Introduction

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. Practitioners 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

$$X = X(u) := \sum_{i=1}^{n} u_iX_i$$ (1)

of a portfolio

$$u = (u_i)_{1\leq i\leq n} \in \mathbb{R}^n$$ (2)

which represents $n\in\mathbb{N}^+=\mathbb{N}\setminus\{0\}$ different payoffs $X_i$ ( $1\leq i \leq n$) with weights $u_i\in\mathbb{R}$. The $X_i$ are assumed to be one-dimensional real-valued random variables. We call $B = (X_1,\ldots, X_n)$ a portfolio base (cf. Fischer, 2003) as any considered portfolio will be described with (2) and (1). As random variables, the components of $B$ do not have to be linearly independent.

As an example, consider a financial market with $n$ numbered securities. Let us assume that the prices of these securities at some time $s\in\mathbb{R}^+_0$ (the positive real numbers including 0) are given by random variables

$$V_1(s), V_2(s), \ldots, V_n(s) .$$ (3)

We assume to have constants for $s=0$, which is the present time. We will also use the notation

$$V_i = V_i(0)$$ (4)

and

$$V = V(u) := \sum_{i=1}^{n} u_iV_i(0)$$ (5)

for the value of the portfolio $u$ at time 0 in the following, and define

$$X_i = V_i(t) - V_i(0),$$ (6)

where the variable $t>0$ is the considered time horizon for which risk management is performed. Hence, the $X_i$ are wins or losses due to the $i$-th asset during the time interval $[0,t]$. Now, the portfolio $u$ of the different assets has a difference in value from time 0 to $t$ which is exactly $X(u)$, and $X > 0$ ($X < 0$) is an increase (decrease) of the portfolio value from time 0 to $t$ and therefore a win (loss) for the portfolio holder.