Session 8: Estimating Hurdle Rates - Regression Betas

In this session eight of a 36 session corporate finance class, we're going to talk about the beta. We're going to look at a conventional way of estimating beta, which is run a regression of returns in a stock against an index. And talk about why that beta might not be the right beta to use in either corporate financial analysis or valuation. We will set up for what's coming next. In the last few sessions, we've laid out the basics for hurdle rates. We've talked about how to estimate the risk-free rate and how to come up with an equity risk premium. In this session, we're going to bring in the third and final input into the process, which is the beta. And in particular, we're going to look at the conventional way in which people estimate betas, which is run a regression of returns in the stock against returns in a market index. So let's set the table. We now have a risk-free rate. We have ways of estimating the equity risk premium. We're trying to measure the relative risk of a company or the beta. And the next few sessions are going to be dedicated to coming up with that measure. When I was taught finance, I was taught that getting a beta was simple. I was told to run a regression of returns in a stock against returns in a market index. The slope of the line, I was told, was the beta. And I have to tell you that this is not a great way to think about betas, but it's a way we're all taught betas. But if you look at the regression, there are two more pieces to that regression that might have value to you if you're interested in a company. The first is the intercept of the regression. And as we're going to see in a few minutes, that intercept tells you something about performance. It tells you whether your stock was, keyword is was, a good or a bad investment during the period of your regression. So the slope tells you something about risk. The intercept tells you something about performance. There's a final piece of output from this regression, which is the R squared. Now statistically, the R squared tells you how good a fit your regression is. But that's not how you should read an R squared from this particular regression. This R squared tells you what proportion of the risk in your dependent variable, which is the stock, is explained by the market. Put another way, the R squared tells you where the risk in your company is coming from, how much of it is coming from the market, and how much of it is firm specific. So let's set the process up. If you think about the intercept in a regression, you don't read much into it. But in this particular regression, here's the way to think about it. Let's go back to the original capital asset pricing model. The return on a stock is the risk-free rate plus beta times RM minus RF, right? That's the equation I've started with. I'm going to do some algebra, move some terms together, and I promise you I'm not changing the equation. What you're going to end up with is the CAPM written slightly differently. The expected return on a stock in the CAPM can actually be written as the risk-free rate times 1 minus beta plus beta times RM. So why are you doing this? Now think about running the regression. You've got returns in the stock is equal to the intercept plus slope times return on the market, right? Compare those two equations. If you compare those two equations, you get the basis for looking at performance in a stock. If you run a regression for two years, five years, you get an intercept from the regression. You compare the intercept to your risk-free rate times 1 minus beta. Here's a way to read that difference. If your stock did exactly as predicted during the period of the regression, your intercept should be roughly equal to the risk-free rate times 1 minus beta. If your stock was one of those super stocks, it did much better than expected, your intercept should be much higher than the risk-free rate times 1 minus beta. And if your stock did much worse than expected, your intercept from the regression should be much lower than the risk-free rate times 1 minus beta. In fact, if you take the difference between the intercept and the risk-free rate times 1 minus beta, that's called Jensen's alpha. Some people just call it alpha. It's a measure of how well or badly your stock did after adjusting for risk and after adjusting for market performance. So when portfolio managers talk about superior alphas, this is what they're talking about. It's a dream of every active investor to earn not just a high return, but a much higher return than what you should expect to make given what the market did and given how risky your investment is. So the intercept is a measure of performance. So let's set up the regression. Saying what do you mean set up the regression? Take a stock like this, it's been listed since 1958. You have lots of choices to make. Here's the first choice to make. Yes, you have to decide how far back in history you want to go. You could go all the way back to the 1950s, but you'd be getting a very different type of company than the company you see today. So here's the trade off. You don't want to go back too far because then the company you're