《投资学原理 Investments》课程教学资源（阅读资料）long-term growth in a short-term market（fm,1974）

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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

erm growth in a Short-Term Market as in the two-parameter model. o Thus, the first portfolio contains the securities with the lowest estimates of Bu, while the twentieth portfolio contains the securi. ties with the highest estimates of B, Let z, be the average through time of the zut. The hypothesis that the market folio is growth-optimal and thus that the pricing of assets is dominated by growth-optimizers can be rejected if the z for some portfolio or for some subset of portfolios can be shown to differ systematically from 1.0. Such a test is provided by Hotelling's T, defined in the present case as Y'S-IY where Y is the vector of Y, =Z-1, p=1,., 9, 11,..., 20, S is the 19 x 19 estimated covariance matrix of the component zut, and n is the number of months used in computing both Y and S One portfolio (we have chosen the tenth) must be omitted from the computations to avoid the singularity in S that would otherwise arise from the fact that for any t the sum of the zpt over the twenty portfolios is always very close to 20 12 Under the(in this case tenuous)assumption that the joint distribution of the zpt is multivariate normal and stationary through time, the statistic n-19 F 19(n-1) has the f distribution with degrees of freedom (DF)19 and n-19 Table 1 presents the results of the Hotelling t- tests for the overall period 1935-6/68 and for various subperiods. There are no F statistics in the table that exceed the 95 fractile of the F distribution, and the f for only one sub- period, 1956-6/68, exceeds the 90 fractile of the F distribution. In short, the vidence in Table 1 is not sufficient to reject the hypothesis that our proxy for the market portfolio is growth-optimal, and thus we cannot reject the hy hesis that the g of assets is dominated by growth-optimizers of the hypothesis E(Rp)=E(Rx), p=1, 2, 20, are presented in Table 2. 13 The results support rejection of the hy pothesis. The F statistic for the overall period 1935-6/ 68 excedes the 95 frac 10. In the two-parameter model, the risk of M is d2(RM), the variance of its one-period centage return, which can be written as 0n)=平 XiarXjM Cov(页,对)= 2xiM(2 xIM cov(郎,瓦) =∑xMov(武,x) Thus cov (R, R) is the risk of asset i in olio m in that it is the contribution of i to the le risk of M, For of this viewpoint, see [51, [7] or Fama and Miller [81, Chapter 7. isis not 11. See, for example, Morrison [19] or Anderson [1 ter that this the case. The number of securities allocated to portfolios in any given month is always less the number available 13. In this case there is no need to delete one of the portfolios from the tests since the covariance matrix of the r, is nonsingular

The Journal of finance tile of the F distribution, and for all subperiods except 1935-40 and 1946-50 the F statistics fall into the extreme right tail of the F distribution. Moreover, for the overall period and for most of the subperiods the values of Rp-r increase from lower to higher values of p. Since the twenty portfolios are formed on the basis of ranked estimates of the B risks of individual securities, these results are consistent with the positive tradeoff of risk for expected one- period percentage return hypothesized by the two-parameter model 14 And this evidence in support of a positive expected return risk tradeoff is actually quite meager relative to that in [3] and [7 In tests on common stock returns for the 1960, s, Roll [21] also finds that to a large extent, both the two-parameter model and the growth-optimal model are consistent with his data. It is well to emphasize again, however, that there is nothing anomolous in this result. It is consistent with a world of risk-averse investors and two-parameter return distributions which in addition has the are well approximated by a log utility function, 1s ers property that the market is dominated by risk-averters whose tastes for wealth V. HISTORICAL GROWTH-OPTIMAL PORTFOLIOS But this view of the world is so specific that further tests are warranted Such checks are especially desirable since the assumption of multivariate normality on which the T tests are based almost certainly does not apply to If the market is dominated by growth-optimizers, then, given complete agreement about return distributions, the market portfolio is growth-optimal Thus for any portfolio p E[In(1+ rotein(1+rot) Assuming a market that also conforms to the two-parameter model, one way to test(6) is to compute average values of observed In(1+ portfolios, and then examine whether the maximum of these averages is ob- tained with an efficient portfolio much different from M. To carry out such tests, however, we must identify the set of efficient portfolios in more concrete terms. Our models for efficient portfolios are taken from the two-parameter models of capital market equilibrium of Sharpe [23], Lintner [16], Black [2] and Vasicek [25] observed in the signs of the Yp = ip-1 of Table 1. Although it is not strong enough values of T2 and F, this pattern may provide some basis fo 15. Ro him lizes that the two-parameter and growth-optimal models are utually consistent. His reasoning, however, is based on the goodness of a quadratic approxima tion to the log utility function oint distribution of security returns is mt normal, which in turn implies that the joint distribution of portfolio returns is multi al. If portfolio returns are multivariate nor the zpt which are ratios of returns, cannot variate normal, so that sumption of the T2 tests on the znt is violated. And appar is known about the effects of nonnormality arious nonparametric methods to test the hypothes He is unable to reject the hypothesis with any of his methods, which all seem to give mparable to the t2 tests

