14 THE REVIEW OF ECONOMICS AND STATISTICS There seems to be a general presumption among uncertainty per se(as distinct from the effects of economists that relative risks are best measured diverse expectations),and to derive further by the standard deviation (or coefficient of implications of such uncertainty.In particular, variation)of the rate of return,*but in the simp-the aggregate market value of any company's lest cases considered-specifically when all equity is equal to the capitalization at the risk- covariances are considered to be invariant (or free interest rate of a uniquely defined certainty- zero)-the indifference functions are shown to equivalent of the probability distribution of the be linear between expected rates of return and aggregate dollar returns to all holders of its stock. their variance,not standard deviation.(With For each company,this certainty equivalent is variances fixed,the indifference function between the expected value of these uncertain returns less the ith expected rate of return and its pooled an adjustment term which is proportional to covariance with other stocks is hyperbolic.)their aggregate risk.The factor of proportion- There is no simple relation between the expected ality is the same for all companies in equilibirum rate of return required to maintain an investor's and may be regarded as a market price of dollar relative holding of a stock and its standard devia- risk.The relevant risk of each company's stock tion.Specifically,when covariances are non- is measured,moreover,not by the standard de- zero and variable,the indifference functions are viation of its dollar returns,but by the sum of the complex and non-linear even if it is assumed that variance of its own aggregate dollar returns and the correlations between rates of return on differ-their total covariance with those of all other stocks ent securities are invariant. The next section considers some of the impli- To this point we follow Tobin [21]and Marko- cations of these results for the normative aspects witz14]in assuming that current security prices of the capital budgeting decisions of a company are given,and that each investor acts on his own whose stock is traded in the market.For sim (perhaps unique)probability distribution over plicity,we impose further assumptions required rates of return given these market prices.In the to make capital budgeting decisions independent rest of the paper,we assume that investors'of decisions on how the budget is financed.The joint probability distributions pertain to dollar capital budgeting problem becomes a quadratic returns rather than rates of returns,and for programming problem analogous to that intro- simplicity we assume that all investors assign duced earlier for the individual investor.This identical sets of means,variances,and covari- capital budgeting-portfolio problem is formula- ances to the distribution of these dollar returns.ted,its solution is given and some of its more However unrealisic the latter assumption may important properties examined.Specifically, be,it enables us,in section IV,to derive a set of the minimum expected return (in dollars of ex- (stable)equilibrium market prices which at pected present value)required to justify the least fully and explicitly reflect the presence of allocation of funds to a given risky project is lent return"classes.Both Propositions I(market value of firm shown to be an increasing function of each of the independent of capital structure)and II(the linear relation following factors:(i)the risk-free rate of return; between the expected return on equity shares and the debt- (ii)the "market price of (dollar)risk";(iii)the equity ratio for firms within a given class)are derived from the above assumptions(and the further assumption that cor- variance in the project's own present value return; porate bonds are riskless securities);they involve no inter- (iv)the project's aggregate present value re- class comparisons,"..nor do they involve any assertion as turn-covariance with assets already held by the to what is an adequate compensation to investors for assuming company,and ()its total covariance with other a given degree of risk...."(p.279). This is,for instance,the presumption of Hirschleifer projects concurrently included in the capital [8,p.I13],although he was careful not to commit himself to budget.All five factors are involved explicitly this measure alone in a paper primarily focussed on other is- in the corresponding (derived)formula for the sues.For an inductive argument in favor of the standard deviation of the rate of return as the best measure of risk,see minimum acceptable expected rale of return on an Gordon [5,especially pp.69 and 761.See also Dorfman in investment project.In this model,all means [3,p.I29 fi.]and Baumol [2]. Except in dominantly "short"portfolios,the constant We also assume that common stock portfolios are not term will be larger,and the slope lower,the higher the (fixed) "inferior goods,"that the value of all other common stocks is level of covariances of the given stocks with other stocks. invariant,and any effect of changes in capital budgets on the sThe dollar return in the period is the sum of the cash covariances between the values of different companies'stocks is dividend and the increase in market price during the period. ignored. This content downloaded from 202.120.21.61 on Mon,06 Nov 2017 02:52:54 UTC All use subject to http://about.jstor.org/terms14 THE REVIEW OF ECONOMICS AND STATISTICS There seems