1-25. Time Series and Cross Sectional Risk Models

Now, let’s see how we can use the Fama French three-factor model to fill in the values we need for the risk model. We have three factors, so we’ll fill in the covariance matrix of factors that is three by three. Notice that the first row has the market factor in all three columns, the … Read more

So, now we’ve seen how Fama and French used theoretical portfolios to create factor returns, time series that represents size and value. What Fama and French did when combining these factors was to define theoretical portfolios that were based on both size and value. Let’s see what I mean by that, first sort by market … Read more

Now let’s look at the third factor, the value factor. There is research to suggest that stocks that have a high book value, relative to their market price, tends to outperform stocks with lower book value. Put another way, stocks that are cheap relative to their fundamentals, tend to outperform stocks that are expensive, relative … Read more

Notice that while we calculated our returns for one particular portfolio, if we chose different stocks, we could create another portfolio to represent the size factor. If we decided to buy in short twice as much, this could potentially be another portfolio representing the size factor. How do we generalize this to other portfolios that … Read more

Now, we’ll build upon the previous risk model by including two additional factors for a total of three. This is the famous Fama-French model, which can be considered the model that started the movement towards multi-factor models. The original three-factor Fama-French model includes the market return, and adds two additional factors, which the other’s called … Read more

If we look again at our risk model, we can see that we’ve got estimates for each of the variables in several of these matrices. In particular, we are able to fill in the matrix of factor variances and covariances, the matrix of factor exposures and the matrix of specific variances. The product of factor … Read more

We can also use the results of the regression to estimate a time series for the specific return. For each time period, the specific return is the residual from taking the actual minus estimated excess return of the stock. The actual return is data that we just used to input into the regression. The estimated … Read more

To estimate the other variables, we’ll also collect a time series of data for each stock and make use of the capital asset pricing model. Remember that the capital asset pricing model assumes that the stock’s excess return above the risk-free rate can be modeled as its exposure to the market’s excess return above the … Read more

It’s important to be familiar with the cross-sectional approach as it’s the one that most commercial risk models are based, and commercial risk models are widely used by institutional investors. In academic research, it’s common to see the time series approach, whereas in industry, practitioners usually either purchase a commercial risk model such as the … Read more

Fundamental factors can also be used in the cross-sectional risk factor model. In fact we’ll discuss two that we saw in the Fama-French time series model, the book to market value and market cap. This is an interesting example because we can see that there are two approaches to working with the same factor. There’s … Read more

Now that we have the matrix of factor variances and covariances, we also want to get values to fill in the matrix of specific variances which contain the variances of the specific returns for each stock. If there are two stocks in the stock universe, then there are two specific risk variances to calculate. Let’s … Read more

Let’s focus on just one country variable, such as the USA country variable and see how we can estimate the values that will plug into the risk model. For the factor exposure matrix and its transpose, we want each stock’s factor exposure to the USA country factor. This can be set to one, if the … Read more

Let’s look at a few examples factors we can use with this cross-sectional risk model. There are categorical factors such as the country in which the company is based or the sector that it is a part of. Intuitively, we can imagine that the country of a company may affect its stock price movement in … Read more

Let’s step back and look at an equation z equals x times y. If we knew both z and x, then we can solve for y. Similarly, if we knew both z and y, then we can solve for x. In other words, if we know any two of the three variables in this equation, … Read more

What we’ve seen so far are called time series risk models. Most practitioners tend to buy commercial risk models that are built and maintained by other companies. These commercial risk models tend to be a different type of risk model, a cross-sectional risk model. The use of cross-sectional risk models is quite common within the … Read more

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 … Read more