Nonlinearly weighted convex risk measure and its application

We propose a new class of risk measures which satisfy convexity and monotonicity, two well-accepted axioms a reasonable and realistic risk measure should satisfy. Through a nonlinear weight function, the new measure can flexibly reflect the investor’s degree of risk aversion, and can control the fat-tail phenomenon of the loss distribution. A realistic portfolio selection model with typical market frictions taken into account is established based on the new measure. Real data from the Chinese stock markets and American stock markets are used for empirical comparison of the new risk measure with the expected shortfall risk measure. The in-sample and out-of-sample empirical results show that the new risk measure and the corresponding portfolio selection model can not only reflect the investor’s risk-averse attitude and the impact of different trading constraints, but can find robust optimal portfolios, which are superior to the corresponding optimal portfolios obtained under the expected shortfall risk measure.     

Two-sided coherent risk measures and their application in realistic portfolio optimization

By using a different derivation scheme, a new class of two-sided coherent risk measures is constructed in this paper. Different from existing coherent risk measures, both positive and negative deviations from the expected return are considered in the new measure simultaneously but differently. This innovation makes it easy to reasonably describe and control the asymmetry and fat-tail characteristics of the loss distribution and to properly reflect the investor’s risk attitude. With its easy computation of the new risk measure, a realistic portfolio selection model is established by taking into account typical market frictions such as taxes, transaction costs, and value constraints. Empirical results demonstrate that our new portfolio selection model can not only suitably reflect the impact of different trading constraints, but find more robust optimal portfolios, which are better than the optimal portfolio obtained under the conditional value-at-risk measure in terms of diversification and typical performance ratios.     

Performance ratio-based coherent risk measure and its application

Utilizing a specific acceptance set, we propose in this paper a general method to construct coherent risk measures called the generalized shortfall risk measure. Besides some existing coherent risk measures, several new types of coherent risk measures can be generated. We investigate the generalized shortfall risk measure’s desirable properties such as consistency with second-order stochastic dominance. By combining the performance evaluation with the risk control, we study in particular the performance ratio-based coherent risk (PRCR) measures, which is a sub-class of generalized shortfall risk measures. The PRCR measures are tractable and have a suitable financial interpretation. Based on the PRCR measure, we establish a portfolio selection model with transaction costs. Empirical results show that the optimal portfolio obtained under the PRCR measure performs much better than the corresponding optimal portfolio obtained under the higher moment coherent risk measure.     

A knowledge‐based framework for incorporating investor's preference into portfolio decision‐making

This paper describes a framework for the integration of a rule‐based system capable of identifying an investor's risk preference into a quantitative portfolio model based on risk and expected return. By inferring rules consisting of an investor's objective and subjective risk preferences, the integrated methodology provides the assets suitable for the preferences. Through investment in the portfolio composed of the assets, the investor is able to obtain the following bene?ts: reduction of costs and time spent to determine target assets, and alleviation of anxiety from ‘out‐of‐favor’ assets. The framework is applied to the development of a knowledge‐based portfolio system for constructing an investor's preference‐oriented portfolio. In the procedure of the system for ?nding an optimal portfolio, the system uses an arti?cial intelligence method of a case‐based reasoning to obtain preference thresholds for an investor when the investor's past investment records are available. Experimental results show that the framework contributes signi?cantly to the construction of a better portfolio from the perspective of an investor's bene?t/cost ratio than that produced by the existing portfolio models. Copyright © 2004 John Wiley & Sons, Ltd.     

Hedging under multiple risk constraints

Motivated by the asset–liability management problems under shortfall risk constraints, we consider in a general discrete-time framework the problem of finding the least expensive portfolio whose shortfalls with respect to a given set of stochastic benchmarks are bounded by a specific shortfall risk measure. We first show how the price of this portfolio may be computed recursively by dynamic programming for different shortfall risk measures, in complete and incomplete markets. We then focus on the specific situation where the shortfall risk constraints are imposed at each period on the next-period shortfalls, and obtain explicit results. Finally, we apply our results to a realistic asset–liability management problem of an energy company, and show how the shortfall risk constraints affect the optimal hedging of liabilities.     

Dynamic portfolio optimization with liquidity cost and market impact: a simulation-and-regression approach

We present a simulation-and-regression method for solving dynamic portfolio optimization problems in the presence of general transaction costs, liquidity costs and market impact. This method extends the classical least squares Monte Carlo algorithm to incorporate switching costs, corresponding to transaction costs and transient liquidity costs, as well as multiple endogenous state variables, namely the portfolio value and the asset prices subject to permanent market impact. To handle endogenous state variables, we adapt a control randomization approach to portfolio optimization problems and further improve the numerical accuracy of this technique for the case of discrete controls. We validate our modified numerical method by solving a realistic cash-and-stock portfolio with a power-law liquidity model. We identify the certainty equivalent losses associated with ignoring liquidity effects, and illustrate how our dynamic optimization method protects the investor's capital under illiquid market conditions. Lastly, we analyze, under different liquidity conditions, the sensitivities of certainty equivalent returns and optimal allocations with respect to trading volume, stock price volatility, initial investment amount, risk aversion level and investment horizon.     

