Abstract

Rubinstein criticizes both the Modigliani & Miller capital structure theorem and dividend policy theorem explained in M&M [6] and M&M [4]. He criticizes the M&M theorems in six points. They are similarity with some former opinions, firm value and stock value confusion, additional assumptions requirement opinion, general rejection of the capital structure irrelevancy, importance of separation opinion between dividend and retained earnings, and conflict between the M&M [6] and M&M [4] articles about dividend irrelevancy opinion. They do not accept the similarity assertion. After my study on Fisher [2], Williams [9], and Morton’s opinions in Rubinstein [8], I believe that, it could be thought that Rubinstein is right. Rubinstein also seems right about the conflict between the M&M [6] and M&M [4] articles. The other assertions of Rubinstein could be seen as a brain training. The M&M accept in advance that the M&M theorem is an innovation and a new opinion.

1. Introduction

In ¹⁰, many critiques to M&M theorems and replies by the Modigliani and Miller together or separately were explored. It was interesting that Rubinstein’s critiques are very much about the theorem. Moreover, all of his critiques are negative critiques. His critiques are about the similarity of the M&M Capital Structure Theorem ⁶ to the Fisher ²’s Separation Theorem, Williams ⁹’ Law of the Conservation of Investment Value and Morton in Rubinstein ⁸. There were other critiques than the similarity critique. They were firm value- stock value dilemma, his assumption requirements for the proposition I and the proposition II, not being their arbitrage proof, his general rejection of the relevance, his discrimination opinion about dividend and retained earnings importance, conflict assertion between the M&M ⁶ and the M&M ⁴ articles about the proposition IV (dividend irrelevance). In this article, I will study whether Rubinstein was right or not on his critiques.

2 The Critiques of Rubinstein on the M&M Theorem

Rubinstein ⁸ says that the M&M ⁶ extends to uncertainty the idea in Fisher ²’s Separation Theorem that the financing and production decisions of a firm can be separated.

Rubinstein ⁷ also thinks that William ⁹ provided the first derivation of the M&M capital structure irrelevancy theorem. In another publication, Rubinstein ⁸ continues to think that the William’s book contains what is probably the first exposition of the M&M ⁶ proposition on the irrelevancy of capital structure. Rubinstein explains his opinions about the similarity of the M&M capital structure theorem and the Williams’ ⁹ Law of the Conservation of Investment Value. Rubinstein says about the M&M capital structure at pp. 122-123 in his same book ⁸ that: “It is also the first formal treatment of the Williams ⁹ Law of the Conservation of Investment Value, showing that in a perfect market, the value of a firm is independent of its capital structure. Although this result was clearly anticipated by Williams, Modigliani and Miller argue that Williams does not really prove his law because he has not made it clear how an arbitrage opportunity would arise if his Law were to fail. To quote Modigliani-Miller’s complete comment with regard to Williams: “A number of writers have stated close equivalents of our Proposition I although by appealing to intuition rather than by attempting a proof and only to insist immediately that the results are not applicable to the actual capital markets”. They say in the footnote that: “See, for example, J.B. Williams ( ⁹, esp. pp.72-73); David Durand ¹; and W.A. Morton in Rubinstein ⁸. None of these writers describe in any detail the mechanism which is supposed to keep the average cost of capital constant under changes in capital structure. They seem, however, to be visualizing the equilibrating mechanism in terms of switches by investors between stocks and bonds as the yields get out of line with their riskiness. This is an agreement quite different from the pure arbitrage mechanism underlying our proof, and the difference is crucial”. Rubinstein comments that M&M’s criticism, it seems to him, is questionable with respect to Williams.

It could be understood from the two sources of the Rubinstein that he thinks the M&M capital structure theorem is not original. Its root is William’s “Law of the Conservation of Investment value”.

