Ever wonder if generating alpha is a zero-sum game? Or do you think quotes like these below hold:

“Active management can generate alpha for investors, and passive investing cannot.”

“In a market with low returns, active management is better, as alpha becomes more important.”

In this post we will establish how much alpha is available in the market, and why statements like the above are simply ridiculous.

To begin, let’s define alpha. *Investopedia* defines it as follows:*

- A measure of performance on a risk-adjusted basis.
- The abnormal rate of return on a security or portfolio in excess of what would be predicted by an equilibrium model like the capital asset pricing model (CAPM). *http://www.investopedia.com/terms/a/alpha.asp#ixzz4Y8m3usBD

For the calculation, we will go ahead and use the ever-popular Capital Asset Pricing Model (CAPM). The formula for alpha (aka Jensen’s Alpha) is below:

To simplify, our market will consist of only three securities (X, Y and Z) and two investors (Jordan and Michael). Each security will have a beginning market capitalization of $100. It doesn’t matter how you think of it, as these could be individual securities like stocks or whole asset classes (stocks, bonds, treasury bills). Each investor will begin with $150 and allocate his/her portfolio based on perceived preferences (valuations). At the end of the year, we will calculate how each investor performed and how much alpha was generated. We will use Excel’s pre-build formulas for all the calculations.

The market portfolio is evenly split by the three securities in the beginning of the measurement period. The far right column shows the return of each security after one year:

Note that every security has to be owned by someone. Our investors have decided on the following portfolio allocations:

Below is a table of the monthly variance of security prices. Technically, in order for the prices to fluctuate, trading will have to occur. We are simplifying this example, but it wouldn’t have an impact on the final calculation. Note the risk statistics–as expected, the market beta is 1.00.

For the finale, let’s calculate the alpha of each portfolio, assuming a risk-free rate of 0.0%.

**Zero-Sum Game**

How much alpha (risk adjusted performance) is there in the overall market? The answer is zero. Notice that Jordan’s negative alpha exactly offsets Michaels’s positive alpha. This fact will hold true even if one expands to thousands of investors and securities. It is what is called “a priori” knowledge– true by definition (observe how alpha is calculated through all the formulas and note it has to sum to zero). It’s important to realize that all risk measures such as beta, alpha, correlation, etc., are relative measures. What follows is that for any security with a beta higher than 1.00, there is a security with a beta smaller than 1.00, and by the same amount. All the securities aggregated together are the market, and the market beta has to be “1.00.” More importantly, generating alpha from the market is a zero-sum game, i.e., for one investor to create alpha, another has to lose it. On the other hand, investing as a whole is not a zero-sum game, as both investors can end up with positive returns. So, active management, overall, cannot generate alpha, but good active managers could extract alpha at the expense of the poorer managers.

*“Alpha is a relative measure; for one to get alpha in the financial markets, someone has to lose it.”*

Concerning the latter part of the quote, alpha actually becomes less important if market returns are expected to be low. If the returns are lower, active managers’ expenses will take a bigger chunk of the return. For example, if US Large Caps are expected to return 2%, but your active manager charges you a 1% fee, then your return will most likely be 1%. Hence, the fee swallows 50% of your expected return. If US Large Caps are expected to generate a return of 20%, and your active manager charges you a 1% fee, then your return will most likely be 19%. In this case, the fee consumes only 5% of your expected return.

*“The risks of using an active manager are lower in higher expected-return markets than in lower-expected return markets.”*

In conclusion, if you want to use an active manager, make sure he/she is able to extract alpha from other active managers. Also, remember that as expected returns decline, fees play a more important role in your portfolio performance.