looking at is very different from the company today. But you want to go back far enough that you get enough data. Most services that I know of use two to five years to get a regression data. The second choice you have to make once you've decided how far back to go is how you're going to slice up the data. What I mean by that is let's say you decide to go back five years. You can look at five years of monthly returns, that's 60 months of data. You can look at five years of weekly returns, that's 260 weeks of data. You can look at five years of daily returns, that might be about 1,250 days of data. You might even break the days down into hours. Now more data is often better than less data, right? But in this case, you gotta be careful. There is a trade off. The more you slice up the data, the more data you have, at least the more observations you have, but the more noisy that data becomes. In fact, I think that once you get below weekly returns, your data's become almost useless. Most services use either weekly or monthly data to get their betas. So how far back do you go? Are you going to use daily, weekly, or monthly returns? Here's the third decision to make. How are you going to compute returns in your stock? Remember, there are two ways you make money on stocks. One is from price appreciation. The price changes on a day-to-day, week-to-week, month-to-month basis. You also get money from dividends, but those dividends don't happen every day, or every week, or even every month. For most US stocks, it might happen once every three months. For some global stocks, it might be only once a year. Don't try to be fair and spread things around. Take the dividend only in the period that you get. It's called the ex-dividend period. So essentially, if there is an ex-dividend date during a month, you get a dividend during that month. So the returns have to be computed with dividends for your stock, though it's often easier to ignore dividends. Finally, you have to choose an index. That's a tough call because there are so many choices you have in front of you. In the US alone, there are almost 120 different market indices. And if you extend that search, you could get hundreds of different indices globally. So let me go through how I made those choices for Disney. But for the index, at least, let me give you my tiebreaker. Remember that all these estimates are for the CAPM, right? And in the CAPM, what you have in your portfolio is a little bit of everything. That's why it's called the market portfolio. A good measure of an index is how close you get to that market portfolio. So you want broader indices rather than narrower indices, more stocks rather than less stocks, and market cap weightings rather than simple averaging. So with Disney, here were my choices. I decided to go back five years. I used monthly returns. And in fact, those are often my defaults. I like betas that I get from five-year monthly returns. I'll use two-year weekly if I don't have much of a choice. I used the S&P 500 as my index. It might be only 500 stocks, but it's the 500 largest market cap stocks. And then I computed returns for Disney every month for my 60 months. What do I mean by every month? Every month, I took the price, and I looked at the change in price. And if there was a dividend during the course of the month, I counted the dividend. So I've given you one example from December of 2009. During that month, the stock price went up about $2.03. I got a dividend during the month of $0.35. My return on Disney during that month was 7.88%. I do the same thing for the index. I look at the change in the index during the course of the month. And if there were dividends on the index, and I have to count them in, I compute the return on the index. I do this each month for 60 months. So think of this as a spreadsheet. At the end of the period, I have returns on Disney every month and returns in a market index. In this case, the S&P 500 every month. Now, before I run my regression, I'm going to show you what's called a scatterplot. In Stat 101, you're taught to do this. Basically, it visualizes the data. In a scatterplot, for each month, you plot a point. So there should be 59 points on this scatterplot. Why only 59? Because remember, you need the second month to actually get returns. In the very first month, because you don't have a price change, you lost the month. So you have 59 months of returns in Disney, 59 months of returns in the S&P 500. This is my scatterplot. The regression line is the best fit line, the line that best fits through those scatter points. And of course, if you have a regression package, this line is fit by minimizing the squared distances from the line. The bottom, you see the output from this regression. It's 0.0071. My intercept is 0.71%. And the slope on my regression is about 1.25. You might wonder what that number in brackets, 0.10, is. That is my standard error. And the R squared for the regression is 73%. I have my regression output. Let's see if we can make sense of it. Let's start by looking at Disney's performance over the five years of the regression. The question we're trying to answer is, was Disney a good investment during that five-year