween the returnon the equity and the returnon the marketwill be zero Thus the riskin the hedgedpositionis zero if the shortpositionin the optionis adjusted continuously If the positionis not adjusted continuously,theriskis small,and consistsentirelyof riskthatcan be diversified away by forming a portfolioof a largenumberof such hedgedpositions In general,since the hedgedpositioncontainsone share of stock long and 11wi optionsshort,thevalue of theequityin the positionis x w w1 (2) The changein thevalue of theequityin a shortintervalAt is Ax Aw w1 (3) Assumingthat the shortpositionis changedcontinuously, we can use stochasticcalculus to expandAw, whichis w(x Ax,t At) w(x,t), as follows Aw w1Ax 2 W11V2X2At w2At (4) In equation (4), the subscriptson w referto partialderivativesand v2 is thevariancerateof the returnon thestock Substituting fromequation (4) intoexpression(3), we findthatthechangein thevalue of theequity in thehedgedpositionis t w1 1v x2 w2V At w1 ((5) Since the returnon the equityin the hedgedpositionis certain,the returnmustbe equal to rAt Even if the hedgedpositionis not changed 5For an expositionof stochasticcalculus,see McKean (1969) B1See footnote1 This content downloaded from 137 52 76 29 on Sat, 06 Jun 2015 17 53 42 UTC All use subject to JSTOR Terms and Conditions OPTIONS 643 AND LIABILITIES continuously, its riskis small and is entirelyriskthat can be diversified away,so theexpectedreturnon the hedgedpositionmustbe at the short terminterestrate 7If this werenot true,speculatorswould tryto profit by borrowing largeamountsof moneyto createsuchhedgedpositions,and wouldin theprocessforcethe returnsdownto theshortterminterestrate Thus the changein the equity (5) mustequal the value of the equity (2) timesrAt w11v2x2 w2 At w1 (x w wi)rAt (6) Droppingthe At fromboth sides,and rearranging, we have a differential equationforthevalue of the option W2 rw rxwl 1 2 22 v2x2w1l (7) Writingt forthematurity date of theoption,and c fortheexerciseprice, we knowthat w(xt ) X ' C x c C, (8) O x

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How do I review the attached articles  I want to have it review them to able to compare with what I have done so far  American Economic Association Corporate Income Taxes and the Cost of Capital  A Correction Author(s)  Franco Modigliani and Merton H  Miller Source  The American Economic Review, Vol  53, No  3 (Jun , 1963), pp  433 443 Published by  American Economic Association Stable URL  http   www jstor org stable 1809167 Accessed  10 09 2009 09 53 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http   www jstor org page info about policies terms jsp  JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non commercial use  Please contact the publisher regarding any further use of this work  Publisher contact information may be obtained at http   www jstor org action showPublisher publisherCode aea  Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission  JSTOR is a not for profit organization founded in 1995 to build trusted digital archives for scholarship  We work with the scholarly community to preserve their work and the materials they rely upon, and to build a common research platform that promotes the discovery and use of these resources  For more information about JSTOR, please contact support jstor org  American Economic Association is collaborating with JSTOR to digitize, preserve and extend access to The American Economic Review  http   www jstor org COMMUNICATIONS 433 equanimity a writing down of the value of their reserves, or unless one is prepared to forego the possibility of exchange rate adjustment, any major extension of the gold exchange standard is dependent upon the introduction of guarantees  It is misleading to suggest that the multiple key currency system is an alternative to a guarantee, as implied by Roosa  6, pp  5 7 and 9 12   IV  Coniclusion The most noteworthy conclusion to be drawn from this analysis is that the successful operation of a multiple key currency system would require both exchange guarantees and continuing cooperation between central bankers of a type that would effectively limit their choice as to the form in which they hold their reserves  Yet these are two of the conditions whose undesirability has frequently been held to be an obstacle to implementation of the alterlnative proposal to create a world central bank  The multiple key currency proposal represents an attempt to avoid the impracticality supposedly associated with a world central bank, but if both proposals in fact depend on the fulfillment of similar conditions, it is difficultto convince oneself that the sacrifice of the additional liquidity that an almost closed system would permit is worth while  Unless, of course, the object of the exercise is to reinforce discipline rather than to expand liquidity  JOHN WILLIAMSON  REFERENCES 1  R  Z  ALIBER,  Foreign Exchange Guarantees and the Dollar  Comment,  Am  Econ  Rev , Dec  1962, 52, 1112 16  2  S  T  1E3ZAAND G  PATTERSON,  Foreign Exchange Guaranteesand the Dollar,  Am  Econ  Rev , June 1961, 51, 381 85  3  AND ,  Foreign Exchange Guaranteesand the Dollar  Reply,  Amn Econ  Rev , Dec  1962, 52, 1117 18  4  F  A  LUTZ,The Problem of International Equilibrium  Amsterdam 1962  5  R  NURKSE, International Currency Experience  Geneva 1944  6  R  V  RooSA,  