Portfolio Selection and Optimization with Higher Moments: Evidence from the Indian Stock Market

The Markowitz portfolio optimization model, popularly known as the Mean-Variance model, assumes that stockreturns follow normal distribution. But when stock returns do not follow normal distribution, this model wouldbe inadequate as it would prescribe sub-optimal portfolios. Stock market literature often deliberates that stock returns are non-normal. In such context the Markowitz model would not be sufficient to estimate the portfolio risks. The purpose of this paper is to expand the original Markowitz portfolio theory (mean-variance) via adding the higher order moments like skewness (third moment about the mean) and kurtosis (fourth moment about the mean) in the return characteristics. The research paper investigates the impact of including higher moments using multi-objective programming model for portfolio stock selection and optimization. The empirical results indicate that the inclusion of higher moments had a considerable impact in estimating the returns behavior of portfolios. The portfolios optimized using all the four moments, generated higher returns for the given level of risk in comparison to the returns of the Markowitz model during the study period 2000–2011. The results of this study would be immensely useful to fund managers, portfolio managers and investors as it would help them in understanding the Indian stock market behavior better and also in selecting alternative portfolio selection models.     

Deciphering robust portfolios

Robust portfolio optimization has been developed to resolve the high sensitivity to inputs of the Markowitz mean–variance model. Although much effort has been put into forming robust portfolios, there have not been many attempts to analyze the characteristics of portfolios formed from robust optimization. We investigate the behavior of robust portfolios by analytically describing how robustness leads to higher dependency on factor movements. Focusing on the robust formulation with an ellipsoidal uncertainty set for expected returns, we show that as the robustness of a portfolio increases, its optimal weights approach the portfolio with variance that is maximally explained by factors.     

MACROECONOMIC VARIABLES AND SEASONAL MEAN REVERSION IN STOCK RETURNS

In this paper I investigate whether seasonal mean reversion in stock portfolio returns is related to common macroeconomic risk factors. I decompose excess returns into explained and unexplained returns using a multifactor pricing model. The explained excess returns exhibit January mean reversion; the unexplained excess returns do not. The mean reversion can be attributed to the components of return related to unexpected inflation, bond default premium, and market risk. The results do not depend on the time-series properties of the portfolio betas. Bond default premia and excess market returns are mean reverting in January.     

Do Beta Pricing Models Explain January Mean Reversion in Stock Returns?

Jegadeesh (1991) finds evidence of January mean reversion in stock returns. In this paper we attempt to distinguish between two competing economic explanations of January mean reversion in returns: (1) mispricing in irrational markets versus (2) predictable time variation in security risk premia. Excess portfolio returns are decomposed into “explained” and “unexplained” components using the Fama-French (1993) pricing model. The explained excess returns exhibit January mean reversion. The unexplained excess returns are not mean reverting. Mean reversion is therefore consistent with rational pricing in the framework of the Fama-French model. Mean reversion can be attributed to the component of return related to a relative distress factor (SMB). A comparison with the Chen, Roll, and Ross (1986) macroeconomic factors reveals that mean reversion is due to the components related to SMB and bond default premium.     

Multivariate Shrinkage for Optimal Portfolio Weights

Abstract

This paper proposes a multivariate shrinkage estimator for the optimal portfolio weights. The estimated classical Markowitz weights are shrunk to the deterministic target portfolio weights. Assuming log asset returns to be i.i.d. Gaussian, explicit solutions are derived for the optimal shrinkage factors. The properties of the estimated shrinkage weights are investigated both analytically and using Monte Carlo simulations. The empirical study compares the competing portfolio selection approaches. Both simulation and empirical studies show that the proposed shrinkage estimator is robust and provides significant gains to the investor compared to benchmark procedures.     

