# Jensen’s Alpha, Revisited

*By Max On February 15, 2011 · Leave a Comment · In Reference*

I introduced Jensen’s alpha in a previous article. One of my readers posted a very insightful comment, which I’ll reproduce here:

What information does Jensen’s alpha add that a simple test of significance would not? For example, if factor X always significantly predicts returns in the cross-section, even accounting for factor Y, then when would X have a negative alpha? More importantly, even if X has a negative alpha, why would you omit X from your model given that it accurately predicts returns?

The answer is that Jensen’s alpha measures *strategy performance *across time, whereas cross-sectional tests only test *predictive power *in a single period. Even if X and Y are great at predicting performance in each given period, it may be that both X and Y actually depend on a common factor Z, where Z fluctuates throughout time, but is the same for each asset in a single period. Then, strategies that bet on both X and Y are actually just betting on Z twice. A cross-section test would say that X and Y are significant, whereas a Jensen’s alpha test would show that X and Y are essentially the same strategy. To illustrate this difference, consider the following example.

Suppose, after much research, you find that companies that produce many gadgets and companies that produce many widgets tend to perform well. There is little correlation between the number of gadgets and the number of widgets produced by a given company, so both of these factors are highly significant in the cross-section. You choose to bet on these trends. You find that betting on high-gadget companies tends to result in high returns when betting on high-widget companies also results in high returns, even though both are on average successful. It turns out that when the economy does well, companies that produce many gadgets and companies that produce many widgets are able to sell more gizmos, and thus have much higher returns than when the economy does poorly.

Note that gadget and widget production are still significant predictors of returns, but betting on gadgets and betting on widgets are both strategies that are essentially betting on the economy! Thus, in the portfolio construction process, it would be unwise to bet on both gadgets and widgets at the same time, since this would simply amount to a double bet on the economy.

At this point, you may wonder: why not account for the economy in the original factor model? The answer is that you *should!* However, it can be difficult, except in the simplest of cases, to determine what the underlying common factor is. More generally, if the common factor fluctuates at a lower frequency than your cross-sectional analysis, then it will be impossible to account for the factor in the cross-sectional analysis.

I hope this helps!

**Update: The following Matlab code snippet shows a very simple way to compute Jensen’s alpha for each factor.**

% ols.m function beta = ols(y, x) beta = (x' * x) \ x' * y end % r = matrix of returns % size(r) = [N, T] % where N = number of firms and T = number of periods % % f = matrix of prediction factors % size(f) = [N, M, T] % where M = number of prediction factors beta = zeros(T, M); for t = 1:T % run a cross-sectional regression each period beta(t,:) = ols( r(:,t), f(:,:,t) ); end % compute a jensen's alpha for each factor alpha = zeros(M,1); for m = 1:M tmp = ols( beta(:,m), [ones(T,1), beta(:, setdiff(1:M, m) )] ); alpha(m) = tmp(1); end

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