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

ABSTRACT

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.

KEYWORDS:

• Bitcoin
• cryptocurrencies
• tail risk
• portfolio optimization
• Monte Carlo

JEL CLASSIFICATION:

• G10
• G11
• G17
• C15
• C61

Acknowledgments

The authors would like to thank an anonymous referee for the valuable suggestions that helped to improve the paper in several respects

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

¹ Shahzad et al. (2022a) provides an excellent general background of bitcoin.

² We thank an anonymous referee for suggesting this important connection between contagion and tail risk.

³ Details on solving and assessing the quality of the solution to Equation (2)(2) $\underset{\omega \in \mathrm{\Omega }}{max}{\bar{u}}_n\left(\omega \right)\equiv \frac{1}{n}\sum { }_{j=1}^{n}u\hspace{0.17em}(\tilde{R}j\omega )$(2) are described in (Morton, Popova, and Popova 2006) and (Popova et al. 2007).

⁴ The left-tailed formulation is available from the authors upon request.

⁵ Histograms for all asset classes are available from the authors upon request.

⁶ Due to the unexpected sizable increase in the auction of US Treasuries in late February 2021, massive sell-off in the bond markets, causing BND prices to plummet. See (Smith, Platt, and Wigglesworth 2021).

⁷ Actual values of the SSD are available from the authors upon request.

⁸ The smaller the number of the percentile, the heavier the attached tail.

⁹ Weights range from 42% BND, 47% SPY, and 11% BTC to 53.5% SPY and 46.5% BTC.

¹⁰ Weights range from 53.5% SPY and −46.5% BTC to 0.2% SPY and −99.82% BTC.

¹¹ Detailed allocation results of the stochastic optimization with tail risk and MVO are available upon request.

¹² Weights range from 42% BND, 47% SPY, and 11% BTC to 53.5% SPY and 46.5% BTC.