This thesis concerns portfolio theory from a Bayesian perspective and it includes two papers related to this theme. In the first paper, optimal portfolio weights are derived from a Bayesian perspective to the problem of minimizing the portfolio risk in terms of value at risk (VaR) or conditional value at risk (CVaR) given a certain level of expected return. The so called efficient frontiers, i.e., the set of optimal portfolios with the highest possible expected return given a certain level of VaR or CVaR, are also presented. Both a noninformative prior and an informative prior are considered and the resulting portfolios are compared to the ones obtained from the conventional method which is based on the sample estimates. Using simulated and empirical stock returns, we conclude that the Bayesian approach outperforms the conventional method in terms of out-of-sample VaR estimation when the global minimum VaR portfolio is considered. Moreover, we show within a simulation study that the efficient frontiers obtained from the Bayesian procedure are generally more conservative and closer to the true efficient frontier when the latter is known.

In the second paper, we consider the problem of determining VaR or CVaR of a portfolio when market conditions are quickly changing. When the market conditions are stable over time, it is possible to model future returns by using an equal belief in all historical observations. However, it is well known that volatility tends to cluster, particularly when considering daily or more frequent data observations. We use a Bayesian approach to create a model where the prior belief about returns resembles the recent period and where the degree of belief depends on how much the recent period deviates from the long term period. The new model is compared to some classical homoscedastic and heteroscedastc models in terms of VaR estimation using both simulated and empirical stock returns. We conclude that the new model performs well in the Basel backtest, particularly during turbulent market conditions where other models struggle.

This thesis contributes to the field of statistical and mathematical finance by introducing novel Bayesian and game-theoretic methods in discrete-time portfolio theory. These methodologies enhance the precision and adaptability of investment strategies and risk management, particularly in complex market environments. The work is structured around five papers.

Paper I introduces a Bayesian framework for optimizing portfolio allocation, utilizing value at risk (VaR) and conditional value at risk (CVaR) as risk measures. This approach leverages the posterior predictive distribution to derive portfolio weights directly from observed data, contrasting with traditional methods that rely on estimates of unobserved variables. The benefit of the Bayesian method is demonstrated through simulations and empirical comparisons, particularly in predicting out-of-sample VaR.

Paper II presents a dynamic Bayesian approach to incorporate volatility clustering into VaR and CVaR estimation, utilizing hyperparameters based on different rolling windows to adapt quickly to changing market conditions. This method shows distinct advantages over existing models by adjusting the certainty and expected values of prior distributions in response to volatility changes, offering improved risk estimates during market turbulence.

Paper III develops a Bayesian inference procedure for tangency portfolios by establishing a new conjugate prior directly for the optimal portfolio weights, integrating high-frequency returns and a market condition metric, such as the CBOE Volatility Index (VIX) or Economic Policy Uncertainty Index (EPU). This approach enables direct inference on portfolio weights, and backtesting suggests potential advantages over traditional strategies in real-world scenarios.

Paper IV addresses the construction of tangency portfolios under short-selling constraints, using the same reparameterized asset return model within a Bayesian context as in Paper III. An innovative prior enforces positive weight constraints. The effectiveness of this method is empirically validated with selected stocks, highlighting its potential to enhance risk-adjusted returns.

Paper V innovates within a game-theoretic framework by introducing a recursive scheme for asset allocation using a mean-semivariance reward functional to better reflect investors' aversion to downside risk. This approach resolves the time-inconsistency problem in multi-period investments through an extended Bellman equation, effectively reaching a Nash equilibrium as demonstrated by an extensive numerical study.

Collectively, these studies provide a cohesive advancement in statistical and mathematical finance, demonstrating the effectiveness of Bayesian methods and game-theoretic approaches in improving the theoretical and practical aspects of portfolio optimization and risk management.