# CAPM - Expected vs. actual returns

I'm trying to calculate alpha in excess of CAPM and have seen a few slightly different calculations for CAPM.

The primary difference I am seeing is that some equations use expected market returns (e.g. CAPM), while others use actual market returns (e.g. Jensen's Alpha).

Which one is correct to use in CAPM? If expected market returns are the correct way to go, how do you estimate this amount?

Based on your comments on other answers, i would like to provide you a summary on the difference of the CAPM-Alpha and Jensen's-Alpha.

### CAPM

The CAPM is an economic model for asset pricing. It states that the equation

$$E[r_i - r_f] = \beta_i E[r_m- r_f]$$

holds for any asset $$i$$. $$r_i$$ denotes the return of asset $$i$$, $$r_f$$ the risk-free rate of interest, $$r_m$$ the market-return and $$\beta_i$$ the beta-factor of asset $$i$$.

I often hear that the CAPM is just a regression, which is not true in fact (see e.g. this excellent answer here). However, we do run the following regression, when we are empirically testing if the CAPM holds:

$$r_{i,t} − r_{t,f}= \alpha_i + \beta_i (r_{t,m} − r_{t,f}) + \epsilon_{i,t}$$

There are several empirical implications for the CAPM, like excess returns are linear in beta, so coefficients on adding a squared beta-term in the above regression should yield in insignificant coefficients. The main implication however is, that $$\alpha_i$$ should be indistinguishable from zero for any asset $$i$$. We test this for multiple assets with an F-test (often called GRS test in finance) or a $$\chi^2$$-test (see this answer here for further information on the test-statistics).

Empirical evidence shows, that the CAPM is a failure. It just does not work, respectively fails to describe asset returns.

### Jensens's alpha

Jensen's alpha is not an economic model, but rather a method to measure portfolio performance. It was first used as a measure in the evaluation of mutual fund managers. How would one measure, if a fund manager has skill or not? Well, let's take a look on the difference of the actual return of a given portfolio and its expected return:

$$\alpha_{i,t} = r_{i,t} - \operatorname{E}[r_{i,t}]$$

That's in fact Jensen's alpha. If it is positive, the fund/portfolio "beats" the expected return and we would assume, that the fund manager has some skill (if the $$\alpha_{i,t}$$ is significantly different from zero over a longer period of time).

### Jensen's alpha and the CAPM together

Well, to calculate Jensen's alpha, one has to figure out the expected return $$\operatorname{E}[r_i]$$ of a portfolio. We may apply an economic model like the CAPM (or alternatively the market-model, the Fama/French Five Factor Model, etc.) to estimate this expected return. It is up to you to decide, what an appropriate model is. If e.g. a managers stock universe is restricted to S&P500 listed stocks, you could also use the simple S&P500-return as a benchmark for the fund performance (i.e. using $$r_t^{S\&P500}$$ instead of $$\operatorname{E}[r_i]$$). Often, the CAPM is used to calculate $$\operatorname{E}[r_i]$$ when applying Jensen's portfolio measurement index, but now you may see, that this has nothing to do with the $$\alpha_i$$ when testing the CAPM.

### References

Elton/Gruber/Brown/Götzmann (2014), Modern Portfolio Theory and Investment Analysis, ed. 9, John Wiley & Sons.

Fama, E. and MacBeth, J. (1973), Risk, return, and equilibrium: Empirical tests., The Journal of Political Economy, 81(3), pp. 607-636.

Jensen, M.C. (1968), The Performance of Mutual Funds in the Period 1945-1964, Journal of Finance (23), pp. 389-416.