Long-Term Growth in a Short-Term Market 865 A. Theory: Eficient Portfolios onsider a world of two-parameter percentage return distributions in which the capital market is perfect and short-selling of all assets is permitted Define a minimum-variance portfolio as a portfolio of only risky assets that minimizes variance at some given level of expected percentage return. Sharpe [24] and Black [2] show that the set of minimum-variance portfolios can be generated as portfolios of any two minimum-variance portfolios. If, in addition to the assumptions above, one assumes that there is complete agreement among in vestors with respect to assessments of the distributions of returns on portfolios then black [2 shows that a market equilibrium implies that the market portfolio M is minimum-variance and efficient. He argues that convenient choices for the two minimum-variance portfolios used to generate the set of minimum-variance portfolios are m and the minimum-variance folio, that is, the minimum-variance portfolio with percentage related with the return on m Note that minimum-variance portfolios are defined to include only risky Assets, that is, assets with positive variances of one- period percentage return. The set of efficient portfolios is a subset of the minimum-variance portfolios when there is no riskless asset. But a minimum-variance portfolio need not be efficient. There may be another minimum-variance portfolio with the same variance of return but higher expected return. For example, the minimum variance zero p portfolio(henceforth referred to a Z) is not efficient If, as in the model of Sharpe [23] and Lintner [16 there is a riskless asset, F, that can be held either short or long, that is, if there is riskless borrowing and lending, then all efficient portfolios are combinations of the riskless asset and the market portfolio. Geometrically, in the familiar mean (ordinate )-standard deviation(abscissa)plane, efficient portfolios are along a raight line from the riskless rate re that is tangent to the curve of minimum variance portfolios at the point corresponding to the market portfolio M If, as in the model of Vasicek [25], there is riskless lending but not borrow g, then efficient portfolios are of two sorts. First, there are combinations of F with the " tangency portfolio'"T, which is the minimum-variance portfolio corresponding to the point where a line from Re is tangent to the curve of minimum-variance portfolios. Only combinations of F and T involving non negative holdings of both are feasible and efficient The remaining efficient rtfolios are those minimum-variance portfolios that have expected return equal to or greater than E(Rr). One such portfolio is the market portfolio M which must either be t or a minimum-variance portfolio with E(Rx)> E(武1).1 Of the component portfolios needed to construct efficient portfolios under he different versions of the two-parameter model, we already have a proxy for the market portfolio M. And since we deal with monthly returns, rates on one-month Treasury Bills provide a proxy for Ret. These two are sufficient construct the efficient portfolios of the Sharpe- Lintner(S-L)model, in which 17. For a discussion of these different versions of two-parameter models of market equilibrium ee Jensen [13]. The points in our summary should become clear in the graphs to be presented later

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