# The optimization of collective investment strategies for heterogenous individuals

In this paper we investigate how collective investment strategies can incorporate the heterogeneity of individuals with respect to risk aversion, the investment horizon, and initial wealth.

Assuming that the preferences of individuals can be compared to each other by means of their certainty equivalents, we will propose three different collective investment strategies: the inequality averse collective investment strategy, the welfare effect averse collective investment strategy, and the welfare loss averse investment strategy.

The inequality averse collective investment strategy that has initially been proposed by Balter et al. (2021) optimizes the collective investor’s power utility given the distribution of individual’s certainty equivalents. We extend this model by allowing for heterogeneity with respect to the investment horizon and initial wealth. Given these individual characteristics, a collective investor who is very inequality averse assigns relatively more weight in his objective to individuals with low certainty equivalents, which corresponds to individuals with high risk aversion, low initial wealth, and/or a short investment horizon. Therefore, the investment strategy of such a very (in)equality averse collective investor heavily depends upon the participants with the most extreme risk aversion, investment horizon and initial wealth.

Thereafter, we extend the existing literature by introducing collective investment strategies that aim at minimizing the extent to which a deviation from the optimal individual investment strategy ’hurts’ individuals. Consequently, a welfare effect averse collective investor aims at finding the investment strategy that maximizes his power utility given the distribution of individuals’ welfare effects. This welfare effect represents the ratio between an individual’s certainty equivalent under the collective investment strategy and the certainty equivalent belonging to the optimal individual investment strategy. Such a collective investor with a high welfare effect aversion assigns relatively more weight in his objective to individuals with a low welfare effect, which corresponds to individuals who have a relatively long investment horizon and risk aversion that is slightly lower than the average risk aversion in the fund. Since such an investor considers that assigning more weight to one individual characteristic automatically yields a lower welfare effect for an individual with the ’opposite’ characteristic, this investment strategy generates more moderate results.

Similarly, a welfare loss averse collective investment strategy minimizes the collective investor’s utility based upon the distribution of individuals’ welfare losses. However, in this case, welfare losses are expressed as the percentage change of individuals’ certainty equivalents. Consequently, this collective investment strategy weights the relative difference in certainty equivalents differently compared to the welfare averse collective investor. As a result, a collective investor with high welfare loss aversion assigns the much weight in his objective to the individual with extreme individual characteristics.

Consequently, this paper proposes several new models and extensions that allow us to set the first steps in designing the optimal collective investment strategy that incorporates the heterogeneity of individuals with respect to welfare effects, risk aversion, initial wealth, and investment horizons.