Reinstatement Premiums and Expected Recovery Cost

Why Reinstatement Terms Matter

An excess-of-loss layer may pay more than its original limit over the treaty period if the contract allows capacity to be reinstated after recoveries. The additional capacity can protect the cedent from later losses, but it may also generate additional premium. Thus, assessing the economics of a layer requires considering both the recovery and the cost of restoring capacity.

This tutorial is aligned with the CAS Exam 9 (Risk Management for Actuaries) syllabus. All companies, contracts, and figures below are hypothetical. Reinstatement wording varies, so calculations must follow the stated contract terms rather than assume that every treaty uses the same premium rate or time basis.

Definition: Consider an excess-of-loss layer attaching at $A$, with per-occurrence limit $L$. For a covered loss $X_i$, the layer loss is $Z_i=\min((X_i-A)^+,L)$, where $(x)^+=\max(x,0)$. A reinstatement restores some or all of capacity consumed by a recovery. A treaty with $n$ full reinstatements can provide at most $(n+1)L$ in total recoveries, assuming no other annual aggregate limit or exclusion restricts payment.

The original limit is the amount available for a covered occurrence, not necessarily the maximum the treaty can pay over the whole year. Under the common arrangement considered here, capacity is restored after a recovery, up to the reinstatement allowance. The cedent pays reinstatement premium on the amount restored. Some contracts instead specify different triggers, rates, or limits; their explicit terms control.

Calculating a Reinstatement Premium

Let $P$ be the original premium for the layer. Suppose the contract charges a fraction $\alpha$ of the original premium for a full limit reinstatement. If $q_j$ is the amount reinstated after recovery $j$, and $\tau_j$ is the applicable time factor, the premium for that restoration is

\text{RP}_j=\alpha P\frac{q_j}{L}\tau_j.

Here $q_j/L$ is the fraction of a full limit restored. For a contract pro rata as to amount but not time, set $\tau_j=1$. For a contract that also prorates by time, use the factor specified in its wording; one possible convention is the fraction of the treaty period remaining when the restoration takes effect. A 100% reinstatement rate means $\alpha=1$, not that the cedent pays a second full premium regardless of the amount restored.

With one or more reinstatements, the total amount of capacity restored cannot exceed $nL$. Under the proportional restoration convention, if $R$ denotes total recoveries actually paid, the total amount restored is $Q=\min(R,nL)$, provided restorations are automatic and the treaty has no other restriction. When every restoration has the same rate and time factor, the total reinstatement premium is

\text{RP}=\alpha P\tau\frac{\min(R,nL)}{L}.

The cap matters: recoveries beyond the reinstatement allowance may still be possible only up to the remaining original capacity, and no further restoration is available after the reinstatement limit is exhausted. More generally, calculate each $q_j$ and $\tau_j$ from the contract and sum the individual premiums.

Worked Example: A Partial Restoration After Each Recovery

A hypothetical treaty has attachment $A=\$1.0$ million, limit $L=\$2.0$ million, one reinstatement, and original premium $P=\$300{,}000$. It charges 100% of the original premium pro rata as to amount, with no time proration. Two covered occurrences have losses of $\$2.2$ million and $\$1.5$ million. Calculate total recoveries, reinstatement premium, and the cedent’s net cost before expenses.

Method. First calculate each occurrence’s layer loss. Then calculate the capacity restored and its premium. Because there is one reinstatement, total restorations are capped at $1\times \$2.0$ million.

For the first occurrence, the loss above attachment is $\$2.2-\$1.0=\$1.2$ million, below the $\$2.0$ million limit. Thus $Z_1=\$1.2$ million. For the second, the loss above attachment is $\$1.5-\$1.0=\$0.5$ million, so $Z_2=\$0.5$ million. Total recovery is $R=\$1.2+\$0.5=\$1.7$ million.

The reinstatement amount is $Q=\min(\$1.7\text{ million},\$2.0\text{ million})=\$1.7$ million. The premium per full reinstated limit is $\$300{,}000$, so

\text{RP}=\$300{,}000\frac{\$1.7\text{ million}}{\$2.0\text{ million}} =\$255{,}000.

The cedent’s net cost before expenses is $\$300{,}000+\$255{,}000-\$1{,}700{,}000=-\$1{,}145{,}000$. A negative net cost means recoveries exceed the two premiums by $\$1{,}145{,}000$. The total recovery is below the treaty’s maximum capacity of $\$4.0$ million, so the result is feasible. Common mistake avoided: charging no reinstatement premium because total recovery is less than the original $\$2.0$ million limit. Under the stated terms, restorations follow recoveries; a paid recovery can trigger a restoration even before total recoveries exceed $L$.

