Learnalog
Reinsurance as a Loss Transformation

Proportional Reinsurance

Calculate quota-share and surplus-share outcomes, including premium and commission economics.

Proportional Reinsurance

Under proportional reinsurance, premium and covered losses are shared in an agreed proportion. The transformation is simple; contract economics are not.

1. Quota share

Let qq be the ceded proportion, 0≤q≤10\le q\le1:

Xced=qX,Xret=(1−q)X.X^{\text{ced}}=qX, \qquad X^{\text{ret}}=(1-q)X.

For a 40% quota share and a £250,000 covered loss:

Xced=0.4(250,000)=£100,000,X^{\text{ced}}=0.4(250{,}000)=£100{,}000,Xret=£150,000.X^{\text{ret}}=£150{,}000.

If the transformation applies to every loss and no other term intervenes,

E[Xret]=(1−q)E[X],Var⁡(Xret)=(1−q)2Var⁡(X),E[X^{\text{ret}}]=(1-q)E[X], \quad \operatorname{Var}(X^{\text{ret}})=(1-q)^2\operatorname{Var}(X),

and positive-homogeneous measures such as VaR scale in the same way. The treaty reduces large and small losses proportionally; it does not target only the tail.

2. Portfolio example

Suppose annual gross earned premium is £12m and ultimate covered loss is £9m. With 40% quota share:

ItemGrossCededRetained
Premium before commission£12.0m£4.8m£7.2m
Loss£9.0m£3.6m£5.4m

If the reinsurer pays a 25% ceding commission on ceded premium, commission is 0.25×4.8=£1.20.25\times4.8=£1.2m. That payment can compensate acquisition and administration costs; it is not a reduction of claim loss.

3. Surplus share

Surplus reinsurance shares a policy according to sum insured. Let the insurer retain rr lines of size MM and let policy sum insured be VV. Ignoring treaty capacity and wording complications, ceded proportion is

q(V)=max⁡(0,1−rMV).q(V)=\max\left(0,1-\frac{rM}{V}\right).

Example: one retention line is £100,000 and a policy has sum insured £400,000:

q=1−100,000400,000=75%.q=1-\frac{100{,}000}{400{,}000}=75\%.

A £200,000 covered loss is then ceded £150,000 and retained £50,000. A policy with sum insured £80,000 remains fully retained.

Surplus participation is commonly determined from the policy's sum insured, then applied proportionally to covered loss. It is not the same as applying an excess layer directly to loss amount.

4. Commissions change incentives

Common structures include:

  • fixed ceding commission;
  • sliding-scale commission linked to loss ratio;
  • profit commission after specified expenses and carry-forward rules.

They affect net economics and incentives even when the gross/ceded loss split is unchanged. State whether a result is loss only or a complete underwriting cash-flow view.

5. When proportional cover is useful

ObjectiveWhy quota/surplus may helpTrade-off
support portfolio growthshares losses and premium from inceptioncedes profitable ordinary business too
manage all-size volatilityscales broad portfolio experienceless targeted than excess cover
provide underwriting capacitysurplus share limits retained linerequires accurate sum-insured administration
align reinsurer with portfolioboth share experiencecommission and reporting terms matter

Practice

  1. A 30% quota share covers a £600,000 loss. Find ceded and retained loss.
  2. Gross premium is £20m, ceded share 30%, and ceding commission 20% of ceded premium. Find ceded premium and commission.
  3. Why does quota share not change the coefficient of variation of loss under the simple proportional transformation?
Answers
  1. Ceded £180,000; retained £420,000.
  2. Ceded premium £6m; commission £1.2m.
  3. Both mean and SD multiply by 1−q1-q, so their ratio is unchanged. Portfolio capital and cash effects can still change.

Next, target layers of loss with Excess-of-Loss Reinsurance.

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Reinsurance as a Loss Transformation

Translate treaty wording into gross, ceded, and retained random variables before pricing or capital analysis.

Excess-of-Loss Reinsurance

Apply limited layers per risk, occurrence, or aggregate and recognise exhaustion and reinstatement.

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