period? Notice, it's not going to answer the question, will Disney be a good investment? It's entirely a postmortem. During these five years, based on the regression, the intercept was 0.71%, right? But to make a judgment, you've got to compare to the risk-free rate times one minus beta over the same period. Here's a measurement issue you've got to deal with. This is a monthly regression. That intercept is a monthly intercept. So if I'm going to use a risk-free rate, it has to be a monthly risk-free rate. To estimate that monthly risk-free rate, I looked at T-build rates every month for those 60 months in an average amount. Why T-build rates? You're looking at one-month holding periods on holding the stock, and I was looking at very short-term rates, and hence the use of T-build rates. The average annualized T-build rate over the five years was half a percent. I want a monthly rate, I divide by 12. I can get fancy and use a compounded average. It's not going to make much of a difference. My average monthly risk-free rate is about 0.041%. When I multiply that by one minus beta, in this case, one minus 1.25, because my beta is 1.25, I end up with a negative number, minus 0.015%. You say, what does that even mean? That is what I would expect the intercept to be if the stock did exactly as predicted during the period of the regression. In other words, my intercept should have been minus 0.01%. It actually was 0.71%. You take the difference between those two numbers, you get 0.72%. That's my Jensen's Alpha per month. Now, if I annualize that number, which it often is, it makes usually more sense to look at an annual number. I come up with 9.02%. You say, what does that tell me? During this five-year period, Disney delivered 9.02% more than expected after adjusting for risk and after adjusting for market performance. Here's a simple way to think about it. Let's say that you have a stock with a beta of two. It's a very risky stock, right? Let's say the market is up 20%, and let's say there's no risk-free rate. You'd expect your stock to be up 40% based upon the fact that it's twice as risky as the average stock. If your Jensen's Alpha is 9%, here's what your stock delivered. It delivered not just 40%, but 40% plus 9%. A Jensen's Alpha is a risk-adjusted, market-adjusted measure of performance. If I did this for every stock in the index, the average Jensen's Alpha, or at least the weighted average Jensen's Alpha across all stocks should be roughly 0%. Stocks collectively can't beat themselves. So positive Jensen's Alpha is a good sign for you as an investor. But does it mean the managers of Disney did a great job? Think about it. It is true Disney beat the market after adjusting for risk and market performance. But is it entirely because of great management? You can have entire sectors outperform the market. For instance, oil companies will outperform the market in a year in which oil prices are up 50%. It has nothing to do with management. To make a judgment whether a Jensen's Alpha is good or bad, whether management is responsible for it, you have to compare it to the average Jensen's Alpha across stocks in that sector. In this case, I compared Disney's Jensen's Alpha to the Jensen's Alpha for the sector. And guess what? The sector didn't do that well. So in this case, at least, you could argue that this Jensen's Alpha is a measure of how well Disney was managed during the period. And that's a good sign for the management of the company. And if you're stockholders, it might be an indicator that you should be rewarding the management for doing a good job. So that's the intercept and the Jensen's Alpha. Let's turn to the other piece of output from the regression, which is the slope. The slope of this regression, at least based on this five-year period, is 1.25. If you trust the regression, the beta for Disney is 1.25, and the standard error is 0.10. If you ask me for a point estimate, a single number to use for Disney's beta, the number is 1.25. But remember, there's a standard error, which means that there's a range on the beta. In fact, with about 67% confidence, which it's in a normal distribution, it's about one standard deviation, your true beta for Disney will be anywhere from 1.15 to 1.35, plus or minus one standard error. With 95% confidence, it's gonna be 1.05 to 1.45, plus or minus two standard errors. Betas are estimates, they're not facts. Much as we tend to think of them as single numbers, they tend to form ranges because that's what you get from a regression. And if you look across all stocks, that 0.10 standard error doesn't look too bad because the median standard error for a beta estimate in the US is between 0.2 and 0.25. The next time somebody gives you a regression beta, remember to ask the follow-up question, what's the range on this beta? Because that range can be a pretty large number, especially for younger companies or companies with volatile histories. Now, if you take the last piece of output from regression, the R-squared of 73%, and ask, what does it tell you? It basically tells you that 73% of the risk in Disney comes from the market, and 27% is firm-specific. Remember we talked about risk that can be diversified