Empirical Properties of Correlation: How Do Correlations Behave in the Real World?: formula sheet

Key formulas

Covariance

Cov(X, Y) = E[(X − μX)(Y − μY)]

Units are the product of the units of X and Y. Sign shows direction only.

Correlation

ρ(X, Y) = Cov(X, Y) ÷ (σX × σY)

Always between -1 and +1. Needs non-zero standard deviations.

Covariance from correlation

Cov(X, Y) = ρ × σX × σY

Use this to move between the two in portfolio questions.

Two-asset portfolio variance

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

Portfolio volatility is the square root of this.

Two-asset diversified VaR

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

Holds for normally distributed returns with zero mean over the horizon. At ρ = 1 it equals VaR1 + VaR2.

Portfolio VaR (normal)

VaR = z × σp × Portfolio value

z is about 1.645 at 95% and 2.326 at 99% one-tailed.

Discrete mean-reverting correlation

ρ(t) − ρ(t−1) = a × (μ − ρ(t−1)) + ε(t)

a is the reversion rate per period, μ the long-run mean, ε a random error.

Expected next-period correlation

E[ρ(t)] = ρ(t−1) + a × (μ − ρ(t−1))

Drop the error term, since its expected value is zero.

Regression form

Δρ = α + β × ρ(t−1) + ε

Estimate by regression of correlation changes on the lagged level.

Recovering parameters

a = −β ; μ = α ÷ a

Mean reversion requires β < 0. Equivalent: μ = −α ÷ β.

Gap after n periods (no shocks)

(ρ(n) − μ) = (1 − a)^n × (ρ(0) − μ)

The distance from the mean shrinks by factor (1 − a) each period.

Two-asset portfolio variance

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

For fixed weights and volatilities, a higher ρ gives a higher portfolio variance. This is how rising correlation reduces diversification.

Correlation from covariance

ρ = Cov(1,2) ÷ (σ₁ × σ₂)

Correlation can rise because covariance rises or because volatilities fall. Check which one moved.

Regime rule (empirical)

ρ(recession) > ρ(expansion); ρ(high volatility) > ρ(low volatility)

A tendency found in equity data, not a law that holds in every sample or asset class.

Correlation

ρ(X,Y) = Cov(X,Y) ÷ (σX × σY)

Always between -1 and +1. It captures linear co-movement only.

Two-asset portfolio variance

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

Higher ρ raises portfolio risk. Use it to show the cost of correlation rising in stress.

Mean reversion of correlation (stylised)

Next correlation ≈ long-run level + persistence × (current − long-run level), with persistence between 0 and 1

Illustrative form. Shows pull toward the long-run mean; it is not a GARP-specified model.

Sample correlation

ρ̂ = Cov(X, Y) ÷ (σX × σY)

Always between -1 and +1. Estimated over a chosen window, so it changes with window length.

Skewness of a correlation series

Skew = E[(ρ − μ)³] ÷ σ³

Negative means a long tail toward lower correlations. Expect this when the mean correlation is high.

Autocorrelation at lag k

AC(k) = Cov(ρt, ρt−k) ÷ Var(ρ)

Applied to a correlation series. Close to +1 means strong persistence. Rolling windows inflate it through overlap.

Fisher z-transform

z = 0.5 × ln[(1 + ρ) ÷ (1 − ρ)]

Makes the sampling distribution of ρ approximately normal, for approximately bivariate normal data.

Standard error of z

SE(z) = 1 ÷ √(n − 3)

n is the number of observations. Shorter windows give wider confidence bands.

Test of equal correlations (two independent samples)

Z = (z1 − z2) ÷ √[1 ÷ (n1 − 3) + 1 ÷ (n2 − 3)]

Compare with the standard normal critical value, such as 1.96 at 5% two-sided.

Two-asset portfolio variance

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

Portfolio risk rises with ρ. Weights w1 and w2 are portfolio proportions.

Parametric VaR (normal, zero mean)

VaR = z × σp × portfolio value

z = 1.645 at 95% and 2.326 at 99% (one-tailed). Higher ρ raises σp and so VaR.

Upper bound on portfolio risk

With ρ = 1: σp = w1σ1 + w2σ2

The no-diversification case. Long-only positions give the maximum risk at ρ = 1.

Diversification benefit

Benefit = (undiversified VaR) − (diversified VaR)

Undiversified VaR is the sum of standalone VaRs. Benefit shrinks as ρ rises.

Quick revision

• Correlation is not constant; it changes over time and with conditions.
• Higher correlation between assets means less diversification and higher portfolio variance.
• Equity correlations show mean reversion: they drift back toward a long-run average.
• Equity correlations tend to be higher when volatility is high and in market downturns.
• Correlations often rise most in stress, when diversification is needed most.
• Correlation behavior differs across asset classes; do not apply equity results to all of them.
• Correlation estimates are noisy, so sample period and window length affect the result.
• Correlations are persistent: recent values help forecast near-term values.
• Correlation distributions can be skewed, so the average may not describe typical behavior.
• Constant-correlation models can understate risk in a crisis.
• Correlation breakdown means historical relationships fail just when you rely on them.
• Use stress testing and stressed correlation assumptions to complement historical estimates.

Common mistakes

• Treating zero correlation as independence. Fix: Correlation captures only linear dependence. Nonlinear relationships can exist with zero correlation.
• Adding stand-alone VaRs to get portfolio VaR. Fix: That is correct only at ρ = 1. Otherwise use the square-root combination.
• Treating the regression slope as the reversion rate with its sign unchanged. Fix: Use a = −β. A negative slope gives a positive reversion rate.
• Computing the long-run mean as α alone. Fix: Use μ = α ÷ a, which equals −α ÷ β.
• Assuming correlation is a constant property of a pair of assets. Fix: Remember that correlation is regime-dependent. It is estimated from data and changes with market conditions.
• Saying correlations fall in a crisis because assets diversify into safe havens. Fix: For equities, correlation rises in downturns. Treat other asset classes separately.
• Assuming stock-bond correlation is always negative. Fix: Remember the sign changes with the regime. Inflation and rate shocks can make it positive.
• Treating all commodities as highly correlated. Fix: Individual commodities have idiosyncratic drivers. Correlation is higher within a sector and in stress.
• Saying correlation distributions are symmetric or normal. Fix: Remember the bounds of -1 and +1. A high mean forces a long lower tail, so skew is negative.
• Linking positive skew to high-mean correlations. Fix: Skew follows the long tail. Mass near +1 with a tail to the left is negative skew.

Exam tips

• Expect applied questions where you compute diversified VaR and then explain what happens if correlation rises toward 1.
• Know the direction of effects: higher correlation means higher portfolio risk and lower diversification benefit.
• Read carefully for covariance versus correlation. Many traps swap one for the other.
• For conceptual items, link correlation instability and stress behavior to understated VaR.
• Check that horizons and units match before you calculate.
• Questions often hand you regression output. Convert slope and intercept to a and μ before anything else.
• A negative slope on the lagged level is the signature of mean reversion. Use it to eliminate options quickly.
• Expected values must sit between the current level and the mean when 0 < a < 1. Use this to check your answer.