Economic Modelling · Role of insurance in reducing or removing risk

Reinsurance and Risk Reduction for Insurers

Q: How does reinsurance reduce insurer risk?

It passes part of the claim cost to a reinsurer, so the insurer's retained claims are smaller or capped. This lowers variance, protects against large or accumulated losses and reduces the capital needed. The insurer pays a premium for this, so expected profit usually falls.

Q: What is the difference between proportional and non-proportional reinsurance?

In proportional reinsurance the reinsurer takes a fixed share of premiums and claims, as in quota share. In non-proportional reinsurance the reinsurer pays only when a claim or total claims exceed a retention, as in excess of loss and stop loss.

Q: What risks remain after reinsurance?

The insurer still bears retained claims, the risk that the reinsurer cannot pay, basis risk where the cover does not match the loss, and the cost of the premium. Errors in assumptions about claims also remain.

Q: Is reinsurance the same as diversification?

No. Diversification reduces risk by spreading it across many different exposures within the insurer's own book. Reinsurance transfers some of that risk to another company. Insurers normally use both, and hold capital for what is left.

Updated 11 October 2026 · Fact-checked

Reinsurance is insurance bought by an insurer. It passes part of the claim cost to a reinsurer in return for a premium. This cuts the variability of claims, protects capital and limits large losses. You solve questions by finding the insurer's retained claim, then comparing mean, variance and residual risks.

Understand Reinsurance and Risk Reduction for Insurers

An insurer takes on many risks. Pooling and the law of large numbers reduce the relative variability of total claims, but they never remove it. Large single claims, catastrophes that hit many policies at once, and errors in the assumed claim distribution all remain.

Reinsurance is the insurer's own risk transfer. The insurer (the cedant) pays a reinsurance premium. The reinsurer pays part of each claim or part of total claims. Reinsurance reduces the insurer's risk, lowers the capital it needs and can let it write more business.

There are two broad types. In proportional reinsurance, the reinsurer takes a fixed share of premiums and claims. Quota share and surplus are the main forms. In non-proportional reinsurance, the reinsurer pays only when claims pass a threshold. Excess of loss (per claim) and stop loss (on aggregate claims) are the main forms.

The two types do different jobs. Proportional cover scales down all risk and mostly reduces total exposure. It reduces the variance by a known factor but does not cut the shape of the distribution. Excess of loss cuts the large claims and so trims the tail directly. Stop loss protects against a bad year in total.

Reinsurance does not remove all risk. Residual risk includes the retained claims, reinsurer credit risk (the reinsurer may fail to pay), basis risk where the cover does not match the loss, the cost of the premium (which reduces expected profit), and risk from wrong assumptions. Insurers also use diversification across lines, regions and reinsurers, and hold capital to absorb the remaining losses.

Key rules to remember

Quota share, retained claim: Retained claim = a × X, where 0 < a ≤ 1; The insurer keeps proportion a of every claim X. The reinsurer pays (1 − a)X. Premium is also shared in the same proportion, before any reinsurance commission.
Quota share moments: E(aX) = a E(X); Var(aX) = a² Var(X); SD(aX) = a SD(X); Variance falls by a², so the coefficient of variation stays the same.
Individual excess of loss, retained claim: Retained = min(X, M); Reinsurer pays max(X − M, 0); M is the retention. The insurer pays all of a claim up to M and the reinsurer pays the excess.
Stop loss, retained aggregate claim: Retained = min(S, d); Reinsurer pays max(S − d, 0); S is total claims in the period and d is the aggregate retention.
Expected reinsurer payment, excess of loss: E[max(X − M, 0)] = ∫ from M to ∞ of (1 − F(x)) dx; For a continuous or general claim X ≥ 0. For Exp(λ), this equals e^(−λM) ÷ λ.
Reinsurer's premium: Reinsurance premium = (1 + θ) × expected reinsurer payment; θ is the reinsurer's loading. If θ is above the insurer's own loading, reinsurance reduces expected profit.

How to solve Reinsurance and Risk Reduction for Insurers questions

Use this method for numerical and discussion questions on reinsurance and residual risk.

1Identify the cover type: quota share, surplus, individual excess of loss or stop loss. Note what the retention or share is.
2Write the insurer's retained claim and the reinsurer's payment as formulas in terms of the claim X or aggregate claim S.
3Find the expected value and variance of the retained amount. Use a and a² for quota share. Use integrals or sums for the excess of loss.
4Work out the reinsurance premium using the stated loading, and subtract it from the insurer's income to find net expected profit.
5Compare the position with and without reinsurance. Look at the mean, variance, and any tail measure or ruin measure asked for.
6State the residual risks that remain: retained claims, reinsurer default, basis risk, cost and assumption error.
7Add context if asked: diversification, capital held and the effect on the amount of business the insurer can write.
8Check units and signs, and answer the exact question asked.

