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Risk Modelling and Survival Analysis · Loss distributions, with and without risk sharing

Proportional and Excess of Loss Reinsurance: Claim Distributions

Updated 11 October 2026 · Fact-checked

Reinsurance splits each claim X between the insurer and the reinsurer. Under proportional (quota share) cover with retention α, the insurer pays αX and the reinsurer (1 − α)X. Under excess of loss with retention M, the insurer pays min(X, M) and the reinsurer pays max(X − M, 0). Find means and variances from these.

Understand Proportional and Excess of Loss Reinsurance

A claim X is a random variable. Reinsurance says who pays which part. The two payments always add up to X. Insurer's payment Y plus reinsurer's payment Z equals X.

Proportional reinsurance (quota share) shares every claim in a fixed proportion. If the insurer keeps a proportion α, then Y = αX and Z = (1 − α)X. The shape of the distribution does not change. It is only scaled. If X is lognormal, αX is still lognormal with μ shifted by ln α. If X is Pareto(a, λ), then αX is Pareto(a, αλ). If X is exponential with rate λ, αX is exponential with rate λ/α.

Excess of loss reinsurance (per claim) has a retention M, also called the deductible or priority. The insurer pays all of a claim up to M. The reinsurer pays only the part above M. So Y = min(X, M) and Z = max(X − M, 0). Small claims stay with the insurer. Large claims are shared. This protects the insurer from big losses, so it cuts the variance more effectively than quota share for the same expected cost.

The reinsurer's payment Z is zero with probability P(X ≤ M). If you only look at claims that actually reach the reinsurer (X > M), the amount paid is X − M given X > M. This is why you must say whether the question asks about all claims or only non-zero payments. Number of claims reaching the reinsurer is also reduced: it is the original number times P(X > M).

For an exam, the work is nearly always the same: write the payment in terms of X, then use the density or survival function to find the distribution, mean and variance. For a Pareto claim, the excess over M is again Pareto with the same shape and a new scale. For an exponential claim, the excess over M is exponential with the same rate (the memoryless property).

Key rules to remember

Claim split
X = Y + Z
Y is the insurer's payment, Z is the reinsurer's payment, for every claim.
Quota share payments
Y = αX, Z = (1 − α)X, with 0 < α < 1
α is the proportion retained by the insurer.
Quota share mean and variance
E(Y) = αE(X); Var(Y) = α²Var(X); E(Z) = (1 − α)E(X); Var(Z) = (1 − α)²Var(X)
Variance scales by the square of the proportion. The coefficient of variation is unchanged.
Excess of loss payments
Y = min(X, M); Z = max(X − M, 0)
M is the retention. Z equals 0 when X ≤ M.
Expected reinsurer payment (all claims)
E(Z) = ∫ from M to ∞ of (x − M) f(x) dx = ∫ from M to ∞ of [1 − F(x)] dx
This averages over all claims, including those where Z = 0.
Insurer's expected payment
E(Y) = E(X) − E(Z) = ∫ from 0 to M of [1 − F(x)] dx
Useful when E(X) is known.
Excess over retention, given a payment
P(Z > z | X > M) = [1 − F(M + z)] ÷ [1 − F(M)]
This gives the distribution of the payment on claims that reach the reinsurer.
Pareto with excess of loss
If X ~ Pareto(a, λ), then X − M | X > M ~ Pareto(a, λ + M)
Same shape a. Scale increases by M. Valid for M ≥ 0.
Exponential with excess of loss
If X ~ Exp(λ), then X − M | X > M ~ Exp(λ)
Memoryless property. Also P(X > M) = e^(−λM).
Pareto mean and moment
E(X) = λ ÷ (a − 1) for a > 1; E(X²) = 2λ² ÷ [(a − 1)(a − 2)] for a > 2
Using the IAI parametrisation with density aλ^a ÷ (λ + x)^(a+1).
Reinsurer's mean on all claims (Pareto)
E(Z) = P(X > M) × E(X − M | X > M) = [λ ÷ (λ + M)]^a × (λ + M) ÷ (a − 1)
Needs a > 1.

