The econometrics of efficient portfolios

It is well known that the standard mean variance approach can be inappropriate when return distributions feature skewness, fat tails or multimodes. This is typically the situation for portfolios including derivatives. In this case, it can be necessary to come back to the basic expected utility approach. In this paper, an efficient portfolio maximizes the expected utility of future wealth. This paper presents an analysis of the efficiency frontier, formed by a set of efficient portfolios corresponding to a parameterized class of utility functions. First, we discuss the estimation of an efficient portfolio and introduce several tests of the efficiency hypothesis, depending on what is known about the utility function and the budget level. Next we analyse the shape of the frontier and develop a procedure for testing the separability of the efficiency frontier into K independent funds. The inference is semi-nonparametric because the return distribution is left unspecified. We illustrate our approach by an application to portfolios including derivatives.     

Portfolio selection with mental accounts and background risk

Das et al. (2010) develop a model where an investor divides his or her wealth among mental accounts with motives such as retirement and bequest. Nevertheless, the investor ends up selecting portfolios within mental accounts and an aggregate portfolio that lie on the mean–variance frontier. Importantly, they assume that the investor only faces portfolio risk. In practice, however, many individuals also face background risk. Accordingly, our paper expands upon theirs by considering the case where the investor faces background risk. Our contribution is threefold. First, we provide an analytical characterization of the existence and composition of the optimal portfolios within accounts and the aggregate portfolio. Second, we show that these portfolios lie away from the mean–variance frontier under fairly general conditions. Third, we find that the composition and location of such portfolios can differ notably from those of portfolios on the mean–variance frontier.     

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.     

Optimal dealer pricing under transactions and return uncertainty

The paper examines the optimal behavior of a single dealer who is faced with a stochastic demand to trade (modeled by a continuous time Poisson jump process) and facing return risk on his stock and on the rest of his portfolio (modeled by diffusion processes). Using stochastic dynamic programming, we derive the optimal bid and ask prices that maximize the dealer's expected utility of terminal wealth as a function of the state in which he finds himself. The relationship of the bid and ask prices to inventory of the dealer, instantaneous variance of return, stochastic arrival of transactions and other variables is examined.     

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 based on the mean–VaR efficient frontier

Value-at-Risk (VaR) has become one of the standard measures for assessing risk not only in the financial industry but also for asset allocations of individual investors. The traditional mean–variance framework for portfolio selection should, however, be revised when the investor's concern is the VaR instead of the standard deviation. This is especially true when asset returns are not normal. In this paper, we incorporate VaR in portfolio selection, and we propose a mean–VaR efficient frontier. Due to the two-objective optimization problem that is associated with the mean–VaR framework, an evolutionary multi-objective approach is required to construct the mean–VaR efficient frontier. Specifically, we consider the elitist non-dominated sorting Genetic Algorithm (NSGA-II). From our empirical analysis, we conclude that the risk-averse investor might inefficiently allocate his/her wealth if his/her decision is based on the mean–variance framework.     

An analysis of portfolio selection with background risk

This paper investigates the impact of background risk on an investor’s portfolio choice in a mean–variance framework, and analyzes the properties of efficient portfolios as well as the investor’s hedging behaviour in the presence of background risk. Our model implies that the efficient portfolio with background risk can be separated into two independent components: the traditional mean–variance efficient portfolio, and a self-financing component constructed to hedge against background risk. Our analysis also shows that the presence of background risk shifts the efficient frontier of financial assets to the right with no changes in its shape. Moreover, both the composition of the hedge portfolio and the location of the efficient frontier are greatly affected by a number of background risk factors, including the proportion of background assets in total wealth and the correlation between background risk and financial risk.     

Portfolio construction incorporating asymmetric dependence structures: a user's guide

We outline a method of portfolio selection incorporating asymmetric dependency structures using copula functions. Assuming normally distributed marginal returns, we illustrate how asymmetric return correlations affect the efficient frontier and subsequent portfolio performance under a dynamic rebalancing framework. Implementing this methodology within the context of tactically allocating a small set of market indices, we demonstrate several key findings. First, we establish the manner by which the efficient frontier constructed under asymmetric dependence differs from a mean‐variance frontier. By establishing a paper portfolio based on these differences, we find that asymmetric correlation structures do have real economic value. The primary source of this economic value is the ability to better protect portfolio value and reduce the size of any erosion in return relative to the normal portfolio when asymmetric return correlations are accounted for.     

