On the Effect of Skewness and Kurtosis Misspecification on the Hedging Error

Using a result in Angelini and Herzel (2009a) , we measure, in terms of variance, the cost of hedging a contingent claim when the hedging portfolio is re‐balanced at a discrete set of dates. We analyse the dependence of the variance of the hedging error on the skewness and kurtosis as modeled by a Normal Inverse Gaussian model. We consider two types of strategies, the standard Black–Scholes Delta strategy and the locally variance‐optimal strategy, and we perform some robustness tests. In particular, we investigate the effect of different types of model misspecification on the performance of the hedging, like that of hedging without taking skewness into account. Computations are performed using a Fast Fourier Transform approach.     

Observational equivalence and nonequivalence of subjective and robust mean–variance preferences

We identify the conditions where robust mean–variance preferences, which capture ambiguity aversion, are observationally nonequivalent to subjective mean–variance preferences. Conversely, we also provide an example showing that observational equivalence holds regardless of the degree of ambiguity aversion.     

Portfolio choice with high frequency data: CRRA preferences and the liquidity effect

This paper suggests a new approach for portfolio choice. In this framework, the investor, with CRRA preferences, has two objectives: the maximization of the expected utility and the minimization of the portfolio expected illiquidity. The CRRA utility is measured using the portfolio realized volatility, realized skewness and realized kurtosis, while the portfolio illiquidity is measured using the well-known Amihud illiquidity ratio. Therefore, the investor is able to make her choices directly in the expected utility/liquidity (EU/L) bi-dimensional space. We conduct an empirical analysis in a set of fourteen stocks of the CAC 40 stock market index, using high frequency data for the time span from January 1999 to December 2005 (seven years). The robustness of the proposed model is checked according to the out-of-sample performance of different EU/L portfolios relative to the minimum variance and equally weighted portfolios. For different risk aversion levels, the EU/L portfolios are quite competitive and in several cases consistently outperform those benchmarks, in terms of utility, liquidity and certainty equivalent.     

SKEWNESS AND KURTOSIS PROPERTIES OF INCOME DISTRIBUTION MODELS

This paper explores the ability of some popular income distributions to model observed skewness and kurtosis. We present the generalized beta type 1 (GB1) and type 2 (GB2) distributions' skewness–kurtosis spaces and clarify and expand on previously known results on other distributions' skewness–kurtosis spaces. Data from the Luxembourg Income Study are used to estimate sample moments and explore the ability of the generalized gamma, Dagum, Singh–Maddala, beta of the first kind, beta of the second kind, GB1, and GB2 distributions to accommodate the skewness and kurtosis values. The GB2 has the flexibility to accurately describe the observed skewness and kurtosis.     

Global idiosyncratic risk moments

We investigate a global cross-sectional relation between idiosyncratic risk moments and expected stock returns by suggesting three global idiosyncratic volatility, skewness, and kurtosis risk factors. We also suggest two global small minus big and high minus low risk proxies for estimating return residuals of the test assets from a global asset pricing model. To perform robustness checks, we suggest other four global risk factors of momentum, leverage, bid-ask spread, and liquidity. We find a significant negative relation between stock portfolio returns and the global moments, and the cross section of stock returns reflects a significant negative price of risk for global idiosyncratic skewness (?0.13%) and idiosyncratic volatility (?1.85%) and a positive and significant price of risk for global idiosyncratic kurtosis. We find that our suggested risk factors are key drivers of risk premia in stock market and are robust to various checks. These factors also can forecast the gross domestic product growth over the sample period.     

Revealed preferences for portfolio selection – does skewness matter?

In this article, we consider the portfolio selection problem as a Bayesian decision problem. We compare the traditional mean–variance and mean–variance–skewness efficient portfolios. We develop bi-level programming problem to investigate the market’s preference for risk by using observed (market) weights. Numerical experiments are conducted on a portfolio formed by the 30 stocks in the Dow Jones Industrial Average. Numerical results show that the market’s preferences are better explained when skewness is included.     

Investment choices: Indivisible non-marketable assets and suboptimal solutions

Several investment decisions deal with non-marketable assets. Non-marketable assets are available only to one investor and are often indivisible. This has relevant consequences on investor investment opportunities. Adhering to a mean–variance representation of the investment space and considering a non-marketable asset (divisible or not), we derive some possible investment scenarios an investor may face. Furthermore, we show how a limited ability to gather and process information affects investor portfolio choices. Our results define a set of conditions under which the non-marketable asset represents a good investment and show that, under certain assumptions, the efficient frontier exhibits non-linearities and intervals of discontinuity.     

