Introduction to copulas: formula sheet

Key formulas

Linear correlation

ρ(X, Y) = Cov(X, Y) ÷ (σ_X σ_Y)

Needs finite, non-zero variances. Measures linear dependence only. Takes values in [-1, 1].

Sample covariance form

r = Sxy ÷ √(Sxx Syy), where Sxy = Σxy - (Σx)(Σy)/n

Use the same form for Sxx and Syy. This is the sample Pearson coefficient.

Spearman's rho (no ties)

ρ_S = 1 - 6 Σd² ÷ (n(n² - 1))

d is the difference between the ranks of each pair. This shortcut holds only when there are no ties. With ties, compute Pearson correlation on the ranks.

Spearman's rho (definition)

ρ_S(X, Y) = ρ(F_X(X), F_Y(Y))

The linear correlation of the probability-integral transforms (the ranks scaled to the unit interval).

Kendall's tau (sample, no ties)

τ = (c - d) ÷ (n(n - 1)/2)

c is the number of concordant pairs and d the number of discordant pairs, out of n(n - 1)/2 pairs.

Kendall's tau (population)

τ = P(concordant) - P(discordant)

For two independent copies (X₁, Y₁) and (X₂, Y₂) of the pair. Pair is concordant if (X₁ - X₂)(Y₁ - Y₂) > 0.

Invariance property

If T₁, T₂ are strictly increasing, then ρ_S(T₁(X), T₂(Y)) = ρ_S(X, Y) and τ(T₁(X), T₂(Y)) = τ(X, Y)

Pearson correlation does not have this property in general.

Definition of a copula

C(u, v) = P(U ≤ u, V ≤ v), with U, V ~ Uniform(0,1)

C is defined on [0,1] × [0,1]. Both marginals must be uniform.

Boundary conditions

C(u, 0) = C(0, v) = 0; C(u, 1) = u; C(1, v) = v

These come from uniform marginals. They are used to check whether a given function is a copula.

2-increasing property

C(u2, v2) − C(u1, v2) − C(u2, v1) + C(u1, v1) ≥ 0 for u1 ≤ u2, v1 ≤ v2

This ensures rectangles have non-negative probability.

Sklar's theorem

F(x, y) = C(F_X(x), F_Y(y))

C is unique if the marginals are continuous.

Inverse form of Sklar

C(u, v) = F(F_X⁻¹(u), F_Y⁻¹(v))

Valid for continuous marginals. Use it to extract the copula from a joint distribution.

Independence copula

C(u, v) = uv

X and Y are independent exactly when their copula is uv.

Fréchet–Hoeffding bounds

max(u + v − 1, 0) ≤ C(u, v) ≤ min(u, v)

Every copula lies between these bounds. The lower bound is the countermonotonic copula W and the upper bound is the comonotonic copula M.

Survival probability from a copula

P(X > x, Y > y) = 1 − F_X(x) − F_Y(y) + C(F_X(x), F_Y(y))

Use inclusion–exclusion with u = F_X(x) and v = F_Y(y).

Sklar's theorem

F(x, y) = C(F_X(x), F_Y(y))

C is unique when the margins are continuous.

Archimedean copula from generator

C(u, v) = φ⁻¹(φ(u) + φ(v))

φ is continuous, strictly decreasing and convex, with φ(1) = 0.

Clayton copula

C(u, v) = (u^(−α) + v^(−α) − 1)^(−1/α), α > 0

Generator φ(t) = t^(−α) − 1. Lower tail dependence λ_L = 2^(−1/α). Upper tail dependence is 0.

Gumbel copula

C(u, v) = exp{ −[(−ln u)^α + (−ln v)^α]^(1/α) }, α ≥ 1

Generator φ(t) = (−ln t)^α. Upper tail dependence λ_U = 2 − 2^(1/α). Lower tail dependence is 0. α = 1 gives independence.

Frank copula

C(u, v) = −(1/α) ln[1 + (e^(−αu) − 1)(e^(−αv) − 1) ÷ (e^(−α) − 1)], α ≠ 0

Generator φ(t) = −ln[(e^(−αt) − 1) ÷ (e^(−α) − 1)]. No tail dependence. α > 0 positive dependence, α < 0 negative.

Kendall's tau for Clayton and Gumbel

Clayton: τ = α ÷ (α + 2). Gumbel: τ = 1 − 1/α

Use to get α from a sample τ.

Tail dependence of the t copula

λ_L = λ_U = 2 × t_(ν+1)( −√((ν + 1)(1 − ρ) ÷ (1 + ρ)) )

Positive for every ρ > −1 and finite ν. The Gaussian copula has λ = 0 for ρ < 1.

Gaussian and t copula link to Kendall's tau

ρ = sin(πτ ÷ 2)

Holds for elliptical copulas, including Gaussian and t.

Upper tail dependence

λU = lim (u→1⁻) P(V > u | U > u) = lim (u→1⁻) [1 − 2u + C(u,u)] ÷ (1 − u)

U and V are uniform(0,1). C is the copula. The limit must exist.

Lower tail dependence

λL = lim (u→0⁺) P(V ≤ u | U ≤ u) = lim (u→0⁺) C(u,u) ÷ u

Evaluate the copula on the diagonal, then divide by u.

Clayton copula

C(u,v) = (u^(−θ) + v^(−θ) − 1)^(−1/θ), θ > 0

λL = 2^(−1/θ) and λU = 0. Larger θ gives larger λL.

