Noise sensitivity of portfolio selection under various risk measures

We study the sensitivity to estimation error of portfolios optimized under various risk measures, including variance, absolute deviation, expected shortfall and maximal loss. We introduce a measure of portfolio sensitivity and test the various risk measures by considering simulated portfolios of varying sizes N and for different lengths T of the time series. We find that the effect of noise is very strong in all the investigated cases, asymptotically it only depends on the ratio N/T, and diverges (goes to infinity) at a critical value of N/T, that depends on the risk measure in question. This divergence is the manifestation of a phase transition, analogous to the algorithmic phase transitions recently discovered in a number of hard computational problems. The transition is accompanied by a number of critical phenomena, including the divergent sample to sample fluctuations of portfolio weights. While the optimization under variance and mean absolute deviation is always feasible below the critical value of N/T, expected shortfall and maximal loss display a probabilistic feasibility problem, in that they can become unbounded from below already for small values of the ratio N/T, and then no solution exists to the optimization problem under these risk measures. Although powerful filtering techniques exist for the mitigation of the above instability in the case of variance, our findings point to the necessity of developing similar filtering procedures adapted to the other risk measures where they are much less developed or non-existent. Another important message of this study is that the requirement of robustness (noise-tolerance) should be given special attention when considering the theoretical and practical criteria to be imposed on a risk measure.     

A Multivariate Linear Regression Test for the Arbitrage Pricing Theory

A test for the arbitrage pricing theory which employs a multivariate linear regression model is developed. Given a sample of return premiums for a set of N assets which includes a subset of k linearly independent portfolios, the k factor APT hypothesis is accepted if the intercept term is zero in the multivariate regression of the $(N?k)$ returns on the k portfolios. The test may be carried out simply, by using univariate multiple regression software. The relation of this test to the concept of performance potential and Sharpe's measure of performance is also discussed. If the performance potential of the k portfolios is not significantly less than the performance potential of the complete set of N assets, then the k factor APT hypothesis is accepted.     

Minimizing CVaR and VaR for a portfolio of derivatives

Value at risk (VaR) and conditional value at risk (CVaR) are frequently used as risk measures in risk management. Compared to VaR, CVaR is attractive since it is a coherent risk measure. We analyze the problem of computing the optimal VaR and CVaR portfolios. We illustrate that VaR and CVaR minimization problems for derivatives portfolios are typically ill-posed. We propose to include cost as an additional preference criterion for the CVaR optimization problem. We demonstrate that, with the addition of a proportional cost, it is possible to compute an optimal CVaR derivative investment portfolio with significantly fewer instruments and comparable CVaR and VaR. A computational method based on a smoothing technique is proposed to solve a simulation based CVaR optimization problem efficiently. Comparison is made with the linear programming approach for solving the simulation based CVaR optimization problem.     

Are Smart Beta strategies suitable for hedge fund portfolios?

In the equity context different Smart Beta strategies (such as the equally weighted, global minimum variance, equal risk contribution and maximum diversified ratio) have been proposed as alternatives to the cap-weighted index. These new approaches have attracted the attention of equity managers as different empirical analyses demonstrate the superiority of these strategies with respect to cap-weighted and to strategies that consider only mean and variance. In this paper we focus our attention to hedge fund index portfolios and analyze if the results reported in the equity framework are still valid. We consider hedge fund index and equity portfolios, the approaches used for portfolio selection are the four ‘Smart Beta’ strategies, mean–variance and mean–variance–skewness. In the two latter approaches the Taylor approximation of a CARA expected utility function and the Polynomial Goal Programing (PGP) have been used. The obtained portfolios are analyzed in the in-sample as well as in the out-of-sample perspectives.     

Mean–variance portfolio selection with ‘at-risk’ constraints and discrete distributions

We examine the impact of adding either a VaR or a CVaR constraint to the mean–variance model when security returns are assumed to have a discrete distribution with finitely many jump points. Three main results are obtained. First, portfolios on the VaR-constrained boundary exhibit (K + 2)-fund separation, where K is the number of states for which the portfolios suffer losses equal to the VaR bound. Second, portfolios on the CVaR-constrained boundary exhibit (K + 3)-fund separation, where K is the number of states for which the portfolios suffer losses equal to their VaRs. Third, an example illustrates that while the VaR of the CVaR-constrained optimal portfolio is close to that of the VaR-constrained optimal portfolio, the CVaR of the former is notably smaller than that of the latter. This result suggests that a CVaR constraint is more effective than a VaR constraint to curtail large losses in the mean–variance model.     

Portfolio selection: An extreme value approach

We show theoretically that lower tail dependence (χ), a measure of the probability that a portfolio will suffer large losses given that the market does, contains important information for risk-averse investors. We then estimate χ for a sample of DJIA stocks and show that it differs systematically from other risk measures including variance, semi-variance, skewness, kurtosis, beta, and coskewness. In out-of-sample tests, portfolios constructed to have low values of χ outperform the market index, the mean return of the stocks in our sample, and portfolios with high values of χ. Our results indicate that χ is conceptually important for risk-averse investors, differs substantially from other risk measures, and provides useful information for portfolio selection.     

Cross-sectional tests of the CAPM and Fama–French three-factor model

Grouping does not produce a wide range of betas. Consequently, cross-sectional tests of the CAPM are bound to lack power. This paper provides a simple way to alleviate the problem by repackaging the data with zero-weight portfolios. When the CAPM is true and the data are repackaged, simulation shows that the average values of the intercept and slope converge to their true values more rapidly and there are striking increases in R² and the power of the tests. Empirical results are dramatically different in datasets with and without the zero-weight portfolios.     

