Insurance and reinsurance · RiskAtoms Academy

By the end you will be able to

1Explain how insurance and reinsurance transfer risk.
2Calculate what is ceded and kept under quota share and excess of loss treaties.
3Compare quota share with excess of loss on a loss distribution: the average, the worst case and the spread.
4Explain the limits of reinsurance: limits that run out, cost, and the reinsurer's own credit risk.

Why it matters

A storm can cause more claims in one night than an insurer collects in premiums in a year. Insurers survive such nights because they have passed part of the risk on to someone else. Who takes it, and on what terms, is the business of reinsurance.

Core explanation

Passing risk along the chain

Insurance. A household or a company pays a premium, and the insurer pays its losses: the transfer response of L2.6.01: Four responses to risk. The insurer can do this because of pooling (L1.2.03: Insurers: life and non-life): one house fire is unpredictable, but the number of fires among a hundred thousand houses is not. The policyholder still keeps part of the risk through the deductible, and anything above the policy limit.

Where pooling stops. Pooling needs risks that are independent. A large factory is too big to pool with houses, and a hurricane hits thousands of houses at once. These are the risks that threaten an insurer (L2.2.06: Insurance risk), and they are why insurers buy reinsurance. The insurer cedes part of its risk to a reinsurer, which pools it with risks from insurers around the world.

Quota share (proportional)

The reinsurer takes a fixed share, say 40%, of every premium and every claim. In return it pays a ceding commission. It shrinks the whole book, small claims and large, and frees capital for growth.

Excess of loss (non-proportional)

The reinsurer pays the part of each loss above a retention, up to a limit. It leaves the small claims alone and cuts off the large ones. Applied to one event's total claims, it is catastrophe cover.

What changes, and what does not. Quota share changes the size of the risk but not its shape: every loss, and so the average, the spread and the worst case, falls by the same percentage. Excess of loss changes the shape: the average falls a little, the worst cases a lot. That is why it is the usual tool against catastrophes, and why it costs more per euro of expected loss ceded: the reinsurer takes the most uncertain part.

What remains. A limit can run out, and losses above it come back to the insurer. The reinsurer may fail to pay, so the insurer swaps insurance risk for counterparty risk. And reinsurance prices rise after big losses, just when cover is most wanted.

Quota share or excess of loss?

4 cards

A young motor insurer is growing fast and needs to support more business than its capital allows.
A property insurer in Florida fears a hurricane causing hundreds of millions of claims in one event.
An insurer has one huge factory on its books that could produce a single claim larger than its annual profit.
An insurer is entering a new line of business and wants a partner to share both its profits and its losses.

The formal version

For a loss $X$ , a quota share with ceded share $q$ , and an excess of loss layer of limit $L$ above retention $R$ (“L xs R”):

\text{Quota share: ceded} = qX \qquad \text{kept} = (1-q)X

\text{Excess of loss: ceded} = \min\big(\max(X - R,\,0),\,L\big) \qquad \text{kept} = X - \text{ceded}

Under quota share the mean and the standard deviation of what is kept both fall by the factor $(1-q)$ . Under excess of loss, what is kept is at most $R$ for any loss up to $R + L$ ; only losses above the top of the layer push it higher.

Conventions here

Amounts in € millions, illustrative; one claim at a time for per-risk cover. The figures use the same reinsurance engine as the site's Reinsurance calculator.

Try it

Eight large claims from one year. Start with excess of loss, €5m xs €2m: the reinsurer only touches the claims above €2m. Raise the limit until the €9.5m claim is fully covered. Then switch to quota share and compare the worst loss kept.

Try itReinsurance layers

Reinsurance calculator →

Treaty

Retention€2.00m

Limit€5.00m

This year's large claims, €m insurer keeps reinsurer pays

This year

Gross claims€20.40m

Reinsurer pays€7.60m

Insurer keeps€12.80m

Largest kept€4.50m

Largest risk, a year

Expected kept (gross 0.385)€0.260m

Worst kept (gross 25)€20.00m

Excess of loss applies to each claim separately. The bottom figures use the loss distribution in the worked example. Illustrative amounts.

