A Multi-Treaty Reinsurance Allocation Workshop

Why Treaty Order Matters

A portfolio can be protected by several reinsurance arrangements at once, but the final allocation depends on what amount each treaty covers. A quota-share treaty divides each covered premium and loss proportionally. A per-risk excess-of-loss treaty then responds to covered losses on individual risks, while stop-loss protection responds to an aggregate amount over a period. The order specified in the contract determines which losses reach each layer.

In this workshop, assume that quota share is applied first to each gross claim, per-risk excess-of-loss is applied to each claim remaining with the insurer, and stop loss is applied to the aggregate remaining after those per-risk recoveries. Treaties have no reinstatements, commissions, exclusions, or other adjustments unless stated. Premiums for the non-proportional treaties are given as quoted annual amounts; they are not inferred from claim recoveries.

Definition: A quota-share treaty with cession proportion $c$ cedes proportion $c$ of each covered premium and loss. A per-risk layer with attachment $A$ and limit $L$, described as $L$ xs $A$, pays $\min(\max(z-A,0),L)$ on a covered loss $z$. A stop-loss layer with aggregate attachment $B$ and limit $M$ pays $\min(\max(S-B,0),M)$ on the covered aggregate $S$.

The word limit means the maximum amount the reinsurer pays, not the maximum loss that can be exposed to the layer. For example, a $25{,}000$ xs $25{,}000$ per-risk layer begins paying above $25{,}000$, and can pay at most $25{,}000$ per risk. Its payment on a post-cession claim of $40{,}000$ is $15{,}000$, not $25{,}000$.

A Formula for the Allocation

Let $X_j$ be gross claim $j$, and let $c$ be the quota-share cession proportion. After quota share, the insurer’s share of that claim is $Z_j=(1-c)X_j$. If the per-risk layer has attachment $A$ and limit $L$, the per-risk recovery is

R_j^{\mathrm{risk}}=\min\bigl(\max(Z_j-A,0),L\bigr).

The amount remaining on risk $j$ after that treaty is $W_j=Z_j-R_j^{\mathrm{risk}}$. For $n$ claims, the aggregate entering the stop-loss treaty is $S=\sum_{j=1}^{n}W_j$. With stop-loss attachment $B$ and limit $M$, its recovery and the final retained aggregate are

R^{\mathrm{SL}}=\min\bigl(\max(S-B,0),M\bigr), \qquad R^{\mathrm{final}}=S-R^{\mathrm{SL}}.

Derivation: A per-risk layer pays nothing while $z\le A$, because the claim has not reached the attachment. Once $z>A$, the excess above attachment is $z-A$. The reinsurer pays that excess only up to the layer limit $L$, giving $\min(z-A,L)$ when $z>A$. Combining the zero-payment and positive-payment cases gives $\min(\max(z-A,0),L)$. The same reasoning applies to the stop-loss layer, replacing an individual claim with the covered aggregate.

For premiums, if gross premium is $G$, the quota-share premium cession is $cG$, assuming the same proportional cession applies to the entire covered premium. If the quoted annual premiums for the per-risk and stop-loss treaties are $Q_{\mathrm{risk}}$ and $Q_{\mathrm{SL}}$, total ceded premium is $cG+Q_{\mathrm{risk}}+Q_{\mathrm{SL}}$, and net premium retained is $G$ minus that total. A simple annual net result before expenses is net premium minus final retained claims.

Worked Portfolio Allocations

Worked Example: Four Claims Through Three Treaties

A hypothetical insurer has annual gross premium of $\$500{,}000$ and four gross claims of $\$50{,}000$, $\$120{,}000$, $\$300{,}000$, and $\$100{,}000$. It cedes 40% under quota share. The remaining claims are covered by a $\$25{,}000$ xs $\$25{,}000$ per-risk layer, followed by a stop-loss layer of $\$60{,}000$ xs $\$180{,}000$. The annual premiums for those two treaties are $\$18{,}000$ and $\$10{,}000$, respectively. Find the recoveries, final retained claims, premium allocation, and annual net result before expenses.

