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IAI Actuarial Core Principles · Risk Modelling and Survival Analysis · Introduction to copulas

An actuary simulates aggregate losses of two lines using a Gaussian copula with correlation ρ. Which sequence of steps correctly generates a pair (X, Y) with marginal CDFs F_X and F_Y?

Simulate correlated standard normals, transform each through Φ to get dependent uniforms, then apply the inverse marginal CDFs to get losses. This preserves the Gaussian copula dependence and gives exactly the specified marginals. The other approaches misapply CDFs or lose the intended dependence.

  1. ASimulate bivariate standard normal (Z1,Z2) with correlation ρ; set U=Φ(Z1), V=Φ(Z2); then X=F_X^(-1)(U), Y=F_Y^(-1)(V)Correct
  2. BSimulate independent uniforms U,V; set X=F_X^(-1)(U), Y=F_Y^(-1)(V); then apply correlation ρ to X and Y
  3. CSimulate bivariate standard normal (Z1,Z2) with correlation ρ; set X=F_X(Z1), Y=F_Y(Z2)
  4. DSimulate bivariate standard normal (Z1,Z2) with correlation ρ; set U=Φ^(-1)(Z1), V=Φ^(-1)(Z2); then X=F_X(U), Y=F_Y(V)
  5. Simulate X and Y from their marginals, then sort both samples in the same order to give correlation ρ

Explanation

Correlated normals are mapped to uniforms via Φ, giving a Gaussian copula sample, and then inverse marginal CDFs transform the uniforms to losses. The other sequences misuse the CDF and inverse, skip the dependence step, or impose comonotonicity by sorting.

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