Assuring the Free World's Liquidity,  Business Review Supplement,Federal Reserve Bank of Philadelphia, Sept  1962    The author is instructor in economics at Princeton University  He acknowledges the helpful comments of Fritz Machlup  Views expressed are those of the author alone  Corporate Income Taxes and the Cost of Capital  A Correction The purpose of this communication is to correct an error in our paper  The Cost of Capital, Corporation Finance and the Theory of Investment  (this Review, June 1958)  In our discussion of the effects of the present method of taxing corporations on the valuation of firms, we said (p  272)  The deduction of interest in computing taxable corporate profits will prevent the arbitrage process from making the value of all firms in a given class proportional to the expected returns generated by their 434 THE AMERICAN ECONOMICREVIEW physical assets  Instead, it can be shown (by the same type of proof used for the original version of Proposition I) that the market values of firms in each class must be proportional in equilibrium to their expected returns net of taxes (that is, to the sum of the interest paid and expectednet stockholderincome)  (Italics added ) The statement in italics, unfortunately, is wrong  For even though one firm may have an expectedreturn after taxes (our Xr) twice that of another firm in the same risk equivalent class, it will not be the case that the actual return after taxes (our X7) of the first firm will always be twice that of the second, if the two firms have different degrees of leverage ' And since the distribution of returns after taxes of the two firms will not be proportional, there can be no  arbitrage  process which forces their values to be proportional to their expected after tax returns 2 In fact, it can be shown and this time it really will be shown that  arbitrage  will make values within any class a function not only of expected after tax returns, but of the tax rate and the degree of leverage  This means, among other things, that the tax advantages of debt financing are somewhat greater than we originally suggested and, to this extent, the quantitative difference between the valuations implied by our position and by the traditional view is narrowed  It still remains true, however, that under our analysis the tax advantages of debt are the only permanent advantages so that the gulf between the two views in matters of interpretation and policy is as wide as ever  I  Taxes, Leverage,and the Probability Distribution of After Tax Returns To see how the distribution of after tax earnings is affected by leverage, let us again denote by the random variable X the (long run average) earnings before interest and taxes generated by the currently owned assets of a given firm in some stated risk class, k 3 From our definition of a risk class it follows that X can be expressed in the form XZ, where X is the expected value of X, and the random variable Z  X X, having the same value for all firms in class k, is a drawing from a distribution, sav fk(Z)  Hence the 1 With some exceptions, which will be noted when they occur, we shall preserve here both the notation and the terminology of the original paper  A working knowledge of both on the part of the reader will be presumed  2 Barring, of course, the trivial case of universal linear utility functions  Note that in deference to Professor Durand (see his Comment on our paper and our reply, this Review,Sept  1959, 49, 639 69) we here and throughout use quotation marks when referring to arbitrage  3 Thus our X corresponds essentially to the familiar EBIT concept of the finance literature  The use of EBIT and related  income  concepts as the basis of valuation is strictly valid only when the underlying real assets are assumed to have perpetual lives  In such a case, of course, EBIT and  cash flow  are one and the same  This was, in effect, the interpretation of X we used in the original paper and we shall retain it here both to preserve continuity and for the considerable simplification it permits in the exposition  We should point out, however, that the perpetuity interpretation is much less restrictive than might appear at first glance  Beforetax cash flow and EBIT can also safely be equated even where assets have finite lives as soon as these assets attain a steady state age distribution in which annual replacements equal annual depreciation  The subject of finite lives of assets will be further discussed in connection with the problem of the cut off rate for investment decisions  COMMUNICATIONS 435 random variable XA, measuring the after tax return, can be expressed as  (1) Xr   (1  r)(X R) where r   R   (1  r)X   rR   (1  r)XZ is the marginal corporate income tax rate (assumed equal to the average), and R is the interest bill  Since E(XT)  XT  X (X (1 T)X TR we can in (1) to obtain  substitute XfTY'rRfor (1 r)X (2)   rR X'( I R)Z   rR ) Z  Thus, if the tax rate is other than zero, the shape of the distribution of XK will depend not only on the  scale  of the stream XTand on the distribution of Z, but also on the tax rate and the degree of leverage (one measure of which is R XT)  For example, if Var (Z) Var (XY)   q2(X)2   f2, (1   we have  r   implying that for given Xr the variance of after tax returns is smaller, the higher r and the degree of leverage 4 