Multi-asset Case

The two-asset case provides important insights regarding the impact of regret on optimal portfolios. However, it does not capture the full richness of potential effects. We therefore look at the multi-asset case now and ask two questions: Do the general effects hold true regardless of the number of assets? What kind of structure is tied to two assets without carrying over to the case of more than two assets?

3.3.2.1 Expected Return

A first answer to these questions can be given for the impact of expected returns on portfolio weights. As is apparent from Figure 3.1, taking regret into consideration leads to a higher sensitivity of optimal portfolio weights with respect to expected returns than the classical Markowitz model. This result also holds generally for N assets. As long as short-sales constraints are not binding, the expected return sensitivity will always be positive in the regret model and larger than in the Markowitz model. A proof is given in the Appendix 3.A.2. We show that the result is not restricted to the case of a multivariate normal distribution but holds for any elliptical distribution. Intuitively, a higher expected return of an asset unambiguously reduces its regret risk and therefore makes it relatively more attractive than in the Markowitz model, regardless of the number of assets in the investible universe.

3.3.2.2 Standard Deviation

With respect to a comparative-static change in the standard deviation of an asset, the two-asset case is somehow special. As we have seen in Figure 3.2, under a bivariate normal distribution and identical expected returns there is no effect of the standard deviation at all in the pure regret case. Figure 3.5 shows the analogous situation for three assets (Part A) and ten assets (Part B), respectively. The plots are again based on multivariate normal distributions with identical means of 10%, identical standard deviations of 25%, and identical correlations of 0.5 between all assets as the base scenario. In this base scenario of homogenous assets the optimal portfolios weights equal 1/N for all assets, both in the Markowitz case and in the pure regret case. As Figure 3.5 shows, increasing the standard deviation of one asset (Asset 1) actually increases the optimal portfolio weight of this asset in the pure regret case, that is, a higher standard deviation lowers the regret risk of the asset relative to the regret risk of the other assets. The reason is that in the multi-asset case, the standard deviation has an impact on the asset’s probability of being the ex-post best one even if the expected returns of all assets are identical. To point out

only one other risky asset with an expected return of 10%, the probability of it being the ex-post best asset will be 0.5, regardless of the standard deviation of this other asset. If there are several risky assets, however, the probability of it being the ex-post best asset can deviate from 1/N. For example, with two uncorrelated risky assets and one risk-free asset, the probability that the risk-free asset will be the ex-post best asset is only 1/4 (the probability that both risky assets have a return below 10%) and not 1/3, which would be the case with three homogeneous assets. This argument, however, suggests that a changing standard deviation can also work in the opposite direction, because the standard deviation effect interacts with the correlation in the multi-asset setting. If the return correlation between the two risky assets approaches one, the probability that the risk-free asset is the ex-post best asset approaches 1/2, which is above the value of 1/3 that is obtained in the homogeneous asset case. For example, if we repeat the simulation from Part A of Figure 3.5 using a correlation of 0.95 instead of a correlation of 0.5, the optimal portfolio weight of Asset 1 indeed drops from a value of about 0.41 for a standard deviation of 10%

to a value of 1/3 for the base scenario standard deviation of 25%. All these results show that a shift in the standard deviation can produce different effects on optimal portfolio weights in the regret model if multiple assets are considered and the simple message from Figure 3.2 does not carry over to the multi-asset case. What does carry over, however, is the economic intuition behind the results. If a comparative-static shift of a parameter changes the probability of an asset to become the ex-post best asset, it is most likely that this shift will also affect optimal portfolio weights.

Figure 3.5: Effects of a Shift in Standard Deviation with Multiple Assets Part A: Portfolio Weight of Asset 1: Three Assets

0.10 0.15 0.20 0.25 0.30 0.35 0.40

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Standard Deviation Asset 1

Weight Asset 1

Markowitz Case Pure Regret Case

Part B: Portfolio Weight of Asset 1: Ten Assets

0.10 0.15 0.20 0.25 0.30 0.35 0.40

0.00.20.40.60.81.0

Standard Deviation Asset 1

Weight Asset 1 Markowitz Case

Pure Regret Case

Note: Figure 3.5 shows the effects of a shift in the standard deviation of an asset (Asset 1) on its optimal portfolio weight in an investible universe with three assets (Part A) and ten assets (Part B). The base scenario uses three (ten) assets with a multivariate normal return distribution, identical expected returns of 10%, identical standard deviations of 25%, and identical correlations of 0.5. The black lines refer to the Markowitz case (α= 0) and the cyan lines to the pure regret case (α= 1).

3.3.2.3 Correlation

Another issue of the multi-asset case is to look at changes in correlations. In the two-asset case a comparative-static change in the correlation has no effect on portfolios weights if the assets have identical expected returns and standard deviations, leaving them fully homogeneous irrespective of their correlation. Only the multi-asset case allows us to introduce some heterogeneity in correlations that potentially has an impact on portfolio weights. To investigate correlation effects, we use the same base scenario as in Figure 3.5 and vary the correlation between Asset 1 and all other assets. Figure 3.6 provides the corresponding results. Again, Part A of the figure refers to the three-asset case and Part B to the ten-asset case. Both parts of the figure show the same tendency: The higher the correlation, the lower the portfolio weight. This result holds both for the Markowitz case and the pure regret case. The well-known intuition for the Markowitz case states that an asset that is highly correlated with other assets offers fewer diversification benefits.

The same intuition works with respect to regret risk. Think of an asset with a low (even negative) correlation. This asset imposes a high regret risk on the other assets if it happens to be the ex-post best asset. To reduce the regret risk of the portfolio, one should therefore overweight this asset in comparison to a weight of 1/N. The correlation effects in the pure regret case, however, are less pronounced than in the Markowitz case, suggesting that the introduction of regret into portfolio selection still mitigates correlation effects to some degree.

Figure 3.6: Effects of a Shift in Correlation

Part A: Portfolio Weight of Asset 1: Three Assets

0.2 0.3 0.4 0.5 0.6 0.7 0.8

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Correlation with Asset 1

Weight Asset 1

Markowitz Case Pure Regret Case

Part B: Portfolio Weight of Asset 1: Ten Assets

0.2 0.3 0.4 0.5 0.6 0.7 0.8

0.00.10.20.30.4

Correlation with Asset 1

Weight Asset 1

Markowitz Case Pure Regret Case

Note: Figure 3.6 shows the effects of a shift in the return correlation of an asset (Asset 1) with all other assets on its optimal portfolio weight in an investible universe with three assets (Part A) and ten assets (Part B). The base scenario uses three (ten) assets with a multivariate normal return distribution, identical expected returns of 10%, identical standard deviations of 25%, and identical correlations of 0.5. The black lines refer to the Markowitz case (α= 0) and the cyan lines to the pure regret case (α= 1).

3.3.2.4 Summary

As we have seen, the case of multiple assets adds complexity to the question of portfolio composition under the regret model. While the direction of an expected return effect is clear, this is not true for a shift in the standard deviation. Even if expected returns are equal among assets, suggesting a pure risk minimization, a higher standard deviation can lead to increasing or decreasing weights, depending on the correlation. Moreover, it has been shown that heterogeneous correlations also have an impact on regret risk and thereby on the resulting portfolio weights.