Quota Share Portfolio Totals and Reconciliation

Why Portfolio Reconciliation Matters

A quota share treaty allocates a fixed percentage of eligible premium and loss to the reinsurer. Portfolio calculations are therefore built from policy-level allocations: determine each policy’s ceded and retained amounts, then sum them. The final check is that the gross total equals the ceded total plus the retained total.

This check is useful both when calculating treaty results and when reviewing a portfolio summary. A mismatch may indicate an arithmetic error, an inconsistent accounting basis, or an amount that was included in the gross total but excluded from the treaty calculation. The examples below assume a hypothetical treaty with no cession cap, exclusions, or other adjustments unless stated.

Definition: Let $q$ be the quota share percentage ceded to the reinsurer, where $0 \le q \le 1$. For policy $j$, let $P_j$ be the gross premium subject to the treaty and $L_j$ the covered loss subject to the treaty. The ceded and retained amounts are $$ C^P_j=qP_j,\qquad R^P_j=(1-q)P_j, $$ $$ C^L_j=qL_j,\qquad R^L_j=(1-q)L_j. $$ Here $C^P_j$ and $C^L_j$ are ceded premium and ceded loss; $R^P_j$ and $R^L_j$ are retained premium and retained loss. These formulas apply to the premium and loss amounts actually subject to the quota share, not automatically to amounts excluded by treaty terms.

From Policy Amounts to Portfolio Totals

For a portfolio of $n$ policies, add the policy-level amounts. If the same quota share percentage applies to every policy, then the portfolio cessions can also be calculated by applying $q$ directly to the corresponding portfolio total.

P_{\mathrm{gross}}=\sum_{j=1}^{n}P_j,\qquad C^P_{\mathrm{total}}=\sum_{j=1}^{n}C^P_j=qP_{\mathrm{gross}},\qquad R^P_{\mathrm{total}}=\sum_{j=1}^{n}R^P_j=(1-q)P_{\mathrm{gross}},

L_{\mathrm{gross}}=\sum_{j=1}^{n}L_j,\qquad C^L_{\mathrm{total}}=\sum_{j=1}^{n}C^L_j=qL_{\mathrm{gross}},\qquad R^L_{\mathrm{total}}=\sum_{j=1}^{n}R^L_j=(1-q)L_{\mathrm{gross}}.

The aggregate identities follow from ordinary addition. They require the same quota share percentage to apply to each amount being aggregated, and the gross, ceded, and retained figures must use consistent definitions and accounting bases. For example, do not compare premium earned over one period with premium written over a different period and expect a reconciliation.

Establish the subject amounts.
Identify the gross premiums and covered losses included in the treaty calculation, using the stated valuation and accounting basis.

Apply the quota share.
For each policy, multiply its subject premium and loss by $q$ to obtain ceded amounts. Multiply by $1-q$ to obtain retained amounts.

Aggregate and reconcile.
Sum the policy-level figures and verify, separately for premium and loss, that gross equals ceded plus retained.

Why the Reconciliation Holds

The key identity is $q+(1-q)=1$. For each policy, the ceded and retained amounts partition the subject amount. Adding these policy-level identities gives the portfolio identity.

C^P_{\mathrm{total}}+R^P_{\mathrm{total}} =\sum_{j=1}^{n}\bigl(qP_j+(1-q)P_j\bigr) =\sum_{j=1}^{n}P_j =P_{\mathrm{gross}}.

Replacing each premium $P_j$ with the covered loss $L_j$ gives $C^L_{\mathrm{total}}+R^L_{\mathrm{total}}=L_{\mathrm{gross}}$. This derivation also explains why a uniform quota share preserves the portfolio loss ratio when it is calculated on the same gross premium and loss basis. If gross premium is positive, then

\frac{C^L_{\mathrm{total}}}{C^P_{\mathrm{total}}} =\frac{qL_{\mathrm{gross}}}{qP_{\mathrm{gross}}} =\frac{L_{\mathrm{gross}}}{P_{\mathrm{gross}}}, \qquad \frac{R^L_{\mathrm{total}}}{R^P_{\mathrm{total}}} =\frac{(1-q)L_{\mathrm{gross}}}{(1-q)P_{\mathrm{gross}}} =\frac{L_{\mathrm{gross}}}{P_{\mathrm{gross}}},

provided $q>0$ for the ceded ratio and $q<1$ for the retained ratio. If a share is zero, the corresponding premium and loss are both zero, so that share’s loss ratio is undefined rather than equal to the gross ratio.

