20 40 60 80 100 120

Training

Figure 12.5 Tracking the EAFE index with a one-year buy-and-hold strategy, with fixed a = 3% per annum: 12-month out-of-sample information ratio as a function of the number of assets and length of training period.

portfolio that is most highly cointegrated with the index plus a% per annum for each choice of a and training period. Each time the a and the training period are changed the choice of assets and the allocations in the portfolio will change. But these allocations are not recorded at this stage. All that will be computed are some of the in-sample and out-of-sample diagnostics that have been described above.

Figure 12.6a shows the 2-month post-sample information ratios, and Figure 12.6b shows the 2-month post-sample differential returns over the index for a 75-stock portfolio in the S&P 100 index that is being optimized at the end of February 2000. Figure 12.6c is similar to Figure 12.6a, but for the 1-month information ratio, and Figure 12.6d shows the 1-month differential return. The maps are shaded so that dark areas indicate better diagnostic test results (much nicer colour plots are on the CD).

These heat maps show a clear hot spot when a is negative but no more than about -7% per annum, and the training period is between 28 and 48 months. Another region that gives promising out-of-sample diagnostics is for a high, positive a and a very long training period. However, the highest differential return and information ratio are obtained within the hot spot when the alpha is approximately -5% and the training period is about 3 years.

12.5.4 Long-Short Strategies

The heat maps in Figure 12.6 also have a cold spot, that is, a region where the parameter choices give rise to rather bad performance measures. In particular, when a is -12% and the training period is 72 months, the 1-month and 2-month out-of-sample information ratios are negative, as are the differential returns. For these parameter choices, the portfolio is (always) chosen to have the highest possible in-sample cointegration with the benchmark, but since alpha is negative it is consistently underperforming the actual index in the post-sample predictive tests. Therefore it should be possible to make money by going short this portfolio.

Note that this short portfolio will itself contain long and short positions, unless the constraint of no short sales has been applied. Similarly, the long portfolio, the one that has the highest information ratios and differential return, will typically also consist of long and short positions. Then a hedged portfolio is obtained by matching the amount invested in the long portfolio with the same amount being shorted with the short portfolio.

Note also that it is not, in fact, necessary for the long portfolio to outperform the index and the short portfolio to underperform the index in the post-sample predictive tests. However, it is necessary for the long portfolio to outperform the short portfolio. If this type of long-short strategy were used with a 75 asset portfolio from the S&P 100 then the heat maps in Figure 12.6 indicate that optimal parameter choices for February 2000 would be as shown in Table 12.8a. The optimal parameter choices will be different every month. For example from Figure 12.7 the optimal parameter choices for October 2000 would be as shown in Table 12.8b.

The portfolio is (always) chosen to have the highest possible in-sample cointegration and it is consistently underperforming the index in the post-sample predictive tests. Therefore it should be possible to make money by going short this portfolio

Table 12.9 shows the parameter choices that were actually used for the 75-asset long portfolio and a 75-asset short portfolio in the S&P 100 index.10 Of course, some of the same assets will be chosen in both portfolios, and the net position in these assets will be determined by the difference between their weight (positive or negative) in the long portfolio and their weight (positive or negative) in the short portfolio.

12.5.5 Backtesting

This section describes two types of model backtests. The first type of backtest is to see how a fixed parameter set, which currently seems optimal according to heat maps of the type just described, actually performs over a historic period. A simple snapshot of portfolio performance at one instance in time, as in Figures 12.6 and 12.7, may not provide sufficient evidence that parameter

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choices are optimal. Therefore, performance measures are obtained by running the model over time, for example month by month. Each month a new set of assets will be chosen and new allocations will be made, but the set of parameters remains fixed.

Table 12.10 reports the in-sample ADF, the turnover percentage, and the 1-, 2-and 3-month out-of-sample information ratios for the long and the short