Nonparametric Portfolio Estimation and Asset Allocation by Julian James Douglass B.Sc.E, Queen’s University, 1992 M.Sc., The University of British Columbia, 1995 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF Doctor of Philosophy in The Faculty of Graduate Studies (Business Administration) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) February, 2009 c© Julian James Douglass 2009 Abstract This thesis comprises two essays that apply nonparametric methods to the es- timation of portfolio allocations. In the first essay, I test the significance to investor welfare of (i) adding additional assets to the portfolio choice set and (ii) conditioning on predictor variables. I estimate unconditional and conditional optimal allocations of a constant relative risk aversion investor by maximizing a nonparametric approx- imation of the expected utility integral. Investors can improve their expected utility significantly over that of an equities and cash investor by adding portfolios based on the value or momentum premiums into their asset allocation decision. In contrast, neither a size premium portfolio nor a long-term bond portfolio improves expected utility. The significance of predictability is increased by si- multaneously conditioning on the two strongest predictors (of eight) studied: the term spread and the gold industry trend. In the second essay, I formulate a nonparametric estimator that permits combining historical data with a qualitative prior. I investigate the impact of an investor belief, motivated by asset-pricing theory, that optimal allocations are positive. In the estimator construction, I use a Bayesian approach to perturb the probabilities associated with each data point in the empirical distribution to reflect qualitative prior beliefs. In a simulation study and in out-of-sample tests, I find that portfolio estimates conditioned on a belief in the positivity of portfolio weights are significantly more stable than those estimated by an uninformed investor, and that the model performs better in out-of-sample tests than a number of plug-in models. However, the out-of-sample performance lags that of the minimum-variance and 1/N policies. ii Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Using Inference to Assess the Benefits of Diversification . . . . 3 1.2 Prior Disbelief in the Optimality of Short Positions . . . . . . . 4 1.3 Nonparametric Portfolio Estimation . . . . . . . . . . . . . . . 5 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2 Asset Allocation with Value Growth Tilts and Predictability 9 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 Empirical Framework . . . . . . . . . . . . . . . . . . . . . . . 16 2.2.1 Investor’s Problem . . . . . . . . . . . . . . . . . . . . . 16 2.2.2 Direct Estimator of Conditional Portfolio Policy . . . . 17 2.2.3 Measure of Portfolio Performance . . . . . . . . . . . . . 19 2.2.4 Data and Utility Assumptions . . . . . . . . . . . . . . 21 2.3 Unconditional Asset Allocation . . . . . . . . . . . . . . . . . . 23 2.4 Conditional Asset Allocation . . . . . . . . . . . . . . . . . . . 28 2.4.1 Predictability of Mean and Variance . . . . . . . . . . . 28 2.4.2 Predictability in an Asset Allocation Framework . . . . 29 2.5 Conditional Asset Allocation with Multiple Assets . . . . . . . 32 2.6 Conditioning on Multiple Predictors . . . . . . . . . . . . . . . 37 2.6.1 Direct Estimator for Multiple Predictors . . . . . . . . . 37 2.6.2 Model Selection . . . . . . . . . . . . . . . . . . . . . . 43 2.6.3 Certainty Equivalent Returns with Multiple Predictors . 44 2.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 iii Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 3 Nonparametric Portfolio Estimation with Prior Belief in the Positivity of Portfolio Weights . . . . . . . . . . . . . . . . . . . 82 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 3.2 Portfolio Choice Framework . . . . . . . . . . . . . . . . . . . . 87 3.3 Discrete-Bayesian Estimator . . . . . . . . . . . . . . . . . . . . 88 3.3.1 Construction of the Discrete-Bayesian Estimator . . . . 90 3.3.2 Sampling from the Posterior . . . . . . . . . . . . . . . 93 3.3.3 Mean-Variance Utility . . . . . . . . . . . . . . . . . . . 93 3.4 Prior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 3.5 Relationship to Previous Portfolio Choice Models . . . . . . . . 97 3.5.1 Plug-in Models . . . . . . . . . . . . . . . . . . . . . . . 97 3.5.2 Portfolio Resampling . . . . . . . . . . . . . . . . . . . . 99 3.5.3 Statistical Refinement versus Regularization . . . . . . . 102 3.6 Portfolios with Positive Weights Prior . . . . . . . . . . . . . . 103 3.6.1 Simulation Setup . . . . . . . . . . . . . . . . . . . . . . 103 3.6.2 Exploratory Analysis of the Prior Assumptions . . . . . 104 3.6.3 Monte-Carlo Study . . . . . . . . . . . . . . . . . . . . . 107 3.6.4 Comparison of Discrete-Bayesian Policy with Other Mod- els . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 3.7 Out-of-Sample Study . . . . . . . . . . . . . . . . . . . . . . . 112 3.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 Appendices A Bandwidth Specification . . . . . . . . . . . . . . . . . . . . . . . 147 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 B Data Tuning for a Portfolio Choice Problem . . . . . . . . . . 150 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 C Sampling from the Posterior . . . . . . . . . . . . . . . . . . . . 153 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 iv List of Tables 2.1 Summary Statistics . . . . . . . . . . . . . . . . . . . . . . . . . 46 2.2 Correlations of Asset Returns and Predictors . . . . . . . . . . . 47 2.3 Volatility Correlations . . . . . . . . . . . . . . . . . . . . . . . 48 2.4 Unconditional Portfolio Weights . . . . . . . . . . . . . . . . . . 49 2.5 Unconditional Certainty Equivalent Returns . . . . . . . . . . . 50 2.6 Bootstrap Test of Expected Utility Improvement . . . . . . . . . 51 2.7 Abbreviations Used for Conditioning Variables . . . . . . . . . . 52 2.8 Returns Regressed on Predictors . . . . . . . . . . . . . . . . . . 53 2.9 Realized Volatility Regressed on Predictors . . . . . . . . . . . . 54 2.10 Portfolio Weights as a Function of Predictor Values . . . . . . . 55 2.11 Certainty Equivalent Returns as a Function of Predictor Values 57 2.12 Increase in Certainty Equivalent Returns from Conditioning on a Predictor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 2.13 Coefficients of Predictive Index . . . . . . . . . . . . . . . . . . 60 2.14 Conditional Certainty Equivalents for Predictive Index . . . . . 61 3.1 Portfolio Choice Models . . . . . . . . . . . . . . . . . . . . . . 116 3.2 Simulation Parameters . . . . . . . . . . . . . . . . . . . . . . . 117 3.3 Posterior Distribution Given a Simulated Sample . . . . . . . . 118 3.4 Monte Carlo Statistics for Predictive Distributions . . . . . . . . 120 3.5 Monte Carlo Averages of Estimated Allocations . . . . . . . . . 121 3.6 Monte-Carlo Average Out-of-Sample Utilities . . . . . . . . . . 123 3.7 Monte-Carlo Average Portfolio Allocations for Alternative Port- folio Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 3.8 Monte-Carlo Average Out-of-Sample Utility for Alternative Port- folio Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 3.9 Out-of-Sample Utility for Discrete-Bayesian Estimator . . . . . 126 3.10 Out-of-Sample Utility for Alternative Models . . . . . . . . . . . 128 v List of Figures 2.1 Single Asset Portfolios Conditioned on Dividend Yield and De- fault Premium . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 2.2 Single Asset Portfolios Conditioned on Term Spread, S&P 500 Trend, and T-bill Rate . . . . . . . . . . . . . . . . . . . . . . . 64 2.3 Single Asset Portfolios Conditioned on Gold Industry Trend, Re- alized Volatility, and Inflation . . . . . . . . . . . . . . . . . . . 65 2.4 Portfolio Allocations versus Dividend Yield . . . . . . . . . . . . 66 2.5 Portfolio Allocations versus Default Premium . . . . . . . . . . 67 2.6 Portfolio Allocations versus Term Spread . . . . . . . . . . . . . 68 2.7 Portfolio Allocations versus S&P 500 Trend . . . . . . . . . . . 69 2.8 Portfolio Allocations versus T-bill Rate . . . . . . . . . . . . . . 70 2.9 Portfolio Allocations versus Gold Industry Trend . . . . . . . . 71 2.10 Portfolio Allocations versus Realized Volatility . . . . . . . . . . 72 2.11 Portfolio Allocations versus Inflation . . . . . . . . . . . . . . . 73 2.12 CER versus Default Premium and Dividend Yield . . . . . . . . 74 2.13 CER versus Term Spread and Index Trend . . . . . . . . . . . . 75 2.14 CER versus Tbill Yield and Gold Industry Trend . . . . . . . . 76 2.15 CER versus versus Inflation and Volatility . . . . . . . . . . . . 77 3.1 Histogram of Posterior Returns for a Simulated Data Set . . . . 131 3.2 Histogram of Posterior Weights . . . . . . . . . . . . . . . . . . 132 3.3 Histogram of Posterior Returns with Prior Against Return Bias 133 3.4 Histogram of Posterior Weights with Prior Against Return Bias 134 3.5 Histogram of Posterior Weights with Strong Prior . . . . . . . . 135 3.6 Histogram of Predictive Return Distribution under Different Priors136 3.7 Histogram of Asset Weights Across Simulations . . . . . . . . . 137 vi List of Algorithms C.1 MCMC Algorithm for Drawing from the Discrete Posterior Distri- bution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 vii Acknowledgements I am greatly indebted to a large number of colleagues, friends, and family who supported me during my years in the Ph.D. program at the Sauder School. First, I would like to thank the faculty and Ph.D. colleagues of the Finance Division who provided a stimulating environment for research. My committee members, Marcin Kacperczyk, Robert Heinkel, and Shinichi Sakata, provided invaluable input and advice. I would especially like to thank my advisor, Adlai Fisher, for continuous encouragement and support throughout my thesis project. I would also like to thank Murray Carlson for many stimulating conversations, and Tan Wang and Alan Kraus for input during the early stages of this project. I also benefited from guidance from Glen Donaldson, Ron Giammarino, and Maurice Levi. I also thank my friends and colleagues, particularly, Issouf Soumaré, Lars- Alexander Kuehn, Kyung Shim, Jeff Colpitts, Jason and Jenny Chen, Harjoat Bhamra, and Ali Lazrak. I owe a great debt to friends and family both in Vancouver and far away. The names are too numerous to list without missing someone. This thesis would not have been possible without their constant support. Finally, I thank Andrea whose companionship lightened the burden even during some of the most difficult challenges. I look forward to future projects and adventures with her at my side. viii To Andrea ix Chapter 1 Introduction The information in historical returns is not sufficient, on its own, to provide useful estimates of optimal portfolio allocations. Previous researchers have doc- umented overwhelming evidence that portfolios estimated by direct substitution of distributions estimated from previous realized returns feature extreme port- folio weights and perform poorly out of sample.1 Furthermore, accounting for this estimation error in a Bayesian framework that accounts for estimation un- certainty without introducing informative prior beliefs only slightly improves expected portfolio performance.2 Portfolio allocation is a problem that is plagued by estimation error. The objective of portfolio allocation is to select the portfolio that yields the optimal balance between future rewards and risk. The theory of choice under uncer- tainty yields an elegant mathematical framework for the portfolio problem.3 In turn, allocation models expressed in this framework are tantalizingly straight- forward to solve, given the investor’s risk aversion and the future distribution of returns.4 Yet, in practice, the future distribution of returns is unknown and must be estimated. Asset returns are highly volatile, and return means esti- mated from time series of past realized returns are very noisy. In addition, optimal allocations are inversely proportional to asset variances for risk averse investors. Thus, optimal portfolios depend on the inverse of the covariance ma- 1See, for example, Jobson and Korkie (1980), Jobson and Korkie (1981), Best and Grauer (1991), and Chopra and Ziemba (1993). 2Early empirical studies of Bayesian approaches to portfolio choice appear in Bawa et al. (1979) and references therein. See Kan and Zhou (2007) for further discussion and results pertaining to the relative performance of plug-in and diffuse Bayesian portfolio estimators. 3Markowitz (1952) developed the mean-variance formulation of this problem that remains the canonical mathematical formulation of the allocation problem. In this framework, rewards as represented by expected portfolio return is balanced against increased risk from portfolio variance. 4A major advantage of the mean-variance model is its analytic tractability. Alternative utility assumptions may lead to less tractable models, but these, too, are readily solved computationally by convex programming algorithms. 1 trix. In applications, this matrix is often ill conditioned because financial asset returns are highly correlated. As a result, errors in variances and correlations are amplified in the portfolio solution. This thesis comprises two essays that examine issues in portfolio estimation. The first essay examines the question of asset selection for the tactical allocation problem. Given that portfolio weights are computed from observed data, they are naturally viewed as statistical estimates. Using inference techniques, I test the significance of gains in expected utility from adding assets to a tactical asset allocation portfolio, and of conditioning on predictor variables in the context of tactical asset allocation. In the second essay, I construct a nonparametric estimator for the portfolio choice problem that incorporates qualitative prior beliefs into the investor’s estimation problem. In an empirical exercise, I consider an industry allocation problem. I examine the value of a prior belief that all optimal portfolio weights are positive. The prior is motivated by economic theory that suggests that, given that all industries have positive weight in the market portfolio, that they are unlikely to be priced such that short positions are optimal. A common theme of both essays is the nonparametric estimation strategy. In the first essay, I evaluate the value of conditioning variables following the nonparametric strategy introduced by Brandt (1999). In the second essay, use Bayesian computational techniques to incorporate qualitative prior beliefs in the investor’s problem. As suggested in the pioneering work of Jobson and Korkie (1980), I treat the investor’s utility maximization as the statistical ob- jective when evaluating conditioning information. I illustrate the high threshold that must be overcome before we can conclude that conditioning information is helpful to an investor in a statistically meaningful way. The remainder of the introduction motivates and reviews the empirical con- tributions, provides background for the estimation strategy, and describes my methodological contribution. 2 1.1 Using Inference to Assess the Benefits of Diversification In the first essay, I use an inference approach to examine the benefits of diver- sification given an optimal portfolio estimated from return data. One of the fundamental corollaries of portfolio theory is that adding additional assets to the set available for diversification improves outcomes.5 In the presence of es- timation error, there is no guarantee that including the diversifying asset will improve subsequent utility. For the investor, portfolio estimates are empirical forecasts of their optimal allocations. If an investor can precisely estimate their optimal portfolio then adding an additional asset always improves expected utility6 However, adding an asset can also reduce the precision of the optimal portfolio estimator. Con- sider the very simple example of an investor who is choosing whether to invest solely in the risk free asset or whether to select an optimal portfolio that com- bines the risky asset and the risk free rate. In the first case the optimal portfolio is trivially and precisely identified since there is only one asset. In the second case, the optimal portfolio depends critically on the investor’s estimate of the probability distribution of returns. There is no guarantee that the estimated optimal portfolio will have a higher expected utility with respect to the under- lying return-generating distribution. I use a hypothesis testing approach to test whether an increase in utility from diversification is large enough to statisti- cally reject the null that the investor would have been better off investing in the original, less diversified portfolio. Following Brandt (1999), I formulate the portfolio allocation problem itself as a statistical estimation problem. The finite sample estimator of expected utility plays the same role as a statistical loss function7, and portfolio weights are parameters to be estimated. I test the significance of the diversification benefits of various assets for a United States investor. The investor has a choice of assets that includes an 5Standard textbook discussions always caution that diversifying beyond one or two dozen equities yields little diversification benefit. 6I am assuming the added asset’s returns are not perfectly correlated with any portfolio formed from the existing assets. 7Actually, statistical optimizations are usually formulated as minimizations so negative utility is the direct analog of a statistical loss function. 3 equity index, a long-term bond index, and an asset formed by going long assets with high book to market ratio with offsetting short position in assets with low book to market ratios. I find that inclusion of the long-term bond asset does not significantly increase the estimated utility. This is surprising since a bond portfolio is a standard component in practical asset allocation problems. By contrast investment in the value premium significantly improves the expected utility estimate. Furthermore, I examine the significance of predictability of index returns. I test against the hypothesis that conditioning allocations on predictor variables does not increase expected utility of the investor. I find that this hypothesis cannot be rejected for most of the predictor variables considered. The excep- tions are the term spread and the gold industry trend which yield marginally significant improvement in unconditional expected utility. 1.2 Prior Disbelief in the Optimality of Short Positions The second essay examines whether portfolio estimates are improved when con- ditioned on additional, economically motivated insights. The idea is to improve the robustness of portfolio estimates by introducing an informative, non-data prior. I accommodate the additional prior by formulating the portfolio estima- tion problem in a Bayesian setting. I consider the prior belief that all asset allocations are positive at the opti- mum. This prior has two motivations. First, the prior is motivated by economic theory. Classic asset pricing models such as the CAPM (Sharpe (1964)) hold that assets are priced such that their market weights reflect the optimal port- folio of an aggregate investor. For assets that have positive market value, the implication is that they will appear with positive weight in the optimal port- folio of individual investors. The second motivation for the no-short-position prior is the importance of short-sales constraints in portfolio allocation applica- tions. While many investment managers impose short-sales constraints due to limitations in their mandates, the presence of such constraints may also reflect the lack of availability of Bayesian technology in standard portfolio selection implementations. As a result, the only convenient method of incorporating a 4 prior belief is to impose it directly via the use of constraints. In a simulation study, I examine the impact of a prior disbelief in the op- timality of short positions. I consider a five-asset universe and simulate return histories from a generating distribution based on the historical distribution of returns for a five-industry breakdown of the United States equity universe. I find that the expected out-of-sample performance under the no-short-sales prior improves over the expected out-of-sample performance under a diffuse prior. The model performs similarly to approaches that impose short-sales con- straints directly. However, the method does not perform as well as the minimum variance and equal weight (or 1/N) rules. For the case considered, both of these rules are asymptotically biased. The true optimal minimum variance portfolio includes negative weights while the weights under the 1/N rule do not equal the true optimal mean variance weights. 1.3 Nonparametric Portfolio Estimation I use a nonparametric estimator of portfolio weights. For the inference analysis in the first essay, I employ the formulation of Brandt (1999).8 In the second essay, I develop a technique for tuning the data to obtain an empirical distrib- ution of returns that reflects an investor’s qualitative (or non-data) priors.9 I use Bayesian methods to adapt the probabilities attached to each data point in the return distribution to the non-data information. Because I discretize the set of possible return outcomes to those observed in the historical data, the data informs the setup of the prior, and the resulting estimator is not strictly Bayesian. However, restricting to return outcomes in the investor’s data set is an effective means of discretizing a high dimensional space of possible return outcomes to a parsimonious grid. Because the actual data generating points is setting the grid, the included points are naturally concentrated near peaks of the data generating distribution. Effectively, the domain of the posterior is restricted to a set of multinomial distributions. However, the set allows for a 8Other applications of Brandt’s (1999) method can be found in Aı̈t-Sahalia and Brandt (2001), Paye (2004), and Chapter 2. 9Previous applications of data tuning focus on minimally perturbing either the probabilities or the data itself until some constraint on the solution is satisfied. See Braun and Hall (2001) for a discussion of these methods and their application. 5 broad range of distributional properties and permits effective incorporation of prior information. I integrate the Bayesian posterior distribution of returns by Markov chain Monte Carlo. In both essays, I combine historical return data with other information. In the first essay, the additional information is the value of a predictor variable while in the second essay the additional information is a qualitative prior. Un- surprisingly, the resulting estimators have a common nonparametric form. In each case, the problem is to choose weights that solve the investor’s problem, max w E [U(r̃, w)|R, I] , (1.1) where w is a vector of portfolio weights, r̃ is a vector of unknown future returns, R is a matrix of past asset returns, and I is the additional information. The resulting nonparametric estimator is given by max w T∑ t=1 atU(r̃t, w), (1.2) where rt is the tth row of the return matrix R, and the at are coefficient weights that depend on the non-return information I. For the unconditional case with no additional information, the at are constant. When conditioning on a predic- tor variable, the at depend on the distance between the historical value of the predictor and the value at time of investment. Finally, in the Bayesian case, the at are obtained following integration over the posterior. The incorpora- tion of prior beliefs into the nonparametric portfolio estimator is an important methodological contribution of the second essay. 6 Bibliography Aı̈t-Sahalia, Y. and Brandt, M. (2001). Variable selection for portfolio choice. Journal of Finance, 56:1297–1351. Bawa, V. S., Brown, S., and Klein, R. (1979). Estimation Risk and Optimal Portfolio Choice. North Holland, Amsterdam. Best, M. J. and Grauer, R. R. (1991). On the sensitivity of mean-variance effi- cient portfolios to changes in asset means: Some analytical and computational results. Review of Financial Studies, 4:315–342. Brandt, M. (1999). Estimating portfolio and consumption choice: A condi- tional Euler equations approach. Journal of Finance, 54:1609–1645. Braun, W. J. and Hall, P. (2001). Data sharpening fo nonparametric inference subject to constraints. Journal of Computational and Graphical Statistics, 10(4):786–806. Chopra, V. K. and Ziemba, W. T. (1993). The effect of errors in means, variances and covariances on optimal portfolio choice. Journal of Portfolio Management, 19:6–11. Jobson, J. D. and Korkie, B. M. (1981). Performance hypothesis testing with the Sharpe and Treynor measures. Journal of Finance, 36(4):889–908. Jobson, J. D. and Korkie, R. (1980). Estimation for markowitz efficient port- folios. Journal of the American Statistical Association, 75:544–554. Kan, R. and Zhou, G. (2007). Portfolio choice with parameter uncertainty. Journal of Financial and Quantitative Analysis, 42(3):621–656. Markowitz, H. (1952). Portfolio selection. Journal of Finance, 7:77–91. 7 Paye, B. (2004). Essays on Stock Return Predictability and Portfolio Allocation. PhD thesis, University of California San Diego. Sharpe, W. F. (1964). Capital asset prices: a theory of market equilibrium under conditions of risk. Journal of Finance, 19:425–442. 8 Chapter 2 Asset Allocation with Value Growth Tilts and Predictability10 2.1 Introduction Researchers document evidence of two empirical features of financial asset re- turns with implications for asset allocation. First, a market index is not suffi- cient to span systematic risk in the cross-section of equity returns.11 Second, equity and bond index returns can be predicted by variables such as the dividend yield, term spread, and short term interest rate.12 An investor is interested in whether conditioning their portfolio policy on either predictability or multi-factor structure improves their expected welfare. More specifically, an investor might ask, “Is the gain in expected utility from conditioning on an empirical feature significant, given the evidence in the data?” A portfolio estimate that conditions on the presence of an additional factor or predictor will always yield a higher expected-utility estimate in-sample, simply from diversification. An important question is whether the utility gain is sig- nificant given that the true optimal portfolio is unknown. The investor cannot be certain that the expected utility for an estimated portfolio with a diversify- ing asset is an improvement over the best estimated portfolio that excludes the additional asset. The actual generating distribution for returns is unknown. In other words, the investor faces a forecasting problem. Adding an asset adds a 10A version of this chapter will be submitted for publication. Douglass, J., Asset Allocation with Value Growth Tilts and Predictability. 11The presence of multiple factors in equity returns can be observed via principle compo- nents analysis (Connor and Korajczyk (1993)). 12See Rey (2004) and Ang and Bekaert (2007) for reviews of predictability evidence. Cochrane (1999) reviews investment implications of multifactor pricing and predictability. 9 parameter to the estimation problem. The extra parameter will permit finding a solution that yields a higher value for the estimation optimand. However, the statistical power is lessened so the out-of-sample performance of the estimator may be lower than under a more parsimonious asset set. In this chapter, I evaluate the significance of estimated expected utility gains from conditioning on multifactor pricing and predictability. For ease of inter- pretation, I express expected utility as a certainty equivalent return (CER).13 I treat the portfolio optimization problem as a statistical optimization. As such, the expected utility achieved at the optimum is a random variable. I employ bootstrap resampling to estimate the distribution of expected utility for a given policy. I compute p-values for the null hypothesis that investing according to an estimated optimal portfolio that conditions on an empirical feature yields no utility gain over a policy that conditions on asset returns only.14 My results build on those of Brandt (1999), Aı̈t-Sahalia and Brandt (2001), and Paye (2004). Following Brandt (1999), I estimate allocations in a single step by direct solution of the Euler equation of the investor’s problem. The estimation is achieved by replacing the expectation integral with an average over the outcomes for each data point. This nonparametric approach has the advantage of bypassing the need to assume and then estimate a parametric distribution of returns. This eliminates a potential source of misspecification, as well as the problem of explicitly determining which moments of the return distribution are of interest. Direct estimation of portfolio weights leads to a transparent interpretation of portfolio allocations as statistical estimates. I assess investment significance based on robustness of expected utility gains to estimation error. Treatment of expected utility as a statistical estimate has its roots in the work of Jobson and Korkie (1980). The investor’s problem is analogous to a statistical optimization. Consider a comparison with the famil- iar problem of linear regression. For estimation in an investment framework, 13A convention that is standard in the portfolio choice literature. See Kandel and Stam- baugh (1996). 14Goetzmann and Jorion (1993) apply a bootstrap approach to the assessment of gains in R2 in linear regressions of returns on the dividend yield. Their bootstrap approach differs from mine in that they focus on comparing the predictive regression R2 with the distribution of regression R2s in bootstrap samples constructed to eliminate conditional correlation. I bootstrap from the original data and use the utility difference between the conditional and unconditional strategies to test the null hypothesis. 10 portfolio weights play the same role as coefficients in a regression, while ex- pected utility is analogous to R2. As in regression studies, in which R2 is rarely accompanied by a measure of precision (Press and Zellner (1978)), estimated expected utility is rarely quoted with a standard error. In general, expected utility estimates based on finite samples have unknown distribution.15 There are two common approaches to estimating standard errors under the finite distribution. One can either use a large sample, asymptotic ap- proximation or compute standard errors by a bootstrap procedure. I primarily employ a bootstrap procedure. I obtain each bootstrap sequence by a resam- pling procedure designed to preserve correlation properties of the data. For each bootstrap sequence, I compute average utility outcomes for optimal poli- cies estimated from the original data. In this manner, I construct an empirical distribution of expected utilities. This becomes an issue for determining stan- dard errors and for hypothesis testing. Wolf (2007), suggests a similar block bootstrap method for testing differences in Sharpe ratios across return series and compares the results with those obtained by asymptotic methods of Job- son and Korkie (1981). Wolf (2007) demonstrates that asymptotic methods are often biased in finite samples and that the bootstrap method performs well in Monte Carlo tests on returns generated by processes with non-IID errors and correlations.16 In the empirical analysis, I consider the asset allocation problem of a do- mestic U.S. investor. I consider two empirical questions. First, I examine the diversification benefit from incorporating portfolios that serve as empirical prox- ies for pricing factors. Multifactor pricing has implications for the role of asset allocation as a mechanism for determining exposure to systematic risk. In a nonparametric factorization of the covariance matrix of equity returns, three or more principle components are required to capture cross sectional variation in returns (Connor and Korajczyk (1993)). The capital asset pricing model suggests the market portfolio as principal factor. However, this portfolio only explains 70-80% of non-diversifiable risk in equity returns. Fama and French 15Jobson and Korkie (1981), updated by Memmel (2003), present asymptotic results for estimating the distribution of a difference in expected utility under the assumption of mean- variance utility and independent and identically distributed returns. 16The Jobson and Korkie (1981) test is applied to comparisons of portfolio strategies in out-of-sample tests by DeMiguel et al. (2007) and DeMiguel and Nogales (2007). In contrast, I perform in-sample testing of differences in expected utility for different strategies. 