to be a general presumption among economists that relative risks are best measured by the standard deviation (or coefficient of variation) of the rate of return, but in the simp- lest cases considered - specifically when all covariances are considered to be invariant (or zero) - the indifference functions are shown to be linear between expected rates of return and their variance, not standard deviation.4 (With variances fixed, the indifference function between the ith expected rate of return and its pooled covariance with other stocks is hyperbolic.) There is no simple relation between the expected rate of return required to maintain an investor's relative holding of a stock and its standard devia- tion. Specifically, when covariances are non- zero and variable, the indifference functions are complex and non-linear even if it is assumed that the correlations between rates of return on differ- ent securities are invariant. To this point we follow Tobin [211 and Marko- witz [ 141 in assuming that current security prices are given, and that each investor acts on his own (perhaps unique) probability distribution over rates of return given these market prices. In the rest of the paper, we assume that investors' joint probability distributions pertain to dollar returns rather than rates of return5, and for simplicity we assume that all investors assign identical sets of means, variances, and covari- ances to the distribution of these dollar returns. However unrealisic the latter assumption may be, it enables us, in section IV, to derive a set of (stable) equilibrium market prices which at least fully and explicitly reflect the presence of uncertainty per se (as distinct from the effects of diverse expectations), and to derive further implications of such uncertainty. In particular, the aggregate market value of any company's equity is equal to the capitalization at the risk- free interest rate of a uniquely defined certainty- equivalent of the probability distribution of the aggregate dollar returns to all holders of its stock. For each company, this certainty equivalent is the expected value of these uncertain returns less an adjustment term which is proportional to their aggregate risk. The factor of proportion- ality is the same for all companies in equilibirum, and may be regarded as a market price of dollar risk. The relevant risk of each company's stock is measured, moreover, not by the standard de- viation of its dollar returns, but by the sum of the variance of its own aggregate dollar returns and their total covariance with those of all other stocks. The next section considers some of the impli- cations of these results for the normative aspects of the capital budgeting decisions of a company whose stock is traded in the market. For sim- plicity, we impose further assumptions required to make capital budgeting decisions independent of decisions on how the budget is financed.6 The capital budgeting problem becomes a quadratic programming problem analogous to that intro- duced earlier for the individual investor. This capital budgeting-portfolio problem is fornmula- ted, its solution is given and some of its more important properties examined. Specifically, the minimum expected return (in dollars of ex- pected present value) required to justify the allocation of funds to a given risky project is shown to be an increasing function of each of the following factors: (i) the risk-free rate of return; (ii) the "market price of (dollar) risk"; (iii) the variance in the project's own presentvalue return; (iv) the project's aggregate present value re- turn-covariance with assets already held by the company, and (v) its total covariance with other projects concurrently included in the capital budget. All five factors are involved explicitly in the corresponding (derived) formula for the minimum acceptable expected rate of return on an investment project. In this model, all means 6We also assume that common stock portfolios are not "inferior goods," that the value of all other common stocks is invariant, and any effect of changes in capital budgets on the covariances between the values of different companies' stocks is ignored. lent return" classes. Both Propositions I (market value of firm independent of capital structure) and II (the linear relation between the expected return on equity shares and the debt- equity ratio for firms within a given class) are derived from the above assumptions (and the further assumption that cor- porate bonds are riskless securities); they involve no inter- class comparisons, ". . . nor do they involve any assertion as to what is an adequate compensation to investors for assuming a given degree of risk. . . ." (p. 279). 3This is, for instance, the presumption of Hirschleifer [8, p. II 31, although he was careful not to commit himself to this measure alone in a paper primarily focussed on other is- sues. For an inductive argument in favor of the standard deviation of the rate of return as the best measure of risk, see Gordon [5, especially pp. 69 and 76I. See also Dorfman in [3, p. I29 ff.] and Baumol [2]. 4Except in dominantly "short" portfolios, the constant term will be larger, and the slope lower, the higher the (fixed) level of covariances of the given stocks with other stocks. 5The dollar return in the period is the sum of the cash dividend and the increase in market price during the period. This content downloaded from 202.120.21.61 on Mon, 06 Nov 2017 02:52:54 UTC All use subject to http://about.jstor.org/terms