Investor preferences and portfolio selection: is diversification an appropriate strategy?

This paper analyzes the relationship between diversification and several distributional characteristics that have risk implications for stock returns. We develop a flexible three-parameter distribution to model the stock returns. Using data on the current 30 DJIA stocks, we show that an investor's strategy on diversification depends on the measures of risk for particular concerns. For example, investors who desire to increase positive skewness would hold a less diversified portfolio, while those who care more about extreme losses would hold a more diversified portfolio. Experimenting with a more general pool of stocks yields the same conclusions.     

MEAN GINI CAPITAL ASSET PRICING MODEL: SOME EMPIRICAL EVIDENCE

The capital asset pricing model of Sharpe (1964) and Lintner (1965) provides a valid approach to portfolio selection if either the distribution of asset returns is jointly normal or the investor's preference function is quadratic. Various authors have questioned the validity of these assumptions, and Roll (1977) raises the question whether the traditional CAPM can be tested. An alternative Capital Asset Pricing Model has been proposed by Shalit and Yitzhaki (1984). In this model the extended mean Gini coefficient is used to measure risk. As little research has been conducted on this model, this paper estimates systematic risk as derived from the extended mean Gini model for a sample of Australian companies and compares the empirical security market line with the predicted extended mean Gini security market line.     

Sufficiency Conditions For Inclusion of One Asset Over Another in Investment Portfolios

By extending Tsiang's (1972) analysis to encompass two risky assets, sufficiency conditions for including one asset over another in any investor's investment portfolio are derived. This derivation stems from the fact that any realistic utility function must have indifference curves with slopes less than one. Using this model's framework, it is found that short-term Treasury bills in addition to cash balances cannot be a component of investor's investment portfolios. The results have implications for both the risk-free rate used in portfolio analysis and provide a partial solution to Mehra and Prescott's (1985) equity premium puzzle.     

Optimal mean-reversion strategy in the presence of bid-ask spread and delays in capital allocations

A portfolio optimization problem for an investor who trades T-bills and a mean-reverting stock in the presence of proportional and convex transaction costs is considered. The proportional transaction cost represents a bid-ask spread, while the convex transaction cost is used to model delays in capital allocations. I utilize the historical bid-ask spread in US stock market and assume that the stock reverts on yearly basis, while an investor follows monthly changes in the stock price. It is found that proportional transaction cost has a relatively weak effect on the expected return and the Sharpe ratio of the investor's portfolio. Meantime, the presence of delays in capital allocations has a dramatic impact on the expected return and the Sharpe ratio of the investor's portfolio. I also find the robust optimal strategy in the presence of model uncertainty and show that the latter increases the effective risk aversion of the investor and makes her view the stock as more risky.     

On the Predictability of Stock Returns: An Asset-Allocation Perspective

Sample evidence about the predictability of monthly stock returns is considered from the perspective of a risk-averse Bayesian investor who must allocate funds between stocks and cash. The investor uses the sample evidence to update prior beliefs about the parameters in a regression of stock returns on a set of predictive variables. The regression relation can seem weak when described by usual statistical measures, but the current values of the predictive variables can exert a substantial influence on the investor's portfolio decision, even when the investor's prior beliefs are weighted against predictability.     

Equilibrium Exchange Rate Hedging

We assume a world like the one that gives the capital asset pricing model, but with many goods and many countries. We assume that investors in a given country have homothetic utility functions with the same weights, and a currency that has a sure end-of-period value using a price index with those weights. Siegel's paradox (derived from Jensen's inequality) makes investors want a positive amount of exchange risk. When average risk tolerance is the same across countries, every investor will hold the same mix of market risk (through the world market portfolio of all assets) and exchange risk (in a diversified basket of foreign currencies). In fact, the ratio of exchange risk to market risk is equal to the average investor's risk tolerance. We can write the ratio of exchange risk to market risk (and the fraction of the market's exchange risk that investors hedge) as depending on an average of world market risk premia, an average of world market volatilities, and an average of exchange rate volatilities. The weights in these averages are the same as the weights of the different countries in the currency basket. Given these averages, the ratio (and the fraction hedged) will not depend directly on exchange rate means or covariances. In equilibrium, we can use the ratio of exchange risk to market risk to measure average risk tolerance: in this model, risk tolerance is observable.     

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.     

