We implement and test the GAMS models of this chapter using two data sets stored in Excel files:

We use the utility GDXXRW to read the content of each file. The GAMS file Read-Data.gms in FINLIB performs the main task of reading data from the Excel files and delivering the manipulated data in GDX format. The file InputData.gdx will then be read for use by the model through the utility GDXIN.

We implement this model using bonds from the three different countries in our data set. We structure the portfolio such that the percentages of bonds in each market are equal to those reported in the reference index. Furthermore, to mirror the risk factors of the target index in more detail we segment the duration of the securities in three classes as low, medium, and high, and then constrain the portfolio to hold in each duration class the same proportion of bonds as the reference index. The model could go on to segment the available securities by, say, market segment, credit rating class, and so on, and match the proportions in each class between the target index and the portfolio. Linear programming theory tells us that the more linear constraints we add the more non-zero variables we have in the optimal solution, and this implies more diversified portfolios. However, if we have as many risk factors as the securities in the index, thus capturing residual risk as well, we end up with a portfolio that is identical to the index - which is not very useful in practice.

We declare the set BxE(BB,EE), defined over the set of bonds (BB) and currencies (EE), to carry the partition of the bonds by currencies. The set BxD(BB,DD), defined over the set of bonds (BB) and duration levels (DD), carries the partition of the bonds by duration levels.

We define by x(i) the fraction of capital to be invested in each bond. The percentage holdings are defined by CurrencyWeights(e) and DurationWeights(k).

The currency and duration constraints are formulated by summing the variable x(i) over the sets BxE and BxD, respectively:

The rest of the equations are similar to those described in Section 4.4.

We use the same set of data as for the structural model. The two-sided tracking model is shaped through an additional constraint to limit the upside potential (see Model PFO-5.3.4 for comparison). The constraints on upside and downside are specified as follows:

As seen in Section 5.3.1, the model is solved for decreasing values of the scalar EpsTolerance . The minimum level of EpsTolerance for which Model PFO-7.3.3 is feasible determines the indexed portfolio. The indexed portfolios obtained with the two approaches are displayed in Figure 7.1.

With respect to hedging exchange rate risk, the portfolio manager can decide to fully hedge the portfolio, to leave currency risk unhedged, or to proceed with partial hedging whereby exchange rate exposure is partly covered by forward contracts. For example, a 50% hedge of the nominal amount invested in foreign countries means that only half of the total capital in foreign investments will be protected by futures. The hedging strategy whereby the hedge ratio is different for each currency included in the indexed portfolio is known as selective hedging. In selective hedging currency risk is hedged, perhaps only partially, for those currencies where data estimates dictate that currency movements are unfavorable, and remains unhedged for those currencies where the exchange rate changes appear favorable; see Chapter PFO-10 for a discussion of these issues.

The scenario optimization model for selective hedging is displayed in Model PFO-10.5.1 (see also Section PFO-7.3).

The model is implemented using data from BroadIndexData.xls for the broad market indices. The risk measure we adopt in building this model is that of MAD, but similar models can be built for CVaR, or expected regret minimization, or for expected utility maximization.

The key modeling concept in implementing a selective hedging model is to split the asset allocation variable x(i) in two parts: the unhedged position variable, u(i), and the hedged position variable, h(i).

Similarly, we define by HedgedReturns(l,i) and UnhedgedReturns(l,i), respectively, the returns of the broad indices when exchange rate is hedged away through forward contracts, and the returns of the same indices when an unhedged position is taken. These are defined in GAMS as:

Note that, for the domestic currency EUR, the exchange rate returns are zero under all scenarios.

The equations for the selective hedging model are obtained by simply substituting in place of x(i) the two new variables u(i) and h(i). The return of the portfolio is now a function of the hedged and unhedged positions, and the MAD and return constraints become:

In Figure 7.2 we compare the efficient frontier obtained with the selective hedging model compared with the fully hedged and unhedged model. Note that the efficient frontier for selective hedging dominates both the fully hedged and unhedged efficient frontiers. This is as it should be, since selective hedging is the most general model, and it encompasses as special cases full hedging, no hedging and, with a small modification, partial hedging.