II  The Valuation of After Tax Returns Note from equation (1) that, from the investor's point of view, the longrun average stream of after tax returns appears as a sum of two components  (1) an uncertain stream (1 r)XZ  and (2) a sure stream rR 5 This suggests that the equilibrium market value of the combined stream can be found by capitalizing each component separately  More precisely, let p7 be the rate at which the market capitalizes the expected returns net of tax of an unlevered company of size X in class k, i e , pr  (1   )X F Vu  or Vu  (I )X 6 PT 4 It may seem paradoxical at first to say that leverage reduces the variability of outcomes, but remember we are here discussing the variability of total returns, interest plus net profits  The variability of stockholder net profits will, of course, be greater in the presence than in the absence of leverage, though relatively less so than in an otherwise comparable world of no taxes  The reasons for this will become clearer after the discussion in the next section  5 The statement that rR the tax saving per period on the interest payments is a sure stream is subject to two qualifications  First, it must be the case that firms can always obtain the tax benefit of their interest deductions either by offsetting them directly against other taxable income in the year incurred  or, in the event no such income is available in any given year, by carrying them backward or forward against past or future taxable earnings  or, in the extreme case, by merger of the firm with (or its sale to) another firm that can utilize the deduction  Second, it must be assumed that the tax rate will remain the same  To the extent that neither of these conditions holds exactly then some uncertainty attaches even to the tax savings, though, of course, it is of a different kind and order from that attaching to the stream generated by the assets  For simplicity, however, we shall here ignore these possible elements of delay or of uncertainty in the tax saving  but it should be kept in mind that this neglect means that the subsequent valuation formulas overstate, if anything, the value of the tax saving for any given Dermanent level of debt  6 Note that here, as in our original paper, we neglect dividend policy and  growth  in the THE AMERICAN ECONOMICREVIEW 436 and let r be the rate at which the market capitalizes the sure streams generated by debts  For simplicity, assume this rate of interest is a constant independent of the size of the debt so that R r   or D D   7 R r Then we would expect the value of a levered firm of size X, with a permanent level of debt DL in its capital structure, to be given by  (I VL   (  (3)   TR T)X )DLX  VU   r Pr In our original paper we asserted instead that, within a risk class, market value would be proportional to expected after tax return XT(cf  our original equation  1 1 ), which would imply  Xr (4) VL (1    pr r TR  T)X     p     VU    TDL  pr pr We will now show that if (3) does not hold, investors can secure a more efficient portfolio by switching from relatively overvalued to relatively undervalued firms  Suppose first that unlevered firms are overvalued or that VL   TDL of the levered firm entitles the holder to the outcome YL      (1 (M SL)   (m SL) (1   T) (XZ   RL)  )XZ   (1   (m SL) T)RL  Consider the following alternative portfolio  (1) borrow an amount for which the interest cost will be (Mn SL)(1 r)RL (M SL)(1  r)DL (assuming, of course, that individuals and corporations can borrow at the same rate, r)  and (2) invest m plus the amount borrowed, i e , m m(1   r)DL   SL    M SL   (1  T)DL   (M SL)1VL' TDL  SL in the stock of the unlevered firm  The outcome so secured will be (MI SL) (VL UTDL) )XZ  (1 Subtracting the interest charges on the borrowed funds leaves an income of Y u   (m SL) (L L) (1   T)XZ   (M SL)(1   T)RL which will dominate YL if (and only if) VL 1 DL  Vu  Thus, in equilibrium, both VL T DL  Vu and VL T DL I if d1 d dl dl   1 r pT Hence the before tax required rate of return cannot be defined without reference to financial policy  In particular, for an investment considered as being financed entirely by new equity capital dD dI O and the required rate of return or marginal cost of equity financing (neglecting flotation costs) would be  S Ps        1  r This result is the same as that in the original paper (see equation  32 , p  294) and is applicable to any other sources of financing where the remuneration to the suppliers of capital is not deductible for tax purposes  It applies, therefore, to preferred stock (except for certain partially deductible issues of public utilities) and would apply also to retained earnings were it not for the favorable tax treatment of capital gains under the personal income tax  For investments considered as being financed entirely by new debt capital dI dD and we find from (7) that  (33 c) pD   p7 which replaces our original equation (33) in which we had  (33)   pS 8 pD   Ir r  1  r 14 Note that we use the term  earnings net of taxes  rather than  earnings after taxes   We feel that to avoid confusion the latter term should be reserved to describe what will actually appear in the firm's accounting statements, namely the net cash flow including the tax savings on the interest (our Xl)  Since financing sources cannot in general be allocated to particular