Spanning Tests for Adding a Strategy

A new strategy shows a Sharpe ratio of 1.0 on its own — respectable. But you already run five other strategies, and it turns out this "new" one behaves almost exactly like a 60/40 blend of two you already hold. Adding it wouldn't actually improve your portfolio at all; it would just be paying to duplicate exposure you already have. A spanning test is the formal way to check this: does the new strategy add anything a combination of your existing strategies couldn't already deliver?

The regression that answers the question

Regress the new strategy's returns, $r_{new}$, on the returns of your existing strategies, $r_1, \ldots, r_k$:

$$r_{new,t} = \alpha + \beta_1 r_{1,t} + \cdots + \beta_k r_{k,t} + \varepsilon_t .$$

In plain English: this asks how well some weighted combination of the strategies you already run can replicate the new one. The $\beta$'s are the best-fit weights of that combination, and $\alpha$ is what's left over — the part of the new strategy's average return that the existing lineup cannot explain at all. The spanning test is simply a test of whether $\alpha$ is significantly different from zero. If $\alpha \approx 0$ and is not statistically significant, the new strategy is "spanned" by the existing set: everything it does, a combination of what you already hold could already do, and adding it brings no genuine diversification or return benefit, however good its standalone Sharpe ratio looks. If $\alpha$ is significantly positive, the new strategy is bringing something the existing lineup genuinely lacks.

Worked example

Existing portfolio: strategies 1 and 2. New strategy regressed on both gives $\hat\beta_1 = 0.6$, $\hat\beta_2 = 0.5$, and an intercept $\hat\alpha = 0.3\%$ per month with a standard error of $0.35\%$ — a t-statistic around 0.86, not significant. Despite the new strategy's own standalone monthly Sharpe looking attractive, this result says a 0.6/0.5 mix of strategies 1 and 2 already replicates it almost completely; the un-explained residual return is indistinguishable from noise. Adding this strategy to the portfolio wouldn't be adding real diversification — it would mostly be re-scaling exposure you already have, at the cost of extra complexity, capacity, and fees if it's an external manager.

The regression itself is an ordinary least-squares fit — drag the points below to get a feel for how a fitted line's residuals (the vertical gaps) are exactly the kind of "leftover" that a spanning test's $\alpha$ captures, just in a one-variable illustration rather than the multi-strategy case above.

What this means in practice

Spanning tests matter most before adding a manager, factor, or systematic strategy that "looks good" on its own numbers — a standalone Sharpe ratio says nothing about marginal value to a portfolio that already holds other things, because correlation with existing holdings can make an individually attractive strategy nearly redundant. They're also the right tool for deciding whether a "new" factor discovered in research is genuinely new information or just a repackaging of well-known factors already in a standard model.