Portfolio performance evaluation with loss aversion

In this paper we consider a loss-averse investor equipped with a specific, but still quite general, utility function motivated by behavioral finance. We show that, under certain concrete assumptions concerning the form of this utility, one can derive closed-form solutions for the investor's portfolio performance measure. We investigate the effects of loss aversion and demonstrate its important role in performance measurement. The framework presented in this paper also provides a sound theoretical foundation for all known performance measures based on partial moments of the distribution.     

CAPM and APT-like models with risk measures

The paper deals with optimal portfolio choice problems when risk levels are given by coherent risk measures, expectation bounded risk measures or general deviations. Both static and dynamic pricing models may be involved. Unbounded problems are characterized by new notions such as (strong) compatibility between prices and risks. Surprisingly, the lack of bounded optimal risk and/or return levels arises for important pricing models (Black and Scholes) and risk measures (VaR, CVaR, absolute deviation, etc.). Bounded problems present a Market Price of Risk and generate a pair of benchmarks. From these benchmarks we introduce APT and CAPM-like analyses, in the sense that the level of correlation between every available security and some economic factors explains the security expected return. The risk level non correlated with these factors has no influence on any return, despite the fact that we are dealing with risk functions beyond the standard deviation.     

The implied Sharpe ratio

In an incomplete market, including liquidly traded European options in an investment portfolio could potentially improve the expected terminal utility for a risk-averse investor. However, unlike the Sharpe ratio, which provides a concise measure of the relative investment attractiveness of different underlying risky assets, there is no such measure available to help investors choose among the different European options. We introduce a new concept—the implied Sharpe ratio—which allows investors to make such a comparison in an incomplete financial market. Specifically, when comparing various European options, it is the option with the highest implied Sharpe ratio that, if included in an investor's portfolio, will improve his expected utility the most. Through the method of Taylor series expansion of the state-dependent coefficients in a nonlinear partial differential equation, we also establish the behaviour of the implied Sharpe ratio with respect to an investor's risk-aversion parameter. In a series of numerical studies, we compare the investment attractiveness of different European options by studying their implied Sharpe ratio.     

Portfolio selection and skewness: Evidence from international stock markets

This paper finds that the returns of the world's 14 major stock markets are not normally distributed, and that the correlation matrix of these stock markets was stable during the January 1988–December 1993 time period. Polynomial goal programming, in which investor preferences for skewness can be incorporated, is utilized to determine the optimal portfolio consisting of the choices of 14 international stock indexes. The empirical findings suggest that the incorporation of skewness into an investor's portfolio decision causes a major change in the construction of the optimal portfolio. The evidence also indicate that investors trade expected return of the portfolio for skewness.     

Recursive risk measures under regime switching applied to portfolio selection

In this paper, we define the conditional risk measure under regime switching and derive a class of time consistent multi-period risk measures. To do so, we describe the information process with regime switching in a product space associated with the product of two filtrations. Moreover, we show how to establish the corresponding multi-stage portfolio selection models using the time consistent multi-period risk measure for medium-term or long-term investments. Take the conditional value-at-risk measure as an example, we demonstrate the resulting multi-stage portfolio selection problem can be transformed into a second-order cone programming problem. Finally, we carry out a series of empirical tests to illustrate the superior performance of the proposed random framework and the corresponding multi-stage portfolio selection model.     

Optimal allocation to hedge funds: an empirical analysis

What percentage of their portfolio should investors allocate to hedge funds? The only available answers to the above question are set in a static mean-variance framework, with no explicit accounting for uncertainty on the active manager's ability to generate abnormal return, and usually generate unreasonably high allocations to hedge funds. In this paper, we apply the model introduced in Cvitanic et al (2002b Working Paper USC) for optimal investment strategies in the presence of uncertain abnormal returns to a database of hedge funds. We find that the presence of the model risk significantly decreases an investor's optimal allocation to hedge funds. Another finding of this paper is that low beta hedge funds may serve as natural substitutes for a significant portion of investor risk-free asset holdings.     

International portfolio selection and efficiency analysis

In the risk-return tradeoff, the traditional mean-variance analysis has been widely used for studies of international portfolio efficiency and diversification. Without prior knowledge about either the parametric structure of assets' return distributions or the form of investors' preference functions, the variance may no longer serve as a suitable risk proxy. This article examines international portfolio efficiency and diversification effects through mean-variance and various distribution-free (or less restrictive) risk-return measures. We show empirically that the mean-variance model is appropriate for large or well-diversified portfolios, but may provide biased results for single assets and less diversified portfolios. While stochastic dominance stands as theoretically the most appropriate method of international portfolio selection and efficiency analysis, the lack of optimal search algorithms reduces its practical usefulness. Very little gain is obtained by using the Gini-mean-difference risk measure as compared to the semivariance measure. The semivariance measure is a powerful and convenient discriminator of risky prospects, while stochastic dominance can serve as a benchmark to justify portfolio efficiency.