Almira Biglova

Different Approaches to Risk Estimation in Portfolio Theory

Some new performance measures may be regarded as alternatives to the most popular criterion for portfolio optimization, the Sharpe ratio. Analysis of some allocation problems here takes into consideration portfolio selection models based on different risk perceptions and sample paths of the final wealth process for each allocation problem. One new performance ratio seems to be suitable for some optimization problems, but we need a thorough classification of the set of performance measures that would be ideal for large classes of financial optimization problems.

Desirable Properties of an Ideal Risk Measure in Portfolio Theory

This paper examines the properties that a risk measure should satisfy in order to characterize an investors preferences. In particular, we propose some intuitive and realistic examples that describe several desirable features of an ideal risk measure. This analysis is the first step in understanding how to classify an investors risk. Risk is an asymmetric, relative, heteroskedastic, multidimensional concept that has to take into account asymptotic behavior of returns, inter-temporal dependence, risk-time aggregation, and the impact of several economic phenomena that could influence an investors preferences. In order to consider the financial impact of the several aspects of risk, we propose and analyze the relationship between distributional modeling and risk measures. Similar to the notion of ideal probability metric to a given approximation problem, we are in the search for an ideal risk measure or ideal performance ratio for a portfolio selection problem. We then emphasize the parallels between risk measures and probability metrics, underlying the computational advantage and disadvantage of different approaches.

THE PROPER USE OF RISK MEASURES IN PORTFOLIO THEORY

This paper discusses and analyzes risk measure properties in order to understand how a risk measure has to be used to optimize the investors portfolio choices. In particular, we distinguish between two admissible classes of risk measures proposed in the portfolio literature: safety-risk measures and dispersion measures. We study and describe how the risk could depend on other distributional parameters. Then, we examine and discuss the differences between statistical parametric models and linear fund separation ones. Finally, we propose an empirical comparison among three different portfolio choice models which depend on the mean, on a risk measure, and on a skewness parameter. Thus, we assess and value the impact on the investors preferences of three different risk measures even considering some derivative assets among the possible choices.

An Empirical Examination of Daily Stock Return Distributions for U.S. Stocks

This article investigates whether the Gaussian distribution hypothesis holds 382 U.S. stocks and compares it to the stable Paretian hypothesis. The daily returns are examined in the framework of two probability models - the homoskedastic independent, identical distributed model and the conditional heteroskedastic ARMA-GARCH model. Consistent with other studies, we strongly reject the Gaussian hypothesis for both models. We find out that the stable Paretian hypothesis better explains the tails and the central part of the return distribution.

Optimal Portfolio Selection and Risk Management: A Comparison between the Stable Paretian Approach and the Gaussian One

This paper analyzes stable Paretian models in portfolio theory, risk management and option pricing theory. Firstly, we examine investor’s optimal choices when we assume respectively either Gaussian or stable non-Gaussian distributed index returns. Thus, we approximate discrete time optimal allocations assuming different distributional assumptions and considering several term structure scenarios. Secondly, we compare some stable approaches to compute VaR for heavy-tailed return series. These models are subject to backtesting on out-of-sample data in order to assess their forecasting power. Finally, when asset prices are log-stable distributed, we propose a numerical valuation of option prices and we describe and compare delta hedging strategies when asset prices are either log-stable distributed or log-normal distributed.

A COMPARISON AMONG PERFORMANCE MEASURES IN PORTFOLIO THEORY

Abstract This paper examines some performance measures to be considered as an alternative of the Sharpe Ratio. More specifically, we analyze allocation problems taking into consideration portfolio selection models based on different performance ratios. For each allocation problem, we compare the maximum expected utility observing all the portfolio selection approaches proposed here. We also discuss an ex-post multi-period portfolio selection analysis in order to describe and compare the sample path of the final wealth processes.