A jackknife-type estimator for portfolio revision

This article proposes a novel approach to portfolio revision. The current literature on portfolio optimization uses a somewhat naïve approach, where portfolio weights are always completely revised after a predefined fixed period. However, one shortcoming of this procedure is that it ignores parameter uncertainty in the estimated portfolio weights, as well as the biasedness of the in-sample portfolio mean and variance as estimates of the expected portfolio return and out-of-sample variance. To rectify this problem, we propose a jackknife procedure to determine the optimal revision intensity, i.e. the percent of wealth that should be shifted to the new, in-sample optimal portfolio. We find that our approach leads to highly stable portfolio allocations over time, and can significantly reduce the turnover of several well established portfolio strategies. Moreover, the observed turnover reductions lead to statistically and economically significant performance gains in the presence of transaction costs.     

Optimal portfolios when stock prices follow an exponential Lévy process

We investigate some portfolio problems that consist of maximizing expected terminal wealth under the constraint of an upper bound for the risk, where we measure risk by the variance, but also by the Capital-at-Risk (CaR). The solution of the mean-variance problem has the same structure for any price process which follows an exponential Lévy process. The CaR involves a quantile of the corresponding wealth process of the portfolio. We derive a weak limit law for its approximation by a simpler Lévy process, often the sum of a drift term, a Brownian motion and a compound Poisson process. Certain relations between a Lévy process and its stochastic exponential are investigated.Received: January 2003Mathematics Subject Classification: Primary: 60F05, 60G51, 60H30, 91B28; secondary: 60E07, 91B70JEL Classification: C22, G11, D81We would like to thank Jan Kallsen and Ralf Korn for discussions and valuable remarks on a previous version of our paper. The second author would like to thank the participants of the Conference on Lévy Processes at Aarhus University in January 2002 for stimulating remarks. In particular, a discussion with Jan Rosinski on gamma processes has provided more insight into the approximation of the variance gamma model.     

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.     

Short Sales Restrictions and Kinks on the Mean Variance Frontier

With a short sales restriction, there may be switching points along the mean variance frontier corresponding to changes in the set of assets held. Traditional wisdom holds that each switching point corresponds to a kink, while Ross has claimed that kinks never occur. This paper shows that the truth lies between the two views, since the efficient frontier may or may not be kinked at a switching point. There is some indication that kinks are rare, since a kink corresponds to a portfolio in which all assets have the same expected return.     

Multi-scale variation,path risk and long-term portfolio management

Mean–variance analysis is constrained to weight the frequency bands in a return time series equally. A more flexible approach allows the user to assign preference weightings to short or longer run frequencies. Wavelet analysis provides further flexibility, removing the need to assume asset returns are stationary and encompassing alternative return concepts. The resulting portfolio choice methodology establishes a reward–energy efficient frontier that allows the user to trade off expected reward against path risk, reflecting preferences as between long or short run variation. The approach leads to dynamic analogues of mean–variance such as band pass portfolios that are more sensitive to variability at designated scales.     

Portfolio selection with mental accounts and delegation

Das et al. (2010) develop an elegant framework where an investor selects portfolios within mental accounts but ends up holding an aggregate portfolio on the mean-variance frontier. This investor directly allocates the wealth in each account among available assets. In practice, however, investors often delegate the task of allocating wealth among assets to portfolio managers who seek to beat certain benchmarks. Accordingly, we extend their framework to the case where the investor allocates the wealth in each account among portfolio managers. Our contribution is threefold. First, we provide an analytical characterization of the existence and composition of the optimal portfolios within accounts and the aggregate portfolio. Second, we present conditions under which such portfolios are not on the mean-variance frontier, and conditions under which they are. Third, we show that the aforementioned analytical characterization is also applicable within the framework of Das et al. and thus improves upon their numerical approach.     