Rubinstein ⁸ also criticizes the Modigliani and Miller’ article written in 1969 that explains their proof to its essentials. He thinks that it seems to him to Williams’ original insights. He says that they assume there are two otherwise identical firms (that is, with the same total future cash flows from assets), one levered and one not. They then show that if the sum of the current values of the stock and bonds of the leveraged firm were not equal to the current value of the stock of the unleveraged firm, there would be a weak arbitrage opportunity (Proposition I). Rubinstein says that he means by “weak” that, among two investments, any investor would prefer one over the other. In this sense, one investment would dominate the other. Unlike arbitrage, this weaker notion of dominance does not require that one be sold short again the other to create arbitrage profits. According to Rubinstein, as Modigliani and Miller construct their new proof, they seem to be using the weaker notion of dominance to circumvent the need to allow short sales.

According to Rubinstein ⁸, it does seem on the mark with respect to Walter A. Morton. Rubinstein says that Morton says that “the essential difference between the obligations of the same company lies in the priority of claim to earnings and assets. If only one security is issued, it bears all the risk whether it be called a bond, preferred stock, or common stock, and would have the same value provided that the security could share in all the earnings. Similarly, if one individual owned all of the various types of securities issued, his risk would be the same. Legal differences in the event of insolvency or reorganization and tax policy will modify this result. If all the securities were sold in packages of bonds, preferred and common, the risk to each owner would be the same as if it were all common stock. It follows accordingly that the overall cost of money would be unaffected by capital structure if individuals could not differentiate risks”. Morton continues to argue that once investors can specialize their portfolios in one type of the firm’s securities, so that one investor owns only its bonds, say, and another its stock, then clientele effect (some value safety, and others value the higher return that inures to higher risk) can cause the some of the values the investors place on their positions to exceed the value had they not been able to specialize. Rubinstein adds his opinion about the Modigliani and Miller theorem that they argued this subject that he calls William’s law to prove using a number of assumptions that their own later work and the work of several others show to be unnecessary. Rubinstein then gives an example of the assumptions that Modigliani and Miller assume the debt of the firm is riskless and that two firms must exist that are in the same “risk class”. This means that the random variables Xu and Xl are equal in all states of the World.

Rubinstein ⁷ and ⁸ asserts that the M&M looks like the opinion that Morton says that the essential difference between the obligations of the same company lies in the priority of claim to earnings and assets. In my opinion, according to the three-similarity assertion, we can conclude that Rubinstein implies that the M&M is not original.

Rubinstein ⁸ criticizes another subject from the M&M that he sees a confusion in the literature. The confusion he thinks is the difference between the irrelevancy of capital structure for (1) firm value and for (2) the stock price (per share). He says Modigliani and Miller assert at the outset that they want to prove the latter, but end up proving only the former. Clearly, from the standpoint of stockholder-centered corporate financial theory, it is the latter proposition that is paramount. It is easy to see how even if (1) is true, (2) need not be. Consider a firm with one risky debt issue, and the firm issues new debt senior to the original debt, using the proceeds to purchase stock. This will increase the risk of the original (now junior) debt holders and transfer value to the stockholders. Fortunately, the jump from (1) to (2) is, with academic hindsight, easy to see: Given (1), as long as with each recapitalization the original debt holders can intelligently renegotiate their debt contracts, then (2) will hold as well.

Rubinstein ⁸ also thinks that some assumptions are necessary to be understood the M&M theorem. These assumptions for the Proposition I are:

1. No arbitrage: That is, “all equal-sized bites of the pie have the same taste”.

2. Operating income from assets is not affected by capital structure: That is, “the total pie is fixed”.

3. The proportion of operating income that is jointly allocated to stocks and bonds is not affected by the firm’s capital structure: That is, “only stockholders and bondholders eat the pie”.

4. The present value function (the economy wide state- prices) is not affected by capital structure: That is, “the taste per bite of the pie is fixed”.

