Examples

The general setup for our numeric examples is a time horizon of one month where the simulation takes 20 steps per month. The number of loops is $1000$. We consider portfolios which have a present value of exactly 1000 EUR. We optimize using a local GS method and a combined GS-SI method. The second one is run with and without constraint $b=0$, i.e. with and without shortselling in the portfolio (cf. (20)). The considered confidence level is $5\%$. Two types of portfolios are examined. The smaller one contains two bonds and two stocks, the bigger one 10 bonds and 10 stocks. In particular, we considered the following bonds and stocks (which are here listed in the same order as in the portfolio vectors):

• 2 zero-coupon bonds: Maturity 1 year and 10 years.

  2 ``stocks'': Xetra DAX and Allianz

• 10 zero-coupon bonds: Maturity 1 year up to 10 years.

  10 stocks: Allianz, BASF, BMW, Bayer, Commerzbank, DaimlerChrysler, Deutsche Bank, Lufthansa, E.ON, Hypovereinsbank

All stocks are elements of the Xetra DAX and had their IPO (Initial Public Offering) at least 10 years ago. Data was taken from http://de.finance.yahoo.com. The estimates are calculated from monthly data from May 2002 to April 2003. We obtain the model parameters listed in Table 1. The same time interval and discretization was taken for the estimation of the term structure model parameters (cf. Subsection 6.1). Maturities from 1 to 10 years were taken into consideration. Results are in Table 2.

The values of the (adjusted) covariance matrix in Table 3 confirm the use of correlations between the interest rate model factors and the stock market dynamics to obtain a more realistic combined model.

For each of our setups we computed the ES-, ES-RORC- and ES-RORAC-optimal portfolio. The mean, VaR, ES, ES-RORC and ES-RORAC for these portfolios are listed in the Tables 4-9 in Appendix B (and the portfolios themselves in the four assets case). The optimized portfolios are compared with ``normed'' portfolios where the same capital is invested in each of the four, respectively 20 assets. As expected, all optimized measures have been improved significantly (see also Figures 3 and 4) and the combined GS-SI method is superior to the local GS method starting at the normed portfolio. Local extreme points seem to exist in the most considered cases. A situation as in Table 6 where the ES of the ES-optimized portfolio is lower than (but close to) the ES of the RORC-optimized portfolio could be a symptom for the need of more (or finer) iterations.

Due to our pragmatic approach we did not invest any time in proofs for the existence or absence of global, respectively local extreme points in our model. Real financial companies are not interested in such questions, especially as portfolios are often optimized in small steps and not a complete restructuring.

An interesting (and reasonable) model output is that the local GS results imply that bonds of longer maturities bear more financial risks. This can be seen in decreasing weights of bonds with higher maturities in the optimized portfolios (this is also true for the portfolios whith 20 assets which are not listed in detail).

Massive short-selling and probable absence of global extreme points (e.g. in the RORAC case, cf. Table 4) motivate the use of constraint $b=0$ (no short-selling). Roughly speaking, the implication seems to be that optimized portfolios under the constraint contain almost no stocks. Optimization under the no-shortselling constraint seems to imply rather similar optimized portfolios for all measures (cf. Tables 6 and 9).

In summary, all obtained results seem to be reasonable from the economic point of view and confirm the proposed methods. We cannot really judge the impact of other models at this stage of our research. However, we guess that reasonable models (e.g. such using Vasicek-2) will imply results close to ours.