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The Black-Litterman model combines the market equilibrium with the investor's personal views and gives optimal portfolio weights. In this paper we will review the original Black-Litterman model, we will modify the model such that it fits in a Bayesian framework by considering the investors' personal views to be a direct prior on the means of the returns and by including a typical Inverse Wishart prior on the covariance matrix of the returns. We will then consider Leonard and Hsu's (1992) idea for a prior on the logarithm of the covariance matrix. We encountered both running time and memory allocation problems when we applied the latter version to the whole S&P 500. To overcome such computational problems, Bayesian factor models are considered for the analysis. This choice was also motivated by the strong connection between Black-Litterman and the Capital Asset Pricing Model, which itself can be seen as a factor model. Sensitivity analyses for the level of confidence that investors have in their own personal views were performed and performance of the models was assessed on a test data set consisting of returns over the month of January 2018.

For 5,500 North American hedge funds following 11 different strategies, we analyse the stand-alone performance of these strategies using a stochastic discount factor approach. Employing the same data, we then consider the diversification benefits of each hedge fund strategy when combined with a portfolio of US equities and bonds. We compute the out-of-sample Black-Litterman portfolios, with Bayes-Stein, higher moments, simulations, desmoothed data and allowance for regimes as robustness checks. All but two hedge fund strategies out-perform the market as stand-alone investments; and all but one provide significant diversification benefits. The higher is an investor’s risk aversion, the more beneficial is diversification into hedge funds.

We develop two dynamic Bayesian portfolio allocation models that address questions of learning and model uncertainty by taking model-specific shortcomings into account.In our first model, we formulate a multi-period portfolio choice problem in which the investor is uncertain about parameters of the model, can learn these parameters over time from observing asset returns, but is also concerned about robustness. To address these concerns, we introduce an objective function which can be regarded as a Bayesian version of relative regret. The optimal portfolio is characterized and shown to involve a ``tilted'' posterior, where the tilting is defined in terms of a family of stochastic benchmarks. We have found this model to perform at least as well as a benchmark given the true market parameters, while outperforming it when the market assets have the same trend.Our next model extends the Black-Litterman portfolio choice model by taking several potential errors into account. We extend Black-Litterman to multiple periods, which allows for us to take into account the pairs of expert forecasts and the realized return. By doing so, we can then perform inference on these experts and discover whether they may have any bias for or against any specific assets. We can also perform similar inference on the market equilibrium distribution, which is typically represented by the capital asset pricing model (CAPM). The result is a model that is analytically intractable but may be solved numerically via Gibbs sampling. Controlled tests show our model performs favorably when Black-Litterman's model assumptions about the market equilibrium and expert views are violated. Backtests shed light on the model's ability to account for CAPM's shortcomings.

In an ever-changing global environment, the ability to adapt to the current economic climate is essential for a company to prosper and survive. Numerous previous re- search state that better risk management and low overall risks will lead to a higher firm value. The purpose of this study is to examine if portfolio theory, made for fi- nancial portfolios, can be used to compose product portfolios in order to minimize risk and optimize returns. The term contingent hedge is defined as an optimal portfolio that can be identified today, that in the future will yield a stable stream of returns at a low level of risk. For companies that might engage in costly hedging activities on the futures market, the benefits of creat- ing a contingent hedge are several. These include creating an optimized portfolio that minimizes risk and avoid trading contracts on futures markets that would incur hefty transaction costs and risks. Using quantitative financial models, product portfolio compositions are generated and compared with the returns and risks profile of individual commodities, as well as the actual product portfolio compositions of publicly traded mining companies. Us- ing Modern Portfolio Theory an efficient frontier is generated, yielding two inde- pendent portfolios, the minimum risk portfolio and the tangency portfolio. The Black-Litterman model is also used to generate yet another portfolio using a Bayesian approach. The portfolios are generated by historic time-series data and compared with the actual future development of commodities; the portfolios are then analyzed and compared. The results indicate that the minimum risk portfolio provides a signif- icantly lower risk than the compositions of all mining companies in the study, as well as the risks of individual commodities. This in turn will lead to several benefits for company management and the firm’s shareholders that are discussed throughout the study. However, as for a return-optimizing portfolio, no significant results can be found. Furthermore, the analysis suggests a series of improvements that could potentially yield an even greater result. The recommendation is that mining companies can use the methods discussed throughout this study as a way to generate a costless contin- gent hedge, rather than engage in hedging activities on futures markets.