The Review of Economic Studies, Ltd.

The Review of Economic Studies, Ltd. Mean-Variance Analysis in the Theory of Liquidity Preference and Portfolio Selection Author(s): M. S. Feldstein Reviewed work(s): Source: The Review of Economic Studies, Vol. 36, No. 1 (Jan., 1969), pp. 5-12 Published by: Oxford University Press Stable URL: http://www.jstor.org/stable/2296337. Accessed: 23/10/2012 23:39 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.. Oxford University Press and The Review of Economic Studies, Ltd. are collaborating with JSTOR to digitize, preserve and extend access to The Review of Economic Studies. http://www.jstor.org

Mean-Variance Analysis in the Theory of Liquidity Preference and Portfolio Selection1 The purpose of this paper is to correct two errors in the theory of economic behaviour under uncertainty and to note some implications for the theory of liquidity preference and portfolio selection. Section 1 shows that Tobin [16], [17], was incorrect in asserting that the 4u-c indifference curves of a risk-averter are convex-downwards whenever the possible investment outcomes are assumed to follow a two-parameter probability distribution. Although Tobin's proof is correct for normal distributions, for a number of economically interesting distributions the indifference curves are not convex. Section 2 shows that portfolios formed by combining a riskless asset (money) and a risky asset (bonds) with a two-parameter probability distribution cannot in general be ranked in the terms of 4u-c indifference curves. The implication of this for the pure theory of liquidity preference is then examined. In particular, it is shown that investors may be more likely to hold a bonds-only portfolio than was suggested by Tobin's analysis. Section 3 shows that when more than one asset has positive variance, an analysis in terms of only,u and ai is not strictly possible unless utility functions are quadratic or the possible subjective probability distributions are severely restricted. 1. INDIFFERENCE CURVE CONVEXITY In his pioneering application of expected utility maximization to the theory of liquidity preference, Professor Tobin considered the implications of the assumption that an investor's preferences among portfolios might be represented in terms of the expected outcome of each portfolio (,u) and its standard deviation (ai). This simplification was justified in two ways. If the investor's utility function is quadratic, the expected utility associated with any probability distributions depends only on,u and ai (Tobin[16], p. 77). Alternatively, regardless of the form of the investor's utility function, if the (subjective) probability distributions of the possible portfolios are all members of a two-parameter family of distributions, preferences can be analyzed in terms of,u and ca.2 The basic conclusions of Tobin's theory of liquidity preference and portfolio choice rest on the properties attributed to these 4u-c indifference curves. When analysis is limited to the risk-averter (the investor whose utility function is characterized by decreasing marginal utility), Tobin claims that the same first-order and second-order properties of 1 I am grateful for discussion with K. Borch, M. Dempster, J. Flemming, W. Gorman, J. Helliwell and M. Nerlove, and for valuable comments from a referee. 2 Of course, preferences can only be represented in terms of, and a if the distributions are assumed to have finite means and variances. Mandelbrot ([11], [12]) and Fama ([3], [4]) have recently argued that share price changes and other investment variables may have infinite variances. But their conclusion rests on the strong assumptions that all observations on any given price-change variable are drawn from the same distribution and that that distribution is a member of the " stable " (Levy-Pareto) class. If these assumptions are accepted, then a finding that the variable is not normally distributed implies that it has infinite variance. There seems no reason to accept both underlying assumptions. We therefore assume throughout this paper that all distributions have finite means and variances. 5

