REGULAR(IZED) HEDGE FUND CLONES

This article addresses the problem of portfolio construction in the context of efficient hedge fund investments replication. We propose a modification to the standard Sharpe “style analysis” by introducing a constraint on the asset weights 1‐norm and 2‐norm. This constraint regularizes the optimization problem, allows efficient selection of relevant factor's and has significant effects on the stability of the resulting asset mix and the risk–return characteristics of the replicating portfolio. The empirical results suggest that the norm‐constrained replicating portfolios exhibit significant correlations with their benchmarks, often higher than 0.9; have a fraction, which is about half to two‐thirds, of active positions relative to those determined through the standard method; and are obtained with turnover, which is in some instances about one‐fourth of that for the standard method.     

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.     

Multiobjective portfolio optimization with non-convex policy constraints: Evidence from the Eurostoxx 50

Our purpose in this paper is to depart from the intrinsic pathology of the typical mean–variance formalism, due to both the restriction of its assumptions and difficulty of implementation. We manage to co-assess a set of sophisticated real-world non-convex investment policy limitations, such as cardinality constraints, buy-in thresholds, transaction costs, particular normative rules, etc., within the frame of complex scenarios, which demand for simultaneous optimization of multiple investment objectives. In such a case, the portfolio selection process reflects a mixed-integer multiobjective portfolio optimization problem. On this basis, we meticulously develop all the corresponding modeling procedures and then solve the underlying problem by use of a new, fast and very effective algorithm. The value of the suggested framework is integrated with the introduction of two novel concepts in the field of multiobjective portfolio optimization, i.e. the security impact plane and the barycentric portfolio. The first represents a measure of each security's impact in the efficient surface of Pareto optimal portfolios. The second serves as the vehicle for implementing a balanced strategy of iterative portfolio tuning. Moreover, a couple of some very informative graphs provide thorough visualization of all empirical testing results. The validity of the attempt is verified through an illustrative application on the Eurostoxx 50. The results obtained are characterized as very encouraging, since a sufficient number of efficient or Pareto optimal portfolios produced by the model, appear to possess superior out-of-sample returns with respect to the underlying benchmark.     

PORTFOLIO OPTIMIZATION UNDER TRACKING ERROR AND WEIGHTS CONSTRAINTS

The performance of active portfolio managers who must comply with a weights constraint is often assessed against a benchmark. The weights constraint is common as the funds are committed by their own prospectus to a minimum (or maximum) portfolio concentration. We characterize the optimal asset allocation and analyze the implications of the weights constraint on the manager's performance and on the relevance of the information ratio. We obtain that because of the weights constraint, at the optimum, the information ratio often decreases when the manager is free to deviate more from the benchmark.     

Multi-asset portfolio optimization and out-of-sample performance: an evaluation of Black–Litterman,mean-variance,and naïve diversification approaches

The Black–Litterman model aims to enhance asset allocation decisions by overcoming the problems of mean-variance portfolio optimization. We propose a sample-based version of the Black–Litterman model and implement it on a multi-asset portfolio consisting of global stocks, bonds, and commodity indices, covering the period from January 1993 to December 2011. We test its out-of-sample performance relative to other asset allocation models and find that Black–Litterman optimized portfolios significantly outperform naïve-diversified portfolios (1/N rule and strategic weights), and consistently perform better than mean-variance, Bayes–Stein, and minimum-variance strategies in terms of out-of-sample Sharpe ratios, even after controlling for different levels of risk aversion, investment constraints, and transaction costs. The BL model generates portfolios with lower risk, less extreme asset allocations, and higher diversification across asset classes. Sensitivity analyses indicate that these advantages are due to more stable mixed return estimates that incorporate the reliability of return predictions, smaller estimation errors, and lower turnover.     

A methodology for index tracking based on time-series clustering

With the increased acceptance of capital market efficiency, there has been a significant increase in the money managed on an indexed basis. Several methodologies are available to replicate the target index. In this paper, we discuss the problems of (1) defining suitable performance objectives and tracking error that scale properly over the entire management period and (2) implementing an optimal investment strategy when full replication of an index is not deemed suitable. We then argue that clustering might be a viable methodology for building parsimonious tracking portfolios. With suitably defined distances between the time series of asset prices, clustering ‘discovers’ the correlation and cointegration structure of an index. Sampling the clusters with appropriate heuristics and optimization techniques, an optimal tracking portfolio can be constructed. One advantage of this approach is that it eschews the difficulties and computational burden of density forecasts and full optimization.     

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.     

