Reinstatements and Reinstatement Premiums

Why Reinstatements Matter

An excess-of-loss treaty can pay its stated limit on one loss, leaving no capacity for a later covered loss during the treaty period. A reinstatement provision can restore some or all of that used capacity, usually in exchange for an additional premium. The provision therefore affects both how much protection remains after a claim and the cedent’s total cost of reinsurance.

A reinstatement does not change the treaty’s attachment point or its limit for any one occurrence. It makes additional limit available during the treaty period, subject to the number and terms of reinstatements purchased. All examples below use hypothetical contracts. They are deterministic contract calculations: no interest rate, mortality assumption, or probability distribution is used, and amounts are undiscounted.

Definition: Let $L$ be an excess-of-loss treaty’s limit for a covered occurrence. A reinstatement restores limit that has been used by reinsurance payments, making that capacity available again for later covered losses. Under the common loss-paid restoration convention used here, the amount restored is the reinsurance recovery paid, not the gross loss. The contract wording determines the number of reinstatements, when restoration takes effect, and how its premium is calculated.

Capacity and the Reinstatement Premium

Suppose a treaty has a limit $L$ and permits $n$ full-limit reinstatements. If every available limit is used, the maximum aggregate recovery over the treaty period is $(n+1)L$: the original limit plus up to $n$ restored limits. This is a potential aggregate total, not a promise that the treaty will pay that amount. Covered losses must still meet the treaty’s terms, and the stated limit continues to apply separately to each occurrence.

In practice, a recovery may use only part of the available limit. Under loss-paid restoration, the available capacity falls by the recovery amount; a reinstatement then restores some or all of that used capacity, depending on the remaining reinstatement allowance. Keep track of recoveries, capacity, and reinstatements used separately. The fact that one reinstatement is available does not mean that every loss automatically receives a fresh full limit.

Premium formula: Let $P_0$ be the original treaty premium, $A$ the amount of limit restored, $L$ the original limit, and $q$ the reinstatement premium rate expressed as a decimal fraction of the original premium. For a pro rata amount-and-time charge, with $T_{\mathrm{rem}}$ representing the fraction of the treaty period remaining at the restoration date, $$ P_{\mathrm{re}}=P_0 q \frac{A}{L}T_{\mathrm{rem}}. $$ For a pro rata amount-only charge, omit $T_{\mathrm{rem}}$: $$ P_{\mathrm{re}}=P_0 q \frac{A}{L}. $$

A 100% reinstatement premium rate means $q=1$; it does not mean that the cedent pays 100% of the original premium for every partial restoration. The amount factor $A/L$ adjusts the charge when less than a full limit is restored. For an amount-and-time charge, $T_{\mathrm{rem}}$ is the fraction of the treaty period remaining, not the fraction elapsed. A 50% rate means $q=0.50$, so the otherwise applicable charge is halved.

Some contracts provide free reinstatements, charge only for reinstatements actually used, or specify another premium basis. Treat the clause’s wording as controlling. The formulas here apply only when the stated pricing terms match the formula’s assumptions.

How the Formula Follows from the Pricing Terms

The formula is a proportional allocation of the original premium. A restoration of $A$ dollars represents the fraction $A/L$ of one full treaty limit. If a full-limit restoration for the full treaty period would cost $qP_0$, then a restoration of that fraction costs $qP_0(A/L)$ under amount-only pricing. If the charge is also proportional to the time remaining, multiply by $T_{\mathrm{rem}}$. Thus,

qP_0 \times \frac{A}{L} \times T_{\mathrm{rem}} =P_0q\frac{A}{L}T_{\mathrm{rem}}.

This reasoning depends on a contract that prices restored capacity linearly in amount and, where applicable, linearly in time. It does not justify using the formula when the wording specifies a fixed charge, a different premium base, a free reinstatement, or another pricing rule.

