If you’ve spent any time around finance folks or MBAs, then you’ve probably heard them utter the term CAPM (it’s pronounced "cap" like baseball cap followed by the letter "M"). CAPM is as its name states a pricing model.
In finance, pricing models are used to price financial assets. But instead of calculating a price, we generally use pricing models to estimate an expected return (which implies a relationship between price and cashflows). CAPM and its cousins tell us what the expected return of an investment should be, given its characteristics.
A Linear Regression Model In Disguise
As complicated as it sounds, CAPM is actually just a single factor linear regression model:
E(Ri) = Rf + B* (E(Rm) – Rf)
Where the intercept term is Rf (the risk free rate), and the slope term is B (beta). E(Rm) is the expected return of the market. In practice, we typically proxy the market with a broad stock index like the S&P 500, so E(Rm) can be thought of as the expected return of the S&P 500.
Risk Free Rate
Let’s go through each piece of the equation. Rf is the risk free rate. That’s the baseline rate of return that we can expect to earn even when we take on no risk. The risk free rate derives from the idea that a dollar today is worth more than a dollar in the future.
Would you prefer to receive 100 dollars today or a year from now? Most everyone would prefer their money today. So in order to incentivize us to wait a year (or longer) for our money, the borrowing institution needs to pay us a return. In the case of FDIC insured banks or the U.S. government (via Treasury bonds), which are considered super safe borrowers, the return they pay is the risk free rate. That is, we give the U.S. government 100 dollars today, confident in the knowledge that Uncle Sam will definitely pay us back in a year. Then in a year, we get back $102 – the $100 we lent and $2 to compensate us for being separated from our money for a year.
Beta
Beta is traditionally seen as an estimate of how risky an investment is. Going back to the CAPM equation, assume that Rf=0 (which it was just a few years ago when the Federal Reserve was setting the Fed Funds Rate at close to 0):
E(Ri) = B* E(Rm)
With the risk free rate zeroed out, we can more clearly see the impact of beta (B) on E(Ri), the expected return of the investment we are analyzing. When beta equals 1, E(Ri)=E(Rm). So if we expect the market to return 10%, then we would expect our investment to do the same.
If beta were instead equal to 0.5, then the expected return of our investment would be 5% (versus a market return of 10%). And if beta were equal to 2, then the expected return of our investment would be 20%. The higher the beta of an investment, the more sensitive its return to that of the market.
The CAPM Way Of Life
At its heart, CAPM is more theoretical than empirical. Typically in statistics or Data Science, we analyze a phenomenon by collecting data on it and using statistical tools like linear regression to quantify the relationships between the phenomenon and other observable factors. That is, we build the model based on the data.
CAPM, on the other hand, did not start with data, but rather with a theory of how markets work and how investors price assets. It’s more a "way of life" and the degree that you believe in it heavily influences how you invest. So in order to understand CAPM, we first need to understand the theory that underlies it.
Not All Risk Pays
CAPM is driven by the belief that not all types of risk pay. When we make an investment, we are exposing ourselves to risk – we don’t know whether or not we will get all of our money back.
CAPM splits this risk into two types – market and idiosyncratic. When we buy stock in Google, we are exposing ourselves to general stock market risk. Google is a U.S. company and its fortunes move in accordance with the overall U.S. economy to an extent. So if the U.S. economy is doing well, we would expect the S&P 500 to do well, and Google stock to produce a positive return. Conversely, if the U.S. economy entered a recession, the S&P 500 would crash, and Google stock would probably not do so hot either. So no matter how amazing its business model, every U.S. company has some sensitivity to the U.S. economy, and therefore some sensitivity to the S&P 500 (the same holds true for foreign stocks relative to their local economies).
But that’s not all. Even in a booming U.S. economy, Google could still misstep. It could suffer a hack to Gmail or a disastrous YouTube product launch. These types of risk are called idiosyncratic because they are unique to Google and not shared.
So an investment in Google means exposure to general stock market risk and unique Google company risk. CAPM belongs to a family of models that assume that there are risks that matter and those that don’t when it comes to expected returns. That is, as investors, we are compensated for bearing certain risks – known as earning a risk premium. But not all risks pay. Those that can be hedged with immaterial effort and cost do not earn investors a risk premium.
By the way, the risk premium is the amount in excess of the risk free rate that we earn for bearing risk (recall that the risk free rate is the amount of return we can earn for taking on zero risk). So the expected return of an investment is equal to the sum of the risk free rate and its exposure to risk premium(s).
Idiosyncratic Risk Is Easily Diversified Away
CAPM is built on the belief that only market risk pays a risk premium. Exposure to idiosyncratic risk (company specific risk) is believed to not produce a positive risk premium over time.
Why you ask? It’s because idiosyncratic risk can be easily diversified away by holding multiple stocks, or better yet just Investing in a market index like the S&P 500. If you hold upwards of 30 stocks in your portfolio (and assuming that the stocks in your portfolio are not all from the same sector), the company specific noise starts to wash out.
In my earlier post on correlation and diversification, I observed that adding more investments to a portfolio has a diversifying effect as long as the correlations between the new investment and the investments already in the portfolio are less than 1. The idiosyncratic components of two stocks’ returns definitely share a low correlation that is significantly less than 1 (assuming they are not closely related businesses like TSMC and Apple, where TSMC is a major chip supplier of Apple). Thus, the more stocks we include in our portfolio, the more the company specific aspects wash out (a.k.a. diversify).
So under the CAPM framework of the world, idiosyncratic risk does not pay a risk premium because investors can easily get rid of it by adding more stocks to their portfolios.
