Catastrophe PML, Reinstatement and the Limits of a Single Catastrophe Layer
A general insurer's commercial fire portfolio in a city in seismic Zone IV has total insurable value of Rs 12,000 crore across 1,800 risks; the largest single risk is a Rs 450 crore petrochemical plant. The catastrophe model gives the following annual exceedance structure for portfolio damage ratio: 45% probability of no damage; 45% of a 2% damage ratio; 8.5% of an 8% damage ratio; 1% of a 20% damage ratio; and 0.5% of a 45% damage ratio. The reinsurance programme is a catastrophe excess of loss of Rs 1,500 crore in excess of Rs 300 crore, with one reinstatement at 100% additional premium pro rata. The layer costs Rs 48 crore. Free assets are Rs 1,800 crore, but the board's risk appetite limits the net loss the company can absorb from a single event to Rs 900 crore. The portfolio's non-catastrophe loss ratio is 55% and its expense ratio is 31%.
1. Compute the average annual loss and the gross loss at the 1-in-100 and 1-in-200 return periods, and state which point of the exceedance structure each corresponds to.
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Average annual loss. Multiply each damage ratio by its probability and apply to total insurable value:
- 0.45 x 2% = 0.90%
- 0.085 x 8% = 0.68%
- 0.01 x 20% = 0.20%
- 0.005 x 45% = 0.225%
Total expected damage ratio = 2.005%, so AAL = 2.005% x 12,000 = Rs 240.6 crore.
Return periods. Exceedance probability is the probability of a loss at least that large, i.e. the annual probability of the damage ratio being reached or exceeded:
- 20% damage ratio: annual probability 1% plus 0.5% = 1.5%, which is the 1-in-67 point, not 1-in-100.
- 45% damage ratio: annual probability 0.5%, which is the 1-in-200 event.
Corrected gross losses: the largest modelled loss on a strict exceedance basis is 45% x 12,000 = Rs 5,400 crore at 1-in-200. The 20% damage ratio gives 2,400 crore, but its exceedance probability is 1.5%, i.e. roughly 1-in-67, or Rs 2,400 crore at about 1-in-67.
Why this matters. Reading the 20% scenario as a 1-in-100 PML understates the 1-in-100 by treating the scenario probability as the exceedance probability. With only five modelled scenarios the curve between them is undefined, so any return period not at a scenario breakpoint is an interpolation, and the interpolation must be disclosed.
2. Calculate the net retained loss at the two largest modelled events, including reinstatement premium, and assess the programme against the board's Rs 900 crore appetite.
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1-in-200 event, gross Rs 5,400 crore. The layer is Rs 1,500 crore xs Rs 300 crore. Gross loss of 5,400 crore exhausts the layer once and the single reinstatement once, so maximum recovery = Rs 3,000 crore. Reinstatement premium at 100% pro rata for full use = the full Rs 48 crore. Net retained = 5,400 - 3,000 + 48 = Rs 2,448 crore. That is 2.7 times the Rs 900 crore appetite and Rs 648 crore more than total free assets of Rs 1,800 crore - a solvency-threatening outcome, not merely a breach of appetite.
1-in-67 event, gross Rs 2,400 crore. Recovery = Rs 1,500 crore (layer fully used once); reinstatement premium = Rs 48 crore; net retained = 2,400 - 1,500 + 48 = Rs 948 crore. Still marginally above the Rs 900 crore appetite, with the second limit now partially consumed and only one reinstatement available for the rest of the year.
Assessment. The programme is inadequate on two counts. First, the retention of Rs 300 crore plus the reinstatement cost leaves the company outside appetite even at a 1-in-67 event. Second, the reinstatement structure exposes the company to a second event in the same year with reduced capacity - and earthquake risk is not evenly spread through the year, since a main shock is normally followed by aftershocks that generate further claims on the same policies. The Rs 48 crore spend (1.6% of the portfolio's roughly Rs 3,000 crore of premium) buys real protection, but it is insufficient at the top of the modelled loss range.
3. Recommend changes to the programme and to the portfolio, ranking them by cost-effectiveness, and say what you would disclose to the board.
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1. Buy a second catastrophe layer. A layer of, say, Rs 3,000 crore xs Rs 1,800 crore sits above the current programme and covers the 1-in-200 gross loss of Rs 5,400 crore down to roughly Rs 2,400 crore. Upper layers are normally priced at a far lower rate on line than the working layer, so each rupee of limit is cheaper. This is the single most effective action because the shortfall is concentrated in a low-frequency, high-severity tail where capital, not earnings, is at risk.
2. Buy a second reinstatement, or reinstate for 100% of the limit rather than 100% pro rata. Aftershock accumulation means one event does not exhaust the year's exposure. The cost is small relative to the protection.
3. Reduce the concentration. The Rs 450 crore petrochemical plant is 3.75% of the portfolio's total insurable value in one location. Impose a maximum line per risk and per zone, decline or co-insure the excess, and make it a rule rather than a case-by-case negotiation. This costs premium but reduces both the model output and the parameter risk of the model itself.
4. Raise the retention only if capital supports it. A higher retention reduces premium cost but the board's Rs 900 crore appetite already sits below the current net retention; raising it is inconsistent with the stated appetite unless the appetite is revised.
Disclosure to the board: show the AAL of Rs 240.6 crore against the Rs 48 crore cover cost; show the net retained loss against the Rs 900 crore appetite at each modelled return period, including reinstatement premium; state that the 1-in-200 scenario as currently worded (20% damage) is a 1-in-67 event, not 1-in-100; and state the model limitations - five scenarios, no exposure growth, no demand surge, no fire-following-earthquake, and no allowance for the non-catastrophe book being simultaneously affected.