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McMaster University, Hamilton, Canada
MATHEMATICAL MODELING AND OPTIMIZATION TECHNIQUES
IN RISK MANAGEMENT
Nowadays mathematical modeling and optimization techniques are used in many areas of science. The main challenge of practical models is minimizing risk in the presence of uncertainty. In the paper the examples of practical risk management problems that demonstrate how mathematical modeling combined with optimization algorithms are shown.
The common feature of risk management models is presence of multiple performance indicators (profit, return on investment, environmental impact) and risk measures (volatility of investment returns, portfolio expected shortfall, etc.). Due to that all these models are multi-objective problems. The solution of a multi-objective optimization problem is the set of Pareto efficient points, known as Pareto efficient frontier. We present our methodology that allows computing Pareto efficient frontier efficiently.
The first risk management problem we use to illustrate our computational algorithms is portfolio optimization. Portfolio optimization problems have similar characteristics whether those are portfolios of oil and gas investment projects or portfolios of financial instruments. Petro-chemical companies and financial institutions are constantly faced with investment decisions in multiple projects or multiple assets.
Portfolio optimization is a computational tool that quantifies selection of multiple investment opportunities such that risk is minimized and value or return is maximized. Risk-return efficient frontier is a 2D piecewise quadratic curve that shows the trade-off between expected return and variance of return (risk) and our algorithm allows computing it efficiently. We also demonstrate identification of the efficient frontier for the portfolio optimization problem when, beside risk and return, additional objectives such as minimizing credit losses, minimizing transaction costs or minimizing price-earnings ratio are considered. Efficient frontier in 3D can be computed using our algorithm as well. We show a practical example demonstrating computation of efficient frontier when multiple objectives include minimizing credit risk, minimizing market risk and maximizing return.
The second illustrative problem we consider is credit risk optimization. Credit risk modeling is a challenging problem for the industry and financial sectors due to the fact that distribution of credit losses is not normal and exhibits fat tails. Because of that, risk measures such as Expected Shortfall or Value-at-Risk are more appropriate than the variance.
Value-at-Risk (VaR) is the loss that is likely to be exceeded with probability (1 – ). Expected Shortfall (ES) is the average loss beyond VaR. VaR and ES are computed by Monte Carlo sampling. We need a huge number of samples to get good estimates of the portfolio loss distribution and consequently of those two risk measures. Sampling techniques for computing portfolio loss distribution require very large number of samples. To decrease a number of samples, sampling technique is combined with approximating conditional portfolio loss in each scenario.
In credit risk optimization we want to adjust the composition of the portfolio to “shrink” the right tail of the portfolio loss distribution. Due to using different sampling and approximation techniques (Monte Carlo sampling, Normal approximation, conditional mean approximation, unconditional model) as well as optimizing different risk measures (Value-at-Risk, Expected Shortfall, variance), we developed a number of optimization formulations for credit risk portfolio optimization.
Our computational results demonstrate relative performance of optimization formulations for a number of large datasets. We describe performance of our optimization formulations with respect to the number of positions in the portfolio and other factors.
Presented models and are aimed for practical implementation and use by risk managers at different financial institutions and industrial enterprises. They were developed by Algorithmics Inc. team in co-operation with McMaster University.
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