Long-Term growth in a Short- Term Market TORIo Eugene F Fama: James D. MacBeth The Journal of Finance, Vol 29, No. 3. Jun, 1974), pp. 857-885 Stable url: http://inks.jstor.org/sici?sici=0022-1082%028197406%2929%3a3%3c857%03algiasm%3e2.0.c0%03b2-1 The Journal of finance is currently published by American Finance Association Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/about/terms.htmlJstOr'sTermsandConditionsofUseprovidesinpartthatunlessyou 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://wwwjstor.org/journals/afina.html 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 an independent not-for-profit organization dedicated to creating and preserving a digital archive of scholarly journals. For more information regarding JSTOR, please contact support@jstor. org http://www」]stor.org Fri mar3122:38:552006
LONG-TERM GROWTH IN A SHORT-TERM MARKET EUGENE F. FAMA AND JAMES D. MACBETH* L. INTRODUCTiON Two MoDELs that receive substantial attention in portfolio theory are the two- period, two-parameter model of Markowitz [17] and the long run, "growth optimal"model of Latane [15] and others. The growth-optimal model is often resented as an appealing alternative to the two-parameter model for those investors whose horizon is distant in the sense that portfolio funds are not needed for consumption for many periods. Thus the impression is left that the growth-optimal and the two-parameter models are not consistent with each other. A minor purpose of this paper is to dispel this impression. Empirical work that attempts to identify growth-optimal portfolios of New York Stock Exchange common stocks is also presented. The major goal of the paper is to examine whether the properties of observed growth-optimal portfolios are con- istent with a market dominated by investors whose primary concern is long run growth II. GRoWTH-OPTIMAL AND TWO-PARAMETER MODELS THEORY In the simplest version of the two-parameter portfolio model, an individual making a consumption- investment decision at time t= l, is assumed to have wealth wi which he must divide between current consumption C, and a port- folio investment W1-CI, the return on which provides his consumption C2 for period 2. The individual is assumed to make his consumption- investment deci sion as if he attempts to maximize expected utility with respect to a utility function for consumption U(Cu, C) that is monotone increasing and strictly concave in (Cu, C2)-in short, the investor is assumed to be risk-averse with respect to consumption uncertainty. The capital market is assumed to be per fect in the sense that investors are price-takers and there are no transactions costs or information costs. Finally, probability distributions of one-period per centage returns on all portfolios are assumed to be normal. a perfect capital arket, investor risk-aversion and normally distributed portfolio returns imply he Efficient Set Theorem: The expected utility maximizing portfolio for any investor is efficient in the sense that no portfolio with the same or higher expected one-period return has lower variance (or standard deviation) of return Graduate School of Business, y of Chicago. Research support from Science Foundation is gratefully ack The comments of F. Black and N the editional atefully acknowledged, And as w ur debt to Roll [21] should become clear 1. See, for example, williams [26 14], and Hakansson [12] 2. See, for example, Hakansson [12]
The journal of finance In Fama [6 the two-parameter portfolio model is generalized to a multi period decision framework Simplifying somewhat the analysis of [6],suppose the individual will cease to consume at the end of time t= T and suppose his tastes for lifetime consumption can be represented by a utility function U(C1, C2, .. Cr)that is monotone increasing and strictly concave in lifetime consumption(C1, c: Thus the individual is assumed to be with respect to lifetime consumption. Assume also that each period the prob ability distributions of one-period return on all portfolios are normal, and, for implicity that the portfolio opportunity set is expected to be the every period. Finally, assume that consumption opportunities(goods available and their relative prices) are likewise expected to be constant through time Then it is shown in [6] that although he has a multiperiod horizon, each period the individuals consumption-investment decision is in conformance with the two-period, two-parameter model. 4 That is, in making his consumption-invest- ment decision for any period t, the individual behaves as if he attempts to maximize expected utility with respect to an induced utility function U, (c Wi+1) that is monotone increasing and strictly concave in consumption for period t, Ct, health at period t+ l, w In simplest terms, the argument of the proponents of the growth-optimal model is that if the individual will not consume out of portfolio funds for many periods, then for each t and c, an appealing strategy is to choose the portfolio that maximizes E(In E(In[wt+1l)=In cr)+E[n(1+ implies higher wealth at the horizon than any alternative policy most 3 26 where R,t +1 is the one-period percentage return on portfolio p from time t to time t +1, and the tilde (" indicates that the return is a random varial The assumed appeal of this portfolio strategy arises from the fact that over an investment horizon of many periods, the policy of maximizing El Rpt+1)] period by period is growth-optimal in the sense that it"alr urely case(Fama [51). But the empirical work of this paper uses monthly returns, and for monthly ne theoretical criticism often directed at the of symmetric stable return distributions is that since they have unbounded tails, such re inconsistent with the limited ability provisions of most securities. But efficient portfol small that it has 29 this point is pre Thus the existence of limited is not an important con ideration in the choice of distribution used to approximate returns. The important consideration is descriptive valid