investments (see below), the after tax or accounting concept is not useful for capital budgeting purposes, although it can be extremely useful for valuation equations as we saw in the previous section  15Remember that when we speak of the minimum required yield on an investment we are referring in principle only to investments which increase the scale of the firm  That is, the new COMMUNICATIONS 441 Thus for borrowed funds (or any other tax deductible source of capital) the marginal cost or before tax required rate of return is simply the market rate of capitalization for net of tax unlevered streaimsand is thus independent of both the tax rate and the interest rate  This required rate is lower than that implied by our original (33), but still considerably higher than that implied by the traditional view (see esp  pp  276 77 of our paper) under which the before tax cost of borrowed funds is simply the interest rate, r  Having derived the above expressions for the marginal costs of debt and equity financing it may be well to warn readers at this point that these expressions represent at best only the hypothetical extremes insofar as costs are concerned and that neither is directly usable as a cut off criterion for investment planning  In particular, care must be taken to avoid falling into the famous  Liquigas  fallacy of concluding that if a firm intends to float a bond issue in some given year then its cut off rate should be set that year at pD  while, if the next issue is to be an equity one, the cut off is ps  The point is, of course, that no investment can meaningfully be regarded as 100 per cent equity financed if the firm makes any use of debt capital and most firms do, not only for the tax savings, but for many other reasons having nothing to do with  cost  in the present static sense (cf  our original paper pp  292 93)  And no investment can meaningfully be regarded as 100 per cent debt financed when lenders impose strict limitations on the maximum amount a firm can borrow relative to its equity (and when most firms actually plan on normally borrowing less than this external maximum so as to leave themselves with an emergency reserve of unused borrowing power)  Since the firm's long run capital structure will thus contain both debt and equity capital, investment planning must recognize that, over the long pull, all of the firm's assets are really financed by a mixture of debt and equity capital even though only one kind of capital may be raised in any particular year  More precisely, if L  denotes the firm's long run  target  debt ratio (around which its actual debt ratio will fluctuate as it  alternately  floats debt issues and retires them with internal or external equity) then the firm can assume, to a first approximation at least, that for any particular investment dD dI  L   Hence, the relevant marginal cost of capital for investment planning, which we shall here denote by p , is  l p TLT     T PS   D  L  ps( 1  L )   p DL  That is, the appropriate cost of capital for (repetitive) investment decisions over time is, to a first approxiimation,a weigllted average of the costs of debt and equity financinig, the weights being the proportions of each in the  target  capital structure 16 assets must be in the sarrme  class  as the old  See in this connection, J  Hirshleifer,  Risk, the Discount Rate and Investment Decisions,  Am  Econ  Rev , May 1961, 51, 112 20 (especially pp  119 20)  See also footnote 16  16 From the formulas in the text one can readily derive corresponding expressions for the required net of tax yield, or net of tax cost of capital for any given financing policy  Specifi  442 THE AMERICAN ECONOMICREVIEW V  Some Concluding Observations Such, then, are the major corrections that must be made to the various formulas and valuation expressions in our earlier paper  In general, we can say that the force of these corrections has been to increase somewhat the estimate of the tax advantages of debt financing under our model and consequently to reduce somewhat the quantitative difference between the estimates of the effects of leverage under our model and under the naive traditional view  It may be useful to remind readers once again that the existence of a tax advantage for debt financing even the larger advantage of the corrected version does not necessarily mean that corporations should at all times seek to use the maximum possible amount of debt in their capital structures  For one thing, other forms of financing, notably retained earnings, may in some circumstances be cheaper still when the tax status of investors under the personal income tax is taken into account  More important, there are, as we pointed out, limitations imposed by lenders (see pp  292 93), as well as many other dimensions (and kinds of costs) in realworld problems of financial strategy which are not fully comprehended within the framework of static equilibrium models, either our own or those of the traditional variety  These additional considerations, which are typically grouped under the rubric of  the need for preserving flexibility,  will normally imply the maintenance by the corporation of a substantial reserve of untapped borrowing power  The tax advantage of debt may well tend to lower the optimal size of that reserve, but it is hard to believe that advantages