Expected Reinstatement Premium Under Multiple Events

Why Multiple Events Require a Sequence

A reinstatement premium depends not just on how much loss occurs, but on how treaty capacity is used over time. An early event may erode the original limit and trigger a reinstatement; a later event then uses the restored capacity and may trigger another premium. The treaty's reinstatement allowance can run out, even if further losses occur.

The calculation therefore has two parts: first determine the recovery and reinstated amount at each event, in order; then calculate the premium for each reinstatement and combine the results across possible contract-period outcomes. This tutorial is aligned with the CAS Exam 8 (Advanced Ratemaking) syllabus.

Definition: Let $L$ be the original limit available for one covered event, $P_0$ the original treaty premium, and $n$ the number of reinstatements permitted. A reinstatement restores some or all of the limit used by a covered loss, subject to the treaty terms. Let $q$ be the paid proportion of reinstatement premium: $q=1$ for a 100% paid reinstatement, $q=0$ for a free reinstatement, and $0<q<1$ for a partially paid reinstatement.

For the calculations here, assume the event loss amounts supplied are eligible demands for this layer, before applying the treaty's remaining aggregate capacity. No event can recover more than $L$. A reinstatement becomes available for the next event after a covered recovery. Each reinstatement can restore capacity up to $L$, and partial restoration is allowed. These are explicit modeling conventions; the actual wording of a treaty determines whether and when reinstatement occurs.

If the treaty also charges reinstatement premium pro rata for the time remaining in the contract, let $\tau_j$ be the fraction of the contract period remaining when the $j$th reinstatement is triggered. If time pro-rating is not specified, use $\tau_j=1$. Under these conventions, the premium for restoring amount $a_j$ is:

R_j=P_0q\frac{a_j}{L}\tau_j

The formula assumes the premium rate per unit of reinstated limit is the original premium divided by the original limit. It does not include discounting or other charges; include those only if the contract terms or problem statement require them.

Track Capacity Before Pricing It

Let $C_j$ be the capacity available immediately before event $j$, and let $x_j$ be that event's eligible layer demand. The recovery $c_j$ is limited by the demand, the per-event layer limit, and the capacity then available. Let $U_j$ be the remaining reinstatement allowance before that event; it starts at $nL$. After a covered recovery, the amount restored is limited both by the recovery and by that remaining allowance:

c_j=\min(x_j,L,C_j), \qquad a_j=\min(c_j,U_j)

The next event's available capacity is $C_{j+1}=C_j-c_j+a_j$, and the remaining reinstatement allowance is $U_{j+1}=U_j-a_j$. Initially, $C_1=L$ and $U_1=nL$. A treaty may specify different mechanics, so these recursions apply only when the stated restoration convention matches its terms.

Read the event path.
Use the losses and their order for each scenario; do not treat losses in a sequence as one undifferentiated total.

Calculate each recovery.
Apply the per-event limit and capacity available at that point in the sequence.

Apply reinstatement limits.
For each recovery, determine how much capacity can be restored and how much reinstatement allowance remains.

Price each restoration.
Apply the paid proportion and, when required by the terms, the time factor.

Weight complete scenarios.
Multiply each scenario's total reinstatement premium by its probability and sum.

Worked Scenarios

Worked Example: One Event With Two Possible Loss Amounts

A hypothetical excess treaty has $L=\$80{,}000$, original premium $P_0=\$24{,}000$, and one 100% paid reinstatement. Time pro-rating does not apply. There is one event: with probability 0.30, its eligible layer demand is $\$40{,}000$; with probability 0.70, it is $\$80{,}000$. Find the expected reinstatement premium.

Method. In each outcome, the event's recovery is its eligible demand, since both amounts are at most the original limit. The reinstated amount is the recovery, up to the one reinstatement allowance of $\$80{,}000$.

For the $\$40{,}000$ outcome, the fraction of the limit reinstated is $\$40{,}000/\$80{,}000=0.5$. Its premium is $ \$24{,}000(1)(0.5)=\$12{,}000$.

For the $\$80{,}000$ outcome, the fraction reinstated is $\$80{,}000/\$80{,}000=1$. Its premium is $ \$24{,}000(1)(1)=\$24{,}000$.