The Optimal Log-Utility Asset Management under Incomplete Information

In this paper, we propose a theory for deriving the optimal portfolio that assures the log-utility investors of maximizing their expected utility. Restricting investors' information at defined levels, we propose the sample path-wise optimal portfolio (SPOP), which is consistent with the back-test framework used in actualinvestment. It is proven that, at any finite terminal time, this SPOP is asymptotically optimal among all the portfolios which are predictable under investors' incompleteinformation. The optimality is guaranteed by the continuous Bayesian updating formula. Finally, we discuss an algorithm for searching the SPOP, based on asset prices at discrete time intervals.     

Portfolio selection with a drawdown constraint

When identifying optimal portfolios, practitioners often impose a drawdown constraint. This constraint is even explicit in some money management contracts such as the one recently involving Merrill Lynch’ management of Unilever’s pension fund. In this setting, we provide a characterization of optimal portfolios using mean–variance analysis. In the absence of a benchmark, we find that while the constraint typically decreases the optimal portfolio’s standard deviation, the constrained optimal portfolio can be notably mean–variance inefficient. In the presence of a benchmark such as in the Merrill Lynch–Unilever contract, we find that the constraint increases the optimal portfolio’s standard deviation and tracking error volatility. Thus, the constraint negatively affects a portfolio manager’s ability to track a benchmark.     

Power mapping with dynamical adjustment for improved portfolio optimization

For financial risk management it is of vital interest to have good estimates for the correlations between the stocks. It has been found that the correlations obtained from historical data are covered by a considerable amount of noise, which leads to a substantial error in the estimation of the portfolio risk. A method to suppress this noise is power mapping. It raises the absolute value of each matrix element to a power q while preserving the sign. In this paper we use the Markowitz portfolio optimization as a criterion for the optimal value of q and find a K/T dependence, where K is the portfolio size and T the length of the time series. Both in numerical simulations and for real market data we find that power mapping leads to portfolios with considerably reduced risk. It compares well with another noise reduction method based on spectral filtering. A combination of both methods yields the best results.     

On <Emphasis Type="Italic">q</Emphasis>-optimal martingale measures in exponential Lévy models

We give a sufficient condition to identify the q-optimal signed and the q-optimal absolutely continuous martingale measures in exponential Lévy models. As a consequence, we find that in the one-dimensional case, the q-optimal equivalent martingale measures may exist only if the tails for upward jumps are extraordinarily light. Moreover, we derive the convergence of q-optimal signed, resp. absolutely continuous, martingale measures to the minimal entropy martingale measure as q approaches one. Finally, some implications for portfolio optimization are discussed. C.N. gratefully acknowledges financial support by UniCredit, Markets and Investment Banking. However, this paper does not reflect the opinion of UniCredit, Markets and Investment Banking, it is the personal view of the authors.     

Comparative issues between linear and non-linear risk measures for non-convex portfolio optimization: evidence from the S&P 500

Abstract

This article focuses on inferring critical comparative conclusions as far as the application of both linear and non-linear risk measures in non-convex portfolio optimization problems. We seek to co-assess a set of sophisticated real-world non-convex investment policy limitations, such as cardinality constraints, buy-in thresholds, transaction costs, particular normative rules, etc. within the frame of four popular portfolio selection cases: (a) the mean-variance model, (b) the mean-semi variance model, (c) the mean-MAD (mean-absolute deviation) model and (d) the mean-semi MAD model. In such circumstances, the portfolio selection process reflects to a mixed-integer bi-objective (or in general multiobjective) mathematical programme. We precisely develop all corresponding modelling procedures and then solve the underlying problem by use of a novel generalized algorithm, which was exclusively introduced to cope with the above-mentioned singularities. The validity of the attempt is verified through empirical testing on the S&P 500 universe of securities. The technical conclusions obtained not only confirm certain findings of the particular limited existing theory but also shed light on computational issues and running times. Moreover, the results derived are characterized as encouraging enough, since a sufficient number of efficient or Pareto optimal portfolios produced by the models appear to possess superior out-of-sample returns with respect to the benchmark.     

Portfolio optimization under a generalized hyperbolic skewed t distribution and exponential utility

In this paper, we show that if asset returns follow a generalized hyperbolic skewed t distribution, the investor has an exponential utility function and a riskless asset is available, the optimal portfolio weights can be found either in closed form or using a successive approximation scheme. We also derive lower bounds for the certainty equivalent return generated by the optimal portfolios. Finally, we present a study of the performance of mean–variance analysis and Taylor’s series expected utility expansion (up to the fourth moment) to compute optimal portfolios in this framework.     

Least-squares approach to risk parity in portfolio selection

The risk parity portfolio selection problem aims to find such portfolios for which the contributions of risk from all assets are equally weighted. Portfolios constructed using the risk parity approach are a compromise between two well-known diversification techniques: minimum variance optimization and the equal weighting approach. In this paper, we discuss the problem of finding portfolios that satisfy risk parity over either individual assets or groups of assets. We describe the set of all risk parity solutions by using convex optimization techniques over orthants and we show that this set may contain an exponential number of solutions. We then propose an alternative non-convex least-squares model whose set of optimal solutions includes all risk parity solutions, and propose a modified formulation which aims at selecting the most desirable risk parity solution according to a given criterion. When general bounds are considered, a risk parity solution may not exist. In this case, the non-convex least-squares model seeks a feasible portfolio which is as close to risk parity as possible. Furthermore, we propose an alternating linearization framework to solve this non-convex model. Numerical experiments indicate the effectiveness of our technique in terms of both speed and accuracy.     

