IAI Actuarial Core Principles · Risk Modelling and Survival Analysis · Introduction to copulas
Two risks each have a standard uniform marginal and are joined by the Clayton copula C(u,v) = (u^(-θ) + v^(-θ) − 1)^(-1/θ) with θ = 1. Calculate the probability that both risks are at or below their 50th percentiles, P(U ≤ 0.5, V ≤ 0.5).
The probability is 1/3, about 0.333. Substituting u = v = 0.5 and θ = 1 into the Clayton copula gives (2 + 2 − 1)^(-1) = 1/3. This exceeds the independence value of 0.25, showing positive lower-tail dependence.
- A0.25
- B0.333Correct
- C0.400
- D0.500
- 0.200
Explanation
With θ = 1, C(0.5,0.5) = (1/0.5 + 1/0.5 − 1)^(-1) = (2 + 2 − 1)^(-1) = 1/3 ≈ 0.333. Independence would give 0.25, so the result shows positive lower-tail dependence. Check: the value lies between 0.25 and the upper bound 0.5.
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