Weight the two scenario premiums by their probabilities:

E[R]=0.30(\$12{,}000)+0.70(\$24{,}000) =\$3{,}600+\$16{,}800 =\$20{,}400

The expected reinstatement premium is $\$20{,}400$. The check is that the answer lies between the two possible scenario premiums. A common mistake is to report $\$21{,}600$; the stated probabilities and scenario premiums instead give $\$20{,}400$.

Worked Example: Two Events With Limited Reinstatement

A hypothetical treaty has $L=\$100{,}000$, $P_0=\$25{,}000$, and one 100% paid reinstatement. There is no time pro-rating. The possible two-event paths are: (A) demands of $\$40{,}000$ then $\$30{,}000$, probability 0.50; (B) $\$80{,}000$ then $\$50{,}000$, probability 0.30; (C) $\$100{,}000$ then $\$40{,}000$, probability 0.20. Calculate the expected reinstatement premium.

Method. Start each path with $\$100{,}000$ of capacity and $\$100{,}000$ of reinstatement allowance. Reinstatement after the first event is available to the second event. Track the allowance separately: capacity restored does not create a new reinstatement allowance.

Path A. The first recovery is $\min(\$40{,}000,\$100{,}000,\$100{,}000)=\$40{,}000$. Restore $\$40{,}000$, leaving $\$60{,}000$ of reinstatement allowance and restoring available capacity to $\$100{,}000$. The second recovery is $\$30{,}000$; restore $\$30{,}000$. Total amount reinstated is $\$70{,}000$, so the premium is $\$25{,}000(\$70{,}000/\$100{,}000)=\$17{,}500$.

Path B. The first recovery is $\$80{,}000$; restore $\$80{,}000$, leaving $\$20{,}000$ of reinstatement allowance. Capacity is again $\$100{,}000$. The second recovery is $\$50{,}000$, but only $\$20{,}000$ can now be restored. Total reinstated is $\$100{,}000$, so the premium is $\$25{,}000$.

Path C. The first recovery is $\$100{,}000$; restore the full $\$100{,}000$ allowance. The second event recovers $\$40{,}000$ from available capacity, but no reinstatement allowance remains. Total reinstated is $\$100{,}000$, and the premium is $\$25{,}000$.

The probabilities sum to $0.50+0.30+0.20=1.00$. Therefore:

E[R]=0.50(\$17{,}500)+0.30(\$25{,}000)+0.20(\$25{,}000) =\$8{,}750+\$7{,}500+\$5{,}000 =\$21{,}250

The expected premium is $\$21{,}250$. The mistake to avoid is restoring more than the one permitted limit over the whole path. A later recovery can still be covered by already-restored capacity, but cannot trigger an additional restoration once the reinstatement allowance is exhausted.

Worked Example: Partial Paid Reinstatements With Time Pro-Rating

A hypothetical treaty has $L=\$200{,}000$, $P_0=\$40{,}000$, and two reinstatements, each 50% paid and pro rata for the remaining contract period. In path A, losses of $\$100{,}000$ occur at one-quarter of the contract year and $\$150{,}000$ at three-quarters of the year; the path probability is 0.40. In path B, losses of $\$200{,}000$ occur at one-half of the year and $\$200{,}000$ at three-quarters of the year; the path probability is 0.60. Find the expected reinstatement premium. Assume no interest discounting.

Method. For each event, use $\tau=1-\text{fraction of contract year elapsed}$. Thus the remaining-time factors at one-quarter, one-half, and three-quarters of the year are 0.75, 0.50, and 0.25. The allowance for two reinstatements is $2L=\$400{,}000$.

Path A. The first recovery is $\$100{,}000$, so $\$100{,}000$ is restored. Its premium is $\$40{,}000(0.50)(\$100{,}000/\$200{,}000)(0.75)=\$7{,}500$. Available capacity is then $\$200{,}000$, and $\$100{,}000$ of reinstatement allowance remains. The second recovery is $\$150{,}000$, but only $\$100{,}000$ can be restored. Its premium is $\$40{,}000(0.50)(\$100{,}000/\$200{,}000)(0.25)=\$2{,}500$. Path A's total is $\$7{,}500+\$2{,}500=\$10{,}000$.

