Quota Share Limits and Per-Risk Caps

Why a Quota Share Needs a Per-Risk Cap

A quota share treaty ordinarily transfers a fixed percentage of each covered loss to the reinsurer. A per-risk cap modifies that rule: the reinsurer still pays the agreed percentage for smaller losses, but its payment for any one risk cannot exceed a stated maximum. The cap therefore matters most for larger losses and can make the effective ceded percentage lower than the treaty’s stated quota share percentage.

This tutorial treats the cap as a maximum ceded loss amount per risk. Treaty wording should always be checked: a stated maximum on a different basis, such as a limit on covered exposure rather than on the reinsurance payment, requires a different calculation. Assume throughout that losses are covered gross losses, amounts are nonnegative, and there are no other treaty adjustments, expenses, or reinstatements. Interest and mortality assumptions do not apply.

Definition: Let $X$ be the gross loss on one risk, $q$ the quota share percentage with $0\le q\le 1$, and $C\ge 0$ the maximum ceded amount for that risk. The ceded recovery is $R(X)=\min(qX,C)$, and the net loss retained by the insurer is $N(X)=X-R(X)$. The cap applies separately to each risk.

Before the cap binds, the reinsurer pays $qX$, and the insurer retains $(1-q)X$. Once $qX$ exceeds $C$, the recovery remains at $C$, so the insurer retains the excess of the gross loss over that fixed recovery. The point at which the cap begins to bind follows by solving $qX=C$, provided $q>0$.

R(X)= \begin{cases} qX, & X\le C/q,\\ C, & X>C/q, \end{cases} \qquad N(X)= \begin{cases} (1-q)X, & X\le C/q,\\ X-C, & X>C/q. \end{cases} \quad (q>0)

This piecewise form is only for $q>0$. If $q=0$, then $qX=0$ for every loss and $R(X)=\min(0,C)=0$, so $N(X)=X$; the threshold $C/q$ is not defined and must not be used. The minimum formula also handles a zero cap: if $C=0$, the recovery is zero for every risk.

How the Cap Changes the Cession

For a positive loss $X$, the effective ceded percentage is $R(X)/X$. It equals $q$ while the cap is not binding. Above the threshold $C/q$, it is $C/X$, which decreases as the loss increases. The treaty percentage is thus not necessarily the percentage of every large loss that is ultimately ceded.

\frac{R(X)}{X}= \begin{cases} q, & 0<X\le C/q,\\ C/X, & X>C/q \end{cases} \qquad (q>0)

The reason for the minimum formula is contractual: the payment must satisfy both restrictions, namely it cannot exceed the proportional amount $qX$ and cannot exceed the per-risk maximum $C$. The largest payment satisfying both restrictions is their lesser value, $\min(qX,C)$. Since the covered gross loss is divided between ceded and retained portions, $X=R(X)+N(X)$, which gives $N(X)=X-R(X)$. These identities provide a direct check on any calculation.

Worked Example: Finding Which Losses Reach the Cap

A hypothetical insurer has a quota share of $35\%$ with a maximum ceded amount of $70,000 per risk. Three covered gross losses are $100,000, $200,000, and $300,000. Find the recovery and retained loss for each, then reconcile the portfolio totals. Assume no other treaty terms or adjustments; no interest, mortality, or probability model is needed because the losses are specified.

Method. For each risk, calculate $qX$, compare it with $C$, and take the lesser amount as the recovery. The threshold is $C/q=\$70{,}000/0.35=\$200{,}000$.

For the $100,000 loss, the proportional amount is $0.35(\$100{,}000)=\$35{,}000$, below the cap. Thus $R=\$35{,}000$ and $N=\$100{,}000-\$35{,}000=\$65{,}000$.

For the $200,000 loss, the proportional amount is $0.35(\$200{,}000)=\$70{,}000$, exactly the cap. Thus $R=\$70{,}000$ and $N=\$200{,}000-\$70{,}000=\$130{,}000$. At the threshold, either branch of the piecewise expression gives the same amount.

For the $300,000 loss, the proportional amount is $0.35(\$300{,}000)=\$105{,}000$, which exceeds the cap. Thus $R=\$70{,}000$ and $N=\$300{,}000-\$70{,}000=\$230{,}000$.

