Welcome back. Today we are going to talk about how we can use factor models to estimate expected returns. In the first factor model that you may want to be using, I mean, the simplest of them all is the Capital Asset Pricing Model the CAPM. If the CAPM is the true asset pricing model, then we have a very simple prescription in the sense that we know that excess expected return for a given stock is proportional to the Beta of the stock as can be seen from this equation. Now, this equation has a very simple implication when you think about it. It says that Mu_i expected return on the stock minus RF the risk-free rate, divided by Beta_i, the Beta of the stock. That ratio excess expected return divided by Beta, which is known as the Treynor ratio, well, the CAPM predicts that the Treynor ratio is going to be exactly the same for all stocks, and the quantity that you're going to get in terms of that Treynor ratio will be the market risk premium Mu_m minus RF, and that quantity will be the same across stocks. Now, if that were true, if we were to live in a world where the CAPM was the true model, then the pricing model, then this would have very profound and simple implications for portfolio construction. Well, to see this just take a look at the Sharpe ratio of the portfolio. What you need to be able to estimate critically, is Mu_i minus RF shows up at the numerator of the Sharpe ratio for the portfolio. Well, Mu_i minus RF according to the CAPM, is nothing but Beta_i times Mu_i minus RF, and the M_i minus RF being the same for all stocks factors out. So in the end, maximizing the Sharpe ratio is just equivalent to maximizing the ratio given by the weighted average of the Betas of individual stocks the sum of w_i Beta_i, which is nothing but the Beta of the portfolio, divided by the portfolio volatility. Well, that's a very simple prescription, and using a good estimate for the covariance matrix, and good estimate for the Betas, then you have everything you need to optimize the portfolio. Well, the problem though, as we know, is that we do not live necessarily in the multi-factor world as we will see in the single factor world as we will see, and therefore we will need to revisit this assumption of the CAPM being the true asset pricing model. Now, before we do so just let's take a look at the estimates you get. So in this context, we are using the 30 industry portfolio from the Ken French's Library, Data Library which is available on his website. So you get 30 sector indices, and we're looking at five years worth of monthly data from 2013-2018. This table actually contains two types of estimates for expected return. The first column is precisely given by the CAPM based estimates for these expected returns using an estimate for Beta based on the sample of data and also an estimate for the risk-free rate and the market risk premium on the same sample. The same way, we are shifting to the column number 2, we are looking at competing estimate for expected return which is given by the historical mean. Simply looking at the sample value, sample mean value of this five-year period of time. Now, clearly as we can see, the CAPM based expected return estimates are much more reasonable. Much more reasonable in particular the range of those estimates is much smaller compared to the range of estimates that you get with the sample mean. With a sample mean you get a negative, I mean the worst or the lowest value that you get is minus 14.88 percent, which just does not make any sense in terms of an expected return estimate. Well, minus 14.88 is just happens to be the realized return on that particular sector index that did particularly poorly over the sample period but of course that does not imply that minus 15 percent would be a reasonable estimate for that sector index expected return looking forward. In contrast if you look at what happens with respect to CAPM based estimate, the range is much smaller, the minimum is around 5.76. So we are looking at things that are much more reasonable numbers. Those more routinely reasonable expected return estimate translate into more reasonable portfolios. So if you start to maximize Sharpe ratios based on either CAPM based expected returns or historical mean expected returns estimates for the same covariance matrix, which is sample-based covariance matrix, then you find portfolio weights that are very different, and in particular the historical weights which show up in orange on this graph show a much broader variation. Some of those weights are as low as minus 50 percent or eve n lower minus 50 percent, and the positive ones are above 50 percent. We are clearly looking at an extreme not well balanced not very reasonable portfolio. Now, if you look at CAPM based estimates because they are more reasonable, they give you a portfolio which is much better behaved. You still see in number of short positions that eventually we may want to deal with, but that's going to come later on. Now, moving on from the CAPM single factor model given that we know that the world is better explained in terms of multi-factor exposures, then we might be tempted to use Steve Ross Arbitrage pricing theory, which is arbitrage pricing model, as a pricing model that gives us is a particular expression for what the expected return should be. That expression is actually generalizes the CAPM equation to the multi-factor setting. This equation tells us that the excess expected return on a given stock is given by the sum of the product of the factor exposures of their stock with respect to all rewarded factors times the reward on those factor. The excess return on those factors as can be seen from these equation. Now, the problem though, if you want to be using that equation in the context of optimizing the Sharpe Ratio the problem is, you need to be able to estimate the excess return of the Sharpe ratios Lambda_k of those factors, which wasn't really the case when you had a single factor because that excess return on the market, you didn't even need to know what it was. It factored out, and we didn't need to estimate it for optimizing the Sharpe Ratio of the portfolio. When you're moving on to multifactors, yes then you need to estimate those excess return and risk premium on factors and that's not good news, because estimating expected return on factors is as difficult as estimating the expected returns on the individual components of the portfolio if you will. So we're back to square one. So what do we do in terms of estimating these expected returns? Well, different options. Option number 1 which is an agnostic approach. If you don't know how to distinguish between these factors in terms of expected returns, you may want to assume that these expected returns are all equal for these factors. Well, that may not be necessarily the most reasonable choice given that some factors are riskier than others. So the agnostic tire will actually tell you that you should assume that all of these factors have to same Sharpe ratio, the reward per unit of risk. So that's you giving up in trying to distinguish factors in terms of which one has the highest reward per unit of risk saying well, the data is not very informative, so I'm going to assume they are all equal. Well, the alternative would be to take a look at the data precisely but take a look at not a small sample but the long sample, the longest possible sample, and then you may want to look at the Sharpe ratio for all of these factors over the longest possible sample. By looking at this, you may come to the conclusion that some of these factors have a higher reward compared to other factors per unit of risk and you may want to use that information or that piece of information to optimize your portfolio. Well, then there's the last approach. Let's call that the active approach, whereby we recognize that it might actually precisely be the value of an active portfolio manager to be able to express meaningful or looking views about the Sharpe ratios of these factors. In which case, then we are going to be using the active use generating by the managers based on some qualitative analysis or perhaps quantitative analysis. Well, that's precisely the skill of active portfolio management to be able to make those calls in terms of forward-looking estimates for active returns. Well, wrapping up factor models can be used to obtain meaningful estimate for expected returns on securities. The bad news though is when there's more than one factor in your factor model, you then need to have to rely on meaningful estimate of expected return on factors, and well if you have at your disposal active managers well, you may kind of want to turn to them and ask them to generate active use for these factors. Alternatively, you may want to be using some agnostic prior assuming all factors have the same reward per unit of risk or do something based on historical data.
