Risk management for mathematical optimization under uncertainty

We present a general multistage stochastic mixed 0-1 problem where the uncertainty appears everywhere in the objective function, constraints matrix and right-hand-side. The uncertainty is represented by a scenario tree that can be a symmetric or a nonsymmetric one. The stochastic model is converted in a mixed 0-1 Deterministic Equivalent Model in compact representation. Due to the difficulty of the problem, the solution offered by the stochastic model has been traditionally obtained by optimizing the objective function expected value (i.e., mean) over the scenarios, usually, along a time horizon. This approach (so named risk neutral) has the inconvenience of providing a solution that ignores the variance of the objective value of the scenarios and, so, the occurrence of scenarios with an objective value below the expected one. Alternatively, we present several approaches for risk averse management, namely, a scenario immunization strategy, the optimization of the well known Value-at-Risk (VaR) and several variants of the Conditional Value-at-Risk strategies, the optimization of the expected mean minus the weighted probability of having a "bad" scenario to occur for the given solution provided by the model, the optimization of the objective function expected value subject to stochastic dominance constraints (SDC) for a set of profiles given by the pairs of threshold objective values and either bounds on the probability of not reaching the thresholds or the expected shortfall over them, and the optimization of a mixture of the VaR and SDC strategies.

Income risk of EU coal-fired power plants after Kyoto

Coal-fired power plants may enjoy a significant advantage relative to gas plants in terms of cheaper fuel cost. Still, this advantage may erode or even turn into disadvantage depending on CO2 emission allowance price. This price will presumably rise in both the Kyoto Protocol commitment period (2008-2012) and the first post-Kyoto years. Thus, in a carbon-constrained environment, coal plants face financial risks arising in their profit margins, which in turn hinge on their so-called "clean dark spread". These risks are further reinforced when the price of the output electricity is determined by natural gas-fired plants' marginal costs, which differ from coal plants' costs. We aim to assess the risks in coal plants' margins. We adopt parameter values estimated from empirical data. These in turn are derived from natural gas and electricity markets alongside the EU ETS market where emission allowances are traded. Monte Carlo simulation allows to compute the expected value and risk profile of coal-based electricity generation. We focus on the clean dark spread in both time periods under different future scenarios in the allowance market. Specifically, bottom 5% and 10% percentiles are derived. According to our results, certain future paths of the allowance price may impose significant risks on the clean dark spread obtained by coal plants.