Gradient Search

Let the risk measure (function) $\rho$ be differentiable on $\mathbb{R}^n$. From standard analysis we obtain for $1 \leq i \leq n-1$

$$\frac{\partial\rho'}{\partial u_i} (u_1,\ldots,u_{n-1}) = \frac{\... ...- \frac{V_i}{V_n}\frac{\partial\rho}{\partial u'_n} (u_1,\ldots,u_{n-1},u'_n) .$$ (21)

Using the partial derivates (21), one can start looking for the (local) extreme points of $\rho'$ in $\mathbb{R}^{n-1}$ by applying Gradient Search (GS) methods. This might be a comfortable approach to solve the optimization problem (19) as long as the considered measures have sufficient differentiability properties. However, the proof of such differentiability properties can be rather difficult (cf. Appendix A or Tasche (2000)). This is one reason for our proposal of Swarm Intelligence optimization methods (see Subsection 3.3).

Outline of a gradient minimum-search (Figure 1):

1. Evaluate the gradient for the current portfolio.
2. If the gradient is zero, exit. We have found a (local or global) minimum.
3. Follow the negative gradient (negative slope) of the current portfolio one small step. Modify the portfolio to satify the constraints. Then continue with step 1.

On a one-processor machine this gradient algorithm has to be started many times with different portfolios (``brute-force''), as one might be stuck in a local minimum. This takes a long time for real-life portfolios.

Figure 1: Slow and simple Ruby-Pseudocode for portfolio optimization using brute-force gradient search. Clever varying choice of epsilon to calculate the gradient and the step-size can give further speed-up. Instead of chosing random portfolios, one can use a grid-based search.

The following three paragraphs derive the respective partial derivatives (21) for Expected Shortfall, ES-RORC and ES-RORAC.