Fama, Eugene F., and James D. MacBeth, Return and Equilibrium: Empirical Tests, Journal of Political Economy 81 (May 1973), 607-636. Jensen, Michael C., Foundations and Current State of Market Theory, in Michael C. Jensen (ed.), Studies in the Theory of Markets (New York: Praeger Publishers, 1972). Johnston, John, Econometric Methods (New York: McGraw-Hill Book Company, 1972), 345-346. Levy, Robert A., On the Short-Term Stationarity of Beta Coefficients, Financial Analysts Journal 27 (Nov. 1971), 55-62. Lintner, John, Valuation of Risk Assets and the Selection of Risky Investments in Stock Portfolios and Budgets, this REVIEW 47 (Feb. 1965), 768-775. Marshall, William J., Jess B. Yawitz, and Edward Greenberg, On the Comparative Statics of Asset Price Adjustments, working paper, Washington University Graduate School of Business, 1977. Miller, Merton H., and Myron Scholes, Rates of Return in Relation to Risk: Re-examination of Some Recent Findings, in Michael C. Jensen (ed.), Studies in the Theory of Markets (New York: Praeger Publishers, 1972). Mossin, Jan, in a Asset Market, Econometrica 34 (Oct. 1966), 768-775. Rao, Potluri, and Roger Leroy Miller, Applied Econometrics (Belmont, CA: Wadsworth Publishing Company, Inc., 1968), 204. The authors attribute the proof to A. S. Merrill, Frequency Distribution of an Index Where Both the Components Follow the Normal Law,; Biometrika 20 (1928), 53-63. Rubinstein, Mark E., A Mean-Variance Synthesis of Corporate Financial Theory, Journal of Finance 28 (Mar. 1973), 167-181. Sharpe, William F., Capital Asset Prices: Theory of Market Equilibrium Under Conditions of Risk, Journal of Finance 19 (Sept. 1964), 425-442.

TN following the mean-variance analysis developed by Markowitz (1952) and Tobin (1958), Sharpe (1964), Lintner (1965a, b) and Treynor (1961) have developed the theory for determination of asset prices under conditions of uncertainty. The equilibrium asset pricing model, and its implication for measuring ex post performance of individual securities, have been empirically tested by Lintner,1 Jensen (1968, 1972), Miller and Scholes (1972), Douglas (1969), Roll (1969) and others. The empirical results obtained by both Douglas and Lintner deviated from the theory of the model. Moreover, Miller and Scholes have run empirical tests similar to those of Douglas and Lintner and found a significant disparity between the theoretical model and the empirical evidence. They maintain that part of this discrepancy can be explained by possible statistical biases, measurement errors, and consideration of the skewness of the distribution of returns. Black, Jensen, and Scholes (1972) (hereafter B-J-S) confirm the systematic bias. Using monthly data covering a 35 year period, they discovered that, on average, high risk securities earned less than the amount predicted by the model. Similarly, earnings on low risk securities exceeded the amount predicted.2 Although these disparities clearly suggest some systematic empirical bias, we are not examining a possible statistical bias but a mathematical bias stemming from one of the assumptions underlying the capital asset pricing model (CAPM). To be more specific, the model assumes that all investors are single period, expected utility of terminal wealth maximizers. There is no particular restriction on the length of this period as long as it is identical for all investors. Clearly, the length of the true investment horizon affects asset prices under conditions of uncertainty. We claim that the disparities noted above may result from using data calculated for an investment horizon that differs from the true investment horizon. In the various empirical tests, the investment horizon has been selected arbitrarily. For example, Lintner and Miller and Scholes use annual data (i.e., they implicitly assume a one-year horizon), Douglas uses quarterly and annual data; Black, Jensen and Scholes, as well as Friend and Blume (1970), use monthly rates of return in their empirical tests, while Roll uses weekly data. It has been shown elsewhere by Levy (1972) that the Reward to Variability index (developed by Sharpe, 1966) is a function of the investment horizon assumed. Hence, there exists a systematic mathematical bias that is a function of the horizon assumed. The above theoretical findings are related to the theory of pricing capital assets, but deal only with efficient portfolios and not individual stocks. In this paper we illustrate that the assumed horizon plays a crucial role in empirical testing. Any deviation from the true horizon causes a systematic bias in the regression coefficient (i.e., in the security systematic risk). This in turn causes a systematic bias in the performance measures of each security, and hence the deviation between the theoretical model and the empirical evidence. The results of this paper are not limited to the theory of pricing capital assets; they are applicable to any econometric study in which the variables have multiplicative rather than additive properties. In such a case the regression coefficients will have a matheReceived for publication February 27, 1975. Revision accepted for publication March 8, 1976. The authors acknowledge the technical assistance of Moshe Smith and two anonymous referees. The first author has been partially financed by the Maurice Falk Foundation, and the second author has been financed by the Ford Foundation. 1 Lintner's paper Security Prices and Risk: The Theory and a Comparative Analysis of A.T.&T. and Leading Industrials was presented at the Conference on Economics of Regulated Public Utilities, June 24, 1965, Chicago. 2 Miller and Scholes show that the presence of certain biases could have accounted for the Douglas and Lintner findings. BJ-S maintain that the assumption of borrowing at riskless interest rates could have accounted for these deviations.

The empirically documented Pareto wealth distribution at high wealth levels implies rather extreme wealth inequality. Is this inequality primarily due to differential talent, or is it due to luck? The answer to this question has profound political, social, and philosophical implications, as well as implications regarding market efficiency. We address this question theoretically and with a unique investment experiment with equal initial endowments and real out-of-pocket money. We show that the empirically observed Pareto distribution implies that luck, rather than differential investment talent, is the main force driving inequality at high wealth levels.

The most commonly employed paradigms for decision making under risk are expected utility, prospect theory, and regret theory. We examine the simple heuristic of maximizing the probability of being ahead, which in some natural economic situations may be in contradiction to all three of the above fundamental paradigms. We test whether this heuristic, which we call probability dominance (PD), affects decisions under risk. We set up head-to-head situations where all preferences of a given class (expected utility, original or cumulative prospect theory, or regret theory) favor one alternative yet PD favors the other. Our experiments reveal that 49% of subjects' choices are aligned with PD in contradiction to any form of expected utility or prospect theory maximization; 73% are aligned with PD as opposed to preferences under risk aversion and under original and cumulative prospect theory preferences; and 68% to 76% are aligned with PD contradicting preferences under regret theory. We conclude that probability dominance substantially affects choices and should therefore be incorporated into decision-making models. We show that PD has significant economic consequences. The PD heuristic may have evolved through situations of winner-take-all competition.