for their helpful comments and suggestions. The views expressed herein are those of the authors and not necessarily those of the National Bureau of Economic Research.

As a consequence of optimal investment choices, a firm's assets and growth options change in predictable ways. Using a dynamic model, we show that this imparts predictability to changes in a firm's systematic risk, and its expected return. Simulations show that the model simultaneously reproduces:~i! the time-series relation between the book-to-market ratio and asset returns;~ii! the cross-sectional relation between book-to-market, market value, and return;~iii! contrarian effects at short horizons;~iv! momentum effects at longer horizons; and~v! the inverse relation between interest rates and the market risk premium. 1560The Journal of FinanceHere J * @r~t!# is defined implicitly by the above identity as the per unit present value of all future growth options. C. The Value of the FirmCombining the value of assets in place and the value of growth options gives the value of the firm:Equation~32! is the analog, in an uncertain environment, of the pricing model under certainty that is often used as a standard introduction to corporate finance~see, e.g., Ross, Westerfield, and Jaffe~1988!, p. 130!. Thewhere J e @r~t!, s Ϫ t, k, r b s * # is, analogously, the expectation of the value of one of these options next period,Optimal Investment, Growth Options, and Security Returns A25!So J e @r~t!, s Ϫ t, k, r b s * # corresponds to the time t expectation of the price at t ϩ 1 of a call option that matures at s, which, upon exercise, delivers I~s!0~E t @I~s!#! riskless discount bonds maturing at time s ϩ k, and has strike price~I~s!0~E t @I~s!# !!B@k, r b s * #~i.e., you have to pay B@k, r b s * # for each bond!. The result then follows by applying Lemma B.10 in Appendix B. ⅢCombining equations~A23!,~A19!, and~A18! yields the following expression for the current expectation of the cum-dividend value of the firm at t ϩ 1:

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We characterize the conditions under which efficient portfolios put small weights on individual assets. These conditions bound mean returns with measures of average absolute covariability between assets. The bounds clarify the relationship between linear asset pricing models and well-diversified efficient portfolios. We argue that the extreme weightings in sample efficient portfolios are due to the dominance of a single factor in equity returns. This makes it easy to diversify on subsets to reduce residual risk, while weighting the subsets to reduce factor risk simultaneously. The latter involves taking extreme positions. This behavior seems unlikely to be attributable to sampling error. THE MEAN-VARIANCE EFFICIENT frontier plays an important role in pedagogy and applications in finance. The properties of efficient portfolios are also central in both static and dynamic asset pricing.1 For all its simplicity and intuitive appeal, however, practical implementation of mean-variance analysis has proved problematic. Portfolios constructed using sample moments of returns often involve very extreme positions. As the number of assets grow, the weights on individual assets do not approach zero as quickly as suggested by naive notions of "diversification."Admittedly, the theoretical links between "diversification," in the sense of putting small weight on any asset, and variance minimization are tenuous. The Arbitrage Pricing Theory (APT) implies that efficient portfolios are well diversified, but the mean-variance Capital Asset Pricing Model (CAPM) requires this only insofar as the value weighted market portfolio is diversified. In applications, however, the connection between mean-variance efficiency and diversification is central. Black and Litterman (1990), for example, point to the inconsistency between naive diversification and the weights generated by asset allocation models as a major obstacle to their implementation. Practitioners are suspicious of portfolios that are not naively * Green is from Carnegie Mellon University and Hollifield is from the University of British Columbia. We wish to thank seminar participants at Carnegie The Journal of Finance diversified, so mean-variance methods are generally implemented with extensive sets of constraints that enforce such diversification. Numerous academic researchers have also documented problems in employing the weights that result from calculating efficient frontiers using sample moments.2 This paperstudies the question of when mean-variance efficiency and "well diversified" coincide, and when they do not. First, we provide a theoretical answer to this question. Then, we use more specialized examples to advance possible explanations for the "poorly diversified" weights found in efficient portfolios calculated using sample return data.Through duality theory, we provide simple and intuitive characterizations of conditions under which minimum-variance portfolios will be well diversified. These characterizations involve bounds on the means of portfolio returns in terms of...