are normalized to sum to 1, thereby giving the portfolio weights. So the problem of finding the optimal replicating portfolio can be solved by finding the best assets with log prices . . ., xn to use in the cointegrating regression, and then defining allocations to give the maximum stationarity in the tracking error s. The more stationary the tracking error, the greater the cointegration between the benchmark and the candidate portfolio.

In practice a high degree of cointegration may be found between the benchmark and the tracking portfolio; the standard ADF unit root test that was described in §11.1.5 should be sufficient to compare different portfolio specifications. When there are a large number of potential assets that could be used in a replicating portfolio it is not at all a trivial problem to test all possible portfolios to find the one that has the most stationary tracking error. If there are N assets in total one has to test N\/n\(N - n)\ portfolios for every n N.

Note that in global asset management models where the benchmark may be a global index such as the Morgan Stanley World Index there will be two stages to the selection-allocation process. First select the country indices to track the global index and assign optimal country allocations, and then either buy/sell the country index futures (if available) or repeat the process for tracking the individual country indices with individual stocks. A single country model could also be approached in two stages: first select the industrial sectors (pr style indices) and assign index weights optimally, then select the stocks within each industry sector (or style index) and optimize portfolios to track the indices.

12.5.2 Constrained Allocations

Examples of constrained allocations include the following:

A fund manager may wish to go long-short in exactly 12 different countries, with the world index as benchmark. The problem then becomes one of selecting the basket of 12 countries that are currently most highly cointegrated with the world index.

A small asset management company might seek a benchmark return of 5% per annum above the S&P 100 index, so in this case the benchmark index will be the S&P 100 plus.

Assets may be selected according to quite specific preferences of investors. For example, 50% of the fund may have to be allocated to the UK, or no more than 5% of capital can be allocated to any single asset.

Equality constraints on allocations, such as 40% in technology related stocks, are simple to implement. The dependent variable just becomes - , ,-, where a fraction ,- of the fund must be assigned to the jth asset, and the other log asset prices are used as unconstrained regressors. Inequality constraints are more difficult to implement. How should one deal with the constraint of no short sales, ,- > 0 for all p. First perform an unconstrained estimation of the model by OLS, because if no constraint is violated there will be no problem.

Equality constraints on allocations, such as 40% in technology related stocks, are simple to implement

But suppose the constraints , > 0 for some j are violated. Then the model is restricted so that all these , are set to zero, and re-estimated to ensure that no other coefficients that were originally positive have now become negative. If that is the case the resulting constrained OLS estimator is obtained, but it will of course be biased. That it is more efficient than the original estimator, because it reflects the value of further information, may be little compensation.

Problems arise when imposing the constraints causes more constraints to be violated, so that other coefficients that were positive in the unconstrained model become negative in the constrained model. The only feasible solution is to set those coefficients to zero, to re-estimate a further constrained model, and to keep shooting coefficients to zero until a purely long portfolio of assets is obtained. If many constraints have to be imposed it is not clear that any co-integrating portfolio will be found for the benchmark. However, when the benchmark is an index there are theoretical reasons for the existence of long-only cointegrating portfolios (§12.4.6) unless the index composition suddenly changes.

12.5.3 Parameter Selection

The basic cointegration index tracking model is defined in terms of certain parameters:

>- any alpha return over and above the index;

>- the time-span of daily data that is used in the cointegrating regression

(12.7) - this is called the training period; >- the number of assets in the portfolio;8

>- any constraints on allocations, which will depend on the preferences of the investor.

The optimal parameter values are chosen by recording a number of in-sample and post-sample performance measures for each set of parameters. The optimal parameter set is that which gives the best performance measures. The most important in-sample performance measures are the following:

>- ADF statistic: This is used to test the level of cointegration between the portfolio and the benchmark during the training period: the larger and more negative the ADF statistic, the greater the level of cointegration. The 1% critical value of the ADF statistic is approximately -3.5, although much greater values than this are normally experienced in practice as, for example, in Table 12.10 and Figures 12.6d, 12.7c and 12.7d.

>- Standard error of the regression: The in-sample tracking error will be stationary if the portfolio is cointegrated with the benchmark. This does not imply that the short-term deviations between the portfolio and the

8In fact the number of non-zero allocations need not be specified. Instead the number of assets chosen can depend on a bound that is set for the tracking error variance.

benchmark are necessarily small. It is also important to choose a portfolio for which the in-sample tracking error has a low volatility, and this is measured by the standard error of the regression.

Turnover: Only those portfolios showing realistic turnover projections as the model is rolled over the backtest period should be considered.9

Having specified the selections and the allocations on the in-sample training period, a fixed period of data immediately following the in-sample data is used to analyse the out-of-sample performance of the portfolio (§A.5.2). This is called the testing period. If the strategy requires monthly rebalancing then it is normal to use a testing period of 1 or 2 months for the post-sample diagnostics. Some typical post-sample diagnostics are as follows:

>- Tracking error variance: This is the variance of the daily tracking errors during the testing period. In fact the tracking error variance, if measured as an equally weighted average, is equivalent to the root mean square forecast error.

>- Differential return: The difference between the portfolio return and the index return.

>- Information ratio: The ratio between the mean daily tracking error and the standard deviation of the daily tracking error over the testing period. In-sample information ratios are zero by design (because the residuals from OLS regression have zero mean) but a positive post-sample information ratio is an important risk-adjusted performance measure.

Consider first a simple example of how to decide which parameters are optimal: for the problem of tracking the EAFE index with a one-year buy-and-hold strategy. The alpha over the EAFE index is fixed at 3% per annum and there are no constraints on allocations. There are only two model parameters to be chosen, the number of country indices in the portfolio (at the time of optimization the maximum number was 23) and the training period for the model. Figure 12.5 shows the 12-month out-of-sample information ratios that are obtained as the number of assets selected varies from 5 to 15 and the length of training period varies from 10 to 130 months. From the figure it seems that the highest post-sample information ratio of 3.8 occurs when the training period is between 100 and 115 months and the number of assets is between 7 and 11.

Instead of fixing the alpha over an index - or indeed under an index - it may be preferable to fix the number of assets in the portfolio. In that case this type of two-dimensional heat map can be used to determine the optimal choice of the alpha over the index and the length of training period. For example, suppose a portfolio of exactly 75 stocks is to be used to track the S&P 100 index plus cx% per annum. A heat map is generated by finding the 75-asset

9In contrast to mean-variance analysis, there should not be a problem with the instability of allocations because cointegration methods are based on common long-run trends.