Accounting for the impact of higher order moments in foreign equity option pricing model

This paper investigates the pricing of foreign equity option whose value depends on foreign equity prices and exchange rate. We assume that the underlying asset returns of foreign equity option is not a Brownian motion, and use the Gram-Charlier series expansion to augment a normal density with two additional terms to capture the effects of skewness and kurtosis. The empirical study shows that the higher order moments (skewness and kurtosis) clearly affect the estimated prices of foreign equity options. This approach enables us to capture more accurately the foreign equity option prices.     

The role of transaction costs and risk aversion when selecting between one and two regimes for portfolio models

Estimation of the inputs is the main problem when applying portfolio analysis, and Markov regime-switching models have been shown to improve these estimates. We investigate whether the use of two-regime models remains superior across a range of values of risk aversion and transaction costs, in the presence of skewness and kurtosis and no short sales. Our results for US data suggest that, due to differences in their risk preferences and transactions costs, most retail investors may prefer to use one-regime models, while investment banks may prefer to use two-regime models.     

On mean-variance portfolio selection under a hidden Markovian regime-switching model

We study a mean-variance portfolio selection problem under a hidden Markovian regime-switching Black–Scholes–Merton economy. Under this model, the appreciation rate of a risky share is modulated by a continuous-time, finite-state hidden Markov chain whose states represent different states of an economy. We consider the general situation where an economic agent cannot observe the “true” state of the underlying economy and wishes to minimize the variance of the terminal wealth for a fixed level of expected terminal wealth with access only to information about the price processes. By exploiting the separation principle, we discuss the mean-variance portfolio selection problem and the filtering-estimation problem separately. We determine an explicit solution to the mean-variance problem using the stochastic maximum principle so that we do not need the assumption of Markovian controls. We also provide robust estimates of the hidden state of the chain and develop a robust filter-based EM algorithm for online recursive estimates of the unknown parameters in the model. This simplifies the filtering-estimation problem.     

The risk-return relations: evidence from the Korean and Taiwan stock markets

This article examines the risk-return relations conditional on up and down market periods in the Korean and Taiwan stock markets. Based on statistical tests adjusted for the effects of heteroskedasticity and autocorrelation of the residuals, beta is found positively (negatively) related to realized returns in up (down) markets. However, the results are sensitive to portfolio aggregation methods. Its role as a risk measure vanishes in down markets for the two-way (beta-size and size-beta) sorted portfolios. Unsystematic risk is significantly and positively priced only in up markets and mainly for beta-sorted portfolios while total risk is correctly priced except in Taiwan during down markets. Moreover, the impact of skewness and kurtosis on realized returns is not only sensitive to portfolio aggregation methods but also different across stock markets. They are found to be more relevant risk characteristics in the Korean than in the Taiwan stock market.     

Systematic skewness in asset pricing: an empirical examination of the Taiwan stock market

This paper is an empirical study of asset pricing with the systematic skewness in the pricing model. We adopt the Fama-French three-factor model, which incorporates the firm-size and book-to-market ratio in asset pricing as the base case, and then includes the skewness factor used by Harvey and Siddique in the pricing model. The evidence shows that systematic skewness is significant and might be important in asset pricing when portfolios are formed by industry, firm-size, book-to-market, or momentum strategies. When portfolios are constructed by momentum or coskewness strategies, lower momentum, or lower coskewness portfolios exhibit higher skewness and higher kurtosis. When portfolios are grouped by excess returns, it is seen that the average excess return is positively correlated with size and coskewness. Thus the systematic skewness is closely related to firm size. And the relationship between systematic skewness and excess return is obscured by the reverse firm-size effect.     

Multinationals and the gains from international diversification

This paper employs mean–variance spanning tests to examine the diversification potential of multinational firms and foreign market indices from the perspective of investors in the G7 countries over the 1984–1995 period. We find evidence that multinational corporations may have provided diversification benefits for investors in Germany and the United States. We find that the addition of foreign market indices to a domestic portfolio—inclusive of multinationals—provided substantial diversification benefits in all countries. The economic importance of the shift of the portfolio frontier varied considerably across markets.     