How to solve Proportional and Excess of Loss Reinsurance questions

Use this order for any question on quota share or excess of loss. It works for exponential, Pareto, lognormal and other claim distributions.

  1. 1Write down the claim distribution of X and its parameters. Check which parametrisation is used.
  2. 2Identify the type of reinsurance and write the insurer's payment Y and the reinsurer's payment Z in terms of X.
  3. 3Decide what the question asks: all claims (Z can be 0) or only claims that reach the reinsurer (X > M).
  4. 4For quota share, scale. Use E(αX) = αE(X) and Var(αX) = α²Var(X), and rescale the parameters.
  5. 5For excess of loss, find P(X > M) from the survival function. Then find the distribution of X − M given X > M.
  6. 6For a mean, either integrate the survival function or use E(Z) = P(X > M) × E(X − M | X > M). For a variance, find E(Z²) and subtract E(Z)².
  7. 7Check: E(Y) + E(Z) must equal E(X). Also check that the insurer's retained mean is lower than E(X).
  8. 8State the answer clearly with units (rupees) and say whether it is per claim or for all claims.

Quickest way: Shortcut using memoryless and Pareto results

When to use it: Use when X is exponential or Pareto and the question asks for the reinsurer's distribution or mean.

  1. Quota share: multiply the mean by α, the variance by α², and rescale the parameter (λ becomes αλ for Pareto, μ becomes μ + ln α for lognormal).
  2. Excess of loss with exponential X: the excess given X > M is Exp(λ) again, so the conditional mean is 1/λ.
  3. Excess of loss with Pareto X: the excess given X > M is Pareto(a, λ + M), so the conditional mean is (λ + M) ÷ (a − 1).
  4. Multiply the conditional mean by P(X > M) to get the mean over all claims. For Pareto, P(X > M) = [λ ÷ (λ + M)]^a. For exponential, it is e^(−λM).
  5. Insurer's mean = E(X) − reinsurer's mean. Do not recompute from scratch.

Common mistakes in Proportional and Excess of Loss Reinsurance

  • Using α instead of α² for the variance under quota share.

    Students copy the rule for the mean.

    Fix: Variance is in squared units. Var(αX) = α²Var(X). The standard deviation scales by α.

  • Writing the reinsurer's payment as X − M for every claim.

    Students forget that claims below M cost the reinsurer nothing.

    Fix: Write Z = max(X − M, 0). Say whether you are conditioning on X > M.

  • Mixing up the conditional mean with the mean over all claims.

    Both are called the reinsurer's expected claim.

    Fix: Conditional mean is E(X − M | X > M). The mean over all claims is that times P(X > M).

  • Keeping the Pareto scale unchanged after an excess of loss.

    Students remember that the shape stays the same and assume the scale does too.

    Fix: The scale becomes λ + M. Only the shape a stays the same. For quota share the scale becomes αλ.

  • Applying the Pareto mean formula when a ≤ 1, or the variance when a ≤ 2.

    The conditions are forgotten.

    Fix: The mean needs a > 1 and E(X²) needs a > 2. Check this before computing. Note the excess over M has the same shape, so the same conditions apply.

  • Claiming that quota share changes the shape of the distribution.

    Students think all reinsurance reshapes the tail.

    Fix: Quota share only rescales. Excess of loss truncates the insurer's payment at M, which changes the shape.

Worked examples

Example 1

Claims X on a portfolio follow a Pareto distribution with a = 3 and λ = ₹60,000. The insurer has an excess of loss treaty with retention ₹40,000 per claim. Find (i) the probability a claim reaches the reinsurer, (ii) the distribution of the reinsurer's payment given that it is non-zero, and (iii) the expected reinsurer payment per claim, and the insurer's expected payment per claim.

