Security Characteristic Line (SCL)
The entire idea behind the __Capital Asset Pricing Model (CAPM)__ is that it is a __single-index model__, which is the simplest of the larger class of __return-generating models__, the set of models used to project expected returns:
$$\displaystyle E[R_i] - R_f = \beta_i(E[R_M] - R_f) $$
Once the CAPM has been built with __beta__ in mind, and represented by the __security market line (SML)__, assets (securities or portfolios) plotted in the same space as the SML can be shown to have some vertical distance between its systematic risk and return profile, and what is predicted. This distance is __Jensen's alpha__.
But perhaps you would like to look at things just a little differently. Instead of looking at an individual asset's location in risk-return space as a single dot, derive that placement in space from a regression of returns against the market. Specifically, return premia, or returns above the risk-free rate, as the equation above shows.
Using the single-index model of the CAPM as shown, what is the relationship between the market's return premium and an individual asset's return premium?
Right! Beta is the value of the closest linear relationship between the two.
This is obtained by using the beta calculation of the covariance of the asset and the market, and dividing by the variance of the market, or by direct estimation using linear regression. The __security characteristic line (SCL)__ is kind of like a picture of that regression:
No, this does not connect the two premia.
No, these values are never subtracted. One is an additive value, and the other is a coefficient.
Now here's the fun thing. If you can comfortably picture the SML and interpret it, then a lot of this falls into place. Imagine that this asset represented was on the SML. A linear relationship between the market premium and the asset premium is shown here, and perhaps this relationship, measured by beta, is 0.8. So it's on the SML to the left of the market portfolio (which always has a beta of 1.0).
How do you suppose this linear relationship can be found on the SCL?
No, the vertical axis is the asset return premium, so the average of this would just be the average asset return premium, disregarding its relationship to the market return premium.
No, the horizontal axis is the market return premium, so the average of this would just be the average market return premium, disregarding its relationship to the asset return premium.
Correct!
Once again, that linear relationship between the market return premium and the asset return premium is shown in the single-index model equation, it is shown in the SML as beta, and it is shown in the SCL as the slope between these exact two variables.
Now imagine that the asset was right on the SML. It had whatever beta you would like to picture.
To the left of the market portfolio on the SML, lower beta, flatter SCL.
To the right of the market portfolio on the SML, higher beta, steeper SCL.
But if it was right on the SML, this linear relationship fits. It equates the two. There's no error. Jensen's alpha would be zero. And on the SCL, this perfect relationship would be shown as the SCL running right through the origin.
But the SCL shown doesn't do that. The security's returns are "too good" for this perfect relationship. What might you assume about Jensen's alpha in this case?
To summarize this discussion:
[[summary]]
No, this would be the case if the SCL passed through the origin.
No, this would be the case if the returns were "too bad" and left an SCL running below the origin.
That's right! And Jensen's alpha is right there in the graph of the SCL as the vertical distance between the origin and the SCL. In other words, it's the vertical axis intercept in the SCL graph.
But this is all a way of getting back to exactly where this started. Recall that the estimation of beta started with a regression of asset premia on market premia. The slope is how beta was estimated to begin with, and there was an alpha that was always Jensen's alpha. What the SCL does is put a picture to that statistical regression to connect all of the pieces.
Beta
Alpha
Beta minus alpha
It must be the average vertical distance of the plots
It must be the average horizontal distance of the plots
It must be the slope of the best fit line relating the market premium to the asset premium
It is negative
It is zero
It is positive
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