6 REVIEW OF ECONOMIC STUDIES the indifference curves can be obtained from the assumption of either a quadratic utility function or a two-parameter probability distribution. The first-order property is that along every indifference curve d4u/dc>0; i.e., an investor is indifferent between two portfolios with different variances only if the portfolio with greater ai also has greater Pu. The second-order property is d24u/dc2>0 along every indifference curve; i.e., the rate at which an individual must be compensated for accepting greater ai (dlu/dc) increases as ay increases. If,u is measured on the vertical axis and ai on the horizontal, the indifference curves are convex.1 The convexity property plays an important part in Tobin's analysis and is fundamental to his explanation of investor diversification: Tobin calls an individual with convex indifference curves a "diversifier" and summarizes his analysis of the risk-averter's indifference curve properties by saying " All risk-averters are diversifiers " ([16], p. 76). A number of other writers have followed Tobin in using convex indifference curves (Hall, [8]; Hicks [9]; Hirshleifer [10]; and Patinkin [14]. This widely accepted assumption about indifference curve convexity is not generally correct. Although the asserted convexity property is correct if the investor does have a quadratic utility function, with other utility functions the indifference curves need not be convex even if the subjective probability distributions are all members of some two-parameter family. If the assumption of a quadratic utility function is rejected, either because it is deemed too restrictive or because of its implausible implications,2 it is no longer possible to assume convexity. It will first be shown that Tobin's proof is not correct for all twoparameter distributions. Of course, this does not establish that his convexity assertion is false. To do this, we present an economically important counterexample. Professor Tobin's proof ([16], pp. 75-76) that risk-averters with two-parameter subjective probability distributions have convex indifference curves may be summarized as follows: (i) The expected utility associated with a distribution f(x; Pu, a), denoted EU(,u, a) is given by EU(,u, a) = S U(x)f(x; j, c)dx -ju(p+ciz)f(z; 0, I)dz. (ii) Let the investor be indifferent between two distributionsf(x; P, a) andf(x;,', a'); i.e., EU(u, a) = EU(u', a') and the two points (,u, a) and (,', a') lie on the same indifference curve. (iii) Diminishing marginal utility implies that for every z, IU(i+Cz)+ U(I + ' c iz)<u 2 + ' Z) (iv) Therefore EU 2 '2?) >EU(yu, a) = EU(4u', c'). (v) Since the point (u +4Lf Ci+Ci lies on a straight line between (y, a) and (,u', ci') and is on a higher indifference curve, the indifference curves are convex. The crucial part of step (i) of this proof assumes that any two-parameter distribution f(x;,u, C) can be put into " standard form ", f(z; 0, 1) with z = (x-#u)/ci. But this is 1 A curve is convex, sometimes called convex-downward, if a chord connecting two points on the curve lies everywhere above the curve. Tobin uses the term concave-upward; convex-downward is more usual and also more compatible with the equivalent statement that the set of,u-u combinations better than those on any given indifference curve is a convex set. 2 A number of writers have criticized or rejected the quadratic utility function because it implies that risk aversion increases with " wealth " and because it limits the concavity that is compatible with a wide range of possible values of wealth; see Arrow [1]; Feldstein [5]; Fellner [6]; Hicks [9]; and Pratt [15].

LIQUIDITY PREFERENCE AND PORTFOLIO SELECTION 7 not a property of all two-parameter probability distributions. For only a limited class of two-parameter distributions are the mean and standard deviation equivalent to measures of location and scale. Such useful distributions as the lognormal and the beta cannot be transformed in the way required for Tobin's proof. This demonstration that Tobin's proof is only valid for a restricted class of probability distributions is, of course, not a proof that a risk-averter's indifference curves are not always convex. But Tobin's general assertion that risk-averters with two-parameter subjective probability distributions always have convex indifference curves is definitely false. This is shown by the following counter-example. The lognormal distribution defines a two-parameter family, f(x) = 1 e-(logx_m)2/2s2 (2ir)4sx where m is the expected value of log x and S2 is the variance of log x. If the " outcome" x of a portfolio investment is the terminal (i.e., at the end of the planning period) value of the assets (including interest and dividends), the lognormal function is perhaps the most plausible distribution to assume.' Although m and s2 are moments of the logarithm of x, there exists a one-to-one mapping between an m, s2_pair and the corresponding moments of x: m + ( ts )2 a 2 (et + (+s2)2(es2 _.1) The value of the positive third moment of the lognormal distribution provides no additional information to an investor who knows i, a and that the distribution is lognormal; i.e., the lognormal distribution has only two independent parameters. We may now derive a set of nonconvex indifference curves most efficiently by assuming that the individual's utility function is of the simple Bernoulli semi-logarithmic form, u(x) = log x. In this case, the expected utility associated with any probability distribution is E(u) = m = f log xf(x)dx, the logarithm of the geometric mean of x. For the lognormal distribution, an expression for the logarithm of the geometric mean is easily obtained from equation 1. Taking logarithms of the first equation yields m = log ju - (f)s2. Dividing the expression for U2 by that for u2 shows U2/u2 = es2-1 or s2 =log (2 +I Combining these results we obtain: E(u) = m = log -!L logqii +1)( Along an indifference curve of constant expected utility, dy - >0.>? (3 as required for a risk-averter. But calculation of the second derivative along the indifference curve shows that sign - = sign (1-2k4- k2)... (4) where k = -, the coefficient of the variation. The sign of d2 /&r2 changes from positive 1 Just as the normal distribution arises as the sum of random variables, the lognormal is generated by a series of independent multiplicative effects. Substantial evidence has accumulated to show that shareprice changes are serially independent and multiplicative; see Cootner [2], Fama [4], and Granger and Morgenstern [7].