Method and quota-share allocation. Apply the 40% cession to each gross claim. The insurer retains 60%, so the post-quota-share claims are $0.60(50{,}000)=\$30{,}000$, $0.60(120{,}000)=\$72{,}000$, $0.60(300{,}000)=\$180{,}000$, and $0.60(100{,}000)=\$60{,}000$. Gross claims total $\$570{,}000$; quota share recovers $0.40(570{,}000)=\$228{,}000$.

Per-risk recoveries. Each claim enters a layer attaching at $\$25{,}000$, with a maximum recovery of $\$25{,}000$. The recoveries are $\min(30{,}000-25{,}000,25{,}000)=\$5{,}000$, $\min(72{,}000-25{,}000,25{,}000)=\$25{,}000$, $\min(180{,}000-25{,}000,25{,}000)=\$25{,}000$, and $\min(60{,}000-25{,}000,25{,}000)=\$25{,}000$. Total per-risk recovery is $5{,}000+25{,}000+25{,}000+25{,}000=\$80{,}000$.

The amounts remaining by risk are $30{,}000-5{,}000=\$25{,}000$, $72{,}000-25{,}000=\$47{,}000$, $180{,}000-25{,}000=\$155{,}000$, and $60{,}000-25{,}000=\$35{,}000$. Their aggregate is $25{,}000+47{,}000+155{,}000+35{,}000=\$262{,}000$.

Stop-loss recovery and final claims. The aggregate exceeds the $\$180{,}000$ attachment by $262{,}000-180{,}000=\$82{,}000$. The limit is $\$60{,}000$, so the stop-loss recovery is $\min(82{,}000,60{,}000)=\$60{,}000$. The insurer’s final retained claims are $262{,}000-60{,}000=\$202{,}000$.

Premiums and result. The quota-share premium cession is $0.40(500{,}000)=\$200{,}000$. Total ceded premium is $200{,}000+18{,}000+10{,}000=\$228{,}000$, so net premium is $500{,}000-228{,}000=\$272{,}000$. The annual net result before expenses is $272{,}000-202{,}000=\$70{,}000$.

Check and common mistake avoided. The claim allocation reconciles: $228{,}000+80{,}000+60{,}000+202{,}000=\$570{,}000$. A common error is to treat the per-risk layer as paying its full $\$25{,}000$ limit on every claim above zero. The first post-quota-share claim is only $\$5{,}000$ above attachment, so its recovery is $\$5{,}000$; using $\$25{,}000$ would overstate recovery and understate the remaining aggregate.

Worked Example: A Per-Risk Limit Followed by an Aggregate Cap

A hypothetical portfolio has gross claims of $\$80{,}000$, $\$160{,}000$, and $\$400{,}000$, and annual gross premium of $\$700{,}000$. The insurer cedes 25% under quota share. It then buys a $\$30{,}000$ xs $\$30{,}000$ per-risk layer and a $\$100{,}000$ xs $\$250{,}000$ stop-loss layer. The annual treaty premiums are $\$24{,}000$ for per-risk cover and $\$15{,}000$ for stop loss. Calculate the final allocation and annual net result.

Method and quota share. The gross claims total $80{,}000+160{,}000+400{,}000=\$640{,}000$. Quota share recovers $0.25(640{,}000)=\$160{,}000$, leaving 75% of each claim: $\$60{,}000$, $\$120{,}000$, and $\$300{,}000$.

Per-risk layer. The layer pays the excess above $\$30{,}000$, up to $\$30{,}000$ per claim. Its recoveries are $\min(60{,}000-30{,}000,30{,}000)=\$30{,}000$, $\min(120{,}000-30{,}000,30{,}000)=\$30{,}000$, and $\min(300{,}000-30{,}000,30{,}000)=\$30{,}000$. Total recovery is $\$90{,}000$. The residual claims are $\$30{,}000$, $\$90{,}000$, and $\$270{,}000$, totaling $\$390{,}000$.

