Optimizing the CVaR via Sampling

Conditional Value at Risk (CVaR) is a prominent risk measure that is being used extensively in various domains. We develop a new formula for the gradient of the CVaR in the form of a conditional expectation. Based on this formula, we propose a novel sampling-based estimator for the CVaR gradient, in the spirit of the likelihood-ratio method. We analyze the bias of the estimator, and prove the convergence of a corresponding stochastic gradient descent algorithm to a local CVaR optimum. Our method allows to consider CVaR optimization in new domains. As an example, we consider a reinforcement learning application, and learn a risk-sensitive controller for the game of Tetris.Comment: To appear in AAAI 201

Asset Allocation under the Basel Accord Risk Measures

Financial institutions are currently required to meet more stringent capital requirements than they were before the recent financial crisis; in particular, the capital requirement for a large bank's trading book under the Basel 2.5 Accord more than doubles that under the Basel II Accord. The significant increase in capital requirements renders it necessary for banks to take into account the constraint of capital requirement when they make asset allocation decisions. In this paper, we propose a new asset allocation model that incorporates the regulatory capital requirements under both the Basel 2.5 Accord, which is currently in effect, and the Basel III Accord, which was recently proposed and is currently under discussion. We propose an unified algorithm based on the alternating direction augmented Lagrangian method to solve the model; we also establish the first-order optimality of the limit points of the sequence generated by the algorithm under some mild conditions. The algorithm is simple and easy to implement; each step of the algorithm consists of solving convex quadratic programming or one-dimensional subproblems. Numerical experiments on simulated and real market data show that the algorithm compares favorably with other existing methods, especially in cases in which the model is non-convex

Robust Performance Attribution Analysis in Investment Management

This dissertation investigates robust optimization models for performance attribution analysis in investment management. Specifically, an investment manager seeks to evaluate the performance of fund managers who manage funds he might invest his clients\u27 money in. A key difficulty for the investment manager is to quantify the fund manager\u27s skill when he may not know the fund manager\u27s allocation precisely. This introduces two main sources of uncertainty for the investment manager: the stock returns and the fund allocations. This dissertation proposes and analyzes robust, quantitative models to address this challenge. We study a robust counterpart to the mean-variance framework when the fund managers\u27 precise allocations are uncertain but belong to known intervals and must sum to one for each manager, present an algorithm to solve the problem efficiently and analyze the investment manager\u27s allocation in the various funds as a function of the benchmark return. Further, we consider the case where the stock returns are also represented as uncertain parameters belonging to a polyhedral set, the size of which is defined by a parameter called the budget of uncertainty, and the investment manager seeks to maximize his worst-case return. We describe how to solve this problem efficiently and analyze how the investment manager\u27s degree of diversification and his specific allocations in the funds vary with the budget of uncertainty