Why Risk Size Matters in Proportional Reinsurance
Quota share and surplus share treaties both divide insured risks proportionally, but they respond differently when policy sizes vary. Under a quota share treaty, the ceded percentage is fixed for every covered policy. Under a surplus share treaty, the ceded amount depends on how much the policy’s sum insured exceeds the insurer’s retention, subject to the treaty’s capacity. Consequently, a treaty can cede a small percentage of one large policy and a larger percentage of another, even when both are covered by the same surplus arrangement.
This comparison is useful when checking policy-level allocations and portfolio totals. The key is to distinguish an absolute amount ceded from the percentage of a particular risk ceded. This tutorial is aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus.
How the Two Treaties Respond to Policy Size
For quota share, the cession percentage \(q\) does not change with \(S\). If the treaty cedes 30%, for example, it cedes 30% of a small covered policy and 30% of a large covered policy. The amount ceded increases in direct proportion to the sum insured.
For surplus share, the cession is determined by the amount above the retention, up to the treaty’s maximum capacity. If \(S\) is at or below \(R\), there is no surplus to cede under this treaty. If \(S\) is between \(R\) and \(R+nR\), the cession is \(S-R\), leaving the insurer with its retention \(R\). If \(S\) exceeds \(R+nR\), the treaty is full: it cedes \(nR\), and the insurer retains the retention plus the amount above treaty capacity.
Here \(C\) denotes the amount ceded and \(T\) the amount retained. A surplus treaty’s cession percentage for a particular policy is \(C_{\mathrm{SS}}/S\), when \(S>0\). That percentage is not generally constant across policies. It is zero for risks at or below the retention, increases as eligible risk size increases, and can decline again for risks above capacity because the ceded amount is capped.
| Policy size | Quota share response | Surplus share response |
|---|---|---|
| At or below retention | Cedes the fixed proportion \(q\) | Cedes zero |
| Above retention but within capacity | Cedes the fixed proportion \(q\) | Cedes the amount above retention; retains \(R\) |
| Above treaty capacity | Cedes the fixed proportion \(q\) | Cedes at most \(nR\); retains the excess above capacity as well as \(R\) |
Why the Surplus Formula Works
A surplus share treaty is designed to cede the part of a risk that exceeds the insurer’s retention, subject to the reinsurance capacity. The positive part of the amount above retention is \(\max(S-R,0)\): it is zero when the risk does not exceed \(R\), and equals \(S-R\) otherwise. The treaty cannot cede more than \(nR\). Taking the smaller of those two amounts therefore gives the actual cession.
If \(S\leq R\), there is no surplus; if \(S>R\), the amount available to cede is \(S-R\). Together these cases are \(\max(S-R,0)\).
The surplus treaty can accept no more than \(nR\), so the cession is the smaller of the amount available and the capacity.
The original sum insured is divided between ceded and retained amounts, so \(T_{\mathrm{SS}}=S-C_{\mathrm{SS}}\).
This derivation also shows why “the insurer retains \(R\)” is not true for every policy under a surplus treaty. It holds only when \(R<S\leq R+nR\). Below the retention, the insurer retains the entire risk; above treaty capacity, it retains more than \(R\).
Worked Example: Comparing Cessions Across Four Policy Sizes
A hypothetical insurer has a 30% quota share treaty. It is also considering a surplus share treaty with a retention of $100,000 and capacity of three lines, meaning \(n=3\) and a maximum cession of $300,000. Compare the ceded and retained amounts for policies with sums insured of $80,000, $150,000, $350,000, and $650,000. Report amounts in thousands of dollars.
Method. Calculate each treaty’s cession and retention separately for each policy. For surplus share, the capacity is \(3(100)=300\) thousand dollars.
Policy of $80,000. Quota share cedes \(0.30(80)=24\) and retains \(80-24=56\). Surplus share cedes \(\min(\max(80-100,0),300)=0\) and retains \(80-0=80\).
Policy of $150,000. Quota share cedes \(0.30(150)=45\) and retains \(150-45=105\). Surplus share cedes \(\min(150-100,300)=50\) and retains \(150-50=100\).
