Maria Bertocchi

Stable distributions in the Black-Litterman approach to asset allocation

The integration of quantitative asset allocation models and the judgment of portfolio managers and analysts (i.e. qualitative view) dates back to a series of papers by Black and Litterman in the early 1990s. In this paper we improve the classical Black–Litterman model by applying more realistic models for asset returns (the normal, the t-student, and the stable distributions) and by using alternative risk measures (dispersion-based risk measures, value at risk, conditional value at risk). Results are reported for monthly data and goodness of the models are tested through a rolling window of fixed size along a fixed horizon. Finally, we find that incorporation of the views of investors into the model provides information as to how the different distributional hypotheses can impact the optimal composition of the portfolio.

A stochastic model for the daily coordination of pumped storage hydro plants and wind power plants

We propose a stochastic model for the daily operation scheduling of a generation system including pumped storage hydro plants and wind power plants, where the uncertainty is represented by the hourly wind power production. In order to assess the value of the stochastic modeling, we discuss two case studies: in the former the scenario tree is built so as to include both low and high wind power production scenarios, in the latter the scenario tree is built on historical wind speed data covering a time span of one and a half year. The Value of the Stochastic Solution, computed by a modified new procedure, shows that in scenarios with low wind power production the stochastic solution allows the producer to obtain a profit which is greater than the one associated to the deterministic solution. In-sample stability of the optimal function values for increasing number of scenarios is reported.

Aggregation issues in operational risk

In this paper we study copula-based models for aggregation of operational risk capital across business lines in a bank. A commonly used method of summation of the value-at-risk (VaR) measures, which relies on a hypothesis of full correlation of losses, becomes inappropriate in the presence of dependence between business lines and may lead to overestimation of the capital charge. The problem can be further aggravated by the persistence of heavy tails in operational loss data; in some cases, the subadditivity property of VaR may fail and the capital charge becomes underestimated. We use α-stable heavy-tailed distributions to model the loss data and then apply the copula approach in which the marginal distributions are consolidated in the symmetric and skewed Student t-copula framework. In our empirical study, we compare VaR and conditional VaR estimates with those obtained under the full correlation assumption. Our results demonstrate a significant reduction in capital when a t-copula is employed. However, the capital reduction is significantly smaller than in cases where a moderately heavy-tailed or thin-tailed distribution is calibrated to loss data. We also show that, when historical weekly data is used, VaR exhibits the superadditivity property for confidence levels below 94% and that, when the loss distribution approach is used, the superadditivity of VaR is observed at a higher confidence level (98%).

Bond portfolio management via stochastic programming

Publisher Summary Stochastic programming is a tool to support bond portfolio management decisions. For a successful application of the stochastic programming methodology, one must choose an adequate model, asses its parameters, generate sensible input scenarios or scenario tree, solve the scenario-based problem using an optimization software and validate the results. This chapter discusses formulation of two-stage and three-stage models for bond portfolio optimization and displays computational results. It presents the comparisons of monthly and quarterly time discretizations and for various topologies of the input scenario trees. It uses Black-Derman-Toy binomial lattice calibrated from market data to generate scenarios and scenario reduction and scenario construction methods are applied using GAMS. The contamination technique is exploited to quantify the impact of including additional (stress or out-of-sample) scenarios and/or additional stages to an already selected scenario tree.

A mixed integer nonlinear optimization model for gas sale company

In this paper the authors propose an optimisation model, called OMoGaS (Optimisation Modelling for Gas Seller), to assist companies dealing with gas retail commercialisation. The model takes into account the limits on price imposed by law on small consumers as well as the gas company policies in order to explore the commercial consequences of different policies. The GAMS framework is used for the optimisation of the defined MINLP model where the profit function is based on the number of contracts with the final consumers, on the tipology of consumers and on the cost supported to meet the final demand while the constraints include information on a maximum daily gas consumption, on yearly maximum and minimum comsumption in order to avoid penalties and on consumption profiles. A case study is presented.

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.

Heavy-tailed distributional model for operational losses

We examine the statistical properties of operational losses obtained from a large European bank using an actuarial-type framework. The simplistic assumption of a Poisson frequency distribution fails and we show that the frequency process follows closely a non-homogeneous Poisson process with a deterministic intensity of the form of a continuous cdf-like function. Further, operational losses are modeled using a variety of distributions. We address the problems of (1) reporting bias; (2) supplementing internal data with external data; (3) tail estimation; and (4) mixing the distributions of the body and the tail, and propose practical solutions to such problems. Finally, our empirical findings are consistent with other studies reporting very heavy-tailed loss distributions with the tail index below unity.

A GAS RETAIL STOCHASTIC OPTIMIZATION MODEL BY MEAN REVERTING TEMPERATURE SCENARIOS

The paper deals with a new stochastic optimization model, named OMoGaS-SV (Optimisation Modelling for Gas Seller-Stochastic Version), to assist companies dealing with gas retail commercialization. Stochasticity is due to the dependence of consumptions on temperature uncertainty. Due to nonlinearities present in the objective function, the model can be classified as an NLP mixed integer model, with the profit function depending on the number of contracts with the final consumers, the typology of such consumers and the cost supported to meet the final demand. Constraints related to a maximum daily gas consumption, to yearly maximum and minimum consumption in order to avoid penalties and to consumption profiles are included. The results obtained by the stochastic version give clear indication of the amount of losses that may appear in the gas seller’s budget.