of the size contemplated under our model could justify any substantial reduction, let alone their complete elimination  Nor do the data cally, let p(L) denote the required net of tax yield for investment financed with a proportion of debt L dD dl  (More generally L denotes the proportion financed with tax deductible sources of capital ) Then from (7) we find  (8)  (L)   (1  r)  (1 Lr)pT and the various costs can be found by substituting the appropriate value for L  In particular, if we substitute in this formula the  target  leverage ratio, L , we obtain  p   p(L)   (1  TL )pr and p  measures the average net of tax cost of capital in the sense described above  Although the before tax and the net of tax approaches to the cost of capital provide equally good criteria for investment decisions when assets are assumed to generate perpetual (i e , non depreciating) streams, such is not the case when assets are assumed to have finite lives (even when it is also assumed that the firm's assets are in a steady state age distribution so that our X or EBIT is approximately the same as the net cash flow before taxes)  See footnote 3 above  In the latter event, the correct method for determining the desirability of an investment would be, in principle, to discount the net of tax stream at the net of tax cost of capital  Only under this net of tax approach would it be possible to take into account the deductibility of depreciation (and also to choose the most advantageous depreciation policy for tax purposes)  Note that we say that the net of tax approach is correct  in principle  because, strictly speaking, nothing in our analysis (or anyone else's, for that matter) has yet established that it is indeed legitimate to  discount  an uncertain stream  One can hope that subsequent research will show the analogy to discounting under the certainty case is a valid one  but, at the moment, this is still only a hope  COMMUNICATIONS 443 indicate that there has in fact been a substantial increase in the use of debt (except relative to preferred stock) by the corporate sector during the recent high tax years 17 As to the differences between our modified model and the traditional one, we feel that they are still large in quantitative terms and still very much worth trying to detect  It is not only a matter of the two views having different implications for corporate financial policy (or even for national tax policy)  But since the two positions rest on fundamentally different views about investor behavior and the functioning of the capital markets, the results of tests between them may have an important bearing on issues ranging far beyond the immediate one of the effects of leverage on the cost of capital  FRANCO MODIGLIANI AND MERTON H  MILLER  17 See, e g , Merton H  Miller,  The Corporate Income Tax and Corporate Financial Policies,  in Staff Reports to the Commission on Money and Credit (forthcoming)    The authors are, respectively, professor of industrial management, School of Industrial Management, Massachusetts Institute of Technology, and professor of finance, Graduate School of Business, University of Chicago  Consumption, Savings and Windfall Gains  Comment In her recent article in this Review  3 , Margaret Reid attempted to answer previous articles by Bodkin  1  and Jones  2  challenging the validity of the permanent income hypothesis  Bodkin and Jones used income and expenditure data for those consumer units who had received the soldiers' bonus (National Service Life Insurance dividends) during 1950, the year of the urban consumption survey  4   These bonuses were regarded as windfall gains for the purposesof their analyses  Professor Reid used data from the same survey, but her windfall gains were represented by  other money receipts   These are defined as  inheritances and occasional large gifts of money from persons outside the family      and net receipts from the settlement of fire and accident policies   4, Vol  1, p  xxix   She assumed that the soldiers' bonus was included, and that it accounted for about one half of other money receipts  Here she made an unfortunate mistake in interpreting the data for the main critical purpose of her article  The soldiers' bonus is not part of  other money receipts  (0) but rather a part of  disposable money income  (Y)  It is the main part of an item in the disposable money income category called  military pay, allotments, and pensions   4, Vol  11, p  xxix   This would appear to alter completely the relationship of Professor Reid's main findings to the Bodkin results and to change the windfall interpretation of the 0 variable  Surely, fire and accident policy settlements are not windfall income, but rather a (partial) recovery of real assets previously lost  Likewise, inheritances are probably best considered as a long anticipated increase in assets not an increase in transitory income  The discovery of this error probably does not affect whatever importance Professor Reid's secondary finding may have         the need, in any study of The Pricing of Options and Corporate Liabilities Author(s)  Fischer Black and Myron Scholes Source  Journal of Political Economy, Vol  81, No  3 (May   Jun , 1973), pp  637 654 Published by  The University of Chicago Press Stable URL  http   www jstor org stable 1831029 Accessed  06 06 2015 17 53 UTC