Dynamic Portfolio Optimization using Sequential Quadratic Programming
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DepartmentMathematics and Statistics
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In Financial Mathematics, classical Markowitz Portfolio theory provides a strategy for optimizing return on a portfolio by considering only the mean and variance of returns. However, recent theoretical and empirical studies show that it has many limitations, some of which are that it assumes a one shot allocation and Gaussianity of returns. We describe a more realistic approach which incorporates probabilities of both large gains and large losses and which is not confined to a particular type of distribution. We use a new approach in the sense that we use probabilities for the constraint and objective functions. Thus, instead of looking only at the expected return, we maximize the probability of the reward, while keeping the probability of loss small. We desire that the portfolio values a predetermined Vu and are averse to portfolio values below a predetermined value Vl. Thus, we maximize the probability of a portfolio value being above Vu subject to the constraint that the probability of portfolio value being below Vl does not exceed a given threshold ?. We introduce and verify a method to optimize the return using sequential quadratic programming. We solve the optimization problem over one period using MATLAB routine fmincon. This approach involves making certain approximations. We verify the validity of these approximations using brute force calculations. We compute the expected values of objective and constraint functions by taking the sample means of large samples generated by Monte Carlo simulations. We expand our model to multi period timeframe by first considering the two period optimization problem. We use dynamic portfolio allocation, fmincon, Monte Carlo simulations, and cubic spline interpolation so solve the optimization problem. To solve the two period problem, a grid of possible portfolio returns after the first period is used. For each of the grid values then the portfolio optimization problem over one period is solved. This then enables us to solve the overall two period problem. When the two period problem is unconstrained, our approach finds a (local) optimal solution. When constraints are introduced, our method is only suboptimal. We test our approach on different types of distributions, including the lognormal, mixture of lognormals and Pareto. While we limit ourselves to two time periods and three assets, comparisons with brute force calculations suggest that our approach using fmincon produces accurate results. Our approach has been developed in a manner that it can be applied to arbitrary number of assets as well as multiple time periods.