8 REVIEW OF ECONOMIC STUDIES to negative at - = (05)+; this is the only positive root of the polynomial. Thus the indifference curves are convex only in the area of the,u-o plane corresponding to points at which the coefficient of variation is less than (05)+ and are concave elsewhere. This is shown in Fig. 1. p FIGURE 1 2. SOME IMPLICATIONS FOR THE PURE THEORY OF LIQUIDITY PREFERENCE In his original paper, Tobin discusses liquidity preference as the theory of optimum portfolio composition in a two-asset (money, bonds) world in which one asset (money) is riskless. This will now be referred to as the pure theory of liquidity preference to contrast it with the analysis which recognizes that price-level uncertainty makes money a risky asset. This generalized liquidity preference theory, developed by Tobin in a later paper [17], raises problems which are present in the theory of optimal portfolio composition whenever there is more than one risky asset even if one asset is assumed to be riskless. This section discusses the pure theory of liquidity preference; section 3 considers the general theory of liquidity preference and portfolio choice. A significant conclusion of Tobin's analysis is that the investor will generally diversify his portfolio between money and bonds, holding all bonds only in the special case when an interest rate is so high that the market opportunity locus touches the highest possible indifference curve at a point where the slope of the opportunity locus is greater. This diversification provides the basis for Tobin's explanation of the shape of the liquidity preference curve. However, this analysis rests on the use of convex P-c indifference curves to represent the investor's preferences among portfolios of bonds and money. But such a portfolio cannot generally be represented by a point in the ji-a space even if the subjective probability distribution of the bond values has only two parameters. A

LIQUIDITY PREFERENCE AND PORTFOLIO SELECTION 9 mixture of such a random variable and a sure variable generally results in a distribution which is in a different class from the original random variable and which has more than two independent parameters. Although the mixture of a normal variable and a sure variable yields a normal variable, the mixture of a lognormal variable and a sure variable does not yield the two-parameter lognormal distribution. This shows that it is not generally possible to determine the characteristics of investor behaviour, and particularly whether "risk-averters are diversifiers," from geometric considerations. However, in the special case considered in section 1 (a lognormal probability distribution of the terminal value of assets and a semilogarithmic utility function), the investor will be a " plunger," holding only bonds unless their variance is very high in relation to their expected yield. To derive a more explicit statement of this result, let: x- 1 be the proportional yield if the entire portfolio is invested in bonds, It -1 and a2 the expected value and variance of that yield, A be the original value of the assets and p be the proportion of the portfolio actually invested in bonds. Then the investor seeks to maximize: E(u) = E{log [(1 -p)a +pax]} = log A + E{log [1 +p(x- 1)]}. * X(5) We now find the conditions on the distribution of x under which the investor will maximize expected utility by holding only bonds, i.e., the conditions which make ae(u)/ap> 0 for 0 < p < 1 and ae(u)/ap > 0 for p = 1. The first derivative of (5), 8E(u) = E x_ 1 (6) DP L+P(X-l)2 is positive atp = 0 and is a strictly decreasing function of p for all admissablep. Therefore, a bonds-only portfolio will be optimal if DE(u)/ap > 0 at p = 1, i.e., if E ( ) = E 1- ) > 0 or, equivalently, E (1) 1. Since (x) is lognormal with mean It and variance a2, E() (+2) (7) Therefore, the bonds-only portfolio is optimal if (ci)2 < jl1,...(8) i.e., if the square of the coefficient of variation of terminal wealth is less than the expected proportional yield. If the expected bond yield is 5 per cent, an optimum portfolio will contain some money (i.e., p< 1) only if (a/u)2 > 0.05, i.e., only if o4y>0 22 or u>0 23. The bonds-only portfolio is thus optimal unless the standard deviation is more than four times the expected bond yield. Considering a much longer time-period and therefore higher value of p leaves this result basically unaffected. For a fifteen-year period, we may take p = 2 and conclude that the investor will hold money only if a> 4, i.e., if the standard deviation is more than twice the percentage yield. In short, even if the subjective probability distribution of the risky asset can be parametized in terms of the mean and variance, the risk-averter may well be a plunger.