Expected Reinstatement Premium

When recoveries are uncertain, reinstatement premium is also uncertain. For the proportional terms above, with a common $\alpha$ and $\tau$, taking expectations gives

E[\text{RP}]=\frac{\alpha P\tau}{L}E[\min(R,nL)].

This follows because $\alpha P\tau/L$ is constant and expectation is linear: $E[c\,g(R)]=cE[g(R)]$ for constant $c$. The cap remains inside the expectation because recoveries above $nL$ do not generate additional reinstated capacity under these terms. In general, one cannot replace $E[\min(R,nL)]$ with $\min(E[R],nL)$; the cap is applied to each possible outcome before averaging.

For a finite set of outcomes $r_k$, with probabilities $p_k$, calculate

E[R]=\sum_k p_k r_k,\qquad E[\text{RP}]=\frac{\alpha P\tau}{L}\sum_k p_k\min(r_k,nL).

This is a direct way to keep the recovery and reinstatement calculations consistent across scenarios. If individual restoration amounts have different time factors, calculate and probability-weight the scenario’s sum of restoration premiums instead.

Worked Example: Expected Cost With a Capped Reinstatement Allowance

A hypothetical layer has $L=\$1.0$ million, one reinstatement, and original premium $P=\$240{,}000$. The contract charges 50% of the original premium pro rata as to amount, with no time proration. The total recovery $R$ is $\$0$ million with probability 0.5, $\$0.6$ million with probability 0.3, and $\$1.4$ million with probability 0.2. Find expected recovery, expected reinstatement premium, and expected net cost before expenses.

Method. Evaluate recovery and reinstatement amount in each outcome before taking expectations. The reinstatement amount in an outcome is $\min(R,\$1.0\text{ million})$, since the treaty permits one limit of reinstatement.

Expected recovery is $0.5(0)+0.3(\$0.6\text{ million})+0.2(\$1.4\text{ million})=\$0.46$ million. The reinstatement amounts in the three outcomes are $\$0$, $\$0.6$ million, and $\$1.0$ million. Their expected value is $0.5(0)+0.3(\$0.6\text{ million})+0.2(\$1.0\text{ million})=\$0.38$ million.

A full limit reinstatement costs $0.5(\$240{,}000)=\$120{,}000$. Therefore expected reinstatement premium is $\$120{,}000(\$0.38/\$1.0)=\$45{,}600$. The expected net cost is

\$240{,}000+\$45{,}600-\$460{,}000=-\$174{,}400.

The negative value means the expected recovery exceeds the base premium plus expected reinstatement premium by $\$174{,}400$, before expenses and other adjustments. The maximum possible recovery is $\$2.0$ million, and the largest stated outcome, $\$1.4$ million, is within that limit. Common mistake avoided: applying the 50% rate to every dollar of recovery without first capping the reinstatement amount. In the $\$1.4$ million outcome, only $\$1.0$ million of reinstatement is available.

Time Proration and Contract-Specific Terms

A time-pro-rated premium can differ materially from an amount-only premium. If restorations occur at different times, each restoration can have its own $\tau_j$. For example, when the contract explicitly defines the factor as the remaining fraction of a one-year treaty period, a restoration with six months left has $\tau_j=6/12$. Do not assume that convention unless the contract says so.

Worked Example: Applying Amount and Time Proration

A hypothetical annual layer has $L=\$3.0$ million, two reinstatements, and original premium $P=\$360{,}000$. It charges 100% of the original premium, pro rata by amount and by the fraction of the year remaining. Three occurrences produce layer payments of $\$1.2$ million with nine months remaining, $\$0.9$ million with three months remaining, and $\$0.6$ million with two months remaining. Calculate total reinstatement premium and net cost before expenses.

Method. Apply the premium formula to each restoration separately. The premium per full reinstated limit is $\$360{,}000$; per $\$1.0$ million restored, it is $\$360{,}000/\$3.0\text{ million}=\$120{,}000$.

The first restoration is $\$1.2$ million with time factor $9/12$, so its premium is $\$120{,}000(1.2)(9/12)=\$108{,}000$. The second is $\$0.9$ million with factor $3/12$, giving $\$120{,}000(0.9)(3/12)=\$27{,}000$. The third is $\$0.6$ million with factor $2/12$, giving $\$120{,}000(0.6)(2/12)=\$12{,}000$.

