A limit is not a permanent stock. A layer pays until the reinsurer's cumulative payments reach its amount, and it then stops responding: any later loss falls back on the cedant, or on the layer above if the structure provides for it. From this follows a distinction that board reports almost always blur. A cedant announcing two hundred million of cover only has it if no layer has been touched, and the only quantity that matters for facing the second event is residual capacity. This is why risk functions track exhaustion in real time through a heavy season, layer by layer, rather than the program's nominal amount.
Reinstatement is the mechanism that lifts that exhaustion. After a loss has eroded or consumed a non-proportional layer, it restores that layer's capacity for the rest of the coverage period against payment of a reinstatement premium. The treaty fixes the number, and that number is the clause to read first: often one or two, sometimes unlimited on a working layer known to be hit frequently, sometimes none at all on a cat layer nobody imagines being hit twice. A so-called free reinstatement is not a gift: its cost already sits in the layer premium.
The price is computed in two ways that must be named separately, because the gap between them is of the same order as the amount itself. Pro rata the amount sets the reinstatement premium against the share of the limit consumed: a ten million layer eroded by six, on a layer premium of 1.2 million, reinstates for 0.72 million. Pro rata temporis adds the time left to run: the same six million consumed five months from expiry then reinstates for 0.72 million times five twelfths, that is 0.30 million. A treaty saying only "payable at 100%" without specifying which of the two applies leaves open a difference of more than double.
A program exhausts in two ways that call for opposite answers, and this is the most useful distinction in the lesson. Vertical exhaustion describes a single but very costly event travelling through the program from the bottom up: it consumes the first layer, erodes and then consumes the second, and stops somewhere above. What it undergoes is a test of height: the quantity that matters is its top compared with the probable maximum loss. Horizontal exhaustion describes one bottom layer hit several times in the year by separate events, until its reinstatements are used up. The program then has no first line left, and the cedant bears net the whole of every following event up to the attachment of the layer above, which nonetheless remains intact and fully paid for.
Horizontal exhaustion is the most frequent and least anticipated failure mode, because sizing attention almost always goes to height. The United States provided a dated case: 2023 counted more than twenty severe convective storm events above one billion dollars, for roughly sixty billion of insured losses, without a single major hurricane landfall. Midwest insurers whose bottom layer carried a single reinstatement had used it up by June and absorbed the summer hail episodes net, while their upper layers, fully paid for, were not touched once. The peril's signature explains everything: hail, flood, convective storm and wildfire produce the repetition of medium events, never the occurrence of an extreme one.
The vertical mode has cases of its own, equally dated. Hurricane Ian, in September 2022, roughly fifty billion dollars of insured losses in Florida, consumed the first two layers of regional insurers in a single event. The Tohoku earthquake and tsunami, in March 2011, produced roughly thirty-five billion dollars of insured losses, far beyond the scenarios several Japanese programs had retained: a cedant whose top stood at eight hundred million dollars saw all its layers exhausted vertically and bore net everything above it. The April 2011 renewals raised program tops by thirty to sixty percent on that market, the answer proper to the vertical mode and one that would have changed nothing about the horizontal one.
Three devices answer frequency, and only one really does. Buying additional reinstatements and lowering the bottom layer's attachment push the threshold back without addressing accumulation. Aggregate cover, which adds events up instead of treating them one by one, answers frequency itself. The top and drop is an efficiency variant: a top layer the first event does not reach drops down to serve as a second line at a low attachment, which converts dormant capacity into frequency protection for far less than an additional reinstatement. The counterpart is complete: a very large first event consumes it at the top and leaves it unavailable at the bottom.
On 1 January 2026 a cedant places a catastrophe program in three layers. L1: 20 million euros excess of 10, premium 4.0 million, two reinstatements at 100% pro rata the amount consumed, no pro rata temporis. L2: 30 million excess of 30, premium 2.4 million, one reinstatement pro rata amount and time. L3: 40 million excess of 60, premium 1.2 million, no reinstatement. Three events: a 38 million windstorm on 18 March, a 26 million hail on 2 June, a 47 million flood on 14 September. On 15 September the finance director tells the board that "the program carries 90 million of capacity". What must be corrected, and what does the house have?
The 90 million is the nominal amount, accurate on 31 December 2025 and wrong since 18 March. The 38 million windstorm leaves 10 million of retention with the cedant, consumes L1's 20 million and erodes L2 by 8 million; L1's reinstatement is payable in full, 4.0 million, and L2's pro rata amount and time, 8 thirtieths of 2.4 million scaled to the 288 days remaining, roughly 0.50 million. The 26 million hail consumes 16 of L1's restored 20 million; L1's second and last reinstatement costs 16 twentieths of 4.0 million, that is 3.2 million. The 47 million flood consumes L1's 20 million, which has no reinstatement left, and 17 million of L2, which has none either. On 15 September the house therefore holds 13 million on L2, responding from 30 to 43, and L3's 40 million excess of 60: 53 million of limit, and above all nothing at all between 10 and 30. What the board needs to hear is not the total but that hole. A fourth flood identical to September's would cost 34 million net, against 10 million in March, with no clause having changed and no reinsurer having failed its obligation. Two further remarks complete the picture and run in opposite directions. The reinstatement premiums paid, 4.0 plus 0.50 plus 3.2, come to 7.7 million against 7.6 million of original premium: the house has bought its program twice and ends up less protected than it started. And yet it has not failed: 81 million recovered out of 111 of gross losses, 30 million net, precisely the service expected of it. Both readings are true together, and that is the substance of the lesson: the open question is not past performance but what is left for the fourth quarter.
- 01A layer is exhausted when the reinsurer's cumulative payments reach its limit: residual capacity, not the nominal amount, is the only quantity that matters for the next loss.
- 02The treaty fixes the number of reinstatements, often one or two, sometimes unlimited on a working layer, sometimes none at the top. A free reinstatement is already paid for in the layer premium.
- 03The price is computed pro rata the amount consumed, sometimes pro rata the time left as well: a treaty that does not say which leaves open a difference of more than double.
- 04Vertical exhaustion tests the program's height, horizontal the number of reinstatements, and a program built for one can fail on the other.
- 052023 in the United States: more than twenty convective storms above one billion dollars, roughly sixty billion of insured losses, no major hurricane. Bottom layers with a single reinstatement were used up by June.
- 06Adding reinstatements or lowering the attachment pushes the threshold back; only aggregate cover answers frequency itself, and the top and drop converts dormant top capacity into a second line.