Stop loss. The amount above the $\$250{,}000$ aggregate attachment is $390{,}000-250{,}000=\$140{,}000$. Since the stop-loss limit is $\$100{,}000$, recovery is $\min(140{,}000,100{,}000)=\$100{,}000$, and final retained claims are $390{,}000-100{,}000=\$290{,}000$.

Premiums and check. The quota-share premium cession is $0.25(700{,}000)=\$175{,}000$. Total ceded premium is $175{,}000+24{,}000+15{,}000=\$214{,}000$, leaving net premium of $700{,}000-214{,}000=\$486{,}000$. The net result before expenses is $486{,}000-290{,}000=\$196{,}000$. Claim recoveries reconcile: $160{,}000+90{,}000+100{,}000+290{,}000=\$640{,}000$. The key mistake avoided is treating the $\$140{,}000$ above attachment as the recovery; the stop-loss limit caps the actual payment at $\$100{,}000$.

Worked Example: Checking Attachments and Aggregate Entry

A hypothetical insurer has gross claims of $\$60{,}000$, $\$140{,}000$, and $\$260{,}000$. It cedes 25% of claims and premiums under quota share. Per-risk cover is $\$40{,}000$ xs $\$40{,}000$, and stop-loss cover is $\$100{,}000$ xs $\$180{,}000$. Gross annual premium is $\$300{,}000$, and the quoted annual premiums are $\$14{,}000$ for per-risk cover and $\$9{,}000$ for stop loss. Find all recoveries, retained claims, and the net result.

Quota-share step. Gross claims total $60{,}000+140{,}000+260{,}000=\$460{,}000$. Quota-share recovery is $0.25(460{,}000)=\$115{,}000$. The insurer’s 75% shares are $\$45{,}000$, $\$105{,}000$, and $\$195{,}000$.

Per-risk step. The first recovery is $\min(45{,}000-40{,}000,40{,}000)=\$5{,}000$. The second is $\min(105{,}000-40{,}000,40{,}000)=\$40{,}000$, and the third is $\min(195{,}000-40{,}000,40{,}000)=\$40{,}000$. Total per-risk recovery is $5{,}000+40{,}000+40{,}000=\$85{,}000$. The residual claims are $\$40{,}000$, $\$65{,}000$, and $\$155{,}000$, so the stop-loss aggregate is $\$260{,}000$.

Stop-loss and premium steps. The aggregate is $260{,}000-180{,}000=\$80{,}000$ above attachment, below the $\$100{,}000$ limit, so the stop-loss recovery is $\$80{,}000$. Final retained claims are $260{,}000-80{,}000=\$180{,}000$. The quota-share premium cession is $0.25(300{,}000)=\$75{,}000$. Total ceded premium is $75{,}000+14{,}000+9{,}000=\$98{,}000$, giving net premium $300{,}000-98{,}000=\$202{,}000$ and net result $202{,}000-180{,}000=\$22{,}000$.

Check and common mistake avoided. Recoveries plus retained claims equal $115{,}000+85{,}000+80{,}000+180{,}000=\$460{,}000$. A common error is to apply stop loss to the aggregate after quota share but before per-risk recoveries. Here the stipulated stop-loss entry is $\$260{,}000$, the aggregate after per-risk recoveries, not the $\$345{,}000$ aggregate immediately after quota share.

Reconciliation and Exam Traps

A multi-treaty calculation is most reliable when each stage is recorded separately. The same claim cannot be ceded twice for the same dollar: quota share removes its agreed proportion first, and later covers act only on the remaining insurer share. Similarly, the stop-loss recovery is calculated from the aggregate specified by the treaty sequence, not from gross claims by default.

