Why the Two Structures Respond Differently
Quota share and surplus share are both proportional forms of reinsurance: the reinsurer takes an agreed proportion of covered loss, rather than paying only losses above a specified attachment point. Their proportions are determined differently. A quota share generally applies the same cession percentage to every covered risk in the contract. A surplus share determines the cession percentage for each risk from its insured amount and the insurer’s retention line, subject to the treaty’s capacity.
This distinction matters when a portfolio contains risks of different sizes. A quota share produces a consistent percentage allocation as insured amounts vary. A surplus share can cede a larger percentage of a larger risk, while the insurer retains a stated line of insured amount, up to the treaty’s capacity. The comparisons below assume that covered losses are allocated in the treaty’s stated proportional share. This topic is aligned with the CAS Exam 9 (Risk Management for Actuaries) syllabus.
Quota Share: A Fixed Proportion of Each Loss
For a quota share with no special limits or exclusions affecting the loss, \(q\) is the same for each covered risk. If the gross covered loss is \(X\), the reinsurer’s recovery and the insurer’s retained loss are:
These equations apply to the covered loss specified by the contract. They do not, by themselves, describe the treatment of premiums, expenses, exclusions, or contract limits. Those terms must be considered separately when they are part of the problem.
Worked Example: Applying One Quota Share Percentage to Different Risks
A hypothetical insurer has a 35% quota share on two policies, with no per-risk or aggregate limit affecting either claim. Policy A has an insured amount of $120,000 and a covered loss of $30,000. Policy B has an insured amount of $480,000 and a covered loss of $120,000. Calculate the recovery and retained loss for each policy.
Method. Apply the treaty’s 35% cession proportion directly to each covered loss. The insured amounts do not change the quota share percentage.
For Policy A, \(q=0.35\) and \(X=\$30{,}000\):
Check: \(\$10{,}500+\$19{,}500=\$30{,}000\).
For Policy B, \(q=0.35\) and \(X=\$120{,}000\):
Check: \(\$42{,}000+\$78{,}000=\$120{,}000\). The larger policy produces larger dollar amounts ceded and retained, but the reinsurer still takes 35% of each loss. The common mistake to avoid is to vary the quota share percentage because the insured amounts differ.
Surplus Share: A Risk-Specific Proportion
For a surplus treaty, first identify the insurer’s retention line \(R\) and the number \(n\) of surplus lines the treaty can accept. Under the standard line convention, the maximum insured amount ceded to the treaty is \(nR\). For a risk with insured amount \(S\), the amount ceded is the amount above the insurer’s line, limited by the treaty’s capacity:
Here \(C(S)\) is the insured amount ceded to the treaty. The total insured amount supported by the insurer’s retention plus the treaty’s capacity is \((n+1)R\). If \(S\) exceeds this amount, the excess is not covered by the surplus treaty; it must be handled as the contract specifies, for example by another placement or by the insurer’s retention.
To turn the ceded insured amount into a loss recovery, use the treaty’s proportional cession share. In the standard surplus-share calculation, the cession proportion for a risk is \(C(S)/S\), provided \(S>0\). Thus, for a covered loss \(X\), the ceded loss is that proportion of \(X\), and the remainder is retained. If \(S=0\), this ratio is not used; a risk with no insured amount has no surplus cession under this calculation.
These formulas assume that the contract allocates covered loss in the same proportion as the insured amount ceded. They are not a claim that every physical part of a partial loss corresponds to a particular part of the insured value. If a contract specifies a different allocation, its terms govern.
Why the Surplus Recovery Formula Holds
The cession proportion follows from the proportional allocation assumption. The insurer and reinsurer share the insured amount in the ratio of their respective amounts. Applying that same ratio to the covered loss gives the reinsurer’s share.
Subtract the insurer’s retention line from \(S\), but do not let the result be below zero or exceed the treaty capacity \(nR\). This gives \(C(S)\).
The treaty takes the fraction \(C(S)/S\) of the insured amount. Under the stated proportional allocation assumption, it therefore takes the same fraction of covered loss \(X\).
The recovery is \(C(S)X/S\); the insurer retains \(X-C(S)X/S\). Adding the recovery and retained loss must give \(X\).
The formula also explains two boundary cases. If \(S\leq R\), then \(C(S)=0\), so the surplus treaty cedes none of the risk under these terms. If \(R<S\leq(n+1)R\), the treaty can take all insured value above \(R\), so the ceded insured amount is \(S-R\). If \(S>(n+1)R\), its cession is capped at \(nR\), leaving a portion of the risk beyond the treaty’s capacity.
Worked Example: Comparing Matched Risks Under Quota Share and Surplus Share
A hypothetical insurer considers two risks under a surplus treaty with a $50,000 retention line and four surplus lines. The risks have insured amounts of $125,000 and $200,000. Each has a covered loss equal to 40% of its insured amount. Compare surplus recoveries and retained losses with those under a 40% quota share. Assume no limits or exclusions affect the losses.
Method. For the surplus treaty, calculate the ceded insured amount and divide it by the risk’s full insured amount. For the quota share, apply 40% directly to each loss. The four-line surplus capacity is \(4(\$50{,}000)=\$200{,}000\).
For the $125,000 risk, the loss is \(0.40(\$125{,}000)=\$50{,}000\). The amount above the retention line is \(\$125{,}000-\$50{,}000=\$75{,}000\), which is within the $200,000 treaty capacity. The surplus cession proportion is \(\$75{,}000/\$125{,}000=0.60\). Therefore:
Under the 40% quota share, recovery is \(0.40(\$50{,}000)=\$20{,}000\), and retained loss is \(0.60(\$50{,}000)=\$30{,}000\). Each pair sums to the $50,000 loss.