11 (1996) and Carhart (1997) find a number of hedge portfolios that serve as prox- ies for priced risk that is not captured by the market portfolio. I consider three long-minus-short equity portfolios constructed to capture value, size, and momentum premiums, respectively.17 I find that including a factor portfolio constructed to reflect the return premium between value and growth stocks significantly improves expected utility. This finding corrobo- rates results of Pástor and Stambaugh (2000) and Avramov (2004). They find that an unconstrained investor who is given the opportunity to diversify across value-premium and size-premium factors will take large positions in the value premium. However, they also report large allocations to a size premium portfo- lio. Furthermore Aı̈t-Sahalia and Brandt (2001) report nontrivial allocations to a long term bond index. I find that neither a factor portfolio based on the size premium nor a long-term bond index provide significant diversification benefit to an equities and cash portfolio. The momentum portfolio provides significant diversification benefit. Finally, I test the null hypothesis that expected utility is not improved by the addition of each factor portfolio in the investor’s available asset set. I find that the hypothesis cannot be rejected at the 99% confidence level unless the value-premium or momentum-premium portfolios are included in the investor’s asset set. Second, I study the impact of conditioning on expected utility. Predictability implies state dependence of an investor’s optimal portfolio. I estimate condi- tional and unconditional expected utility from following optimal policies condi- tioned on a set of eight predictors. I compare the results to conditional and un- conditional expected utilities estimated for the unconditionally optimal policy. I find that conditioning on individual predictors leads to a significant improve- ment in unconditional expected utility. However, utility gains that are condi- tional on the predictor value are statistically insignificant. My unconditional results are in line with a large literature on the significance of predictability on return regressions.18 In addition, the unconditional results corroborate results of asymptotic tests by Brandt (1999) for a subset of the predictors studied in this paper. 17These three portfolios are widely cited factor proxies proposed by Fama and French (1993) (value and size premiums) and Carhart (1997) (momentum premium). As of October 2007, the former had over 2400 citations on Google scholar, while the latter had over 1400. 18See, for example, Campbell and Yogo (2006). 12 My results show that optimal portfolios vary substantially with predictor value. However, as noted previously, I find that conditioning on individual pre- dictors does not yield a significant improvement in conditional expected utility. The implication is that conditioning on individual predictors is not economically significant to a myopic investor. This result contrasts with previous literature in which economic significance is linked to conditional variability of portfolio weights. However, following the optimal conditional policy does not increase conditional expected utility significantly over that attained following the opti- mal unconditional strategy. Of the eight predictors I consider, seven are taken from the literature.19The exception is the gold industry trend which measures recent returns on the gold industry. Despite being known for its low correlation with other industries, gold industry returns have not received much attention in previous literature on pre- dictability. The lack of interest might be explained by the lack of significance of this variable in a linear regression versus next period returns. Linear re- gression is the standard approach to analyzing potential predictors.20 However, gold industry trend proves to be one of the two strongest predictors in a non- parametric estimation of portfolio weights and expected utility. Examination of the portfolio policy as a function of gold industry trend reveals a strong non- linear relationship that smooths out to a constant under a linear assumption. The significance of gold industry trend is corroborated by findings of Makarov and Papanikolaou (2008). They find evidence that a latent factor that weights heavily on base metal industries helps explain market equity returns. The result demonstrates that not accounting for potential nonlinearity in the relationship between predictors and expected returns can have important implications for model assessment. This point is also emphasized by Ferson and Siegel (2000) in the context of testing asset pricing models. Rejection of the hypothesis that predictability improves expected utility based on an in- sample study requires a framework in which conditioning information is used with maximum efficiency. The nonparametric model places minimal restrictions on the portfolio policy and is therefore likely to result in a higher expected utility improvement over the unconditional policy when predictability is used 19See Ang and Bekaert (2007) and references therein. 20Predictors should also have an economic rationale to mitigate data snooping. 13 in-sample. Thus rejection of the hypothesis that the unconditional policy yields the same expected utility is more challenging, and, therefore, p-values that suggest rejection are more convincing. My approach to the question of investment significance differs from much of the recent literature, where two approaches are prevalent. First, many studies abstract from estimation uncertainty by calculating optimal allocations from parameters of a model of the expected distribution of returns that are fixed at sample values (i.e., Brennan et al. (1997), Campbell and Viceira (1999), Balduzzi and Lynch (1999), Campbell et al. (2003), and Jurek and Viceira (2005)). These contributions yield insight into potential implications of empirical features for an asset allocator, but do not establish whether the computed impacts are statistically significant. The second approach seen in the recent literature assumes a Bayesian model of the investor in which estimation uncertainty is incorporated into the pre- dictive distribution of returns. Kandel and Stambaugh (1996) pioneer this ap- proach in a study of the investment impact of predictability evidence.21 They conclude that the impact of predictability on investment decisions is significant even though the empirical evidence for predictability is weak. Their conclusions are based on the substantial variability of portfolio weights with predictor value, but they do not provide evidence on the significance of the expected utility gains from using predictor variables. My results have different implications for the importance of predictor vari- ables when compared to those of Kandel and Stambaugh (1996) and subsequent Bayesian studies.22 The discrepancy can be attributed to differences in the cri- teria used to assess significance. I treat the portfolio problem as a statistical estimation problem and assess significance in terms of welfare gains using a hy- pothesis testing approach. I examine whether expected utility gains of adopting optimal strategies are robust to estimation error. In contrast, Kandel and Stam- baugh (1996) ask whether investment policies are influenced by predictability but only obtain point values for expected utility gains without addressing the 21Kandel and Stambaugh (1996) is a concept study. Barberis (2000) and Wachter and Warusawitharana (2005) apply the ideas to empirical data. Barberis (2000) examine the importance of predictability in the face of estimation risk. Wachter and Warusawitharana (2005) study allocations of an investor with priors biased against predictability. 22See, for example, Barberis (2000) and Wachter and Warusawitharana (2005). 14 statistical significance of these gains. However, expected utility estimated in- sample must increase when the investor is given additional information. Thus, whether or not an empiricist uses a Bayesian model of the investor, it is im- portant to address the robustness of estimated expected utility gains before ascribing investment significance. Finally, I extend the predictability analysis to allow for simultaneous con- ditioning on multiple predictors. I remain in the nonparametric environment, but follow Aı̈t-Sahalia and Brandt (2001) and reduce the dimensionality of the nonparametric problem by constructing indices from linear combinations of pre- dictors. This semiparametric approach reduces the dimension of the nonpara- metric estimation problem to one, thereby mitigating problems with the curse of dimensionality that arise in multidimensional nonparametric estimation. I find that predictability can be significant for an investor with a single- month horizon when multiple predictors are used. I estimate optimal index values for all combinations of four predictors that can be formed from a set of eight predictor variables. This model selection exercise suggests that the best and most robust predictor combination is any combination of predictors that includes the term spread and the gold industry trend. The important predictors differ from those ordinarily employed in studies that focus on the dynamics of investment with time varying investment opportunities. For example, the dividend yield is not a statistically robust component of the optimal predictive index, but is often used in calibrated dynamic models. Of the two variables that play an important role in predicting optimal allocations, the term spread has the longest history in studies of predictability (see Campbell (1987)). Unlike in the single predictor case, the improvement in expected utility for the best multivariate index is marginally significant in some states of the world. It is important to clarify that the tests in this paper are not primarily aimed at verifying the presence of predictability. The objective of my tests is to eval- uate the relative significance of portfolio estimation risk versus the benefits of diversification or conditioning using a metric, expected utility, that reflects in- vestor preferences. In other words, I assume that diversification or conditioning would add value if the true optimal portfolio was known. The testing approach in this paper would also be applicable to tests of predictability. However, the standard errors and p-value statistics would have to be adjusted to address the 15 data-snooping/joint-testing problem. This analysis is beyond the scope of the this thesis.23 The estimated expected-utility gains permit weighing the benefit of incor- porating factor portfolios into the asset allocation decision versus the benefit of conditioning on predictability. For an equities and cash investor, I find that point estimates of expected utility gains from adding the value premium proxy to the portfolio choice set are effectively equal to the gains from conditioning on the term spread or gold industry trend. The magnitude of the improvement in CER is approximately 50 basis points per month or six percent per year. 2.2 Empirical Framework I employ an econometric framework based on the portfolio choice problem of a single period investor. The investor’s problem is set up as a statistical decision problem with asset allocations as parameters and expected utility as the objec- tive. The allocations are estimated by direct maximization of expected utility. The remainder of this section describes the investment framework and presents the empirical estimator. 2.2.1 Investor’s Problem Consider a single period investor who maximizes the expected value of utility u(Wt+1) over next period’s wealth Wt+1. The investor has access to a set of N portfolios for investment. The investor’s choice variable is an N -vector of port- folio weights αt. Expected utility is conditional on a set of predictor variables Zt. Hence, the investor solves max αt E [ u(Wt+1)|Zt ] , (2.1) where u(·) is the investor’s expected utility over wealth. The above formulation of the portfolio problem does not explicitly include a budget constraint. Instead of requiring that portfolio weights add to one, the 23The predictability of stock returns and the precise nature of additional pricing factors remains a topic of current research(Ang and Bekaert (2007)). White (2000) and Dudoit and van der Laan (2007) describe frameworks for testing multiple hypotheses. 16 asset set is defined such that each asset is a hedging portfolio. Hedging portfolios consist of long positions in some assets and offsetting short positions in other assets such that the net investment is zero. A net long or short position in risky assets is made possible by the inclusion of at least one asset that represents a long position in a portfolio of risky assets and an offsetting short position in the risk-free asset. Given this construction of the asset set, and assuming no other constraints, the vector of portfolio weights α can take on any value in RN . Let random vector Rt be the vector of gross returns on securities from t to t + 1. Next period wealth Wt+1 depends on this period’s wealth and portfolio weights along with the intervening period’s vector of returns Rt+1, Wt+1(Wt, αt, Rt+1) = Wt [ Rf + αᵀtRt+1 ] (2.2) where Rf is the risk-free rate. The predictor variables are realizations of a random vector Zt from an M - dimensional predictor space Λ. The distribution of expected returns is depen- dent on Zt. This along with the dependence of utility on returns explains the conditional form of the expectation in the investor’s objective function. The investor’s optimal portfolio policy is the solution α(Z) to (2.1). A portfolio policy is a mapping from the predictor space to the space of allowed portfolio weights, α : Λ → RN . The first order conditions of the investor’s problem are E [ u′ (Wt(Rf + α ᵀ tRt+1))Rt+1|Zt ] = 0. (2.3) Hence, the portfolio policy is given by α(Z) = { αt : E [ u′(Wt(Rf + α ᵀ tRt+1))Rt+1|Zt = Z ] = 0 } . (2.4) The estimator developed in the following section is based on this equation. 2.2.2 Direct Estimator of Conditional Portfolio Policy Portfolio policies can be estimated without explicitly modeling return dynam- ics24. Brandt (1999) adapts the method of moments approach of Hansen and 24The need to model the entire return distribution considerably complicates estimation of optimal portfolio policies by adding an additional set of assumptions beyond those already 17 Singleton (1982) to this problem. The empirical approach entails replacing the expectation in the first order condition or Euler equation of the investor’s prob- lem (2.3) with a consistent estimator. The expectation E[·|Zt] on the right-hand side of the Euler equation is replaced with a nonparametric estimate Ê[·|Zt] that converges to the true expectation as T →∞. Upon replacing the expectation in (2.3) with its empirical counterpart, an empirical moment condition is obtained and α can be estimated by method of moments. Operationally, the expectation on the left-hand side of the first-order condi- tion (2.3) is replaced with a historic average Ê [u′(WsαᵀRs+1)Rs+1 |Zt = z] = 1 Tz ∑ {s:Zs=Z,s<t} u′(WsαᵀRs+1)Rs+1 = 0, (2.5) where Tz is the number of observations at which Zt = z. For Zt on a continuous domain, the above estimator is infeasible for finite samples. Brandt (1999) shows that a standard nonparametric estimator converges for moment functions that obey reasonable properties. The nonparametric estimator of α in some state z is obtained by weighting each observation according to the similarity of its state with z. Define the weighting function ω(·) and bandwidth hT . The nonparametric estimator of the expectation is Ê [u′(WsαᵀRs+1)Rs+1 |Zt = z] = 1 τ(hT , z) T∑ s=1 ω ( z − zt hT ) u′(WsαᵀRs+1)Rs+1, (2.6) where τ(hT , z) is equal to the sum of the weights applied to each observation. Substituting the empirical expectation into the Euler equation yields a set of empirical moment conditions that are satisfied by the optimal portfolio policy. required to model investor preferences. For most preference models, the precise shape of the portfolio policy is a complicated function of multiple moments of the conditional distribution of returns. This point is discussed extensively in Aı̈t-Sahalia and Brandt (2001). They evaluate the portfolio policy for a variety of utility specifications, including expected utility, ambiguity and loss aversion, and prospect theory preferences. For example, consider the case of an investor with mean-variance utility. In this case, there is an analytic solution for the optimal conditional policy. The optimal policy depends on the ratio of the first two conditional moments of the return distribution. Thus, even in this simple case, in order to study the effects of predictability on portfolio choice by traditional means, one would have to effectively model both the first and second moments as functions of the predictor variables. The problem is exacerbated if allocations depend on higher moments (Harvey et al. (2003), Kacperczyk (2003)) or if allocating across a large number of assets (Brandt et al. (2005)). 18 The problem of solving the Euler equations can be formulated as a method of moments minimization whereby α(z) is chosen to equate Ê [ u′(Wtα ᵀ tRt+1)Rt+1|Zt = z ]′ Ê [ u′(Wtα ᵀ tRt+1)Rt+1|Zt = z ] (2.7) to zero. 2.2.3 Measure of Portfolio Performance Kandel and Stambaugh (1996) suggest certainty equivalent returns as a useful metric for comparing portfolio performance.25 The certainty equivalent CE is the certain wealth outcome that would be accepted by the investor as an even trade for the risky bet, i.e., U(CE) = E [u (Wt+1)]. When comparing two certainty equivalents, the fractional change is a useful metric. This is defined as δCE = CEA/CEB − 1; CEA is the certainty equivalent of interest and CEB is certainty equivalent value of a reference portfolio. Following convention, I refer to δCE as the certainty equivalent return (CER) when the reference portfolio is a 100% investment in the riskless asset. For unconditional portfolio allocation, the certainty equivalent can be esti- mated by replacing the expectation with a sample average: ĈE = U−1 ( 1 T T∑ t=1 u(αᵀRt) ) (2.8) When using conditioning information, the estimated certainty equivalent is state dependent. Adopting the kernel averaging approach described in the previous section, the conditional certainty equivalent for an investor with unit wealth is ĈE(z) = U−1 ( 1 τ(hT , z) T∑ s=1 ω ( z − zt hT ) U (α(Zt) ᵀRs+1) ) . (2.9) I calculate p-values for hypothesis tests and compute standard errors using a bootstrap analysis. I recalculate equivalent statistics for 1000 bootstrap sam- ples. I use the stationary bootstrap procedure of Politsis and Romano (1994) 25The use of certainty equivalent to evaluate portfolio performance is a standard approach (Aı̈t-Sahalia and Brandt (2001), Avramov (2004), Brennan et al. (1997)). 19 to preserve autocorrelation properties of the data in the bootstrap samples. Many of the hypothesis tests involve comparing expected utility gains achieved by different strategies. I first compute the optimal policies for the two strategies. I then calculate expected utility estimates for each bootstrap draw. If the null hypothesis is that strategy A achieves as high an expected utility as strategy B, then I compute a p-value based on the fraction of draws for which average utility achieved under policy A equals or exceeds that achieved under policy B. An investor’s ability to exploit the increase in CER from diversification will depend on the amount of data available to the investor and potential CER gain given full knowledge of the generating distribution of returns. To guage the relative importance of these two factors, I conduct a simulation experiment. I consider the investment benefit of adding an additional risky asset to a two risky asset portfolio. The correlation between the returns of the three assets are based on correlations between the equity index, bond index and HML asset described in section 2.2.4. I vary the Sharpe ratio of the third added asset so that the full-information increase in CER varies between 1 basis point and 100 basis points per period. I simulate 400 return series of length T equal to 120, 240 and 600. For each return series, I estimate an optimal portfolio for the two risky asset and three risky asset cases. I also compute a bootstrap p-value against the hypothesis that the two asset portfolio will achieve a higher CER out of sample. The results demonstrate that an investor seeking to reject the dominance of the estimated two-asset portfolio would need to be adding an asset with the potential to add greater than 1% return per period given a data set of 120 peri- ods. For a data set of length 600, the null that the estimated portfolio achieves a higher out-of-sample CER can be rejected in the majority of cases as long as the CER is greater than 25 basis points. The latter result is relevant to the unconditional results because the data set used in this study includes over 600 monthly return periods. For results that are conditioned on predictor variables, the effective number of samples is reduced by the kernel averaging method. As a result, larger conditional improvements in CER are required to achieve diversification benefits given the available data for portfolio estimation.26 26Specifically, the Monte-Carlo Average p-values are, given a potential diversification benefit of 1 basis point, 0.30, 0.28, and 0.20 for data series lengths T = 120, 240, and 600 respectively. For potential diversification benefit of 25 basis points, the corresponding Monte-Carlo average 20 In addition to bootstrap errors, I also compute asymptotic standard errors where possible. The asymptotic properties of the nonparametric conditional es- timator were given by Brandt (1999). The asymptotic distribution of estimates of portfolio weights are directly analogous to standard convergence results for the method of moments. For conditional and unconditional certainty equivalent returns estimated via (2.8) and (2.9), I compute asymptotic standard errors via the delta method. 2.2.4 Data and Utility Assumptions I obtain monthly data for asset returns and predictors for the time period Jan- uary 1952 to December 2006.27 I consider an investor who allocates wealth between an equity index, a bond index, and cash. The investor also has ac- cess to hedge portfolios that have been recognized in empirical studies as good proxies for priced risk in the equity market that is not captured by the mar- ket portfolio (Fama and French (1996), Carhart (1997)). These consist of a portfolio long high book-to-market (value) stocks and short low book-to-market (growth) stocks, a portfolio long small market capitalization stocks and short large capitalization stocks, and a portfolio positive high momentum stocks and short negative momentum stocks. The market index is the value-weighted in- dex of NYSE, AMEX, and Nasdaq stocks from CRSP. The bond portfolio is an index of long-term government bonds. Returns on this portfolio are obtained from CRSP. Factor portfolios based on value minus growth, small minus big and upward momentum minus downward momentum are the HML (High-book-to- market Minus Low-book-to-market), SMB (Small-stock Minus Big-stock), and UMD (Upward-momentum Minus Downward-momentum) portfolios from Ken French.28 The return of the riskless asset is the yield on treasuries matching p-values are 0.21, 0.14, and 0.05. For potential diversification benefit of 50 basis points, corresponding Monte-Carlo average p-values are 0.14, 0.06, and 0.009. Finally, for potential diversification benefit of 100 basis points, the corresponding Monte-Carlo average p-values are 0.07, 0.02, and 0.0003. 27This start date is commonly used in calibration studies of predictability and portfolio choice (e.g. Brandt (1999), Campbell and Viceira (1999), Brennan et al. (1997), Aı̈t-Sahalia and Brandt (2001)), and coincides with availability of data for a large set of predictor variables. The time period is subsequent to the 1951 Treasury-Fed accord that allowed independent conduct of monetary policy. 28http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data library.html 21 the investment horizon. For the monthly return horizon, this is the 30-day Treasury-bill yield. Summary statistics for the asset returns are provided in Table 2.1. Investment opportunities are functions of eight predictor variables that have been cited as able to forecast market returns and volatility. The predictors include the NYSE dividend yield, default premium, term premium, trend in the S&P 500 composite, trend in the 30-day Treasury-bill yield, gold industry returns, value spread, and inflation29. The dividend yield on NYSE stocks is imputed from cum- and ex-dividend returns from CRSP. All interest rate data is obtained from Global Insight with the exception of the T-bill yield. The latter is obtained from CRSP. The default premium is the difference in yield between an index of BAA and AAA rated corporate bonds. The term spread is the difference between the yield on long-term government bonds and three- month Treasury bills. As suggested by Campbell (1991) and Hodrick (1992), I stochastically detrend the 30-day Treasury-bill yield used for prediction. I take the current yield minus a six-month moving average centered six months previously. This eliminates the long-term secular trend in this variable, rising prior to and descending since 1980. The 30-day Treasury-bill yield serves a dual role as both predictor and yield on the risk-free asset for investments with one-month horizon. The trend in the S&P 500 composite is an average over trailing returns of the previous twelve months. Realized volatility is calculated using daily returns over the previous month. Inflation is the monthly change in consumer price index (CPI) for the previous month. Summary statistics for the predictors are provided in Table 2.1. Correlations between returns and lagged predictors are listed in Table 2.2. Correlations between realized volatilities and lagged realized volatilities and predictor variables are listed in Table 2.3. All the above predictors have received attention in the literature on pre- dictability with the exception of gold industry trend. This variable is con- structed from returns on the gold industry portfolio within the 48 industry breakdown of Ken French. This breakdown is based on industry definitions 29Other predictors that have received attention include the ratio of consumption to in- come (Lettau and Ludvigson (2001)) and measures of market sentiment (Baker and Wurgler (2006)). I restrict attention to predictors derived from market observables (with the excep- tion of inflation). This mitigates possible interpretation issues associated with the timing and availability of non-market based variables. 22 that begin in June 1964. To extend the series backwards, I construct a gold industry based on membership in the gold industry in June 1964. While there is danger of introducing survivorship bias into the early part of the series, this is overshadowed by the likely loss of information from discarding the returns entirely. For use as predictor, gold industry returns are smoothed by taking a moving average over the trailing twelve months. For the empirical analysis, I assume an investor with power utility, U(W ) = W 1−γ/(1 − γ). I use a power utility model because of its appealing properties with respect to the effect of wealth on risk aversion. Power utility engenders decreasing absolute risk aversion. This is more reasonable than the increasing absolute risk aversion of mean-variance utility30. Mean-variance utility, which is often used in practice, can be a good approximation to other utility functions (including power utility). Mean variance is obtained by truncating a power series expansion of the utility at the second power so the quality of the approximation depends on the size of return magnitudes. I consider a monthly investment horizon so some returns in the sample are quite large. In addition, optimal portfolios may be leveraged thereby magnifying portfolio returns. I vary the risk aversion coefficient γ between one (log utility) and ten. Most of the experiments are performed with risk aversion set to five. 2.3 Unconditional Asset Allocation I begin the empirical analysis with an examination of the investment gains derived from incorporating portfolios that serve as empirical proxies for pricing factors into the asset allocation portfolio. The section presents an in-sample analysis of the unconditional portfolio choice and certainty equivalent gains for various combinations of assets. To obtain estimates of unconditional policies, I compute solutions to the unconditional version of (2.7). Table 2.4 lists estimates of unconditional portfolio choices of an investor with power utility. The first row lists portfolio allocations for investors whose investment universe is limited to the equity index. The results are broadly 30Many of the experiments of the next section were repeated with mean-variance utility. Essential features of the results (not reported) are similar to the power utility case. Markowitz (1952) and Arrow (1971) provide theoretical arguments for utility with decreasing absolute risk aversion. 23 consistent with those of Brandt (1999).31 The optimal weights are large. For an investor with risk aversion of five, optimal investment in the index is 58%, rising to 142% at risk aversion of two. These results are consistent with the well- known tendency for estimated allocations to indicate large positions in equities for reasonable levels of risk aversion (Mehra and Prescott (1985)). The second row of Table 2.4 lists portfolio allocation for an investor who invests in HML only. While this is not a realistic portfolio choice for most in- vestors as it implies short positions in the growth portfolio, the results illustrate the investment appeal of HML’s historical return properties. The allocation is almost double that selected by the equities investor. This can be attributed to HML’s higher Sharpe ratio (Table 2.1). The mean-variance solution illus- trates the source of the difference between the two allocations. For a mean- variance investor, the optimal allocation to a risky asset is proportional to the ratio of mean return to its variance. Using this fact, we expect the ratio of the portfolio weights for the index-only portfolio and the HML portfolio to be αIndex/αHML ≈ µIndexσ2HML/(σ2IndexµHML). Based on mean and variances listed in Table 2.1, αIndex/αHML = 0.58. The ratios between the estimated weights in rows 1 and 2 of Table 2.4 are all within rounding error of 0.56. The remaining rows of Table 2.4 present allocations across different port- folios. The next two rows show allocation estimates for an equity and bond index portfolio. The allocation to bonds is less than the standard error of the allocation estimate. The boundaries of a 95% confidence interval around the optimal bond allocation are -0.41 and 1.03. In addition, equity allocations in equity and bond portfolios are almost unchanged from their optimum without the bond asset. An equities-only investor achieves little diversification benefit by adding bonds to the asset set. Combining the equity index portfolio with HML leads to higher allocations to the equity index and HML relative optimal holdings in their single asset portfolios. When compared to the allocation for an equities-only investor, the allocation to equities increases by approximately 40% when HML is added to the asset set. These results reflect the large positive premium accorded to value stocks over the fifty-year period encompassed by this data set along with the significant negative correlation between the index and 31Brandt (1999) considers a simultaneous portfolio allocation and consumption decision. However, the consumption decision has little effect on portfolio allocation. 24 HML. Combining the bond asset with the equity index and HML combination has little effect on the portfolio weights of the equity portfolios. In all cases, the allocation to the bond portfolio is less than one standard error from zero and the allocations to the equity portfolios effectively are unchanged. The large standard errors reported in Table 2.4 reflect considerable uncer- tainty in return estimates obtained from data. Brandt (1999) suggests using the mean-variance solution for intuition on the source of uncertainty. If we assume known variance then the standard error of the return estimate for the mean- variance solution, and consequently, of the portfolio weights, will be inversely proportional to (γ ∗√T ∗ σ), where σ is the standard deviation of returns. The errors calculated using this approximation and sample values in Table 2.1 are of similar size to those given in Table 2.4. Table 2.5 lists estimated CERs for an investor who follows the optimal un- conditional policies for various portfolios. I compute standard errors for CERs using the stationary bootstrap. For each bootstrapped realization of the data, I compute a CER estimate under the assumption that the investor holds the esti- mated optimal portfolio.32 The certainty equivalent return of an index investor is less than two standard errors above the certainty equivalent return of follow- ing a risk-free strategy. A 95% confidence interval around the CER estimate for an equities-only portfolio runs from -5 to positive 37 basis points. The results suggest that an equity index investor may not be able to reject the possibility that the CER of investing in the equity index is no greater than investing solely in the risk-free asset. This level of uncertainty in the CERs might give pause to advocates of straight indexing as an optimal allocation policy. I also compute the asymptotic standard errors for the CERs reported in Table 2.5. The CER is a function of the realized utilities ut = w ′ ∗ rt. The delta method approximation linearizes the variations to approximate the stan- dard error by (CER′(ū))2var(u). I use a Newey-West procedure to account for autocorrelation in the realized utilities when computing the variance. The re- sults are very similar to those obtained by the bootstrap analysis. For example, the standard error of the CER for the equities, bonds, and HML computed by the bootstrap is 0.24 basis points compared per month whereas the asymptotic 32Alternatively, asymptotic standard errors can be computed by the delta method applied to (2.8). The results are similar and not reported. 