Three-Part Alphas
of Active Management

by Ronald N. Kahn

In the previous issue we learned that information ratios, as the key to active management, depend on both skill and breadth. In this issue we will learn the constituent parts of alphas, the fundamental inputs for active management.

Raw signals like analysts' earnings forecasts or broker buy/sell recommendations hopefully contain information useful in forecasting returns. But these raw data are not alphas: expected residual returns. They do not even necessarily have the units of return.

A basic forecasting formula governs the connection between these raw signals and alphas. This formula refines the raw signals into alphas by controlling for expectations, skill, and volatility. In many cases we can simplify this formula to a particularly intuitive form.

Converting raw signals into alphas

The basic forecasting formula provides the best linear unbiased estimate of the residual return q, given the raw signal g:

According to Equation 1, the expectation of q conditional on g equals the unconditional expectation of q, plus a term dependent on the difference between the observed signal and its unconditional expectation. Reordering terms, we see that:

So this formula controls for expectations. Only if the raw signal g differs from its unconditional expectation will the expectation of q differ from its unconditional expectation.

This result is intuitive. If company earnings exactly match expectations, we do not expect the stock to move. That happens only when earnings do not match expectations.

Now let's simplify Equation 2 into a more intuitive form that reveals how alphas include controls for skill and volatility. First, the unconditional expected residual return is zero. In the absence of information, expected returns match their consensus (CAPM) values. Second, E{q | g}-E{ q} is the alpha, the expected residual return given signal g. Third, the covariance term includes a correlation term and two standard deviations.

Substituting:

We commonly denote the correlation of raw signals and realizations as the information coefficient IC, and the standard deviation of the residual return as w. We will refer to the standardized raw signal as the score S, because it has mean zero and standard deviation 1. It ranges roughly from -2 to +2. Hence:

So there are three parts to every alpha: an information coefficient, a volatility, and a score.

Equation 4 clearly shows how alphas control for skill and volatility. The information coefficient is a measure of skill. With no skill the IC is zero, and Equation 4 sets the alpha to zero, as it should. The greater the skill, the greater the alpha, other things equal.

Volatility serves two purposes in Equation 4. First, it provides the units of return. The IC and score are dimensionless. Second, it controls the alpha for volatility. For a given skill level, imagine two stocks with equal bullish scores of +1. We think both stocks will go up. Equation 4 says that the higher volatility stock will go up more. If both Pacific Gas & Electric and Netscape achieve earnings one standard deviation above expectations, then Netscape should rise more. Both stocks will rise, but the more volatile stock will rise more.

Providing structure

Understanding the three constituent parts of an alpha can provide intuition. It can also provide structure in unstructured situations, where the connections between raw signals and alphas are unclear.

The ultimate example of an unstructured situation is the stock tip. Even in this case, Equation 4 can provide structure. Imagine that the stock in question has residual volatility of 20%. Then Table 1 shows the range of possible alphas as a function of IC and score.

Table 1. Analyzing a stock tip.

Very,     Very Very     Positive Positive IC   S = 1 S = 2 Great 0.10 2% 4% Good 0.05 1% 2% Average 0.00 0% 0%

Since stock tips are always presented as very, very positive ("I make only one or two recommendations a year; you are the first person I called . . .," etc.), converting from the tip to an alpha only requires estimating the tipper's IC. Is Warren Buffet on the line, or someone you have never heard of?

For an institutional money manager, a more relevant example involves converting broker buy/sell recommendations into alphas. This common situation has relatively little structure, but understanding three-part alphas can help.

Table 2 shows an example, assuming that the broker has a good information coefficient of 0.05.

Table 2. Converting BUY/SELL signals to alphas

Stock w Rec. Score µ A 15% BUY +1 0.75% B 20% BUY +1 1.00% C 15% SELL -1 -0.75% D 30% SELL -1 -1.50% E 25% SELL -1 -1.25%

Our conversion from recommendations into scores is straightforward. Notice that Stocks A and B, both recommended, have different alphas. Stock B has the higher volatility: We expect it to go up more than Stock A.

Summary

In summary, to convert from raw signals to alphas requires controlling for expectations, skill, and volatility. Understanding this conversion provides insight and structure in many situations.

Footnotes

¹ If we ignore constraints and active return correlations, optimal active holdings are proportional to the ratio of alpha to active variance.