Diana Roman

Mean-risk models using two risk measures: A multi-objective approach

This paper proposes a model for portfolio optimization, in which distributions are characterized and compared on the basis of three statistics: the expected value, the variance and the CVaR at a specified confidence level. The problem is multi-objective and transformed into a single objective problem in which variance is minimized while constraints are imposed on the expected value and CVaR. In the case of discrete random variables, the problem is a quadratic program. The mean-variance (mean-CVaR) efficient solutions that are not dominated with respect to CVaR (variance) are particular efficient solutions of the proposed model. In addition, the model has efficient solutions that are discarded by both mean-variance and mean-CVaR models, although they may improve the return distribution. The model is tested on real data drawn from the FTSE 100 index. An analysis of the return distribution of the chosen portfolios is presented.

Portfolio construction based on stochastic dominance and target return distributions

Mean-risk models have been widely used in portfolio optimization. However, such models may produce portfolios that are dominated with respect to second order stochastic dominance and therefore not optimal for rational and risk-averse investors. This paper considers the problem of constructing a portfolio which is non-dominated with respect to second order stochastic dominance and whose return distribution has specified desirable properties. The problem is multi-objective and is transformed into a single objective problem by using the reference point method, in which target levels, known as aspiration points, are specified for the objective functions. A model is proposed in which the aspiration points relate to ordered outcomes for the portfolio return. This concept is extended by additionally specifying reservation points, which act pre-emptively in the optimization model. The theoretical properties of the models are studied. The performance of the models on real data drawn from the Hang Seng index is also investigated.

Enhanced indexation based on second-order stochastic dominance

Second order Stochastic Dominance (SSD) has a well recognised importance in portfolio selection, since it provides a natural interpretation of the theory of risk-averse investor behaviour. Recently, SSD-based models of portfolio choice have been proposed; these assume that a reference distribution is available and a portfolio is constructed, whose return distribution dominates the reference distribution with respect to SSD. We present an empirical study which analyses the effectiveness of such strategies in the context of enhanced indexation. Several datasets, drawn from FTSE 100, SP 500 and Nikkei 225 are investigated through portfolio rebalancing and backtesting. Three main conclusions are drawn. First, the portfolios chosen by the SSD based models consistently outperformed the indices and the traditional index trackers. Secondly, the SSD based models do not require imposition of cardinality constraints since naturally a small number of stocks are selected. Thus, they do not present the computational difficulty normally associated with index tracking models. Finally, the SSD based models are robust with respect to small changes in the scenario set and little or no rebalancing is necessary.

An enhanced model for portfolio choice with SSD criteria: a constructive approach

We formulate a portfolio planning model that is based on second-order stochastic dominance as the choice criterion. This model is an enhanced version of the multi-objective model proposed by Roman et al. [Math. Progr. Ser. B, 2006, 108, 541–569]; the model compares the scaled values of the different objectives, representing tails at different confidence levels of the resulting distribution. The proposed model can be formulated as a risk minimization model where the objective function is a convex risk measure; we characterize this risk measure and the resulting optimization problem. Moreover, our formulation offers a natural generalization of the SSD-constrained model of Dentcheva and Ruszczyński [J. Bank. Finance, 2006, 30, 433–451]. A cutting plane-based solution method for the proposed model is outlined. We present a computational study showing: (a) the effectiveness of the solution methods and (b) the improved modeling capabilities: the resulting portfolios have superior return distributions.

An algorithm for moment-matching scenario generation with application to financial portfolio optimisation

We present an algorithm for moment-matching scenario generation. This method produces scenarios and corresponding probability weights that match exactly the given mean, the covariance matrix, the average of the marginal skewness and the average of the marginal kurtosis of each individual component of a random vector. Optimisation is not employed in the scenario generation process and thus the method is computationally more advantageous than previous approaches. The algorithm is used for generating scenarios in a mean-CVaR portfolio optimisation model. For the chosen optimisation example, it is shown that desirable properties for a scenario generator are satisfied, including in-sample and out-of-sample stability. It is also shown that optimal solutions vary only marginally with increasing number of scenarios in this example; thus, good solutions can apparently be obtained with a relatively small number of scenarios. The proposed method can be used either on its own as a computationally inexpensive scenario generator or as a starting point for non-convex optimisation based scenario generators which aim to match all the third and the fourth order marginal moments (rather than average marginal moments).