25 standard error given a Newey-West adjustment with 20 lags is 0.25 basis points. Given the similarity of the results obtained by the two methods, I only report bootstrap standard errors in Table 2.5 and subsequent tables. In contrast to the equity index-only case, the CER of portfolios that include both the equity index and HML portfolios are significantly larger than twice the standard errors (Table 2.5). The portfolio of an HML-only investor has a higher Sharpe ratio in-sample than the equity index, but the CER of an investor who allocates exclusively to this portfolio is less than two standard deviations from the risk-free CER of zero. In contrast, the investor whose portfolio includes equities and HML achieves CER greater than three times the standard error. A 95% confidence interval for CER for the equities and HML investor runs from 0.2 to 1.1 basis points. The results in Table 2.5 have important implications for the bond asset. Inclusion of the bond portfolio in the asset set has an insignificant impact on CERs of both the equities-only investor and the equities and HML investor. In both cases, the estimated CERs remain unchanged. In addition, there is no shift in the bootstrap confidence intervals. Referring to Table 2.4, the estimated allocations to the bond asset are quite large, but remain insignificant. Consider the case of risk aversion γ equal to five. The estimated bond weight is 30% and 17% in the equities and bonds, and equities, bonds and HML portfolios repectively. The CER results demonstrate that there is no in-sample evidence that an investor achieves any diversification by including the long-term bond asset in their asset allocation portfolio. Table 2.6 lists bootstrap p-values based on the null hypothesis that adding a particular asset does not improve expected utility. Each p-value is based on 1000 resamples of the joint return series. The left column lists the asset set that is assumed optimal under the null while the middle column lists the asset under consideration for addition to the portfolio. The first and third rows examine the importance of utility gains from investing in the equity index and HML respectively versus the null of holding only the risk-free asset. The p-values are approximately 0.05 in each case, indicating that investing in either portfolio is an improvement over simply holding the risk-free asset. As suspected given the insignificant holdings in the portfolios listed in Table 2.4, the null hypothesis that investing fully in the risk-free asset is preferred to investing in a mix of 26 bond index and risk-free asset cannot be rejected. The sixth row lists the p-value of adding the bond index to a portfolio that includes the equity index and HML. The seventh row states the p-value of adding HML to the traditional asset allocation portfolio. The null that the bond index is not additive to an investor’s CER cannot be rejected. In contrast, the corresponding null in the HML case is strongly rejected in favor of adding HML to the asset allocation portfolio. Other portfolios have been suggested as proxies for risk captured by stock returns. Table 2.5 also lists in-sample CERs for an investor who adds two other portfolios to the asset allocation decision. The first is SMB: a portfolio long large stocks and short small stocks. SMB is a member of the oft-cited three factor set of Fama and French (1993). If the investor believes that SMB serves as a proxy for a priced risk factor, then the portfolio choice results (not shown) indicate that the investor will take a marginally significant long position in this asset. However, the increase in CER over that achieved by the optimal equities, bonds, and HML portfolio is insignificant. The p-value for the bootstrap test against the hypothesis that SMB has no diversification benefit is 0.24 (eighth line of Table 2.6), providing further evidence against SMB as a useful diversification factor. The increase in CER is much larger when the momentum factor UMD is added to the asset set. The increase in CER for the equities and HML investor who includes UMD in their asset set is just under twice the standard error. The p-value on the last line of Panel A of Table 2.6 corroborates this observation. The portfolios that include HML can only be implemented if short-sales are permitted. The short sales occur when short positions on the growth side of the HML portfolio are larger than long positions in the same stocks held in the equity index portfolio. Table 2.4 also list portfolio weights for an investor who cannot hold assets short. Weights in equities and bonds must be greater than zero. To ensure that short positions in the long-short factors are covered, positions in these portfolio are limited to one third of the equity holding. The asset weights listed in Table 2.4 suggest an increased role for bonds in the equities, bonds, and HML portfolio. With the short sale constraint restricting the investor’s ability to leverage the HML portfolio, the optimal bond weight is 42% for an investor with risk aversion of five. This compares to 17% in the unconstrained case. However, the certainty equivalent gain achieved by moving 27 from the optimal equities and HML portfolio to the optimal portfolio with bonds included is inconsequential. The p-value listed in Table 2.6 for this case (line 5 of Panel B) is close to 0.5. The hypothesis that bonds do not increase CER cannot be rejected even when short sales are not permitted. 2.4 Conditional Asset Allocation This section examines predictability evidence in an asset allocation framework. The standard approach to predictability evidence is to estimate a statistical model in a linear regression or vector autoregression framework. Returns are regressed on lagged predictor variables. This section considers an alternative approach. I examine the implications of individual predictors on investor wel- fare. I consider the portfolio problem of a power utility investor whose portfolio choice is between the asset of interest and a risk-free asset. The first subsection is a brief examination of predictability evidence based on a simple regression analysis. The second subsection looks at predictability evidence through the lens of the optimal asset allocation problem. I consider the allocation problem of an investor with access to a single, risky asset. I evaluate the impact of predictors on portfolio choice by examining the significance of differences between conditional and unconditional allocations estimated by the method of section 2.2.2. 2.4.1 Predictability of Mean and Variance Panel A of Table 2.8 lists results of univariate least squares regressions of individ- ual asset returns on individual predictor variables, i.e., rt = a + b ′Zt. Because the choice of predictor variables is primarily based on previous literature on market predictability, it is no surprise that levels of significance are reasonably high for regressions involving the equity index return. In each case, a number of predictors yield t-statistics greater than two33. The least predictable asset is HML; at short horizons, only the Treasury-bill rate yields a t-statistic greater 33Care must be taken in interpreting t-statistics obtained from regressing a time series on a second, highly persistent time series (Stambaugh (1999)). See Campbell and Yogo (2006) and references therein for a discussion of inference issues associated with regression evidence for stock return predictability. 28 than one. The value spread does not show evidence of much correlation with future HML returns. Panel B of Table 2.8 presents results of multivariate regressions. For each asset, next month’s return is regressed on the full vector of predictor variables. The results are broadly consistent with those of the previous table, but with dampened significance for individual coefficients due to the simultaneous nature of the regression. For the index portfolio, no one predictor appears particularly significant. By contrast, the regression results indicate term spread and bonds have significant impact for predicting bond returns. Additionally, The Treasury- bill rate stands out as a significant predictor for the HML portfolio. The R2 values for the multivariate regressions are highest for the equities and bond indices. Once again, this result is unsurprising given that the predictor variables used are from the long literature on stock market predictability. Thus, while the index and bond portfolios may be more predictable than HML, a data snooping bias in the choice of regressors must also be acknowledged. Panel A of Table 2.9 displays results of univariate regressions of realized volatility of equity index and HML portfolios respectively on each lagged pre- dictor variable. A number of variables appear to predict realized volatility at the one-month horizon. For example, the t-statistic is greater than two for realized volatility of the equity index regressed on dividend yield, default premium, S&P 500 trend, and value spread. The same holds for HML regressed on dividend yield, S&P 500 trend, and gold industry trend. Panel B of Table 2.9 shows results of regressing realized volatilities of the equity portfolios on the full set of predictors. Significant regression coefficients are observed for both the equity index and HML. The high R2 values reflect the high short-term persistence of volatility. 2.4.2 Predictability in an Asset Allocation Framework In this section, I analyze the impact of predictability on portfolio choice through the lens of the investor’s portfolio problem. The analysis provides direct insight into the significance of predictability to portfolio choice. I employ the nonpara- metric estimator of portfolio policy (Section 2.6) to examine the evidence for predictability of different assets. Consider the portfolio policy of an investor who is able to invest in only one risky asset. The investor’s problem is to 29 choose optimal allocations to the risky asset as a function of predictor value. In this context, the significance of a predictor for portfolio choice is a function of the ratio of the maximum deviation of the conditional allocation from the unconditional allocation and the standard error of the allocation estimates. I apply this approach to analysis of predictability of the equity index and HML portfolios. Table 2.10 lists conditional allocations for an investor choosing between one risky asset and a risk-free asset. Results in this table can be used to assess the relevance of any one predictor variable for portfolio allocation. Opti- mal allocations are listed for values at the 20th, 50th and 80th percentiles of the empirical distribution of the predictor variable. Based on differences between al- locations at the 20th and 80th percentiles, the greatest in-sample predictability is obtained for the equity index conditioned on the term spread, Treasury-bill rate, inflation rate, and gold industry trend. Interestingly, the gold industry trend does not show up as significant in either univariate or multivariate linear regression results given in Table 2.8. In agreement with regression results, pre- dictability of HML does not appear significant by this portfolio choice metric. The largest difference between allocations at the 80th and 20th percentiles is smaller than those observed for equities. Comparison of policies at three points of the predictor distribution may miss large variations in the optimal allocations if portfolio policies are nonlin- ear. Figures 2.1-3 provide a graphical comparison of optimal conditional and unconditional single-asset policies as a function of predictor values. Each fig- ure plots optimal portfolio weight versus standardized predictor value34. The portfolios are estimated for a non-uniform range of percentiles of the empirical distribution. The calculation points are denoted by dots in the figures. Condi- tional policies are reported for single-asset decisions when the risky asset is the equity index (left panels) or HML (right panels). The equity index policies that condition on the dividend yield, default premium, and term spread are consis- tent with those documented by Brandt (1999) and, with the T-bill rate added, Paye (2004). The second column presents results for the HML portfolio. The conditional allocation to the equity index as a function of dividend yield (top left panel of Figure 2.1) is broadly consistent with results of Barberis (2000).35 34Predictor values are standardized by subtracting their means and normalizing by their standard deviations. 35See left side of his Figure 5, one period horizon case. 30 The shaded confidence regions are interpolated between two standard devi- ation error bars estimated by bootstrap. The vertical bar in the center of each plot depicts bootstrap two standard deviation confidence interval of the uncon- ditional portfolio. Overall, the width of the confidence regions confirms the insignificance of predictability evidence when conditioned on predictor value. The conditional policy rarely moves beyond the limits of the vertical bar on any panel of the three figures. The term spread and gold industry trend are the only two predictors that appear qualitatively significant for equity investment. For each of these predictors, there is a range of predictor values over which the optimal policy is greater than two standard errors from the unconditional policy. Of interest is the strength of predictability of the gold industry trend relative to the regression results presented in Table 2.7. Conditional policies show significant variability even though the coefficient on a linear regression is below standard significance levels. Estimated portfolio policies for gold industry trend vary nonlinearly with predictor value such that a linear fit as is used in the regression would damp out much of the conditional variability. Evidence for predictability of HML is weaker. This is not unexpected given that the majority of predictors that I consider are sourced from the literature on predictability of the equity index. Gold industry trend, inflation and real- ized volatility appear marginally significant as predictors for an HML investor. For the other predictors, the conditional variation in the portfolio policies is swamped by the standard errors of the estimates. The data suggest high hold- ings of HML when the trend in gold industry returns is more than one standard deviation above or below its average trend, and lower holding of HML when trend is near its average. The results for realized volatility suggest decreasing emphasis on value stocks in high volatility scenarios. In contrast, conditional significance of inflation occurs at inflation values close to two standard devia- tions above its mean. A glance at the histogram for inflation in Figure 2.11 shows low data density for high inflation environments. Hence, the significance in this region is likely to be spurious. 31 2.5 Conditional Asset Allocation with Multiple Assets In this section, I study the consequences of predictability for the conventional asset allocation problem – that of choosing between an equity index, long-term bond index and cash. In addition, I examine the importance of predictability when the asset set is expanded to include HML.36 Section 2.3 highlights the significant impact of incorporating the HML portfolio into the asset set used for asset allocation. The results in this section permit a comparison of the utility gain achieved by adding HML as a risk factor proxy to the gain from conditioning on individual predictors. Figures 2.4-11 depict the portfolio policies of a tactical allocation investor who conditions on a single predictor. There are five panels in each figure laid out in three rows and two columns. Each column depicts results for a differ- ent set of assets. First columns show optimal conditional portfolio policies for an investor with access to equity and bond indices. This corresponds to the traditional problem of tactical asset allocation. Second columns show optimal conditional policies for an investor who achieves risk factor diversification by investing in the HML portfolio. All results are obtained assuming risk aversion equal to five. The predictor values are standardized. The shaded regions are two standard deviation error estimates obtained by interpolating between two standard deviation error bars for each point estimate.37 Unconditional portfolio weights are indicated by horizontal lines, with a single cross bar on each plot showing the two standard error range of the unconditional weight. For the asset allocation problem of allocating across equities, bonds, and cash, conditional variability of portfolio weights is marginally significant for some of the predictors. Based on Figures 2.4-11, the largest deviations of opti- mal conditional weights from their unconditional level occurs when conditioning on the term spread, S& P 500 trend, T-bill rate, gold industry trend and infla- tion. Addition of the HML portfolio to the set of assets included in the portfolio 36I focus on HML rather than UMD. While the UMD portfolio also provides significant utility gains, the practical implementation of UMD strategies requires high asset turnover that leads to high transaction costs. 37Error bars are estimated by stationary bootstrap. 32 yields two observations. First, as was true for single asset policies for HML (right panels of Figures 2.1-3) the variability in conditional allocations to HML does not appear significant. Second, the presence of the HML asset increases allocations to the equity index. By comparing the top two rows of Figures 2.4 to 2.11, I observe that the effect of adding the HML to the asset set of a index and bond investor is similar to that observed in the unconditional case. The addition of a second, predictable asset increases the total investment in risky assets, but does not influence the conditional variation in asset allocations. Allocations to bonds remain essentially unchanged. The allocations to HML are similar to those obtained by Jurek and Viceira (2005). They calculate a VAR model that includes two of the predictors considered here: the term premium and T-bill rate38. However, they employ a calibration framework and do not evaluate the uncertainty of computed optimal portfolios. Neither the dividend yield (Figure 2.4) or the default premium (Figure 2.5) yield conditional policies that show significant variation with respect to predic- tor value. Results in Figure 2.6 suggest underweighting equities when the term spread is low and overweighting bonds when term spread is high. The optimal policy with respect to the S&P 500 trend is to underweight equities and over- weight bonds when the trend is greater than one standard deviation below its mean and to underweight bonds when the trend is greater than one standard deviation above its mean. The downward slope of the relationship between the T-bill rate and optimal allocation to the index (Figure 2.8) shows that post-war U.S. experience vali- dates the common wisdom that equity investment should decrease when short term interest rates rise. The effect of the T-bill rate on HML investment is statistically insignificant, but hints that an investor would have benefited dur- ing the in-sample period from holding larger positions in value stocks during periods of very high and very low short term interest rates. Figures 2.12-15 illustrate the in-sample gain in CERs from conditioning on predictor variables. Results are shown for an investor with risk aversion of five. The solid line is the conditional CER of the conditionally optimal policy as a function of predictor value. The conditional CER is the return equivalent 38Jurek and Viceira (2005) find weak statistical significant evidence of predictability. They employ a parameterized VAR model of returns and an approximate solution to the portfolio choice problem. 33 of a kernel weighted average over monthly utility outcomes (see equation (9)). Three other curves are plotted for comparison. The dashed curve is the con- ditional CER of the unconditionally optimal strategy. The dotted line is the unconditional CER that obtains when following the unconditionally optimal strategy. The dash dot line is the unconditional CER from following the op- timal conditional strategy. The unconditional CERs are return equivalents of simple time-series averages of realized utilities for each strategy. By comparing the conditional CER of the conditional and unconditional strategies, we can qualitatively examine the conditional significance of pre- dictability to the asset allocation investor. First, consider results for the bonds and equities portfolio with dividend yield as predictor (left panels of Figure 2.12). The difference between conditional CERs of the conditional strategy (solid line) and unconditional strategy (dashed line) are very small relative to the uncertainty in the conditional CER estimate (shaded region). CER is a di- rect proxy for expected utility. Thus conditioning on the dividend yield does not improve conditional expected utility when conditioning on the dividend yield. Similar results hold for all predictors for both the equities and bonds investor (left panels of Figures 2.12-15) and the equities, bonds and HML investor (right panels of Figures 2.12-15). With HML in the asset set, the conditional CER of the unconditional portfolio never leaves the two standard deviation region (right panels of Figures 2.12-15). There are some exceptions near the extremes of the empirical distribution for the case of an equities and bonds investor. At very low term spreads, the conditional CER of the unconditional strategy drops below the two standard error region around the conditional CER of the conditional strategy (top left panel of Figure 2.6). This implies potential benefits from reducing exposures to risky assets when the term spread is small or negative. The CER of the unconditional portfolio also drops below the error region at low values of S%P 500 trend and high values of the gold industry trend and consumer price index. However, these results must be interpreted with caution since the data density at the margins of the empirical distribution is small. This can lead to bias in a bootstrap framework. An empirical distribution is a noisy model of the true distribution in the tail area. Table 2.11 shows conditional in-sample estimates of CERs attained by fol- lowing the conditional and unconditional strategies for the portfolios: equities 34 and bonds; and equities, bonds, and HML. Bootstrap standard errors are given in brackets below each CER. The third line of each entry is a bootstrap esti- mate of the conditional p-value for the null hypothesis that conditioning does not improve CER. The CER improvement over the unconditional strategy is rarely more than one or two basis points. P-values near 0.5 reflect the lack of significant improvement of conditional expected utility from conditioning on single predictor variables. Gold industry trend and term spread are the only predictors that consistently yield p-values below 0.3. The difference between significance results for conditional and unconditional hypotheses has consequences for evaluating predictability in an investment con- text. For an investor with a one-month horizon, the only concern is the signif- icance in the conditional problem. While this paper focuses on the one-month investment horizon, the high persistence of the predictor series means that long horizons are required to ensure a high probability of visiting all predictor states. As a result, even for investment horizons of moderate length, investors will wish to condition on the starting value of the predictor when evaluating predictability evidence. Table 2.12 lists increases in unconditional CERs from conditioning on indi- vidual predictors. The CERs are computed from realized returns from following optimal conditional policies depicted in Figures 2.4 to 2.11. As one might an- ticipate given the conditional CER results, the greatest CER improvements are attained by conditioning on gold industry trend or term spread. The results can be directly compared to the CERs estimated for the unconditionally opti- mal policy that are given in Table 2.5. For the conventional asset allocation problem the unconditional CER is 0.19. Only gold and term spread result in unconditional CERs that are more than two standard deviations greater than the unconditional portfolio. For the equities, bonds, and HML case, the uncon- ditional CER estimate is 0.67. In terms of multiples of the standard deviation of the CER estimates, CER gains are less significant than for the case without HML. Only the gold industry trend yields an unconditional certainty equivalent gain of more than two standard deviations. The results in the first column of Table 2.12 permit comparison of the CER gain from conditioning on individual predictors with that achieved by simply adding HML to the conventional asset allocation. From Table 2.5, the diver- 35 sification benefit of the HML portfolio is 0.48 basis points per month. CERs achieved by conditioning the conventional asset allocation portfolio on individ- ual conditioning variables are almost all less than the CER estimate for the optimal unconditional portfolio with HML added to the asset set. The lone ex- ception is the CER of the portfolio conditioned on gold industry trend for which the unconditional improvement in CER is equal to the diversification benefit of the HML portfolio. Beyond indicating whether conditioning improves investment outcomes, the results in Figures 2.12-15 can also be used to determine whether predictors can be used to differentiate between good and poor investment states. Even if conditioning on predictors does not significantly improve investment prospects can we at least identify whether our prospects are good or poor in the current state? Good states feature conditional expected utility significantly greater than the unconditional expected utility of the strategy, and vice versa for bad states. Using Figures 2.12-15, we can identify potentially good or poor states by identifying predictor values for which the unconditional expected utility of the conditional strategy is outside the error region for the conditional expected utility. For the equities and bonds investor (left panels), some predictors can be used to identify poor investment states. For example, a high up trend in gold industry returns corresponds to low conditional CER relative to unconditional levels. The same is also true for states with negative term spread. Similar regions obtain for a number of the other predictors. When HML is added to the asset set, the predictors lose any ability to differentiate good and poor states, with the notable exception of gold industry trend. Moderately positive upward trend in gold industry return corresponds to poor investment outlook for the asset allocation investor (bottom right panel of Figure 2.14). The results for the term spread and trend (Figure 2.13), gold industry re- turns (Figure 2.14), and inflation (Figure 2.15) suggest that conditional policies have significantly higher conditional CERs relative to unconditional policies at the extremes of predictor distributions. For example, the conditional policy indicated for negative term spreads (or inverted yield curve) has significantly higher CER than the corresponding conditional returns of the unconditional policy. However, these results have to be interpreted with caution because the density of the data set decreases towards the extremes of the in-sample range 36 of the predictor variables. Whether or not the conditional policy is adopted, the CER in negative term spread (inverted yield curve) environments is significantly less than the un- conditional expected utility. This result suggests that an inverted yield curve indicates a state of the economy in which investment outlook is gloomy. How- ever, following a conditional strategy does not mitigate the negative impact of being in this poor state. 2.6 Conditioning on Multiple Predictors Analysis of the previous section demonstrates that conditioning on individual predictors does not significantly improve investment expectations. In this sec- tion, I examine whether this result continues to hold when the investor is able to condition on multiple predictors simultaneously. 2.6.1 Direct Estimator for Multiple Predictors The nonparametric estimator (2.6) rapidly loses statistical power when applied to a multi-dimensional conditioning vector, i.e. when Zt is a vector. Given a finite sample, the data density drops exponentially with number of dimensions. As a consequence, there is an exponential drop in rate of convergence with number of dimensions. Motivated by the implementation of Aı̈t-Sahalia and Brandt (2001), I con- sider a structural assumption that reduces the dimensionality of the nonpara- metric estimation. However, in forming the estimator, I make a slightly less restrictive modeling assumption. Aı̈t-Sahalia and Brandt (2001) reduce the dimensionality of the estimation problem through a partial parameterization of the model of section 2.2. With Zt representing anM -vector of conditioning variables, the portfolio policy is as- sumed to depend on a parametric function Z(Zt; β) of the conditioning variables where β is a vector of parameters. The parametric function projects realiza- tions of Zt onto a lower dimensional space. Hence, the dimensionality of the nonparametric estimation is reduced at the expense of requiring the estimation of a finite number of unconditional (or global) parameters β. Instead of model- 37 ing portfolio allocations as nonparametric functions of a vector Zt defining the state at time t, portfolio allocations are modeled as a nonparametric function of a parametric function Z(Zt; β) of the predictor variables, where β is a vector of parameters. The function Z(·; β) : RM → RL reduces the dimension of the nonparametric problem (i.e., L < M). The semiparametric approximation mitigates problems with the curse of di- mensionality because the nonparametric part of the estimation is with respect to a lower dimension variable, Z(Zt; β). If we further assume that Z(Zt; β) = β′Zt where β is an L×M matrix, then we obtain a model analogous to the semipara- metric index model that has received attention in the regression literature.39 In the empirical application to follow, I assume β has one dimension, i.e., L = 1. Under the semiparametric specification, additional predictors are incorpo- rated into the model via the addition of a finite number of parameters per dimension. These parameters add to model complexity, but at a much slower rate than if an additional dimension were added for nonparametric estimation. These parameters, index coefficients β, are global parameters. As a result, they are estimated by optimizing an unconditional objective. Unlike nonparametric portfolio policies, the estimate converges with √ n efficiency once an appropri- ate decision criteria has been specified. For example, in the standard regression framework, index coefficients might be chosen to minimize the squared differ- ences between observed data and the best fit curve for the chosen bandwidth. I first follow Aı̈t-Sahalia and Brandt (2001) in assuming that the investor’s optimal conditional portfolio is independent of information in Zt that is not captured by Z(Zt, β). In other words, the hypothesis is that an investor with an information set of infinite size would infer an optimal portfolio that depends only on Z(Zt; β), i.e., argmax αt E [ u ( Wtα ′ tRt+1 )∣∣∣Zt] ≡ αt(βᵀZt), (2.10) where ≡ indicates equality by assumption. Assuming that Z(Zt, β) = β′Z, the policy of the investor depends only on the subspace Λ of RM spanned by the 39In the regression literature, models in which dimensionality of right-hand side vector is reduced through partial parameterization are called semiparametric models. The term single- index model is used when the dimensionality of the parametric problem is reduced to one (see e.g. Hardle et al. (2004)). 38 column space of β. Of course, β is unknown and must be estimated. Aı̈t-Sahalia and Brandt (2001) note that a GMM estimator for β can be constructed based on assump- tion (2.10). The following first order conditions are a direct consequence of (2.10). E [ u′ ( Wtα ′ t(β ᵀZt)Rt+1 ) Rt+1 ∣∣∣Zt] = 0. (2.11) Upon multiplying each of the above conditions by a vector of predetermined functions of the conditioning variables g(Zt) and taking unconditional expecta- tions, we obtain a set of moment conditions. These conditions can be used to estimate β by solving the GMM problem min β E [ u′(Wtα(β′Zt)′Rt+1)Rt+1 ⊗ g(Zt) ]ᵀ WE [ u′(Wtα(β′Zt)′Rt+1)Rt+1 ⊗ g(Zt) ] (2.12) β′β = 1 (2.13) whereW is a weighting matrix, g : RM → RMg is a predetermined function of the vector of predictor variables. To ensure identifiability, the number of moment conditions must exceed the number of independent index coefficients; i.e., Mg× N > L(M − 1). The constraint (2.13) arises because the optimal directions are only identified up to scale. As a result, the number of free parameters to be estimated is equal to one less than the number of predictor variables M . 