Your use of the JSTOR archive indicates your acceptance of the Terms   Conditions of Use, available at http   www jstor org page  info about policies terms jsp JSTOR is a not for profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive  We use information technology and tools to increase productivity and facilitate new forms of scholarship  For more information about JSTOR, please contact support jstor org  The University of Chicago Press is collaborating with JSTOR to digitize, preserve and extend access to Journal of Political Economy  http   www jstor org This content downloaded from 137 52 76 29 on Sat, 06 Jun 2015 17 53 42 UTC All use subject to JSTOR Terms and Conditions The Pricing of Options and Corporate Liabilities FischerBlack Universityof Chicago MyronScholes MassachusettsInstituteof Technology If optionsare correctlypricedin the market,it shouldnot be possible to make sure profitsby creatingportfoliosof long and shortpositions in options and theirunderlyingstocks  Using this principle,a theoreticalvaluationformulafor optionsis derived Since almost all corporateliabilitiescan be viewedas combinationsof options,the formula and the analysisthatled to it are also applicableto corporateliabilities such as commonstock, corporatebonds, and warrants In particular, the formulacan be used to derive the discountthat shouldbe applied to a corporatebond because of the possibilityof default  Introduction An option is a securitygiving the right to buy or sell an asset, subject to certain conditions,withina specifiedperiod of time  An  American option  is one that can be exercised at any time up to the date the option expires  A  European option  is one that can be exercised only on a specified future date  The price that is paid for the asset when the option is exercisedis called the  exercise price  or  strikingprice   The last day on which the option may be exercised is called the  expiration date  or  maturitydate   The simplest kind of option is one that gives the right to buy a single share of common stock  Throughout most of the paper, we will be discussing this kind of option, which is often referredto as a  call option   Received for publication November 11, 1970  Final version received May 9, 1972  The inspirationfor this work was provided by Jack L  Treynor (1961a, 1961b)  We are grateful for extensive comments on earlier drafts by Eugene F  Fama, Robert C  Merton, and Merton H  Miller  This work was supported in part by the Ford Foundation  637 This content downloaded from 137 52 76 29 on Sat, 06 Jun 2015 17 53 42 UTC All use subject to JSTOR Terms and Conditions 638 JOURNAL OF POLITICAL ECONOMY In general,it seems clear that the higherthe price of the stock,the greaterthe value of the option When the stock price is much greater thantheexerciseprice,theoptionis almostsureto be exercised The currentvalue of the optionwill thus be approximately equal to the priceof the stockminusthe priceof a pure discountbond that matureson the samedate as theoption,witha facevalue equal to thestriking priceof the option  On the otherhand, if the price of the stock is much less than the exerciseprice,the optionis almostsure to expirewithoutbeingexercised, so its value willbe nearzero  If the expirationdate of the optionis veryfarin the future,thenthe priceof a bond thatpays the exercisepriceon the maturitydate will be verylow, and the value of the optionwill be approximately equal to the priceof the stock  On the otherhand,if the expirationdate is verynear,the value of the optionwill be approximately equal to the stock price minusthe exercise price,or zero,if the stockpriceis less than the exerciseprice Normally, thevalueof an optiondeclinesas its maturity date approaches,if thevalue of thestockdoes not change  These generalpropertiesof the relationbetweenthe optionvalue and thestockpriceare oftenillustrated in a diagramlikefigure1  Line A representsthemaximumvalue of theoption,sinceit cannotbe worthmorethan the stock Line B represents the minimumvalue of the option,since its value cannotbe negativeand cannotbe less thanthestockpriceminusthe exerciseprice Lines T1, T2, and T3 representthe value of the optionfor shortermaturities  successively Normally,thecurverepresenting the value of an optionwill be concave upward Since it also lies below the 45 line, A, we can see that the $40  A    $3    TI                  c$20 oa     $10    0l o           '     48  Add        ' r   0 T     $10 $20 1 $30 $40 Stock Price (ExercisePrice   $20) FiG  1  The relationbetweenoptionvalue and stock price This content downloaded from 137 52 76 29 on Sat, 06 Jun 2015 17 53 42 UTC All use subject to JSTOR Terms and Conditions OPTIONS 639 AND LIABILITIES option will be more volatile than the stock  A given percentage change in the stock price,holding maturityconstant,will result in a largerpercentage change in the option value  The relative volatility of the option is not constant,however  It depends on both the stock price and maturity  Most of the previous work on the valuation of options has been expressed in termsof warrants For example, Sprenkle (1961), Ayres (1963), Boness (1964), Samuelson (1965), Baumol, Malkiel, and Quandt (1966), and Chen (1970) all produced valuation formulas of the same general form Their formulas,however, were not complete, since they all involved one or more