We’ve previously seen how volatility is a common measure of risk. Variance is just the volatility squared. If we were to estimate the variance of a portfolio of two stocks, we would calculate this as the weighted sum of the variance of the returns of each stock as well as the covariance between the returns of these two stocks. If we were looking at the US stock market, we may have 9,000 stocks in the stock universe. Recall that we can organize these variances and covariances into a covariance matrix, which contains the pairwise covariances of each stock with itself and with all the other stocks in the portfolio. This means we have a covariance matrix that has 9,000 rows and 9,000 columns. Multiplying 9,000 by 9,000 results in 81 million elements. However, since the covariance matrix is symmetric, we will have about 14.5 million unique values to fill in the matrix. That’s still too many values to estimate and maintain. Notice that this issue of having too many values to estimate is a general problem referred to as the Curse of Dimensionality. In one dimension, there are a couple of thousand stock returns to estimate. In two dimensions, to fill in the two dimensional matrix, we end up with a couple million values to estimate. This process of calculating the covariance matrix of assets is called a historical measure of a portfolio’s risk. It becomes difficult to do when there are many stocks in the portfolio. The challenge with estimating the covariance matrix of assets motivates the need for a different approach. That approach is the Risk Factor Model.