Dynamic Investment Strategy with Factor Models Under Regime Switches

A model for dynamic investment strategy is developed where assets’ returns are represented by multiple factors. In a mean–variance framework with factor models under regime switches, we derive a semi-analytic solution for the optimal portfolio with transaction costs. Due to the existence of transaction costs, the optimal portfolio is characterized as a linear combination of current and target portfolios, the latter of which maximizes the value function in the current regime. For some special cases of interest, we also derive simplified analytical solutions. To see the effect of regime switches, the proposed model is applied to US equity market in which small minus big and high minus low are employed as factors. Investment strategy based on our model demonstrates empirically that the regime switching models exhibit superior performance over the single regime model for such performance measures as realized utility and Sharpe ratio which are of particular interest in practice. Taking a close look at the time series of portfolio returns, the result shows the usefulness of the regime switching model as investors flexibly optimize asset allocations depending on the state of the market.     

Focusing on the worst state for robust investing

Despite its shortcomings, the Markowitz model remains the norm for asset allocation and portfolio construction. A major issue involves sensitivity of the model's solution to its input parameters. The prevailing approach employed by practitioners to overcome this problem is to use worst-case optimization. Generally, these methods have been adopted without incorporating equity market behavior and we believe that an analysis is necessary. Therefore, in this paper, we present the importance of market information during the worst state for achieving robust performance. We focus on the equity market and find that the optimal portfolio in a market with multiple states is the portfolio with robust returns and observe that focusing on the worst market state provides robust returns. Furthermore, we propose alternative robust approaches that emphasize returns during market downside periods without solving worst-case optimization problems. Through our analyses, we demonstrate the value of focusing on the worst market state and as a result find support for the value of worst-case optimization for achieving portfolio robustness.     

Portfolio selection using hierarchical Bayesian analysis and MCMC methods

This paper contributes to portfolio selection methodology using a Bayesian forecast of the distribution of returns by stochastic approximation. New hierarchical priors on the mean vector and covariance matrix of returns are derived and implemented. Comparison’s between this approach and other Bayesian methods are studied with simulations on 25 years of historical data on global stock indices. It is demonstrated that a fully hierarchical Bayes procedure produces promising results warranting more study. We carried out a numerical optimization procedure to maximize expected utility using the MCMC (Monte Carlo Markov Chain) samples from the posterior predictive distribution. This model resulted in an extra 1.5 percentage points per year in additional portfolio performance (on top of the Hierarchical Bayes model to estimate μ and Σ and use the Markowitz model), which is quite a significant empirical result. This approach applies to a large class of utility functions and models for market returns.     

MODELS OF STOCK MARKET PREDICTABILITY

I briefly review the success of past studies purporting to explain equity valuations and predict future equity returns. The Campbell‐Shiller mean reversion models are contrasted with an expanded version of the so‐called Federal Reserve model. At least from 1970 to 2003, Federal Reserve–type models did somewhat better at predicting long‐horizon returns than did a mean reversion model based on dividend yields and price‐earnings multiples. However, timing investment strategies based on any of these prediction models do no better than a buy‐and‐hold strategy. Although some predictability of returns exists, there is no evidence of any systematic inefficiency that would enable investors to earn excess returns.     

Investment decisions when utility depends on wealth and other attributes

The problem of optimal investment under a multivariate utility function allows for an investor to obtain utility not only from wealth, but other (possibly correlated) attributes. In this paper we implement multivariate mixtures of exponential (mixex) utility to address this problem. These utility functions allow for stochastic risk aversions to differing states of the world. We derive some new results for certainty equivalence in this context. By specifying different distributions for stochastic risk aversions, we are able to derive many known, plus several new utility functions, including models of conditional certainty equivalence and multivariate generalisations of HARA utility, which we call dependent HARA utility. Focusing on the case of asset returns and attributes being multivariate normal, we optimise the asset portfolio, and find that the optimal portfolio consists of the Markowitz portfolio and hedging portfolios. We provide an empirical illustration for an investor with a mixex utility function of wealth and sentiment.     

What really happens if the positive definiteness requirement on the covariance matrix of returns is relaxed in efficient portfolio selection?

