Why Reinstatement Provisions Matter
An excess treaty usually provides a stated limit for covered losses during its term. A sufficiently large loss can use some or all of that limit. A reinstatement provision can restore capacity after it is used, allowing the treaty to respond to later losses. Whether that restoration is free or subject to additional premium—and how much capacity is restored—depends on the contract.
This distinction matters when reviewing a treaty’s protection and pricing. The stated per-occurrence limit alone does not reveal the maximum amount that could be recovered over the treaty period. The number of reinstatements, the trigger for restoring capacity, and the premium terms also affect the treaty’s operation. The examples below use hypothetical excess-of-loss treaties and deterministic losses; no probability distribution or mortality assumption is used, and the assumed interest rate is \(i=0\), so no interest adjustment applies.
Capacity, Erosion, and Restoration
Write a treaty as \(L\) excess of \(A\), where \(A\) is the attachment point and \(L\) is the per-occurrence limit. For a covered loss \(Z\), the treaty’s payment before considering remaining aggregate capacity is
This expression gives the layer loss for one occurrence when the full per-occurrence limit is available. If previous losses have reduced available capacity, the actual payment is also constrained by that remaining capacity. A reinstatement makes some capacity available again, subject to the treaty’s wording and the number of reinstatements available.
A reinstatement is not automatically a fresh, unlimited treaty. It is bounded by the number of reinstatements and the amount that can be restored under the contract. For example, one full reinstatement of a \(L\)-limit treaty can permit up to \(2L\) of total recoveries over the term, if the treaty’s other terms allow both limits to be used. A partial restoration may produce less than this maximum.
Treaties differ in their restoration triggers. Wording may restore capacity as covered losses are paid, or may specify restoration only after the limit is exhausted. It may also specify whether restoration can apply to a loss that exhausts the remaining limit or only to later losses. These distinctions affect the order in which losses are allocated and should not be inferred from the word “reinstatement” alone.
Worked Example: Tracking a Free Reinstatement
A hypothetical treaty covers \(3\) million excess of \(2\) million per occurrence. It provides one free reinstatement, but restoration occurs only after the current limit is exhausted and applies to later occurrences. Three covered occurrence losses are \(4.4\) million, \(2.6\) million, and \(5.0\) million. Find the treaty payment for each occurrence and the total recovery. Assume no other limits or exclusions apply.
Method and assumptions. Calculate each occurrence’s layer loss, then cap payment at the capacity available at that point. The treaty is deterministic for this example; no probability distribution or mortality assumption is used. Set \(i=0\), so there is no interest adjustment.
First occurrence. Its layer loss is \(\min(4.4-2,3)=2.4\) million. The treaty pays \(2.4\) million, leaving \(3-2.4=0.6\) million of the initial limit.
Second occurrence. Its layer loss is \(\min(2.6-2,3)=0.6\) million. The treaty pays \(0.6\) million, exhausting the initial \(3\)-million limit. The one free reinstatement restores up to \(3\) million for later occurrences. There is no additional premium for this restoration.
Third occurrence. Its layer loss is \(\min(5.0-2,3)=3\) million. The restored capacity is \(3\) million, so the treaty pays \(3\) million.
Total and check. Total recovery is \(2.4+0.6+3=6.0\) million. This equals the initial \(3\)-million limit plus one full \(3\)-million reinstatement, so the result respects the treaty’s maximum capacity under the stated assumptions. The common mistake is to treat the first \(2.4\)-million payment as triggering an immediate full restoration; the wording here allows restoration only once the initial limit is exhausted.
Paid Reinstatements and Premium Bases
A paid reinstatement adds a premium obligation when capacity is restored. The amount cannot be determined from “paid reinstatement” alone. The contract must specify the premium base and the applicable rate or factor. A common structure relates the charge to the original treaty premium and the fraction of the limit restored. Some wordings also prorate for the time remaining in the treaty period.
For example, a contract may express the premium as the original treaty premium multiplied by (1) the fraction of limit reinstated, (2) a stated reinstatement rate, and, if required, (3) the fraction of the treaty period remaining. Other contracts may use a stated rate applied to restored capacity. These are contractual pricing conventions, not interchangeable formulas. A loss amount is not automatically the premium base: the wording must say how the amount of restored capacity relates to the premium calculation.
A reinstatement rate of \(100\%\) in a specified formula does not mean that the cedent pays the entire original premium again in every circumstance. The amount depends on the formula’s other factors. Similarly, a \(50\%\) reinstatement rate does not, by itself, mean that half the original premium is due; the fraction of limit restored and any time proration may also apply.
Worked Example: Identifying the Amount-Based Premium Factor
A hypothetical treaty has a \(4\)-million limit and one paid reinstatement. Its wording restores capacity as covered losses are paid and charges \(100\%\) of the original treaty premium, prorated by the fraction of the limit restored. The original premium is \(64{,}000\). A covered occurrence produces a \(2.5\)-million layer payment. Find the restored limit and the reinstatement premium under the stated formula. Assume there is no time proration.
Method and assumptions. The treaty expressly restores capacity in proportion to the covered payment, up to the \(4\)-million limit. The hypothetical occurrence is deterministic; no probability distribution or mortality assumption is used. Assume \(i=0\), with no interest adjustment.
Fraction of the limit restored. The payment uses \(2.5/4=0.625\), or \(62.5\%\), of the limit. Thus, \(2.5\) million of capacity is restored, leaving the treaty with \(4\) million available after restoration under this wording.
Premium calculation. The amount-based factor is \(0.625\). The reinstatement rate is \(100\%=1.00\), and the original premium is \(64{,}000\). Therefore, the additional premium is \(64{,}000\times0.625\times1.00=40{,}000\).