Show the solution
  1. (i) P(X > M) = [λ ÷ (λ + M)]^a = [60,000 ÷ 100,000]^3 = 0.6^3 = 0.216.
  2. (ii) Given X > M, the excess X − M is Pareto with the same a = 3 and scale λ + M = ₹1,00,000.
  3. (iii) Conditional mean = (λ + M) ÷ (a − 1) = 1,00,000 ÷ 2 = ₹50,000.
  4. Expected reinsurer payment per claim = 0.216 × 50,000 = ₹10,800.
  5. E(X) = λ ÷ (a − 1) = 60,000 ÷ 2 = ₹30,000.
  6. Insurer's expected payment = 30,000 − 10,800 = ₹19,200.

Answer: (i) 0.216. (ii) Pareto(3, ₹1,00,000). (iii) Reinsurer's expected payment is ₹10,800 per claim and insurer's is ₹19,200 per claim.

Example 2

Claim amounts X are exponential with mean ₹20,000. Compare (a) a 25% quota share (insurer keeps 75%) with (b) excess of loss with retention ₹20,000. For (a), find the insurer's mean and variance. For (b), find the reinsurer's expected payment per claim and the insurer's expected payment per claim.

Show the solution
  1. X ~ Exp(λ) with λ = 1/20,000. E(X) = 20,000 and Var(X) = 20,000² = 40,00,00,000 (4 × 10⁸).
  2. (a) Insurer keeps α = 0.75. Mean = 0.75 × 20,000 = ₹15,000.
  3. Variance = 0.75² × 4 × 10⁸ = 0.5625 × 4 × 10⁸ = 2.25 × 10⁸, which is 22,50,00,000.
  4. (b) P(X > M) = e^(−λM) = e^(−1) ≈ 0.3679.
  5. By the memoryless property, X − M given X > M is Exp(λ) with mean ₹20,000.
  6. Reinsurer's expected payment per claim = 0.3679 × 20,000 ≈ ₹7,358.
  7. Insurer's expected payment = 20,000 − 7,358 ≈ ₹12,642. Check: E(min(X, M)) = 20,000 × (1 − e^(−1)) = 20,000 × 0.6321 ≈ 12,642. This agrees.

Answer: (a) Insurer's mean ₹15,000 and variance ₹² 22,50,00,000. (b) Reinsurer's expected payment ≈ ₹7,358 per claim and the insurer's ≈ ₹12,642 per claim.

Exam tips

  • Always write Y and Z in terms of X first. Marks are given for setting up the payments correctly.
  • Read whether the question asks for payments on all claims or only non-zero ones. State your choice in the answer.
  • For Pareto and exponential questions, use the closed-form conditional results. Do not integrate unless asked to derive them.
  • Show the check E(Y) + E(Z) = E(X). It catches arithmetic slips.
  • In computer-based papers, define M and α as named variables, and calculate the survival probability and mean in separate lines so the working is clear.

Practice questions from Loss distributions, with and without risk sharing

Proportional and Excess of Loss Reinsurance in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Proportional and Excess of Loss Reinsurance: frequently asked questions

What is the main difference between proportional and excess of loss reinsurance?

Proportional reinsurance shares every claim in a fixed ratio, so the reinsurer pays part of each claim, big or small. Excess of loss reinsurance pays only the part of a claim above a retention. Excess of loss gives more protection against large claims.

How do I find the reinsurer's claim distribution under excess of loss with Pareto claims?

First find P(X > M) = [λ ÷ (λ + M)]^a. Given X > M, the excess X − M is Pareto with the same shape a and scale λ + M. Combine these for the mean over all claims.

How does quota share affect the mean and variance of claims?

The insurer's mean is multiplied by α and its variance by α². The reinsurer's mean and variance use 1 − α in the same way. The distribution keeps its family, with a rescaled parameter.

Does excess of loss reduce variance more than quota share?

It usually does for the same expected reinsurance cost, because it removes the largest claims from the insurer's account. Say this only when comparing treaties of equal expected cost; do not claim it for every pair of treaties.