what are the differences between CAPM and Single index model? (specially their alphas)

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hirastikanah

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hi
what is the difference between alpha in single index model, and the one in capm?
which one is jensen’s alpha?!
i am so confused with these two models!!
any answer would be appreciated
thanks in advance
 
CAPM is a “general equilibrium model” that is based on microeconomic ideas such as concave utilities, costless diversification etc. It makes an exact prediction about expected return, i.e.:
E(Ri) = Rf + beta-of-i * (Rm - Rf)
or equivalently:
Ri = Rf + beta-of-i * (Rm - Rf) + ei where ei is an error term and E(ei) = 0
CAPM may or may not be true, it depends upon the validity of its assumptions (which almost certainly are not true).
A single index model is simply a form of a correlation equation between 2 variables, which are (Ri - Rf) and (Rm - Rf). It must always be true. And of course it tells you a lot less than CAPM, specifically it does not say anything about the magnitude of the expected return (which of course is of great interest to finance professionals).
Ri - Rf = alpha-of-i + beta-of-i * (Rm - Rf) + ei
with E(ei) = 0
Note the above must be true for any two random variables, all it says is that the return to a random variable is the correlation (beta-of-i) of it with another variable multiplied by the return to the other variable plus a trend (alpha-of-i) plus an unbiased error. If the error was not unbiased, simply remove the bias and add it to alpha-of-i.
Adding all the economic theory (Markovitz’s diversification, Von Neumann and Morgenstern expected utilities etc.) leads one to CAPM which says alpha-of-i = 0.
CAPM was the first general equilibrium model that gave a theory of discount rates (and hence prices). It also made a very strong prediction: Discount rates depend only upon the asset’s correlation with the market, and all other characteristics such as size, value etc. are irrelevant. Empirical tests from the 1970s onwards have refuted this prediction, but its popularity (possibly due to its simplicity) remains.
Best,
Jayanta Sen
 
thanks a lot
it helped me so much
so the alpha in single index model is not jensen’s alpha, right?
the one that may appear in CAPM (when we have under/over priced securities) is the jensen alpha, right?
 
Jensen’s alpha is the difference between the actual (real world) expected return of an asset, and the expected return of the asset according to a theoretical model. The theoretical model most commonly used is the CAPM.
To empirically calculate the actual expected return, you cannot simply take one observation, you need a large number of observations.
Best,
Jayanta Sen
 
SenFinance wrote:… the correlation (beta-of-i) of it with another variable … .
Click to expand...
Just to be clear, βi is not a correlation coefficient. It is the product of the correlation of the security’s returns and the market’s returns and the relative volatilities of the security’s returns and the market’s returns:
βi = ρ(i,mkt) × (σi/σmkt).
SenFinance wrote:[CAPM] also made a very strong prediction: Discount rates depend only upon the asset’s correlation with the market … .
Click to expand...
Again, this is incorrect. The prediction is that discount rates depend on the correlation of the asset’s returns with the market’s returns, and their relative volatilities of returns.
 
Indeed… I was careful not to use the word “correlation coefficient” because beta-of-i is not that, etc.
Jayanta Sen
 
I can of course simply give you the technical terms, such as:
βi = cov(i, mkt) / var(mkt) etc.
But that doesn’t give the intuition about correlation, correlation coefficient, beta etc.
I will try to give you the intuition.
Think of an asset that has an average excess-return of, say 4 * mkt-excess-return.
Would you say this asset is highly correlated with the market? Intuitively you would think, yes. Beta would be 4, so that would indeed be a high sensitivity to the return of the market.
The “correlation coefficient” of this asset with the market however can be anywhere from very small to very large. If the asset has a lot of idisyncratic risk, then the correlation coefficient could be, for example, as low as 0.01.
So for a beta of 4, the correlation coefficient could be 0.01. It could also be 0.99 if the idisyncratic risk was very small.
One foundation of CAPM is that the idiosyncratic part does not matter at all, as it can be diversified away costlessly.
So when thinking about CAPM, it is best not to think of the “correlation coefficient”, which forces the volatility of the asset also to be considered. Remember in CAPM, beta rather than the asset’s volatility determines risk. This is the major development of CAPM over prior microeconomic thinking in which it was understood that individual utilities are concave, so individuals dislike risk.
Instead just think “if the market has an excess return of X%, then at an average the asset has an excess return of βi * X%”.
Now the contribution of asset i to the variance of the investor’s portfolio is determined by βi rather than the volatility of asset i (due to costless diversification of the idiosyncratic part of volatility, and assuming the investor has optimized by diversifying). Hence the market only cares about βi and ignores the asset’s volatility, which is CAPM.
You may think of βi as a measure of the asset’s “correlation” with the market, “sensitivity” with the market, whatever, but it is not the “correlation coefficient”, a phrase I did not use in my initial post. I hope you have an intuition about βi and CAPM from the above discussion.
Another assumption necessary to get the linear form of the equation CAPM is that utlities are quadratic (which does not produce moments higher than order 2) or that return distributions are normal (which can be completely characterized by the first and second order moments and does not require higher moments). Add assumptions such as zero transaction costs, perfect information etc. and you have the CAPM.
Best,
Jayanta Sen
 
I got what u meant; but thanks for describing it
And thanks a lot for the time u spend for replying
:-)
 
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