Rubinstein adds some additional opinion about his four assumptions about the M&M Proposition I. He thinks that Assumption I ensure the existence, but not necessarily the uniqueness, of state-prices. Assumption 2 rules out (1) bankruptcy costs, (2) differential transactions costs in issuing or trading stocks and bonds, (3) managerial incentives to alter operating income that are changed by capital structure such as occurs with employee stock options or capital structure effects on managerial salaries and perks, (4) stockholder incentives to accept high-risk, negative net present value projects that shift value from bondholders to stockholders, and (5) conveyance of information about the operating income of the firm to the market by signaling via capital structure. Assumption 3 rules out differential taxes for income from stock and bonds. Assumption 4 disallows the possibility of creating or destroying desired patterns of returns not otherwise existing in the market by changing capital structure.

Rubinstein ⁸ also thinks that the second related proposition (the proposition II) that capital structure does not affect the stock price requires an additional assumption. This assumption is that “there are no pure transfers between bondholders and stockholders, or between new and old shareholders. This assumption rules out 3 things below:

1. Pure asset substitution, that is, ex-post risk changing projects that can shift value between bondholders and stockholders even though they leave the total market value of the firm unchanged.

2. Use of contractual arrangements between bondholders and stockholders that permit ex-post transfers between them.

3. Violations of “strong-form market efficiency” that permit stockholders or managers to use inside information to issue bonds or stock in response to the failure of the market to reflect that information in their relative prices such as issue stock when it is overpriced and bonds when the stock is underpriced.

Rubinstein ⁸ thinks about the M&M Capital Structure theorem generally that the M&M ⁶ is not seemed as a realistic proof that capital structure is irrelevant but it is seemed relevant. He also claims that the Modigliani& Miller did not even invent arbitrage reasoning or proof as it is sometimes claimed. As it could be seen, his general opinion about the M&M theorem is negative. He does not believe the irrelevance of capital structure.

Rubinstein ⁸ qualifies the reason of M&M dividend theorem ⁴ as a rough intuition. Rubinstein criticizes that after 100 per cent dividend payment, the corporation’s stock price should be zero. On the one hand, shareholders are better-off since they receive the dividend; on the other, they are worse off because the firm now has less to invest by the amount of the dividend, so the stock price falls by the amount of the dividend. Taking these together, the shareholder is no better off as a result of the dividend. For these reasons, the Miller and Modigliani’s dividend irrelevancy theorem is transparent. He also adds that these arguments seem to hold the investment policy of the firm fixed. However, that is not really required. Rubinstein also asserts a paradox between the two articles of Modigliani and Miller ⁶ and ⁴ about dividend irrelevancy theorem explanation. He says that, in M&M ⁶, dividend policy irrelevancy can be interpreted as an immediate and simple consequence of present value additivity. From this perspective, it is easy to see that if a firm reduces its dividend in one period and reinvests the extra retained earnings in a zero net present value project that provides increased future dividends, then (provided state-prices are not affected) the present value of the firm’s dividends, and hence its current stock price, will remain unchanged.

3. The Replies of Modigliani and Miller to Rubinstein

M&M ⁶ says that a number of writers have stated close equivalents of the Proposition I although by appealing to intuition rather than by attempting a proof and only to insist immediately that the results were not applicable to the actual capital markets. The M&M continues their explanation in the same page at the footnotes 13. The Modigliani and Miller say at the footnote that none of these writers such as J.B. Williams ⁹ especially pp. 72-73, Durand ¹, and Morton in Rubinstein ⁸ describe in any detail the mechanism which is supposed to keep the average cost of capital constant under changes in capital structure. However, they seem to be visualizing the equilibrating mechanism in terms of switches by investors between stocks and bonds as the yields get out of line with their riskiness. The M&M think that this is an argument quite different from the pure arbitrage mechanism underlying our proof, and the difference is crucial. Regarding Proposition I as resting on investors’ attitudes toward risk leads inevitably to a misunderstanding of many factors influencing relative yields such as limitations on the portfolio composition of financial institutions. It could be thought that this sentences from the M&M ⁶ are like the answer to Rubinstein ⁷ and ⁸ to Modigliani and Miller. Modigliani and Miller also point out the Proposition II (cost of capital irrelevancy) that the other writers have no opinion about that. As a result, they think that the proposition is original.

