FRM Exam Part II · Correlation Basics: Definitions, Applications, and Terminology

Pearson Correlation vs Spearman and Kendall Tau

Updated 11 October 2026 · Fact-checked

Pearson correlation measures only the strength of linear dependence between two variables. It can be near zero even when variables are strongly dependent in a nonlinear way, and outliers distort it. Spearman and Kendall are rank-based measures that capture monotonic dependence and are less sensitive to outliers. To solve questions, match the measure to the data pattern.

Understand Pearson Correlation vs Other Dependence Measures

Correlation summarises how two variables move together in one number between -1 and +1. The most common version is Pearson correlation. It is the covariance of the two variables divided by the product of their standard deviations. It tells you how well a straight line describes the relationship.

The key point for FRM: Pearson measures linear dependence only. Zero correlation does not mean independence. If Y = X² and X is symmetric around zero, Y depends completely on X, yet the Pearson correlation is zero. Independence does imply zero correlation, but the reverse is not true in general. A big exception: for jointly normal variables, zero correlation does imply independence.

Pearson also has other weaknesses. It is sensitive to outliers, so one extreme day can move it a lot. It is not invariant to nonlinear transformations: the correlation of prices differs from the correlation of log prices. It is also defined only when variances are finite, which can fail for heavy-tailed returns. And in a crisis, a single number cannot show that dependence is stronger in the tails.

Rank correlations fix some of this. They use the ordering of the data, not the raw values. Spearman's rho is the Pearson correlation of the ranks. Kendall's tau compares pairs of observations: it counts concordant pairs (both variables move in the same direction) against discordant pairs (opposite directions). Both detect monotonic relationships, not just linear ones, and both are unchanged by any strictly increasing transformation of either variable. This is why they link to copulas: rank correlations depend only on the copula, not on the marginal distributions.

Rank measures are not perfect. They can still miss non-monotonic dependence, such as Y = X². And none of the three measures describes tail dependence fully. For that you need tail-dependence measures or a copula model.

Key formulas to remember

Pearson correlation: ρ(X,Y) = Cov(X,Y) ÷ (σX × σY); Lies between -1 and +1. Measures linear dependence only.
Spearman rank correlation (no ties): ρS = 1 − [6 Σ dᵢ²] ÷ [n (n² − 1)]; dᵢ is the difference between the ranks of X and Y for observation i. Equals Pearson correlation of the ranks.
Kendall's tau: τ = (nc − nd) ÷ [n (n − 1) ÷ 2]; nc = concordant pairs, nd = discordant pairs, no ties. Total pairs = n(n − 1) ÷ 2.
Independence and correlation: Independent ⇒ ρ = 0, but ρ = 0 does not imply independent; The reverse holds for jointly normal variables.
Invariance property: Spearman and Kendall unchanged under strictly increasing transforms; Pearson changes under nonlinear transforms such as taking logs.

How to solve Pearson Correlation vs Other Dependence Measures questions

Use this routine for any question comparing Pearson, Spearman and Kendall, or asking what a correlation figure really means.

1Identify what the question gives you: raw data, ranks, pair counts, or a stated relationship such as Y = X².
2Decide what type of dependence is described: linear, monotonic but nonlinear, non-monotonic, or tail-driven.
3Pick the measure that fits. Linear and well-behaved data: Pearson. Monotonic or outlier-prone data: Spearman or Kendall.
4If computing, rank the data (average ranks for ties), then apply the Spearman or Kendall formula carefully.
5Check the result lies between -1 and +1 and that its sign matches the pattern in the data.
6Interpret in risk terms: say what the number does not capture, such as non-monotonic dependence or tail co-movement.
7Eliminate answer options that claim zero correlation means independence or that rank measures capture all dependence.

Quickest way: Match the measure to the pattern

When to use it: Use for conceptual multiple-choice questions where you must choose the best statement or measure in under a minute.

Ask: is the relationship a straight line? If yes, Pearson is fine.
Ask: is it always rising or always falling but curved, or are there outliers? Choose Spearman or Kendall.
Ask: is it U-shaped or otherwise non-monotonic? All three may show near zero, so none is adequate.
Reject any option saying zero correlation proves independence, unless variables are jointly normal.
For calculations, Kendall needs pair counts and Spearman needs rank differences. Do not mix the formulas.