How does the choice of Value-at-Risk estimator influence asset allocation decisions?

Considering the growing need for managing financial risk, Value-at-Risk (VaR) prediction and portfolio optimisation with a focus on VaR have taken up an important role in banking and finance. Motivated by recent results showing that the choice of VaR estimator does not crucially influence decision-making in certain practical applications (e.g. in investment rankings), this study analyses the important question of how asset allocation decisions are affected when alternative VaR estimation methodologies are used. Focusing on the most popular, successful and conceptually different conditional VaR estimation techniques (i.e. historical simulation, peak over threshold method and quantile regression) and the flexible portfolio model of Campbell et al. [J. Banking Finance. 2001, 25(9), 1789–1804], we show in an empirical example and in a simulation study that these methods tend to deliver similar asset weights. In other words, optimal portfolio allocations appear to be not very sensitive to the choice of VaR estimator. This finding, which is robust in a variety of distributional environments and pre-whitening settings, supports the notion that, depending on the specific application, simple standard methods (i.e. historical simulation) used by many commercial banks do not necessarily have to be replaced by more complex approaches (based on, e.g. extreme value theory).     

When do improved covariance matrix estimators enhance portfolio optimization? An empirical comparative study of nine estimators

The use of improved covariance matrix estimators as an alternative to the sample estimator is considered an important approach for enhancing portfolio optimization. Here we empirically compare the performance of nine improved covariance estimation procedures using daily returns of 90 highly capitalized US stocks for the period 1997–2007. We find that the usefulness of covariance matrix estimators strongly depends on the ratio between the estimation period T and the number of stocks N, on the presence or absence of short selling, and on the performance metric considered. When short selling is allowed, several estimation methods achieve a realized risk that is significantly smaller than that obtained with the sample covariance method. This is particularly true when T/N is close to one. Moreover, many estimators reduce the fraction of negative portfolio weights, while little improvement is achieved in the degree of diversification. On the contrary, when short selling is not allowed and T?>?N, the considered methods are unable to outperform the sample covariance in terms of realized risk, but can give much more diversified portfolios than that obtained with the sample covariance. When T?<?N, the use of the sample covariance matrix and of the pseudo-inverse gives portfolios with very poor performance.     

Weighing asset pricing factors: a least squares model averaging approach

Empirical evidence has demonstrated that certain factors in asset pricing models are more important than others for explaining specific portfolio returns. We propose a technique that evaluates the factors included in popular linear asset pricing models. Our method has the advantage of simultaneously ranking the relative importance of those pricing factors through comparing their model weights. As an empirical verification, we apply our method to portfolios formed following Fama and French [A five-factor asset pricing model. J. Financ. Econ., 2015, 116, 1–22] and demonstrate that models accommodated to our factor rankings do improve their explanatory power in both in-sample and out-of-sample analyses.     

Pareto‐archived evolutionary wavelet network for financial constrained portfolio optimization

The multi‐objective portfolio optimization problem is too complex to find direct solutions by traditional methods when constraints reflecting investor's preferences and/or market frictions are included in the mathematical model and hence heuristic approaches are sought for their solution. In this paper we propose the solution of a multi‐criterion (bi‐objective) portfolio optimization problem of minimizing risk and maximizing expected return of the portfolio which includes basic, bounding, cardinality, class and short sales constraints using a Pareto‐archived evolutionary wavelet network (PEWN) solution strategy. Initially, the empirical covariance matrix is denoised by employing a wavelet shrinkage denoising technique. Second, the cardinality constraint is eliminated by the application of k‐means cluster analysis. Finally, a PEWN heuristic strategy with weight standardization procedures is employed to obtain Pareto‐optimal solutions satisfying all the constraints. The closeness and diversity of Pareto‐optimal solutions obtained using PEWN is evaluated using different measures and the results are compared with existing only solution strategies (evolution‐based wavelet Hopfield neural network and evolution‐based Hopfield neural network) to prove its dominance. Eventually, data envelopment analysis is also used to test the efficiency of the non‐dominated solutions obtained using PEWN. Experimental results are demonstrated on the Bombay Stock Exchange, India (BSE200 index: period July 2001–July 2006), and the Tokyo Stock Exchange, Japan (Nikkei225 index: period March 2002–March 2007), data sets. Copyright © 2010 John Wiley & Sons, Ltd.