Distance and Similarity Measures for Assets

You want to group similar assets together — for diversification, for pairs trading, for building a hierarchical risk model — and the first step is deciding what "similar" even means, mathematically. It's tempting to reach for correlation directly, but correlation isn't itself a distance: two perfectly correlated assets (correlation 1) should have distance zero, but two perfectly anti-correlated assets (correlation −1) are also, in a real sense, maximally informative about each other — they're not "randomly unrelated," they're opposite twins. A naive use of $1 - \rho$ as a distance would treat anti-correlated assets as maximally far apart (distance 2), lumping them in with assets that share no relationship at all, which loses real information a clustering or grouping algorithm could use.

An analogy: sorting people by how they answer, not just whether they agree

Imagine grouping people by their answers to yes/no survey questions, intending "similar" people to answer the same way often. If two people answer oppositely to every single question — one always says yes when the other says no — they're not "unrelated" to each other; they're perfectly predictable from one another, just in reverse. Grouping them with truly random strangers (who agree with each about half the time, no pattern at all) would throw away a strong, structured relationship. A well-designed distance measure should place perfect agreers close together, perfect disagreers also meaningfully close together (in a mirrored sense), and only truly unpredictable pairs far apart.

The mechanics, one symbol at a time

Let $\rho_{ij}$ be the correlation between assets $i$ and $j$. The commonly used correlation-based distance (from Mantegna's work on financial market structure) is

$$d_{ij} = \sqrt{2(1-\rho_{ij})},$$

which ranges from $d_{ij}=0$ (when $\rho_{ij}=1$, perfectly correlated — zero distance) to $d_{ij}=2$ (when $\rho_{ij}=-1$, perfectly anti-correlated — maximum distance) with $d_{ij}=\sqrt2$ at $\rho_{ij}=0$ (uncorrelated). In plain English: this metric treats perfect anti-correlation as the most dissimilar possible relationship, the same conceptual mistake as the survey example — it doesn't recognize that perfectly opposite behavior is still a strong, structured relationship, not "no relationship." An alternative, when you specifically want to group assets that move together regardless of direction (useful for hedging or long/short pair construction), uses $d_{ij} = \sqrt{2(1-|\rho_{ij}|)}$, which correctly places both perfectly correlated and perfectly anti-correlated pairs at distance zero.

Two other common choices matter for different reasons: Euclidean distance on raw return vectors, $d_{ij} = \|r_i - r_j\|$, is sensitive to each asset's overall volatility level, not just how the two co-move — a low-vol and a high-vol asset can look "far apart" in Euclidean terms even if they're highly correlated, because the raw magnitude of their moves differs. Rank-based (Spearman) correlation distance replaces returns with their rank order before computing correlation, which is far more robust to outliers and nonlinear (but monotonic) relationships than ordinary Pearson correlation, at some cost in sensitivity to genuinely linear relationships.

Worked example 1: the anti-correlation trap

Take three assets: A and B have correlation $\rho_{AB} = 0.95$ (near-identical movers, perhaps two share classes of the same company). C and A have $\rho_{AC} = -0.95$ (C is a near-perfect inverse — say, a short ETF on A's sector). Using the signed distance $d=\sqrt{2(1-\rho)}$: $d_{AB} = \sqrt{2(0.05)} \approx 0.32$ (very close), but $d_{AC} = \sqrt{2(1.95)} \approx 1.97$ (nearly maximally far) — C is clustered with unrelated random assets, even though knowing A tells you almost exactly what C will do (just flipped in sign). A clustering algorithm using this metric would completely miss that C is one of the most informative, structurally related assets to A in the whole universe, simply of opposite sign.

Worked example 2: fixing it with absolute correlation distance

Same three assets, now using $d_{ij} = \sqrt{2(1-|\rho_{ij}|)}$: $d_{AB} = \sqrt{2(1-0.95)} \approx 0.32$ (unchanged, still close) and $d_{AC} = \sqrt{2(1-0.95)} \approx 0.32$ as well — now A and C are recognized as equally "close" in the sense of predictive relationship, just with opposite sign. A hierarchical clustering run on this corrected distance would group A, B, and C together as a tightly linked cluster (same underlying driver, different sign exposures), which is the economically meaningful grouping for hedging or pairs-trading purposes, whereas the signed distance in worked example 1 would have split them apart.

What this means in practice

The right distance metric depends on what "similar" should mean for the task: use signed correlation distance when direction of movement genuinely matters (you want long-only diversifiers, not inverse plays lumped in); use absolute correlation distance when you care about strength of relationship regardless of sign (hedge candidates, pairs-trading universes, or building a hierarchical risk-parity tree where anti-correlated assets are just as useful for diversification as uncorrelated ones); use rank-based distance when returns have heavy tails or outliers that would distort ordinary correlation. This choice feeds directly into clustering, hierarchical risk parity, and dimensionality-reduction techniques like multidimensional scaling, all of which only see the distances you hand them.