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 目自含 888888a88a88888888 |R=国 日 P/2
Long-Term Growth in a Short-Term Market 225否含 2到8活 丙菌营自园园 /习一 营营营营营
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