Worked Calculations

Worked Example: Partial Restoration Priced by Amount and Time

A hypothetical annual treaty provides a limit of $5,000,000 excess of $2,000,000 per occurrence and one reinstatement. Its original premium is $480,000. The reinstatement rate is 100% of the original premium, pro rata as to amount and time. At a date when exactly eight of twelve months remain, a covered gross loss is $5,000,000. Find the recovery, the restoration premium, and the recovery on a later covered gross loss of $7,000,000.

Assumptions and method. This is a deterministic calculation using the stated contract terms. No interest rate, mortality assumption, or loss distribution applies. The recovery is the portion of gross loss above the $2,000,000 attachment, capped at the $5,000,000 occurrence limit.

First recovery. The loss above the attachment is $5,000,000 − $2,000,000 = $3,000,000. This is below the $5,000,000 limit, so the recovery is $3,000,000. The amount restored is therefore $A=$3,000,000$, or $3/5=0.60$ of the limit.

Reinstatement premium. The time remaining is $8/12=2/3$, and the rate is $q=1$. Therefore,

P_{\mathrm{re}} =\$480{,}000(1)\left(\frac{\$3{,}000{,}000}{\$5{,}000{,}000}\right)\left(\frac{8}{12}\right) =\$480{,}000(0.60)\left(\frac{2}{3}\right) =\$192{,}000.

After restoration, the available limit is $2,000,000 + $3,000,000 = $5,000,000. On the later gross loss, the amount above attachment is $7,000,000 − $2,000,000 = $5,000,000, so the recovery is $5,000,000. Total recoveries are $3,000,000 + $5,000,000 = $8,000,000, matching the original $5,000,000 plus the $3,000,000 restored. The calculation avoids charging for a full-limit restoration when only $3,000,000 was paid.

Worked Example: Two Amount-Only Reinstatements

A hypothetical treaty has a $4,000,000 occurrence limit excess of $1,000,000, two reinstatements, and an original premium of $300,000. Each reinstatement is charged at 100% of the original premium, pro rata as to amount only. Three covered gross losses are $5,500,000, $3,400,000, and $6,000,000. Find each recovery and the total additional premium.

Assumptions and method. No interest rate, mortality assumption, or loss distribution applies. Each recovery equals the amount of that occurrence above $1,000,000, capped at $4,000,000. A paid recovery restores the same amount, provided a reinstatement remains.

First occurrence. The amount above attachment is $5,500,000 − $1,000,000 = $4,500,000. Capping at the limit gives a $4,000,000 recovery. This uses the original limit in full and the first reinstatement restores $A=$4,000,000$. The premium is

\$300{,}000(1)\left(\frac{\$4{,}000{,}000}{\$4{,}000{,}000}\right)=\$300{,}000.

Second occurrence. After restoration, $4,000,000 is available. The amount above attachment is $3,400,000 − $1,000,000 = $2,400,000, so the recovery is $2,400,000. The second reinstatement restores that amount, with premium

\$300{,}000(1)\left(\frac{\$2{,}400{,}000}{\$4{,}000{,}000}\right) =\$300{,}000(0.60)=\$180{,}000.

Third occurrence and check. After the second restoration, $4,000,000 is again available. The amount above attachment is $6,000,000 − $1,000,000 = $5,000,000, capped at $4,000,000. Total recoveries are $4,000,000 + $2,400,000 + $4,000,000 = $10,400,000. That is less than the maximum $3 \times $4,000,000 = $12,000,000. Both reinstatements have been used, and the total additional premium is $300,000 + $180,000 = $480,000. The amount-only term means no time factor is included.

Worked Example: A Reduced Reinstatement Rate

A hypothetical annual treaty provides $2,000,000 excess of $1,000,000 and one reinstatement. The original premium is $240,000. The reinstatement premium rate is 50% of the original premium, pro rata as to amount and time. At a date with nine months remaining, a covered gross loss of $3,500,000 exhausts the limit. A later covered gross loss is $2,200,000. Calculate the reinstatement premium and both recoveries.

Assumptions and method. The calculation is deterministic; no interest rate, mortality assumption, or loss distribution applies. There is one $2,000,000 occurrence limit, and the restoration amount is the recovery paid.