HMM based scenario generation for an investment optimisation problem

The Geometric Brownian motion (GBM) is a standard method for modelling financial time series. An important criticism of this method is that the parameters of the GBM are assumed to be constants; due to this fact, important features of the time series, like extreme behaviour or volatility clustering cannot be captured. We propose an approach by which the parameters of the GBM are able to switch between regimes, more precisely they are governed by a hidden Markov chain. Thus, we model the financial time series via a hidden Markov model (HMM) with a GBM in each state. Using this approach, we generate scenarios for a financial portfolio optimisation problem in which the portfolio CVaR is minimised. Numerical results are presented.

Portfolio Choice Models Based on Second-Order Stochastic Dominance Measures: An Overview and a Computational Study

In this chapter we present an overview of second-order stochastic dominance-based models with a focus on those using dominance measures. In terms of portfolio policy, the aim is to find a portfolio whose return distribution dominates the index distribution to the largest possible extent. We compare two approaches, the unscaled model of Roman et al. (Mathematical Programming Series B 108: 541–569, 2006) and the scaled model of Fabian et al. (Quantitative Finance 2010). We constructed optimal portfolios using representations of the future asset returns given by historical data on the one hand, and scenarios generated by geometric Brownian motion on the other hand. In the latter case, the parameters of the GBM were obtained from the historical data. Our test data consisted of stock returns from the FTSE 100 basket, together with the index returns. Part of the data were reserved for out-of-sample tests. We examined the return distributions belonging to the respective optimal portfolios of the unscaled and the scaled problems. The unscaled model focuses on the worst cases and hence enhances safety. We found that the performance of the unscaled model is improved by using scenario generators. On the other hand, the scaled model replicates the shape of the index distribution. Scenario generation had little effect on the scaled model. We also compared the shapes of the histograms belonging to corresponding pairs of in-sample and out-of-sample tests and observed a remarkable robustness in both models. We think these features make these dominance measures good alternatives for classic risk measures in certain applications, including certain multistage ones. We mention two candidate applications.

Mixture Distribution Scenarios for Investment Decisions with Downside Risk

Recently considerable attention has been given to downside risk control in the context of portfolio choice; see Sortino and Satchell (2005). We propose an integrated model for portfolio choice in which downside risk is considered explicitly at the stage of the scenario generation which describes asset price behaviour and in the subsequent step of portfolio construction. The scenario generation method is deliberately chosen to create a mixture discrete distribution for future asset returns. The behaviour and dependence of asset returns in the lower tails are captured using a discrete approximation to a multivariate copula model. The upside returns are modelled using a discrete approximation to a multivariate generalized autoregressive conditional heteroskedasticity (MGARCH) model. The asset allocation problem is based on an extension of the mean-variance approach, with an additional conditional value at risk (CVaR) constraint. The implementation of the model is that of a long short portfolio of exchange traded funds (ETF) of equity indices. Computational results of validating the model and backtesting are presented.

Employees’ Provident Funds of Singapore, Malaysia, India and Sri Lanka: a Comparative Study

Demographic changes affect social and economic performance all over the world. Current demographic trends, such as declining fertility rates, declining mortality rates and increasing life expectancies, are causing an aging population, in which the proportion of elderly people to the total popu-lation is increasing (Long, 2008). In 2000, less than one in ten people were over 60 years old, but estimates indicate that by year 2050 one in every five people will be over 60 years old (United Nations, 2000). As an example, in Japan, which is one of the fastest aging nations in the world, there were 9.3 people under 20 for every person over 65 in 1950; for 2025, this ratio is forecasted to be 0.59 people under 20 for every person older than 65 (United Nations, 2000).

ALM models based on second order stochastic dominance

We propose asset and liability management models in which the risk of underfunding is modelled based on the concept of stochastic dominance. Investment decisions are taken such that the distribution of the funding ratio, that is, the ratio of asset to liabilities, is non-dominated with respect to second order stochastic dominance. In addition, the funding ratio distribution is close in an optimal sense to a user-specified target distribution. Interesting results are obtained when the target distribution is degenerate; in this case, we can obtain equivalent risk minimisation models, with risk defined as expected shortfall or as worst case loss. As an application, we consider the financial planning problem of a defined benefit pension fund in Saudi Arabia.