Policy of $350,000. Quota share cedes \(0.30(350)=105\) and retains \(350-105=245\). Surplus share cedes \(\min(350-100,300)=250\) and retains \(350-250=100\).
Policy of $650,000. Quota share cedes \(0.30(650)=195\) and retains \(650-195=455\). Surplus share cedes \(\min(650-100,300)=300\) and retains \(650-300=350\).
| Sum insured ($000) | Quota ceded ($000) | Quota retained ($000) | Surplus ceded ($000) | Surplus retained ($000) |
|---|---|---|---|---|
| 80 | 24 | 56 | 0 | 80 |
| 150 | 45 | 105 | 50 | 100 |
| 350 | 105 | 245 | 250 | 100 |
| 650 | 195 | 455 | 300 | 350 |
Check. In every row, ceded plus retained equals the sum insured. The surplus cession never exceeds the $300,000 capacity. The risk of $350,000 is within surplus capacity above retention, so the insurer retains exactly $100,000; the $650,000 risk exceeds that range, so the insurer retains the $100,000 retention plus $250,000 beyond the treaty capacity.
Common mistake avoided. Applying the surplus formula as \(S-R\) without the capacity limit would give a $550,000 cession on the $650,000 risk. The treaty can cede only $300,000, so the correct retained amount is $350,000.
Applying the Cession to Claims
For a proportional treaty, a policy’s claim is allocated using that policy’s cession percentage, provided the claim is shared in the same proportion as the insured risk. Under quota share, that percentage is \(q\). Under surplus share, it is \(C_{\mathrm{SS}}/S\). Thus a surplus treaty may allocate different percentages of claims for policies with different sums insured. This allocation assumes the claim is otherwise covered and that the treaty’s proportional terms apply; it does not introduce a separate claim limit or other coverage adjustment.
Worked Example: Allocating Claims Under Both Treaties
Use the same hypothetical 30% quota share and three-line surplus treaty with \(R=\$100{,}000\) and maximum surplus cession \(\$300{,}000\). Four covered policies have sum insureds of $80,000, $150,000, $350,000, and $650,000, with claims of $20,000, $60,000, $210,000, and $520,000, respectively. Find the ceded and retained claim under each treaty. Amounts are in thousands of dollars.
Policy of $80,000. The surplus cession percentage is \(0/80=0\), so the $20,000 claim has no surplus cession and is retained in full. Quota share cedes \(0.30(20)=6\) and retains \(20-6=14\).
Policy of $150,000. The surplus cession is \(50\), so its percentage is \(50/150=1/3\). The surplus treaty cedes \(60(1/3)=20\) of the claim and retains \(60-20=40\). Quota share cedes \(0.30(60)=18\) and retains \(60-18=42\).
Policy of $350,000. The surplus cession is \(250\), so its percentage is \(250/350=5/7\). The surplus treaty cedes \(210(5/7)=150\) and retains \(210-150=60\). Quota share cedes \(0.30(210)=63\) and retains \(210-63=147\).
Policy of $650,000. The surplus cession is capped at \(300\), so its percentage is \(300/650=6/13\). The surplus treaty cedes \(520(6/13)=240\) and retains \(520-240=280\). Quota share cedes \(0.30(520)=156\) and retains \(520-156=364\).
Check. Each pair of ceded and retained claim amounts sums to the original claim. For example, the largest risk’s surplus allocation is \(240+280=520\). The surplus claim cession is not automatically \(300\): $300,000 is the cession capacity for the sum insured, not a fixed claim payment.
Common mistake avoided. Using the $300,000 surplus capacity as a claim cession confuses an insured amount with a loss amount. First calculate the policy’s ceded proportion, \(300/650\) here, and then apply that proportion to the claim.
Portfolio Totals and Risk Mix
Portfolio totals are obtained by summing policy-level ceded and retained amounts. A portfolio’s overall ceded percentage is the total ceded amount divided by total sum insured; it is not generally the simple average of individual policy cession percentages. Large and small policies contribute different amounts of exposure, and surplus cession depends on each policy’s position relative to retention and capacity.