Worked example

Two treaties, one loss distribution

A commercial property insurer looks at the loss from its largest risk in a year. It compares a 40% quota share with an excess of loss of €18m xs €2m. Illustrative figures:

Loss	Chance	Kept: no treaty	Kept: 40% quota share	Kept: €18m xs €2m
€0m	90.0%	0.0	0.0	0.0
€1m	6.0%	1.0	0.6	1.0
€4m	2.5%	4.0	2.4	2.0
€10m	1.0%	10.0	6.0	2.0
€25m	0.5%	25.0	15.0	7.0

Average, spread and worst case

Step 1 of 5

1
Expected loss
Gross: 0.06 × 1 + 0.025 × 4 + 0.01 × 10 + 0.005 × 25 = €0.385m a year.

The excess of loss treaty cedes a little more expected loss, but it cuts the spread and the worst case far more. For protecting capital against large losses, it is the sharper tool; for sharing a whole book, the quota share is.

Regulatory lens · looking ahead

Under Solvency II, reinsurance reduces the capital an insurer must hold for underwriting and catastrophe risk, but adds a charge for the risk that the reinsurer defaults (M4.6: Solvency II and insurance regulation). Under IFRS 17, reinsurance contracts held are measured separately from the policies they protect (M4.8: Insurance accounting: IFRS 17 (overview)). How insurers design their reinsurance programmes returns in M3.7: Insurance risk.

Sparks

SparkCatastrophe bonds

Some insurers pass catastrophe risk straight to investors. A catastrophe bond pays a high coupon, and the money raised is held in a trust. If a defined event happens, such as a hurricane above a set size in a set region, investors lose some or all of their principal and the insurer receives it. Cat bonds appeared in the mid-1990s, after Hurricane Andrew in 1992 showed how much catastrophe risk the reinsurance market could absorb. See the catastrophe return period calculator for how such events are measured.

SparkThe reinsurer's reinsurer

Reinsurers buy reinsurance too, called retrocession. Risk can travel along a chain from a homeowner to an insurer, a reinsurer and a retrocessionaire. The chain spreads risk widely, but it can also make it hard to see who finally holds it.

Key takeaways

Insurance transfers risk from policyholders to insurers, which rely on pooling independent risks.
Reinsurance transfers what pooling cannot handle: very large single risks and catastrophes.
Quota share cedes the same share of every premium and claim; it shrinks the risk without changing its shape.
Excess of loss pays min(max(X − R, 0), L); it cuts the tail, so it protects capital against large losses.
Limits run out, cover costs more after big losses, and reinsurance adds counterparty risk.

Reaction

Check what stuck

Glossary

Pooling: Combining many similar, independent risks so that the total loss becomes far more predictable than any one of them.
Reinsurance: Insurance for insurers: an insurer (the cedant) passes part of its risk to a reinsurer for a premium.
Cede: To pass risk to a reinsurer. The insurer that does so is the cedant; the part passed on is ceded.
Proportional reinsurance: Reinsurance in which the reinsurer takes a fixed share of each policy's premiums and claims.
Quota share: A proportional treaty in which the reinsurer takes the same percentage of every premium and every claim.
Ceding commission: What a quota share reinsurer pays the insurer back out of the ceded premium, to cover the cost of selling and running the policies.
Excess of loss: Non-proportional reinsurance that pays the part of a loss above a retention, up to a limit.
Retention: The amount of each loss the insurer pays itself before an excess of loss treaty pays. Also called the priority or attachment point.
Limit: The most an excess of loss treaty pays on one loss. A layer is written 'limit xs retention', for example €5m xs €2m.
Catastrophe cover: Excess of loss cover for the total of many claims caused by one event, such as a storm or an earthquake.
Retrocession: Reinsurance bought by a reinsurer.

Deep Dives & Origins

Deep Dives

Swiss Re, The essential guide to reinsurance (2013 edition). Introductory.
Rob Kaas, Marc Goovaerts, Jan Dhaene and Michel Denuit, Modern Actuarial Risk Theory, 2nd edition (Springer, 2008), the chapters on reinsurance. Advanced.

Origins

Swiss Re, The essential guide to reinsurance, 2013.
Directive 2009/138/EC (Solvency II), Articles 101 and 105, on the Solvency Capital Requirement and risk mitigation.
IASB, IFRS 17 Insurance Contracts (2017), the requirements for reinsurance contracts held.

Watch this lesson

Video version

Prefer to watch? The same lesson as a 10-minute video will sit here, made from this page's script.