Keywords

• Portfolio optimization
• Risk constraints
• Dynamic risk measures
• Stochastic optimal control
• Markov decision problems

Project description

Portfolio optimization is one of the central tasks of modern financial mathematics. The goal of portfolio optimization is to invest an investor's available capital optimally in securities on the financial market. Furthermore, the investor has the option of consuming parts of his assets. The portfolio problem for an investor is then to determine an optimal investment and consumption strategy.

Since the last financial crisis and the accompanying collapse of large financial institutions, there has been great interest in quantifying and limiting the risk of losing parts of the invested assets. In practice, the risk measure “value at risk” has become the standard risk measure for financial risks. However, the use of value at risk is not entirely uncontroversial, as the calculation only takes into account the probability of a loss, but not its amount. For this reason, various alternative risk measures are considered in addition to value at risk, such as tail conditional expectation and expected loss, as well as coherent and convex risk measures.

A natural extension is the consideration of dynamic risk measures. A dynamic risk measure quantifies the risk of loss of an investment strategy at any given point in time, taking into account the information available up to that point.  To determine the risk measures, the usual assumption is that the proportion invested in securities and the consumption rate are kept constant over a given time horizon (e.g., a day or a week). For the classic (time-continuous) portfolio optimization problem, this assumption is typically not fulfilled. An investor acting according to the optimal strategy will continuously adjust the proportion invested in securities and the consumption rate. Accordingly, the risk measure is only determined approximately. For this reason, we alternatively consider an investor who can only adjust his portfolio proportions and consumption rate at discrete points in time. For this time-discrete investor, the risk measures can be determined explicitly.

To solve the portfolio problems, these are formulated as stochastic control problems and solved using dynamic programming methods. In the continuous-time case, the Hamilton-Jacobi-Bellman (HJB) equation associated with the control problem is examined for the value function of the problem. The pointwise optimization problem that arises in this equation and contains the risk constraint is solved using the Lagrange multiplier method. A policy improvement algorithm and numerical methods for solving partial differential equations are used to determine the optimal investment strategies.

 In the discrete-time case, the theory of Markov decision problems is used to solve the portfolio optimization problem. These lead to a recursive method for determining the value function and the optimal investment strategy.

Finally, solving the control problems allows the quantification of the loss in portfolio performance resulting from the limitation of the risk of loss as well as the loss resulting from the limitation to time-discrete investment strategies.

Project-related publications

I. Redeker, R. Wunderlich: Portfolio optimization under dynamic risk constraints: Continuous vs. discrete time trading. Statistics & Risk Modeling, 2017 Preprint