Total restorations are $\$1.2+\$0.9+\$0.6=\$2.7$ million, below the reinstatement allowance of $2L=\$6.0$ million. Total recovery is also $\$2.7$ million. Total reinstatement premium is $\$108{,}000+\$27{,}000+\$12{,}000=\$147{,}000$. The cedent’s net cost before expenses is $\$360{,}000+\$147{,}000-\$2{,}700{,}000=-\$2{,}193{,}000$. The total recovery is within the maximum treaty capacity of $\$9.0$ million. Common mistake avoided: using one time factor for all restorations when the contract’s remaining-term factor changes with the restoration date.

Interpreting Expected Economics and Avoiding Traps

For the cedent, a useful expected-value comparison before expenses is

\text{Expected net cost}=P+E[\text{RP}]-E[R].

This is a cash-flow summary, not a complete profitability or value measure. It does not by itself allow for payment timing, discounting, expenses, taxes, counterparty default, or the value of reducing volatility. For the reinsurer, the same components have the opposite cash-flow signs: premium income is received and recoveries are paid.

Check the contract and the calculation in a consistent order:

Identify whether $L$ is a per-occurrence limit, an annual aggregate limit, or both.
Calculate the layer payment from the loss above attachment, capped at the per-occurrence limit.
Determine whether reinstatement is automatic or subject to a stated condition, and identify the maximum number or amount allowed.
Apply the stated premium percentage, amount basis, and time factor to the amount actually reinstated.
Calculate expected recovery and expected reinstatement premium from the same set of outcomes.

A frequent exam trap is to treat “one reinstatement” as an extra premium equal to the whole original premium in every outcome. That is correct only when the full limit is restored and the contract’s rate and time terms produce that charge. A partial restoration generally produces a partial premium under pro-rata amount wording. Another trap is to apply the reinstatement rate to the gross loss rather than to the layer amount restored. The contract’s definition of reinstated capacity determines the premium base.

Key takeaway: Reinstatement premium is tied to restored capacity, not automatically to gross losses or to recoveries exceeding the original limit. Calculate recoveries under the layer and aggregate terms, apply the reinstatement allowance and stated premium factors, and then combine expected premium with expected recovery.

Check Your Understanding

For each problem, use the contract terms stated and calculate amounts in dollars unless millions are shown.

A hypothetical layer has limit $\$500{,}000$, original premium $\$80{,}000$, and one reinstatement. The reinstatement rate is 50% of original premium, pro rata as to amount, with no time proration. If $\$200{,}000$ is restored, what is the reinstatement premium? (Answer: $\$16{,}000$)
A layer attaches at $\$1.0$ million and has limit $\$1.5$ million, one reinstatement, and original premium $\$240{,}000$. Three covered losses are $\$1.8$, $\$2.2$, and $\$1.4$ million. The rate is 100% pro rata as to amount, with no time proration. Find total recovery and reinstatement premium. (Answer: total recovery $\$2.4$ million; reinstatement premium $\$240{,}000$)
A layer has $L=\$1.0$ million, one reinstatement, and $P=\$150{,}000$. It charges 100% pro rata as to amount, with no time proration. Recovery is $\$0$ million with probability 0.4, $\$0.8$ million with probability 0.4, and $\$1.6$ million with probability 0.2. Find expected recovery and expected reinstatement premium. (Answer: expected recovery $\$640{,}000$; expected reinstatement premium $\$78{,}000$)
A hypothetical layer has $L=\$1.0$ million, one reinstatement, and $P=\$200{,}000$. The reinstatement rate is 75% pro rata as to amount, with no time proration. Before applying treaty capacity, annual layer demand is $\$0.7$ million with probability 0.5, $\$1.4$ million with probability 0.3, and $\$2.4$ million with probability 0.2. Find expected actual recovery, expected reinstatement premium, and expected net cost before expenses. (Answer: expected recovery $\$1{,}170{,}000$; expected reinstatement premium $\$127{,}500$; expected net cost $-\$842{,}500$)
A layer has $L=\$2.0$ million, one reinstatement, and $P=\$300{,}000$. Under the stated contract, the rate is 100%, pro rata by amount and by the fraction of the year remaining. Two restorations are $\$0.5$ million with six months remaining and $\$0.8$ million with two months remaining. Calculate total reinstatement premium and verify the reinstatement allowance is not exceeded. (Answer: reinstatement premium $\$57{,}500$; total restored $\$1.3$ million, below the $\$2.0$ million allowance)

What You'll Learn