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120 portfolio of models. As these models generally do not have the same clustering of good and bad periods, the overall risk is then reduced. But this is true only if the composition of the trading model portfolio is not changed too often. Dynamic modifications of the trading model portfolio, which keep a reasonable risk profile, are very hard to obtain and at O&A we advocate choosing static portfolio strategies where the dynamic behavior is left to the trading models themselves. The optimal trading model portfolio strategy depends on certain decisions of the investor such as the choice of the investment assets, frequency of changes, and limits of risk and exposure. One of the main problems is the selecdon of the trading models to be used in such a portfolio. It is easy to test many different combinations and to select the best one, but such a procedure can produce undesirable results. Tn fact, during this selection procedure, the risk of overfitting particular historical data may again occur from the back door. To overcome these types of problems, it is often desirable to use an equally weighted portfoliothat is, a portfolio where the same proportion of capital or credit limit is invested in each trading model. Another possibility is to select the optimal trading model portfolio using a robust optimization procedure, like in trading model optimization (Section 11.5.1), but we will not discuss such an approach here. Table 11.12 compares the performance obtai ned for a trading model portfolio, which corresponds to an equally weighted portfolio strategy, to the performance of the individual trading models on the same period. The analysis period is from January 1993 to December 1997. During this period, all these trading models were running in the realdme O&A information system with no reoptimizations. On this table we observe very well that for the same annualized total return the risk of the portfolio is considerably lower than the average risk on the individual models. The maximum drawdown of the portfolio is about half of the average maximum drawdown of the models and the annualized Xeg is one of the largest. The variation of the total return of the portfolio over the years is plotted on Figure 11.4. Portfolios of trading models can be used as dynamic investment strategies in many financial markets, but the complexity of the optimization of such portfolios based on a very large number of trading models (needed for a good diversification) would be extremely hard to control. As we observed in the previous sections, the optimization of the different trading strategies is in itself a complicated process that needs to be done at regular intervals to take into account the nonstationarity of the underlying time series. In the case of foreign exchange, an interesting application of portfolio trading models is the dynamic hedging of currency risk. In this case, the number of models to optimize is reduced and it is reasonable to consider a dynamic hedging strategy based on the trading recommendations. Tn the next section we will provide a brief description of this approach.
TABLE 11.12 Portfolio performance of O&A trading models. Performance comparison between 10 O&A class RTT, RTM trading models, and an equally weighted portfolio of the same models. The different performance measures displayed are the annualized total return R, the risksensitive performance measure Xeff> the maximum drawdown D, and the annualized Sharpe ratio S. FX rate  Model   Xeff    USDDEM   6.8%  2.3%  10.2%  0.73    3.4%  1.3%  9.8%  0.51  DEMJPY   1.7%  2.6%  16.1%  0.19    9.3%  6.4%  8.9%  1.32  USDCHF   3.7%  1.2%  10.9%  0.38    2.5%  0.3%  13.7%  0.36  USDFRF   9.2%  4.7%  9.9%  1.03    5.6%  2.5%  10.0%  0.71  GBPDEM   5.6%  2.2%  14.3%  0.69    5.0%  2.8%  8.1%  0.79  Average values   5.2%  1.8%  11.2%  0.71  Portfolio   5.2%  4.1%  5.5%  1.09 
11.8 currency risk hedging Hedging problems arise whenever an investor, for example, a fund manager or a commercial organization, is holding foreign assets such as foreign securities over a period of time. The foreign assets are denominated in a foreign currency. The investor measures the performance of his/her investment in terms of the investors home currency. The foreign assets have a degree of volatility in terms of their own currency. Due to the foreign exchange rate movements the volatility is, however, higher when expressed in terms of the investors home currency. This implies additional risk. By additionally taking a short position in the foreign currency, this implicit foreign currency exposure can be compensated and the risk can be reduced; this is the basic idea of hedging. Whereas a constant short position is referred to as static hedging, this section deals with dynamic hedging where the foreign currency positions vary over time. In this section, a strategy of hedging the foreign exchange (FX) risk associated with foreign investment is specified. As an innovative element of this strategy, realtime trading models are used. The whole strategy can then be called a dynamic overlay. To be successful, we need profitable trading models that are only weakly
30.00 20.00 10.00 0.00 1993 1994 1995 1996 1997 Time (years) FIGURE I 1.4 Total return of a portfolio of 10 O&A trading models over 5 years. correlated or anticorrelated to the usual primary investments, because positive correlation would imply an increased risk. An investors risk/return decisions must be matched to the set of all possible investments (including dynamic allocation of capital to the foreign currencies), the feasible set. Figure 11.5 shows this feasible set as a shadowed region. The upperleft border of this set is termed the efficient frontier, those investment portfolios lying along this frontier deliver the maximum possible return for the minimum possible risk or the minimum risk for the set of best possible returns. The point at which an investors indifference curve has a common tangent with the efficient frontier represents the best possible match between the investors preferences and the possible investment portfolios. The right, dashed vertical line in Figure 11.5 indicates the expectation for the risk of a primary investment, which is left completely unhedged. A circle is drawn where this vertical line intersects with the horizontal line indicating the expected return of the primary investment. The dashed vertical line to the left in Figure 11.5 indicates the reduction of risk achieved through an optimal static hedge of the primary investment, which usually implies short positions in all foreign currencies in which the foreign assets
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