The problem

In this section we explain from a general point of view how to optimize a portfolio with respect to ES or with respect to the performance measures ES-RORC or respectively ES-RORAC.

We assume a fixed budget/portfolio value of $V(u)$ which must be fully invested at the present time $t=0$. Otherwise, the considered problems become trivial or unsolvable as the ES is scalable (positive homogeneous of degree 1) and the considered performance measures are invariant due to scaling.

Let us assume that a portfolio $u\in\mathbb{R}^n$ is given. Furthermore, this portfolio has to be optimized with respect to a risk or performance measure $\rho$ on $\mathbb{R}^n$. For convenience, we assume that the risk $\rho$ has to be minimized. As mentioned, exactly the fixed amount $V$ has to be invested in the market at time 0. As $V_i$ is the price of asset $i$ at time 0, this implies the following constraint for the portfolios $u$ which must be satisfied:

$$V = \sum_{i=1}^{n} u_iV_i .$$ (16)

A possible solution of the optimization problem is given by a portfolio $u^*\in\mathbb{R}^n$, such that $\rho(u^*)$ is minimal (on $\mathbb{R}^n$) under the constraint (16). Defining

$$u'_n := (V - \sum_{i<n} u_iV_i)/V_n ,$$ (17)

and $\rho'$ as

$$\rho'(u_1,\ldots,u_{n-1}) := \rho(u_1,\ldots,u_{n-1},u'_n)$$ (18)

it follows from (16) that we can express the solution $u^*$ by

$$(u_1^*,\ldots ,u_{n-1}^*) = \rho'(u_1,\ldots,u_{n-1}) ,$$ (19)

together with $u^*_n = (V - \sum_{i<n} u^*_iV_i)/V_n$.

Working with real data, we discovered that portfolios which are candidates for extremal points due to the considered risk or performance measures can contain tremendous amounts of short-sold assets, i.e. the portfolio as a vector of real numbers contains huge negative components. For this reason we introduce a further constraint: For $a, b \geq 0$ we require

$$- b \frac{V}{V_i} \leq u_i \leq a \frac{V}{V_i} 1\leq i \leq n .$$ (20)

For instance, $b=0$ implies portfolios which allow no short-selling. The values $a=b=1$ guarantee that the amount of capital or debts in no asset is bigger than the total value $V$ of the portfolio.

As an optimization (e.g. for a 1-year horizon) can be driven daily or hourly, one could also think of self-financing ES-, ES-RORC or ES-RORAC-optimal strategies in this context.