Worked Examples

Worked Example: Reconciling Premium and Loss by Policy

A hypothetical insurer has three policies under a 25% quota share treaty. The subject gross premiums are $1,200, $1,800, and $1,000; the covered losses are $0, $600, and $2,400. Calculate the ceded and retained portfolio premium and loss, first by policy and then in total. Check the reconciliation.

Method. Apply $q=0.25$ and $1-q=0.75$ separately to each policy’s subject premium and covered loss.

Policy	Gross premium	Ceded premium	Retained premium	Gross loss	Ceded loss	Retained loss
1	$1,200	$300	$900	$0	$0	$0
2	$1,800	$450	$1,350	$600	$150	$450
3	$1,000	$250	$750	$2,400	$600	$1,800

For example, policy 2 has ceded premium $0.25(1{,}800)=450$ and retained premium $0.75(1{,}800)=1{,}350$. Its ceded loss is $0.25(600)=150$, leaving $0.75(600)=450$ retained.

Premium totals. Gross premium is $1{,}200+1{,}800+1{,}000=4{,}000$. Ceded premium is $300+450+250=1{,}000$, and retained premium is $900+1{,}350+750=3{,}000$. The check is $1{,}000+3{,}000=4{,}000$.

Loss totals. Gross loss is $0+600+2{,}400=3{,}000$. Ceded loss is $0+150+600=750$, and retained loss is $0+450+1{,}800=2{,}250$. The check is $750+2{,}250=3{,}000$.

Check and common mistake. Applying 25% to each gross total gives the same ceded amounts: $0.25(4{,}000)=1{,}000$ premium and $0.25(3{,}000)=750$ loss. A common error is to treat the ceded percentage as the retained percentage; that would wrongly report $3,000 ceded premium. The policy calculations and both total checks confirm that the correct ceded share is 25%.

Worked Example: Adding a Ceding Commission Without Misallocating Loss

A hypothetical portfolio has gross subject premium of $10,000 and covered losses of $6,000 under a 35% quota share. The reinsurer also pays a ceding commission equal to 20% of ceded premium. Calculate gross, ceded, and retained premium; the commission; the net premium retained by the insurer and received by the reinsurer after commission; and the ceded and retained losses.

Method. First allocate gross premium and loss by the quota share. Then calculate the commission as a separate premium-related transfer. The commission does not change the loss allocation.

Ceded premium is $0.35(10{,}000)=\$3{,}500$. Gross retained premium before commission is $0.65(10{,}000)=\$6{,}500$. The premium allocation reconciles: $\$3{,}500+\$6{,}500=\$10{,}000$.

The commission is $0.20(\$3{,}500)=\$700$. After receiving the commission, the insurer’s net premium is $\$6{,}500+\$700=\$7{,}200$. The reinsurer’s net premium after paying the commission is $\$3{,}500-\$700=\$2{,}800$. These net amounts also reconcile: $\$7{,}200+\$2{,}800=\$10{,}000$.

Ceded loss is $0.35(\$6{,}000)=\$2{,}100$, and retained loss is $0.65(\$6{,}000)=\$3{,}900$. The loss check is $\$2{,}100+\$3{,}900=\$6{,}000$.

Check and common mistake. The commission is 20% of ceded premium, not 20% of gross premium or of losses. Subtracting the $700 commission from ceded loss would produce an incorrect $1,400 loss recovery; the correct ceded loss remains $2,100. The gross premium, net premium, and loss totals each reconcile on their own stated basis.

Worked Example: Testing Portfolio Loss-Ratio Consistency

A hypothetical treaty summary reports gross premium of $12,000 and gross covered losses of $7,200. The treaty cedes 40% of both. A separate report lists ceded premium of $4,800 and retained loss of $4,320. Determine the missing ceded loss and retained premium, and check whether the reported figures are consistent. Compare the gross, ceded, and retained loss ratios.

Method. Apply the treaty percentage to the gross totals, calculate the complementary shares, and check both the allocation identities and the ratios.