Common mistakes in Pearson Correlation vs Other Dependence Measures

Concluding that zero correlation means the variables are independent.
The two ideas are taught together and feel equivalent.
Fix: Remember Y = X² with symmetric X: perfect dependence, zero Pearson correlation. Only for jointly normal variables does the conclusion hold.
Saying Spearman and Kendall capture all dependence.
Rank measures are described as more general than Pearson.
Fix: They capture monotonic dependence only. Non-monotonic and tail-specific dependence can still be missed.
Using Spearman's formula 1 − 6Σd² ÷ [n(n² − 1)] with raw values instead of ranks.
Candidates rush and subtract the data directly.
Fix: Rank each variable first, then take rank differences d, then square and sum.
Expecting Pearson to stay the same after a log transform.
Candidates assume correlation is a property of the pair only.
Fix: Pearson is invariant only to positive linear transforms. Rank correlations are invariant to any strictly increasing transform.
Assuming Kendall tau and Spearman rho must be equal.
Both are rank-based and both lie between -1 and +1.
Fix: They use different calculations and usually give different values for the same data. Their signs generally agree, but magnitudes differ.
Thinking an outlier affects Pearson and rank measures equally.
Candidates forget ranks discard magnitude.
Fix: An outlier changes a rank by at most the number of places it moves, so its influence is limited. In Pearson, its size enters squared.

Worked examples

Example 1

Five assets have the following ranks on two variables X and Y (rank of X, rank of Y): (1,2), (2,1), (3,3), (4,5), (5,4). Compute Spearman's rank correlation.

Show the solution

Find rank differences d: 1−2 = −1; 2−1 = 1; 3−3 = 0; 4−5 = −1; 5−4 = 1.
Square them: 1, 1, 0, 1, 1. Sum Σd² = 4.
n = 5, so n(n² − 1) = 5 × 24 = 120.
ρS = 1 − (6 × 4) ÷ 120 = 1 − 24 ÷ 120 = 1 − 0.2 = 0.8.

Answer: Spearman's rank correlation = 0.80.

Example 2

A risk analyst observes X taking values −2, −1, 0, 1, 2 with equal probability, and Y = X². Which statement is correct?

Show the solution

Compute E[X] = 0 by symmetry.
Compute Cov(X,Y) = E[X·X²] − E[X]E[X²] = E[X³] − 0.
E[X³] = (−8 − 1 + 0 + 1 + 8) ÷ 5 = 0, so Cov(X,Y) = 0 and Pearson ρ = 0.
But Y is fully determined by X, so the variables are strongly dependent.
Spearman and Kendall also give values near zero here because the relationship is non-monotonic: Y falls then rises as X increases.

Answer: Pearson correlation is zero even though Y is perfectly dependent on X. Zero correlation does not imply independence, and rank measures do not fix this non-monotonic case.

Exam tips

Expect conceptual statements asking which is true about zero correlation and independence. The trap is nearly always the false converse.
Know that Spearman and Kendall are rank-based and invariant to strictly increasing transforms. This is the usual reason given for preferring them.
For Kendall, count total pairs as n(n − 1) ÷ 2 first. It stops arithmetic slips.
In case-style questions about crises, note that all three measures may understate tail co-movement. Name tail dependence or copulas as the next step.
Check the sign and range of your answer before choosing an option.

Practice questions from Correlation Basics: Definitions, Applications, and Terminology

Pearson Correlation vs Other Dependence Measures in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Pearson Correlation vs Other Dependence Measures: frequently asked questions

What are the main limitations of Pearson correlation?

It captures only linear dependence, is sensitive to outliers, and changes under nonlinear transformations. It also requires finite variances and can hide stronger dependence in the tails. Zero Pearson correlation does not prove independence.

What is the difference between Spearman and Kendall rank correlation?

Spearman is the Pearson correlation of the ranks and uses squared rank differences. Kendall counts concordant and discordant pairs. Both measure monotonic dependence, but they give different values for the same data.

Why does correlation not imply dependence in finance?

Correlation measures only linear co-movement. Two assets can be strongly dependent, for example through an option payoff, and still show near-zero correlation. Risk models therefore need more than one correlation number.

Are rank correlations better than Pearson for FRM exam questions?

They are better for monotonic nonlinear data and for data with outliers. They are not better for everything, since they still miss non-monotonic and tail-specific dependence. Choose by the pattern described.