Modigliani ⁵ replied the question whether he and Miller had produced the M&M theory using another person’s or people’ opinion or not. Modigliani says that “I had been intrigued by the subject ever since attending a National Bureau conference on Business Finance at which I gave a fairly conventional paper. But mostly I listened to a paper by David Durand ¹ in which the possibility that financial structure would not affect the market valuation or the cost of capital was suggested, only to be rejected as not relevant to the actual capital markets”. It could be seen frankly that he has been affected from Durand’s proceeding. He does not hide it. But, it is not Fisher ², Williams ⁹ or Morton in Rubinstein ⁸. It is Durand ¹. The Modigliani’s Conference story implies that during the M&M theorem is being created, Fisher ², Williams ⁹, and, Morton in Rubinstein ⁸ have not been considered.

Miller ³ says that though departing substantially from the conventional views about capital structure, their propositions were certainly not without links to what had gone before. He says their distinction between the real value of the firm and its financial packaging raised many issues long familiar to economists in discussions of the money illusion and money neutrality. Even some of the particular financial illusions to which they were directing attention had themselves already been noted by others as they duly cited in their paper. These earlier statements from which they were aware of were not followed by finance writers. He points out that the only prior treatment similar in spirit to their own was by Durand ¹. Miller says about Fisher effect assertion that they opted for a Fisherian rather than the standard Marshallian representation of the firm. Fisher’s view of firm focuses on the underlying net cash flow. The firm for Fisher was just an abstract engine transforming current consumable resources, obtaining by issuing securities, into future consumable resources payable to the owners of the securities. Even so, what did it mean to speak of firms or cash flow streams being different, but still similar enough to allow for arbitrage or anything close to it? He finds their (Modigliani and himself) irrelevance opinions closer to Fisher’s opinions than that of Marshall’s.

Both Modigliani ⁵ and Miller ³ give importance to Durand ¹ in their capital structure irrelevance opinion. Miller ³ also points out Modigliani and himself were Fisherians rather than Marshallians in their firm insights.

3.1.1.1.1. Fisher’s Capital Value Insight

Fisher ² says that capital value is simply future income discounted or capitalized. The value of any property or rights to wealth is its value as a source of income and is found by discounting that expected income. He continues his opinion about that as follow: “We may, if we so choose, for logical convenience, include as property the ownership in ourselves, or we may, conformably to custom, regard human beings as in a separate category”. Then, Fisher defines wealth as consisting of material objects owned by human beings including human beings themselves. The ownership may be divided and parceled out among different individuals in the form of partnership rights, shares of stock, bonds, mortgages, and other forms of property rights. In whatever ways the ownership, be distributed and symbolized in documents, the entire group of property rights are merely meaning to an end income which is the “alpha and omega” of economics. Fisher frankly accepts that what is important is only end income rather than its distribution. Of course, the distribution is determined by capital structure. His opinion could be explained that the firm value is created by income not by capital structure.

Fisher ² also says that capital value is merely capitalized income. Moreover, capital value is itself dependent on a preexisting rate of interest. Borrowing and lending are in form a transfer of capital, but they are in fact a transfer of income of which that capital is merely the present value. It reflects the M&M’s basic logic. Present value is real value, not its distribution between bond and equity owners.