Worked Example: Comparing Portfolio Totals for a Mixed Book
A hypothetical insurer writes two policies of $90,000, one policy of $240,000, and one policy of $500,000. Compare total cessions and retentions under a 40% quota share and a surplus treaty with a $100,000 retention and two lines of capacity. Report amounts in thousands of dollars.
Step 1: Find total sum insured. The total is \(2(90)+240+500=920\).
Step 2: Calculate quota share totals. The total cession is \(0.40(920)=368\). The total retained amount is \(920-368=552\). The overall ceded percentage is \(368/920=0.40\), or 40%.
Step 3: Calculate each surplus cession. The capacity is \(2(100)=200\). Each $90,000 policy is below retention and has a cession of 0. The $240,000 policy has a cession of \(\min(240-100,200)=140\). The $500,000 policy has a cession of \(\min(500-100,200)=200\).
Step 4: Sum and reconcile. Total surplus cession is \(0+0+140+200=340\). Total retained amount is \(920-340=580\). The overall surplus cession percentage is \(340/920=0.369565\ldots\), or 36.96% to two decimal places.
Check. The surplus treaty retains \(90+90+100+300=580\), agreeing with \(920-340=580\). The two $90,000 policies generate no cession, while the $500,000 policy reaches the $200,000 treaty capacity.
Common mistake avoided. Applying a single average surplus percentage to every policy would disregard the two risks below retention and the cap on the largest risk. Calculate each risk’s cession first, then sum.
Exam Traps and Practical Checks
When comparing treaty results, check both the amount and the basis of the calculation. A quota share percentage is applied to the full covered sum insured. A surplus share cession is based on the amount above retention, constrained by capacity. Then, if the question asks for a claim allocation, apply the policy-level cession percentage to the claim rather than reusing a sum-insured amount.
- Forgetting the capacity ceiling: the surplus cession cannot exceed \(nR\), even when \(S-R\) is larger.
- Assuming surplus always leaves exactly \(R\) retained: that is true only when the risk is above retention and within treaty capacity.
- Mixing amount and percentage: a cession of $200,000 does not mean a 20% cession; divide by that policy’s sum insured to find the percentage.
- Averaging policy percentages without weights: use total ceded amount divided by total sum insured for the portfolio ceded percentage.
- Using the wrong treaty’s rule: a fixed quota share percentage does not replace the retention-and-capacity calculation for surplus share.
Check Your Understanding
For each problem, calculate policy-level amounts before combining them where needed. Amounts are in thousands of dollars unless stated otherwise.
- A 25% quota share covers policies with sums insured of 40, 120, and 280. Find the total ceded and retained amounts. (Answer: Total sum insured \(=440\); ceded \(=110\); retained \(=330\).)
- A surplus treaty has retention \(R=75\) and two lines of capacity. Find the ceded and retained amounts for risks of 50, 140, 300, and 400. (Answer: Ceded \(=0, 65, 150, 150\); retained \(=50, 75, 150, 250\), respectively.)
- For a sum insured of 360, compare a 35% quota share with a surplus treaty having retention 120 and two lines of capacity. Give the cession and retention under each, and the surplus cession percentage. (Answer: Quota share ceded \(=126\), retained \(=234\); surplus ceded \(=240\), retained \(=120\), cession percentage \(=66.67\%\).)
- A surplus treaty has retention 80 and three lines of capacity. A policy has sum insured 260 and a covered claim of 52. Find the ceded and retained claim. (Answer: Sum-insured cession \(=180\), cession percentage \(=180/260=9/13\); ceded claim \(=36\), retained claim \(=16\).)
- A portfolio has policies with sums insured 90, 240, and 500. Compare total cession and retention under a 40% quota share and a surplus treaty with retention 100 and two lines of capacity. (Answer: Total sum insured \(=830\); quota share ceded \(=332\), retained \(=498\); surplus cessions \(=0,140,200\), total ceded \(=340\), retained \(=490\).)