It is relevant to test the practical impact from the use of conditional volatility forecasts.
Simulating a trade strategy based on the forecasted volatilities is an easily implementable test that may allow for a quantifiable and easily comprehensible result.
7.3.1 Simulation setup
The approach chosen as described in Section 4.6.3, tests the applicability of the volatility fore
casts generated by the three conditional models in a portfolio optimization scheme. In order to isolate the effects from volatilities, the objective function minimizes the portfolio variance, whereby the only external factor will be the covariances. As comparison parameter, the same procedure will be conducted for the unconditional variances.
The variancereturn frontier consists of all possible combinations of the 20 emerging market indices. At each of the 1828 OOS days, the portfolio is calibrated. With 4 models this results in 7312 “20x20” variance/covariance matrices which makes the operation rather data intensive.
All covariances are unconditional and based on past realized returns whereby the compara
bility of the results can be secured83.
83 This means that the covariances are predicted unconditionally. Recent advances in covariance predic
tion show significant improvements in covariance predictability. Especially the Dynamic conditional correlation estimates prove empirically well-performing. See Engle (2002), Marshall et al (2008).
The calibrations are subject to the following constraints:
• The sum of the portfolio weights must be 1, whereby full investment is forced.
• No short-selling is allowed by restricting the weights w to be larger or equal to 0.84
• No trading costs are introduced85.
The covariance matrices used are attached in Appendix B7.4. Precausion should be exhibited when concluding based on the results. Whereas the theoretical loss functions evaluate on the ability of the volatility models in predicting the errors for each country separately, the portfolio approach is a measure of the ability of a model over the full range of countries in the portfolio. Further, the covariances are unconditional and affect the evolution of the portfolios through time86.
7.3.2 Simulation results
Figure 7.1 shows a 260 days moving window graph of the portfolio variances calculated according to equation 4.42, and illustrates how the unconditional portfolio variance in most periods is larger than those generated by the heteroskedastic models. Appendix B7.5 contain the portfolio weights in each period as predicted by the respective unconditional and conditional volatility models.
As in previous evaluations it is found that the GARCHtype models in most periods are closely related and only diverge marginally from each other. As previously, the results can be scrutinized by regarding the performance in the full sample and in subsamples. The results are presented in Table 7.4. The weights in each period for all models are also attached in Appendix B7.5.
84 Shortselling may not be practically available for index trading and is banned in a number of countries.
85 While this deviates from practice is helps to encourage changing weights during re-calibrations whereby the potential of the conditional forecasts will be clearer. Further, trading costs facing institutional investors are nontransparent and not available.
86 Although this is a potential error source, it has a minor impact on the result only, as the covariances are equal under all models.
0.00005.0001.00015.0002.00025Variance MA(260)
3500 4000 4500 5000 5500
t
UNCOND GARCH
TGARCH EGARCH
Figure 7.1. The figure shows the development of the portfolios calibrated by the unconditional and conditional models respectively in the OOS period.
Source: Own calculations.
Table 7.4
Portfolio variances
Sample periods Unconditional GARCH TGARCH EGARCH
Full OOS 8.29147E05 6.5862E05 6.61887E05 6.96112E05
Subsample 1 4.91481E05 3.0904E05 3.10476E05 3.14377E05
Subsample 2 1.95282E04 1.7207E04 1.73566E04 1.89652E04
Subsample 3 7.34842E05 6.6797E05 6.66931E05 6.75341E05
The table shows portfolio variances in the full OOS and three sub-samples.
The results support the impression, that the GARCHtype models consistently outperform the unconditional volatility measure by generating portfolios with significantly lower variance in the full sample as well as in each individual sub-sample87.
For the full sample, the GARCH-type models returns portfolios with variances 16.04 to 20.57 percent lower than that of the unconditional measure. The result is mainly driven by
sub-87 For the full sample and subsample 1, the gains in variance minimization are significant on the 1% level, whereas for subsample 2 and 3 the gains are only borderline significant at the 5% level.
sample 1 where the variance was 36.34 to 37.12 percent below the unconditional followed by 2.88 to 11.88 percent in subsample 2 and 8.10 to 9.24 percent in subsample 3. The smallest advance over the unconditional measure is in all cases found by the EGARCH model.
Remarkably, the symmetric GARCH model generates the portfolio with the lowest variance in all cases except from the last sub-sample, although the difference between the performances of the GARCH-type models is largely insignificant88.
The magnitude of the results is close to what could be expected following the OLS results, which demonstrated that up to around a fifth of the volatility, could be explained by the GARCH-type models although with large variation. Even so, the gain is large, when taking into consideration the large amount of the portfolio variance that is carried by the covariances as the number of assets grows (Elton et al 2007).
The EGARCH model features the lowest gains of the GARCH-type models. The inefficient portfolios it produces are especially inferior in the high-volatility period around the finan-cial crisis. As the portfolio optimizations are products of the full range of countries, this means that faulty estimations for separate indices will impact on the over-all result.
The sign-flipping behavior found for a few countries when applying the EGARCH model may be what causes the over-all performance of the EGARCH model to deteriorate.
88 With one exception; the full sample, the EGARCH models performs significantly worse than the GARCH and TGARCH at the 1% significant level.
In the following, the conclusions derived from the paper will be summarized and discussed briefly accompanied by a few suggestions for further research.