For the $200,000 risk, the loss is \(0.40(\$200{,}000)=\$80{,}000\). The amount above the $50,000 retention line is \(\$200{,}000-\$50{,}000=\$150{,}000\), also within capacity. The surplus cession proportion is \(\$150{,}000/\$200{,}000=0.75\). Therefore:
Under the 40% quota share, recovery is \(0.40(\$80{,}000)=\$32{,}000\), and retained loss is \(0.60(\$80{,}000)=\$48{,}000\). Each pair sums to the $80,000 loss.
The loss ratio is 40% for both risks, but that does not make the insurer’s retained proportion of loss 40% under the surplus treaty. It retains 40% of the $50,000 loss on the $125,000 risk and 25% of the $80,000 loss on the $200,000 risk. The retention line is $50,000 of insured amount in both cases; it is not a fixed $50,000 retained loss. The common mistake is to confuse the loss ratio with the treaty’s retained proportion.
Surplus Capacity and a Larger Risk
A larger risk can exceed the amount a surplus treaty is willing to accept. In that case, the ceded percentage is limited by capacity, not simply calculated as all insured value above the retention line. The insurer’s retained proportion then includes both its retention line and the part beyond treaty capacity.
Worked Example: Applying the Surplus Treaty Capacity Limit
A hypothetical insurer has a $60,000 retention line and a surplus treaty that accepts three lines. A risk has an insured amount of $300,000 and a covered loss of $90,000. Calculate the treaty recovery and retained loss. Assume proportional allocation of covered loss and no other coverage.
Method. Calculate the treaty’s maximum ceded insured amount, then compare it with the risk amount above the retention line.
The treaty capacity is \(3(\$60{,}000)=\$180{,}000\). The risk amount above the retention is \(\$300{,}000-\$60{,}000=\$240{,}000\). Since $240,000 exceeds the $180,000 capacity, the ceded insured amount is \(C(S)=\$180{,}000\). The cession proportion is \(\$180{,}000/\$300{,}000=0.60\).
Check: \(\$54{,}000+\$36{,}000=\$90{,}000\). The treaty supports the insurer’s $60,000 line plus $180,000 of surplus, for a total of $240,000 of insured amount. The remaining $60,000 of the risk is beyond the treaty’s capacity. It is not ceded by this treaty. The common mistake is to treat all $240,000 above the line as covered, which would ignore the three-line limit.
Comparison and Exam Traps
| Feature | Quota share | Surplus share |
|---|---|---|
| Basis for cession proportion | Stated percentage \(q\), generally unchanged from risk to risk | Ceded insured amount divided by total insured amount, subject to treaty capacity |
| Effect of a larger insured amount | Changes dollar recovery when loss changes, but not the stated percentage | Can increase the ceded proportion as more insured amount exceeds the retention line |
| Risk at or below the retention line | Still ceded according to \(q\), subject to contract terms | No surplus insured amount is ceded under the stated formula |
| Capacity issue | Consider any separately stated limits | Surplus cession is capped at \(nR\); excess insured amount is outside this treaty |
For surplus share calculations, keep three quantities distinct: the retention line \(R\), the cession proportion \(C(S)/S\), and the loss ratio \(X/S\). The retention line is an insured-amount measure. The cession proportion describes the treaty’s share of the risk and, under proportional allocation, of the loss. The loss ratio describes the loss relative to insured amount. They are not interchangeable.
A reliable check is to calculate the gross loss first, then calculate the ceded and retained shares, and finally verify that recovery plus retained loss equals gross covered loss. Also check the treaty capacity before treating the entire amount above the retention as ceded. These checks are useful even when arithmetic is simple because the main errors are often conceptual: applying a quota share percentage to insured amount instead of loss, treating a surplus retention line as a fixed retained loss, or overlooking the surplus-line limit.
Check Your Understanding
For each problem, state the recovery and retained loss under the specified terms. Assume covered losses are allocated in the treaty’s proportional share and that no other limits or exclusions apply.
- A hypothetical 30% quota share covers a policy with a $280,000 insured amount and a $84,000 covered loss. Calculate the recovery and retained loss. (Answer: recovery $25,200; retained loss $58,800.)
- A surplus treaty has a $60,000 retention line and five surplus lines. A risk has a $210,000 insured amount and a $63,000 covered loss. Calculate its cession proportion, recovery, and retained loss. (Answer: cession proportion \(150{,}000/210{,}000=5/7\); recovery $45,000; retained loss $18,000.)
- A surplus treaty has a $75,000 retention line and four surplus lines. A risk has a $400,000 insured amount and a $120,000 covered loss. Calculate the treaty capacity, cession proportion, recovery, and retained loss. (Answer: capacity $300,000; cession proportion \(300{,}000/400{,}000=0.75\); recovery $90,000; retained loss $30,000.)
- A hypothetical policy has a $250,000 insured amount and a $75,000 covered loss. Compare a 40% quota share with a surplus treaty having a $100,000 retention line and four surplus lines. (Answer: quota share recovery $30,000 and retained loss $45,000; surplus recovery $45,000 and retained loss $30,000.)
- A risk has an $80,000 insured amount and a $32,000 covered loss. Compare a surplus treaty with a $100,000 retention line and three surplus lines against a 25% quota share. (Answer: surplus recovery $0 and retained loss $32,000; quota share recovery $8,000 and retained loss $24,000.)