1 – M4 L2A 12 Time Series Risk Model Factor Variance V2

So, you’ve seen the structure of the risk factor model and how it attempts to explain a portfolio’s risk in terms of the common risk factors. Let’s see how this works with a factor that we’ve seen before, the market return which was introduced when discussing the capital asset pricing model. Our goal is to use the market return as a single factor in our Risk Factor Model and then fill in the values that we need to calculate the portfolio variance. First, there is the covariance matrix of factors. Since there’s only one factor, it has one value which is the variance of the market’s excess return above the risk-free rate. To make our example more manageable, we’ll work with just two stocks. For the matrix of factor exposures, we want the factor exposure of each stock to the single factor, which in our model, is the market return. So the factor exposure matrix is a column vector with two rows, and the transpose is a row vector with two columns. We’ll also fill in the matrix of specific returns for each stock. Let’s see how we get the values that we’ll put into the matrices of this risk model. For the variance of the market factor, we can collect a time series of an index. For instance, the S and P 500 if the stock universe is focused on the US stock market. We can choose a time window of four weeks, eight weeks, or 12 weeks. Then calculating the variance of this time series gives an estimate for this value. There’s a bit of a judgement call as to what time window to use. Too short of a sample, and the data is mostly noise. Too long of a sample, and the earlier data points may no longer be useful for estimating the current variance of the market. Also, the time window used for analysis scales with a holding period and the frequency at which we expect to trade. So, for daily trading, we may try different windows of several weeks. For strategies with longer holding periods, we may use a longer time window such as one, two, or three years. So now we’ve estimated one value in the risk model.

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