A two-period model with portfolio choice: Understanding results from different solution methods

Using a stylized two-period model we compare portfolio solutions from two local solution approaches–the approach of Judd and Guu (2001) and the approach of Devereux and Sutherland (2010, 2011)–with the true nonlinear portfolio solution.     

Modelling the risk–return relation for the S&P 100: The role of VIX

A significantly positive risk–return relation for the S&P 100 market index is detected if the implied volatility index (VIX) is allowed for as an exogenous variable in the conditional variance equation. This result holds for 4 alternative GARCH specifications, irrespective of the conditional distribution, and regardless of whether the conditional mean equation includes a constant term. This finding is robust to sub-samples, and to using VIX innovations to control for dividend yield and trading volume effects. Monte Carlo evidence suggests that if VIX is not included, the risk–return relation is more likely to be negative or weak, in line with several previous studies. If VIX is included, the distribution of the risk–return parameter has more than 99% of its mass in the area of positive values. We conclude that VIX carries important forward-looking information which improves the precision of the conditional variance estimation and, subsequently, reveals a significantly positive relation.     

Robust goal programming for multi-objective portfolio selection problem

This paper proposes a robust optimization model for the portfolio selection problem that uses a goal programming (GP) approach. In GP, decision makers can achieve more than one objective function. Some uncertain coefficients exist in both single and multi-objective models of the portfolio selection problem, which affects the feasibility and optimality of solutions. Robust optimization is an approach that deals with the uncertainty parameters in mathematical models, and guarantees the feasibility of the solutions. This paper tries to address the uncertainty parameters with robust optimization approach. This paper presents GP for the portfolio selection problem and addresses the uncertainty of the parameters by use of robust optimization approach. The approach is illustrated by a numerical example.     

Measuring and Optimizing Portfolio Credit Risk: A Copula‐based Approach*

In this work, we present a methodology for measuring and optimizing the credit risk of a loan portfolio taking into account the non‐normality of the credit loss distribution. In particular, we aim at modelling accurately joint default events for credit assets. In order to achieve this goal, we build the loss distribution of the loan portfolio by Monte Carlo simulation. The times until default of each obligor in portfolio are simulated following a copula‐based approach. In particular, we study four different types of dependence structure for the credit assets in portfolio: the Gaussian copula, the Student's t‐copula, the grouped t‐copula and the Clayton n‐copula (or Cook–Johnson copula). Our aim is to assess the impact of each type of copula on the value of different portfolio risk measures, such as expected loss, maximum loss, credit value at risk and expected shortfall. In addition, we want to verify whether and how the optimal portfolio composition may change utilizing various types of copula for describing the default dependence structure. In order to optimize portfolio credit risk, we minimize the conditional value at risk, a risk measure both relevant and tractable, by solving a simple linear programming problem subject to the traditional constraints of balance, portfolio expected return and trading. The outcomes, in terms of optimal portfolio compositions, obtained assuming different default dependence structures are compared with each other. The solution of the risk minimization problem may suggest us how to restructure the inefficient loan portfolios in order to obtain their best risk/return profile. In the absence of a developed secondary market for loans, we may follow the investment strategies indicated by the solution vector by utilizing credit default swaps.     

Economic benefits of using realized covariance forecasts in risk-based portfolios

This article examines the economic benefit of using the realized covariance matrix forecasts, for constructing the risk-based portfolios. We use the two-scale realized covariance estimator (TSC), the jump robust two-scale realized covariance estimator (RTSC) and the realized bipower covariance estimator (BPC), to forecast the daily realized covariance matrix. Using these covariance matrix forecasts, we implement three risk-based portfolios: the global minimum variance portfolio, the equal risk contribution portfolio and the most diversified portfolio. There is evidence that the portfolio performance improves by using TSC or RTSC estimators as compared to the daily-returns-based estimator. The performance gains are robust to the choice of risk-based portfolio strategy, the degree of investor’s relative risk-aversion, the market conditions and the choice of time intervals.     

Market moves and the information content of option prices

In this paper we estimate risk-neutral returns distributions using the prices of options written on S&P 500 index futures and investigate whether or not specific characteristics of the returns distributions might be useful information for the purpose of predicting changes in market direction. The key distributional characteristics we focus on are skewness, kurtosis, and the probability weight in the extreme tails of the implied risk-neutral returns distributions. We find that, with one possible exception, the characteristics we considered are unlikely to improve a trader's ability to predict market moves.