3.1.1.1.2. Williams’ Law of the Conservation of Investment Value

Rubinstein ⁸ comments that while MM’s criticism, it seems to him, is questionable with respect to Williams. Let’s look at Williams ⁹. Willams ⁹ says that if the investment value of an enterprise as a whole is by definition the present worth of all its future distributions to security holders, whether on interest or dividend account, then this value in no wise depends on what the company’s capitalization is. Clearly if a single individual or a single institutional investor owned all the bonds, stocks and warrants issued by a corporation, it would not matter to this investor what the company’s capitalization was, except for details concerning the income tax. Any earnings collected as interest could not be collected as dividends. To such an individual it would be perfectly obvious that total interest and dividend paying power was in no wise dependent on the kind of securities issued to the company’s owner. Furthermore, no change in the investment value of the enterprise as a whole would result from a change in its capitalization. Bonds could be retired with stock issues, or two classes of junior securities such as common stock and warrants could be combined into one, without changing the investment value of the company as a whole. Actually, the opinion is very similar with the M&M theorem, the Proposition I. It could be understood that there was irrelevance opinion before the M&M theory. However, it was not open and detailed by the M&M theorem. Maybe, the Modigliani and Miller could had informed about the concept irrelevance via this publication. In my opinion, especially, William’s explanation of the irrelevance opinion via present value of future cash distribution to the owners and creditors is very logical. If the company pays more interest it can pay less dividend to the owners, the stockholders. Again, if the company pays less interest it can pay more dividend. If same person has the all securities, the money will be collected will be same for the two condition. For this reason, Rubinstein seems right about the similarity to the William’s capital irrelevancy opinion for the Proposition I of the M&M theorem.

In the conclusion section of the M&M ⁶, Modigliani and Miller replied in advance to Rubinstein’s some critiques. They say that much remains to be done before the cost of capital can be put away on the shelf among the solved problems. They say that their approach has been that of static, partial equilibrium analysis. It has assumed among other things a state of atomistic competition in the capital markets and an ease of access to those markets which only a relatively small (though important) group of firms even come close to possessing. These and other drastic simplifications have been necessary in order to come to grips with the problem at all. Having served their purpose they can now be relaxed in the direction of greater realism and relevance, a task in which they hope others interested in this area will wish to share. They already expect from other financial scientists to complete the theorem by cancelling some assumptions and adding some other necessary things such as different data etc. The M&M offered some propositions, not rules. Of course, the propositions would be discussed. All offerings are discussed in the life. Financial propositions are not excluded from this reality. Rubinstein’s critiques and opinions could be thought in this framework.

Rubinstein criticizes that dividend irrelevancy is transparent and there is a conflict between the M&M ⁶ and the M&M ⁴ about dividend policy irrelevancy.

The Modigliani& Miller ⁶ says that caution is indicated especially with regard to their test of Proposition II, partly because of possible statistical pitfalls and partly because not all the factors that might have a systematic effect on stock yields have been considered. In particular, no attempt was made to test the possible influence of the dividend payout ratio whose role has tended to receive a great deal of attention in current research and thinking. The M&M say that there are two reasons for this omission. First, their main objective has been to assess the prima facie tenability of their model, and in this model, based as it is on rational behavior by investors, dividend per se play no role. Second, in a world in which the policy of dividend stabilization is widespread, there is no simple way of disentangling the true effect of dividend payment on stock prices from their apparent effect, the latter reflecting only the role of dividends as a proxy measure of long-term measure anticipations. The difficulties just mentioned are further compounded by possible interrelations between dividend policy and leverage. In the footnote 43 at p.289 they suggest that failure to appreciate this difficulty is responsible for many fallacious, or at least unwarranted, conclusions about the role of dividends. In the footnote 44 at the same page they point out that in the sample of electric utilities, there is a substantial negative correlation between yield and pay-out ratios, but also between pay-out ratios and leverage, suggesting that either the association of yields and leverage or of yields and pay-out ratios may be (at least partly) spurious. However, these difficulties do not arise in the case of the oil industry sample. A preliminary analysis indicates that there is here no significant relation between leverage and pay-out ratios and also no significant correlation (ether gross or partial) between yields and payout ratios.