Start with gross amounts.
Record the gross premium and each gross claim. Sum claims independently as a check.

Apply proportional cessions.
Calculate quota-share premium and claim cessions, then retain the unc ceded proportion of each claim.

Apply per-risk layers claim by claim.
Use the attachment and limit on each post-quota-share claim. Sum recoveries and subtract them from the claims entering the layer.

Apply aggregate protection.
Sum the post-per-risk residual claims, calculate the amount above the stop-loss attachment, and apply the aggregate limit.

Reconcile and report.
Check that quota-share recovery, per-risk recovery, stop-loss recovery, and final retained claims equal gross claims. Separately reconcile premiums and calculate the requested annual result.

The two reconciliations answer different questions. For claims, total recoveries plus final retained claims must equal the covered gross claims. For premiums, gross premium less all ceded premiums must equal net premium. A positive net result before expenses means net premium exceeds retained claims for the year; it is not a measure of profit after expenses or investment income.

Key takeaway: Keep the treaty sequence explicit, apply each layer to its stated input, and respect both attachments and limits. A final claim reconciliation is an effective way to detect an allocation error before reporting the annual result.

Check Your Understanding

For each problem, follow the stated order: quota share, per-risk excess-of-loss, then stop loss.

A hypothetical portfolio has gross claims of $\$50{,}000$, $\$150{,}000$, and $\$300{,}000$. Quota share cedes 20%. The per-risk layer is $\$40{,}000$ xs $\$40{,}000$, and stop loss is $\$100{,}000$ xs $\$250{,}000$. Find the total quota-share recovery, per-risk recovery, stop-loss recovery, and final retained claims. (Answer: Quota share $\$100{,}000$; per risk $\$80{,}000$; stop loss $\$70{,}000$; retained claims $\$250{,}000$.)
Gross claims are $\$70{,}000$, $\$130{,}000$, and $\$200{,}000$. Quota share cedes 50%, followed by a $\$25{,}000$ xs $\$25{,}000$ per-risk layer and a $\$50{,}000$ xs $\$120{,}000$ stop-loss layer. Find the final retained aggregate. (Answer: $\$120{,}000$.)
Gross claims are $\$100{,}000$, $\$200{,}000$, and $\$300{,}000$; gross premium is $\$420{,}000$. Quota share cedes 30%. Per-risk cover is $\$50{,}000$ xs $\$50{,}000$, and stop loss is $\$60{,}000$ xs $\$200{,}000$. Treaty premiums are $\$12{,}000$ and $\$8{,}000$, respectively. Find net premium, final retained claims, and annual net result before expenses. (Answer: Net premium $\$274{,}000$; retained claims $\$240{,}000$; net result $\$34{,}000$.)
Gross claims are $\$120{,}000$ and $\$280{,}000$. Quota share cedes 25%, followed by a $\$60{,}000$ xs $\$60{,}000$ per-risk layer and a $\$40{,}000$ xs $\$160{,}000$ stop-loss layer. Find the recoveries under each treaty and final retained claims. (Answer: Quota share $\$100{,}000$; per risk $\$90{,}000$; stop loss $\$40{,}000$; retained claims $\$170{,}000$.)
Gross claims are $\$90{,}000$, $\$210{,}000$, and $\$300{,}000$, with gross premium $\$500{,}000$. Quota share cedes 20%; per-risk cover is $\$40{,}000$ xs $\$40{,}000$; stop loss is $\$75{,}000$ xs $\$250{,}000$. The annual per-risk and stop-loss premiums are $\$16{,}000$ and $\$11{,}000$. Find total recoveries, retained claims, net premium, and net result before expenses. (Answer: Quota share $\$120{,}000$; per risk $\$112{,}000$; stop loss $\$75{,}000$; retained claims $\$293{,}000$; net premium $\$373{,}000$; net result $\$80{,}000$.)

What You'll Learn