Check and common mistake. The restored fraction is \(2.5/4\), not \(2.5\) divided by the occurrence’s ground-up loss. The calculation restores \(2.5\) million of a \(4\)-million treaty limit and charges \(62.5\%\) of the original premium, consistent with the stated amount-based rule. Without that explicit wording, one could not assume this premium formula.
Amount Proration and Time Proration
When the contract specifies pro rata as to amount, the premium factor reflects the fraction of the treaty limit restored. When it specifies pro rata as to time, the factor reflects the fraction of the treaty period remaining, according to the contract’s timing convention. If both apply, the factors are combined as specified. For a simple twelve-month treaty with three months remaining, a straight-line time factor is \(3/12\); a treaty may define a different convention or the relevant dates.
The “amount” factor should be tied to the quantity the contract identifies. If the wording says the premium is prorated by the amount of limit reinstated, use restored limit divided by the original limit. Do not substitute the ground-up loss, the loss above attachment, or the amount of original premium already earned unless the contract expressly makes that quantity relevant. Likewise, time proration normally uses time remaining—not time elapsed—when the wording calls for a factor based on the unexpired treaty period.
Worked Example: Applying Both Proration Factors
A hypothetical twelve-month treaty has a \(5\)-million limit, an original premium of \(100{,}000\), and one paid reinstatement. The wording specifies a reinstatement rate of \(80\%\), prorated both by the amount of limit restored and by the fraction of the treaty period remaining. A loss restores \(1.2\) million of limit exactly when three months remain. Calculate the reinstatement premium. Assume straight-line time proration and no other premium adjustments.
Method and assumptions. Multiply the original premium by the amount fraction, the time-remaining fraction, and the reinstatement rate. The loss and timing are deterministic; no probability distribution or mortality assumption is used. Set \(i=0\), so no interest adjustment applies.
Amount factor. The fraction of the limit restored is \(1.2/5=0.24\).
Time factor. Three months remain in a twelve-month term, so the fraction of time remaining is \(3/12=0.25\).
Premium. The reinstatement premium is \(100{,}000\times0.24\times0.25\times0.80=4{,}800\).
Check and common mistake. If the contract had specified amount proration only, the charge would instead be \(100{,}000\times0.24\times0.80=19{,}200\). The actual wording requires both factors, giving \(4{,}800\). Using nine months elapsed instead of three months remaining would produce the wrong time factor for this contract.
Contract Terms to Confirm
Before evaluating a reinstatement provision, identify the terms that govern its operation. The label “one paid reinstatement” leaves important questions unanswered. A careful review should establish:
- Number and size. How many reinstatements are available, and can each restore the full per-occurrence limit?
- Trigger and timing. Does restoration occur as losses are paid, only upon exhaustion, or at another specified point? When does the restored capacity become available?
- Application to losses. Can restored capacity apply to the occurrence that exhausts the remaining limit, or only to subsequent occurrences?
- Premium base. Is the charge based on the original treaty premium, a stated rate applied to capacity, or another expressly defined basis?
- Proration. Is the charge prorated by amount, time remaining, both, or neither? What exact amounts and dates enter those factors?
- Free or paid status. Is a restoration free, subject to premium, or treated differently across successive reinstatements?
- Other limits and conditions. Do aggregate limits, exclusions, or other treaty provisions restrict the recovery or restoration?
Common Exam Trap
A frequent error is to treat every paid reinstatement as a fixed percentage of the original premium without checking what that percentage applies to. For a contract requiring amount and time proration, a calculation that uses only the amount fraction omits a required factor. Conversely, adding time proration when the contract specifies amount proration only understates the premium. The correction is to write the contract’s factors explicitly before substituting values:
This is a template for organizing a formula, not a universal contractual rule. If the wording defines a different base or calculation, follow that wording rather than this illustrative structure. This careful separation of capacity restoration from premium calculation is aligned with the CAS Exam 8 (Advanced Ratemaking) syllabus.
Check Your Understanding
For each hypothetical treaty, use the stated wording; do not assume a restoration or premium rule that is not provided.
- A \(2\)-million excess of \(1\)-million treaty has one free reinstatement. Restoration occurs only after exhaustion and applies to later occurrences. Covered losses are \(2.5\) million, \(1.5\) million, and \(3.0\) million. What is the payment for each occurrence and the total recovery? (Answer: \(1.5\) million, \(0.5\) million, and \(2.0\) million; total \(4.0\) million.)
- A \(6\)-million limit treaty restores capacity as losses are paid. Its paid reinstatement is prorated by amount only, at \(75\%\) of the original \(120{,}000\) premium. A covered payment restores \(2\) million of limit. What is the reinstatement premium? (Answer: \(120{,}000\times(2/6)\times0.75=30{,}000\).)
- A treaty has a \(3\)-million limit and a \(90{,}000\) original premium. It restores \(1.5\) million, and its wording specifies amount proration only at a \(100\%\) reinstatement rate. What is the premium, and should the fraction of the year remaining be applied? (Answer: \(90{,}000\times(1.5/3)\times1.00=45{,}000\); no time factor is specified.)
- A twelve-month treaty has a \(4\)-million limit, a \(72{,}000\) original premium, and an \(80\%\) reinstatement rate. Its premium is prorated by both restored amount and time remaining. It restores \(1\) million with two months remaining. Using straight-line time proration, calculate the charge. (Answer: \(72{,}000\times(1/4)\times(2/12)\times0.80=2{,}400\).)
- A \(5\)-million excess treaty has one reinstatement, but the wording provided does not say whether restoration occurs on payment or only on exhaustion, nor whether it applies to the exhausting occurrence. Can the payment for a loss that exhausts the remaining limit be determined from the reinstatement count alone? (Answer: No; the trigger and application wording are needed to determine whether restored capacity is available for that occurrence.)