Note that CAPM is not saying that you can’t earn a positive return from taking on company specific risk once in a while. There’s an important difference between risk premiums and regular returns. When we invest in Google stock and get a positive regular return, that’s because we were correct (or lucky) in our read of Google’s business and future prospects. So to make money from idiosyncratic risk, we need to be correct (or lucky). Unfortunately, we can’t all be correct – across the entire investor population, some are right and some are wrong so that in aggregate, its approximately zero sum. In other words, while subsets of the population at times might be able to earn a positive return, the population as a whole cannot systematically earn a positive return from taking on idiosyncratic risk.
Risk premiums are different in that everyone who participates CAN earn a positive return in excess of the risk free rate – it’s not zero sum (though the level will vary over time depending on the number of participants and the exuberance of each participant). And we don’t need to be correct or lucky to earn it (assuming that the risk premium in question actually exists). Of course, this does not mean that your year to year returns are guaranteed to always be positive. Rather, it means that over a long period of time, you should expect your stock portfolio to earn a return in excess of the risk free rate (from the exposure to market risk), similar to how an insurance company expects to earn a profit for bearing hurricane risk.
The Market (Equity) Risk Premium
So in a CAPM world, idiosyncratic risk doesn’t pay a risk premium, but market risk does. Why does the market pay a risk premium? Well, there are several reasons:
- Market and economic risk cannot be easily diversified away. When an economy experiences a slowdown, corporate profits suffer, people lose their jobs, weak companies go out of business, and stocks decline in price. If you’re a participant in the economy or an investor in stocks, there’s no way to completely avoid the pain.
- Given that bearing market risk can be painful at times, there must be a reason investors are willing to do so (assuming that they are rational). Bearing market risk over time must pay us a return in excess of the risk free rate. If it did not, why would any rational person be willing to bear that risk when they could just earn the risk free rate and bear no risk at all. It’s like the hurricane insurance company – if there were no profits to be earned, then there would be absolutely no reason for the insurance company to bear the risk of hurricanes.
- There is utility created by bearing market risk. Taking on market risk means buying stocks, which give companies the money they need to make capital expenditures, hire employees, and expand their operations. All companies fund themselves through a mix of debt (borrowing from banks or bond investors) and equity (selling stock to investors). So when investors buy and hold stocks they are in effect greasing the gears of capitalism, and deserve to be paid for doing so.
And over time, the stock market does tend to outperform Treasury bonds (we traditionally consider U.S. Treasury bonds the risk free asset because the U.S. government is considered a very safe credit). I only plot the past 10 years here, but longer periods show the same general trend of stocks outperforming bonds by a significant margin.
So how can we find the part of a stock’s risk that is attributed to the market (the only part that pays)? Good old linear regression to the rescue. Recall from my earlier post, if you read it ;), that the variance of an investment is used as a proxy for its risk. And linear regression models are used to analyze and explain variance – the X variables are used to explain the variance in the Y variable. So by setting the X variable to be the market’s (S&P 500’s) return and the Y variable to be the stock’s return, we can figure out how much market exposure a stock has:
Stock Return = B* Market Return + alpha
The beta, B, in the above linear regression equation is the same as the beta (B) in the CAPM equation. In linear regression, beta is a measure of the sensitivity of the Y variable to changes in the X variable. So in this case, beta is the sensitivity of the stock’s return to changes in the market. That sounds like exactly what we want. The equation for beta gives even more intuition:
B = Corr(Stock, Market)* Stdev(Stock)/Stdev(Market)
Beta equals the correlation between stock and market multiplied by the ratio of the stock’s volatility (a.k.a. standard deviation) to the market’s volatility. Let’s think about the second part first – the ratio of volatilities gives us a sense of how risky the stock is. A high number means the stock’s returns are much more volatile than the overall market, and thus the stock is probably very risky. But under CAPM, not all risk pays. So we multiply the ratio of volatilities by the correlation between the stock’s returns and the market’s returns. If the correlation is low, then beta will be low as well – meaning that most of the stock’s risk is idiosyncratic and not attributable to the market.
Conversely, the higher the beta, the more market risk a stock has. And according to CAPM, the higher the expected return of the stock needs to be in order to compensate for all the market risk we are taking on.
Putting It All Together
Almost there! Let’s take a look at the CAPM equation again:
E(Ri) = Rf + B* (E(Rm) – Rf)
OK, now that we know how B is calculated (and the intuition behind it), let’s fill in the rest of the numbers for a stock where B=2 (lots of market risk). Rf, the risk free rate, is usually proxied with the 10 year U.S. Treasury yield, which as of this writing is hovering around 1.82%.
The market risk premium, E(Rm) – Rf, is usually approximated by the average excess return of the S&P 500 over 10 year U.S. Treasury bonds (though some cooler models attempt to calculate a forward looking implied market risk premium). We will be lazy and grab it from Professor Damodaran who estimates the market risk premium to be 4.8%. Now we have everything we need to calculate the expected return of our stock:
E(Ri) = 1.82% + 2* 4.80% = 11.42% annually
So how is this number actually used? It would be foolhardy to just assume that our stock will earn us the 11.42% annual return we calculated. Rather, think of this as the return that we should require from our stock in a CAPM world (where only market risk pays and bearing market risk earns us returns in excess of the risk free rate). Thus, if you don’t think you can get 11.42%, you probably shouldn’t invest.
Finally, I want to note that multiple studies have observed how low beta stocks tend to outperform high beta stocks. Wait what? In a CAPM world that shouldn’t happen – stocks with more market risk (higher betas) should outperform stocks less market risk. But we probably don’t live in a 100% CAPM world. So take the model with a grain of salt. While it is a useful framework for thinking about markets, risk, and expected returns, it does make many assumptions about the world. And during times when those assumptions are not valid, the model’s output will not be either.
This post is Part 4 of my primer on investment portfolio optimization. If you haven’t already, check out Parts 1, 2, and 3 as well:
