Ruin Probability and Proportional Reinsurance for Motor Claims
An Iranian motor insurer has initial surplus IRR 1,500,000 million and no reinsurance. Claim arrivals follow a Poisson process with annual intensity 80. Individual claim amounts are modelled as exponential with mean IRR 40,000 million. Premium income is received continuously at IRR 4,000,000 million per year, and the insurer uses the Cramér-Lundberg surplus process. The board has a risk appetite of at most a 0.5% ultimate ruin probability. The actuary proposes 50% proportional reinsurance. The reinsurer charges a 20% loading on the expected ceded claims; all other assumptions are unchanged. The CEO opposes reinsurance because the company has positive premium loading and says ruin is a theoretical concept. The actuary must advise the board whether the current position meets the risk appetite and whether the reinsurance is justified.
1. Compute the adjustment coefficient R and the Lundberg upper bound on ultimate ruin probability for the current no-reinsurance position. Does the insurer meet the 0.5% risk appetite under this bound?
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Use amounts in IRR million. Premium rate c = 4,000,000; Poisson intensity λ = 80; exponential claim mean μ = 40,000.
For exponential claim sizes the adjustment coefficient solves λ + cR = λ/(1 − μR), which gives R = (c − λμ)/(cμ) = (4,000,000 − 80 × 40,000) / (4,000,000 × 40,000) = (4,000,000 − 3,200,000) / 160,000,000,000 = 800,000 / 160,000,000,000 = 5.0 × 10^−6 per IRR million.
For initial surplus u = 1,500,000: Ru = 5.0 × 10^−6 × 1,500,000 = 7.5. Lundberg bound: ψ(u) ≤ e^−7.5 ≈ 0.000553 = 0.0553%.
This is below the 0.5% risk appetite. Under the model and the bound, the current position meets the appetite, although the bound is an upper bound and the model assumptions are strong.
2. Compute R and the Lundberg bound after 50% proportional reinsurance, including the reinsurer's 20% loading. Compare with the current position.
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With 50% proportional reinsurance, the insurer retains half of each claim. Retained claim mean = 20,000. Expected ceded claims = 0.5 × 3,200,000 = 1,600,000. Reinsurance premium = 1,600,000 × 1.20 = 1,920,000. Net premium rate c_net = 4,000,000 − 1,920,000 = 2,080,000.
R_net = (c_net − λ μ_net)/(c_net μ_net) = (2,080,000 − 80 × 20,000)/(2,080,000 × 20,000) = (2,080,000 − 1,600,000)/(41,600,000,000) = 480,000 / 41,600,000,000 = 1.1538 × 10^−5 per IRR million.
Ru_net = 1.1538 × 10^−5 × 1,500,000 = 17.31; bound ≈ e^−17.31 ≈ 3.0 × 10^−8. The model ruin bound improves dramatically, but the insurer pays 320,000m per year in reinsurance loading (1,920,000 − 1,600,000).
3. Advise the board. Address the CEO's claim and any non-model risks such as parameter error, dependence, and reserving.
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Advice: The current position already meets the 0.5% target under the Lundberg bound, so the CEO is wrong to dismiss ruin as purely theoretical; capital and risk appetite are real constraints. But the model is a simplification: exponential claims may understate tail risk, claim arrivals may be dependent (storms, riots, or legal changes), and parameters λ and μ are estimated with error. Proportional reinsurance buys a much lower bound but at a high loading, and it may not be the most cost-effective way to buy tail protection.
Recommendation: do not rely on the bound alone. Stress test with a heavier-tailed claim distribution, a 20% increase in claim frequency, and a single large claim. Compare 50% proportional reinsurance with an excess-of-loss layer sized to protect the surplus. If the board keeps no reinsurance, set a hard capital trigger and monitor. If reinsurance is purchased, disclose the cost and the residual basis risk. The actuary should not present the bound as an exact ruin probability.