10 REVIEW OF ECONOMIC STUDIES 3. PORTFOLIO CHOICE AND GENERALIZED LIQUIDITY PREFERENCE THEORY A further problem arises if it is recognized that money is not riskless or, more generally, if more than one asset with positive variance may be included in the portfolios. In this case, it is not generally possible to define a preference ordering of portfolios in terms of,u and a alone if each asset in the portfolio has a distribution with only two independent parameters. If we exclude linear and quadratic utility functions, such a preference ordering can be defined if and only if each asset has a distribution such that any linear combination of these variables (assets) has a distribution with only two independent parameters.' This is a very restrictive requirement. If each asset is normally distributed, the requirement is satisfied; the uniform, lognormal, beta and other " standard " two-parameter distributions are inadmissible.2 Although there may be distributions other then the normal which satisfy the requirement, it is unreasonable to assume that investors' subjective probability distributions are necessarily of such a form. Moreover, only the normal would have the additional property that the portfolio distribution would have the same form as its constituents. This simple point, which appears to have been overlooked in previous discussions of generalized liquidity preference and portfolio choice, shows that it is in general impossible within the framework of von Neumann-Morgenstern utility theory to treat the problems of portfolio choice and liquidity preference in terms of mean and variance when there is more than one risky asset. The quadratic utility function and the normal probability distributions are each a sufficient assumption to permit mean-variance analysis and may be the only sufficient assumptions with any credibility. But the quadratic utility function is very restrictive and the normal distribution of investment outcomes appears contrary to available evidence (e.g., Cootner [2]). Some modification of the von Neumann-Morgenstern approach is needed if meanvariance analysis is to be admissible. Perhaps an acceptable assumption is that the investor makes the statistical mistake of assuming that all distributions are stable-i.e., of assuming that the portfolio has the same two-parameter distribution as its constituents. Although such an assumption is a " statistical mistake" unless the assets are normally distributed (or have infinite variance), it corresponds to the implicit assumption that underlies all current mean-variance portfolio analysis. In addition to the plausibility, it has the virtue of allowing the investor to act according to von Neumann-Morgenstern axioms of expected utility maximization-although using incorrect and mutually inconsistent subjective probability distributions.3 I Of course, if the investor does not seek to maximize the expected utility of the investment outcome but has a utility function defined directly in terms of the moments of the portfolio distribution, analysis in terms of I and a may be possible regardless of the actual portfolio distribution. But this is completely alien to the spirit of the von Neumann-Morgenstern approach. 2 Although a sum of gamma variables is a gamma variable, this is actually only a one parameter distribution with equal mean and variance. Similarly, the chi-square distribution is reproductive but the variance always equals twice the mean. When the mean and variance are related in this way, all possible portfolios are represented by a single curve in the p-a plane. 3 An alternative justification of mean-variance analysis is possible if money is a riskless asset and the risky assets are (nearly) independent. Tobin's separation theorem states that if money is a riskless asset and is held in the portfolio in any amount, the utility function is relevant only to the determination of the optimal proportion of money and not of the optimal mix of risky assets (Tobin [16], pp. 82-85). If the number of risky assets in the optimal portfolio is large and the dependence between asset outcomes is sufficiently weak, the portfolio outcome will be approximately normally distributed. If we disregard the difference from normality, a mean-variance description of the non-money component of the portfolio is appropriate and therefore, since riskless monev has a degenerate distribution, all relevant portfolios can be preference-ordered in terms of mean and variance. But the separation theorem rests on the assumed risklessness of money and is sufficient to justify mean-variance analysis only if risky assets are sufficiently numerous and independent. These assumptions are much more in conflict with the opinions that guide investors' actions than the " statistical mistake " approach suggested above.

