# Capital Asset Pricing Model (CAPM Equation)

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Financial literature is riddled with support for β, as well as doubt surrounding its validity, due to it nearly being impossible to prove or refute the model when the very data being used for this purpose is extracted directly from the marketplace itself (i.e., it is difficult to isolate variables to study within the marketplace).  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.  These models convert the linear CAPM model into a non-linear model via the use of multiple linear factors instead of a single factor, β, and thus theoretically improve the accuracy of the model.  Both of these models are probably a step in the right directions since non-linearity seems more plausible, and can be implemented easily via multiple linear regression (curvilinear regression).  Future posts to this blog will address this topic in greater detail.

If we step back and examine the CAPM and its application, we would quickly find that there is confusion about how to properly apply this model, even though it is a simple and widely used linear regression model (i.e., y=mx + b equation picture below):

$R_{S} = R_{RF,C} + \beta \times MRP$

Where:

$MRP = \frac{\displaystyle\sum_{i=1}^{n} (R_{M,i} - R_{RF,i})}{n}$
RM,i = Historical market index returns (S&P 500)
RRF,i = Historical risk-free rate (30-Yr T-Bond or 30-Day T-Bill)
RS = Required return for stock
RRF,C = Current risk-free rate (30-Year T-Bond or 30-Day T-Bill)
β = Slope of historical monthly market index real returns fit against stock’s real returns

Even though the CAPM is a simple linear regression model, the choice of y, x and b is very subjective, with sound and logical reasoning supporting numerous variations of each variable.  Outline below are the methods that should be used, and why, when calculating β via the CAPM model.

• Should one use nominal returns or real returns (i.e., the spread above the risk-free rate) for their calculation of β?  The answer should be real returns.  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 work best with both the SML theory and the CAPM model.  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).   Rearrangement of the CAPM model clearly shows that real interest rates should be used:

(RS – RRF,C) = α = β × (RM,i – RRF,i)

• 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).  Using 5 years of monthly returns is the preferred method 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 an 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.  But some would argue that the long-term nature of the stock market dictates a geometric return.  Since the CAPM is the required cost of stock, the more conservative number would be the arithmetic return since it in inherently larger, and thus provides a larger discount (hurdle) rate.
• The next question is what “risk-free” rate should be used?  Some people suggest the short-term 30 to 90-Day U.S. 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 U.S. Treasury TIPS is even better (Treasury Inflation-Protected Securities).  And still others recommend using a long-term U.S. Treasury such as the 30-Year U.S. 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 U.S. Treasury such as the 30-Year U.S. T-Bond makes the most sense since we are matching the maturity of the risk-free asset with the ongoing company.  Despite this, in the Excel model attached with this blog post, both methods have been used showing that if the MRP is calculated using the same term of treasury that is also used for the risk-free-rate, then the variance between these two risk-free rates (30-Day U.S. T-Bill and the 30-Yr U.S. T-Bond) is minimal.
• What market index should be used for your market return?  The standard used in the United States 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, then you should still at least calculate the former, because market risk and return is based on a completely diversified portfolio, and if your portfolio is not completely diversified, 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 that you hold 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 is needed?, as far back as your historical 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 and the great recession – 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 historical 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, but preferably the 30-Year U.S. T-Bond.

Below is a file that actually calculates β for Apple (AAPL), and also shows its application using 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 10.4% versus 8.4% respectively (arithmetic is the proper choice, and the more conservative choice as well).  The variance between a T-Bill and T-Bond based model was only 0.3%, so if applied properly, either risk-free rate should suffice (but the preferred is the 30-Yr T-Bond).

On a side note, the methods used in this post match Google Finances’ betas almost exactly, so Google must be using very similar methods in determining their β.  In this example for AAPL, the two differed by only 0.3%, and typically the variance is even less than this using the methods recommended in this blog post.

Access model from here