We study the optimal portfolio allocation problem from a Bayesian perspective using value at risk (VaR) and conditional value at risk (CVaR) as risk measures. By applying the posterior predictive distribution for the future portfolio return, we derive relevant quantities needed in the computations of VaR and CVaR, and express the optimal portfolio weights in terms of observed data only. This is in contrast to the conventional method where the optimal solution is based on unobserved quantities which are estimated. We also obtain the expressions for the weights of the global minimum VaR (GMVaR) and global minimum CVaR (GMCVaR) portfolios, and specify conditions for their existence. It is shown that these portfolios may not exist if the level used for the VaR or CVaR computation are too low. By using simulation and real market data, we compare the new Bayesian approach to the conventional plug-in method by studying the accuracy of the GMVaR portfolio and by analysing the estimated efficient frontiers. It is concluded that the Bayesian approach outperforms the conventional one, in particular at predicting the out-of-sample VaR.

This paper introduces Bayesian inference procedures for tangency portfolios, with a primary focus on deriving a new conjugate prior for portfolioweights. This approach not only enables direct inference about the weightsbut also seamlessly integrates additional information into the prior specification. Specifically, it automatically incorporates high-frequency returns and amarket condition metric (MCM), exemplified by the CBOE Volatility Index(VIX) and Economic Policy Uncertainty Index (EPU), significantly enhancing the decision-making process for optimal portfolio construction. While theJeffreys prior is also acknowledged, emphasis is placed on the advantages andpractical applications of the conjugate prior. An extensive empirical studyreveals that our method, leveraging this conjugate prior, consistently outperforms existing trading strategies in the majority of examined cases.

We address the challenge of constructing tangency portfolios in the context of short-selling restrictions. Utilizing Bayesian techniques, we reparameterize the asset return model, enabling direct determination of priors for the tangency portfolio weights. This facilitates the integration of non-negative weight constraints into an investor’s prior beliefs, resulting in a posterior distribution focused exclusively on non-negative values. Portfolio weight estimators are subsequently derived via the Markov Chain Monte Carlo (MCMC) methodology. Our novel Bayesian approach is empirically illustrated using the most significant stocks in the S&P 500 index. The method showcases promising results in terms of risk-adjusted returns and interpretability.

This paper introduces a novel recursive scheme for optimal asset allocation based on a mean–semivariance reward functional and a game-theoretic approach in a discrete-time setting. Unlike established frameworks that can handle variance as a risk measure, this study shifts focus to semivariance, which cannot be handled by existing theory due to aspects of its definition, including the use of an indicator function. To address this problem and the corresponding challenges of time inconsistency in multi-period investment decisions, we propose an extended Bellman equation to find a Nash equilibrium. The main contribution of this paper is a computational framework and a numerical investigation of a semivariance-based allocation strategy, based on an extended Bellman equation. Our analysis is restricted to the two-asset case — one risky and one risk-free asset — as a proof of concept, leaving multi-asset extensions for future work. The results of the numerical study indicate that our proposed method shows potential in achieving favorable investment outcomes.