Fishing for Alpha and catching Beta

We examine the implications of rewarding portfolio managers on the basis of Alpha, and how this can lead to an increase in the Beta of the portfolio

A common measure of portfolio performance is alpha. The idea comes from the fundamental equation of the Capital Asset Pricing Model (CAPM). The CAPM is a general equilibrium model that (in theory) allows the pricing of all assets in the economy as a function of only two assets, the risk-free asset (think Treasury bonds) and the market portfolio (think S&P500).

The fundamental equation of the CAPM states that the expected returns of portfolio i at time t should be a linear combination of the expected returns of the risk-free rate and the expected excess return of the market portfolio over the risk-free rate.

In this model, there is only one source of systematic risk: the market. And, the holder of a portfolio is compensated with a premium over the risk-free rate that is proportional to the amount of market risk that the portfolio carries. The amount of risk is captured by beta.

In other words, a portfolio should generate a return that is equal to the cost of time (= risk free rate) plus a premium that is equal to the product between the price of risk (= the excess return of the market portfolio over the risk-free rate) and the amount of risk that the portfolio holds (= beta).

The CAPM is commonly employed to assess the performance of a portfolio, using the following procedure. We rewrite the CAPM equation in the form of a regression, which we estimate over some historical return (for example, over the daily return generated in the previous three years).

From the regression, we obtain an estimate of alpha and beta. The beta coefficient tells us how much systematic risk the portfolio carries (for example beta = 1 would imply that the portfolio carries exactly 100% of market risk).

The alpha of the regression tells us whether portfolio i has generated returns that are in excess or below what the CAPM predicts, given the beta of portfolio i. If alpha is positive, then portfolio i has generated a return that is over and above the expected premium captured by beta. Vice versa, if alpha is negative, the portfolio has underperformed with respect to what the CAPM would predict, given the portfolio's beta.

The manager of portfolio i will claim success if alpha is positive, on the basis that the portfolio has generated a premium after explicitly accounting for the amount of systematic risk that the portfolio carries.

Well....that is one way of interpreting this...

What if the CAPM did not capture exposure to systematic risk correctly?

Another way to look at the above result is that the CAPM does not fully capture the exposure of the portfolio to systematic risk, because the market is not a good enough proxy for systematic risk.

Suppose that the Value Factor (= returns of a portfolio that is long on firms with high book-to-market and short on low book-to-market) and Size Factor (= returns of a portfolio that is long on small firms and short on large ones) also captured the exposure of portfolio i to systematic risk.

Then, the following model should be used to assess the performance of the portfolio, instead of the CAPM.

What happens if one uses the CAPM equation as a way to compute alpha, instead of the above equation?

Because the "correct" model is the one that includes the Value and Size factors, the CAPM is incorrect (or to be precise misspecified because of omitted variables). This means that potentially the CAPM alpha is not capturing true over-performance, but rather some exposure to systematic risk (via exposure to Value and Size) that is not correctly accounted for by the CAPM model (because it does not include Value and Size).

the CAPM alpha is not capturing true over-performance, but rather some exposure to systematic risk

In other words, the originally estimated CAPM alpha is not capturing over-performance but rather incorrectly measured exposure to systematic risk (i.e. the beta of Value and Size).

What came in as alpha, goes out as beta...

Notice that this argument works as long as there are factors that capture the exposure to systematic risk. That is to say that as long as one can come up with factors (John Cochrane's Zoo of Factors) that capture systematic risk, and these factors are not included in the model, the interpretation of alpha is potentially flawed.

The implications of the above reasoning are important: a portfolio manager that is outperforming the market in terms of alpha (on the basis of a misspecified model) might be just loading on the beta of the factors that are not included in the model, thus taking more systematic risk than what may appear.

a portfolio manager that is outperforming the market in terms of alpha might be just loading on the beta of other factors

Does alpha exist?

Yes, it does, if you assume that markets are not in equilibrium.

If you assume that markets are in equilibrium (meaning "efficient") than prices reflect fair values of securities. In this case, true alpha does not exist. By "true", we mean the alpha computed on the basis of a correctly specified model, i.e. a model that perfectly captures the exposure of a portfolio to all sources of systematic risk.

The combination of fair prices + correct model of systematic risk leaves no room for alpha. The only way to generate returns is by loading the portfolio with systematic risk. This is the spirit of Factor Investing.

But what if markets were not in equilibrium? If some securities in the market are not fairly priced, the market is not in equilibrium. In this case, alpha can be created, even if the correct model of systematic risk is employed.

Informed investors can generate alpha by translating their information into trades, buying underpriced securities and selling overpriced securities. This will lead prices towards their fair value. Through this process, the market converges towards its equilibrium. Informed investors can make money through these trades, without increasing the exposure of the portfolio to systematic risk. This is true alpha. This is Value Investing (meaning Warren Buffet style of investing, not to be confused with investing in the Value Factor, as sometimes is done, see here).

Let's revisit our initial argument that the CAPM alpha is not a correct measure of the true alpha. Does it still hold if markets were out of equilibrium? Yes it does. Even with out-of-equilibrium markets (i.e. a context in which true alpha could potentially arise), it is impossible to assess whether the CAPM alpha is capturing exposure to systematic risk (via omitted factors) or true alpha.