# On the use of conditional expectation in portfolio selection problems

## Abstract

In this paper, we examine the use of conditional expectation, either to reduce the dimensionality of large-scale portfolio problems or to propose alternative reward–risk performance measures. In particular, we focus on two financial problems. In the first part, we discuss and examine correlation measures (based on a conditional expectation) used to approximate the returns in large-scale portfolio problems. Then, we compare the impact of alternative return approximation methodologies on the ex-post wealth of a classic portfolio strategy. In this context, we show that correlation measures that use the conditional expectation perform better than the classic measures do. Moreover, the correlation measure typically used for returns in the domain of attraction of a stable law works better than the classic Pearson correlation does. In the second part, we propose new performance measures based on a conditional expectation that take into account the heavy tails of the return distributions. Then, we examine portfolio strategies based on optimizing the proposed performance measures. In particular, we compare the ex-post wealth obtained from applying the portfolio strategies, which use alternative performance measures based on a conditional expectation. In doing so, we propose an alternative use of conditional expectation in various portfolio problems.

## Keywords

Conditional expectation Large-scale portfolio selection Performance measures Return approximation Heavy tailed distribution## Notes

### Acknowledgements

This paper was supported by the Italian funds MURST 2016/2017 and by STARS Supporting Talented Research—Action 1—2017. The research was also supported by the Czech Science Foundation (GACR) under Project 17-19981S, and by VSB-TU Ostrava under the SGS Project SP2018/34.

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