Georg Ch. Pflug

Some Remarks on the Value-at-Risk and the Conditional Value-at-Risk

The value-at-risk (VaR) and the conditional value-at-risk (CVaR) are two commonly used risk measures. We state some of their properties and make a comparison. Moreover, the structure of the portfolio optimization problem using the VaR and CVaR objective is studied.

A branch and bound method for stochastic global optimization

A stochastic branch and bound method for solving stochastic global optimization problems is proposed. As in the deterministic case, the feasible set is partitioned into compact subsets. To guide the partitioning process the method uses stochastic upper and lower estimates of the optimal value of the objective function in each subset. Convergence of the method is proved and random accuracy estimates derived. Methods for constructing stochastic upper and lower bounds are discussed. The theoretical considerations are illustrated with an example of a facility location problem.

Value-at-risk in Portfolio Optimization: Properties and Computational Approach

Value-at-Risk (VAR) is an important and widely used measure of the extent to which a given portfolio is subject to risk present in financial markets. In this paper, we present a method of calculating a portfolio that gives the optimal VAR among those which yield at least some specified expected return. This method allows us to calculate the mean-VAR-efficient frontier. The method is based on the approximation of historical VAR by smoothed VAR (SVAR), which filters out local irregular behavior of the historical VAR function. Moreover, we compare VAR as a risk measure to other well-known measures of risk, such as conditional value-at risk (CVAR) and the standard deviation. We show that the resulting efficient frontiers are quite different. An investor who wants to controls his or her VAR should not look at portfolios lying on other than the VAR efficient frontier, although the calculation of this frontier is algorithmically more complex than other frontiers. We support this conjecture by presenting the results of a large-scale experiment with a representative selection of stock and bond indices from developed and emerging markets that involved the computation of many thousand VAR-optimal portfolios.

Stochastic approximation and optimization of random systems

I Foundations of stochastic approximation.- 1 Almost sure convergence of stochastic approximation procedures.- 2 Recursive methods for linear problems.- 3 Stochastic optimization under stochastic constraints.- 4 A learning model recursive density estimation.- 5 Invariance principles in stochastic approximation.- 6 On the theory of large deviations.- References for Part I.- II Applicational aspects of stochastic approximation.- 7 Markovian stochastic optimization and stochastic approximation procedures.- 8 Asymptotic distributions.- 9 Stopping times.- 10 Applications of stochastic approximation methods.- References for Part II.- III Applications to adaptation algorithms.- 11 Adaptation and tracking.- 12 Algorithm development.- 13 Asymptotic Properties in the decreasing gain case.- 14 Estimation of the tracking ability of the algorithms.- References for Part III.

Simulated Annealing for noisy cost functions

We generalize a classical convergence result for the Simulated Annealing algorithm to a stochastic optimization context, i.e., to the case where cost function observations are disturbed by random noise. It is shown that for a certain class of noise distributions, the convergence assertion remains valid, provided that the standard deviation of the noise is reduced in the successive steps of cost function evaluation (e.g., by repeated observation) with a speed O(k-γ), where γ is an arbitrary constant larger than one.

Ambiguity in portfolio selection

In this paper, we consider the problem of finding optimal portfolios in cases when the underlying probability model is not perfectly known. For the sake of robustness, a maximin approach is applied which uses a ‘confidence set’ for the probability distribution. The approach shows the tradeoff between return, risk and robustness in view of the model ambiguity. As a consequence, a monetary value of information in the model can be determined.

Sovereign financial disaster risk management: The case of Mexico

Abstract In 2006, Mexico became the first transition country to transfer part of its public-sector natural catastrophe risk to the international reinsurance and capital markets. The Mexican case is of considerable interest to highly exposed transition and developing countries, many of which are considering similar transactions. Risk financing instruments can assure governments of sufficient post-disaster capital to provide emergency response, disaster relief to the affected population and repair public infrastructure. The costs of financial instruments, however, can greatly exceed expected losses, and for this reason it is important to closely examine their benefits and alternatives. This paper analyzes the Mexican case from the perspective of the risk cedent (the Ministry of Finance and Public Credit), which was informed by analyses provided by the International Institute for Applied Systems Analysis (IIASA). The rationale for a government to insure its contingent liabilities is presented along with the fiscal, legal and institutional context of the Mexican transaction. Using publicly available data, the paper scrutinizes the choice the authorities faced between two different risk-transfer instruments: reinsurance and a catastrophe bond. Making use of IIASAs catastrophe simulation model (CATSIM), this financial risk management decision is analyzed within the context of a public investment decision.

Dynamic Stochastic Optimization

Optimization problems arising in practice involve random parameters. For the computation of robust optimal solutions, i.e., optimal solutions being insensitive with respect to random parameter variations, deterministic substitute problems are needed. Based on the distribution of the random data, and using decision theoretical concepts, optimization problems under stochastic uncertainty are converted into deterministic substitute problems. Due to the occurring probabilities and expectations, approximative solution techniques must be applied. Deterministic and and stochastic approximation methods and their analytical properties are provided: Taylor expansion, regression and response surface methods, probability inequalities, First Order Reliability Methods, convex approximation/deterministic descent directions/efficient points, stochastic approximation methods, differentiation of probability and mean value functions. Convergence results of the resulting iterative solution procedures are given.

Asymptotic ruin probabilities for risk processes with dependent increments

In this paper, we derive a Lundberg type result for asymptotic ruin probabilities in the case of a risk process with dependent increments. We only assume that the probability generating functions exist, and that their logarithmic average converges. Under these assumptions we present an elementary proof of the Lundberg limiting result, which only uses simple exponential inequalities, and does not rely on results from large deviation theory. Moreover, we use dependence orderings to investigate, how dependencies between the claims affect the Lundberg coefficient. The results are illustrated by several examples, including Gaussian and AR(1)-processes, and a risk process with adapted premium rules.

A Distance For Multistage Stochastic Optimization Models

We describe multistage stochastic programs in a purely in-distribution setting, i.e., without any reference to a concrete probability space. The concept is based on the notion of nested distributions, which encompass in one mathematical object the scenario values as well as the information structure under which decisions have to be made. The nested distance between these distributions is introduced and turns out to be a generalization of the Wasserstein distance for stochastic two-stage problems. We give characterizations of this distance and show its usefulness in examples. The main result states that the difference of the optimal values of two multistage stochastic programs, which are Lipschitz and differ only in the nested distribution of the stochastic parameters, can be bounded by the nested distance of these distributions. This theorem generalizes the well-known Kantorovich-Rubinstein theorem, which is applicable only in two-stage situations, to multistage. Moreover, a dual characterization for the nested distance is established. The setup is applicable both for general stochastic processes and for finite scenario trees. In particular, the nested distance between general processes and scenario trees is well defined and becomes the important tool for judging the quality of the scenario tree generation. Minimizing—at least heuristically—this distance is what good scenario tree generation is all about.