40 Given assumption (2.10), for known β the investor’s optimal portfolio policy is a solution to αt ( βᵀZt ) = argmax αt E [ u ( Wtα ′ tRt+1 )∣∣∣β′Zt]. (2.14) Thus, with known β, the estimation problem for the portfolio policy is equiva- 40I follow Aı̈t-Sahalia and Brandt (2001) in setting the weighting matrix (2.12) proportional to the inverse covariance matrix of the moment conditions W = Cov[u′(Wtα(β′Zt)′Rt+1)Rt+1g(Zt)]−1. Aı̈t-Sahalia and Brandt (2001) write out the uncon- ditional utility maximization problem for selecting β and show that the optimal choices of g(zt) would be the gradient of α(Z ′tβ) with respect to β. While theoretically optimal, these instruments are not ideal in practice since they require estimates of the derivative of the policy function α with respect to each element of β. Instead, I use instruments that are linear in each predictor variable. 39 lent to that for a single predictor with β′Z taking the role of the single predictor. Consequently, the methods of the previous section can be used for policy esti- mation. Aı̈t-Sahalia and Brandt (2001) note that the assumption (2.10) may be sub- optimal. In a second approach, I make a less restrictive assumption. Instead of directly assuming a restricted functional dependence of portfolio policy on predictor variables, I arrive at the same functional form for the portfolio policy by assuming the investor is limited in the number of independent shocks that they have the capacity to condition on. The limitation could arise either due to a bounded information gathering capacity or a bounded ability to process infor- mation. In the current context, this assumption limits the investors conditioning information to βᵀZt where, as before, β projects Zt onto a lower dimensional subspace of RM . Given β, the optimal policy under the current assumption is α(Zt) ≈ α(β′Zt) = max αt E [ u ( Wtα ᵀ tRt+1 )∣∣∣βᵀZt]. (2.15) This assumption does not limit the information set, just the amount of infor- mation that the investor has the ability to use. Hence, the index matrix β is selected by the investor. This is a weaker assumption than that of the previous paragraph. In the previous paragraph, the investor is assumed to condition on all information, but that the resulting portfolio only depends on the projection of predictor shocks onto a lower dimensional subspace. Here, the true optimal policy may depend on the full information subset, but the investor must choose an optimal subspace of the set of possible predictor outcomes upon which to condition their portfolio policy. The two assumptions differ in their implications for index identification. Assumption (2.15) does not imply that the investor can gain no information from observing shocks along directions perpendicular to those included in β. As a consequence, the conditioning variables are not necessarily instruments of the first-order conditions. Thus, the moment conditions used to construct the GMM objective (2.12) do not necessarily hold. This precludes using the estimator described for the previous case. Instead, the investor chooses directions, the columns of β, that span the subspace of the space of possible predictor vector 40 outcomes that yields the largest gain in expected utility. max β E [ max αt E [u(Wtα ′ tRt+1) | β′Z] ] = max β E [ u (Wtα(β ′Z)′Rt+1) ] , (2.16) β′β = 1. (2.17) In essence, the investor is choosing the most informative index coefficients given the objective of maximizing utility. In the following I shall refer to (2.15) as the limited-information case, and (2.10) as the restricted case. To provide further illustration of the differences between the formulations, consider the first-order conditions of the index optimization problem (2.16) E [u′(Wtα (βᵀZt) ᵀRt+1) (α ′(βᵀZt)ᵀRt+1)Zt] = 0. (2.18) Evaluation of the above first-order condition requires not only knowledge of the policy function, but its derivative as well. By contrast the first-order conditions of the restricted case depend only on the policy function itself. Hence, despite the strong assumption upon which it is based, the restricted case may be more efficient in finite samples since the first order condition of its objective does not depend on derivatives of an estimated nonparametric function, making it more amenable to solution by standard optimization techniques. In addition, the restricted approach may be easier to implement in practice given the analytic form of the first-order condition. Two factors diminish the practical advantage of the restricted case. First, the normalization constraint confines possible index values to a finite domain. As such, grid search is a feasible approach to the index optimization problem. Because grid search can be based on direct evaluation of the objective function, the differentiability of the objective in the restricted case offers less advantage. Second, neither the objective function in (2.16) nor in (2.12) is concave. Because α(·) is a nonparametric function, there is no guarantee that the maximands are concave in β. Consequently, in general some form of grid search is necessary whether or not the optimal portfolio policy is restricted by assumption. For implementation purposes, the index optimization problems present some solution challenges. For a grid search to be successful, the estimation problem must satisfy two criteria: the space to be searched must be of low dimension 41 and the curvature of the objective function must be constrained. Data den- sity drops exponentially with the number of grid dimensions. As a result, even on a finite domain, grid search becomes impractical for problems with more than a few dimensions. However, the identification constraint (2.13) reduces the dimensionality of the problem by one. As such, grid search over the objec- tive functions in (2.16) and (2.12) is feasible for problems including up to four predictor variables. Limitations on the curvature of the objective function with respect to β are required to ensure that a sparse sampling of possible index values will locate the desired extremum. However, reasonable smoothness of these derivatives can be assured by choice of a sufficiently large bandwidth for the portfolio policy. In the applications to follow, this proves to be a non-issue if the bandwidth selection procedure described in Appendix A is employed in determining port- folio policies for given index values. This approach yields objectives that are of sufficient smoothness for estimation on reasonably sparse grids in both the limited information and restricted cases.41 In addition to dealing with a non-concave objective function, evaluation of the objective functions of the index optimization problems requires nonparamet- ric estimation of portfolio policies for each data point. This renders function evaluation computationally expensive. I deal with this issue by evaluating the α(β′Z) at a reasonably small subset of points in the range of β′Z. Realiza- tions at intermediate points are then obtained by interpolation. Because the smoothness of the policy function is determined by the bandwidth used in the point-by-point optimization, two to four points per bandwidth length along with a polynomial interpolation is sufficient to ensure that numerical errors that are introduced are within statistical uncertainty error. The global search is implemented over a finite grid. The dimension of the volume to be searched in β space is one less than the dimension of the pre- dictor variable. To keep the global search computationally feasible, I restrict the number of predictors included in the index to four.42 This is perhaps less of a limitation than it might appear. Previous research on model selection for return predictability in a linear regression framework has shown that standard 41For a three dimensional problem, I obtain stable estimated for a grid with O(103) points. 42Aı̈t-Sahalia and Brandt (2001) also examine an index of four predictor variables in their empirical application. 42 model selection procedures rarely select more than four predictors (Bossaerts and Hillion (1999)). In the empirical analysis that follows, I estimate utilities of the optimal strategies under the assumptions of both the restricted case and the limited information case. In the current application, I do not find a significant difference between the results obtained under the two assumptions. For brevity, I only report results obtained under the limited information assumption. 2.6.2 Model Selection Turning now to the evaluation of conditional significance of predictability in a multi-predictor setting, I evaluate optimal index values for all seventy combi- nations of four predictor variables. Index coefficients are estimated by the grid search procedure described in section 6.1. Table 2.13 displays coefficients estimated for the six combinations of four variables that yield the highest in-sample CER for the equities, bonds, and HML portfolio. The standard errors of the CERs for the equities, bonds, and HML portfolio are between 26 and 29 basis points, and for the case without HML, the standard errors range from 16 to 19 basis points. The difference between the best and sixth best CER is 5 basis points in the no-HML case and 2 basis points for the case with HML. The term spread and gold industry trend are the dominant predictors. The predictor combinations that produce the top six CERs all include both the term spread and gold industry trend. Of the six other predictors, all but realized equity index volatility appear in at least one of the top six combinations. In all indices, the highest index coefficients are associated with the gold industry trend and term spread predictors. This evidence provides robust support for use of an index consisting of term spread and gold industry trend for estimating allocations. The seventh line of Table 2.13 shows index coefficients estimated for the two-variable index consisting of term spread and gold industry trend. The in-sample CERs achieved using this latter index are only 0.06 and 0.09 less than the maximum CERs achieved with four predictor variables for the no-HML and HML cases respectively. These differences are within one bootstrap standard error and are insignificant. The last line of Table 2.13 lists index coefficients for an index made up of 43 the four predictors studied by Aı̈t-Sahalia and Brandt (2001). The term spread dominates this latter index. For the no-HML case, the coefficient on the term spread is four times larger than that on the other three variables. The term spread also boasts the largest coefficient in the case with HML. In agreement with results reported by Aı̈t-Sahalia and Brandt (2001), the term spread and S&P 500 trend variables are the largest index components. 2.6.3 Certainty Equivalent Returns with Multiple Predictors Table 2.14 lists conditional CERs for strategies conditioned on the particular indices listed in Table 2.13 along with conditional CERs from following the corresponding unconditional policies. Results are calculated for the case with and without HML. For each set of predictors, results are given for index values at the 20th, 50th and 80th percentiles of the index distributions. The first six sets of predictors correspond to the first six predictor combinations listed in Table 2.13. In addition, results are given for the two-predictor index combining term spread and gold industry trend and the four predictor index of Aı̈t-Sahalia and Brandt (2001). The results can be contrasted with those obtained in the single predictor case. The improvement in conditional CERs from following the conditional policies as opposed to the unconditional policy are significant for the predictor indices that include the term spread and gold industry trend predictors. In contrast, in the single predictor case, the hypotheses that the unconditional policy yielded as high a conditional CER as the conditional policy could not be rejected (see Table 2.10). 2.7 Conclusion The empirical analysis of this chapter yields three observations: i) the benefits of incorporating a proxy for an additional priced risk factor (the value pre- mium) outweigh the gains from conditioning on individual predictor variables; ii) short horizon investors do not benefit from conditioning on single predictor variables in their portfolio decisions; and iii) short horizon investors may be able 44 to improve expected outcomes by conditioning portfolio decisions on multiple predictors simultaneously. In prior studies of predictability, the significance of predictability is based on averaging over long time periods. For predictors with high persistence, the investment significance of predictability is dampened for investors with short horizons. The conditional results presented in this paper help to distinguish pre- dictors that are likely to be of interest to that large contingent of investors that evaluates their performance (or have their performance evaluated) at shorter horizons. In evaluating the predictability results, the risk of data snooping must be borne in mind. Many of the predictor variables that I examine are extracted from an extensive literature, of which the majority is based on data from the United States covering periods that either include or substantially overlap the sample period considered in this paper. The variables can be legitimately clas- sified as snooped with regard the their ability to predict index returns. Where my results are additive is in the assessment of investment significance of predictability under the assumption that the predictability is real but that the underlying conditional distribution of returns is unknown. Under this as- sumption, I demonstrate that simultaneous conditioning on the term spread and gold industry trend increase an investor’s expected utility by approximately 50 basis points per month - an amount which is approximately equal to the ddi- versification benefit of incorporating a mechanical portfolio based on the vale premium. A topic for further study is the implications for hedging demands. As hori- zon increases, the effective sample size shrinks, and statistical power decreases. However, at infinite horizon, a single policy function holds. The sensitivity of this policy to data realization is a promising problem for future exploration using techniques discussed in this paper. 45 Table 2.1: Summary Statistics This table shows the following summary statistics for monthly data on asset returns and forecasting variables for the time period January, 1952 to November, 2004: mean, median, standard deviation, skewness, kurtosis, Sharpe ratio, and lag one autocorrelation ρ1. The assets are the value weighted CRSP index minus the risk-free interest rate (Index), the HML, SMB and UMD portfolios of Ken French, and returns on an index of long-term bonds minus risk-free rate. The predictor variables are dividend yield (div), default premium (def), term spread (term), S&P 500 trend (trend), the 30-day T-bill rate (tbill), gold industry trend (gold), log of previous month realized volatility (realvol) and inflation (cpi). The mean, median, and standard deviation are annualized. mean median std skewness kurtosis sharpe ρ1 equities 0.071 0.116 0.147 -0.510 5.033 0.136 0.062 SMB 0.022 0.006 0.103 0.577 9.326 0.061 0.063 HML 0.051 0.047 0.094 0.053 5.915 0.152 0.133 RF 0.050 0.048 0.008 1.068 4.643 1.748 0.960 UMD 0.105 0.112 0.129 -0.662 9.199 0.225 -0.034 bonds 0.013 0.000 0.073 0.283 4.604 0.069 div 3.377 3.244 1.101 0.228 2.385 0.986 def 0.934 0.810 0.412 1.463 5.382 0.972 term 1.305 1.210 1.407 -0.126 3.309 0.964 trend 3.617 4.700 8.968 -0.706 3.572 0.923 tbill 0.002 0.012 0.098 -0.492 8.211 0.760 gold 1.129 0.878 2.451 0.695 4.894 0.899 realvol -2.260 -2.294 0.455 0.349 3.585 0.668 cpi 0.312 0.259 0.231 1.422 4.956 0.992 46 T ab le 2. 2: C or re la ti on s of A ss et R et u rn s an d P re d ic to rs T h is ta b le sh ow s th e co rr el at io n m at ri x fo r th e p re d ic to r va ri ab le s an d su b se q u en t m on th as se t re tu rn s. T h e p re d ic to r va ri ab le s ar e d iv id en d y ie ld (d iv ), d ef au lt p re m iu m (d ef ), te rm sp re ad (t er m ), S & P 50 0 tr en d (t re n d ), th e 30 -d ay T -b il l ra te (t b il l) , go ld in d u st ry tr en d (g ol d ), lo g of p re v io u s m on th re al iz ed vo la ti li ty (r ea lv ol ) an d in fl at io n (c p i) . eq u it ie s S M B H M L U M D b on d s d iv d ef te rm tr en d tb il l go ld re al vo l cp i eq u it ie s 1. 00 S M B 0. 26 1. 00 H M L -0 .3 6 -0 .2 6 1. 00 U M D -0 .0 6 -0 .0 1 -0 .1 4 1. 00 b on d s 0. 16 -0 .0 8 0. 02 0. 04 1. 00 d iv 0. 09 0. 05 -0 .0 5 -0 .0 1 0. 04 1. 00 d ef 0. 06 0. 09 -0 .0 0 -0 .0 4 0. 07 0. 39 1. 00 te rm 0. 11 0. 04 -0 .0 2 0. 00 0. 11 -0 .2 3 0. 10 1. 00 tr en d 0. 01 -0 .0 5 0. 03 0. 06 -0 .1 2 -0 .1 6 -0 .0 8 0. 12 1. 00 tb il l -0 .0 7 -0 .0 7 0. 07 -0 .0 6 -0 .0 1 -0 .0 4 -0 .2 5 -0 .4 6 -0 .0 1 1. 00 go ld -0 .0 7 0. 07 -0 .0 3 0. 01 -0 .2 0 -0 .0 3 0. 06 -0 .0 3 0. 14 0. 07 1. 00 re al vo l 0. 01 0. 04 -0 .0 4 -0 .0 3 0. 07 -0 .1 2 0. 34 0. 02 -0 .3 6 -0 .0 4 0. 05 1. 00 cp i -0 .0 7 0. 06 -0 .0 1 0. 00 -0 .0 3 0. 44 0. 55 -0 .4 2 -0 .2 7 0. 09 0. 20 0. 26 1. 00 47 T ab le 2. 3: V ol at il it y C or re la ti on s T h is ta b le sh ow s th e co rr el at io n s of re al iz ed vo la ti li ti es w it h la gg ed re al iz ed vo la ti li ti es an d p re d ic to r va ri ab le s. T h e re al iz ed vo la ti li ti es ar e ca lc u la te d m on th ly u si n g w it h in m on th d ai ly re tu rn s. T h e p re d ic to r va ri ab le s ar e d iv id en d y ie ld (d iv ), d ef au lt p re m iu m (d ef ), te rm sp re ad (t er m ), S & P 50 0 tr en d (t re n d ), th e 30 -d ay T -b il l ra te (t b il l) , go ld in d u st ry tr en d (g ol d ), lo g of p re v io u s m on th re al iz ed vo la ti li ty (r vo l) an d in fl at io n (c p i) . eq u it ie s S M B H M L eq u it ie s − 1 S M B −1 H M L −1 eq u it ie s 1. 00 S M B 0. 81 1. 00 H M L 0. 69 0. 57 1. 00 eq u it ie s − 1 0. 58 0. 43 0. 49 1. 00 S M B −1 0. 50 0. 52 0. 43 0. 81 1. 00 H M L −1 0. 49 0. 38 0. 74 0. 70 0. 57 1. 00 d iv -0 .0 9 -0 .1 5 -0 .2 7 -0 .0 6 -0 .1 3 -0 .2 6 d ef 0. 20 0. 11 0. 05 0. 23 0. 16 0. 07 te rm 0. 01 0. 14 -0 .0 6 0. 03 0. 15 -0 .0 4 tr en d -0 .3 9 -0 .1 3 -0 .2 7 -0 .3 9 -0 .1 5 -0 .2 7 tb il l -0 .0 4 -0 .0 1 -0 .0 5 -0 .1 2 -0 .0 8 -0 .1 0 go ld 0. 04 0. 03 -0 .1 8 0. 08 0. 06 -0 .1 5 re al vo l 0. 92 0. 70 0. 65 0. 62 0. 51 0. 50 cp i 0. 11 -0 .0 3 -0 .0 4 0. 11 -0 .0 3 -0 .0 5 48 T ab le 2. 4: U n co n d it io n al P or tf ol io W ei gh ts T h is ta b le sh ow s op ti m al p or tf ol io al lo ca ti on s as a fr ac ti on of w ea lt h . T h e in ve st or h as p ow er u ti li ty w it h ri sk av er si on gi ve n b y γ . A ll o ca ti on s ar e li st ed fo r an u n co n st ra in ed in ve st or (P an el A ), an d fo r an in ve st or su b je ct to lo n g- on ly co n - st ra in ts (P an el B ). L on g- on ly co n st ra in ts re st ri ct in ve st m en t in eq u it ie s an d b on d s to b e p os it iv e, an d re st ri ct in ve st m en t in th e H M L p or tf ol io to le ss th an 1/ 3 th e al lo ca ti on to eq u it ie s in ab so lu te va lu e. O p ti m al al lo ca ti on s ar e li st ed fo r p or tf ol io s m ad e u p of th e ri sk -f re e as se t p lu s, in or d er fr om to p to b ot to m , i) th e eq u it y in d ex , ii ) th e H M L p or tf ol io , ii i) th e eq u it y an d b on d in d ic es , iv ) th e eq u it y in d ex an d th e H M L p or tf ol io , v ) th e eq u it y an d b on d in d ic es p lu s th e H M L p or tf ol io . T h e w ei gh t in th e ri sk -f re e as se t is eq u al to on e m in u s th e su m of th e al lo ca ti on s to th e eq u it y an d b on d in d ic es . T h e es ti m at es ar e ob ta in ed b y m et h o d of m om en ts . S ta n d ar d er ro rs ar e ar e ca lc u la te d b as ed on 10 00 st at io n ar y b o ot st ra p re sa m p li n gs of th e d at a se t, an d ar e li st ed in b ra ck et s. P a n e l A : U n co n st ra in e d P a n e l B : N o S h o rt S a le s γ = 1 γ = 2 γ = 5 γ = 10 γ = 1 γ = 2 γ = 5 γ = 10 e q u it ie s 2. 82 1. 50 0. 61 0. 31 1. 00 1. 00 0. 61 0. 31 [0 .8 3] [0 .4 8] [0 .2 0] [0 .1 0] [0 .0 6] [0 .1 2] [0 .1 9] [0 .1 0] H M L 5. 15 2. 72 1. 11 0. 56 [1 .5 4] [0 .8 7] [0 .3 6] [0 .1 8] e q u it ie s 2. 81 1. 47 0. 60 0. 30 1. 00 1. 00 0. 60 0. 30 [0 .8 7] [0 .4 9] [0 .2 0] [0 .1 0] [0 .0 7] [0 .1 3] [0 .2 0] [0 .1 0] b o n d s 1. 63 0. 78 0. 30 0. 15 0. 00 0. 00 0. 30 0. 15 [1 .8 0] [0 .9 4] [0 .3 8] [0 .1 9] [0 .0 6] [0 .1 1] [0 .2 0] [0 .1 6] e q u it ie s 4. 50 2. 50 1. 03 0. 52 1. 00 1. 00 0. 61 0. 30 [0 .8 1] [0 .5 2] [0 .2 2] [0 .1 1] [0 .1 0] [0 .1 6] [0 .2 1] [0 .1 1] H M L 7. 52 4. 24 1. 75 0. 88 0. 33 0. 33 0. 20 0. 10 [1 .9 1] [1 .0 8] [0 .4 4] [0 .2 2] [0 .1 8] [0 .0 9] [0 .0 7] [0 .0 4] e q u it ie s 4. 54 2. 49 1. 02 0. 51 1. 00 1. 00 0. 58 0. 29 [0 .8 5] [0 .5 2] [0 .2 2] [0 .1 1] [0 .1 2] [0 .1 8] [0 .2 2] [0 .1 1] b o n d s 1. 24 0. 49 0. 17 0. 08 0. 00 -0 .0 0 0. 42 0. 21 [1 .7 2] [0 .9 5] [0 .3 9] [0 .2 0] [0 .1 2] [0 .1 7] [0 .2 2] [0 .1 8] H M L 7. 56 4. 23 1. 74 0. 87 0. 33 0. 33 0. 19 0. 10 [1 .9 2] [1 .0 8] [0 .4 4] [0 .2 2] [0 .1 8] [0 .0 9] [0 .0 7] [0 .0 4] 49 T ab le 2. 5: U n co n d it io n al C er ta in ty E q u iv al en t R et u rn s E ac h ro w of th is ta b le li st s in -s am p le es ti m at es of m on th ly ce rt ai n ty eq u iv al en t re tu rn s (C E R s) at ta in ed b y al lo ca ti n g to es ti m at ed op ti m al p or tf ol io s. O p ti m al p or tf ol io s ar e es ti m at ed fo r an in ve st or w it h C R R A u ti li ty . R es u lt s ar e co m p u te d fo r fo u r va lu es of ri sk av er si on p ar am et er γ . C E R s ar e gi ve n in p er ce n t. A ll o ca ti on s ar e li st ed fo r an u n co n st ra in ed in ve st or (P an el A ), an d fo r an in ve st or su b je ct to lo n g- on ly co n st ra in ts . L on g- on ly co n st ra in ts re st ri ct in ve st m en t in eq u it ie s an d b on d s to b e p os it iv e, an d re st ri ct in ve st m en t in th e H M L p or tf ol io to le ss th an 1/ 3 th e al lo ca ti on to eq u it ie s in ab so lu te va lu e. T h e le ft co lu m n li st s as se ts in ea ch p or tf ol io . P or tf ol io s in cl u d e a ri sk -f re e as se t p lu s a su b se t of : an eq u it y in d ex m in u s ri sk -f re e as se t (e q u it ie s) , a lo n g- te rm b on d in d ex m in u s th e ri sk -f re e as se t (b on d s) , H M L , S M B an d U M D p or tf ol io s p ro v id ed b y K en F re n ch . C E R s ar e es ti m at es gi ve n op ti m al p or tf ol io p ol ic ie s ob ta in ed b y m et h o d of m om en ts . S ta n d ar d er ro rs ar e ca lc u la te d b as ed on 10 00 st at io n ar y b o ot st ra p re sa m p li n gs of th e d at a se t, an d ar e li st ed in b ra ck et s. P a n e l A : U n co n st ra in e d P a n e l B : N o S h o rt S a le s A ss e t S e t γ = 1 γ = 2 γ = 5 γ = 10 γ = 1 γ = 2 γ = 5 γ = 10 {e q u it ie s} 0. 86 0. 44 0. 18 0. 09 0. 49 0. 39 0. 18 0. 09 [0 .5 1] [0 .2 7] [0 .1 1] [0 .0 5] [0 .1 7] [0 .1 7] [0 .1 1] [0 .0 5] {H M L } 1. 12 0. 57 0. 23 0. 12 [0 .6 3] [0 .3 3] [0 .1 3] [0 .0 7] {e q u it ie s, b o n d s} 0. 93 0. 47 0. 19 0. 10 0. 49 0. 39 0. 19 0. 10 [0 .5 4] [0 .2 8] [0 .1 1] [0 .0 6] [0 .1 7] [0 .1 7] [0 .1 1] [0 .0 6] {e q u it ie s, H M L } 3. 15 1. 64 0. 66 0. 33 0. 63 0. 55 0. 28 0. 14 [1 .0 4] [0 .5 7] [0 .2 3] [0 .1 2] [0 .1 6] [0 .1 6] [0 .1 0] [0 .0 5] {e q u it ie s, b o n d s, H M L } 3. 19 1. 65 0. 67 0. 33 0. 63 0. 55 0. 28 0. 14 [1 .0 7] [0 .5 8] [0 .2 4] [0 .1 2] [0 .1 6] [0 .1 6] [0 .1 1] [0 .0 5] {e q u it ie s, b o n d s, H M L , S M B } 3. 49 1. 79 0. 72 0. 36 0. 56 0. 55 0. 29 0. 15 [1 .1 8] [0 .6 2] [0 .2 5] [0 .1 3] [0 .1 7] [0 .1 6] [0 .1 1] [0 .0 5] {e q u it ie s, b o n d s, H M L , U M D } 4. 95 2. 87 1. 23 0. 63 0. 76 0. 66 0. 35 0. 17 [1 .0 1] [0 .6 2] [0 .2 8] [0 .1 4] [0 .1 7] [0 .1 7] [0 .1 2] [0 .0 6] 50 Table 2.6: Bootstrap Test of Expected Utility Improvement This table gives bootstrap p-values for the null hypothesis that adding an asset to an investor’s asset set improves expected utility. Starting asset sets are listed in the left column. The second column is the asset being added. Results are computed for a power utility investor with risk aversion of five. The p-values are computed based on expected utility differences computed for 1000 bootstrap samples. Asset Set Added Asset p-value Panel A: Unconstrained {} {equities} 0.054 {} {bonds} 0.26 {} {HML} 0.046 {equities} {bonds} 0.33 {equities} {HML} 0.009 {equities, HML} {bonds} 0.417 {equities, bonds} {HML} 0.009 {equities, bonds, HML} {SMB} 0.237 {equities, bonds, HML} {UMD} 0.001 Panel B: No Short Sales {} {equities} 0.054 {} {bonds} 0.26 {equities} {bonds} 0.33 {equities} {HML} 0 {equities, HML} {bonds} 0.479 {equities, bonds} {HML} 0.001 {equities, bonds, HML} {SMB} 0.171 {equities, bonds, HML} {UMD} 0.059 51 Table 2.7: Abbreviations Used for Conditioning Variables div dividend yield (log) def default premium term term spread trend S&P 500 trend tbill T-bill rate gold gold index return realvol CRSP index realized volatility (log) cpi inflation 52 Table 2.8: Returns Regressed on Predictors This table shows coefficient estimates for predictive regressions of realized re- turns on lagged conditioning variables. Full names of the conditioning variables are given in Table 7. Panel A shows coefficients of linear regressions of next month’s returns on a individual predictor variables. Panel B presents coeffi- cients of multivariate regressions of asset returns on the full set of predictor variables. The last column lists R2 values for the multivariate regressions. div def term trend tbill gold realvol cpi R2 Panel A: univariate regression results equities 0.353 0.185 0.376 -0.012 -0.347 -0.333 0.224 -0.229 - [0.180] [0.187] [0.188] [0.208] [0.164] [0.216] [0.175] [0.211] bonds 0.050 0.122 0.217 -0.183 -0.029 -0.348 0.232 -0.054 - [0.102] [0.110] [0.108] [0.089] [0.096] [0.115] [0.076] [0.116] HML -0.126 0.047 0.009 0.041 0.157 -0.061 -0.047 -0.021 - [0.131] [0.119] [0.118] [0.115] [0.120] [0.145] [0.129] [0.128] Panel B: multivariate regression results equities 0.694 -0.057 0.272 0.167 -0.129 -0.258 0.496 -0.412 0.055 [0.194] [0.268] [0.212] [0.224] [0.192] [0.211] [0.190] [0.271] bonds 0.135 0.037 0.291 -0.068 0.156 -0.341 0.240 -0.042 0.060 [0.089] [0.139] [0.105] [0.100] [0.093] [0.108] [0.087] [0.147] HML -0.225 0.256 0.027 -0.025 0.226 -0.083 -0.150 -0.024 0.035 [0.144] [0.162] [0.151] [0.130] [0.134] [0.146] [0.154] [0.178] 53 Table 2.9: Realized Volatility Regressed on Predictors This table shows coefficient estimates for linear regressions of realized monthly volatilities on lagged predictors. Full names of the conditioning variables are given in Table 7. Panel A shows coefficients of linear regressions of realized one-month volatilities of each asset on individual predictor variables. Panel B lists coefficients of multivariate linear regressions of realized volatilities of each asset on the full set of predictor variables. The last column lists R2 values for the multivariate regressions. div def term trend tbill gold realvol cpi R2 Panel A: univariate regression results equities -0.809 1.137 -0.081 -1.744 -0.458 0.463 3.973 0.745 - [0.394] [0.275] [0.379] [0.405] [0.262] [0.452] [0.255] [0.325] HML -1.059 0.082 -0.279 -0.722 -0.199 -0.585 1.797 -0.138 - [0.267] [0.154] [0.181] [0.195] [0.164] [0.243] [0.227] [0.166] Panel B: multivariate regression results equities -1.445 1.519 -0.458 -1.868 -0.528 0.620 2.898 -0.327 0.457 [0.423] [0.296] [0.245] [0.367] [0.244] [0.311] [0.327] [0.371] HML -1.692 0.926 -0.864 -0.734 -0.453 -0.504 1.221 -0.144 0.408 [0.302] [0.203] [0.188] [0.175] [0.149] [0.159] [0.200] [0.220] 54 Table 2.10: Portfolio Weights as a Function of Predictor Values This table shows portfolio allocations conditioned on individual predictor vari- ables. Portfolio weights are estimated by solving a nonparametric approxima- tion to the investor’s Euler equation. The optimal strategies are estimated for a power utility investor with risk aversion equal to five and a one-month invest- ment horizon. Portfolio weights are shown for four assets sets: equities only; HML only; equities and bonds; and equities, bonds, and HML. Asymptotic standard errors are given in square brackets. {equities} {bonds} {equities,bonds} {equities,bonds,HML} div 0.2 0.49 0.35 0.46 0.35 0.90 0.10 1.77 [0.31] [0.58] [0.27] [0.56] [0.31] [0.57] [0.58] 0.5 0.44 0.22 0.48 0.06 0.93 -0.14 2.12 [0.27] [0.52] [0.25] [0.46] [0.26] [0.46] [0.47] 0.8 0.79 0.10 0.79 -0.22 1.17 -0.18 2.10 [0.30] [0.59] [0.27] [0.50] [0.27] [0.47] [0.59] def 0.2 0.53 0.04 0.53 -0.04 0.97 -0.21 1.73 [0.30] [0.58] [0.28] [0.53] [0.27] [0.51] [0.53] 0.5 0.46 0.18 0.47 0.13 0.92 -0.02 1.68 [0.26] [0.49] [0.25] [0.47] [0.26] [0.47] [0.50] 0.8 0.51 0.89 0.49 0.68 0.86 0.52 1.58 [0.25] [0.52] [0.22] [0.47] [0.26] [0.47] [0.48] term 0.2 0.38 0.03 0.46 -0.10 0.86 -0.22 1.39 [0.32] [0.59] [0.30] [0.53] [0.30] [0.51] [0.47] 0.5 1.07 0.46 0.98 0.09 1.30 -0.02 1.55 [0.27] [0.54] [0.24] [0.49] [0.26] [0.47] [0.57] 0.8 1.10 0.59 1.08 0.50 1.46 0.48 1.82 [0.39] [0.58] [0.32] [0.58] [0.34] [0.55] [0.63] 55 {equities} {bonds} {equities,bonds} {equities,bonds,HML} trend 0.2 0.34 0.65 0.34 0.56 0.90 0.29 1.83 [0.43] [0.53] [0.40] [0.53] [0.36] [0.51] [0.52] 0.5 0.93 0.41 0.85 0.03 1.23 -0.15 1.84 [0.24] [0.56] [0.23] [0.51] [0.25] [0.49] [0.60] 0.8 0.77 -0.09 0.79 -0.16 1.16 -0.27 1.83 [0.26] [0.62] [0.26] [0.54] [0.31] [0.52] [0.65] tbill 0.2 0.76 0.84 0.71 0.53 1.03 0.34 1.64 [0.25] [0.61] [0.23] [0.53] [0.24] [0.51] [0.49] 0.5 0.60 0.25 0.59 0.21 0.91 0.12 1.62 [0.22] [0.48] [0.22] [0.46] [0.24] [0.47] [0.51] 0.8 0.46 0.25 0.46 0.15 0.84 0.07 1.63 [0.24] [0.45] [0.23] [0.43] [0.25] [0.44] [0.50] gold 0.2 1.26 0.90 1.15 0.35 1.55 0.22 1.67 [0.28] [0.60] [0.26] [0.57] [0.27] [0.54] [0.54] 0.5 1.11 0.20 1.07 -0.22 1.37 -0.24 1.56 [0.26] [0.56] [0.25] [0.53] [0.26] [0.51] [0.51] 0.8 0.49 -0.50 0.58 -0.56 0.97 -0.54 1.71 [0.36] [0.57] [0.33] [0.54] [0.31] [0.52] [0.51] realvol 0.2 1.06 0.58 0.91 0.09 1.25 -0.25 2.69 [0.32] [0.64] [0.28] [0.56] [0.28] [0.53] [0.58] 0.5 0.59 0.22 0.61 -0.07 1.12 -0.28 2.50 [0.27] [0.49] [0.24] [0.43] [0.25] [0.43] [0.52] 0.8 0.46 0.00 0.49 -0.01 0.94 -0.05 1.78 [0.25] [0.43] [0.25] [0.41] [0.28] [0.41] [0.54] cpi 0.2 0.97 0.62 0.95 0.69 1.33 0.59 1.76 [0.27] [0.49] [0.25] [0.48] [0.27] [0.47] [0.55] 0.5 0.76 0.85 0.74 0.70 1.18 0.54 1.74 [0.26] [0.45] [0.25] [0.43] [0.27] [0.44] [0.51] 0.8 0.40 0.99 0.37 0.76 0.90 0.46 1.79 [0.32] [0.58] [0.30] [0.50] [0.30] [0.50] [0.51] 56 Table 2.11: Certainty Equivalent Returns as a Function of Predictor Values This table displays certainty equivalent returns (CERs) portfolio strategies con- ditioned on a single predictor variable. The optimal strategies are estimated for a power utility investor with risk aversion equal to five and a one-month in- vestment horizon. CERs are shown for three possible assets sets: equities only; equities and bonds; and equities, bonds, and HML. The equity and bond as- sets are hedge funds that are long the appropriate index and short the risk-free asset. The first row shows CER of unconditional strategies. Subsequent rows show conditional CER calculated as kernel averages of utility outcomes at dif- ferent (standardized) values of predictor variables. Two columns are shown for each set of assets. The first is the conditional CER of an optimal conditional strategy. The second is the conditional CER of the optimal unconditional strat- egy. Standard errors are calculated by the delta method and given in square brackets. {equities,bonds} {equities,bonds,HML} CER CERunc CER CERunc {div} 0.2 0.10 0.09 0.60 0.59 [0.11] [0.14] [0.28] [0.29] (0.38) (0.39) 0.5 0.10 0.09 0.66 0.62 [0.10] [0.14] [0.26] [0.25] (0.38) (0.26) 0.8 0.29 0.25 0.81 0.78 [0.19] [0.17] [0.33] [0.29] (0.28) (0.34) {def} 0.2 0.12 0.11 0.59 0.58 [0.12] [0.15] [0.26] [0.27] (0.38) (0.36) 0.5 0.11 0.10 0.56 0.56 [0.10] [0.14] [0.25] [0.27] (0.37) (0.39) 0.8 0.21 0.18 0.58 0.55 [0.14] [0.14] [0.26] [0.28] (0.32) (0.31) {term} 0.2 0.09 0.07 0.42 0.38 [0.09] [0.13] [0.21] [0.26] (0.28) (0.29) 0.5 0.41 0.35 0.73 0.68 [0.17] [0.10] [0.27] [0.27] (0.21) (0.22) 0.8 0.60 0.48 1.06 0.95 [0.25] [0.13] [0.37] [0.30] (0.15) (0.17) 57 {equities,bonds} {equities,bonds,HML} CER CERunc CER CERunc {trend} 0.2 0.11 0.08 0.64 0.62 [0.12] [0.18] [0.27] [0.27] (0.33) (0.40) 0.5 0.29 0.26 0.79 0.77 [0.15] [0.11] [0.29] [0.27] (0.28) (0.31) 0.8 0.28 0.24 0.78 0.75 [0.15] [0.12] [0.33] [0.31] (0.28) (0.29) {tbill} 0.2 0.29 0.27 0.68 0.67 [0.17] [0.13] [0.27] [0.27] (0.38) (0.43) 0.5 0.17 0.17 0.59 0.58 [0.12] [0.12] [0.25] [0.27] (0.46) (0.40) 0.8 0.11 0.10 0.54 0.53 [0.09] [0.13] [0.24] [0.27] (0.35) (0.34) {gold} 0.2 0.56 0.44 0.96 0.84 [0.23] [0.12] [0.31] [0.27] (0.14) (0.13) 0.5 0.42 0.33 0.74 0.67 [0.18] [0.11] [0.26] [0.25] (0.17) (0.20) 0.8 0.15 0.08 0.55 0.49 [0.13] [0.13] [0.25] [0.26] (0.20) (0.23) {realvol} 0.2 0.27 0.24 0.97 0.87 [0.16] [0.12] [0.35] [0.25] (0.29) (0.19) 0.5 0.16 0.14 0.90 0.81 [0.11] [0.13] [0.31] [0.25] (0.34) (0.17) 0.8 0.14 0.12 0.66 0.65 [0.13] [0.17] [0.28] [0.30] (0.32) (0.35) {cpi} 0.2 0.41 0.34 0.87 0.82 [0.17] [0.11] [0.30] [0.27] (0.20) (0.20) 0.5 0.30 0.27 0.78 0.75 [0.15] [0.11] [0.28] [0.26] (0.28) (0.28) 0.8 0.16 0.12 0.65 0.63 [0.15] [0.16] [0.28] [0.27] (0.23) (0.32) 58 Table 2.12: Increase in Certainty Equivalent Returns from Conditioning on a Predictor This table reports the difference between unconditional CER of optimal policies conditioned on individual predictor variables minus the CER of the optimal unconditional strategy. The standard deviation of CER differences is given in parentheses. P-values reflect the probability of rejecting the null hypothesis that the unconditional certainty equivalent of following the conditional strategy leads to lower certainty equivalent return than that achieved by following the optimal unconditional strategy. Standard deviations and p-values are calculated based on 1000 stationary bootstrap samples. {equities,bonds} {equities,bonds,HML} div 0.10 0.11 [0.05] [0.06] (0.018) (0.022) def 0.06 0.11 [0.05] [0.05] (0.077) (0.016) term 0.34 0.36 [0.09] [0.10] (0.000) (0.000) trend 0.21 0.20 [0.07] [0.08] (0.000) (0.000) tbill 0.09 0.14 [0.05] [0.08] (0.024) (0.016) gold 0.49 0.55 [0.12] [0.14] (0.000) (0.000) realvol 0.16 0.29 [0.06] [0.10] (0.002) (0.001) cpi 0.19 0.22 [0.08] [0.09] (0.004) (0.002) 59 T ab le 2. 