Path B. The first recovery is $\$200{,}000$, restoring $\$200{,}000$ and using one reinstatement. Its premium is $\$40{,}000(0.50)(1)(0.50)=\$10{,}000$. The second recovery is also $\$200{,}000$; the second permitted reinstatement can restore the full amount. Its premium is $\$40{,}000(0.50)(1)(0.25)=\$5{,}000$. Path B's total is $\$15{,}000$.

The expected premium is $0.40(\$10{,}000)+0.60(\$15{,}000)=\$4{,}000+\$9{,}000=$ $\$13{,}000$. The check is that path B has the larger premium: it restores more limit, and its second reinstatement is still chargeable under the stated terms, though at a shorter remaining-time factor.

Why Scenario Weighting Works—and When It Does Not

Let $R_s$ be the total reinstatement premium in scenario $s$, and let $p_s$ be that scenario's probability. The expected premium is:

E[R]=\sum_s p_sR_s, \qquad \sum_s p_s=1

This follows directly from the definition of the expectation of a discrete random variable: multiply each possible value by the probability of that value and add. Here each value is the total premium for a complete event path, not the premium for an isolated event. If outcomes are described by conditional probabilities, first form the joint probability of each path, then weight that path's total premium.

In particular, do not assume event losses are independent unless the problem states that they are or provides enough information to justify it. Marginal probabilities for individual events do not, by themselves, determine the probabilities of all multi-event paths. Nor is it generally safe to calculate a premium from expected aggregate losses: limits, reinstatement caps, timing, and paid percentages can make the premium a nonlinear function of the path.

Common Errors and Practical Checks

Ignoring event order. A recovery changes the capacity available to later events. Apply the treaty terms sequentially.
Reinstating more than permitted. Track remaining reinstatement allowance separately from available capacity.
Charging on the full original limit every time. For a partial reinstatement, the premium is proportional to the amount restored.
Using the wrong time factor. If the terms say pro rata as to time, use the fraction of the contract period remaining at the trigger date, not the elapsed fraction.
Weighting event premiums instead of path totals. First add the premiums within each scenario, then multiply by that scenario's joint probability.

Key takeaway: Expected reinstatement premium is found by pricing the capacity actually restored along each complete event path, respecting reinstatement limits and any paid or time-pro-rating terms, and then probability-weighting those path totals.

Check Your Understanding

For each problem, track the reinstatement allowance as well as the available capacity. Unless a problem says otherwise, there is no time pro-rating or interest discounting.

A hypothetical treaty has $L=\$50{,}000$, $P_0=\$15{,}000$, and one 100% paid reinstatement. A single event has eligible demand $\$20{,}000$ with probability 0.25 or $\$50{,}000$ with probability 0.75. Find the expected reinstatement premium. (Answer: $\$12{,}750$)
A treaty has $L=\$100{,}000$, $P_0=\$30{,}000$, and one 50% paid reinstatement. Two events produce eligible demands of $\$40{,}000$ and $\$30{,}000$. Find the reinstatement premium for the path. (Answer: $\$10{,}500$)
A treaty has $L=\$80{,}000$, $P_0=\$16{,}000$, and one 100% paid reinstatement. With probability 0.40, two successive demands are $\$30{,}000$ and $\$30{,}000$; with probability 0.60, they are $\$80{,}000$ and $\$80{,}000$. Find the expected premium. (Answer: $\$14{,}400$)
A treaty has $L=\$100{,}000$, $P_0=\$20{,}000$, and one 50% paid reinstatement pro rata for the remaining time. A $\$50{,}000$ recovery triggers reinstatement at one-quarter of the contract year. Find its premium. (Answer: $\$3{,}750$)
A treaty has $L=\$60{,}000$, $P_0=\$12{,}000$, and one 100% paid reinstatement. The two-event path is $(\$30{,}000,\$20{,}000)$ with probability 0.50 or $(\$60{,}000,\$60{,}000)$ with probability 0.50. Find the expected premium. (Answer: $\$11{,}000$)

What You'll Learn