arbitraryparameters  For example  Sprinkle's formulaforthe value of an option can be written as follows  kxN(b1)   lnkx c   by 1  v2(t  2 t) vv (t  t) lnkx c  b2 k cN(b2) 2 v2(t  tAv(t  t) t) In this expression,x is the stock price, c is the exercise price, t  is the maturitydate, t is the currentdate, v2 is the variance rate of the returnon the stock,' In is the natural logarithm,and N(b) is the cumulative normal density function But k and k  are unknown parameters  Sprenkle (1961) definesk as the ratio of the expected value of the stock price at the time the warrantmatures to the currentstock price, and k  as a discount factor that depends on the risk of the 'stock  He tries to estimate the values of k and k  empirically,but finds that he is unable to do so  More typically, Samuelson (1965) has unknown parameters x and (, where x is the rate of expected returnon the stock, and ( is the rate of expected returnon the warrant or the discount rate to be applied to the warrant 2He assumes that the distributionof possible values of the stock when the warrant matures is log normal and takes the expected value of this distribution,cuttingit offat the exercise price  He then discounts this expected value to the present at the rate (  Unfortunately,there seems to be no model of the pricingof securitiesunder conditions of capital market 1 The variance rate of the return on a securityis the limit, as the size of the intervalof measurementgoes to zero, of the variance of the returnover that interval divided by the length of the interval  2The rate of expected returnon a securityis the limit,as the size of the interval of measurementgoes to zero, of the expected return over that interval divided by the lengthof the interval  This content downloaded from 137 52 76 29 on Sat, 06 Jun 2015 17 53 42 UTC All use subject to JSTOR Terms and Conditions 640 JOURNAL OF POLITICAL ECONOMY equilibrium thatwouldmakethisan appropriate procedurefordetermining thevalueof a warrant  In a subsequentpaper,Samuelsonand Merton(1969) recognizethe fact thatdiscounting the expectedvalue of the distribution of possiblevalues of the warrantwhenit is exercisedis not an appropriateprocedure They advancethe theoryby treatingthe optionpriceas a functionof the stock price They also recognizethat the discountratesare determined in part by the requirement thatinvestorsbe willingto hold all of the outstanding amountsof both the stockand the option But theydo not make use of thefactthatinvestors mustholdotherassetsas well,so thattheriskof an optionor stockthat affectsits discountrate is only that part of the risk thatcannotbe diversified away  Their finalformuladependson the shape of theutilityfunctionthattheyassumeforthe typicalinvestor  One of theconceptsthatwe use in developingour modelis expressedby Thorp and Kassouf (1967)  They obtain an empiricalvaluationformula forwarrants by fitting a curveto actual warrantprices Then theyuse this formulato calculatetheratioof sharesof stockto optionsneededto create a hedgedpositionby goinglong in one securityand shortin the other  Whattheyfailto pursueis thefactthatin equilibrium, theexpectedreturn on such a hedgedpositionmustbe equal to the returnon a risklessasset  What we show below is that this equilibriumconditioncan be used to derivea theoretical valuationformula  The Valuation Formula In derivingour formulaforthevalue of an optionin termsof the priceof the stock,we will assume  ideal conditions in the marketforthe stock and fortheoption  a) The short term interestrateis knownand is constantthroughtime  b) The stockprice followsa randomwalk in continuoustime with a variancerateproportional to the square of the stockprice Thus the distribution of possiblestock pricesat the end of any finiteintervalis lognormal The variancerateof the returnon the stockis constant  c) The stockpays no dividendsor otherdistributions  d) The option is  European,  that is, it can only be exercisedat maturity  e) There are no transactioncosts in buyingor sellingthe stockor the option  f) It is possibleto borrowany fractionof the price of a securityto buy it or to hold it, at the short term interestrate  g) There are no penaltiesto shortselling A sellerwho does not own a securitywillsimplyacceptthepriceof thesecurityfroma buyer,and will agree to settle with the buyer on some futuredate by payinghim an amountequal to thepriceof the securityon thatdate  This content downloaded from 137 52 76 29 on Sat, 06 Jun 2015 17 53 42 UTC All use subject to JSTOR Terms and Conditions OPTIONS AND LIABILITIES 64I Under these assumptions,the value of the option will depend only on the price of the stock and time and on variables that are taken to be known constants  Thus, it is possible to create a hedged position, consistingof a long position in the stock and a short position in the option, whose value will not depend on the price of the stock, but will depend only on time and the values of known constants  XWriting w(x, t) for the value of the option as a functionof the stock price x and time t, the number of options that must be sold short against one share of stock long is  1 wI(x,t)  (1) In expression (1), the subscript refersto the partial derivative of w(x,t) with respect to its firstargument  To see that the value of such a hedged position does not depend on the price of the stock, note that the