Unstructured Covariance Matrices
This chapter considers the estimation and use of return covariance matrices without any factor structure imposed upon them. Following this, in chapters 3–6, we consider structured covariance matrices.
Section 2.1 discusses the estimation of return covariance matrices using classical statistical methods. Section 2.2 describes the errormaximization problem in portfolio management and its implications for risk modeling. Section 2.3 treats portfolio risk management as a problem in decision making under uncertainty, and considers the Bayesian approach to covariance matrix estimation.
This chapter is concerned with the estimation of the n × n covariance matrix of returns or excess returns. Since, by definition, the riskless rate has no variance, there is theoretically no difference between the total and excess return covariance matrices:
Although the riskless rate is known one period ahead, there is variation in its value over time, so that time-series-based estimates of cov(ri, rj) and cov(xi, xj) can differ. In most circumstances it is best practice to use excess returns in the estimation of C. The resulting estimate can be thought of as measuring the covariance matrix of total returns, conditional upon knowing the riskless return.
For notational simplicity, we will not differentiate between the return frequency of the data set and the risk horizon used for risk analysis. In practice, the return frequency used in estimation can differ from the return frequency used in risk analysis. So, for example, the analyst may use a daily return frequency to measure the covariance matrix even though the analyst mostly cares about portfolio risk at the monthly return frequency. Or the analyst may use monthly returns to analyze risk