Marine cargo excess-of-loss: pricing a heavy-tailed layer
Daehan General's marine cargo account carries 90 million KRW of annual subject premium. Over the last three years the book produced about four liability claims a year, modelled as Poisson. Severity was fitted as a Pareto distribution with tail index alpha = 2.5 and scale theta = 10 million KRW, giving a mean severity of about 16.7 million KRW. The company retains losses up to 50 million KRW per claim and has been offered a per-claim excess-of-loss layer of 50 million in excess of 50 million. The reinsurer quotes 2.4% of subject premium, 2.16 million KRW, for the layer. The CFO argues the layer must be bad value because the expected annual recovery looks like a small fraction of the quoted premium; the CRO argues the opposite and points at the tail index. The board wants a recommendation before the 1 January renewal and also asks whether an attachment at 20 million with a 30 million limit would be better value on the same premium.
1. Calculate the expected annual recovery from the 50 xs 50 layer and express the quote as an expected loss ratio.
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For a Pareto severity with parameters alpha and theta, E[X capped at d] = theta/(alpha-1) x [1 - (theta/(theta+d))^(alpha-1)].
With theta = 10m and alpha = 2.5, theta/(alpha-1) = 6.667m.
- d = 50: (10/60)^1.5 = 0.0680, so E[X capped at 50] = 6.667 x 0.9320 = 6.213m.
- d = 100: (10/110)^1.5 = 0.0274, so E[X capped at 100] = 6.667 x 0.9726 = 6.484m.
- Expected layer severity = 6.484 - 6.213 = 0.271m per claim.
- Annual expected recovery = 4 x 0.271 = 1.08m KRW.
Expected loss ratio = 1.08 / 2.16 = 50%. The loading over expected loss is 100%.
So on a pure expected-value test the CFO is right: the treaty is quoted at twice the expected cost. But that comparison is only decisive if the risk being transferred is irrelevant, which it is not. Note also that the expected recovery is not small relative to the premium - it is half of it - so the CFO's phrasing misstates the arithmetic even though the conclusion (expensive on average) is correct. The real question is what the tail is worth, and that is the CRO's point.
2. Assess the tail using the fitted distribution and explain why the expected loss ratio alone is an inadequate decision rule here.
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Tail probabilities: P(X > 50) = (10/60)^2.5 = 1.13% per claim; P(X > 100) = (10/110)^2.5 = 0.25% per claim. With four claims a year the probability of at least one layer claim in a year is 1 - (1 - 0.0113)^4, about 4.4%.
The 99.5th percentile of single-claim severity solves (10/(10+x))^2.5 = 0.005, giving x of about 73m. A claim at that level recovers 23m - roughly ten years of premium - and the layer's 50m limit is never exhausted by a single claim, which is itself a design weakness.
Why the expected loss ratio is insufficient: with alpha = 2.5 the severity variance is infinite, so any mean-variance or standard-deviation capital charge understates the risk. A treaty can have a 50% expected loss ratio and still be worth buying if it removes a loss the capital model charges heavily for, and a treaty with a 70% expected loss ratio can be poor value if it sits above the capital model's effective attachment. The decision rule should compare the reinsurance cost with the reduction in required capital at the company's confidence level plus the reduction in earnings volatility, not with the expected recovery alone.
Practical recommendation: do not accept 2.16m unchanged. Either negotiate toward something closer to 1.6-1.8m (a 50-65% expected loss ratio is defensible for a heavy-tailed marine layer), or restructure - a narrower layer with a lower attachment buys more capital relief per won of premium.
3. Evaluate the proposed 30 xs 20 alternative and state how the two quotes should be compared.
Read the worked answer
Expected layer severity for 30 xs 20 = E[X capped at 50] - E[X capped at 20].
E[X capped at 20] = 6.667 x [1 - (10/30)^1.5] = 6.667 x 0.8076 = 5.384m. So expected layer severity = 6.213 - 5.384 = 0.829m per claim, and annual expected recovery = 4 x 0.829 = 3.32m KRW.
At a 2.16m premium that is an expected loss ratio of 154%, which looks far better value than the 50 xs 50 layer. But the comparison is invalid as posed: a reinsurer will not charge the same premium for a layer that attaches at 20m as for one that attaches at 50m. The correct comparison metrics are rate-on-line (premium / limit: 4.3% for 50 xs 50, 7.2% for 30 xs 20) and expected loss ratio on each layer's own quoted premium.
The two layers also buy different things. P(X > 20) = (10/30)^2.5 = 6.4% per claim, so there is roughly a 23% chance of at least one recovery in a year - the 30 xs 20 layer is an earnings-smoothing purchase. The 50 xs 50 layer responds about four times less often and is a genuine tail purchase. If the objective is K-ICS capital relief, the lower layer is inefficient because it spends limit on losses the company can absorb; if the objective is protecting the underwriting result from a bad year, it is the better buy. Decide the objective first, then negotiate price.