The Modigliani & Miller ⁶ says that in the case of retaining earnings, suppose that in the course of its operations the firm acquired I dollars of cash without impairing the earning power of its assets. If the cash is distributed as a dividend to the stockholders their wealth W0, after the distribution will be:

(1)

where X ¯0 represents the expected returns from the assets exclusive of the amount I in question. If, however the funds are retained by the company and used to finance new assets whose expected rate of return is p*, then the stockholders’ wealth would become:

(2)

They say that clearly W1 ≤ ⦥ W0 as p*≤ ⦥ pk so that an investment financed by retained earnings raises the net worth of the owners if and only if p*˃ pk. Miller &Modigliani ⁴ says that the dividend irrelevancy proposition is under perfect capital markets, rational behavior, and perfect certainty, the valuation of all shares would be governed by the following fundamental principle: the price of each share must be such that the rate of return (dividends plus capital gain per dollar invested) on every share will be same throughout the market over any given interval of time.

The M&M ⁶ means that if there is no dividend payment because of retained earnings, the firm value increases. The condition of p*˃ pk is valid for all the investment of the firm. It does not affect the opinion of the M&M that not to distribute the profit and transfer it to the retained earnings (self-financing) increases the firm value. It is against to the dividend irrelevancy opinion of M&M. In this point, Rubinstein seems right.

The M&M ⁴ accepts that the dividend and retained earnings is equal and what is important is their total, that is rate of return. For this reason, distributing dividend or not distributing it not important for firm value. The important thing about that is the return.

I think, it seems that, as if Rubinstein is right about the conflict between the M&M ⁶ and M&M ⁴. The first publication separates effects of dividends and retained earnings. The second publication does not separate them. This means an adoption of the irrelevancy about dividend policy. It could be thought that the M&M’s opinion is different in 1958 and 1961.

Rubinstein ⁷ and ⁸ criticizes the M&M theorems. His brain storming in some points of the M&M and answers or replies of Rubinstein and this writer are shown at the Table 2 below:

General summary of Rubinstein’s rightfulness about the M&M theorems is shown at the Table 3 below:

4. Conclusion

Rubinstein has much critiques on the Modigliani and Miller’s capital structure and dividend policy theorems. When the critiques are studied, it could be seen that some of them are useful critiques. From the point of view of continuity of science, similarity critique of Rubinstein could be seen acceptation of opinions of Fisher, Williams and Morton. Similar opinions can be improved every time in the life. Modigliani and Miller have done this, too. They have improved some financial opinions to contribute the financial science.

The source list of the M&M ⁶ does not cover any article from Fisher. However, Modigliani and Miller accept together and separately that they were affected from Fisher’s opinions generally. Miller ⁵ also points out that Modigliani and Himself were Fisherians rather than Marshallians in their firm insights. This could show Fisher effect on the M&M capital structure theorem indirectly.

Debt value could be important in firm value depending on debt ratio. The M & M have studied unleveraged firms and leveraged firms together to fix their three propositions. Debt means risk increase after some level. This risk affects stock price, so the firm value is affected. At the same, the M&M already proposed their second proposition that risk of debt increase affects stock value, so firm value is not affected. It means that if debt ratio increase, cost of capital seems as if it decreases, but because of risk increase with debt increase, cost of equity increases, too. As a result, the firm value does not change. Stock value represents only equity, but firm value represents both of equity and debt of firm. I think expansion of research of Modigliani and Miller caused to change the target of explaining stock value to firm value. It could be thought differently from other financial scientists. This is only my opinion.

