Alpha, Beta and the Linear Regression - Investments 101
Team Latte
August 19, 2007
It seems that in the last couple of weeks everyone - from the quants and fund managers to regulators and journalists - is talking about alpha and beta, two of the key math constructs in portfolio engineering. All talk and discussions are centered on the alpha strategies of the fund managers (especially, hedge fund managers) and how hedge funds, including the funds run by Goldman Sachs, trying to exploit alpha have recently fallen prey to severe market turbulence. A lot of old school quantitative fund managers are saying that beta - the coefficient of the capital market line and the measure of the systematic risk in CAPM - is after all the holy grail of investment and there is nothing more to investment analysis than beta.
What are alpha and beta? Alpha is the excess return over CAPM and beta is the systematic risk measure in CAPM.
Let's forget about investment theory for a moment, go back to Math 101 and see how we look at these concepts in a quintessential way. Say 
is a dependent variable whose value depends linearly on the independent variable
. We can write the linear relationship between these two variables as:
The above equation says that if we plot the values of and in a scatter plot, it will be a straight line. The slope of that line will be beta and the intercept will be alpha. Here alpha and beta are both purely mathematical concepts. The coefficient of determination, 
will tell us how good the fit or the linear relationship is.
The slope of the line, beta, is the first mathematical derivative of with respect to . That is, 
. Therefore, beta is a sensitivity measure. It tells us that by how much will change is there is a very small change in the value of . And since it is a sensitivity measure, in investment parlance this becomes a risk measure.
Alpha, on the other hand, is simply an intercept of the line on the axis. Note that if we make the value of alpha as zero the equation is still that of a straight line. If 
then 
and this is still an equation of a straight line, the only difference being that this straight line passes through the origin. The intercept can therefore be looked upon as the value of when 
.
Now going back to the theory of investments if we say that the return on a stock is linearly dependent on the return on a certain market index (a stock market index such as S&P500 or the Nikkei225) then we can write the return equation in the same math form as above:
In the above equation beta is once again the slope of the line - the 
scatter plot of the return of the stock and the return of the market index - and it represents the "systematic risk" of the stock. By "systematic risk" we mean what amount of the risk of the stock is explained by the market (index). This is the same as a sensitivity measure as we say above. If the market moves up by one unit and the stock also moves up by one unit then beta of the stock will be one. If the market moves up by one unit and the stock moves down by one unit then the beta of the stock will be -1 (minus one).
Actually, beta can be calculated using the volatility (standard deviation) and and (the return of the market and that of the stock respectively) and the correlation between them.
Alpha, the intercept on the stock return axis, 
, is not a sensitivity measure but it is a return measure. It is a risk adjusted excess return on the stock. If the market (index) return is zero then the stock will still have a residual value and that value is given by alpha.
If 
then the stock return will still be given by 
. CAPM tells us that alpha is zero which is the other way of saying that any excess risk that is not linked with the risk of the market (index) can be diversified away by a fund manager. Even if alpha exists - a positive alpha for portfolios above the SML and negative alpha for portfolios below the SML - the efficiency of the market and the actions of rational investors will make it zero before anyone can make profit - excess return - from exploiting it.
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