Size matters: Optimal calibration of shrinkage estimators for portfolio selection

We carry out a comprehensive investigation of shrinkage estimators for asset allocation, and we find that size matters—the shrinkage intensity plays a significant role in the performance of the resulting estimated optimal portfolios. We study both portfolios computed from shrinkage estimators of the moments of asset returns (shrinkage moments), as well as shrinkage portfolios obtained by shrinking the portfolio weights directly. We make several contributions in this field. First, we propose two novel calibration criteria for the vector of means and the inverse covariance matrix. Second, for the covariance matrix we propose a novel calibration criterion that takes the condition number optimally into account. Third, for shrinkage portfolios we study two novel calibration criteria. Fourth, we propose a simple multivariate smoothed bootstrap approach to construct the optimal shrinkage intensity. Finally, we carry out an extensive out-of-sample analysis with simulated and empirical datasets, and we characterize the performance of the different shrinkage estimators for portfolio selection.     

A dynamic model of active portfolio management with benchmark orientation

This paper studies optimal dynamic portfolios for investors concerned with the performance of their portfolios relative to a benchmark. Assuming that asset returns follow a multi-linear factor model similar to the structure of Ross (1976) [Ross, S., 1976. The arbitrage theory of the capital asset pricing model. Journal of Economic Theory, 13, 342–360] and that portfolio managers adopt a mean tracking error analysis similar to that of Roll (1992) [Roll, R., 1992. A mean/variance analysis of tracking error. Journal of Portfolio Management, 18, 13–22], we develop a dynamic model of active portfolio management maximizing risk adjusted excess return over a selected benchmark. Unlike the case of constant proportional portfolios for standard utility maximization, our optimal portfolio policy is state dependent, being a function of time to investment horizon, the return on the benchmark portfolio, and the return on the investment portfolio. We define a dynamic performance measure which relates portfolio’s return to its risk sensitivity. Abnormal returns at each point in time are quantified as the difference between the realized and the model-fitted returns. Risk sensitivity is estimated through a dynamic matching that minimizes the total fitted error of portfolio returns. For illustration, we analyze eight representative mutual funds in the U.S. market and show how this model can be used in practice.     

Small caps in international equity portfolios: the effects of variance risk

We show that predictable covariances between means and variances of stock returns may have a first order effect on portfolio composition. In an international asset menu that includes both European and North American small capitalization equity indices, we find that a three-state, heteroskedastic regime switching VAR model is required to provide a good fit to weekly return data and to accurately predict the dynamics in the joint density of returns. As a result of the non-linear dynamic features revealed by the data, small cap portfolios become riskier in bear markets, i.e., display negative co-skewness with other stock indices. Because of this property, a power utility investor ought to hold a well-diversified portfolio, despite the high risk premium and Sharpe ratios offered by small capitalization stocks. On the contrary, small caps command large optimal weights when the investor ignores variance risk, by incorrectly assuming joint normality of returns.      

SYMMETRIC VERSUS ASYMMETRIC CONDITIONAL COVARIANCE FORECASTS: DOES IT PAY TO SWITCH?

Volatilities and correlations for equity markets rise more after negative returns shocks than after positive shocks. Allowing for these asymmetries in covariance forecasts decreases mean‐variance portfolio risk and improves investor welfare. We compute optimal weights for international equity portfolios using predictions from asymmetric covariance forecasting models and a spectrum of expected returns. Investors who are moderately risk averse, have longer rebalancing horizons, and hold U.S. equities benefit most and may be willing to pay around 100 basis points annually to switch from symmetric to asymmetric forecasts. Accounting for asymmetry in both variances and correlations significantly lowers realized portfolio risk.     

Active portfolio management with benchmarking: A frontier based on alpha

Active portfolio management often involves the objective of selecting a portfolio with minimum tracking error variance (TEV) for some expected gain in return over a benchmark. However, Roll (1992) shows that such portfolios are generally suboptimal because they do not belong to the mean-variance frontier and are thus overly risky. Our paper proposes an appealing method to lessen this suboptimality that involves the objective of selecting a portfolio from the set of portfolios that have minimum TEV for various levels of ex-ante alpha, which we refer to as the alpha-TEV frontier. Since practitioners commonly use ex-post alpha to assess the performance of managers, the use of this frontier aligns the objectives of managers with how their performance is evaluated. Furthermore, sensible choices of ex-ante alpha lead to the selection of portfolios that are less risky (in variance terms) than the portfolios that active managers would otherwise select.     

Constructing inverse factor volatility portfolios: A risk-based asset allocation for factor investing

In this study, we investigate risk-based asset allocation approaches for factor investing strategies by constructing a multifactor portfolio based on the inverse weighting method. We propose the inverse factor volatility (IFV) strategy, which is the simplified variant of a factor risk parity, assuming constant factor correlation. In IFV portfolio construction, the portfolio's weights are determined by using scaled inverse factor volatility treated as a proxy for a targeted exposure in the optimization. Based on daily stock and index returns on global markets from 2002 to the end of 2017, we implemented the empirical analysis of IFV portfolios among three stock markets: Japan, Euro, and the US. The results obtained reveal that the IFV portfolios significantly outperformed market capitalization weighted portfolios by successfully acquiring factor risk premiums.     

The diversification cost of large,concentrated equity stakes. How big is it? Is it justified?

While the hypothesis that ownership concentration can affect the value of a company has seen a lot of empirical study, little light has been shed on a complementary problem, that these concentrated owners have a cost of their position due to an undiversified portfolio. Using a unique data set of the actual diversification of all Norwegian equity owners, we show that the largest owners of a corporation in fact have very undiversified equity portfolios, and that such owners have significant costs to their concentrated portfolios. At the level of risk of a benchmark portfolio, if they were to move from their present portfolio composition in risky assets to a well diversified portfolio, their returns would have increased by about 13 percentage points in annual terms. We ask whether this cost can be explained by estimated benefits of ownership concentration (private benefits), and show that extant estimates of private benefits are too low to offset our cost estimates.     

Robustness of the inference procedures for the global minimum variance portfolio weights in a skew-normal model

In this paper, we study the influence of skewness on the distributional properties of the estimated weights of optimal portfolios and on the corresponding inference procedures derived for the optimal portfolio weights assuming that the asset returns are normally distributed. It is shown that even a simple form of skewness in the asset returns can dramatically influence the performance of the test on the structure of the global minimum variance portfolio. The results obtained can be applied in the small sample case as well. Moreover, we introduce an estimation procedure for the parameters of the skew-normal distribution that is based on the modified method of moments. A goodness-of-fit test for the matrix variate closed skew-normal distribution has also been derived. In the empirical study, we apply our results to real data of several stocks included in the Dow Jones index.     

Mean–variance and mean–semivariance portfolio selection: a multivariate nonparametric approach

While univariate nonparametric estimation methods have been developed for estimating returns in mean-downside risk portfolio optimization, the problem of handling possible cross-correlations in a vector of asset returns has not been addressed in portfolio selection. We present a novel multivariate nonparametric portfolio optimization procedure using kernel-based estimators of the conditional mean and the conditional median. The method accounts for the covariance structure information from the full set of returns. We also provide two computational algorithms to implement the estimators. Via the analysis of 24 French stock market returns, we evaluate the in-sample and out-of-sample performance of both portfolio selection algorithms against optimal portfolios selected by classical and univariate nonparametric methods for three highly different time periods and different levels of expected return. By allowing for cross-correlations among returns, our results suggest that the proposed multivariate nonparametric method is a useful extension of standard univariate nonparametric portfolio selection approaches.     

Using industry momentum to improve portfolio performance

Minimum-variance portfolios, which ignore the mean and focus on the (co)variances of asset returns, outperform mean–variance approaches in out-of-sample tests. Despite these promising results, minimum-variance policies fail to deliver a superior performance compared with the simple 1/N rule. In this paper, we propose a parametric portfolio policy that uses industry return momentum to improve portfolio performance. Our portfolio policies outperform a broad selection of established portfolio strategies in terms of Sharpe ratio and certainty equivalent returns. The proposed policies are particularly suitable for investors because portfolio turnover is only moderately increased compared to standard minimum-variance portfolios.     

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.     

International portfolio diversification: Is there a role for the Middle East and North Africa?

We examine the issue of possible portfolio diversification benefits into seven Middle East and North African (MENA) stock markets. We construct international portfolios in dollars and local currencies. Ex ante weights are obtained by plugging five optimization models and two risk measures into a rolling block-bootstrap methodology. This allows us to derive 48 monthly rebalanced ex post portfolio returns. We analyze the out-of-sample performance based on Sharpe and Sortino ratios and the Jobson–Korkie statistic. Our results highlight outstanding diversification benefits in the MENA region, both in dollar and local currencies. Overall, we show that these under-estimated, under-investigated markets could attract more portfolio flows in the future.     

Asset pricing models: implications for expected returns and portfolio selection

When a risk factor is missing from an asset pricing model, theresulting mispricing is embedded within the residual covariancematrix. Exploiting this phenomenon leads to expected returnestimates that are more stable and precise than estimates deliveredby standard methods. Portfolio selection can also be improved.At an extreme, optimal portfolio weights are proportional toexpected returns when no factors are observable. We find thatsuch portfolios perform well in simulations and in out-of-samplecomparisons.     

The Effect of Shortfall as a Risk Measure for Portfolios with Hedge Funds

Abstract:  Current research suggests that the large downside risk in hedge fund returns disqualifies the variance as an appropriate risk measure. For example, one can easily construct portfolios with nonlinear pay-offs that have both a high Sharpe ratio and a high downside risk. This paper examines the consequences of shortfall-based risk measures in the context of portfolio optimization. In contrast to popular belief, we show that negative skewness for optimal mean-shortfall portfolios can be much greater than for mean-variance portfolios. Using empirical hedge fund return data we show that the optimal mean-shortfall portfolio substantially reduces the probability of small shortfalls at the expense of an increased extreme crash probability. We explain this by proving analytically under what conditions short-put payoffs are optimal for a mean-shortfall investor. Finally, we show that quadratic shortfall or semivariance is less prone to these problems. This suggests that the precise choice of the downside risk measure is highly relevant for optimal portfolio construction under loss averse preferences.     

The explanatory role of factor portfolios for industries exposed to foreign competition: evidence from the Swedish stock market

Our purpose is to find factors that are important for expected returns and risk of Swedish industrial portfolios during 1980–1997. The tested factors are supposed to be essential for a small open economy. We take into account the small sample problem that surfaces in the form of firms dominating the value weighted test portfolios. An extreme bound analysis (EBA) investigates the robustness of the estimated parameters. Principal component analysis is used to assess the importance of the factors in explaining return covariances. Our overall conclusion is that the market portfolio, which refers to the world as well as the Swedish market portfolio, is almost sufficient for explaining expected returns and risk.     

The effect of downside risk reduction on UK equity portfolios included with Managed Futures Funds

The concept of asymmetric risk estimation has become more widely applied in risk management in recent years with the increased use of Value-at-risk (VaR) methodologies. This paper uses the n-degree lower partial moment (LPM) models, of which VaR is a special case, to empirically analyse the effect of downside risk reduction on UK portfolio diversification and returns. Data on Managed Futures Funds are used to replicate the increasingly popular preference of investors for including hedge funds and fund-of-funds type investments in the UK equity portfolios. The result indicates, however that the potential benefits of fund diversification may deteriorate following reductions in downside risk tolerance levels. These results appear to reinforce the importance of risk (tolerance) perception, particularly downside risk, when making decisions to include Managed Futures Funds in UK equity portfolios as the empirical analysis suggests that this could negatively affect portfolio returns.     