The Markowitz critical line method for mean–variance portfolio construction has remained highly influential today, since its introduction to the finance world six decades ago. The Markowitz algorithm is so versatile and computationally efficient that it can accommodate any number of linear constraints in addition to full allocations of investment funds and disallowance of short sales. For the Markowitz algorithm to work, the covariance matrix of returns, which is positive semi-definite, need not be positive definite. As a positive semi-definite matrix may not be invertible, it is intriguing that the Markowitz algorithm always works, although matrix inversion is required in each step of the iterative procedure involved. By examining some relevant algebraic features in the Markowitz algorithm, this paper is able to identify and explain intuitively the consequences of relaxing the positive definiteness requirement, as well as drawing some implications from the perspective of portfolio diversification. For the examination, the sample covariance matrix is based on insufficient return observations and is thus positive semi-definite but not positive definite. The results of the examination can facilitate a better understanding of the inner workings of the highly sophisticated Markowitz approach by the many investors who use it as a tool to assist portfolio decisions and by the many students who are introduced pedagogically to its special cases.     

Industry momentum strategies and autocorrelations in stock returns

In this study, we examine the sources of profits to momentum strategies of buying past winner industry portfolios and selling short past loser industry portfolios. We decompose the profit into (1) own-autocovariances in industry portfolio returns, (2) cross-autocovariances among industry portfolio returns, and (3) cross-sectional dispersion in mean portfolio returns. Our empirical results show that the industry momentum effect is mainly driven by the own-autocorrelation in industry portfolio returns, not by return cross-autocorrelations or by cross-sectional differences in mean returns. Indeed, the industry momentum strategy generates statistically significant profits only when own-autocorrelations are positive and statistically significant. The evidence is consistent with several behavioral models (e.g. Journal of Financial Economics 45 (1998) 307; Journal of Finance 53 (1998) 1839; Journal of Finance 54 (1999) 2143) that suggest positive own-autocorrelations in stock returns and hence the price momentum.     

Portfolio selection subject to experts' judgments

Since Markowitz [Markowitz, H. M. (1952). Portfolio selection. The Journal of Finance, 7, 77-91.], mean-variance theory has assumed that risky-asset returns to be random variables. The theory deals with this uncertainty by further assuming that investors hold homogeneous beliefs regarding the probability distribution governing return uncertainty. While the theory deals with return uncertainty, it fails to address measurement imprecision. In his original work, Markowitz recognized the need to combine randomness with heterogeneous expert judgment resulting in such imprecision. The main objective contributions of the paper are (i) to explore the implications of fuzzy return indeterminacy on mean-variance optimal portfolio choice, (ii) to use bid-ask spread as a proxy measure of the indeterminacy or “fuzzy” nature of random returns, and (iii) to introduce a brief, self-contained glimpse of empirical representations to practitioners unfamiliar with the fuzzy modeling field. Exposition, such as this one, is expected to open new collaborations between other branches of fuzzy mathematics and asset-pricing theories.     

Dynamic Portfolio Selection by Augmenting the Asset Space

We present a novel approach to dynamic portfolio selection that is as easy to implement as the static Markowitz paradigm. We expand the set of assets to include mechanically managed portfolios and optimize statically in this extended asset space. We consider “conditional” portfolios, which invest in each asset an amount proportional to conditioning variables, and “timing” portfolios, which invest in each asset for a single period and in the risk‐free asset for all other periods. The static choice of these managed portfolios represents a dynamic strategy that closely approximates the optimal dynamic strategy for horizons up to 5 years.     

Markowitz efficiency and size effect: evidence from the UK stock market

Academics and practitioners have frequently debated the relationship between market capitalization and expected return. We apply the Markowitz efficient frontier approach to develop a portfolio performance measure that compares the return of a portfolio to its optimal return, using data from the UK stock market over the period 1985–2012. Our results show that there is a negative relationship between portfolio size and portfolio return during the period under study. When comparing actual portfolio return with achievable return for the same level of risk, we find that as the portfolio size expands, underperformance of the portfolio increases, i.e. the larger the portfolio size, the greater the underperformance. This indicates that Markowitz efficiency is difficult to achieve, particularly in large portfolios. Changing model parameters leads to alternative efficient frontiers that impact upon the measurement of performance. However, the use of alternative efficient frontiers does not affect our result of the size effect on the relative performance of portfolios. Our study shows that the size effect is present over the full period. Our findings also suggest that the excess returns found in small portfolios are likely to be associated with higher levels of diversifiable risk in comparison with larger portfolios. Furthermore, in contrast to other studies, we find no evidence to support the size reversal effect in the data.     