13 : C o effi ci en ts of P re d ic ti ve In d ex T h is ta b le d is p la y s es ti m at es of op ti m al in d ex co effi ci en ts fo r d iff er en t co m b in at io n s of p re d ic to r va ri ab le s. E ac h ro w is la b el ed w it h ab b re v ia ti on s fo r th e p re d ic to r va ri ab le s in cl u d ed in ea ch in d ex . A b b re v ia ti on s ar e d efi n ed in T ab le 7. T h e in d ex co effi ci en ts , {β i}, ar e es ti m at ed fo r a p ow er u ti li ty in ve st or w it h ri sk av er si on eq u al to fi ve . In d ex co effi ci en ts ar e li st ed fo r tw o as se t se ts : eq u it ie s an d b on d s; an d eq u it ie s, b on d s an d H M L . T h e fi ft h co lu m n fo r ea ch as se t se t sh ow s in -s am p le es ti m at es of C E R s ac h ie ve d b y fo ll ow in g th e op ti m al p or tf ol io co n d it io n ed on th e es ti m at ed in d ex . {e q u it ie s, b on d s} {e q u it ie s, b on d s, H M L } β 1 β 2 β 3 β 4 C E β 1 β 2 β 3 β 4 C E R d iv te rm tr en d go ld 0. 08 0. 46 -0 .1 5 -0 .3 1 0. 69 0. 23 0. 38 -0 .0 8 -0 .3 1 1. 25 d ef te rm go ld cp i 0. 23 0. 38 -0 .3 8 0. 00 0. 68 0. 08 0. 38 -0 .3 8 0. 15 1. 24 te rm tr en d go ld cp i 0. 38 -0 .2 3 -0 .3 8 0. 00 0. 69 0. 38 -0 .0 8 -0 .3 8 0. 15 1. 24 d iv te rm tb il l go ld 0. 08 0. 46 -0 .0 0 -0 .4 6 0. 69 0. 23 0. 46 -0 .0 0 -0 .3 1 1. 23 d iv d ef te rm go ld 0. 15 0. 15 0. 38 -0 .3 1 0. 69 0. 23 -0 .0 0 0. 46 -0 .3 1 1. 23 d ef te rm tr en d go ld 0. 15 0. 38 -0 .1 5 -0 .3 1 0. 70 0. 15 0. 38 -0 .1 5 -0 .3 1 1. 23 te rm tr en d tb il l go ld -0 .3 8 0. 15 0. 00 0. 46 0. 69 -0 .3 1 0. 08 -0 .1 5 0. 46 1. 21 te rm go ld 0. 54 -0 .4 6 0. 68 -0 .2 4 0. 76 1. 21 d iv d ef te rm tr en d 0. 08 0. 08 0. 69 -0 .1 5 0. 51 -0 .0 8 0. 15 0. 46 -0 .3 1 0. 99 60 Table 2.14: Conditional Certainty Equivalents for Predictive Index This table shows certainty equivalent returns (CERs) of portfolio strategies conditioned on predictor indices. Each index is a linear combination of up to four predictor values. The index coefficients are given in Table 12. The optimal strategies are estimated for a power utility investor with risk aversion equal to five and a one-month investment horizon. CERs are shown for two assets sets: equities and bonds; and equities, bonds and HML. The equities and bond assets are hedge funds that are long the appropriate index and short the risk-free asset. Each row shows conditional CER calculated as kernel averages of utility outcomes at different (standardized) values of predictor variables. Standard errors are calculated by the delta method and given in square brackets. Two columns are shown for each set of assets. The first is the conditional CER of an optimal conditional strategy. The second is the conditional CER of the optimal unconditional strategy. Bootstrap p-values are given in parentheses. The p- values are based on the null hypothesis that the unconditional strategy is as efficient in terms of CER as the conditional strategy. {equities,bonds} {equities,bonds,HML} CER CERunc CER CERunc {div,term,trend,gold} 0.2 0.11 -0.20 0.31 -0.06 [0.20] [0.25] [0.31] [0.46] (0.05) (0.04) 0.5 0.09 0.06 0.30 0.17 [0.15] [0.23] [0.27] [0.43] (0.28) (0.17) 0.8 0.74 0.44 1.51 1.20 [0.56] [0.27] [0.92] [0.53] (0.06) (0.07) {term,gold,vs,cpi} 0.2 0.09 -0.16 1.14 0.92 [0.18] [0.25] [0.66] [0.47] (0.07) (0.08) 0.5 0.08 0.05 0.72 0.68 [0.14] [0.23] [0.46] [0.43] (0.31) (0.27) 0.8 0.72 0.31 0.47 -0.02 [0.58] [0.27] [0.51] [0.52] (0.03) (0.02) 61 Table 2.14: (Continued) {equities,bonds} {equities,bonds,HML} CER CERunc CER CERunc {def,term,gold,cpi} 0.2 0.22 -0.22 0.32 -0.13 [0.31] [0.27] [0.35] [0.49] (0.02) (0.03) 0.5 0.09 0.06 0.35 0.27 [0.15] [0.23] [0.30] [0.43] (0.26) (0.23) 0.8 0.78 0.48 1.40 1.13 [0.58] [0.27] [0.82] [0.51] (0.07) (0.08) {term,trend,gold,cpi} 0.2 0.10 -0.23 0.31 -0.09 [0.20] [0.26] [0.35] [0.48] (0.06) (0.03) 0.5 0.10 0.08 0.36 0.29 [0.16] [0.23] [0.30] [0.43] (0.31) (0.24) 0.8 0.70 0.40 1.43 1.13 [0.52] [0.26] [0.82] [0.51] (0.07) (0.07) {term,gold,,} 0.2 0.08 -0.02 0.92 0.84 [0.15] [0.24] [0.57] [0.44] (0.15) (0.19) 0.5 0.28 0.24 0.45 0.40 [0.28] [0.23] [0.35] [0.43] (0.30) (0.29) 0.8 1.33 0.64 0.77 -0.24 [1.02] [0.36] [1.13] [1.03] (0.02) (0.04) {div,def,term,trend} 0.2 0.03 -0.16 0.41 0.23 [0.10] [0.25] [0.37] [0.46] (0.13) (0.11) 0.5 0.16 0.14 0.46 0.41 [0.21] [0.24] [0.36] [0.43] (0.33) (0.28) 0.8 0.53 0.41 1.28 1.12 [0.50] [0.29] [0.81] [0.55] (0.20) (0.17) 62 Figure 2.1: Single Asset Portfolios Conditioned on Dividend Yield and Default Premium Figures 2.1-3 depict conditional, single risky asset allocations as functions of predictor variables. Each row of panels corresponds to a predictor variable. Column one shows index allocation for an equities and cash investor. Column two shows allocations to HML for an investor who otherwise only holds cash. The shaded regions show plus or minus two standard deviation regions. The dotted lines correspond to the unconditional policy. Predictor values are stan- dardized to have mean zero and standard deviation 1. 63 Figure 2.2: Single Asset Portfolios Conditioned on Term Spread, S&P 500 Trend, and T-bill Rate Conditional, single risky asset allocations as a function of predictor variables. See caption to Figure 2.1 for details. 64 Figure 2.3: Single Asset Portfolios Conditioned on Gold Industry Trend, Real- ized Volatility, and Inflation Conditional, single risky asset allocations as functions of predictor variables. See caption to Figure 2.1 for details. 65 Figure 2.4: Portfolio Allocations versus Dividend Yield Figures 2.4-11 depict portfolio allocations versus predictor value. Predictor values are standardized to have mean zero and standard deviation one. Each figure shows results for a different predictor. Figure 1 shows results for the dividend yield. Each column shows results for a different set of assets. The first column plots results for an investor who trades in the value weighted CRSP index as well as the CRSP bond index. The second column plots allocations for an investor who also trades in the HML portfolio. The conditional allocations are plotted for risk aversion equal to 5. The horizontal dotted lines in each plot depict unconditional allocations. The solid vertical line segment on each plot marks the two standard error band for the unconditional weights. The lower 66 left panel of each figure shows a histogram of predictor variable occurrences. Figure 2.5: Portfolio Allocations versus Default Premium This figure plots conditional portfolio allocations versus default premium stan- dardized to have mean zero and standard deviation one. See Figure 2.4 for complete description. 67 Figure 2.6: Portfolio Allocations versus Term Spread This figure plots conditional portfolio allocations versus term spread standard- ized to have mean zero and standard deviation one. See Figure 2.4 for complete description. 68 Figure 2.7: Portfolio Allocations versus S&P 500 Trend This figure plots conditional portfolio allocations versus S&P 500 trend stan- dardized to have mean zero and standard deviation 1. See figure 2.4 for complete description. 69 Figure 2.8: Portfolio Allocations versus T-bill Rate This figure plots conditional portfolio allocations versus 3 month T-bill rate standardized to have mean zero and standard deviation one. See Figure 2.4 for complete description. 70 Figure 2.9: Portfolio Allocations versus Gold Industry Trend This figure plots conditional portfolio allocations versus gold industry trend standardized to have mean zero and standard deviation one. See Figure 2.4 for complete description. 71 Figure 2.10: Portfolio Allocations versus Realized Volatility This figure plots conditional portfolio allocations versus realized volatility stan- dardized to have mean zero and standard deviation one. See Figure 2.4 for complete description. 72 Figure 2.11: Portfolio Allocations versus Inflation This figure plots conditional portfolio allocations versus inflation standardized to have mean zero and standard deviation one. See Figure 2.4 for complete description. 73 Figure 2.12: CER versus Default Premium and Dividend Yield Conditional certainty equivalent returns (CERs) as a function of predictor value. The solid curve is the CER for an investor who allocates to the optimal condi- tional portfolio. The dashed curve is the conditional CER of an investor who allocates to the optimal unconditional portfolio. The dash-dot horizontal line is the unconditional CER attained by following the conditional portfolio. 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Working Paper. White, H. (2000). A reality check for data snooping. Econometrica, 68:1097– 1126. Wolf, M. (2007). Robust performance hypothesis testing with the sharpe ratio. 81 Chapter 3 Nonparametric Portfolio Estimation with Prior Belief in the Positivity of Portfolio Weights43 3.1 Introduction Short-sales constraints are a common technique for stabilizing portfolio esti- mates based on historical returns. The imposition of short sales constraints has the effect of penalizing variability in portfolio weights. A disadvantage of this approach is that it results in a biased portfolio estimator. The reward is a more robust estimate that performs better out of sample. However, what if historical returns do not capture all information available to the investor? Any additional information would decrease the variance of portfolio estimates and, hence, de- crease the need to bias the portfolio estimator by explicitly constraining the weights vector. A source of information beyond that provided by historical returns is eco- nomic intuition. If investors have beliefs about the properties of portfolio weights, then those beliefs should be incorporated into the investor’s information set. One candidate belief can be imputed from the very popularity of short-sales constraint. Given that many assets can be readily shorted either directly or indi- rectly via exchange traded funds or derivative trades, it seems plausible that this constraint is more reflective of investor bias or beliefs. Furthermore, for many asset allocation problems, the positivity of portfolio allocations is supported by equilibrium asset pricing theory. For example, an investor choosing an allo- 43A version of this chapter will be submitted for publication. Douglass, J., Nonparametric Portfolio Estimation with Prior Belief in the Positivity of Portfolio Weights. 82 cation across industry portfolios might reflect that all industries have positive market weight and therefore ought to be priced accordingly at equilibrium. In this chapter, I investigate the impact of conditioning portfolio estimates on a belief in the nonnegativity of portfolio weights. I use Bayesian techniques to construct a nonparametric estimator that can incorporate qualitative investor beliefs. Then, in a simulation study, I examine the properties of estimated weights and evaluate the performance of a portfolio rule based on the conditional estimator. I construct a conditional nonparametric method-of-moments estimator that incorporates qualitative prior beliefs. The final form of the estimator is analo- gous to the conditional estimator developed by Brandt (1999).44 The difference is that Brandt (1999) conditions portfolio estimates on predictor variables rather than qualitative beliefs. Both estimators approximate the expectation integral in the investor’s problem with a probability weighted sum over historical return outcomes. The weights can be viewed as a perturbation of the empirical dis- tribution of returns that leads to the most likely predictive distribution given the conditioning information. The advantages of a nonparametric framework for portfolio choice are discussed by Brandt (1999) and Aı̈t-Sahalia and Brandt (2001). Briefly, the nonparametric framework does not require the assump- tion of a parametric model of the underlying distribution thereby eliminating a potential source of specification error. The conditional estimator introduced here is also related to data tuning applications developed by Peter Hall et. al.45 Data tuning methods seek to minimally perturb the data set until a nonparametrically estimated quantity satisfies a set of restrictions or constraints. The data set can be perturbed by shifting the data points themselves or by perturbing the relative probabil- ity of each data point. These approaches impose prior views on the estimator using constraints. The estimator developed in this paper is akin to data tun- ing approaches that perturb probabilities. The objective is to obtain a data set that accommodates prior views with a minimal shift of these probabilities. Rather than imposing constraints, I use Bayesian techniques to obtain a dis- crete predictive distribution that reflects investor priors. The approach allows 44Brandt’s (1999) approach is applied by Aı̈t-Sahalia and Brandt (2001), Paye (2004) and the author (see Chapter 2). 45See, for example, Choi and Hall (1999), Choi et al. (2000) and Braun and Hall (2001). 83 for a probabilistic specification of prior beliefs. In addition, unlike data tuning algorithms, the resulting discrete-Bayesian estimator is consistent in the sense that the estimator converges in the large-sample limit even if the prior is wrong. The goal of the estimator construction is to translate qualitative views into a discrete predictive distribution of returns. To accomplish this, I apply Bayes’ rule on a discrete subset of the candidate probability distribution functions for returns. I start from a standard formulation of the investor’s problem in a Bayesian statistical decision framework (Zellner and Chetty (1965)). In this framework, investor beliefs are modeled as a prior density over a space of can- didate probability distributions. The investor combines their prior information with the data likelihood to obtain posterior probabilities for each candidate distribution. The predictive distribution for future returns is then obtained by integrating over the domain of the posterior distribution. The domain of the posterior is a space of probability distribution functions. In a nonparametric setting, integration over a domain of functions is not com- putationally tractable without some specification assumptions. The standard approach is to restrict the candidate distributions to a parametric family such as the set of multivariate Gaussians. To retain the nonparametric character of the estimator, I instead restrict the domain of the posterior by discretizing the set of possible return outcomes. I restrict the domain of the posterior to the set of multinomial distributions with nonnegative probabilities on the chosen set. The final step in the estimator design is the designation of the set of return outcomes. The set must be large enough to allow for a rich set of candidate distributions, yet parsimonious enough to permit sampling. A good candidate is the set of the return realizations in the historical data. This set is a random sample from the return domain that has the useful property of being generated by the data generating process of interest. As a result, this set will likely place a greater number of points where the density is concentrated and the data provides the most information. Furthermore, by discretizing to the return outcomes in the data, the resulting predictive distribution takes the form of a perturbation of the empirical distribution of returns. One caveat is that the resulting discrete-Bayesian estimator is not strictly Bayesian. By using the data to approximate the posterior domain, I am implic- itly restricting the prior domain as well. Despite this, the prior probabilities 84 attached to each candidate distribution do reflect prior beliefs, and these prior beliefs are effectively incorporated into the posterior. In a simulation study, I investigate the implications of investor belief in the nonnegativity of portfolio weights on portfolio choice. I model prior views as exponential functions that decay as the conflict between investor views and candidate distributions increases. I examine the implications of the no short views on the predictive distribution of returns and on the portfolio weights. I consider the example problem of an investor seeking to estimate alloca- tions across a set of five assets that are highly correlated and that have similar distribution characteristics.46 As in Kan and Zhou (2007), I use simulations to compare the expected out-of-sample performance of different models. In ad- dition, I conduct out-of-sample experiments on historical data, and compare investment outcomes with a set of portfolio choice models from the literature. I set the parameters of the underlying distribution such that the true optimal allocations are all positive. I find that, even when blessed with a prior belief in the positivity of portfolio weights that is true for the generating distribution, the expected out-of-sample performance of the estimates struggles to match the performance of the mini- mum variance and 1/N portfolios. The weak performance of the Bayesian model with informative prior attests to the usefulness of regularization in portfolio forecasting. Regularization is a technique that places smoothness criteria on the vector of estimated weights. The 1/N solution is an extreme example of a regularized solution. The 1/N policy is optimal if an extremely large penalty on cross-sectional variation in portfolio weights.47 For history lengths of up to two-hundred and forty months, the data does not contain enough information to estimate robustly portfolio weights. This is true even after imposing a prior that is both correct and leads to less variation in portfolio weights across assets. This study adds to several strands of the portfolio choice literature. First, I build on work that has incorporated Bayesian analysis into the portfolio choice problem. The Bayesian framework not only allows for possible prior views, but has the additional benefit of incorporating the impact of distribution uncertainty 46The distribution characteristics are based on the historical returns distribution of the five-industry breakdown of the United States equity market provided by Ken French. 47Additionally, portfolio weights are assumed to sum to one (DeMiguel et al. (2008)). 85 on investor decisions. Numerous applications of Bayesian analysis in portfolio choice examine the influence of distribution uncertainty when an investor has uninformative prior beliefs.48 Kandel and Stambaugh (1996), and Wachter and Warusawitharana (2005) examine the impact of degrees of belief in return pre- dictability on portfolio decisions.49 Pástor (2000) and Pástor and Stambaugh (1999) examine the implications of investor beliefs in an asset pricing model. This chapter is closely related to that of Chevrier and McCulloch (2008) who also consider the impact of investor views on portfolio composition. This study differs with that of Chevrier and McCulloch (2008) in several significant respects. I employ a modeling approach that is nonparametric with respect to the underlying return distribution. I am able to construct a nonparametric posterior by introducing a simple restriction on the space of possible probability distributions. In contrast, Chevrier and McCulloch (2008) assume the return distribution is normal. While I do assume an investor whose utility depends only on the first two moments of return, I do not restrict the set of underlying distributions to the normal family. As such, my approach could be applied to models with different investor utility such as those considered by Aı̈t-Sahalia and Brandt (2001). In addition, as well as reporting out-of-sample performance tests, I also conduct a simulation study which permits comparing expected out-of-sample performance of the model. The disadvantage of out-of-sample studies is that performance results are based on outcomes for a single sequence of returns. With a simulation experiment, the performance of a portfolio rule can be estimated to arbitrary precision given enough simulated return histories. Finally, I describe the potential for positive weight constraints to introduce upward bias in the predictive return distribution and introduce an adjustment to the prior that mitigates this issue. This remainder of the chapter is divided into seven sections. In section 3.2, I formulate the portfolio choice problem in a Bayesian decision framework. In section 3.3, I construct a discrete estimator for the investor’s optimand. In section 3.4, I quantify the prior beliefs that I incorporate into the investor’s problem. Section 3.5 describes the relationship between the portfolio choice model developed in section 3.3 with models from literature. The empirical 48See, for example, Brown (1976) and Bawa et al. (1979). 49Avramov (2002) and Cremers (2002) use Bayesian methods to help with model selection given a set of predictors. 86 results are presented and discussed in sections 3.6 and 3.7. Section 3.8 concludes. In Appendix B, I formulate two data tuning approaches to portfolio choice. Appendix C details the Markov chain Monte Carlo (MCMC) algorithm used to resample from the posterior distribution. 3.2 Portfolio Choice Framework In this section, I formulate the investor’s problem as a utility maximization in a Bayesian decision framework. The Bayesian decision framework will be the starting point for development of the resampling estimator developed in section 3.3. The Bayesian formulation of the investor’s decision problem has its roots in the work of Zellner and Chetty (1965). Other applications can be found in Brown (1976), Bawa et al. (1979) and Barberis (2000), amongst others. Consider an investor who must select a portfolio from a universe of N risky assets plus a riskless asset. Define the N vector of risky asset returns rrt and the risk-free rate rft from time t−1 to time t. Excess returns on the risky assets are given by rt = r r t − rft . The investor does not know the probability distribution of returns. Instead the investor is Bayesian, and bases portfolio decisions on an information set comprised of historical return data plus prior beliefs. The prior beliefs may be based on economic insights or other non-data knowledge. The prior beliefs and the non-data knowledge are combined to obtain a posterior distribution. The domain of the posterior distribution is a function space that includes all probability distributions that could possibly describe the data set (prior to introducing any information). This is a very large set. For the portfolio problem, we are concerned with the future distribution of returns. The unrestricted set of potential future distributions includes all N-dimensional functions that integrate to one on RN . In applications, one must integrate over this set to evaluate the investor’s expectation. This functional integral is an essential feature of Bayesian applications. The evaluation of functional integrals is challenging in applications. Unlike with classical integration of volumes50, there is no universal fundamental definition for functional integrals. Thus, in typical applications the set of possible distributions is restricted to a subset that permits evaluation of 50Volume is a generalized concept here defined on N dimensions. 87 a standard integral. I return to this issue and discuss the restriction used in this paper in the next section. Let Ω ⊂ RN be the space of possible next period return outcomes rτ+1, and let ∆ be the unrestricted space of possible probability distributions of returns. The investor uses his or her available information set to assign posterior prob- abilities to each possible probability distribution δ ∈ ∆. The posterior qpost is the product of a likelihood distribution given the available data and a prior distribution qprior that quantifies investor views, i.e., qpost(δ|R) ∝ l(R|δ)qprior(δ), (3.1) where R is a T by N matrix of historical return data, and the outer integral is a functional integral over the function space ∆. I assume the investor’s utility can be expressed as a von-Neumann-Morgenstern utility function that includes a parameter γ that increases in risk aversion. The investor’s problem is to maximize expected utility given the available informa- tion. max w E [ u(W ) ] = ∫ ∆ ∫ Ω u(r, w) p(r|δ) dr qpost(δ|X) dδ, (3.2) where p(r|δ) is the probability of return r under probability distribution δ. Introducing the notation E∆ for the expectation integral over ∆ and Eδ for the expectation integral over Ω for a given δ ∈ ∆, the investor’s problem can be written as max w Eq [ u(r) ] = max w E∆ [ Eδ [ u(r) ] ] , (3.3) where q is the predictive distribution of returns, i.e., q(r) = ∫ ∆ p(r|δ)qpost(δ)dδ. 3.3 Discrete-Bayesian Estimator As in many Bayesian applications, solution of the investor’s decision problem requires computing an expectation over the space of possible distributions ∆. To render this integration feasible, the domain of possible distributions ∆ can be restricted by a model specification. A standard approach is to restrict the set of distributions to a parametric model. For example, the investor might restrict 88 the space of possible return distributions to the set of multivariate normal dis- tributions. In this case, the posterior is a function of the model parameters: the mean returns and the covariance matrix. In essence, the model specification is a meta-prior that places zero probability on all distributions outside the model space. In this section, I develop an estimator based on a model specification that does not rely on a parametric specification to achieve a computationally fea- sible algorithm. In the portfolio choice context, the method extends Brandt’s (1999) nonparametric method of moments to the case with non-data priors. The method also can also be related to a class of “data tuning” (Braun and Hall (2001)) methods developed for conditional smoothing and inference applications in nonparametric statistics. To provide some intuition for the linkage between the estimator discussed here and previous literature, I anticipate the final form of the estimator con- structed below. The approximation to the expectation in the investor’s problem (3.3) reduces to the form, Eq [ u(rt+1, w) ] ≈ T∑ t=1 u(rt, w)qt, (3.4) where the qt, (t ∈ {1 . . . T}), are weights that depend on the prior. Like classical nonparametric estimators, the estimator is a sum over historical realizations of the integrand. The estimator departs from the classic estimator in that it incor- porates observation weights. These weights represent a discrete approximation to the predictive distribution of returns q that depends on investor priors. The estimator (3.4) is analogous to the nonparametric estimator of the in- vestor’s problem developed by Brandt (1999). Brandt (1999) also estimates the investor’s expected utility by a weighted sum over historical return outcomes. However, whereas the weights in Brandt (1999) reflect conditioning on return predictors such as the dividend yield, the weights applied here depend on non- data priors. In either case, each of the estimation weights reduces to 1/T when no additional information is available. Incorporation of non-data priors into a nonparametric estimation framework also motivates data tuning methods. Data tuning methods incorporate non-data information by perturbing the empirical distribution of the data subject to a 89 distance constraint on the perturbation. This is accomplished either by altering the weights applied to each data point in the estimator or by perturbing the data themselves subject to a distance constraint.51 While the estimator developed in this paper (3.4) has a similar structure to data tuning methods that perturb the probabilities attached to each data point, the methods of arriving at the final estimator differ. The perturbed probabilities used in data tuning are obtained by selecting the highest likelihood distribution for which the optimal decision variables satisfy a constraint.52 By contrast, the data weights for the nonparametric Bayesian approach are a reflection of qualitative or non-data prior views, and are obtained by numerical integration over the investor’s posterior distribution by Markov chain Monte Carlo. The estimator takes the familiar nonparametric form as a result of the discretization scheme used to model the domain of distributions spanned by the posterior. 3.3.1 Construction of the Discrete-Bayesian Estimator To construct the discrete-Bayesian estimator (3.4), I start by assuming that we have a finite sample of B draws from the posterior qpost. Under this assumption, the finite sample equivalent of the investor’s objective in (3.3) is Eδ [ u(rt+1, w) ] ] ≈ 1 B B∑ b=1 Tb∑ t=1 u(rbt, w), (3.5) where {δb}Bb=1 is the sample of B distributions drawn from the posterior, and {rbt}Tbt=1 is a draw from δb for each b. To implement the above estimator, we require a feasible means of drawing a set of distributions from the posterior qpost. The expression (3.5) is a Monte Carlo approximation to the investor’s expectation integral. In theory, the ap- proximation will converge as the number of draws goes to infinity. The challenge 51Hall and Presnell (1999) introduce a method of perturbing the probability weights applied to each observation. Methods that perturb the data are referred to as data sharpening. (Choi and Hall (1999) and Choi et al. (2000) develop data sharpening methods applicable to nonparametric curve smoothing, density estimation and inference. See Braun and Hall (2001) for further references to related techniques. 52To illustrate the contrast between the Bayesian resampling approach developed below and data tuning methods, I include two data tuning formulations for a sample investment problem with a non data prior. See Appendix B. 90 is to develop a scheme that is stable and computationally feasible. Given the limited number of historical returns available to the investor, and the high di- mensionality of the underlying return distribution, some form of restriction on the space of return distributions is warranted. The usual approach is to restrict the set of possible distributions ∆ by introducing a parametric model for the return distribution – often a multivariate normal.53 Here, I depart from the standard approach. I modify the set of candidate distributions ∆ by restricting the possible return outcomes to a finite set. This discretizes ∆ to a subspace of distributions with nonnegative probabilities at a finite set of points, i.e., a set of multinomial distributions. Unlike paramet- ric models, the approximation places no restriction on the functional form of the distribution. However, the approximation does introduce the problem of selecting a set of return outcomes to form the domain of the candidate distrib- utions. The set must be sufficiently parsimonious for computational feasibility, yet must be dense enough to allow for a rich array of posterior distribution characteristics. Ideally, the set of possible return outcomes would be chosen to place a greater density of points near peaks of the underlying distribution because the data are more likely to be informative on distribution shape in those regions. But we do not know the distribution in advance. I deal with this issue by setting the discrete set of possible return outcomes equal to the returns realized in the historical data. The underlying data generating process is itself the ideal randomizer for selecting a parsimonious grid of returns with the desired point distribution. By restricting the set of possible distributions to multinomial distributions with nonnegative probabilities at the realized returns, I obtain an approxima- tion that is analogous to classical nonparametric estimation in which inference is based on the empirical distribution. In the presence of an uninformative prior, the resulting discrete-Bayesian estimator is equivalent to the classical nonpara- metric estimator, i.e., the probability weights assigned to each observation are equal. One caveat is that the resulting estimator is not strictly Bayesian. The approximation used to allow integration over the posterior domain also estab- 53An exception is Kacperczyk (2003) who studies a Bayesian model in which third and fourth moments of returns are unrestricted. 91 lishes the domain of the prior. Thus, while the prior likelihoods associated with any distribution are not affected by the use of the return outcomes to set the discretized grid, the prior and the data are no longer independent. To formalize the construction of the discrete-Bayesian estimator, I define a subset ∆R of the set of possible return distributions ∆. Each member of ∆R has finite domain consisting of the T realized returns in the set of realized returns R. Thus the subspace ∆R is the space of multinomial distributions with nonnegative probabilities pt at all points in the set of realized returns R and zero probability elsewhere, i.e., δ ∈ ∆R ⇒ δ(r) = pt r ∈ {r1, . . . , rT}0 otherwise∑T t=1 pt = 1. (3.6) For the purpose of evaluating the investor’s problem, I restrict ∆ to ∆R. Each member δ of ∆R is parameterized by T − 1 independent probabilities. The likelihood l(δ) is the probability of drawing the return sample given a distribution δ. Hence, l(δ|R) ∝ p1p2 · · · pT−1 ( 1− T∑ t=1 pt ) , (3.7) where pt is the probability of draw t under distribution δ. Multiplying by the prior qprior yields the posterior q(δ) = ( T−1∏ t=1 pt )( 1− T−1∑ t=1 pt ) qprior(δ). (3.8) Given a sample of B draws δb = {pb1, . . . , pbt} from ∆R and noting that p(rt|δb) = pbt, the estimator (3.5) reduces to 1 B B∑ b=1 T∑ t=1 u(rt, w)pbt. (3.9) Summing over B simplifies (3.9) to the form given in (3.4). The components qt of the predictive distribution are given by qt = (1/B) ∑B b=1 pbt. The pre- 92 dictive distribution reduces to a set of weights applied to each of the sample observations. 