Why Reinstatement Premiums Need Careful Reading
An excess treaty can pay a covered recovery that uses some or all of its limit. A reinstatement provision restores some or all of that capacity, subject to the treaty’s wording and any limit on the number of reinstatements. If the reinstatement is paid, the cedent owes an additional premium. Calculating that premium requires identifying the reinstatement rate, the amount being restored, the applicable time period, and the contract’s proration basis.
The premium is not automatically a fixed percentage of the original premium. For example, a treaty may charge a stated percentage of the original premium, then prorate that charge for only part of the limit being restored and for only the unexpired portion of the treaty period. A free reinstatement has no additional reinstatement premium, even though capacity is restored.
Use the following notation for the calculations in this tutorial. Let \(P\) be the original treaty premium, \(L\) the treaty limit, \(r\) the paid reinstatement rate expressed as a decimal, and \(A\) the amount of capacity restored. If the treaty prorates by time, let \(T\) be the treaty period and \(t\) the time remaining when the reinstatement takes effect. All examples are hypothetical annual treaties. No investment income or discounting is included; the examples calculate the contractual premium, not its present value.
The Basic Premium Calculation
When the contract charges pro rata to the limit and pro rata to time, the premium for a paid reinstatement is the original premium multiplied by the reinstatement rate, the fraction of the limit restored, and the fraction of the treaty period remaining. The amount fraction is \(A/L\), and the time fraction is \(t/T\). Use this formula only when the treaty specifies these proration bases.
A rate of 100% means \(r=1.00\); a rate of 50% means \(r=0.50\). A stated rate is applied to the original premium, not to the amount of loss, unless the treaty explicitly defines a different premium basis. If the treaty does not prorate by time, omit the time factor. If it does not prorate by amount or limit, do not insert that factor without contractual support.
Some contracts describe the amount basis as pro rata to the limit restored; others refer to the amount of covered recovery or loss paid. Read the wording carefully. If the recovery paid and the capacity restored are the same amount, both bases produce the same amount fraction. If the contract defines them differently, calculate the factor from the basis it actually specifies.
Why the Factors Multiply
The formula follows from proportional charging. A full reinstatement at rate \(r\), with no time proration, costs \(Pr\). If only the fraction \(A/L\) of the limit is restored, a pro-rata-to-limit provision charges that same fraction of the full reinstatement premium. If only the fraction \(t/T\) of the treaty period remains, a pro-rata-to-time provision charges that same fraction of the amount-prorated premium. Applying both reductions gives the product of the factors.
This reasoning assumes the reinstatement premium scales linearly with restored capacity and with the remaining period, exactly as the stated pro-rata provisions require. It does not determine the premium where the treaty imposes a minimum charge, uses a different premium base, or says that a reinstatement is charged in full regardless of timing or amount. In those cases, follow the contract rather than imposing this formula.
Worked Calculations
Worked Example: Pro Rata to Limit and Time
A hypothetical annual excess treaty has a $4,000,000 limit and an original premium of $480,000. It permits one paid reinstatement at 100%, pro rata to the limit and to time. Three months into the treaty year, $1,000,000 of capacity is restored. Calculate the reinstatement premium.
Method. Apply the paid reinstatement rate to the original premium, then prorate by the fraction of limit restored and the fraction of the year remaining.
The limit fraction is \(A/L=\$1,000,000/\$4,000,000=0.25\). The time remaining is \(12-3=9\) months, so \(t/T=9/12=0.75\). The reinstatement rate is \(r=1.00\).
The premium is $90,000. Check: the full annual reinstatement premium would be $480,000; one-quarter of the limit costs $120,000 before time proration, and three-quarters of that is $90,000. A common mistake is to use an incorrect remaining-time fraction or to omit the time proration. Here, three months have elapsed, so nine months remain.
Worked Example: A Partial Reinstatement at 50%
A hypothetical annual treaty has a $5,000,000 limit and an original premium of $600,000. One paid reinstatement is available at 50%, pro rata to the limit and to time. When six months remain in the treaty year, $2,500,000 of capacity is restored. Find the additional premium.
Method. Identify each factor before multiplying: original premium \(P=\$600,000\), reinstatement rate \(r=0.50\), limit fraction \(A/L=\$2,500,000/\$5,000,000=0.50\), and time fraction \(t/T=6/12=0.50\).