away? Again, to the extent that you trust this regression, it looks like 27% of the risk in this company can be eliminated through diversification. Put differently, if you put all your money in Disney, 27% of the risk you're exposed to, you should not expect to get rewarded for because you made the choice not to be diversified. So as a diversified investor, you might not care what the R-squared is because you're gonna eliminate the risk that is firm-specific. But if you're not that diversified, take a look at the R-squared. You want higher R-squared stocks if you have only four or five stocks in your portfolio. Now, there are services that allow you to look up betas. One of my favorites is Bloomberg. You won't find it online, but if you have access to a Bloomberg terminal, you can print the beta page off for a company. And I did for Disney using the same five-year period that I ran the regression using the raw data. And the output I got was fairly similar. In fact, the beta I got was 1.25. The R-squared was about 0.73. The standard error was 0.10. The only number where there was a significant difference was the intercept. And part of the reason for that is my decision to include dividends in both returns on Disney and returns in a market index. It's the right thing to do. But many services often ignore either dividends on one or both when they run regressions because it's often easier to do these regressions without the dividends built in. That's why the intercept on the Bloomberg regression is only 0.60% in contrast with my 0.71% in my estimate. Now that we have all the numbers that we need, let's bring them together in a hurdle rate because that's where we've been going for the last few sessions. I have a beta for Disney, or at least based on the regression I do, 1.25. My risk-free rate is the U.S. 10-year T-bond rate because I'm doing my analysis in U.S. dollars of 2.75%. The equity risk premium for Disney based on where it operates is 5.76%. 2.75%, the risk-free rate, plus beta times my equity risk premium gives me an expected return of 9.95%. You say, what does that tell me? If I'm an investor in Disney and I'm a purist, here's what it tells me. If I believe markets price stocks efficiently, this expected return is what I can expect to make given the risk in Disney. In other words, given the beta and my expected return on stocks, 9.95% is the return I can reasonably expect to make in the long term if the stock is correctly priced and the CAPM is the right model for risk. That strikes me as a sterile way of thinking about the 9.95%. Here's an alternative and better way of thinking about it. 9.95% is the return you need to make on Disney to break even. You say, why do I need to make 9.95%? The risk-free rate is only 2.75%. You need to make 9.95% because Disney is a risky investment. You need to make far more than 2.75% to break even. In fact, one way to think about good stocks to buy is let's say you do your own research on Disney and conclude that you can make 11, 12, or 13% on the stock. That's my definition of a good stock. The 9.95% is my benchmark. I need to make more than that for a stock to be a good investment. So that's the reason we compute expected returns because that expected return now plays a different role if you're looking at this number from the viewpoint of the CFO or the management of Disney. Disney raises equity, right? We're equity investors. We need to make 9.95% to break even, not even to be happy. This now becomes the cost of equity for Disney. Of course, Disney managers might wonder, what'll happen if I don't deliver the 9.95%? It's not like debt, where if you don't make the interest payment, you go bankrupt. If you don't deliver the 9.95%, in other words, you go out and take projects that make less than 9.95% consistently, the first thing that'll happen is your stock price will drop because that's the only way stockholders can react. And if it drops for a long enough period, perhaps stockholders will get mad enough at you that they try to replace you as managers. You can see why this view of cost of equity is tied to your view of corporate governance. If you truly believe that a company belongs to its stockholders, this is a good way to think about cost of equity. If you don't buy into that notion, you could argue that equity costs you very little, that all you have to do is make your dividend payment to make investors happy. So here's an exercise that you could try. If you can find a regression beta page or even the regression output for your company, try to break it down. Look at the beta, look at the R squared, which will tell you where the risk comes from. See if you can use the intercept to get a measure of how good or bad an investment your stock was during the period of the regression. And then finally, bring it all together in an expected return. You'll need a risk-free rate based on the currency you're doing your assessment in and an equity risk premium. But we visited those concepts in the last two or three sessions. So if you get a chance, go back and review those sessions because this is central to thinking about cost of equity and hurdle rates in corporate finance. Thank you very much for listening.