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Thursday, May 29, 2008

Alpha and Beta Explained

I liked this post on Bill Rempel. There is a concept in investing called "alpha". If you say "My trading system has an alpha of 1% relative to the S&P 500", that means that, in the long run, you will outperform the S&P 500 by 1%. Alpha is ALWAYS the rate of one investment compared to another investment.

What should the benchmark be? The S&P 500 is most commonly used, but the S&P 500 has its own problems.

No investment site has ever suggested using M2 or M3 as the benchmark! If you say "My investment strategy returns 5% more than the growth rate of M3", that's impressive indeed!

After all, when the money supply increases, your investment system should AT A MINIMUM match the rate of money supply expansion. When you consider taxes and fees, even M2 (around 7%) is a high hurdle to overcome. Some sources estimate the rate of growth in M3 to be over 15%. Very few trading systems can validly claim to return over 15%.

How Alpha and Beta are Calculated

When you talk about alpha and beta, you always are talking about the return of one trading system relative to another. If you say "This stock has a beta of 2.0", that's like saying "Two is greater than". The standard benchmark is the S&P 500. A correct statement is "This stock has a beta of 2.0 relative to the S&P 500". It could be "relative to the Dow" or "relative to the price of gold" or "relative to investing in 10 year Treasury Bonds".

Alpha and beta have a precise mathematical meaning. Suppose you are trying to calculate the alpha and beta of trading system A relative to trading system B. Let {P_i} be the prices observed of system A, and {Q_i} be the prices observed of system B. You can use daily price points or whatever interval you choose.

Let {X_i} by {log (Q_i+1)/(Q_i)}. Let {Y_i} be {log (P_i+1)/(P_i)}. You now plot {X_i}, {Y_i} and do a least-squares fit, finding the line of best fit.

Alpha is the y-intercept of this line. Beta is the slope of this line.

You can calculate alpha and beta for ANY pair of trading systems.

Alpha is your expected gain relative to another investment, and beta is your risk.

For example, if your alpha is 1%, then the expected return of system A is 1% higher than system B.

If your beta is 2.0, then whenever system B gains 5%, system A gains 10%. Similarly, whenever system B loses 5%, system A loses 10%.

If you want to know beta for a stock, you should calculate it yourself. Many other sources don't do the mathematically correct calculation. They don't reveal enough details for you to properly doublecheck their calculation.

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