Correlation Basics: Definitions, Applications, and Terminology: formula sheet

Key formulas

Pearson correlation

ρ(X,Y) = Cov(X,Y) ÷ (σX × σY)

Bounded between -1 and +1. Defined only when both standard deviations are non-zero.

Covariance from correlation

Cov(X,Y) = ρ × σX × σY

Use it to rebuild covariance when the question gives correlation and volatilities.

Two-asset portfolio variance

σp² = w1²σ1² + w2²σ2² + 2 w1 w2 ρ σ1 σ2

Weights are portfolio weights. Lower ρ gives lower portfolio variance.

Two-position VaR aggregation

VaRp = √(VaR1² + VaR2² + 2 ρ VaR1 VaR2)

Holds for normally distributed (elliptical) returns with zero mean. ρ = 1 gives the simple sum; ρ = 0 gives the square root of the sum of squares.

Variance of a sum

Var(X + Y) = σX² + σY² + 2 Cov(X,Y)

For a difference, the covariance term is subtracted.

Sample correlation

r = Σ(xi - x̄)(yi - ȳ) ÷ √[Σ(xi - x̄)² × Σ(yi - ȳ)²]

Estimated from data, so it has sampling error and depends on the window used.

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.

Correlation swap payoff (fixed-rate payer, long correlation)

Payoff = N × (ρ_realized − K)

N is notional per unit of correlation, K is the strike. The short side receives the negative of this amount.

Average realized pairwise correlation

ρ_realized = [2 ÷ (n² − n)] × Σ(i<j) ρ(i,j)

For n assets there are n(n − 1) ÷ 2 distinct pairs. This is an equally weighted average.

Basket variance (equal weights, equal volatility σ, common correlation ρ)

σ_basket² = σ² × [1/n + (1 − 1/n) × ρ]

Higher ρ raises basket volatility. This is why index volatility relative to single-stock volatility reveals implied correlation.

Two-asset portfolio variance

σ_p² = w₁²σ₁² + w₂²σ₂² + 2w₁w₂ρσ₁σ₂

Portfolio risk rises with ρ, other things equal.

Correlation range

−1 ≤ ρ ≤ +1

A strike or realized value outside this range is not possible for Pearson correlation.

Correlation

ρ(X,Y) = Cov(X,Y) ÷ (σX × σY)

Unit-free and always between -1 and +1. Defined only when both standard deviations are non-zero.

Covariance from correlation

Cov(X,Y) = ρ × σX × σY

Covariance has units of X times units of Y and no fixed bound.

Two-asset portfolio variance

σp² = w1²σ1² + w2²σ2² + 2 w1 w2 ρ σ1 σ2

Higher ρ gives higher portfolio risk, all else equal. Use it to show the effect of rising correlation.

Correlation matrix properties

Diagonal = 1; ρij = ρji; -1 ≤ ρij ≤ 1; matrix positive semi-definite

Entries chosen one by one can produce an invalid matrix.

Mean reversion (concept)

Correlation tends to move toward its long-run mean over time

A tendency, not a guarantee. Speed varies by asset class and period.

Quick revision

• Pearson correlation = Cov(X,Y) ÷ (σX × σY), always between −1 and +1.
• Correlation is covariance scaled by both standard deviations, so it has no units.
• Pearson captures linear dependence only. Zero correlation does not imply independence.
• Independence implies zero correlation, but only if the correlation exists (finite variances).
• In a two-asset portfolio, lower correlation means lower portfolio volatility, other things equal.
• At ρ = +1 there is no diversification benefit; at ρ = −1 a perfect hedge is possible with the right weights.
• Pearson correlation can be distorted by outliers and is not invariant to nonlinear transformations.
• Rank-based measures, such as Spearman and Kendall, capture monotonic dependence rather than only linear.
• Correlation risk is the risk that correlations change adversely from the values assumed in a model or position.
• A correlation swap exchanges realised correlation for a fixed strike correlation.
• Correlations are not stable over time and tend to rise in market stress, reducing diversification when it is needed most.

Common mistakes

• Saying zero correlation means independence. Fix: Independence implies zero correlation, not the reverse. Zero correlation implies independence only in special cases such as jointly normal variables.
• Adding VaRs directly for a portfolio. Fix: Add VaRs only when ρ = 1. Otherwise use the square-root aggregation formula, under its normality assumption.
• Concluding that zero correlation means the variables are independent. 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. Fix: They capture monotonic dependence only. Non-monotonic and tail-specific dependence can still be missed.
• Getting the payoff sign wrong Fix: Always compute realized − strike for the long side, then flip for the short side.
• Counting pairs as n² or n instead of n(n − 1) ÷ 2 Fix: For 4 assets the number of distinct pairs is 6, not 16 or 12.
• Treating correlation as a constant input. Fix: Remember ρ is an estimate that varies with time and market state.
• Confusing mean reversion with correlation breakdown. Fix: Mean reversion is a gradual pull toward the average. Breakdown is a sharp departure from the historical relationship.

Exam tips

• Expect applied questions that ask what happens to portfolio risk or VaR when correlation rises or falls. Answer with the direction and the bounds.
• Watch for traps on 'zero correlation' versus 'independence', and on correlation versus causation.
• Know that VaR aggregation by the square-root formula relies on normality or elliptical returns; with other distributions subadditivity can fail.
• Link correlation to crises: diversification benefits are weakest when you need them most.
• Do quick bound checks at ρ = 1, 0 and -1 before heavy arithmetic.
• 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.