In the first part of this paper we introduce a general concept of portfolio optimization with respect to the risk measure Expected Shortfall (ES) and also (but not at the same time) with respect to the performance measures RORC and RORAC (Return On Risk/Risk-Adjusted Capital) due to ES. Later, a specific financial market model that enables optimization of portfolios consisting of bonds and stocks is proposed. This model combines and correlates a stock market model of geometric brownian motions with a two-factor Cox-Ingersoll-Ross (CIR-2) model for the interest rates, respectively bonds. Numeric optimization is done using recent results from the theory of risk capital allocation, performance measurement and (Particle) Swarm Intelligence (SI). Explicit formulas and methods for the model and algorithms like Gradient Search (GS) and SI optimization are provided. Examples of optimized portfolios for German market data are given. Furthermore, we analyze the scalability of the solution (with respect to multi-processor machines, clusters or Global Grid environments) to assure fast run-times for large real-life portfolios.
The idea of portfolio optimization with respect to modern risk and performance measures like ES, RORC or RORAC and also taking correlations between interest rates and stocks using an enhanced interest rate model into account seems to be new. Fast developing computer technology enables to solve optimization problems numerically even for complex market models and big portfolios.
Our approach is pragmatic in the sense that some of the theoretical
problems which can emerge are (although not considered in depth) solved
by unorthodox methods that seem to work well with real-life portfolios.
The focus of the paper lies on the presented methods and
not on the testing of these methods. From an academic point
of view, this might be unsatisfying, but the mentioned models
have (separately and sometimes in unfortunately simplified forms)
been considered in some of the biggest German insurance companies.
should keep in mind that the paper shows what is actually
possible in optimizing portfolios, but not how good the presented
methods are from the historical point of view or compared to other market
models. Such questions are not part of this paper.
The paper is organized as follows. Section 2 introduces the
considered risk and performance measures. Section 3 explains
the general approach to the optimization problem. Furthermore, the Gradient
Search, the Swarm Intelligence optimization method and
the role of stochastic simulation in these approaches are described.
After a general introduction to optimization methods,
Section 4 introduces and discusses the proposed financial
market model. Section 5 is dedicated to the stochastic
simulation part, i.e. the generation of market scenarios.
In Section 6, information on parameter estimation can
be found. Section 7 gives a brief chronological overview
of all steps concerning our proposed optimization methodology.
In Section 8 we present first results for
German market data. Section 9 shows the
scalability of our solution. In Section 10 we conclude.
Finally in the Appendix
we give two results on the form
of the derivatives of Value-at-Risk and
Expected Shortfall expressions.
We introduce some notation. Let us define the total payoff
As an example, consider a financial market with numbered securities. Let us assume that the prices of these securities at some time (the positive real numbers including 0) are given by random variables