3.3.2 Sampling from the Posterior The posterior distribution (3.8) is a mixture of a multinomial distribution and an as yet unspecified prior. With T − 1 parameters, obtaining a draw from the distribution that achieves statisfactory convergence of the investor’s expectation is a challenge. One viable approach for such a high dimensional problem is Markov chain Monte Carlo.54 To construct the Markov chain, I employ a Metropolis Hastings algorithm. At each step of the chain, I draw a distribution δ from a tractable proposal distribution. The proposal is conditional on the current distribution. This pro- posal is either accepted or rejected with an acceptance probability that depends on i) the ratio of the value of the posterior distribution at the proposal and the current value, and ii) the ratio of the conditional proposal probabilities. The accept-reject algorithm ensures that as the length of the chain goes to infinity, the number of draws from any region of the posterior’s domain is proportional to that region’s posterior probability. I base the calibration of the MCMC algorithm on the convergence with number of draws of the posterior distributions of the mean returns and of the optimal portfolio weights implied by each distribution drawn. I obtain good convergence by combining draws from five independent chains of length 5000. The first half of the draws from each chain are dropped to allow for convergence of the chain to the posterior distribution. Thus, the posterior estimates are based on a total of 12500 draws. I provide full details and an algorithm chart in Appendix C. 3.3.3 Mean-Variance Utility In the empirical study in section 3.6, I assume quadratic utility over wealth. Under this assumption, the portfolio estimator developed in the previous section simplifies to mean variance form. Working in the mean variance framework allows for comparison of results with those of previous research, as it is the 54Gamerman and Lopes (2006) contrast MCMC with more direct simulation methods. 93 standard model in the portfolio choice literature.55 However, application of the portfolio estimator discussed in the previous section is not dependent on this assumption. The fact that the portfolio allocations only depend on the first two moments of the return distribution is purely a function of the utility assumption. The return distribution is modeled without moment assumptions. Let µ(δ) and Σ(δ) be functions that return the mean and covariance matrix respectively of the distribution δ. Under quadratic utility, the expected utility estimator (3.4) is u(r, w) = w′µ(q)− γ 2 w′Σ(q)w, (3.10) where q is the discrete predictive distribution of returns defined in the previous section. 3.4 Prior I formulate the investor’s beliefs as a quantitative prior. My objective is to examine the consequences of incorporating prior beliefs in the portfolio decision problem. The discrete nonparametric estimator described in section (3.3) allows for specification of arbitrary priors. This is liberating for Bayesian analysis as priors are not limited by analytical requirements for conjugacy with a particular return model. In addition, since the distributions are nonparametric, priors do not have to be formulated in terms of specific distribution parameters. In theory priors should take the form of probability distributions. However the estimator is insensitive to multiplicative constants. Hence, the priors must satisfy two restrictions: they must be positive functions defined on the set of candidate distributions, and multiplication with the data likelihood must result in an integrable posterior distribution. I model four priors. The first two priors are suggested by Chevrier and McCulloch (2008). The first is a restriction on the global minimum variance portfolio implied by a candidate distribution. The second models the belief that optimal portfolios are positive. 55Proposed by Markowitz (1952). Brandt (2005) and DeMiguel et al. (2007) review subse- quent developments. 94 The third prior mitigates a side effect of the second prior. As will be demon- strated in section 3.6.2, the second prior biases posterior mean returns upward. It is implausible that a belief in the positivity of portfolio weights is associated with a belief that historical return means are biased downward. To mitigate this effect, I introduce a prior that models the belief that the mean of the return means or “grand mean” is close to the sample value. Finally, I consider a prior that models the belief that all portfolio weights are smaller than one. The first prior q1 is a step function that places zero probability on return distributions that imply a return on the global minimum variance portfolio that is less than the risk-free rate. This is a no-arbitrage condition that states that risky portfolios dominate the risk-free asset in terms of expected return. I quantify the prior by setting q1(δ) = 1rGMV >rf (3.11) where rGMV = w ′ GMV µ(δ) is the return on the global minimum variance portfolio wGMV = Σ −1(δ)1 for distribution δ. The second prior expresses investor belief that portfolio weights, given the underlying or true distribution of returns, are uniformly positive. This prior is motivated by both asset pricing theory and modeling precedents in practice and research. Asset pricing theory holds that the assets in positive net supply are priced such that the aggregate investor would have positive allocations in an optimal portfolio. If we assume the investor’s preferences are similar to those of the aggregate market, then it follows that the investor would expect optimal portfolio holdings to reflect the holdings of the market. Short-sales constraints are often employed by investors and practitioners in implementations of portfolio choice models. Mean-variance portfolio weights computed by plug-in methods often feature large negative positions. These po- sitions are attributable to high estimation error and the dominance of one or few eigenvalues in the covariance matrix of returns (Green and Hollifield (1992)). Whether negative weights are justified or not, mean variance methods are often implemented with a constraint against short sales. The constraints might re- flect restrictions on investment mandates or trading restrictions. However, the number of assets available for taking short positions is large and continues to grow. In addition, a restrictive investment mandate, while telling us nothing 95 about the beliefs of an asset manager, indicates bias against short positions on the part of the investor’s clients. Prior skepticism regarding short positions is implemented by the following function: q2(δ, c2) = exp ( −c(neg(wtan)′wtan)2,) (3.12) where c is a decay parameter, wtan is the N vector of portfolio weights in the actual optimal portfolio given distribution δ, and neg(·) is a vector function that returns a vector of the same length with zeros substituted for positive values and −1 substituted for strictly negative values.56 The larger the value of c2, the greater is the investor’s belief that optimal portfolio weights are greater than zero. The third prior expresses the belief that the grand mean, or mean of the return means, is unlikely to exceed its sample value. While intuitive, the prior q2 biases the posterior towards distributions in ∆ with expected return vectors that are uniformly higher than the sample mean. The high variance of returns creates regions of the posterior domain characterized by high likelihood and high return means. Distributions with lower return means are discounted by the prior, and these distributions receive lower weight in the posterior. The result is an upward bias in optimal weights that is difficult to justify. The primary motivation for the belief in positive weights is to focus the posterior on candidate distributions with means that are similar to the sample, but whose returns and covariances do not combine to create unrealistic positions. Unrealistic positions are those with high leverage that arise when estimating optimal portfolios for highly correlated assets.57 The third prior mitigates this effect. The prior is implemented as q3(δ) = exp (−c3 (max(µg(δ)− µ̂g), 0)2) , (3.13) where µg(δ) = ( ∑N i=1 µi(δ) 1)/N, µ̂g = ( ∑N i=1 µ̂i)/N , and c3 is a parameter. I also consider a fourth prior that expresses the view that large deviations 56wtan is computed as the solution of maxw Eδ[U(w′r)]. To eliminate dependence on risk aversion in the prior, tangency weights are scaled to sum to one. 57See Green and Hollifield (1992) and Jagannathan and Ma (2003) for studies of gearing in mean variance portfolios. 96 from the grand mean from the sample value are unlikely whether positive or negative. The prior is expressed as q3(δ) = exp (−c4(µg(δ)− µ̂g)2) . (3.14) Prior q4 is related to Bayes-Stein estimation. Jorion (1986) shows that a Bayes- Stein estimator that shrinks return means dominates traditional plug-in esti- mation. An investor can have one or more of these priors. The investor’s overall prior distribution is expressed as qprior(δ; c) = ∏ i∈I qi(δ, ci) where I is the investor’s subset of priors and c is a vector of the relevant prior coefficients. 3.5 Relationship to Previous Portfolio Choice Models In this section, I discuss a selection of models for portfolio choice that are related to the estimator employed in the current study. Table 3.1 lists the models considered. Researchers have proposed a vast number of innovative approaches to portfolio choice. The list considered here is by no means comprehensive. See, for example, DeMiguel et al. (2007) and Garlappi et al. (2007) for comparative empirical studies of large sets of models that overlap the set considered here. The subsections below describe two groups of models. The first group in- cludes plug-in rules. These rules are variants of the classical approach that makes use of sample estimates of return means and variances. The second group consists of portfolio resampling models. I conclude with a brief note on regularization in portfolio choice. 3.5.1 Plug-in Models Perhaps the most straightforward approach to portfolio estimation is to directly substitute estimated distribution parameters into the investor’s problem. For the mean variance model, this amounts to approximating the investors problem max w w′µ− γ 2 w′Σw (3.15) 97 by max w w′µ̂− γ 2 w′Σ̂w (3.16) where µ̂ = 1 T ∑T t=1 rt and Σ̂ = 1 T ∑T t=1(rt − µ̂)(rt − µ̂)′ are sample moments. Solutions obtained by the direct plug-in model (3.16) are notoriously unstable and lead to poor out-of-sample performance.58 In efforts to achieve more robust performance, researchers have proposed numerous alternative estimators for the distribution moments used in the mean variance model. A large class of these models take the form of plug-in rules that combine the risk-free asset, a sample estimate of the mean variance portfolio, and a sample estimate of the minimum variance portfolio formed without the risk-free asset. Plug-in models assign a portfolio weight vector of the general form w = pi1 γ Σ̂−1µ̂+ pi2 γ Σ̂−11 (3.17) where pi1 and pi2 are constants. The classic mean variance solution (3.16) is obtained with pi1 = 1 and pi2 = 0. 59 A number of plug-in rules that take the form (3.17) are motivated by statisti- cal considerations. Kan and Zhou (2007) provide a detailed analysis of portfolio models whose solutions take this form. For the most part, these models shrink the estimated portfolio towards the risk-free asset (i.e., pi1 < 1) or towards the minimum variance portfolio (i.e., pi2 > 0). I list a subset of these plug-in rules in Table 3.1. The first model is the direct substitution mean variance model. I consider the direct substitution approach because it is a longstanding standard in the literature. The second plug-in model is obtained following a parametric Bayesian analy- sis in which the investor assumes normally distributed returns and has a diffuse Jeffrey’s prior over the distribution parameters. Kan and Zhou (2007) demon- strate that this model is dominated in terms of out-of-sample performance by some of the plug-in rules that follow. I include this model because it is interest- ing to compare the parametric Bayesian solution to the discrete, nonparametric 58See Best and Grauer (1991). 59If the unbiased estimator of Σ is used then pi1 = (T − 1)/T . 98 Bayesian approach developed in this paper. The third through fifth models are developed in Kan and Zhou (2007). Kan and Zhou (2007) evaluate the efficiency of plug-in rules based on their expected out-of-sample utilities under the assumption of normally distributed returns. The third model is the most efficient plug-in rule with a coefficient on the sample portfolio pi1 that is independent of the sample. The fourth and fifth models are plug-in rules that aim to improve efficiency by allowing the coefficients in (3.17) to depend on the sample. The fourth is derived under the restriction pi2 = 0, and the fifth allows for investment in the sample minimum variance portfolio. The sixth model is the classical mean variance model with short sales con- straints imposed on the investor’s problem (3.16). As discussed in the intro- duction, a common approach is to solve (16) under a constraint against short sales. Solutions obtained with short sale constraints imposed are closely related to those obtained by statistical approached that shrink portfolios towards the minimum variance portfolio. Because they are straightforward to implement in many optimization packages, models with short sale constraints are very com- mon in investment practice. The last two models back away entirely from relying on the data to estimate return means. The seventh model is the global minimum variance portfolio. The global minimum variance portfolio has a statistical advantage in small samples in that it is independent of return estimates, and has proven to be a strong performer in many out-of-sample studies, despite being an ad hoc rule. The final entry is the 1/N model discussed extensively in Garlappi et al. (2007). This model takes data skepticism to the extreme. The model assumes portfolio weights sum to one and assigns equal weight to each asset. 3.5.2 Portfolio Resampling Portfolio resampling refers to the recomputation of optimal portfolios for re- peated, randomly generated return samples. The return samples can be gen- erated by drawing from a parametric return model estimated from the sample data60 or bootstrapping from the empirical distribution of returns. Resampling is a standard technique in statistics for estimating sampling distributions and 60See Jorion (1992), Michaud (1998), Markowitz and Usmen (2003) and Harvey et al. (2003). 99 addressing hypothesis tests (Efron (1979)). Jobson and Korkie (1981) suggest the technique for statistical analysis of portfolio estimates. The technique has been adopted in a number of subsequent studies.61 Michaud (1998) considers extending the application of portfolio resampling beyond statistical hypothesis testing to decision analysis. He suggests choosing portfolios by averaging across optimal allocations computed for each resam- pled return draw.62 Scherer (2002) and Harvey et al. (2003) point out that the resampling of return distributions is conceptually equivalent to drawing from a Bayesian posterior distribution with an uninformed prior.63 Viewed from a Bayesian perspective, construction of portfolio decisions from averages of resam- pled portfolios switches the order of the maximization and the integration over the posterior in the investor’s portfolio problem (3.2). Using the notation from section 3.3, weights computed by portfolio resampling are an approximation to the integral w = ∫ ∆ [ argmax wb ∫ Ω u(r, wδ)p(r|δ)dr ] qpost(δ)dδ. (3.18) Given B resampled return draws, the finite sample approximation is ŵ ≈ 1 B B∑ b=1 [ argmax wb T∑ t=1 u(rbt, wb)pbt ] , (3.19) where δb is the empirical distribution associated with the bth resampled return series. Under mean-variance utility, the portfolio resampling computation (3.18) 61For example, Jorion (1992) uses resampling to illustrate portfolio sensitivity to estimation risk. Scherer (2002) suggests application of resampling techniques for hypothesis tests of the statistical difference between two portfolios. I apply resampling to test the difference in investment benefits of different asset allocation portfolios in Chapter 2 of this thesis. 62See Scherer (2002) or Meulli (2006) for a complete description of Michaud’s algorithm. Herold and Maurer (2006) implements portfolio resampling using the bootstrap. 63For example, as T −→ ∞ likelihood properties of distribution moments computed from bootstrap distributions will approach those of moments based on distributions sampled from the posterior discussed in section 3. 100 simplifies to 1 B B∑ b=1 argmax wb ( µ(δb) ′wb − γw′bΣ(δb)wb ) . (3.20) In this case, the solution to the portfolio resampling problem is wPS = 1 Bγ T∑ t=1 Σ(δb) −1µ(δb). (3.21) The portfolio resampling solution is an average of optimal weights implied by a set of distributions drawn from the posterior. While portfolio resampling procedures are not usually associated with Bayesian analysis, the portfolio resampling heuristic can be readily extended to allow for informed priors. The solution (3.21) can be computed as long as a sampling scheme is available to obtain draws from the resulting posterior distribution. The primary model implemented by Chevrier and McCulloch (2008) is an ex- ample of such a scheme.64 They analyze portfolio weights obtained as solutions to (3.21) where the set of distributions is drawn from a posterior that incorpo- rates prior beliefs. In the empirical examples of section 3.6, I compare portfolio allocations es- timated via the Bayesian approach to two portfolio resampling models. Both implement the mean variance solution (3.21). The first is a standard portfolio resampling model with uninformed prior and the second incorporates prior be- liefs. In both models, I employ the MCMC technique developed in section 3.3 to draw from the posterior. I find that the performance of portfolio resampling with prior is very similar to that of the discrete-Bayesian model. Finally, I compute results for a heuristic proposed by Michaud (1998) that combines the placement of short sales constraints with portfolio resampling. For this model, each of the resampled portfolio weights are computed with constraints on short sales. For consistency with previous applications, I draw the returns for this last model from a posterior that does not include prior information. 64Chevrier and McCulloch (2008) describe the method as the “posterior mean over the implied weights”. 101 3.5.3 Statistical Refinement versus Regularization It is useful to distinguish models that are statistical refinements of the clas- sic plug-in approach from those that give up consistency in a statistical sense to regularize (i.e., impose smoothness) on the solution vector. Of the models listed in Table 3.1, the regularizing solutions are those that impose short sale constraints and the 1/N rule. The other models, with the exception of the min- imum variance portfolio, converge to the true optimal portfolio as the length of the available data set approaches infinity. Statistically motivated models seek to improve allocation estimates by accounting for estimation uncertainty or ad- ditional non-data information. Regularization seeks to compensate for data insufficiency by imposing smoothness on the portfolio weights vector.65 A major challenge in portfolio estimation is that the amount of data re- quired to achieve a stable estimate is many orders of magnitude larger than is usually available to the investor. As such, even the best statistical estimates of the return distribution yield unstable portfolio estimates. As a result, despite introducing bias, regularized solutions such as those based on constraining short sales (Jagannathan and Ma (2003) and 1/N (DeMiguel et al. (2007)) perform well in many applications. While regularization is effective, ideally an investor would be able to estimate optimal allocations from available information. Previous studies indicate that statistical approaches have a very hard time outperforming the regularized so- lutions. The simulation analysis of the next section further tests this conclusion by comparing short sales constrained and 1/N performance with performance of a Bayesian model that incorporates additional non-data priors in a simulation setup in which the priors are correct. 65In general, regularization models result in a Lagrangian with additional terms that pe- nalize variation of the solution vector. For example, a typical regularized Lagrangian for the portfolio problem might have the form E [U (r̃, w)]− κ||Φw||p (3.22) where κ is a scaling parameter, p is the degree of the vector norm, and Φ is an NxN matrix. Jagannathan and Ma (2003) show that the no short sales solution is equivalent to (3.22) with p = 1 and Φ equal to a diagonal matrix with the Lagrange multipliers of the short sales constraints on the diagonal. Sufficient conditions for the 1/N solution have Φ equal to a demeaning operator, Φ = I − 11′/N , p = 2 and κ→∞. 102 3.6 Portfolios with Positive Weights Prior In this section, I compare the properties and performance of portfolios estimated using the discrete-Bayesian model for portfolio choice developed in section 3.3 with models from the literature. The empirical analysis is divided into two parts. In the first, I compare the expected out-of-sample performance of the models in a simulation experiment. In the second, I conduct an out-of-sample study using historical returns. Simulation results are a useful tool for comparing the performance of portfo- lio models under controlled conditions. Because the underlying return distrib- ution is known to the experimenter, out-of-sample performance for a simulated trial can be computed exactly for any portfolio weight policy. This study differs from previous simulation studies in that some of the rules I consider assume investors have information beyond that contained in historical data. See Kan and Zhou (2007), Herold and Maurer (2006), and Garlappi et al. (2007), amongst others, for simulation results that examine outcomes for portfolio rules estimated from historical data. This study most directly extends results of Kan and Zhou (2007). I use a simulation procedure that closely parallels their procedure. In addition they provide analytic results for a number of plug-in rules that can be exploited for verifying simulation convergence. 3.6.1 Simulation Setup In the simulations, I assume a mean-variance investor with relative risk aversion γ equal to five. I generate independent return series from a normal distribution. I consider a five-asset example. I calibrate the covariance matrix to sample values computed for the five-industry data set. In setting the return means, I partially shrink the sample mean vectors towards their grand mean. The degree of shrinkage is high enough to ensure that the optimal portfolio weights of a mean variance investor with full knowledge of the underlying distribution would be strictly positive.66 The parameters used in the simulation trial for the five-asset case are listed in Table 3.2. 66I set the vector of return means of the data generating process for the simulations to µ = ²µ̂+ (1− ²)(∑Ni=1 µ̂i)/N , with 0 < ² < 1. I choose a shrinkage coefficient ² small enough to ensure that all optimal weights implied by the underlying distribution are greater than zero; i.e., Σ̂−1µ > 0. 103 I employ a simulation strategy similar to that employed by Kan and Zhou (2007). I randomly generate S return series from the underlying distribution. These return series become the “historical” data for S hypothetical investors. Each investor estimates optimal portfolios using the twelve models listed in Table 3.1. I then compute the mean and standard deviation of the out-of-sample utility realizations for each estimated portfolio given their portfolio estimates and the actual distribution parameters. I calibrate the number of simulations S required to effectively estimate the out-of-sample utility. Given the computational burden of the Markov chain Monte Carlo methods, I wish to minimize S. Kan and Zhou (2007) provide analytic expressions for the out-of-sample utility of the plug-in rules for which pi1 and pi2 are independent of the data. Using these results, I choose an S that is large enough to yield estimates that are precise to within two significanct digits when compared with the theoretical result. I find that S = 250 is sufficient for this purpose. 3.6.2 Exploratory Analysis of the Prior Assumptions An important preliminary step in a Bayesian investigation is characterization of the posterior. A prior model designed to quantify an investor belief may have unintended consequences for the posterior. This can result in a posterior model that, while ostensibly capturing the investor belief that is of primary interest, is decidedly unrealistic on another dimension. Here I contrast the posterior distributions for several different priors. The posteriors are based on a return series of length T = 120. The return series is drawn from the five-asset distribution parameterized in Table 3.2. For the analysis, I employ 12500 distributions drawn from the posterior distribution by the MCMC method described in the appendix. I compute summary statistics for each of the 12500 draws. The summary statistics include return means, return standard deviations, and the implied portfolio weights of a mean variance investor with risk aversion coefficient equal to five. The first panel of Table 3.3 shows return means, return standard deviations, and portfolio weights estimated directly from sample data. The remainder of Table 3.3 reports averages and standard deviations of summary statistics across posterior draws under different prior specifications. The second panel shows the 104 summary statistics averaged across 12500 distributions drawn without prior. As expected, given that the prior adds no information, the means, returns, and weights are close (within 1%) to values obtained under a diffuse prior assumption on µ and Σ. For a normal model, and given a diffuse prior, the optimal weights are the standard mean variance weights multiplied by (T −N − 2)/(T + 1). In this case, this implies average weights of 0.81, slightly below the value predicted by simulation. The third panel of Table 3.3 displays results obtained with investor bias against negative portfolio weights incorporated into the prior. For this initial case, the model only incorporates the economically motivated priors described in Section 3.4: the no-arbitrage prior q1, and the positivity prior q2. I select the parameter c2 following a qualitative tuning exercise. I consider the posterior distribution obtained when the informed prior is paired with a return sample of length T = 120. I find that c2 = 20 yields a set of posterior draws that places less than 5% weight on distributions that imply negative weights in a mean variance optimized portfolio. Hence, this choice of parameter reflects a strong, but not completely dogmatic belief that all portfolio weights are positive. Under priors q1 and q2, the variability of the means and standard deviations are similar to those observed with no prior. However, comparison of the bottom lines of panels 2 and 3 reveals that implied portfolio weights are far less vari- able across posterior draws for the informed investor. The prior q2 informs the implied weights of the mean variance investor without increasing the precision of the individual moment estimates. Figure 3.1 compares histograms of portfolio weights obtained with no prior with those obtained when priors q1 and q2 are implemented. Under the informed prior, the posterior probability that the optimal portfolio weights are less than zero is clearly reduced relative to the no-prior case, as is the variance of the pos- terior distribution of portfolio weights. The standard deviations of the portfolio weights implied by the posterior distribution are reduced by two to three times upon incorporating a belief in the positivity of portfolio weights. While the prior q2 succeeds in incorporating a belief in positive weights, there is a noticeable upward bias in the overall average of the return means. Averaging across the first row of any panel in Table 3.3 yields a cross-asset mean or “grand” mean of the asset returns. The grand mean based on the first 105 row of panel 3 is 1.15 while that for the second panel is 0.82, a difference of 0.33 percentage points per month. Given a prior that favors positive weights, the posterior places greater weight on distributions with higher return means. The return bias is illustrated in Figure 3.2. Each panel shows a histogram of the posterior distribution of one asset’s return mean (vertical bars). The solid line is a normal distribution fitted to the data. The average posterior return means are shifted to the right relative to their no prior counterparts. The return bias is a side effect of the positive weights prior. Mean variance analysis tells us that portfolio weights are functions of the mean variance ratio for each asset and the correlation matrix. The positive weights prior informs the relative magnitudes of the mean variance ratios as well as the correlation matrix, but does not directly restrict the absolute magnitudes of either the means of the variances. The upward shift in return means is not warranted by theory. Skepticism about negative portfolio weights is not usually associated with optimism with respect to return means. Investors are skeptical about negative portfolio weights because of noise in the relative returns of highly correlated assets that create opportunities for long-short trades that are not warranted by the data evidence. To correct the posterior bias of return means, I activate prior q3 that reflects skepticism that the grand mean of returns is larger than the grand mean of the sample returns. The fourth panel of Table 3.3 lists the posterior characteristics under the revised prior. The averages of the posterior mean returns are shrunk towards the grand mean in the sample, and the standard deviation of these quantities is reduced relative to the posteriors computed without the prior. The primary effect of the shrinkage of the return means is to reduce the overall investment in risky assets. The impact on the posterior distribution of implied portfolio weights is muted. Figure 3.3 shows the distribution of posterior returns when priors q1, q2, and q3 are incorporated into the model. A comparison with Figure 3.2 shows that prior q3 has a noticeable effect on the return distribution. The information in the prior acts to restrict the posterior distributions of return means. The distributions are shifted towards the grand mean of the sample, and the posterior distribution of the return means has visibly less variance. Incorporating prior q3 has an inconsequential effect on the precision of port- 106 folio weight estimates. Figure 3.4 compares histograms of portfolio weights obtained with no prior with those obtained under prior beliefs that include the no-arbitrage q1, positive-weights q2, and mean-restriction q3 priors. The his- tograms are similar to those shown in Figure 3.2 for the case under priors q1 and q2 only. In addition, differences in the averages and standard deviations of the portfolio weights listed in the third and fourth panels of Table 3.3 are inconsequential. Thus, while prior q3 mitigates return bias introduced by the positive weights prior, it has little impact on the precision of the final allocation estimates. 3.6.3 Monte-Carlo Study For each of the 250 simulated data sets, I compute the first two moments of the Bayesian predictive distribution of returns under different prior specifica- tions. The top and bottom panels of Table 3.4 list return means and standard deviations, respectively, averaged across simulations. The first and second lines of each panel show results computed for an investor with no prior and with prior that reflects belief in the positivity of portfolio weights, i.e., with prior q2 activated.67 Comparison of the two lines reveals the upward return bias asso- ciated with this prior. The mean predictive returns are at least one standard deviation higher than those computed without prior. While not significant in the conventional sense, the upward bias is economically significant. The bias represents an approximate 5% per annum upward shift in expected returns. The introduction of prior q3 or q4 eliminates the upward bias in return means. Results given these priors are listed in lines 3-5 of each panel of the table. Examination of the means of the return standard deviations computed for each simulation (reported in the second panel) reveals the limited impact of priors on second moments of the predictive return distribution. Some reduction in the variance of predictive distributions is expected given that the priors introduce additional information. However, when compared to the no-prior case, the reduction in the standard deviations are small (between 1% and 8%). Table 3.5 shows simulation averages of portfolio weights estimated under different priors. The first panel shows results estimated using the discrete- 67I include the no-arbitrage prior in all cases. 