On the robustness of risk-based asset allocations

Since the subprime crisis, portfolios based on risk diversification are of great interest to both academic researchers and market practitioners. They have also been employed by several asset management firms and their performance appears promising. Since they do not rely on estimates of expected returns, they are assumed to be robust. The same argument holds for minimum variance and equally weighted portfolios. In this paper, we consider a Monte Carlo simulation, as well as an empirical global portfolio dataset, to study the effect of estimation errors on the outcomes of two recently proposed asset allocations, the equally weighted risk contribution (ERC) and the principal component analysis (PCA) portfolio. The ERC portfolio is more robust to changes in the input parameters and has a smaller estimation error than the Markowitz approaches, whereas the PCA portfolio is even more unstable than the classical approaches. In the worst-case scenario, neither approach delivers what it promises. However, in every case the resulting return?Crisk relationship is dominated by the Markowitz approaches.     

Short Positions in the First Principal Component Portfolio

Insurance companies and pension plans typically hold well-diversified equity portfolios. These institutions are also often restricted from taking short positions. The diversification requirement operates on the portfolio level, while the short sale constraint is at the individual security level. We examine an investment strategy that exposes a tension between these two requirements. This strategy uses the first principal component to construct the portfolio and by design meets the first requirement. Empirical portfolios based on the first principal component do an excellent job of capturing market exposure and minimizing diversifiable risk. However, in practice such portfolios sometimes contain a few short positions. So this strategy does not always meet the second requirement. We examine which features of stock returns give rise to short positions when a portfolio is based on the first principal component, and we are able to identify the characteristics of the stocks that are responsible for the short positions. These stocks tend to have negative correlations with the majority of other stocks. In contrast such stocks would typically be held long in a Markowitz portfolio. We discuss and explain this puzzle.     

Mean-Variance Versus Direct Utility Maximization

Levy and Markowitz showed, for various utility functions and empirical returns distributions, that the expected utility maximizer could typically do very well if he acted knowing only the mean and variance of each distribution. Levy and Markowitz considered only situations in which the expected utility maximizer chose among a finite number of alternate probability distributions. The present paper examines the same questions for a case with an infinite number of alternate distributions, namely those available from the standard portfolio constraint set.     

Portfolio selection with higher moments

We propose a method for optimal portfolio selection using a Bayesian decision theoretic framework that addresses two major shortcomings of the traditional Markowitz approach: the ability to handle higher moments and parameter uncertainty. We employ the skew normal distribution which has many attractive features for modeling multivariate returns. Our results suggest that it is important to incorporate higher order moments in portfolio selection. Further, our comparison to other methods where parameter uncertainty is either ignored or accommodated in an ad hoc way, shows that our approach leads to higher expected utility than competing methods, such as the resampling methods that are common in the practice of finance.     

Cross‐region and cross‐sector asset allocation with regimes

Cross‐region and cross‐sector asset allocation decisions are one of the most fundamental issues in international equity portfolio management. Equity returns exhibit higher volatilities and correlations, and lower expected returns, in bear markets compared to bull markets. However, static mean–variance analysis fails to capture this salient feature of equity returns. We accommodate the nonlinearity of returns using a regime switching model across both regions and sectors. The regime‐dependent asset allocation potentially adds value to the traditional static mean–variance allocation. In addition, optimal allocation across sectors provide greater benefits compared to international diversification, which is characterized by higher returns, lower risks, lower correlations with the world market and a higher Sharpe ratio.