It is well-known that the Efficient Set Theorem can be obtained with the assumption that investor utilities are well approximated by functions that are quadratic in c. But the problems with this approach are also well known. For a discussion see Fama and Miller [81, Chapter 6 4.The assumptions that consumption and investment opportunities are constant through time than required by the model of [61, but this simplified version of the model is con sufficient for our purposes. A detailed discussion of the conditions under whic behavior in a multiperiod setting is in 5. The growth-optimal model has often been criticized, most recently by Samuelson [22]. with rious examples, Samuelson shows that although over the very long run, the growth-optima policy almost surely provides higher wealth and thus higher realized utility than alternativ strategies, this does not imply that the growth-optimal policy maximizes expected utility. In
Long-Term Growth in a Short-Term Market In terms of the expected utility model of [6], the policy of maximizing E[In(1+Ro, t+1] period by period implies that for each t and ct, U(c, Wt+1) is well approximated by a function that is logarithmic in w+1. Since it is monotone increasing and strictly concave in wt+1, the log utility function is consistent with the two-parameter model Given normally distributed one- period percentage portfolio returns, the Efficient Set Theorem applies. The growth-optimal portfolio is just the specific mean-variance efficient portfolio that is optimal for the log utility function. Thus the two-parameter and growth optimal models are mutually consistent o The main goal henceforth is to test whether the process of price formation the New York Stock Exchange is dominated by growth-optimizers. The first step is to develop the characteristics of such a"growth-optimizers'market III. A GROWTH-OPTIMIZERS, CAPITAL MARKET: THEORY For portfolio decisions at time t, the growth-optimal portfolio maximizes [In(1+Ro,t+i)1, the expected value of the continuously compounded rate of growth of portfolio funds from t to t+ 1. If portfolio G maximizes E[ln(1+ Rp, t+1)l, then it can be shown that for all assets i and j in g R R E E 1+R (1) In mathematical terms equation (1) is just a necessary condition for a maximum of E[In(1+Ro, t+1]. But it provides a testable implication of the hypothesis that the market is dominated by growth-optimizers when that hy- pothesis is structured into a theory of market equilibrium. In particular, sup pose that all investors are growth-optimizers and that there is complete agreement among them with respect to the distribution of the return on any portfolio. Then every investor has the same view of which portfolio is growth- optimal and all will hold that portfolio, It follows that if the market is to clear the growth-optimal portfolio G must be the market portfolio, henceforth with each asset weighted in proportion to the total market value of a.x referred to as M. By definition, M is the portfolio of all assets in the mark Since M is growth-optimal, it follows from (1) that for any two asset d ) that ve strategies have higher expected utilities than the growth-optimal polio shows that any uniform policy such as maximizing E[In(1 by period is un likely to be optimal. For example, as the n he consumes from portfolio funds, long-run growth considerations are likely to become less impressive, so that the log utility function is no longer a good approximation to his taste But it is well to note that early advocates of the growth-optimal model (e. g, Williams [26] eed lata d the model as an alternative to expected utility maximization, Among later advocates, Hakansson [121 does present the growth-optimal model in an expected utility framework, and it is primarily arguments like his that Samuelsons criticisms apply 6. See Roll [21]. This development of the characteristics of a growth-optimizers'market is du 7. The " complete agreement'assumption is usually called "homogeneous expectations
The Journal of finance 十R Moreover, for any portfolio p, defined by the proportions x,p of portfolio funds invested in individual securities j E(1+B+1)=E( (1+氮t+) tR. .+RM 1+ R R E 1+Rut From(2)and (3), it follows directly that E(1+1+ +R E 0.(4) 1+ In words, in a market dominated by growth-optimizers, the expected value of the ratio of one plus the percentage return on any security or portfolio to one plus the return on the market portfolio is equal to 1.0. Assuming the equired correspondence between ex ante assessments of return distribution and ex post return distributions, we can test this condition by testing for differences of average values of the return ratios for securities or portfolios from the theoretical value of 1.0. We work with portfolios and in particular with the time series of returns on twenty portfolios available from our earlier study [71 IV. HOTELLING T TESTS Let +rpt 1+RM/P=1,2, 20;t=1/35,2/35,,,6/68, where the Rpt are monthly percentage returns on the twenty portfolios for the period January 1935 to June 1968, and where the proxy for Rit is "Fishers Arithmetic Index"[9], a simple average of the returns from month t-1 to month t on all common stocks on the New York Stock Exchange. The twenty folios to be discussed in more detail later, are likewise formed from stock on this Exchange. For present purposes, suffice it to say that securities are allocated in approximately equal numbers to each of the twenty portfolios, and the allocation is according to ranked values of estimates of P, where COV (R1,R3 2(Rx) is the risk of security i in the market portfolio M, and where risk is measured 8. The theory, of course, calls for a value weighted index rather than an index where each stock,s return equally, In tests similar to some of those to be carried out here how- er, Roll [2 that his conclusions are insensitive to whether a value weighted or an equally weighted returns is used 9. The data at he Center for Research in Security Prices of the University of chicago