First recovery and premium. The first loss exceeds the $1,000,000 attachment by $2,500,000, but the recovery is capped at $2,000,000. Thus $A=$2,000,000$, $A/L=1$, $q=0.50$, and $T_{\mathrm{rem}}=9/12=0.75$. The premium is

P_{\mathrm{re}} =\$240{,}000(0.50)(1)(0.75) =\$90{,}000.

The reinstatement makes $2,000,000 available for the later loss. That loss exceeds the attachment by $2,200,000 − $1,000,000 = $1,200,000, which is below the limit, so the second recovery is $1,200,000. Total recoveries are $2,000,000 + $1,200,000 = $3,200,000, within the maximum $4,000,000. The 50% term reduces the premium rate; it does not restore only half the limit.

Common Errors and Practical Checks

A reliable calculation follows the treaty’s terms in sequence: determine the recovery, identify how much limit that recovery uses, check whether a reinstatement is available, calculate the amount restored, and then apply the stated premium basis. This order keeps the coverage calculation distinct from the premium calculation.

Using gross loss as the restoration amount. Under the loss-paid convention, restore the reinsurance recovery, not the underlying gross loss. A loss partly below the attachment may produce a much smaller recovery than its gross amount.
Using elapsed time instead of remaining time. For an amount-and-time charge, a restoration with eight months remaining uses $8/12$, not $4/12$, if four months have elapsed in a twelve-month treaty period.
Confusing premium rate with limit restored. A 50% reinstatement premium rate affects the charge. It does not, by itself, mean that only half the paid recovery is restored.
Applying the reinstatement to the same occurrence limit incorrectly. Reinstatement does not make the treaty pay more than its stated limit on one occurrence. It makes used capacity available for other covered losses.
Assuming every clause uses the same convention. The timing of restoration, premium basis, and availability of partial reinstatements depend on the wording. Confirm these before calculating.

These distinctions matter when evaluating how much protection remains after a large loss and what additional premium is owed. A calculation that tracks only total losses, without tracking treaty recoveries and restored capacity, can overstate or understate both. The methods here are aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus.

Check Your Understanding

For each hypothetical contract, apply the stated attachment, limit, reinstatement allowance, and premium terms. No interest, mortality, or probability model is needed.

A treaty has a $3,000,000 limit excess of $1,000,000, an original premium of $150,000, and one 100% amount-only reinstatement. A covered gross loss is $3,200,000. Find the recovery and additional premium. (Answer: recovery $2,200,000; premium $110,000.)
An annual treaty has a $2,500,000 limit excess of $500,000, a $200,000 original premium, and one reinstatement priced at 100% pro rata as to amount and time. A loss occurs with exactly six of twelve months remaining; its gross amount is $2,000,000. Find the recovery and reinstatement premium. (Answer: recovery $1,500,000; premium $60,000.)
A treaty has a $2,000,000 limit excess of $1,000,000 and one reinstatement. Two covered gross losses are $4,500,000 and $3,000,000. Assuming the first recovery is restored before the second loss, find both recoveries and their total. (Answer: $2,000,000 and $2,000,000; total recovery $4,000,000.)
An annual treaty has a $4,000,000 limit excess of $1,000,000, an original premium of $400,000, two reinstatements, and a 100% amount-and-time premium rate. A $5,000,000 gross loss occurs with six months remaining, followed by a $3,000,000 gross loss with three months remaining. Find the recovery from each loss and the total additional premium. (Answer: recoveries $4,000,000 and $2,000,000; premiums $200,000 and $50,000; total premium $250,000.)
A treaty has a $5,000,000 limit excess of $2,000,000, an original premium of $600,000, and one reinstatement charged at 75% pro rata as to amount and time. A gross loss of $8,000,000 occurs with ten of twelve months remaining. After restoration, a later gross loss is $4,000,000. Find both recoveries, the reinstatement premium, and total recoveries. (Answer: recoveries $5,000,000 and $2,000,000; premium $375,000; total recovery $7,000,000.)

What You'll Learn