ratio of the change in the option value to the change in the stock price, when the change in the stock price is small, is w1(x,t)  To a first approximation, if the stock price changes by an amount Ax, the option price will change by an amount w1(xt) Ax, and the number of options given by expression (1) will change by an amount Ax  Thus, the change in the value of a long position in the stock will be approximately offsetby the change in value of a short position in 11w options  As the variables x and t change, the number of options to be sold short to create a hedged position with one share of stock changes  If the hedge is maintained continuously,then the approximationsmentionedabove become exact, and the return on the hedged position is completely independent of the change in the value of the stock  In fact, the returnon the hedged position becomes certain 3 To illustrate the formationof the hedged position, let us refer to the solid line (T ) in figure1 and assume that the price of the stock starts at $15 00, so that the value of the option starts at $5 00  Assume also that the slope of the line at that point is 1112 This means that the hedged position is created by buying one share of stock and selling two options short  One share of stock costs $15 00, and the sale of two options brings in $10 00, so the equity in this position is $5 00  If the hedged position is not changed as the price of the stock changes, then there is some uncertaintyin the value of the equity at the end of a finiteinterval Suppose that two options go from$10 00 to $15 75 when the stock goes from$15 00 to $20 00, and that they go from $10 00 to $5 75 when the stock goes from $15 00 to $10 00  Thus, the equity goes from $5 00 to $4 25 when the stock changes by $5 00 in either direction  This is a $ 75 decline in the equity for a $5 00 change in the stock in either direction 4 3 This was pointed out to us by Robert Merton  4 These figuresare purely for illustrativepurposes They correspondroughlyto the way figure1 was drawn, but not to an option on any actual security  This content downloaded from 137 52 76 29 on Sat, 06 Jun 2015 17 53 42 UTC All use subject to JSTOR Terms and Conditions 642 JOURNAL OF POLITICAL ECONOMY In addition,the curve shifts (say fromT2 to T3 in fig  1) as the maturityof the optionschanges The resultingdecline in value of the optionsmeansan increasein theequityin thehedgedpositionand tendsto offsetthepossiblelossesdue to a largechangein thestockprice  Note that the declinein the equityvalue due to a large changein the stockpriceis small  The ratio of the declinein the equity value to the magnitudeof the changein the stockpricebecomessmalleras the magnitudeof thechangein thestockpricebecomessmaller  Note also that the directionof the changein the equityvalue is independentof the directionof the changein the stockprice This meansthat underour assumptionthat the stock price followsa continuousrandom walk and that the returnhas a constantvariancerate, the covariance betweenthe returnon theequityand the returnon the stockwill be zero  If the stockpriceand the value of the  marketportfolio followa joint continuousrandomwalk withconstantcovariancerate,it means that the covariancebetween the returnon the equity and the returnon the marketwill be zero  Thus the riskin the hedgedpositionis zero if the shortpositionin the optionis adjusted continuously  If the positionis not adjusted continuously,theriskis small,and consistsentirelyof riskthatcan be diversified away by forming a portfolioof a largenumberof such hedgedpositions  In general,since the hedgedpositioncontainsone share of stock long and 11wi optionsshort,thevalue of theequityin the positionis  x w w1  (2) The changein thevalue of theequityin a shortintervalAt is  Ax Aw w1  (3) Assumingthat the shortpositionis changedcontinuously, we can use stochasticcalculus to expandAw, whichis w(x   Ax,t   At)   w(x,t), as follows  Aw  w1Ax   2 W11V2X2At   w2At  (4) In equation (4), the subscriptson w referto partialderivativesand v2 is thevariancerateof the returnon thestock  Substituting fromequation (4) intoexpression(3), we findthatthechangein thevalue of theequity in thehedgedpositionis     t w1 1v x2  w2V At w1 ((5) Since the returnon the equityin the hedgedpositionis certain,the returnmustbe equal to rAt  Even if the hedgedpositionis not changed 5For an expositionof stochasticcalculus,see McKean (1969)  B1See footnote1  This content downloaded from 137 52 76 29 on Sat, 06 Jun 2015 17 53 42 UTC All use subject to JSTOR Terms and Conditions OPTIONS 643 AND LIABILITIES continuously, its riskis small and is entirelyriskthat can be diversified away,so theexpectedreturnon the hedgedpositionmustbe at the short terminterestrate 7If this werenot true,speculatorswould tryto profit by borrowing largeamountsof moneyto createsuchhedgedpositions,and wouldin theprocessforcethe returnsdownto theshortterminterestrate  Thus the changein the equity (5) mustequal the value of the equity (2) timesrAt    w11v2x2  w2 At w1   (x   w wi)rAt  (6) Droppingthe At fromboth sides,and rearranging, we have a differential equationforthevalue of the option  W2  rw   rxwl     1 2 22 v2x2w1l  (7) Writingt  forthematurity date of theoption,and c fortheexerciseprice, we knowthat  w(xt ) X ' C x c C, (8)  O x

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