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IAI Actuarial Core Principles · Risk Modelling and Survival Analysis · Loss distributions, with and without risk sharing

Ground-up losses are Pareto with alpha = 3 and lambda = 20,000 rupees (S(x) = (20,000/(20,000+x))^3). An excess-of-loss reinsurance treaty covers each loss above a retention of Rs 10,000. What is the expected reinsurer payment per loss (ground-up loss, zeros included)?

The expected reinsurer payment is Rs 4,444 per loss. Integrating the Pareto survival function from the retention of Rs 10,000 gives 20,000³ divided by 2 × 30,000², which equals 4,444.

  1. ARs 10,000
  2. BRs 5,926Correct
  3. CRs 8,889
  4. DRs 4,444
  5. Rs 6,667

Explanation

E[(X-d)+] = integral from d of S(x)dx = lambda^3/((alpha-1)(lambda+d)^(alpha-1)) = 20,000^3/(2×30,000^2) = 8×10^12/(1.8×10^9) = 4,444? Compute: 8e12/1.8e9=4,444. Check via mean: mean=10,000; E[min(X,d)] = 10,000×[1-(20/30)^2]=5,556; mean minus this = 4,444.

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