Portfolio Management
1 Portfolio Management
Portfolio management is the problem of choosing how to divide capital among the financial assets available to an investor, and of monitoring and adjusting that choice over time. A portfolio is described by its weights \(w=(w_1,\dots,w_n)\), the fractions of wealth held in each asset, with \(\sum_i w_i = 1\). Its return is the weighted average of the asset returns,
\[ R_p = \sum_{i=1}^n w_iR_i, \]
so its expected return is \(\mu_p = w^\top\mu\) and its variance is \(\sigma_p^2 = w^\top\Sigma w\), where \(\mu\) and \(\Sigma\) are the mean vector and covariance matrix of the asset returns. The covariances, and not only the individual variances, determine the risk of the portfolio. This is what makes diversification possible.
The sections below introduce the main topics. Each links to its own note.
2 Risk and Return
Investors are compensated for bearing risk, so higher expected returns come with higher risk. The Risk-Return Tradeoff quantifies this relationship and the measures of risk used to describe it.
3 Allocation and Diversification
Combining assets whose returns are not perfectly correlated lowers the risk of a portfolio without lowering its expected return by the same amount. Asset Allocation and Diversification covers how capital is split across asset classes and securities, and what diversification can and cannot remove.
4 Portfolio Optimization
Portfolio Optimization chooses the weights \(w\) that give the best trade-off between \(\mu_p\) and \(\sigma_p^2\), as in Markowitz mean-variance theory, leading to the efficient frontier. The Black-Litterman Model extends this by combining market equilibrium returns with the views of the investor.
5 Asset Pricing and Factor Models
CAPM and Factor Models explain the expected return of an asset through its exposure to systematic risk factors, from the single factor of the CAPM (beta) to multi-factor models such as Fama-French and Carhart. They also define alpha, the return not explained by these exposures.
6 Performance Evaluation
Performance Evaluation measures the return of a portfolio relative to the risk taken, using ratios such as the Sharpe ratio, the information ratio and Jensen’s alpha.
7 Managing a Portfolio
- to outperform it.
- investor assumed by the models above.
Measuring and limiting the risk of a portfolio (VaR, stress testing) is covered under Risk Management in Quantitative Finance.