Differences in Perstistence in Aggregated and Disaggregated UK Stock Returns: A Reconciliation

Studies of the persistence in the returns series of UK stocks, using inter alia variance ratios, have documented clear differences between the relatively low levels of persistence in individual security returns and the relatively high levels of persistence in the returns of portfolios composed of these same securities. In this paper, I reconcile this contrast by showing that portfolio return variance ratios should not be expected to reflect (own) persistence levels in the component security returns, but instead should reflect a 'cross-persistence' between the securities. I calculate synthetic portfolio variance ratios from measures of security return 'cross-persistence' and find that they replicate closely the observed portfolio return variance ratios, which provides empirical support for the theoretical results.     

Optimal Asset-Liability Management for an Insurer Under Markov Regime Switching Jump-Diffusion Market

This paper considers an asset-liability management problem under a continuous time Markov regime-switching jump-diffusion market. We assume that the risky stock’s price is governed by a Markov regime-switching jump-diffusion process and the insurance claims follow a Markov regime-switching compound poisson process. Using the Markowitz mean-variance criterion, the objective is to minimize the variance of the insurer’s terminal wealth, given an expected terminal wealth. We get the optimal investment policy. At the same time, we also derive the mean-variance efficient frontier by using the Lagrange multiplier method and stochastic linear-quadratic control technique.     

An analysis of risk-based asset allocation and portfolio insurance strategies

This paper compares traditional portfolio insurance strategies with modern risk-based dynamic asset allocation strategies within a currency portfolio context for reserve management. Given the objective of preserving reserve value, the evaluation of the hedging performances of various strategies focuses on four perspectives regarding, in particular, the return distribution of the hedged portfolio. In terms of the Sharpe Ratio, the constant proportional portfolio insurance is the best performer due to having the lowest volatility, while the Value at Risk strategy based upon the normal distribution is the worst due to its having the smallest return. From the perspective that the return distribution of the hedged portfolio is shifted to the right, the synthetic put performs the best, with the expected shortfall strategy the second best. In terms of the cumulative portfolio return across years, the expected shortfall strategy using the historical distribution ranks first, as a result of its participation in upward markets. Furthermore, the expected shortfall-based strategy results in a lower turnover within the investment horizon, thereby saving transaction costs.     

The effects of dividend yield and dividend policy on common stock prices and returns

This paper suggests that it is not possible to demonstrate, using the best available empirical methods, that the expected returns on high yield common stocks differ from the expected returns on low yield common stocks either before or after taxes. A taxable investor who concentrates his portfolio in low yield securities cannot tell from the data whether he is increasing or decreasing his expected after-tax return by so doing. A tax exempt investor who concentrates his portfolio in high yield securities cannot tell from the data whether he is increasing or decreasing his expected return. We argue that the best method for testing the effects of dividend policy on stock prices is to test the effects of dividend yield on stock returns. Thus the fact that we cannot tell, using the best available methods, what effects dividend yield has on stock returns implies that we cannot tell what effect, if any, a change in dividend policy will have on a corporation's stock price.     

Options Trading and the CAPM

This article studies equilibrium asset pricing when agents facenonnegative wealth constraints. In the presence of these constraintsit is shown that options on the market portfolio are nonredundantsecurities and the economy's pricing kernel is a function ofboth the market portfolio and the nonredundant options. Thisimplies that the options should be useful for explaining riskyasset returns. To test the theory, a model is derived in whichthe expected excess return on any risky asset is linearly related(via a collection of betas) to the expected excess return onthe market portfolio and to the expected excess returns on thenonredundant options. The empirical results indicate that thereturns on traded index options are relevant for explainingthe returns on risky asset portfolios.     

Asset allocation using flexible dynamic correlation models with regime switching

The asset allocation decision is often considered as a trade-off between maximizing the expected return of a portfolio and minimizing the portfolio risk. The riskiness is evaluated in terms of variance of the portfolio return, so that it is fundamental to consider correctly the variance of its components and their correlations. The evidence for the heteroskedastic behaviour of the returns and the time-varying relationships among the portfolio components have recently shifted attention to the multivariate GARCH models with time varying correlation. In this work we insert a particular Markov Switching dynamics in some Dynamic Correlation models to consider the abrupt changes in correlations affecting the assets in different ways. This class of models is very general and provides several specifications, constraining some coefficients. The models are applied to solve a sectorial asset allocation problem and are compared with alternative models.