The additional premium is $75,000. A useful check is to apply the factors in stages: 50% of $600,000 is $300,000; half of that for the partial limit is $150,000; half again for the six months remaining is $75,000. A common error is to stop after applying the reinstatement rate and limit fraction, reporting $150,000 without applying the time factor.
Worked Example: Free Capacity Followed by a Paid Reinstatement
A hypothetical annual treaty has a $3,000,000 limit and an original premium of $360,000. The contract provides one free reinstatement, followed by one paid reinstatement at 80%, pro rata to the amount of covered recovery and to time. A first covered recovery uses $1,200,000 of limit, which is restored under the free provision. Later, a second covered recovery of $900,000 is restored when four months remain. Calculate the premium due for the second restoration.
Method. Apply the free reinstatement to the first restoration: no additional premium is due for that restoration. The paid provision applies to the second restoration. Its amount factor is the covered recovery divided by the treaty limit, and its time factor is the four months remaining out of twelve.
The paid reinstatement rate is \(r=0.80\). The amount fraction is \(\$900,000/\$3,000,000=0.30\), and the time fraction is \(4/12=1/3\). Therefore:
The premium for the second restoration is $28,800; the first restoration has no reinstatement charge. Check: 80% of the original premium is $288,000; 30% of that is $86,400; one-third of $86,400 is $28,800. The common mistake is to charge the first restoration simply because it restores limit. The contract makes that reinstatement free, so only the later paid reinstatement is charged.
Reading the Wording and Checking the Result
Before calculating, extract the operative terms from the treaty. In particular, distinguish the reinstatement rate from the fraction of the limit restored. A 50% reinstatement rate does not mean that 50% of the limit has been restored; those are separate factors. Likewise, “one reinstatement at 100%” usually describes the rate for a full reinstatement, not an instruction to charge the full original premium for every partial restoration.
A free reinstatement has no additional reinstatement premium. Confirm the number and order of free and paid reinstatements allowed.
Use the original premium and the reinstatement rate as specified. Convert a percentage such as 75% to 0.75 for calculation.
For pro rata to limit, divide restored capacity by treaty limit. For pro rata as to amount, use the covered amount specified in the wording.
For a treaty-year basis, calculate time remaining divided by the full treaty period. Do not use elapsed time as the numerator.
Confirm that the restored amount and reinstatement count are permitted, and that each factor matches the contract terms.
A common exam trap is to use gross loss rather than the treaty recovery as the amount basis. For an excess layer, the treaty recovery may be limited by the layer attachment and limit; the gross loss can be larger than the amount paid by the treaty. If wording says pro rata to amount of recovery, use that recovery amount. Another trap is to apply time proration when the contract states a full reinstatement premium without time adjustment, or to omit time proration when it is expressly required.
A quick reasonableness check is to compare the answer with the full reinstatement charge \(Pr\). Under both amount and time proration, each factor is between zero and one, so the result cannot exceed \(Pr\). If a calculation does exceed that benchmark, check for a percentage entered as a whole number, an incorrect denominator, or an unintended extra factor.
Check Your Understanding
For each hypothetical treaty, identify the applicable basis and calculate the premium due.
- An annual treaty has a $6,000,000 limit and a $420,000 original premium. A paid reinstatement is at 100%, pro rata to limit and time. It restores $1,500,000 when four months remain. What is the premium? (Answer: $35,000.)
- An annual treaty has a $10,000,000 limit and an $800,000 original premium. A 75% paid reinstatement is pro rata to limit and time. It restores $3,000,000 when nine months remain. What is the premium? (Answer: $135,000.)
- A treaty with a $4,000,000 limit and a $500,000 original premium provides one free reinstatement followed by a paid reinstatement at 50%, pro rata to amount and time. After the free restoration, a covered recovery of $1,200,000 is restored under the paid provision with three months remaining in the annual period. What is the paid reinstatement premium? (Answer: $18,750.)
- A treaty has a $5,000,000 limit and a $250,000 original premium. A paid reinstatement at 80% is pro rata to limit, but not to time. It restores $2,000,000 with two months remaining. What premium is due? (Answer: $80,000.)
- A treaty has a $4,000,000 limit and a $400,000 original premium. Its wording says to prorate the paid reinstatement at 100% by recovery amount and time. A $600,000 recovery is paid, but the contract specifies that $1,000,000 of limit is restored. Six months remain in the annual period. What amount basis applies, and what premium is due? (Answer: Recovery amount; $30,000.)