107 Bayesian model. These results illustrate the additional stability of portfolio weight estimates when priors are incorporated. For example, comparison of lines 1 and 3 of either panel reveals that a model that incorporates priors q1, q2 and q3 reduces standard deviations of portfolio weights across simulations by a factor of four. The second panel shows means of allocations for the portfolio resampling approach with prior. For each simulation, I calculate allocation estimates by averaging the optimal weights computed for each distribution drawn from the posterior by MCMC. Comparison of results listed in the first and second panels reveals very little difference in either the average or the variability of portfolio weights across simulations. Thus, despite being an ad hoc decision rule, portfolio weights computed by portfolio resampling are as stable as those obtained by solving the full Bayesian predictive problem when the posterior is based on the same priors. Table 3.6 shows out-of-sample performance given estimated portfolio weights averaged across simulations. Performance is reported as percent certainty equiv- alent return per month. As in Table 3.5, I report results for the discrete-Bayesian model and the portfolio resampling approach. The three columns summarize model performance for three sizes of data window: T ∈ {60, 120, 240}. The results demonstrate the difficulty of estimating portfolios given just 60 data points. The paucity of information in such a small data set is not sufficient to achieve a risk adjusted portfolio return that exceeds the risk-free rate whether or not the investor has prior information. The certainty equivalent returns in the first column are uniformly negative. For the longer data histories, the expected risk adjusted return is improved under the positive weight prior. Reading line 2 of Table 3.6, the monthly risk adjusted returns are -0.03 and 0.13 in the T = 120 and T = 240 cases respec- tively compared to −0.27 and 0.02 in the no-prior case. However, the upward return bias associated with the positive weights prior is costly. The investor’s out-of-sample utility is further improved if the prior also downweights distrib- utions with grand mean greater than that of the sample. For the T = 120 and T = 240 case, the positive returns are highest when prior q3 is implemented, as in lines 3 and 4 of either panel of Table 3.6. 108 3.6.4 Comparison of Discrete-Bayesian Policy with Other Models Next, I compare the portfolio policies and the expected performance of the discrete-Bayesian model to those of the allocation models listed in Table 3.1. Table 3.7 lists simulation summary statistics while Table 3.8 mirrors Table 3.6 in reporting expected out-of-sample performance of estimated policies. The first panel of Table 3.7 shows optimal policies of an investor with full knowledge of the first two moments of the underlying distribution. Needless to say, these policies are unachievable in practice. However their properties and performance are a useful reference when evaluating the performance of estimated optimal policies. As in Table 3.5, the average weight across assets is listed in column six. For the full information mean variance portfolio, the average weight assigned across assets is 0.15. This is essentially equal to the simulation mean of the average weights for each of the discrete-Bayesian models examined in Table 3.5 with the exception of the case with positive weights prior instituted without the prior against return bias. In the latter case, the average weight across assets is 5% higher, corresponding to a 25% greater allocation to risky assets. The second panel of Table 3.7 shows portfolio weights averaged across sim- ulations for eight plug-in rules. The last column show the simulation average of the allocation magnitudes averaged across assets. The difference between mean allocation magnitudes and mean weights reveals the importance of neg- ative weights in the portfolios suggested by the various policies. The first four policies are in decreasing order of the degree to which they account for estima- tion error by shrinking towards the zero portfolio. As expected, both the mean weight and mean weight magnitude decrease with degree of shrinkage. However, the mean policy weights obtained using the estimated two-fund rule of Kan and Zhou (2007) are substantially lower than those of the full information portfolio. Kan and Zhou (2007) provide an analytic foundation and simulation results that support the conclusion that their best two-fund and estimated two-fund rules will exhibit better expected out-of-sample performance. However, there is a clear upper bound on their ability to improve performance, because they do not alter the relative magnitudes of the estimated weights. Shrinkage to- wards the zero portfolio results in smaller negative weights only insofar as the 109 magnitude of all weights are reduced. The estimated three-fund rule is an optimal combination of the expected mean variance portfolio, the zero portfolio, and the estimated minimum variance portfolio. This rule allows for shrinkage in the relative magnitudes of portfolio weights by including the estimated minimum variance portfolio as an additional shrinkage target. This effectively allows for some rotation of the covariance ma- trix and reduction in the relative magnitudes of negative weights. The contrast is best illustrated by comparing the results for the best two-fund and estimated three-fund portfolios (lines 3 and 5 of the second panel of Table 3.7). The aver- age weights for these two portfolios are equal, but the average weight magnitude is 25% lower in the estimated three-fund portfolio. The estimated three-fund policy allocates less to negative positions on average. However, having the min- imum variance portfolio as a target portfolio does have disadvantages. The average allocation to the fifth asset is negative and the average allocation to the second asset is well above that of the full information mean variance portfolio. This bias is attributable to the fact that the full information global minimum variance portfolio in this example places extreme weights on these two assets; -0.45 on asset 5 and 0.83 on asset 2, relative to the true mean variance optimal portfolio. Table 3.8 lists expected out-of-sample performance of the alternative policies for data windows of length 60, 120, and 240. Results for the plug-in rules are listed in the second panel. The expected out-of-sample performance improves with horizon for all policies (except the mechanical 1/N rule). The performance results for the first five mean-variance based rules reflect the conclusions of Kan and Zhou (2007). The two and three-fund policies pro- posed by Kan and Zhou (2007) show better performance than the classical mean- variance and diffuse Bayesian policies, with the estimated three-fund portfolio showing best performance. The significance of the performance improvement decreases with horizon. None of the policies has expected out-of-sample returns greater than the risk-free rate for the 60 period case. The performance of the estimated three-fund policy is essentially equal to that of the mean variance policy with short sale constraints. The expected out-of-sample performance of the discrete-Bayesian model with the no bias positive weights prior is higher than that of the estimated 110 three-fund rule of Kan and Zhou (2007). The difference in performance is ob- tained by comparing line 3 of Table 3.6 with the results in Table 3.8. For a 120 period horizon, the improvement is seven basis points, dropping to three basis points at the 240 period horizon. The expected performance improvement is small and insignificant relative to the performance standard errors. The in- crease in expected risk adjusted return is approximately one third of its standard error at both the 60 and 120 period horizons. Based on the improvement in out-of-sample performance alone, the discrete-Bayesian approach is not over- whelmingly superior to the best mean-variance based models of Kan and Zhou (2007). Stability of estimated weights is also an important consideration. Here, the discrete-Bayesian model is a significant improvement over the estimated two- fund and three-fund rules of Kan and Zhou (2007). The standard deviations of the estimated weights are two to three times greater for the mean variance based policies than they are for the discrete-Bayesian model. Even the mean variance policy with short sales constraints imposed exhibits standard errors in its weight estimates that are approximately one and a half times the standard deviation of the discrete-Bayesian weights. The last two lines of the second panel of Table 3.8 show expected out-of- sample utility for the minimum variance and 1/N policies. Both of these policies exhibit expected out-of-sample performance that is marginally higher than that of the discrete-Bayesian model. The minimum variance and 1/N policies miti- gate the impact of estimation error by ignoring estimated return means. This approach trades off a major source of estimation error (because return means are difficult to estimate precisely given the high variance of asset returns) for estimator bias. The bias arises because the portfolio weights obtained by min- imum variance do not converge to the true optimal weights as sample length approaches infinity. The difference between the realized risk-adjusted return of the full information portfolio without bias and the full information portfolios under the two biased estimates serves as a measure of the potential lost opportu- nity. For the example problem considered here, an investor gives up twenty-nine or twenty-seven basis points in the full information limit, depending on whether they employ the minimum variance or 1/N approaches. Despite the potential costs, the biased estimators dominate the unbiased 111 mean variance estimators in the five-asset example. The convergence of the ex- pected out-of-sample utility to the full information value is significantly slower for the unbiased mean variance estimators. In the five-asset example, the ex- pected out-of-sample utility of the minimum variance strategy reaches its limit given a sample length of 120, and is not significantly below the full information limit with a sample length as low as 60. Of course, the expected out-of-sample utility of the 1/N strategy is independent of sample size because the data play no role in setting the portfolio allocations. Even for sample sizes as high as T = 240, the expected out-of-sample utility of the mean variance estimators included in Table 3.8 does not surpass the values for the minimum variance and 1/N portfolios. The discrete-Bayesian portfolio does better, but only matches the expected out-of-sample performance of the minimum variance portfolio for sample length T = 240. The last panel of Tables 3.7 and 3.8 shows portfolio results and expected out- of-sample utility for the portfolio resampling model with short sales restrictions. This model differs from the portfolio resampling models listed in Table 3.6. In both cases, a final portfolio estimate is obtained by averaging over a larger number of estimated portfolios. 3.7 Out-of-Sample Study In this section, I examine the out-of-sample performance of the discrete-Bayesian portfolio estimates described in section 3.3. I consider the performance of in- vestments in three asset baskets. The data set for each basket consists of a set of excess returns over the risk-free rate. The first two data sets are monthly returns of five and ten-industry portfolios for the period January 1952 to De- cember 2006. The industry definitions and data are from the web site of Ken French. The third consists of international returns on four country indices from Morgan Stanley Capital International for the period January 1975 to December 2006. The study design is based on a rolling window framework. The investor is assumed to re-balance their portfolio based on data for the previous T months, where T is either 120 or 240. The rolling window setup is rather arbitrary as 112 it sets an arbitrary date-based cutoff on the information in historical returns. My main objective with the out-of-sample experiment design is to maintain a connection with the fixed-window simulation analysis, and to ensure that there is no look-ahead bias. I compute certainty-equivalent returns for the discrete-Bayesian policy, the resampling policy with prior, as well as seven policies of the policies from the literature listed in Table 3.1. Table 3.9 lists the out-of-sample risk-adjusted returns for the two policies based on draws from the discrete-Bayesian posterior, and Table 3.10 lists parallel results for the policies from the literature. The first and second panels of Tables 3.9 show results for the T = 120 and T = 240 estimation window respectively. For the T = 120 case, the portfolio policy never achieves positive ex-post risk adjusted returns. The five-industry results (column 1) are poor when compared to the results obtained for the simulated case. In the simulation experiment, the expected out-of-sample utility is significantly greater than the realized results over the four time periods listed in Table 3.9. Even with the estimation window expanded to 240 months, the realized risk-adjusted returns are not significantly different from zero (second panel of Table 3.9). The risk-adjusted returns listed in Table 3.9 can be compared with those obtained for suggested approaches from the previous literature listed in Table 3.10. The discrete-Bayesian policy outperforms the mean variance policy68, but does not do as well as the statistically motivated policies of Kan and Zhou (2007). Both the estimated two-fund and the estimated three-fund policies outperform the discrete-Bayesian policy. Consistent with results in DeMiguel et al. (2007), the best performing poli- cies are the minimum variance and 1/N policies. The outperformance of the minimum variance portfolio is an indicator that historical returns data yields exploitable information on return correlations. However, there is a cost to this outperformance as average position magnitudes are approximately twice the size of estimated for the models that constrain short sales (a set that includes the 1/N portfolio) as well as for the models listed in Table 3.9 that incorporate priors against short selling. 68The poor performance of standard mean variance is well known, and this poor perfor- mance extends to the mean variance method with diffuse prior (not reported). 113 The 1/N models and the policies that explicitly forbid short sales (mean variance with no short sales and the direct averaging of resampled weights with no short sales) introduce bias into the portfolio estimation problem. The 1/N policy achieves superior risk adjusted returns across asset sets and time pe- riods than the discrete-Bayesian model with informed prior. The short sales constrained policies achieve higher risk adjusted returns for most cases, but the improvement was not statistically significant. These models differ from the discrete-Bayesian policy in that the problems are formulated with exoge- nous constraints designed to smooth, or ‘regularize’, the variability of the port- folio weights vector. In contrast, the Bayesian approach imposes priors that restrict the space of possible distributions to weights that obey certain prop- erties. In each case, there is a tradeoff of expected bias and variance of the weight estimates. The better performance of the 1/N policy relative to the discrete-Bayesian policy demonstrates that costs of estimation error outweigh the smoothing bias for the three asset baskets during the time periods considered in this study. The out-of-sample tests attest to the usefulness of shrinkage approaches in portfolio forecasting. One could increase the smoothness of the informed Bayesian solution by increasing the confidence in the in prior. However, that approach implicitly conflates the problem of distribution estimation and the problem of selecting the best solution given the available information. 3.8 Conclusion This paper develops a simple and flexible modeling approach for incorporat- ing general priors into the portfolio estimation problem without requiring the specification of a parametric distribution family. I construct a nonparametric approximation to the predictive distribution of returns by restricting the set of possible return outcomes in candidate distributions under the posterior to returns realized in the sample. This allows me to sample from the posterior by MCMC. Given a resampled set of draws from the posterior, the estimator for the investor’s utility expectation reduces to a simple weighted sum over the sample outcomes. I apply the discrete-Bayesian estimator to an analysis of a portfolio estima- 114 tion problem under the prior belief that all portfolio weights are positive. In the simulation analysis, I examine the problem of allocating across five assets with asset correlations greater than 0.5. Such problems are particularly thorny for traditional mean variance analysis as estimation error in the returns can result in solutions that leverage similar assets against each other. In a simulation analysis, I examine the performance of the discrete-Bayesian model for a five asset problem given an investor with prior belief that positive weights are positive. I find that the discrete-Bayesian policy with informed prior performs on par with models that impose short sale constraints and un- derperforms the 1/N portfolio. Thus, despite introducing bias, smoothing type estimators such as the short-sales constrained and 1/N models are as useful to the investor as conditioning on a prior that is true. The benefits of regularizing are further demonstrated in an out-of-sample study on three historical asset sets. This paper does not extend the analysis of the portfolio allocation problem to the analysis of regularization of estimators in the presence of additional infor- mation. The analysis of regularization versus informed Bayesian and statistical estimators is a fertile and open area for future research. The implications of more general forms of regularization is, itself, a new area of research in the portfolio choice field. It will be interesting to address the question of whether portfolio forecasts can be improved by combining portfolio forecasts that incor- porate non-data information with regularization of the optimization problem. 115 Table 3.1: Portfolio Choice Models This table lists portfolio allocation models that are evaluated in the simulation and out of sample tests. Model Description Discrete Bayesian Models Bayesian Full Prior Portfolio estimated from empirical distribution with probability weights tuned to reflect prior beliefs. Bayesian techniques are used to condi- tion observation weights on priors. Bayesian No Prior Portfolio estimated assuming an investor with no prior beliefs. Plug-in Models Mean variance Portfolio estimated using sample estimates for mean returns and covariances. Mean variance with diffuse prior Mean variance model with estimation uncer- tainty accounted for via the introduction of dif- fuse priors for mean and variance. Best two fund Portfolio estimated using sample estimates for means and covariance matrix with fixed shrink- age coefficient chosen to optimize expected out of sample performance of the model. Estimated two fund Portfolio estimated using sample estimates for means and covariance matrix with shrinkage co- efficient estimated from data. Estimated three fund Portfolio estimated using sample estimates of means and covariance matrix. Portfolio formed by combining standard mean variance portfolio, minimum variance portfolio and risk-free asset with coefficients estimated from data. Mean variance - no short Mean variance portfolio with no-short-sales con- straint. Minimum variance Global mimimum variance portfolio. 1/N Portfolio formed by placing equal weights in each asset. Portfolio Resampling Models Averaged weights Portfolio formed by averaging over resampled weights. Each vector of resampled weights is computed by mean variance for using resampled data. Averaged weights - no short sales As above but with no-short-sales constraints ap- plied to each weights calculation. 116 Table 3.2: Simulation Parameters This table lists parameters of the multivariate normal distribution used to gen- erate return samples for the simulations. Asset: 1 2 3 4 5 Means 0.64 0.58 0.66 0.68 0.69 Standard Deviations 4.44 4.08 5.22 4.99 4.89 Correlation Matrix 1.00 0.81 0.73 0.70 0.88 0.81 1.00 0.71 0.68 0.87 0.73 0.71 1.00 0.63 0.73 0.70 0.68 0.63 1.00 0.70 0.88 0.87 0.73 0.70 1.00 Mean Var. Wts. (γ = 5) 0.21 0.25 0.08 0.22 0.01 117 Table 3.3: Posterior Distribution Given a Simulated Sample: Summary Statis- tics This table lists summary statistics for the posterior distribution of asset returns given a simulated data set of length T = 120 and prior information. The data set consists of asset returns for five assets and is generated from the multivariate normal distribution described in Table 2. The first panel shows summary statis- tics of the random return sequence. Subsequent panels show averages of return means and standard deviations as well as implied portfolio weights. The three distribution summary statistics are averaged over 12500 distributions drawn from the posterior. Each panel corresponds to a different prior specification. The second panel is the no prior case. In the next three panels, the prior co- efficients {c1, c2, c3} reflect investor belief that i) the global minimum variance portfolio has return greater than the risk-free asset, ii) optimal portfolio weights are positive, and iii) the grand mean of returns is not higher than the grand mean of the sample or 5% per annum - whichever is higher. 118 Table 3.3: (continued) Asset: Avg. 1 2 3 4 5 Sample Statistics mean 0.75 0.77 0.78 0.96 1.07 0.87 std. dev. 4.33 4.24 4.92 4.98 4.64 4.62 weight -0.96 -0.20 0.08 0.48 1.56 0.19 Prior: none mean 0.71 0.73 0.76 0.94 1.01 0.83 [0.29] [0.27] [0.30] [0.31] [0.30] std. dev. 4.28 4.17 4.87 4.89 4.60 4.56 [0.19] [0.18] [0.22] [0.23] [0.21] weight -0.96 -0.22 0.12 0.53 1.52 0.19 [0.70] [0.65] [0.44] [0.37] [0.75] Prior: c1 = Yes c2 = 20 c3 = 0 mean 1.13 1.09 1.11 1.16 1.26 1.15 [0.24] [0.23] [0.24] [0.26] [0.28] std. dev. 4.24 4.17 4.76 4.85 4.60 4.52 [0.19] [0.16] [0.22] [0.19] [0.18] weight 0.17 0.20 0.26 0.33 0.52 0.30 [0.21] [0.23] [0.23] [0.27] [0.34] Prior: c1 = Yes c2 = 20 c3 = 100 mean 0.80 0.76 0.74 0.84 0.87 0.80 [0.10] [0.09] [0.11] [0.14] [0.11] std. dev. 4.20 4.11 4.66 4.86 4.50 4.47 [0.15] [0.20] [0.20] [0.19] [0.17] weight 0.15 0.18 0.15 0.27 0.32 0.21 [0.18] [0.19] [0.17] [0.22] [0.25] Prior: c1 = Yes c2 = 200 c3 = 100 mean 0.81 0.78 0.79 0.84 0.89 0.82 [0.08] [0.08] [0.11] [0.12] [0.10] std. dev. 4.16 4.06 4.66 4.82 4.50 4.44 [0.15] [0.19] [0.19] [0.17] [0.17] weight 0.16 0.19 0.21 0.24 0.32 0.22 [0.15] [0.17] [0.17] [0.20] [0.23] 119 Table 3.4: Monte Carlo Statistics for Predictive Distributions This table lists the average across simulations of summary statistics for the predictive distributions of returns for investors with different prior specifications. The first panel shows predictive return means averaged across simulations. The second panel shows return standard deviations. Simulation standard deviations are listed in square brackets. For each simulation, return series are drawn from the multivariate normal distribution specified in Table 2. The prior coefficients {c1, c2, c3} reflect investor beliefs that i) the global minimum variance portfolio has return greater than the risk-free asset, ii) optimal portfolio weights are positive, and iii) the grand mean of returns is not higher than the grand mean of the sample or 5% per annum - whichever is higher. Asset: AVG Prior Model 1 2 3 4 5 Average Return Means No prior 0.66 0.61 0.66 0.70 0.73 0.67 [0.39] [0.36] [0.46] [0.44] [0.43] c1 = Y c2 = 20 1.10 1.01 1.18 1.14 1.23 1.13 [0.28] [0.26] [0.30] [0.31] [0.31] c1 = Y c2 = 20 c3 = 100 0.67 0.61 0.72 0.70 0.75 0.69 [0.26] [0.24] [0.28] [0.29] [0.29] c1 = Y c2 = 200 c3 = 100 0.68 0.62 0.73 0.70 0.76 0.70 [0.26] [0.24] [0.29] [0.29] [0.30] c1 = Y c2 = 20 c3 = 0 c4 = 100 0.66 0.61 0.70 0.69 0.74 0.68 [0.35] [0.32] [0.39] [0.37] [0.39] Average Return Standard Deviations No prior 4.41 4.07 5.18 4.96 4.86 4.70 [0.28] [0.27] [0.35] [0.31] [0.30] c1 = Y c2 = 20 4.35 4.02 5.12 4.91 4.80 4.64 [0.29] [0.27] [0.36] [0.32] [0.32] c1 = Y c2 = 20 c3 = 100 4.25 3.92 5.01 4.81 4.68 4.53 [0.27] [0.26] [0.35] [0.31] [0.30] c1 = Y c2 = 200 c3 = 100 4.25 3.92 5.00 4.80 4.68 4.53 [0.28] [0.26] [0.35] [0.31] [0.31] c1 = Y c2 = 20 c3 = 0 c4 = 100 4.14 3.83 4.92 4.71 4.56 4.43 [0.27] [0.26] [0.35] [0.31] [0.29] 120 Table 3.5: Monte Carlo Averages of Estimated Allocations This table lists estimated allocations averaged across return simulations. Re- sults are displayed for different prior specifications. For each simulation, T = 120 returns are drawn from the multivariate normal distribution specified in Table 2. A risk aversion coefficient of five is assumed for the portfolio calcu- lations. The first panel shows allocations estimated using the discrete Bayesian model. The portfolio weights are the mean variance optimal weights given the estimated predictive distribution of returns. The second panel shows allocations estimated by averaging portfolio weights implied by a set of distributions sam- pled from the posterior. Mean variance optimal allocations are computed for each distribution in the investor’s posterior draw, and allocations are set to the average of these allocations. Standard deviations across simulations are listed in square brackets. The prior depends on three parameters. c1 indicates investor belief that the global minimum variance portfolio has higher return than the risk-free rate. c2 determines the strength of investor belief in the positivity of portfolio weights. c3 determines the strength of the investor’s prior view that the grand mean of the asset returns does not exceed that of the sample. 121 Table 3.5: (Continued) Asset Avg. Avg. Prior Model 1 2 3 4 5 w |w| Discrete Bayesian No prior 0.21 0.33 0.03 0.22 0.06 0.14 0.67 [0.92] [1.04] [0.64] [0.61] [1.06] c1 = Y c2 = 20 0.29 0.34 0.22 0.30 0.23 0.23 0.28 [0.23] [0.25] [0.21] [0.27] [0.19] c1 = Y c2 = 20 0.18 0.22 0.14 0.20 0.14 0.15 0.18 c3 = 100 [0.15] [0.16] [0.13] [0.19] [0.13] c1 = Y c2 = 200 0.18 0.20 0.15 0.20 0.15 0.15 0.18 c3 = 100 [0.13] [0.13] [0.11] [0.17] [0.12] c1 = Y c2 = 20 0.20 0.26 0.13 0.22 0.15 0.16 0.20 c3 = 0 c4 = 100 [0.20] [0.27] [0.23] [0.27] [0.18] Portfolio Resampling No prior 0.23 0.34 0.03 0.22 0.05 0.14 0.70 [0.96] [1.08] [0.66] [0.62] [1.09] c1 = Y c2 = 20 0.29 0.34 0.22 0.30 0.23 0.23 0.28 [0.24] [0.26] [0.21] [0.28] [0.20] c1 = Y c2 = 20 0.19 0.22 0.14 0.20 0.15 0.15 0.18 c3 = 100 [0.15] [0.16] [0.14] [0.20] [0.13] c1 = Y c2 = 200 0.18 0.20 0.15 0.20 0.16 0.15 0.18 c3 = 100 [0.13] [0.13] [0.11] [0.17] [0.12] c1 = Y c2 = 20 0.20 0.26 0.13 0.22 0.15 0.16 0.20 c3 = 0 c4 = 100 [0.20] [0.29] [0.23] [0.32] [0.21] 122 Table 3.6: Monte-Carlo Average Out-of-Sample Utilities This table shows out-of-sample utility averaged across simulations. A risk aver- sion coefficient of five is assumed for the portfolio calculations. The first panel shows results for allocations estimated using the discrete Bayesian model. The portfolio weights are the mean variance optimal weights given the estimated predictive distribution of returns. The second panel shows results for alloca- tions estimated by portfolio resampling. Mean variance optimal alocations are computed for each distribution in the investor’s posterior draw, and allocations are set to the average of these allocations. Standard deviations across simula- tions are listed in square brackets. The prior depends on three parameters. c1 indicates investor belief that the global minimum variance portfolio has higher return than the risk-free rate. c2 determines the strength of investor belief in the positivity of portfolio weights. c3 determines the strength of the investor’s prior view that the grand mean of the asset returns does not exceed that of the sample. ŪOOS Prior Model T = 60 T = 120 T = 240 Discrete Bayesian No prior -1.01 -0.27 0.02 [1.01] [0.43] [0.17] c1 = Y c2 = 20 -0.03 0.13 [0.32] [0.15] c1 = Y c2 = 20 c3 = 100 -0.01 0.16 0.20 [0.57] [0.18] [0.09] c1 = Y c2 = 200 c3 = 100 0.16 0.20 [0.18] [0.09] c1 = Y c2 = 20 c3 = 0 c4 = 100 -0.28 0.08 0.16 [0.95] [0.30] [0.14] Portfolio Resampling No prior -1.16 -0.30 0.01 [1.12] [0.45] [0.18] c1 = Y c2 = 20 -0.04 0.13 [0.34] [0.15] c1 = Y c2 = 20 c3 = 100 -0.02 0.16 0.20 [0.60] [0.19] [0.09] c1 = Y c2 = 200 c3 = 100 0.16 0.20 [0.18] [0.09] c1 = Y c2 = 20 c3 = 0 c4 = 100 -0.31 0.06 0.15 [1.02] [0.37] [0.14] 123 Table 3.7: Monte-Carlo Average Portfolio Allocations for Alternative Portfolio Estimates This table lists means across simulations of investment weights computed using different allocation models. The data set consists of 120 monthly returns for all five assets in the simulated portfolio. Standard deviations are reported in square brackets. Asset Avg. Avg. Model 1 2 3 4 5 w |w| Full information Mean variance 0.21 0.25 0.08 0.22 0.01 0.15 Global minimum 0.40 0.83 0.05 0.18 -0.45 0.20 variance Plug in Mean variance 0.21 0.32 0.03 0.21 0.07 0.17 0.67 [0.92] [1.02] [0.63] [0.59] [1.04] Mean variance: 0.19 0.30 0.03 0.20 0.06 0.16 0.62 diffuse prior [0.86] [0.95] [0.59] [0.56] [0.97] Best two fund 0.19 0.29 0.02 0.19 0.06 0.15 0.60 [0.82] [0.91] [0.57] [0.53] [0.93] Est. two fund 0.08 0.14 0.01 0.09 0.04 0.07 0.27 [0.44] [0.50] [0.31] [0.29] [0.53] Est. three fund 0.25 0.49 0.04 0.15 -0.18 0.15 0.35 [0.41] [0.53] [0.27] [0.25] [0.50] Mean variance: 0.19 0.23 0.14 0.22 0.15 0.19 0.19 no short sales [0.33] [0.36] [0.24] [0.32] [0.31] Minimum variance 0.41 0.82 0.06 0.18 -0.46 0.20 0.39 [0.17] [0.17] [0.11] [0.10] [0.19] 1/N 0.20 0.20 0.20 0.20 0.20 0.20 0.20 [0.00] [0.00] [0.00] [0.00] [0.00] Portfolio resampling Resampling: 0.20 0.25 0.17 0.24 0.15 0.20 0.20 no short sales [0.29] [0.31] [0.21] [0.28] [0.26] 124 Table 3.8: Monte-Carlo Average Out-of-Sample Utility for Alternative Portfolio Estimates This table shows out-of-sample utility averaged across simulations for different portfolio selection models listed in Table 1. ŪOOS Model T = 60 T = 120 T = 240 Full information Mean variance 0.49 0.49 0.49 Global minimum variance 0.20 0.20 0.20 Plug in Mean variance -0.97 -0.25 0.03 [0.98] [0.42] [0.17] Mean variance diffuse prior -0.66 -0.19 0.04 [0.73] [0.36] [0.16] Best two fund -0.50 -0.15 0.05 [0.60] [0.33] [0.15] Estimated two fund -0.04 0.04 0.10 [0.25] [0.17] [0.09] Estimated three fund -0.05 0.09 0.17 [0.34] [0.19] [0.10] Mean variance - no short sales -0.13 0.09 0.16 [0.57] [0.21] [0.11] Minimum variance 0.18 0.20 0.20 [0.06] [0.06] [0.06] 1/N 0.22 0.22 0.22 [0.06] [0.06] [0.06] Portfolio resampling Resampling: no short sales -0.17 0.09 0.17 [0.67] [0.23] [0.11] 125 Table 3.9: Out-of-Sample Utility for Discrete-Bayesian Estimator This table shows out-of-sample utilities that would have been realized by an investor using the discrete-Bayesian model with informed prior to set their al- location strategy. The investor has prior belief in the positivity of the portfolio weights. The investor does not believe that the mean of the asset return means is higher than that observed in the return sample. The investor rebalances every three months. The investor’s data set at each investment month includes the previous T monthly returns. The three panels correspond to three data sets: the five- and ten-industry portfolios of Fama and French, and a portfolio of four-country indices from Morgan Stanley Capital International. T = 120 Five Ten Four Industries Industries Countries Jan1962-Jan2007 Nonparametric Bayes -0.47 -0.86 Resampling with prior -0.50 -1.06 Jan1972-Jan2007 Nonparametric Bayes -0.26 -0.48 Resampling with prior -0.21 -0.74 Jan1985-Dec2006 Nonparametric Bayes -0.22 -0.39 0.03 Resampling with prior -0.19 -0.49 -0.01 Jan1995-Dec2006 Nonparametric Bayes -0.23 -0.60 -0.20 Resampling with prior -0.30 -0.66 -0.33 126 Table 3.9: (Continued) T = 240 Five Ten Four Industries Industries Countries Jan1972-Jan2007 Nonparametric Bayes -0.12 -0.16 Resampling with prior -0.07 -0.22 Jan1985-Dec2006 Nonparametric Bayes 0.04 -0.00 Resampling with prior 0.10 -0.08 Jan1995-Dec2006 Nonparametric Bayes 0.04 0.08 0.03 Resampling with prior 0.10 0.05 0.07 127 Table 3.10: Out-of-Sample Utility for Alternative Models This table shows out-of-sample utilities that would have been realized by an investor using the discrete-Bayesian model with informed prior to set their al- location strategy. The investor has prior belief in the positivity of the portfolio weights. The investor does not believe that the mean of the asset return means is higher than that observed in the return sample. The investor rebalances every three months. The investor’s data set at each investment month includes the previous T monthly returns. The three panels correspond to three data sets: the five- and ten-industry portfolios of Fama and French, and a portfolio of four-country indices from Morgan Stanley Capital International. 