The total gross loss is $\$100{,}000+\$200{,}000+\$300{,}000=\$600{,}000$. Total recoveries are $\$35{,}000+\$70{,}000+\$70{,}000=\$175{,}000$, and total net loss is $\$65{,}000+\$130{,}000+\$230{,}000=\$425{,}000$. The check holds: $\$175{,}000+\$425{,}000=\$600{,}000$. The method avoids assuming that every risk receives the full $35\%$ cession.

Apply the Cap Risk by Risk

A per-risk cap is not a single limit on the portfolio’s total recovery. If there are $n$ risks with gross losses $X_1,\ldots,X_n$, the total recovery is the sum of the capped recoveries on those risks. No independence assumption is needed for this accounting identity. In particular, applying one cap to the sum of the gross losses generally gives a different result.

R_{\mathrm{total}}=\sum_{j=1}^{n}\min(qX_j,C), \qquad N_{\mathrm{total}}=\sum_{j=1}^{n}\left[X_j-\min(qX_j,C)\right]

The cap is applied after calculating the quota share amount for each risk, not after adding the risks together. A useful check is $R_{\mathrm{total}}+N_{\mathrm{total}}=\sum_{j=1}^{n}X_j$. The equality follows risk by risk from $R(X_j)+N(X_j)=X_j$, then by summing both sides.

Worked Example: A Per-Risk Cap Is Not a Portfolio Cap

A hypothetical treaty cedes $60\%$ of each covered loss, subject to a $90,000 maximum recovery per risk. Three risks produce losses of $80,000, $150,000, and $250,000. Calculate each recovery, the portfolio recovery and net loss, and compare the result with mistakenly applying $90,000 once to the portfolio. Assume the stated losses are certain; no interest, mortality, or probability model is used.

For the $80,000 risk, $qX=0.60(\$80{,}000)=\$48{,}000$, so $R=\min(\$48{,}000,\$90{,}000)=\$48{,}000$, and $N=\$80{,}000-\$48{,}000=\$32{,}000$.

For the $150,000 risk, $qX=0.60(\$150{,}000)=\$90{,}000$, so $R=\$90{,}000$ and $N=\$150{,}000-\$90{,}000=\$60{,}000$.

For the $250,000 risk, $qX=0.60(\$250{,}000)=\$150{,}000$, so the per-risk cap reduces the recovery to $R=\$90{,}000$. The net loss is $N=\$250{,}000-\$90{,}000=\$160{,}000$.

The correct total recovery is $\$48{,}000+\$90{,}000+\$90{,}000=\$228{,}000$. Total gross loss is $\$80{,}000+\$150{,}000+\$250{,}000=\$480{,}000$, so total net loss is $\$480{,}000-\$228{,}000=\$252{,}000$. The check is $\$228{,}000+\$252{,}000=\$480{,}000$.

If the $90,000 cap were incorrectly applied once to the aggregate, the recovery would be $\min(0.60(\$480{,}000),\$90{,}000)=\$90{,}000$. That is not the treaty described: the contract has a separate $90,000 maximum for each of three risks. The difference, $\$228{,}000-\$90{,}000=\$138{,}000$, demonstrates why the unit to which a cap applies must be identified.

Expected Recovery Under a Capped Quota Share

When a risk is random, the cap must be applied to each possible loss before taking the expectation. For a nonnegative random loss $X$, the expected recovery is $E[\min(qX,C)]$. It is generally incorrect to calculate a quota share of expected loss and then cap that amount: in general, $E[\min(qX,C)]\ne\min(qE[X],C)$. The two expressions differ because the cap acts on each realized loss.

Worked Example: Expected Recovery From a Discrete Loss Model

A hypothetical portfolio has one covered risk with loss distribution $P(X=\$0)=0.20$, $P(X=\$100{,}000)=0.30$, $P(X=\$250{,}000)=0.30$, and $P(X=\$500{,}000)=0.20$. The quota share is $40\%$, with a $80,000 per-risk cap. Find the expected gross loss, expected recovery, and expected net loss. No interest or mortality assumptions apply; the full discrete probability model is specified here.

