In a world where non-linearity and randomness are the norm, the capital asset pricing model (CAPM) is widely accepted despite it being a linear model, and this is probably due to the simplicity of the model and its pre-computer age birth (see equation below). A well recognized and utilized metric in finance is beta (β), which is the slope in the linear CAPM. To derive β one simply plots the returns (capital gains plus dividend yields) of an individual stock (y-axis) against the returns of a well diversified portfolio of stocks ( x-axis), with the resulting slope being called β. Thus β represents the risk associated with an individual stock, as it is compared to a well diversified portfolio, and since the market portfolio theoretically only contains market risk, a β above (below) one reflects the degree of company-specific risk of the individual stock that should be diversified away as it is added to the market portfolio.

Finance literature is riddled with support for β, as well as doubt surrounding its validity, but why do these mixed reviews exist?, probably because it is nearly impossible to prove or refute the model when the very data being used for this purpose is extracted from within the impure and intertwined marketplace. Recent efforts have been made to go beyond β via multiple linear regression models, with two such examples being: (1) the arbitrage pricing theory (APT), and (2) the more widely accepted Famma French model. The impetus behind these models is the desire to bridge the linear CAPM model into the realm of multiple linear factors instead of a single factor, β. And both of these models are definitely a step in the right directions since non-linearity can be captured via their curvilinear regression (multiple linear regression). This post will not expand further on curvilinear regression, but a future post to this blog will address this topic in great detail.

If we step back though and examine the CAPM and its application, we would quickly find that a lot of people are confused about how to properly apply this model, even though it is such a simplistic and widely used model. To coin the phrase – there is still “a lot of low hanging fruit” around educating people to the nuances surrounding the CAPM. The following is a list of things to consider when applying the CAPM, and as you will see there are more options than initially meet the eye, and the choices one makes are quite subjective:

- Should one use nominal returns or the spread above the risk free rate for their calculation of β? The answer is
. The CAPM model itself assumes a market risk premium spread over the risk-free rate, and the intercept of the security market line (SML) is the risk-free rate (i.e., the SML is the theory behind the CAPM) – so spreads fit both with the SML theory and the CAPM model. If the risk-free rate is steady however over your sampling period, then it matters less whether you use spreads or nominal returns in determining β. But another benefit of using spreads is that your intercept is then alpha (α), the expected return of the stock above the return of the SML (the market average).**the spread above the risk free rate**

- When fitting data to determine β, what should be the sample size and frequency of the data? A standard rule is 3-5 years of monthly returns (so 36 to 60 data points), but some recommend using 1-3 years of weekly returns (so 52 to 156 data points). I personally recommend using
**5 years of monthly returns**for several reasons: (1) to allow ample trading time to transpire between measurements especially for illiquid stocks, (2) to provide enough data points to make the measurement statistically sound, and (3) to provide a ample history of the stock’s returns to better reflect the true potential risk of the stock. With a typical 10-year business cycle, one could even argue that 5 years is not long enough, but then you run the risk of the data not reflecting the true nature of the company’s current asset base and strategy. Since we are using historical returns to predict future returns, a date range that more or less reflects the current asset base and strategy would be best to capture in the β. This would suggest that 1-year of weekly returns would be best, but then we are missing the business cycle effects. The compromise is a 5-year monthly sampling.

- Should arithmetic or geometric returns be used?
**Arithmetic**since each period return on the stock market is independent of one another – there is no compounding on the stock market.

- The next question is what “risk-free” rate should be used? Some people suggest the short-term 90-day US Treasury T-Bill since it is the most risk-free rate available (i.e., it does not include maturity, liquidity, or default risk). Some people point out that this still contains inflation risk and a that 90-Day TIPS is even better (Treasury Inflation-Protected Securities). And still others recommend using a long-term US Treasury such as the 30-year T-Bond, because companies are ongoing entities, and a 30-year treasury better reflects their true risk-free rate. A principal in finance is to match the return of an asset with an appropriate discount rate that is based on the maturity and risk of the cash flow being discounted. Likewise, when calculating equity-treasury spreads for use in the CAPM model, and calculation of β, the long-term US Treasury such as the
**30-year US T-Bond**makes the most sense since we are matching the maturity of the risk-free asset with the the ongoing company.

- What market index should be used for your market return? The standard used in the US is the S&P 500, but really any index that matches your investment portfolio strategy, and is well diversified should work. And theoretically, once you have a diversified portfolio of your own, you could even use this as your market index since this is what you are really comparing the new stock against (i.e., should I add this stock to my existing portfolio or not?). If you choose to do the later, I would still do the former, because market risk and return is based on a completely diversified portfolio, and if your portfolio is not this, then your risk/return analysis will be skewed, and you might not end up assessing the correct return that you require for given level of risk in the equity. Studies have been done on the various types of market indexes (price-weighted, value-weighted, and unweighted) to see what index methodology produces the best results, but the results reported are at present inconclusive and contradictory to one another. The recommend weighting method is the value-weighted index, which is simply the ratio of current market cap to base market cap, times the base index value. Since the
**S&P 500**is clearly well diversified and is also a value-weighted index, it is the recommended index to use.

- A frequent mistake of people when applying the CAPM is to use the current risk-free rate and a calculated average market return to calculate their market risk premium (MRP). Realize that the MRP is meant to be the spread of the equity index over the risk-free rate, and over time! But how much time?,
**as far back as your data goes**, with ibbotson being a good source of data. Some people will refute this, and will instead suggest using a more current sampling of spreads, but it is important to capture the great depression, as well as the tech bubble – otherwise you wouldn’t have the proper spread of potential outcomes to protect yourself for a future event, such as the most recent great recession. So the MRP should be based on historic spreads dating as far back as records go, but the risk-free rate used for the intercept of the equation should be the current risk-free rate. And the type of risk-free rate used for both should be the same, and preferably the 30-year US T-Bond.

Below is a file that actually calculates β for Apple (AAPL), and also shows its application then in the CAPM to estimate Apple’s required rate of return. As can be seen from this file, β is essentially the same whether derived using the S&P500 or the NASDAQ Composite Index. Also, the assumption of the arithmetically or geometrically derived MRP has a significant impact on the CAPM, yielding 12% versus 10% respectively (arithmetic is the proper choice, and the more conservative choice as well) :

**Download Excel Example Here: Derivation of AAPL’s Beta in Excel**

**Download PDF of this Blog Post Here: PDF of this blog post on CAPM and Beta**

©2011 Ben Etzkorn