Systematic Risk and Empirical Research

We show here that risky asset returns generating processes stated in terms of factors which include both accounting and non-accounting based measures of risk (e.g. book to market ratios) imply, under fairly standard regularity conditions, that the Sharpe-Lintner-Black asset pricing model beta is a 'sufficient' statistic in the sense that it captures all important attributes of the returns generating process in a single number. We then derive the parametric relationship between betas based on inefficient index portfolios and betas based on the market or tangency portfolio. We demonstrate that the relationship between risky asset expected returns and betas computed on the basis of inefficient index portfolios is both consistent with the predictions of the Capital Asset Pricing Model and the multi-factor asset pricing models of Fama and French (1992, 1993, 1995 and 1996). The 'trick' is to realise that inefficient index portfolios are composed of the market portfolio and a collection of inefficient but self financing 'kernel' or 'arbitrage' portfolios. It then follows that there is a perfect linear cross sectional relationship between risky asset expected returns, betas based on inefficient index portfolios and the arbitrage portfolios. Hence, if we happen to stumble across variables that span the same subspace as the vectors representing the arbitrage portfolios, it is easy to create the illusion that risky asset expected returns depend on variables other than 'beta'.     

Risk parity portfolio optimization under a Markov regime-switching framework

We formulate and solve a risk parity optimization problem under a Markov regime-switching framework to improve parameter estimation and to systematically mitigate the sensitivity of optimal portfolios to estimation error. A regime-switching factor model of returns is introduced to account for the abrupt changes in the behaviour of economic time series associated with financial cycles. This model incorporates market dynamics in an effort to improve parameter estimation. We proceed to use this model for risk parity optimization and also consider the construction of a robust version of the risk parity optimization by introducing uncertainty structures to the estimated market parameters. We test our model by constructing a regime-switching risk parity portfolio based on the Fama–French three-factor model. The out-of-sample computational results show that a regime-switching risk parity portfolio can consistently outperform its nominal counterpart, maintaining a similar ex post level of risk while delivering higher-than-nominal returns over a long-term investment horizon. Moreover, we present a dynamic portfolio rebalancing policy that further magnifies the benefits of a regime-switching portfolio.     

An analysis of the impact of timberland, farmland and commercial real estate in the asset allocation decisions of institutional investors

We examine the effects of including timberland, farmland and commercial real estate in a mixed asset portfolio with stocks, government bonds and T-Bills. Using both smoothed and unsmoothed returns (as per Geltner [Geltner, D. (1993). Estimating market values from appraised values without assuming an efficient market. Journal of Real Estate Research, 8, 25-345.]) and both constrained and unconstrained allocation assumptions (as per Eichhorn, Gupta and Stubbs [Eichhorn, D., Gupta, F., & Stubbs, E. (1998). Using constraints to improve the robustness of asset allocation. Journal of Portfolio Management, Spring, 41-48.]), we employ Markowitz portfolio optimization and find widely varying allocation outcomes. However, timberland entered nearly all portfolios, accounting for large percentages in several scenarios, while farmland entered only low-risk portfolios. At lower risk levels, commercial real estate dominates the real estate allocation but as acceptable risk levels rise, timberland supplants commercial real estate as the primary component of the portfolio's real estate allocation.     

Disentangling the role of variance and covariance information in portfolio selection problems

The covariation among financial asset returns is often a key ingredient used in the construction of optimal portfolios. Estimating covariances from data, however, is challenging due to the potential influence of estimation error, specially in high-dimensional problems, which can impact negatively the performance of the resulting portfolios. We address this question by putting forward a simple approach to disentangle the role of variance and covariance information in the case of mean-variance efficient portfolios. Specifically, mean-variance portfolios can be represented as a two-fund rule: one fund is a fully invested portfolio that depends on diagonal covariance elements, whereas the other is a long-short, self financed portfolio associated with the presence of non-zero off-diagonal covariance elements. We characterize the contribution of each of these two components to the overall performance in terms of out-of-sample returns, risk, risk-adjusted returns and turnover. Finally, we provide an empirical illustration of the proposed portfolio decomposition using both simulated and real market data.