Ceded loss is $0.40(7{,}200)=\$2{,}880$. Retained premium is $0.60(12{,}000)=\$7{,}200$. The ceded premium check is $0.40(12{,}000)=\$4{,}800$, matching the report. The retained loss check is $0.60(7{,}200)=\$4{,}320$, also matching the report.

The premium totals reconcile: $\$4{,}800+\$7{,}200=\$12{,}000$. The loss totals reconcile: $\$2{,}880+\$4{,}320=\$7{,}200$.

The gross loss ratio is $\$7{,}200/\$12{,}000=0.60$, or 60%. The ceded loss ratio is $\$2{,}880/\$4{,}800=0.60$, or 60%. The retained loss ratio is $\$4{,}320/\$7{,}200=0.60$, or 60%.

Check and common mistake. All reported amounts are consistent with a uniform 40% quota share. A common mistake is to divide ceded loss by gross premium, which mixes two different shares. The ceded loss ratio uses ceded loss divided by ceded premium; on that consistent basis, it matches the gross and retained ratios.

Common Reconciliation Traps

A reconciliation is only meaningful when the figures being compared refer to the same subject amounts and accounting basis. Review these points before concluding that a portfolio total is wrong:

Mixing gross and net premium. Ceded premium is part of the gross premium allocation. A ceding commission is a separate transfer that changes net premium after commission; it is not an additional gross premium or a reduction in loss.
Using different populations. If some policies or losses are excluded from the treaty, apply the quota share only to the subject amounts. Do not compare cessions on eligible business with gross totals that include ineligible business.
Using different time bases. Written premium and earned premium are not interchangeable. Likewise, paid losses and incurred losses may differ. Reconcile figures measured on a consistent basis.
Rounding each policy too early. Keep the full calculated amounts through aggregation when possible, and round only the reported totals. Policy-level rounding can create a small apparent difference from applying the quota share to the unrounded gross total.
Confusing the two shares. The ceded percentage is $q$; the retained percentage is $1-q$. They sum to 100%, but they are not the same amount unless the share is 50%.

Key takeaway: Under a uniform quota share, aggregate ceded premium and loss are the treaty percentage of the corresponding subject gross totals, and retained amounts are the complementary share. Verify premium and loss separately; treat any ceding commission as a distinct adjustment to net premium, not as a change to the loss split.

Check Your Understanding

For each problem, use the stated quota share on the specified subject amounts and check the relevant total.

A hypothetical two-policy portfolio has subject premiums of $900 and $1,100, and covered losses of $0 and $1,500. Under a 20% quota share, find total ceded and retained premium and loss. (Answer: Gross premium $2,000; ceded premium $400; retained premium $1,600; gross loss $1,500; ceded loss $300; retained loss $1,200.)
A hypothetical portfolio has gross subject premium of $12,000 and covered losses of $7,800. A 35% quota share applies. Find ceded and retained premium and loss. (Answer: Ceded premium $4,200; retained premium $7,800; ceded loss $2,730; retained loss $5,070.)
A hypothetical portfolio has policy premiums of $400, $600, and $1,000, and covered losses of $200, $800, and $0. A 25% quota share has a ceding commission of 10% of ceded premium. Find ceded and retained loss, the commission, and the insurer’s net premium after commission. (Answer: Ceded loss $250; retained loss $750; ceded premium $500; gross retained premium $1,500; commission $50; insurer’s net premium $1,550.)
A treaty summary for a hypothetical portfolio shows gross premium of $7,200, ceded premium of $2,160, gross covered loss of $7,500, and retained loss of $5,250. Determine the implied quota share and the ceded loss, and state whether the figures are consistent with a uniform quota share. (Answer: The implied ceded share is 30%; ceded loss is $2,250; the figures are consistent because retained loss is 70% of gross loss.)
A hypothetical insurer reports gross subject premium of $25,000 and covered losses of $18,000 under a 40% quota share. The ceding commission is 15% of ceded premium. Find the ceded and retained amounts for premium and loss, the commission, and both net premium amounts after commission. (Answer: Ceded premium $10,000; gross retained premium $15,000; ceded loss $7,200; retained loss $10,800; commission $1,500; insurer’s net premium $16,500; reinsurer’s net premium $8,500.)

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