• Thanks very much for this summary! I would like to confirm that for the portion of the right-hand side of the CAPM equation [E(Rm - Rf)] means: the actual return of the benchmark, minus the risk-free return at time t? – steicher Aug 23 '19 at 11:13
• The CAPM is an economic model. The right-hand side just expresses what we call the expected market excess return. That's all; as a theoretical model, it does not care about actual implementation and usage in the real world. What you state is just that: Yes, we commonly use the actual market return minus the risk-free rate because that's what comes closest to the model requirements. – skoestlmeier Aug 23 '19 at 11:49
• Thanks for the clarification. Do you have any insight into the question here: quant.stackexchange.com/questions/47276/… I'm trying to understand the appropriate frequency to use for the Beta calculation (not done via regression) – steicher Aug 23 '19 at 12:00
• Well, what are you trying to do, i.e. what is your research question? I tried to make it clear, that these two measurements are used in different context. CAPM as an economic model, assuming that $\alpha_i$ is zero for any tested asset/portfolio. Jensen's measurement is just an ex-post portfolio measurement index, i.e. what is the excess-return with regards to a certain benchmark. – skoestlmeier Aug 26 '19 at 7:29
• With all respect, your research question as stated is nonsense, as long as it is not clear if you are testing the validity of the CAPM or analyzing e.g. the skill of actively managed funds. For short, you may apply the CAPM to estimate the expected return, which is used to calculate the abnormal return, i.e. Jensen's alpha. But as pointed out several times, both CAPM-alpha and Jensens' are two different stages (but you may use the first for the latter approach). – skoestlmeier Aug 26 '19 at 13:24

You compute both Alpha and Beta in a single step. You do a linear regression using past data for $$R_F,R_M$$ and $$R_S$$. The slope coefficient in the regression is Beta, and the intercept is Alpha. No further calculations needed.

(If you take your Alpha, your Beta, your average returns $$\bar{R}_F,\bar{R}_M,\bar{R}_S$$ you will be able to verify that the following equation holds:

$$\alpha=(\bar{R}_S-\bar{R}_F)-\beta(\bar{R}_M-\bar{R}_F)$$

but all this does is verify that the regression worked correctly. It is not really a useful calculation, you already had Alpha in the previous step.)

• Maybe what I am confused about is the Alpha intercept vs. Jensen's Alpha. Does the intercept from the regression represent the same concept of Alpha (excess market returns)? – steicher Aug 23 '19 at 0:48

You should calculate the average return of the market for a certain period that corresponds to your time frame, lets say if your benchmark is S&P500 you calculate the average returns on that period that's your market expected returns

• So then the alpha calculation is the difference between actual returns and the expected return of the stock (calculated with CAPM using the avg returns during the period)? – steicher Aug 22 '19 at 12:27

Maybe I'm misunderstanding the question - but the beta in the CAPM is calculated using historical returns (it's the slope of the regression line between the asset returns and market returns). That beta can then be used to calculate expected future return for an asset.

Alpha, though, is the actual return in excess of this expected return. So for future returns, alpha is always zero.

• Meaning - I would multiply the beta by the benchmark return (minus the risk free rate) and the difference between this value and the actual asset returns for some time period equates to alpha? – steicher Aug 23 '19 at 0:52
• correct. R= Rf + B(Rm-Rf) + a – D Stanley Aug 23 '19 at 1:03
• In this instance, Rm is the actual return of the benchmark over the time period, correct? Not the average? And that equation represents Jensen's Alpha? Trying to learn the ins and outs of Alpha here – steicher Aug 23 '19 at 1:05
• Correct........... . – D Stanley Aug 23 '19 at 1:06
• Jensen's alpha is a generic term to indicate the return over (or below) the return expected by some model. When that model is CAPM it's also called "CAPM alpha". – D Stanley Aug 23 '19 at 1:11

The beta for CAPM is usually calculated using historical stock market returns. The beta calculated is the CAPM equation to determine the future expected return on the asset. See this CAPM calculator.

However, the question is whether this is the right approach theoretically. The future does not necessarily follow from the past and the historical beta is not necessarily reflective of the future beta.

The beta theoretically is really supposed to give you a sense of how much this asset will move relative to the market in the future, or at least the future period that you are interested in investing in.

In my view, you should use historical data to calculate the beta, but then adjust it to reflect your views on the future when investing.

"CAPM is just a regression, which is not true in fact" from skoestlmeier is correct. However, we are not able to test CAPM

1. CAPM is neither a cross sectional model, nor a time series model
2. Time series return is not the return in CAPM. CAPM is a one period model (two time points), it can not be rolled on
3. In CAPM, Beta is not a systemic risk or a characteristic of any security. The beta value is calculated from the equilibrium return, using beta value to explain the expected return is a circular argument. For more on CAPM, see CAPM: Absolute Pricing, or Relative Pricing? or Arbitrage Opportunity, Impossible Frontier, and Logical Circularity in CAPM Equilibrium

The Jensens's alpha is inspired by CAPM, but is wrongly inspired.