5.3 Portfolio aspired moments
6.1.1 Portfolio wealth
Figure 9 to 12 illustrate the daily cumulative return of a $1 investment in each of the GMVP, and the MV-, MVS- and MVSK portfolio, given different sets of constraints and strategies. The portfolios are ranked based on terminal wealth for a given set of constraints, as if the investor is subject to a set of constraints and chooses the portfolio that gives him the highest wealth at the end of the period.
Given basic constraints the MVS portfolio obtains the highest terminal wealth, under both the buy-hold and rebalancing strategy, as shown in Figure 9. The MVSK portfolio obtains higher terminal wealth than GMVP, under both strategies, yet it loses to the MV portfolio. All of the portfolios obtain higher returns when rebalanced compared to buy-hold, and the MV- and MVS portfolio benefits the most being rebalanced.0
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Global Minimum Variance portfolio Mean−Variance portfolio Mean−Variance−Skewness portfolio Mean−Variance−Skewness−Kurtosis portfolio
Figure 9: Cumulative returns for portfolios with basic constraints
The mild and strong diversification constraint have different implications for the higher-moment portfolios, as shown in Figure 10 and Figure 11. While the mild diversification constraint reduces the dominance of the MVS portfolio, and reduces the relative performance of the MVSK portfolio, the strong diversification constraint improves the relative performance of the
higher-moments portfolios. In fact, the MVSK portfolio obtains a higher terminal wealth than the MV portfolio under the strong diversification constraint, but vice versa under the mild diversification constraint and the basic constraints, regardless of strategy.0
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BH portfolios w/ mild div. constraint
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RB portfolios w/ mild div. constraint
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BH portfolios w/ mild div. constraint
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Global Minimum Variance portfolio Mean−Variance portfolio Mean−Variance−Skewness portfolio Mean−Variance−Skewness−Kurtosis portfolio
Figure 10: Cumulative returns for portfolios with mild div. constraint
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BH portfolios w/ strong div. constraint
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RB portfolios w/ strong div. constraint
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BH portfolios w/ strong div. constraint
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RB portfolios w/ strong div. constraint
Global Minimum Variance portfolio Mean−Variance portfolio Mean−Variance−Skewness portfolio Mean−Variance−Skewness−Kurtosis portfolio
Figure 11: Cumulative returns for portfolios with strong div. constraint
From Figure 12 we observe that the higher-moment portfolios obtain the highest terminal wealth with the turnover constraint imposed. The MVSK porfolio obtains the highest terminal wealth, for both the mild and the strong turnover constraint, and is positively affected compared to its counterpart with basic constraints. The MVS portfolio is negatively affected by the turnover constraints, and we also observe that the MV portfolio obtains lower terminal wealth than the GMVP.
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BH portfolios w/ mild turn. constraint
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RB portfolios w/ strong turn. constraint
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Global Minimum Variance portfolio Mean−Variance portfolio Mean−Variance−Skewness portfolio Mean−Variance−Skewness−Kurtosis portfolio
Figure 12: Cumulative returns for portfolios with turn. constraint
The higher-moment portfolios obtain higher terminal wealth given the strong diversification-and strong turnover constraint, as shown in Figure 13. While GMVP is unaffected by this constraint set, the MV portfolio performs relatively worse and even obtains the lowest terminal wealth. The MVS portfolio ends up with the highest wealth, followed by the MVSK portfolio, and this indicates that the strong diversification constraint is the more important driver as the MVSK portfolio dominates MVS when only the additional turnover constraint is imposed.0
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BH portfolios w/ strong div.− and turn. constraint
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BH portfolios w/ strong div.− and turn. constraint
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BH portfolios w/ strong div.− and turn. constraint
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Global Minimum Variance portfolio Mean−Variance portfolio Mean−Variance−Skewness portfolio Mean−Variance−Skewness−Kurtosis portfolio 0
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BH portfolios w/ strong turn. constraint
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BH portfolios w/ strong turn. constraint
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Global Minimum Variance portfolio Mean−Variance portfolio Mean−Variance−Skewness portfolio Mean−Variance−Skewness−Kurtosis portfolio
Figure 13: Cumulative returns for portfolios with strong div. and turn. constraint
Figure 14 shows the cumulative returns of the MVS portfolio given different sets of constraints, under the buy-hold- and the rebalancing strategy. The cumulative returns of the MVS portfolio are particularly affected by the diversification constraint due to the occasional concentrated
allocations. The scenario where the MVS portfolio obtains the highest terminal wealth is with the strong diversification constraint and the rebalancing strategy. Imposing only the turnover constraint, reduces the terminal wealth of the MVS portfolio, compared to the buy-hold version with basic constraints. The MVS portfolio also performs well when both the strong diversification- and the strong turnover constraints are imposed.
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BH w/ mild div. constraint BH w/ strong div. constraint
BH w/ mild turn. constraint BH w/ strong turn. constraint
BH w/ strong div.− and turn. constraint
Figure 14: Wealth of mean-variance-skewness portfolios
Figure 15 shows the cumulative returns of the MVSK portfolio given different sets of constraints under the buy-hold- and the rebalancing strategy. The strong diversified MVSK portfolios end up with the highest terminal wealth regardless of strategy, and we observe that the rebalanced MVSK portfolio with the strong diversification constraint performs the best out of all of the portfolios in Figure 15. Both the turnover and the strong diversification constraint have a positive impact on the MVSK portfolio, in fact the wealth of the MVSK portfolio with any of these constraints have been the best versions of this portfolio since 1997.
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BH w/ mild div. constraint BH w/ strong div. constraint
BH w/ mild turn. constraint BH w/ strong turn. constraint
BH w/ strong div.− and turn. constraint
Figure 15: Wealth of mean-variance-skewness-kurtosis portfolios