The threshold is $C/q=\$80{,}000/0.40=\$200{,}000$. Evaluate the recovery at each possible loss: at $0 it is $0; at $100,000 it is $0.40(\$100{,}000)=\$40{,}000$; at $250,000 the proportional amount is $100,000, so recovery is capped at $80,000; and at $500,000 the proportional amount is $200,000, so recovery is also $80,000.

The expected gross loss is $0.20(\$0)+0.30(\$100{,}000)+0.30(\$250{,}000)+0.20(\$500{,}000)=\$30{,}000+\$75{,}000+\$100{,}000=\$205{,}000$.

Weight the capped recoveries by their probabilities: $E[R(X)]=0.20(\$0)+0.30(\$40{,}000)+0.30(\$80{,}000)+0.20(\$80{,}000)=\$12{,}000+\$24{,}000+\$16{,}000=\$52{,}000$.

By $N(X)=X-R(X)$, expected net loss is $\$205{,}000-\$52{,}000=\$153{,}000$. The check is that expected recovery plus expected net loss equals expected gross loss: $\$52{,}000+\$153{,}000=\$205{,}000$.

A tempting shortcut gives $qE[X]=0.40(\$205{,}000)=\$82{,}000$, then $\min(\$82{,}000,\$80{,}000)=\$80{,}000$. That is not the expected capped recovery; it exceeds the correct $52,000 because the shortcut applies the cap to an average rather than to each possible outcome. The method is to calculate the recovery at each loss value, then probability-weight those recoveries.

Common Mistakes and Practical Checks

The main exam trap is to use the stated quota share percentage as if it applied without limit to every loss. Once $qX>C$, the recovery is $C$, not $qX$, and the insurer’s net loss is $X-C$. Other errors arise from applying the cap to the wrong unit, or from taking an expectation before applying the cap.

Common mistake: With $q=40\%$, $C=\$80{,}000$, and $X=\$250{,}000$, the incorrect uncapped calculation is $R=0.40(\$250{,}000)=\$100{,}000$, $N=\$150{,}000$. The correction is $R=\min(\$100{,}000,\$80{,}000)=\$80{,}000$, $N=\$250{,}000-\$80{,}000=\$170{,}000$. The conservation check is $\$80{,}000+\$170{,}000=\$250{,}000$.

For a reliable calculation, identify whether the maximum is per risk or aggregate, compute the proportional amount for each applicable risk, take the lesser of that amount and the cap, and subtract the recovery from gross loss. If using a threshold, state that it requires $q>0$; if $q=0$, calculate directly from the minimum formula. These checks keep boundary cases and treaty structure consistent.

Key takeaway: Under a quota share with a per-risk cession cap, each risk’s recovery is $\min(qX,C)$. Apply this rule risk by risk, then add recoveries; for random losses, calculate capped recoveries before taking expectations.

Check Your Understanding

For each problem, use the stated cap as a maximum ceded amount per risk, and assume there are no other treaty terms or adjustments.

A hypothetical treaty cedes $25\%$ of each loss, capped at $30,000 per risk. Two risks have gross losses of $80,000 and $140,000. Find each recovery and net loss, and the portfolio totals. (Answer: Recoveries $20,000 and $30,000; net losses $60,000 and $110,000; total recovery $50,000 and total net loss $170,000.)
A treaty cedes $50\%$, subject to a $75,000 cap per risk. Three losses are $100,000, $160,000, and $240,000. Find total gross loss, total recovery, and total net loss. (Answer: Gross loss $500,000; recovery $200,000; net loss $300,000.)
A risk has loss distribution $P(X=\$50{,}000)=0.40$, $P(X=\$100{,}000)=0.40$, and $P(X=\$200{,}000)=0.20$. For $q=30\%$ and $C=\$45{,}000$, find expected recovery and expected net loss. (Answer: Expected recovery $27,000; expected net loss $73,000.)
A hypothetical portfolio has three covered losses: $60,000, $140,000, and $80,000. A quota share of $40\%$ has a $50,000 cap per risk. Find total recovery and total net loss, and explain why one portfolio cap of $50,000 would be incorrect. (Answer: Total recovery $106,000 and total net loss $174,000; the stated cap applies separately to each risk.)
A treaty has $q=0$, a $12,000 per-risk cap, and one covered gross loss of $40,000. Find recovery and net loss, without using a threshold expression. (Answer: Recovery $0 and net loss $40,000.)

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