Quickest way: Retained claim first, then mean and variance

When to use it: Use this for MCQs and short calculations where you must compare the insurer's risk before and after reinsurance.

Write retained = what the insurer pays. For quota share, retained = aX. For excess of loss, retained = min(X, M).
For quota share, scale: mean by a, variance by a², SD by a.
For excess of loss, compute the reinsurer's mean first, then retained mean = E(X) − reinsurer's mean.
Net profit = premium received − retained expected claims − reinsurance premium paid, plus any commission if given.
For a discussion, name one benefit and one residual risk for each type.

Common mistakes in Reinsurance and Risk Reduction for Insurers

Scaling variance by a instead of a² under quota share.
Students remember that the mean scales by a and copy this to the variance.
Fix: Variance is of squared units. Var(aX) = a² Var(X). Only the SD scales by a.
Treating excess of loss as paying the whole claim once it exceeds the retention.
The word 'excess' is read loosely as 'if above M'.
Fix: The reinsurer pays only the part above M: max(X − M, 0). The insurer always keeps the first M.
Claiming reinsurance removes all risk.
Students focus on the transfer and ignore what stays.
Fix: Always list residual risk: retained claims, reinsurer credit risk, basis risk, cost and model error.
Ignoring the reinsurance premium when judging profit.
The calculation stops at the lower variance.
Fix: Subtract the reinsurance premium. A loading above the insurer's own loading reduces expected profit, so state the trade-off.
Mixing up proportional and non-proportional features.
Both reduce claims, so the names blur.
Fix: Proportional: fixed share of premiums and claims. Non-proportional: payment depends on claim size or total, with a retention.
Saying diversification and reinsurance are the same thing.
Both lower risk.
Fix: Diversification spreads risks inside the insurer's own portfolio. Reinsurance moves risk to another party. Capital is the buffer for what is left.

Worked examples

Example 1

An insurer faces individual claims X with E(X) = ₹40,000 and SD(X) = ₹30,000. It takes a 70% quota share, keeping 30% of every claim. (a) Find the mean and SD of the retained claim. (b) State whether the coefficient of variation changes.

Show the solution

The retained proportion is a = 0.3. Retained claim = 0.3X.
Mean = 0.3 × 40,000 = ₹12,000.
SD = 0.3 × 30,000 = ₹9,000. Variance = 0.09 × 30,000² = 8,10,00,000 (rupees squared), whose root is 9,000, which is consistent.
CV before = 30,000 ÷ 40,000 = 0.75. CV after = 9,000 ÷ 12,000 = 0.75.

Answer: Retained mean is ₹12,000 and retained SD is ₹9,000. The coefficient of variation stays at 0.75, so quota share cuts the scale of risk but not its relative variability.

Example 2

Claims X are exponential with mean ₹50,000. The insurer buys individual excess of loss with retention M = ₹1,00,000. The reinsurer charges 30% above its expected payment. Find the expected claim paid by the reinsurer per claim, the reinsurance premium per claim and the insurer's expected retained claim.

Show the solution

For Exp(λ) with mean 50,000, λ = 1 ÷ 50,000.
Expected reinsurer payment = e^(−λM) ÷ λ = 50,000 × e^(−2), since λM = 1,00,000 ÷ 50,000 = 2.
e^(−2) = 0.135335, so the expected payment = 50,000 × 0.135335 = ₹6,766.76.
Reinsurance premium = 1.3 × 6,766.76 = ₹8,796.79.
Expected retained claim = 50,000 − 6,766.76 = ₹43,233.24.

Answer: Expected reinsurer payment is about ₹6,767 per claim. Reinsurance premium is about ₹8,797 per claim. Expected retained claim is about ₹43,233.

Exam tips

For calculations, always define retained claim and reinsurer payment as formulas before you use numbers. This earns method marks even if arithmetic slips.
In discussion questions, give both sides: the benefit (lower variance, tail protection, capital relief) and the residual risk (default, basis risk, cost).
Know which cover fits which problem: quota share for overall scale, excess of loss for large claims, stop loss for a bad total year.
In computer-based questions, show the simulated retained claims, the formula used and the summary statistic, and state your assumptions about the claim distribution.
Link to ruin theory when asked: reinsurance changes the premium income net of cost and the claim distribution, so it changes the adjustment coefficient.

Practice questions from Role of insurance in reducing or removing risk