LIQUIDITY PREFERENCE AND PORTFOLIO SELECTION 11 4. CONCLUSION In his discussion of liquidity preference and portfolio choice as behaviour towards risk, Tobin has been careful to emphasize that the mean-variance analysis must be regarded as an approximation (e.g., Tobin [17], pp. 12-14). This paper has shown that even his rather plausible simplifying assumption that the investor's subjective probability distribution has only two parameters is insufficient for deriving the indifference curve convexity properties that facilitate the analysis of investor behaviour. Moreover, if there is more than one asset, an analysis in terms of only mean and variance is not strictly possible unless utility functions are assumed quadratic or probability distributions are severely restricted. But the inability to obtain unambiguous a priori conclusions about liquidity preference or portfolio choice behaviour does not diminish Tobin's contribution in emphasisizing the importance of analyzing this behaviour as optimization under uncertainty. Rather it emphasizes the need to develop empirical research in this direction. Harvard University M. S. FELDSTEIN First version received 6.3.67; final version received 27.6.68 REFERENCES [1] Arrow, K. J. " Comment" on J. Duesenberry. " The Portfolio Approach to the Demand for Money and Other Assets ", Review of Economics and Statistics (1963), Supplement, p. 34. [2] Cootner, P. H. The Random Character of Stock Market Prices (Cambridge, Mass., M.I.T. Press, 1964). [3] Fama, E. F. "Portfolio Analysis in a Stable Paretian Market ", Management Science (1965), p. 404. [4] Fama, E. F. "The Behaviour of Stock-Market Prices ", The Journal of Business (1965), p. 34. [5] Feldstein, M. S. " The Effects of Taxation on Risk-Taking ", forthcoming, Journal of Political Economy. [6] Fellner, W. Probability and Profit: A Study of Economic Behaviour Along Bayesian Lines (Homewood, Ill., Richard D. Irwin, 1965). [7] Granger, C. W. J. and Morgenstern, 0. "Spectral Analysis of New York Stock Market Prices ", Kyklos (1963), p. 1. [8] Hall, C. A., Jr. Fiscal Policy for Stable Growth (New York, Holt, Rinehart and Winston, 1960). [9] Hicks, J. R. " Liquidity ", The Economic Journal (1962), p. 787. [10] Hirshleifer, J. " Investment Decision under Uncertainty: Choice-Theoretic Approaches ", Quarterly Journal of Economics (1965) p. 509. [11] Mandelbrot, B. " New Methods in Statistical Economics ", Journal of Political Economy (1963), p. 421. [12] Mandelbrot, B. " The Variation of Certain Speculative Prices ", Journal of Business (1963), p. 394. [13] Miller, M. and Orr, D. " A Model of the Demand for Money by Firms ", Quarterly Journal of Economics (1966) p. 413.

12 REVIEW OF ECONOMIC STUDIES [14] Patinkin, D. Money, Interest and Prices. Second Edition (New York, Harper and Row, 1965). [15] Pratt, J. " Risk Aversion in the Small and in the Large ", Econometrica (1964), p. 122. [16] Tobin, J. E. " Liquidity Preference as Behaviour towards Risk ", Review of Economic Studies (1958), p. 65. [17] Tobin, J. E. "The Theory of Portfolio Selection ", in F. H. Hahn and F. P. R. Brechling (eds.), The Theory of Interest Rates (London, Macmillan, 1965).