Gumbel copula

C(u,v) = exp{ −[(−ln u)^θ + (−ln v)^θ]^(1/θ) }, θ ≥ 1

λU = 2 − 2^(1/θ) and λL = 0. θ = 1 is independence.

Student t copula

λU = λL = 2 × t(ν+1)( −√[ (ν+1)(1 − ρ) ÷ (1 + ρ) ] )

t(ν+1) is the cdf of a t distribution with ν+1 degrees of freedom. ν is the degrees of freedom, ρ the correlation parameter. Positive for every finite ν and ρ > −1.

Gaussian and Frank copulas

λU = λL = 0

For the Gaussian copula this holds when |ρ| < 1.

Survival copula link

λU of C = λL of the survival copula of C, and the reverse

Reversing the direction of both variables swaps the two tails.

Sklar's theorem

F(x₁, …, xₙ) = C(F₁(x₁), …, Fₙ(xₙ))

C is a copula. If the marginals are continuous, C is unique.

Inverse transform

Xᵢ = Fᵢ⁻¹(Uᵢ)

Uᵢ is uniform on (0, 1). Xᵢ then has distribution Fᵢ.

Aggregate loss

S = X₁ + X₂ + … + Xₙ

Add the simulated losses from the same simulation run.

Independence copula

C(u, v) = u × v

Use as the benchmark with no dependence.

Gaussian copula simulation

Z ~ N(0, Σ), then Uᵢ = Φ(Zᵢ)

Σ is a correlation matrix. Φ is the standard normal distribution function.

Clayton copula

C(u, v) = (u^(−θ) + v^(−θ) − 1)^(−1/θ), θ > 0

Lower tail dependence. Independence is the limit as θ → 0.

Gumbel copula

C(u, v) = exp(−[(−ln u)^θ + (−ln v)^θ]^(1/θ)), θ ≥ 1

Upper tail dependence. θ = 1 gives independence.

Simulated percentile estimate

VaR at level α = the α-quantile of the simulated S values

Sort N simulated totals. Take the value at position about αN.

Quick revision

• Linear correlation measures only linear dependence. Independent variables have zero correlation, but zero correlation does not imply independence.
• A copula is a joint distribution function on [0,1]² with uniform marginals.
• Sklar's theorem: F(x, y) = C(F₁(x), F₂(y)) for any joint distribution F with marginals F₁ and F₂.
• If the marginals are continuous, the copula is unique.
• A copula is unchanged when each variable is transformed by a strictly increasing function.
• Independence copula: C(u, v) = uv.
• Gaussian copula: no tail dependence when the correlation is below 1.
• t copula: symmetric upper and lower tail dependence, which falls as the degrees of freedom rise.
• Clayton copula has lower tail dependence. Gumbel copula has upper tail dependence.
• Tail dependence measures the chance of one variable being extreme given that the other is extreme.
• Frank copula has no tail dependence and is symmetric.
• To simulate: draw from the copula to get uniforms, then apply the inverse marginal distribution functions.

Common mistakes

• Saying zero correlation means the variables are independent. Fix: State that independence implies zero correlation, not the reverse. Quote the X and X² example when asked for a counter-example.
• Using the 6Σd² shortcut when there are tied values. Fix: Check for ties first. If there are ties, use average ranks and compute the Pearson correlation of the ranks.
• Thinking the copula contains the marginal distributions. Fix: Remember that C only describes dependence. The marginals are fed into C through u = F_X(x) and v = F_Y(y).
• Putting x and y directly into C instead of F_X(x) and F_Y(y). Fix: Always convert to uniforms first. Write u = ... and v = ... as the first line of working.
• Saying the Gaussian copula has tail dependence when correlation is high. Fix: Remember that for ρ < 1 the Gaussian copula has λ = 0. Only the t copula (and Clayton, Gumbel) give tail dependence.
• Mixing up which tail Clayton and Gumbel affect. Fix: Clayton is lower tail, Gumbel is upper tail. Link Clayton with joint falls and losses at the bad end.
• Saying the Gaussian copula has tail dependence because it has high correlation. Fix: For |ρ| < 1 the Gaussian copula has λU = λL = 0, however large ρ is. Only the limit in the tail matters.
• Giving Clayton an upper tail dependence value, or Gumbel a lower one. Fix: Clayton clusters at low values, so λL = 2^(−1/θ) and λU = 0. Gumbel clusters at high values, so λU = 2 − 2^(1/θ) and λL = 0.
• Applying the inverse marginal to a normal value instead of a uniform. Fix: Always check that the input to Fᵢ⁻¹ lies between 0 and 1. Convert normals with Φ first.
• Saying the Gaussian copula has tail dependence. Fix: State that the Gaussian copula has no tail dependence for correlation below 1. Use a t or Gumbel copula when losses cluster in the tail.

Exam tips

• Learn the standard list of limitations of linear correlation. Written questions often ask you to 'discuss' them, so give four or five points with a brief reason each.
• In calculations, show the rank table. Marks are given for correct ranks even if the final arithmetic slips.
• Always state whether you used the no-ties shortcut and that it is valid for the data given.
• MCQs often test invariance. Remember: Spearman and Kendall are unchanged by strictly increasing transformations of either variable; Pearson is not.
• Link the topic to copulas. Rank correlations depend only on the copula, which is why they are used to fit and compare copulas.
• Write u = F_X(x) and v = F_Y(y) as your first line. Examiners award method marks for this conversion.
• Quote Sklar's theorem with its condition: the copula is unique when the marginals are continuous.
• When asked to show a function is a copula, check the boundary conditions and the 2-increasing property, and say that you have checked both.