Additional assumption advices of Rubinstein could be seen as brain storming. For instance, his additional assumptions for the Proposition I such as operating income from assets is not affected by capital structure, the proportion of operating income that is jointly allocated to stocks and bonds is not affected by the firm’s capital structure, and the present value function is not affected by capital structure explain same thing. This thing is “firm value is created in asset side of balance sheet”. His “no arbitrage” assumption suggestion for the Proposition I am opposite to the M&M’ assumption “investors use arbitrage opportunity to eliminate the discrepancy between the market values of the firms”. It summarizes the change of capital among different financing sources. However, the M&M tries to prove the hypothesis “even if the capital structure change, the firm value will not change”. This could be seen an effort by Rubinstein to prove that “if no arbitrage, the capital structure will not change, so the M&M capital structure theorem could not be proved”.

His additional “no pure transfer between stock and debt” assumption for the Proposition II helps his rejection of the M&M capital structure irrelevancy theorem.

Rubinstein’s the M&M dividend policy theorem (the Proposition IV) critiques seem more realistic and consistent with logic. Much dilemmas could be seen if the theorem is studied carefully. For this reason, it seems that Rubinstein is pretty right about his critiques on the M&M dividend policy irrelevance theorem. The M&M’s opinion is different in 1958 and 1961. The first publication separates dividend and retained earnings. The second publication does not separate them. The M&M ⁶ means that if there is no dividend payment because of retained earnings, the firm value increases. The M&M ⁴ accepts that the dividend and retained earnings is equal and what is important is their total value, that is amount of return. For this reason, distributing dividend or not distributing it is not important for the firm value. The important thing about that is the return.

I think the discussion about the M&M theorems will continue in the future, too.

Notes

1. Williams, John Burr, The Theory of Investment Value, North Holland Publishing Company, Amsterdam, Netherland, 1938.

2. Rubinstein gives these sentences from MM (1958:271) about the subject. He also gives some other sentences in the footnote 13 at the same page of the M&M. These sentences are: “None of these writers [See, for example, J.B. Williams ⁹, esp. pp. 72–73]; David Durand ¹; and W.A. Morton ⁸ describe in any detail the mechanism which is supposed to keep the average cost of capital constant under changes in capital structure. They seem, however, to be visualizing the equilibrating mechanism in terms of switches by investors between stocks and bonds as the yields get out of line with their “riskiness.” This is an argument quite different from the pure arbitrage mechanism underlying our proof, and the difference is crucial”.

3. Rubinstein (2006) says that Morton writes these opinions in [Morton (1954:442)] “The Structure of the Capital Market and the Price of Money,” American Economic Review 44, No. 2 (May 1954), pp.440-454.

4. Rubinstein provides a proof at (2006:130) that the Modigliani and Miller did not provide in their study. This proof is: The proof starts with a firm without debt VU = nSU. Then, the company buys back m shares and replaces them with debt. As a result, VL= (n – m) SL + DL = (n – m) SL + mSL = nSL. In this formula, the stock of the leveraged firm may sell at a different price per share SL. Then, VL = nSL, VU = nSU, and VL = VU, then SL = SU.

5. A state-price security is a contract that agrees to pay one unit of a numeraire (a currency or a commodity) if a particular state occurs at a particular time in the future and pays zero numeraire in all the other states (https://en.wikipedia.org/wiki/State_prices).

6. This subsection was prepared by using some knowledge from Yilmaz, H. (2020), “The Modigliani &Miller Capital Structure Theorem After 62 Years”, Journal of Business and SocialScienceReview, Vol. 1, No.11; November 2020, pp.36-56.

7. Morton W.A.,” The Structure of the Capital Market and the Price of Money”, American Economic Review, May 1954, 44, 440-454.

8. He means Modigliani and Ziman, 1952.

9. “Alpha and omega” is an idiom and it means “the basic or essential element or elements” (https://www.dictionary.com/browse/alpha-and-omega).

10. This table was produced from the Table V of Yilmaz, 2020: 51 ¹⁰.

11. It is the Equation 25 at the M&M (1958).

12. p* shows the rate of return on investment.

13. It is the Equation 26 at the M&M (1958).

References