MODIGLIANI AND MILLER:THEORY OF INVESTMENT 269 its capital structure and is egual to the capitalization rate of a pure equity stream of its class. To establish Proposition I we will show that as long as the relations (3)or(4)do not hold between any pair of firms in a class,arbitrage will take place and restore the stated equalities.We use the term arbitrage advisedly.For if Proposition I did not hold,an investor could buy and sell stocks and bonds in such a way as to exchange one income stream for another stream,identical in all relevant respects but selling at a lower price.The exchange would therefore be advantageous to the inves- tor quite independently of his attitudes toward risk.As investors exploit these arbitrage opportunities,the value of the overpriced shares will fall and that of the underpriced shares will rise,thereby tending to eliminate the discrepancy between the market values of the firms. By way of proof,consider two firms in the same class and assume for simplicity only,that the expected return,X,is the same for both firms. Let company 1 be financed entirely with common stock while company 2 has some debt in its capital structure.Suppose first the value of the levered firm,V2,to be larger than that of the unlevered one,Vi.Con- sider an investor holding sa dollars'worth of the shares of company 2, representing a fraction a of the total outstanding stock,S2.The return from this portfolio,denoted by Y2,will be a fraction a of the income available for the stockholders of company 2,which is equal to the total return X2 less the interest charge,rDa.Since under our assumption of homogeneity,the anticipated total return of company 2,X2,is,under all circumstances,the same as the anticipated total return to company 1,Xi,we can hereafter replace X2 and Xi by a common symbol X. Hence,the return from the initial portfolio can be written as: (5) Y2=a(X-rD2). Now suppose the investor sold his aS:worth of company 2 shares and acquired instead an amount s=(S2+D)of the shares of company 1. He could do so by utilizing the amount aS2 realized from the sale of his initial holding and borrowing an additional amount aDa on his own credit,pledging his new holdings in company 1 as a collateral.He would thus secure for himself a fraction s1/S1=a(S2+D2)/S1 of the shares and earnings of company 1.Making proper allowance for the interest pay- ments on his personal debt aDa,the return from the new portfolio,Yi,is given by: u In the language of the theory of choice,the exchanges are movements from inefficient points in the interior to efficient points on the boundary of the investor's opportunity set;and not movements between efficient points along the boundary.Hence for this part of the analysis nothing is involved in the way of specific assumptions about investor attitudes or behavior other than that investors behave consistently and prefer more income to less income,ceferis paribus. This content downloaded from 202.120.21.61 on Thu,30 Nov 201707:07:36 UTC All use subject to http://about.jstor.org/termsMODIGLIANI AND MILLER: THEORY OF INVESTMENT 269 its capital structure and is equal to the capitalization rate of a pure equity stream of its class. To establish Proposition I we will show that as long as the relations (3) or (4) do not hold between any pair of firms in a class, arbitrage will take place and restore the stated equalities. We use the term arbitrage advisedly. For if Proposition I did not hold, an investor could buy and sell stocks and bonds in such a way as to exchange one income stream for another stream, identical in all relevant respects but selling at a lower price. The exchange would therefore be advantageous to the inves- tor quite independently of his attitudes toward risk.1' As investors exploit these arbitrage opportunities, the value of the overpriced shares will fall and that of the underpriced shares will rise, thereby tending to eliminate the discrepancy between the market values of the firms. By way of proof, consider two firms in the same class and assume for simplicity only, that the expected return, X, is the same for both firms. Let company 1 be financed entirely with common stock while company 2 has some debt in its capital structure. Suppose first the value of the levered firm, V2, to be larger than that of the unlevered one, Vi. Con- sider an investor holding S2 dollars' worth of the shares of company 2, representing a fraction a of the total outstanding stock, S2. The return from this portfolio, denoted by Y2, will be a fraction ac of the income available for the stockholders of company 2, which is equal to the total return X2 less the interest charge, rD2. Since under our assumption of homogeneity, the anticipated total return of company 2, X2, is, under all circumstances, the same as the anticipated total return to company 1, XI, we can hereafter replace X2 and Xi by a common symbol X. Hence, the return from the initial portfolio can be written as: (5) Y2- a(X - rD2). Now suppose the investor sold his aS2 worth of company 2 shares and acquired instead an amount Sl= a(S2+D2) of the shares of company 1. He could do so by utilizing the amount aS2 realized from the sale of his initial holding and borrowing an additional amount aD2 on his own credit, pledging his new holdings in company 1 as a collateral. He would thus secure for himself a fraction sl/S = a(S2+?D2)/S, of the shares and earnings of company 1. Making proper allowance for the interest pay- ments on his personal debt aD2, the return from the new portfolio, Y1, is given by: 11 In the language of the theory of choice, the exchanges are movements from inefficient points in the interior to efficient points on the boundary of the investor's opportunity set; and not movements between efficient points along the boundary. Hence for this part of the analysis nothing is involved in the way of specific assumptions about investor attitudes or behavior other than that investors behave consistently and prefer more income to less income, ceteris paribus. This content downloaded from 202.120.21.61 on Thu, 30 Nov 2017 07:07:36 UTC All use subject to http://about.jstor.org/terms