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Assets with tail risk may produce a suboptimal portfolio under mean-variance optimization when asset returns are not normally distributed. We provide a new Monte Carlo simulation method for computing and attaching tails to observed empirical return distributions. We find that a combination of stochastic optimization and the new method for simulating tails in returns with expected shortfall utility function produces optimal portfolios that have better return and risk characteristics than those of mean-variance optimal portfolios. Results from this study suggest that bitcoin can be a diversifier in a multi-asset portfolio when optimization takes all moments of return into consideration.

Computing optimal portfolios of multi-assets with tail risk: the case of bitcoin