128 Table 3.10: (Continued) T = 120 Five Ten Four Industries Industries Countries Jan1962-Jan2007 Mean variance -0.81 -1.68 Estimated two fund -0.20 -0.22 Estimated three fund -0.09 -0.06 Mean variance - no short sales -0.37 -0.60 Minimum variance 0.04 0.09 1/N 0.04 0.09 Average - no short sales -0.36 -0.62 Jan1972-Jan2007 Mean variance -0.72 -1.40 Estimated two fund -0.10 -0.07 Estimated three fund -0.09 -0.03 Mean variance - no short sales -0.19 -0.30 Minimum variance 0.16 0.25 1/N 0.07 0.13 Average - no short sales -0.19 -0.31 Jan1985-Dec2006 Mean variance -0.35 -1.15 -0.19 Estimated two fund -0.03 -0.04 -0.01 Estimated three fund -0.00 0.03 0.01 Mean variance - no short sales -0.07 -0.21 -0.06 Minimum variance 0.35 0.39 0.35 1/N 0.34 0.36 0.29 Average - no short sales -0.08 -0.24 -0.08 Jan1995-Dec2006 Mean variance -0.40 -1.06 -0.30 Estimated two fund -0.08 -0.08 -0.09 Estimated three fund -0.07 -0.00 -0.03 Mean variance - no short sales -0.20 -0.45 -0.19 Minimum variance 0.34 0.27 0.24 1/N 0.40 0.41 0.11 Average - no short sales -0.22 -0.47 -0.22 129 Table 3.10: (Continued) T = 240 Five Ten Four Industries Industries Countries Jan1972-Jan2007 Mean variance -0.30 -0.57 Estimated two fund -0.11 -0.10 Estimated three fund -0.10 -0.03 Mean variance - no short sales -0.10 -0.14 Minimum variance 0.17 0.27 1/N 0.07 0.13 Average - no short sales -0.07 -0.08 Jan1985-Dec2006 Mean variance 0.06 -0.35 Estimated two fund 0.03 -0.01 Estimated three fund 0.02 0.05 Mean variance - no short sales 0.16 0.13 Minimum variance 0.43 0.44 1/N 0.34 0.36 Average - no short sales 0.19 0.20 Jan1995-Dec2006 Mean variance -0.00 -0.48 0.07 Estimated two fund 0.03 -0.05 0.08 Estimated three fund 0.00 0.05 0.06 Mean variance - no short sales 0.20 0.07 0.05 Minimum variance 0.52 0.35 0.16 1/N 0.40 0.41 0.11 Average - no short sales 0.24 0.19 0.05 130 0.5 1 1.5 2 0 200 400 600 800 0.5 1 1.5 2 0 200 400 600 800 0.5 1 1.5 2 0 200 400 600 800 0.5 1 1.5 2 0 200 400 600 0.5 1 1.5 2 0 200 400 600 0 50 100 150 200 250 0 100 200 300 0 50 100 150 200 0 50 100 150 200 0 50 100 150 200 250 a) Asset 1 b) Asset 2 c) Asset 3 d) Asset 4 e) Asset 5 Figure 3.1: Histogram of Posterior Returns for a Simulated Data Set This figure shows histograms of mean returns across 12500 Markov chain Monte Carlo draws from the posterior distribution of an informed investor. The return data is a simulated series of returns of length T = 120 from a five dimensional multivariate normal distribution. The parameters of the underlying distribution are chosen to reflect historical values for the five-industries data set of Fama and French. The investor has a prior that reflects a belief in the positivity of returns. The investors prior coefficients are c1 = 20, c2 = 0, and c3 = 0. The line plots are probability density functions of the returns of the generating distribution (dotted) and the maximum likelihood normal distribution given the sample (solid). 131 −2 −1 0 1 2 0 100 200 300 −2 −1 0 1 2 0 100 200 300 −2 −1 0 1 2 0 100 200 300 −2 −1 0 1 2 0 100 200 300 −2 −1 0 1 2 0 100 200 300 a) Asset 1 b) Asset 2 c) Asset 3 d) Asset 4 e) Asset 5 Figure 3.2: Histogram of Posterior Weights This figure shows histograms of optimal portfolio weights implied by 12500 distributions drawn from the posterior distribution with prior belief in the pos- itivity of portfolio weights. The return data is a simulated series of returns of length T = 120 from a five dimensional multivariate normal distribution. The parameters of the underlying distribution are chosen to reflect historical values for the five industries data set of Fama and French. The posterior is sampled via a Markov chain Monte Carlo algorithm. The investor has a prior that re- flects a belief in the positivity of returns. The investor’s prior coefficients are c1 = 1, c2 = 20, and c3 = 0. The rust colored histograms in the background depict histograms of optimal portfolio weights under the no-arbitrage prior ony (c1 = 1, c2 = 0, and c3 = 0). 132 0.4 0.6 0.8 1 1.2 1.4 0 500 1000 1500 2000 0.4 0.6 0.8 1 1.2 1.4 0 500 1000 1500 2000 0.4 0.6 0.8 1 1.2 1.4 0 500 1000 1500 0.4 0.6 0.8 1 1.2 1.4 0 200 400 600 800 0.4 0.6 0.8 1 1.2 1.4 0 500 1000 1500 0 50 100 150 200 250 0 100 200 300 0 50 100 150 200 0 50 100 150 200 0 50 100 150 200 250 a) Asset 1 b) Asset 2 c) Asset 3 d) Asset 4 e) Asset 5 Figure 3.3: Histogram of Posterior Returns with Prior Against Return Bias This figure shows histograms of mean returns across 12500 Markov chain Monte Carlo draws from the posterior distribution of an informed investor. The return data is a simulated series of returns of length T = 120 from a five dimensional multivariate normal distribution. The parameters of the underlying distribution are chosen to reflect historical values for the five industries data set of Fama and French. The investor has a prior that reflects a belief in the positivity of returns. The investors prior coefficients are c1 = 1, c2 = 20, and c3 = 100. The line plots is the maximum-likelihood normal distribution given the sample. 133 −2 −1 0 1 2 0 100 200 300 −2 −1 0 1 2 0 100 200 300 −2 −1 0 1 2 0 100 200 300 −2 −1 0 1 2 0 100 200 300 −2 −1 0 1 2 0 100 200 300 a) Asset 1 b) Asset 2 c) Asset 3 d) Asset 4 e) Asset 5 Figure 3.4: Histogram of Posterior Weights with Prior Against Return Bias This figure shows histograms of weights implied by distributions drawn from the posterior. 12500 Markov chain Monte Carlo draws from the posterior distribu- tion of an informed investor. The return data is a simulated series of returns of length T = 120 from a five dimensional multivariate normal distribution. The parameters of the underlying distribution are chosen to reflect historical values for the five-industries data set of Fama and French. The investor has a prior that reflects a belief in the positivity of returns. The investor’s prior coefficients are c1 = 1, c2 = 20, and c3 = 100. 134 −2 −1 0 1 2 0 100 200 300 −2 −1 0 1 2 0 100 200 300 −2 −1 0 1 2 0 100 200 300 −2 −1 0 1 2 0 100 200 300 −2 −1 0 1 2 0 100 200 300 a) Asset 1 b) Asset 2 c) Asset 3 d) Asset 4 e) Asset 5 Figure 3.5: Histogram of Posterior Weights with Strong Prior This figure shows histograms of weights implied by distributions drawn from the posterior. 12500 Markov chain Monte Carlo draws from the posterior distribu- tion of an informed investor. The return data is a simulated series of returns of length T = 120 from a five dimensional multivariate normal distribution. The parameters of the underlying distribution are chosen to reflect historical values for the five industries data set of Fama and French. The investor has a prior that reflects a belief in the positivity of returns. The investor’s prior coefficients are c1 = 200, c2 = 1000 and c3 = 0. 135 −0.1 −0.05 0 0.05 0.1 0.15 0 2 4 6 8 10 12 14 16 −0.1 −0.05 0 0.05 0.1 0.15 0 200 400 600 800 1000 1200 1400 1600 −0.1 −0.05 0 0.05 0.1 0.15 0 200 400 600 800 1000 1200 1400 1600 −0.1 −0.05 0 0.05 0.1 0.15 0 200 400 600 800 1000 1200 1400 1600 a) Empirical Distribution b) $c1 = 0$ $c2 = 20$ $c3 = 0$ c) $c1 = 0$ $c2 = 20$ $c3 = 0$ d) $c1 = 0$ $c2 = 200$ $c3 = 0$ Figure 3.6: Histogram of Predictive Return Distribution under Different Priors This figure shows histograms of predictive distributions under different prior assumptions for a simulated return series. The predictive distributions are based on observation weights computed by summing over 12500 distributions drawn from the discrete-Bayesian posterior. 136 Figure 3.7: Histogram of Asset Weights Across Simulations This figure shows histograms of estimated optimal allocations across a set of 250 simulated historical return series. The figures show an investor’s allocation to the first asset in a five asset problem. The returns are simulated from a multivariate normal distribution with mean and covariance parameters that are reflective of those of United States industry portfolios. The investor’s risk aversion parameter is 5. For the resampling methods, 12500 distributions are drawn by Markov Chain Monte Carlo from the set of discrete distributions whose domain is the sample returns. Plot (a) shows portfolio weights estimated by an informed Bayesian who has prior views against the presence of negative portfolio weights, and believes the average of the expected returns across assets is close to the average return across assets in the sample. The Bayesian computes expectations by averaging over resampled distributions that take into account their priors. Plot (b) shows portfolio estimated by an uninformed Bayesian. Plots (c) and (d) shows portfolio weights estimated by resampling. For plot(c), the investor uses the same set of distributions as the informed Bayesian, and computes portfolio weights for each draw with short sale constraints imposed, and then takes on average over the ’resampled’ weights. For plot(d), the investor uses the same set of distributions as the uninformed Bayesian. Plot (e) shows portfolio weights estimated by an investor who use sample values to proxy for expected returns and means, but imposes short sale constraints on the risky asset weights. 137 Figure 3.7 (continued) 0 0.2 0.4 0.6 0.8 1 1.20 20 40 60 80 −3 −2 −1 0 1 2 30 20 40 60 80 0 0.2 0.4 0.6 0.8 1 1.20 20 40 60 80 0 0.5 1 1.50 20 40 60 80 0 0.2 0.4 0.6 0.8 1 1.20 20 40 60 80 a) Informed Bayes b) Uninformed Bayes c) Portfolio Resampling with Prior d) Portfolio Resampling with Constraint e) Mean Variance with Short Sale Constraints 112 162 138 Bibliography Aı̈t-Sahalia, Y. and Brandt, M. (2001). Variable selection for portfolio choice. Journal of Finance, 56:1297–1351. Avramov, D. (2002). Stock return predictability and model uncertainty. Jour- nal of Financial Economics, 64:423–458. Barberis, N. (2000). Investing for the long run when returns are predictable. Journal of Finance, 55:225–264. Bawa, V. S., Brown, S., and Klein, R. (1979). Estimation Risk and Optimal Portfolio Choice. North Holland, Amsterdam. Best, M. J. and Grauer, R. R. (1991). On the sensitivity of mean-variance effi- cient portfolios to changes in asset means: Some analytical and computational results. Review of Financial Studies, 4:315–342. Brandt, M. (1999). Estimating portfolio and consumption choice: A condi- tional Euler equations approach. Journal of Finance, 54:1609–1645. Brandt, M. W. (2005). Portfolio choice problems. In Ait-Sahalia, Y. and Hansen, L. P., editors, Handbook of Financial Econometrics. Elsevier Science, Amsterdam. Braun, W. J. and Hall, P. (2001). Data sharpening fo nonparametric inference subject to constraints. Journal of Computational and Graphical Statistics, 10(4):786–806. Brown, S. J. (1976). Optimal Portfolio Choice under Uncertainty: A Bayesian Approach. PhD thesis, University of Chicago. Chevrier, T. and McCulloch, R. (2008). Using economic theory to build optimal portfolios. Working Paper. 139 Choi, E. and Hall, P. (1999). Data sharpening as a prelude to density estima- tion. Biometrika, 86:941–947. Choi, E., Hall, P., and Rousson, V. (2000). Data sharpening methods for bias reduction in nonparametric regression. Tha Annals of Statistics, 28(5):1339– 1355. Cremers, K. J. M. (2002). Stock return predictability: a Bayesian model selection perspective. Review of FInancial Studies, 15(4):1223–1249. DeMiguel, V., Garlappi, L., Nogales, F., and Uppal, R. (2008). Improving per- formance by constraining portfolio norms: A generalized approach to portfolio optimization. Working Paper. DeMiguel, V., Garlappi, L., and Uppal, R. (2007). Optimal versus naive diver- sification: How inefficient is the 1/n portfolio strategy? Review of Financial Studies, Forthcoming. Efron, B. (1979). Bootstrap methods: another look at the jackknife. Annals of Statistics, 7(1):1–26. Gamerman, D. and Lopes, H. F. (2006). Markov Chain Monte Carlo. Chapman and Hall, New York, second edition edition. Garlappi, L., Uppal, R., and Wang, T. (2007). Portfolio selection with para- meter and model uncertainty: A multi-prior approach. Review of Financial Studies, 20(1):41–81. Green, R. C. and Hollifield, B. (1992). When will mean-variance efficient portfolios be well diversified? Journal of Finance, 47(5):1785–1809. Hall, P. and Presnell, B. (1999). Intentionally biased bootstrap methods. Jour- nal of the Royal Statistical Society Series B, 61:143–158. Harvey, C. R., Liechty, J. C., Liechty, M. W., and Muller, P. (2003). Portfolio selection with higher moments. Working Paper. Herold, U. and Maurer, R. (2006). Portfolio choice and estimation risk. a comparison of bayesian and heuristic approaches. Astin Bulletin, 36(1):135– 160. 140 Jagannathan, R. and Ma, T. (2003). Risk reduction in large portfolios: Why imposing the wrong constraints helps. Journal of Finance, 58:1651–1684. Jobson, J. D. and Korkie, B. M. (1981). Performance hypothesis testing with the Sharpe and Treynor measures. Journal of Finance, 36(4):889–908. Jorion, P. (1986). Bayes-stein estimation for portfolio analysis. Journal of Financial and Quantitative Analysis, 21:279–292. Jorion, P. (1992). Portfolio optimization in practice. Financial Analysts Jour- nal, 48(1):68–74. Kacperczyk, M. (2003). Asset allocation under distribution uncertainty. Work- ing Paper. Kan, R. and Zhou, G. (2007). Portfolio choice with parameter uncertainty. Journal of Financial and Quantitative Analysis, 42(3):621–656. Kandel, S. and Stambaugh, R. F. (1996). On the predictability of stock returns: An asset allocation perspective. Journal of Finance, 51:385–424. Markowitz, H. (1952). Portfolio selection. Journal of Finance, 7:77–91. Markowitz, H. M. and Usmen, N. (2003). Resampled frontiers versus diffuse bayes: An experiment. Journal of Investment Management, 1(4):9–25. Meulli, A. (2006). Risk and Asset Allocation. Springer, Berlin-Heidelberg-New York. Michaud, R. O. (1998). Efficient Asset Management. Harvard Business School Press, Boston. Pástor, L̆. (2000). Portfolio selection and asset pricing models. Journal of Finance, 55:179–223. Pástor, L̆. and Stambaugh, R. (1999). Costs of equity capital and model mispricing. Journal of Finance, 54:67–121. Paye, B. (2004). Essays on Stock Return Predictability and Portfolio Allocation. PhD thesis, University of California San Diego. 141 Scherer, B. (2002). Portfolio resampling: Review and critique. Financial Analysts Journal, 58:98–109. Wachter, J. A. and Warusawitharana, M. (2005). Predictable returns and asset allocation: Should a skeptical investor time the market? Working Paper. Zellner, A. and Chetty, V. K. (1965). Prediction and decision problems in regression models from the bayesian point of view. Journal of the American Statistical Association, 60(310):608–616. 142 Chapter 4 Conclusion In this thesis, I examine whether supplementing historical returns data yields a significant improvement in risk-adjusted expected return for an investor. In Chapter 2, I find that return variability can severely limit expected gains from diversification and conditioning on predictor variables. In Chapter 3, I demon- strate that, even when granted a prior that is correct, an investor may have trouble attaining risk-adjusted out-of-sample return that matches that of biased estimators such as the 1/N model (DeMiguel et al. (2007)). In this chapter, I discuss my results in relation to recent research and describe some promising avenues for future research. The results of both Chapters 2 and Chapters 3 reflect the difficulty of es- timating optimal portfolios from limited information available for return fore- casting. Even with correct additional information, it is difficult to outperform biased estimates such as short-sales constrained, minimum-variance and 1/N portfolios. However, the results should not discourage an investor from incor- porating as much information as they have available into their portfolio choice problem. It is important to differentiate the underlying motivation and objective of the conditional estimation models of Chapters 2 and 3 and smoothing estima- tors such as the 1/N model. The Bayesian approach aims to obtain the best estimate possible of the underlying distribution of returns by supplementing the information provided by the data. In contrast, the 1/N model and models that impose short sales constraints are examples of smoothing or regularization techniques. Regularization tech- niques aim to improve the robustness of estimates by smoothing the optimiza- tion solution. This smoothing is achieved by adding functions that penalize solution variability to the Lagrangian of the optimization problem.69 Statis- 69Or, equivalently, placing constraints on the solution (see Jagannathan and Ma (2003)). Hence, short sales constraints are an example of a regularization technique. 143 ticians use smoothing techniques to deal with estimation problems for which data is limited or highly collinear. Portfolio allocation qualifies as just such a problem.70 Asset returns are often highly collinear. For example, DeMiguel et al. (2007) demonstrate that a return history of 250 years would be required to match the 1/N strategy for an example problem with 25 assets. A key distinguishing feature of Bayesian and regularization methods is that Bayesian estimators are consistent whereas regularization may be biased even in large sample. Also, the aim of Bayesian methods is to improve the distribution estimate, whereas regularization aims to improve the robustness of small sample estimates by constraining the solution vector. Recent research has focused on implementations based on one or the other of these approaches. For example, Chevrier and McCulloch (2008), Tu and Zhou (2008), and Chapter 3 of this thesis, consider economically motivated priors in a Bayesian context, whereas DeMiguel et al. (2008) focus on improving estimates by constraining norms of the vector of portfolio weights. This division between Bayesian portfolio estimates and those based on in- formed priors is artificial. The objective of the model should be to first obtain the best estimate of the underlying return distribution that is possible given the information at hand, and, second, to then apply an appropriate regularization if the effective sample size given the data and the other information remains too small to obtain consistently robust solutions. To illustrate, we can add a regularization term to the estimator first dis- cussed in Chapter 1 (see equation 1.2) and applied throughout this thesis. max w T∑ t=1 atU(r̃t, w)− κ||w||p, (4.1) where κ is a parameter that determines the strength of the regularization and p > 0 determines the norm.71 It seems likely that the optimal portfolio rule will be one that incorporates all the best available information plus some regularization. This suggests an interesting avenue for future research into both the impact of predictability and 70Optimal portfolio weights can be expresses as parameters of a least squares regression (Britten-Jones (1999)). 71A p-norm regularization is used for illustration. Other regularization functions are possi- ble. See equation 1.2 and surrounding text for definitions of other notation. 144 the importance of the positive-weights prior. Would out-of-sample performance be improved by an estimator of the form 4.1? On a related note, the estimation formulation 4.1 suggests an alternative means of assessing the impact of new information on portfolio estimates. The current paradigm for assessing a portfolio rule that incorporates new information is to compare its performance to various ad hoc regularized portfolios such as minimum variance, 1/N , or short-sales constrained. Instead, one might ask the more subtle question of whether incorporation of the information reduces the degree of regularization required to obtain an admissible solution, i.e., does the new information allow us to obtain equivalent or improved out-of-sample performance as the previous regularized solution but with a reduced weight on the regularization penalty function. Given the inherent instability of the portfolio problem, this softer criterion is likely to be a more suitable basis for inference when analyzing the economic significance of information beyond that provided by historical returns. 145 Bibliography Britten-Jones, M. (1999). The sampling error in estimates of mean-variance efficient portfolio weights. Journal of Finance, 54(2):655–671. Chevrier, T. and McCulloch, R. (2008). Using economic theory to build optimal portfolios. Working Paper. DeMiguel, V., Garlappi, L., Nogales, F., and Uppal, R. (2008). Improving per- formance by constraining portfolio norms: A generalized approach to portfolio optimization. Working Paper. DeMiguel, V., Garlappi, L., and Uppal, R. (2007). Optimal versus naive diver- sification: How inefficient is the 1/n portfolio strategy? Review of Financial Studies, Forthcoming. Jagannathan, R. and Ma, T. (2003). Risk reduction in large portfolios: Why imposing the wrong constraints helps. Journal of Finance, 58:1651–1684. Tu, J. and Zhou, G. (2008). Incorporating economic objectives into bayesian priors: Portfolio choice under parameter uncertainty. Working Paper. 146 Appendix A Bandwidth Specification There is considerable leeway in the specification of the weighting function and bandwidth used in the non-parametric estimator. However, the properties of the solution are much more sensitive to choice of bandwidth than they are to choice of weighting function72. I set the weighting function equal to the density function of the normal distribution73. Bandwidth choice is a classic example of the tradeoff between bias and vari- ance. If the bandwidth is very narrow then few observations receive any weight in the estimator for a given value z of the conditioning variable. This leads to high variance. As the bandwidth is broadened, variance of the estimator drops as more data points receive significant weights, but bias increases as points from states farther away from the z of interest gain more weight in the estimator. A standard approach to balancing this tradeoff is to minimize mean squared error of the estimates. Consistent with Brandt (1999) I consider bandwidths of the form hT = λσT ( −1K+4), (A.1) where σ is the unconditional variance of the state variable, and λ is a compli- cated function of unobservable functions of the moment conditions and condi- tioning variable. In practice, there are a number of methods of choosing λ, and none is ideal. For example Brandt (1999) uses leave-one-out cross validation. For every obser- vation zt, α(zt) is estimated both with and without including zt in the data set. The bandwidth parameter λ is selected that minimizes the sum of the squared differences between the two estimates. My experiments (not reported) indicate that the λ selected by leave-one-out cross validation tends to be overly conser- vative. Instead I make use of a rule of thumb proposed by Silverman (1986). 72See Hardle (1990) for a summary and discussion of the relative importance of weighting function and bandwidth to asymptotic convergence of nonparametric regression estimators. 73This is a special case of the product of normal densities used by Brandt (1999). 147 The standard deviation σ in the above formula is replaced by the interquartile range R, and bandwidths are set to hT = λRT −1/(K+1). I report results for λ = 0.79 which is consistent with Paye (2004). Recalculation of results of, for example, figures 1-11, for bandwidths within 25% does not significantly change the results. 148 Bibliography Brandt, M. (1999). Estimating portfolio and consumption choice: A condi- tional Euler equations approach. Journal of Finance, 54:1609–1645. Hardle, W. (1990). Applied Nonparametric Regression. Cambridge University Press, New York. Paye, B. (2004). Essays on Stock Return Predictability and Portfolio Allocation. PhD thesis, University of California San Diego. Silverman, B. W. (1986). Density Estimation for Statistics and Data Analysis. Chapman and Hall, New York. 149 Appendix B Data Tuning for a Portfolio Choice Problem This appendix presents two formulations that make use of data tuning to impose the restriction that all portfolio weights are positive. Assume an investor seeking to choose allocations to N assets plus a risk free asset. The standard investor’s problem is max w E [U(r̃, w)] , (B.1) where r̃ is a random return over the investment period and w is a vector of portfolio allocations. Suppose we wish to find a solution to the above problem under the constraint that all weights are positive. The standard approach is to impose the constraint w ≥ 0. Data tuning is an alternative approach that seeks to adjust the data set until the solution to the unconstrained problem satisfies the nonnegativity requirement for the portfolio allocations. Two approached have been proposed. The first is to shift the probabilities associated with each observation until the most likely multinomial distribution is found that satisfies the constraint. This approach is applied to bootstrap inference by Hall and Presnell (1999). The formulation is given by max T∑ t=1 log pt (B.2) 150 subject to w∗ = argmax T∑ t=1 ptU(rt, w) > 0. T∑ i=1 pt = 1 (B.3) The second approach is to perturb the data vector the minimum distance required to yield a weight vector that satisfies the required constraint. Choi and Hall (1999) and Choi et al. (2000) refer to this latter approach as data sharpening and provide analysis and applications. In this case, the perturbed data set S = [s1 : . . . : sT ] is found by solving min S D(R− S) (B.4) subject to w∗ = argmax T∑ t=1 U(st, w) > 0, where D(·) is a distance metric and R = [r1 : . . . : rT ] is the N × T matrix of return data. 151 Bibliography Choi, E. and Hall, P. (1999). Data sharpening as a prelude to density estima- tion. Biometrika, 86:941–947. Choi, E., Hall, P., and Rousson, V. (2000). Data sharpening methods for bias reduction in nonparametric regression. Tha Annals of Statistics, 28(5):1339– 1355. Hall, P. and Presnell, B. (1999). Intentionally biased bootstrap methods. Jour- nal of the Royal Statistical Society Series B, 61:143–158. 152 Appendix C Sampling from the Posterior I sample from the posterior (3.8) using an implementation of the Metropolis Hastings algorithm.74 As described in Gamerman and Lopes (2006), an arbi- trary posterior can be sampled by drawing from a Markov chain that has the posterior as a limiting distribution. The Metropolis Hastings is a means of con- structing such a chain for arbitrary posteriors. The Markov chain is generated by stepping through the parameter space. At each step a proposal distribution is drawn from a transition kernel pi(ξ|δ). The chain steps to the proposed draw with probability given by the acceptance level a(ξ, δ) = min { 1, qpost(δ|X)pi(ξ|δ) qpost(ξ|X)pi(δ|ξ) } . (C.1) If the transition is rejected, the chain remains at the current distribution δ until the next step. The transition kernel must allow for efficient spanning of a large dimensional parameter space, and be computationally feasible. I construct such a kernel as a mixture of a discrete uniform distribution over the set of observation indices {1, . . . , T} and a Beta distribution. The discrete draw i UNIFORM({1, . . . , T}) identifies an individual pi. The proposal is then formed by drawing a new value of for pi from a Beta distribution, and adjusting the remaining probabilities in δ such that the summation condition is satisfied. The transition kernel has some similarity to sequential transition schemes that update individual parameter either one at a time or in blocks. However, while a single pi the focus of the proposed jump, all the remaining components of the probability vector are also altered to satisfy the summation condition. As such, the kernel describes a standard Metropolis Hastings algorithm. Let α and β be the parameters of the Beta distribution. The parameters 74The high dimensionality of the parameter space renders more traditional direct sampling approaches computationally infeasible. 153 are set to achieve a distribution whose mode is equal to the current pi. This is achieved by setting α = 1 + pi/v and β = 1 − pi/v + 1/v. When α > 0 and β > 0, the mode of the β distribution is equal to (α− 1)/(α+ β − 2) = pi. The variance of the Beta distribution is also a function of its parameters, αβ (α + β)2(α+ β + 1) = ( 1 + pi v ) ( 1 + pi v + 1 v )( 2 + 1 v ) ( 3 + 1 v ) ≈ v (C.2) where the approximate equality holds as long as both v and pi are small. The latter condition is likely to hold if T is large since, in that case, any one pi is likely to be small. The choice of v determines the average size of jumps in the pa- rameter space and, by extension, the acceptance rate for proposal distributions. The acceptance rate must be small enough to ensure that the Markov chain does not get stuck at a single point, while at the same time the jumps must be large enough to ensure that the chain visits spans enough of the parameter space to ensure convergence in a reasonable number of iterations. Following recommen- dations in Gamerman and Lopes (2006), I choose v to obtain an acceptance rate between 0.35 and 0.5. I use a standard sampling scheme. I generate a small number of Markov chains m. I sample every k step of each chain until I accumulate 2n samples from each chain. I drop samples from the first half of each chain to allow convergence of the chain to the limiting distribution. The end result is nm draws that can be used for inference based on the posterior and for computing the predictive distribution. I list pseudocode for the Markov chain Monte Carlo implementation in Algorithm 1. Using convergence diagnostics described by Gamerman and Lopes [2006, p.196-7], I find that convergence is achieved for the portfolio choice problems considered in Section 3.6 with m = 5 and n = 2500. 154 for c = 1 : # of chains do Select a random start distribution δ0 = {p01, . . . , p0T} ; b = 0; for n = 1 : kb do Select a random index i ∈ {1, . . . , T} ; Draw p∗i ∼ Beta(²n−1i , 1 + ²n−1i , 1− ²n−1i + ²n−1i /pn−1i ) where ²si = p s i/v ; Compute m = 1−p∗i 1−pn−1i ; Set p∗−i = p n−1 −i m ; Compute posterior at proposal: qpost(δ ∗) ; Compute acceptance probability r = qpost(δ ∗) qpost(δn−1) = mT−1 p∗i pn−1i qpost(δ ∗) qpost(δn−1) τ(i, n− 1, ∗) τ(i, ∗, n− 1) where τ(i, s, l) = Betapdf(pli, 1 + ² s i , 1− ²si + ²si/psi) ; Draw u ∼ U(0, 1) ; if u < r then δb = δ∗ end else δn = δn−1 end if (n mod k) = 0 then b = b + 1; Store δb = δn end end end Algorithm 1: MCMC Algorithm for Drawing from the Discrete Posterior Distribution The above lists the Markov Chain Monte Carlo algorithm for drawing from the posterior distribution of the informed Bayesian investor. The domain of the pos- terior encompasses discrete distributions that assign positive probabilities to T return outcomes. The notation is as follows. Beta(α, β) is the beta distribution with parameters α and β and Betapdf(x, α, β) is its associated probability dis- tribution function. δn is the vector of posterior parameters {pn1 , . . . , pni , . . . , pnT} indexed by n. pn−i is the set of parameters of δ n with pi excluded. b is total number of draws to be saved, and k is the frequency that the chain is sampled. 155 Bibliography Gamerman, D. and Lopes, H. F. (2006). Markov Chain Monte Carlo. Chapman and Hall, New York, second edition edition. 156
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Nonparametric portfolio estimation and asset allocation Douglass, Julian James 2009
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Title | Nonparametric portfolio estimation and asset allocation |
Creator |
Douglass, Julian James |
Publisher | University of British Columbia |
Date Issued | 2009 |
Description | This thesis comprises two essays that apply nonparametric methods to the estimation of portfolio allocations. In the first essay, I test the significance to investor welfare of (i) adding additional assets to the portfolio choice set and (ii) conditioning on predictor variables. I estimate unconditional and conditional optimal allocations of a constant relative risk aversion investor by maximizing a nonparametric approximation of the expected utility integral. Investors can improve their expected utility significantly over that of an equities and cash investor by adding portfolios based on the value or momentum premiums into their asset allocation decision. In contrast, neither a size premium portfolio nor a long-term bond portfolio improves expected utility. The significance of predictability is increased by simultaneously conditioning on the two strongest predictors (of eight) studied: the term spread and the gold industry trend. In the second essay, I formulate a nonparametric estimator that permits combining historical data with a qualitative prior. I investigate the impact of an investor belief, motivated by asset-pricing theory, that optimal allocations are positive. In the estimator construction, I use a Bayesian approach to perturb the probabilities associated with each data point in the empirical distribution to reflect qualitative prior beliefs. In a simulation study and in out-of-sample tests, I find that portfolio estimates conditioned on a belief in the positivity of portfolio weights are significantly more stable than those estimated by an uninformed investor, and that the model performs better in out-of-sample tests than a number of plug-in models. However, the out-of-sample performance lags that of the minimum-variance and 1/N policies. |
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Language | eng |
Date Available | 2009-03-03 |
Provider | Vancouver : University of British Columbia Library |
Rights | Attribution-NonCommercial-NoDerivatives 4.0 International |
DOI | 10.14288/1.0067008 |
URI | http://hdl.handle.net/2429/5414 |
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Doctor of Philosophy - PhD |
Program |
Business Administration |
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Business, Sauder School of |
Degree Grantor | University of British Columbia |
GraduationDate | 2009-05 |
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