Sergio Ortobelli

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.

Portfolio Choice Theory with Non-Gaussian Distributed Returns

This chapter discusses the parametric distributions of asset returns and proposes portfolio choice models consistent with the maximization of the expected utility. We analyze multi-parameter models to select nonstochastically dominated portfolios when short sales are allowed and when short sales are not allowed. We also concentrate our attention on the stable distributional approach in order to derive optimal portfolios with heavy-tailed distributed financial returns. Finally, we examine and compare optimal allocations obtained with the multivariate normal model and the sub-Gaussian stable one.

Portfolio selection with stable distributed returns

Abstract. This paper analyzes and discusses the stable distributional approach in portfolio choice theory. We consider different hypotheses of portfolio selection with stable distributed returns and, more generally, with heavy-tailed distributed returns. In particular, we examine empirical differences among the optimal allocations obtained with the Gaussian and the stable non-Gaussian distributional assumption for the financial returns. Finally, we compare performances among stable multivariate models.

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.

Delta hedging strategies comparison

Abstract In this paper we implement dynamic delta hedging strategies based on several option pricing models. We analyze different subordinated option pricing models and we examine delta hedging costs using ex-post daily prices of S&P 500. Furthermore, we compare the performance of each subordinated model with the Black–Scholes model.

A Stochastic Model for Mortality Rate on Italian Data

A new stochastic model for mortality rate is proposed and analyzed on Italian mortality data. The model is based on a stochastic differential equation derived from a generalization of the Milevesky and Promislow model (Milevesky, M.A., Promislow, S.D.: Insur. Math. Econ. 29, 299–318 (2001)). We discuss and present a methodology, based on the discretisation approach by Wymer (Wymer, C.R.: Econometrica 40(3), 565–577 (1972)) to evaluate the parameters of our model. The comparison with the Milevesky and Promislow model shows the relevance of our proposal along an horizon, which includes periods of time with a different volatility of mortality rates. The estimate of the parameters turns out to be stable over time with the exception of the mean reverting parameter, which shows, for a person of a fixed age, an increase over time.

Moment based approaches to value the risk of contingent claim portfolios

Abstract In this paper we describe and apply the estimating function methodology to value the risk of asset derivative portfolios. We first implement the Li’s model based on the first four moments and then we show the limits of this model in forecasting the maximum loss of contingent claims. In addition, we show that four moments are not enough to describe the behavior of the lower percentiles of derivatives. Finally, we propose a model that considers the first six moments and we compare the performances of these models proposing a backtest analysis on several historical and truncated asset derivative portfolios.

Non-Gaussian Distribution for Var Calculation: An Assessment for the Italian Market

Abstract In this paper we compare different approaches to computing VaR (Value-at-Risk) for heavy tailed return series. Using data from the Italian market, we show that almost all the return series present statistically significant skewness and kurtosis. We implement (i) the stable models proposed by Rachev et al . (2000), (ii) an alternative to the Gaussian distributions based on a Generalized Error Distribution and (iii) a non-parametric model proposed by Li (1999). All the models are then submitted to backtest on out-of-sample data in order to assess their forecasting power. We observe that when the percentiles are low, all the models tested produce results that are dominant compared to the standard RiskMetrics model.