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.     

Mutual fund performance persistence: Factor models and portfolio size

We re-examine US mutual fund performance persistence. We investigate persistence (i) using both “academic” factor models and “practitioner” index models, (ii) using decile-size recursive portfolios and also portfolios formed from smaller numbers of funds, (iii) using nonparametric bootstrap p-values as well as conventional t-tests and (iv) using both net-of-fee fund returns (net alphas) and gross alphas. Our key result is that positive net alpha performance persistence can be found using small portfolios of funds together with a holding period of 6 months or less, for both practitioner index models and academic factor models.     

Abstract, classic, and explicit turnpikes

Portfolio turnpikes state that as the investment horizon increases, optimal portfolios for generic utilities converge to those of isoelastic utilities. This paper proves three kinds of turnpikes. In a general semimartingale setting, the abstract turnpike states that optimal final payoffs and portfolios converge under their myopic probabilities. In diffusion models with several assets and a single state variable, the classic turnpike demonstrates that optimal portfolios converge under the physical probability. In the same setting, the explicit turnpike identifies the limit of finite-horizon optimal portfolios as a long-run myopic portfolio defined in terms of the solution of an ergodic HJB equation.     

Which demands affect optimal international portfolio choices?

This study analyzes the asset allocations of simple international portfolios that include domestic risky assets, foreign risky assets, and domestic risk-free bonds, through a theoretical analysis. A close-form solution for the optimal holding rates is derived, and can be further sub-divided into three categories of demand: speculative demand, diversified demand, and hedging demands. We carefully explore the essential problem of identifying the underlying reasons for asset allocations, which in turn allows us to answer the question of which of these demands are critical in influencing holding changes.     

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.     

Adaptive basket liquidation

We consider the infinite-horizon optimal basket portfolio liquidation problem for a von Neumann–Morgenstern investor in a multi-asset extension of the liquidity model of Almgren (Appl. Math. Finance 10:1–18, 2003) with cross-asset impact. Using a stochastic control approach, we establish a “separation theorem”: the sequence of portfolios held during an optimal liquidation depends only on the (co-)variance and (cross-asset) market impact of the assets, while the speed with which these portfolios are reached depends only on the utility function of the trader. We derive partial differential equations for both the sequence of portfolios reached during the execution and the trading speed.     

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.     

Stock market liquidity and macro-liquidity shocks: Evidence from the 2007–2009 financial crisis

We develop an empirical framework that links micro-liquidity, macro-liquidity and stock prices. We provide evidence of a strong link between macro-liquidity shocks and the returns of UK stock portfolios constructed on the basis of micro-liquidity measures between 1999 and 2012. Specifically, macro-liquidity shocks, which are extracted on the meeting days of the Bank of England Monetary Policy Committee (MPC) relative to market expectations embedded in 3-month LIBOR futures prices, are transmitted in a differential manner to the cross-section of liquidity-sorted portfolios, with liquid stocks playing the most active role. We also find that there is a significant increase in shares' trading activity and a rather small increase in their trading cost on MPC meeting days. Finally, our results emphatically document that during the recent financial crisis the shocks–returns relationship has reversed its sign. Interest rate cuts during the crisis were perceived by market participants as a signal of deteriorating economic prospects and reinforced “flight to safety” trading.     

Mean-variance vs. stochastic dominance some empirical findings on efficient sets

The study attempts to shed additional light on the issue of the costs and benefits of using the mean-variance criterion as opposed to stochastic dominance criteria for investment decisions. Relevant probabilities which facilitate measurement of these costs and benefits are identified. The mean-variance criterion is shown to be useful to some extent in identifying potentially optimal portfolios. However, it is shown that the informationally less demanding mean-variance criterion admits two types of errors: (i) including portfolios that no expected utility maximizing risk averters would choose, and (ii) excluding portfolios which some risk averters would find optimal. The empirical investigation also indicates that although the composition of the efficient sets appears to be unstable over time, the relationships between the efficient sets are persistent over time.     

Liquidity costs,idiosyncratic volatility and expected stock returns

This paper considers liquidity as an explanation for the positive association between expected idiosyncratic volatility (IV) and expected stock returns. Liquidity costs may affect the stock returns, through bid-ask bounce and other microstructure-induced noise, which will affect the estimation of IV. We use a novel method (developed by Weaver, 1991) to eliminate microstructure influences from stock closing price-based returns and then estimate IV. We show that there is a premium for IV in value-weighted portfolios, but this premium is less strong after correcting returns for microstructure bias. We further show that this premium is driven by liquidity in the prior month after correcting returns for microstructure noise. The pricing results from equally-weighted portfolios indicate that IV does not predict returns either before or after controlling for liquidity costs. These findings are robust after controlling for common risk factors as well as analysing double-sorted portfolios based on IV and liquidity.     

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.     

Optimal delegated portfolio management with background risk

Most investors delegate the management of a fraction of their wealth to portfolio managers who are given the task of beating a benchmark. However, in an influential paper [Roll, R., 1992. A mean/variance analysis of tracking error. Journal of Portfolio Management 18, 13–22] shows that the objective functions commonly used by these managers lead to the selection of portfolios that are suboptimal from the perspective of investors. In this paper, we provide an explanation for the use of these objective functions based on the effect of background risk on investors’ optimal portfolios. Our main contribution is to provide conditions under which investors can optimally delegate the management of their wealth to portfolio managers.