The area of finance poses many challenging problems to the decision maker. One of them is the modelling of the expected return on stocks and the covariance matrix of returns. This thesis approaches the decision problem of choosing an optimum portfolio of stocks in which to invest from the point of view of statistical decision theory. We use regression methods to predict the expected monthly return on stocks and the covariance matrix between returns, the predictor variables being a company's 'fundamentals', such as dividend yield and the history of previous returns. Predictions are evaluated out of sample for shares traded on the London Stock Exchange from 1976 to 2005. Many modelling and inferential approaches are examined and evaluated, the main ones being shrinkage of regression coefficients, and transforming predictor variables to near normality. It is important to use suitable statistics to make a fair comparison of the out-of-sample performance of rival methodologies. We review well-known measures of assessing investment performance, including Sharpe, Sortino and Omega ratios, and derive a new statistic from the exponential utility function. We also suggest a graphical aid which could be used as a useful summary of investment performance.