From a Single Optimum to a Distribution of Portfolios: A Bayesian Approach to Cryptocurrency Portfolio Optimization

The rise of cryptocurrencies presents a number of challenges for the classical toolkit of portfolio optimization. This study examines how a Bayesian statistical framework can be used to construct multi-asset portfolios based on weekly returns of the BUX, DAX, and FTSE equity indices, together with Bitcoin and Ethereum. Parameter uncertainty is incorporated directly into the optimization through Markov Chain Monte Carlo (MCMC) sampling, while the long-only constraints are imposed using the SLSQP algorithm for each posterior draw. The in-sample results show that portfolio optimization produces distributions of asset weights rather than a single optimal allocation, thereby reflecting the substantial uncertainty surrounding asset allocation. Under a return-maximizing decision rule, cryptocurrencies assume dominant positions, whereas risk-minimizing allocations are tilted towards traditional equity indices. Maximizing the Sharpe ratio results in a more balanced portfolio structure. An in-sample prior-sensitivity analysis of the Sharpe-ratio-maximizing portfolio confirms the statistical stability of the model and indicates that the empirical data exert a stronger influence than the choice of prior distributions. However, a multivariate Student-t robustness check applied to this strategy leads to a substantial reallocation of capital from the BUX and DAX towards Ethereum and the FTSE index. An out-of-sample rolling-window backtest further highlights the vulnerability of the in-sample return- and Sharpe-maximizing strategies to changes in market regimes. In contrast, the Bayesian Global Minimum Variance strategy delivers the most stable capital growth and the lowest maximum drawdown, while requiring negligible portfolio turnover. These findings suggest that the Bayesian framework provides a more realistic representation of the role of cryptocurrencies in diversified portfolios and, importantly, that out-of-sample performance is better supported by Bayesian-adjusted variance minimization than by strategies optimized solely on in-sample return or Sharpe-ratio objectives.