What makes a risk insurable

CM2 · Financial Engineering & Loss Reserving · 13 min read

The question

An insurer is asked to cover something. Before any pricing can start there is a prior question: can this be insured at all? Some risks cannot, and not because nobody has thought of a price. They fail a condition that has to hold before a price means anything.

This is the half of the risk topic that carries no formulae, which makes it tempting to skip and makes it reliable marks. The conditions are short, the interesting part is the cases that nearly satisfy them, and the arithmetic below shows that "nearly" can be worth a great deal of money.

The conditions

1. The policyholder must have an interest in the risk. They must stand to lose if it happens. Without this, insurance is a bet, and insuring a building you do not own gives you a reason to want it to burn down. This is the condition that makes insurance a different activity from gambling, and in most countries it is also a legal requirement rather than only a commercial one.

2. The risk must be of a financial and reasonably quantifiable nature. The loss has to be measurable in money. Grief, reputation and inconvenience are real losses and are not insurable as such, which is why policies pay for named, countable things — a funeral cost rather than a bereavement, a day's revenue rather than embarrassment.

3. The amount payable must bear some relationship to the loss. Otherwise the policyholder either gains from the event, which is condition 1 again, or is left short, which makes the cover worth less than it appears. Indemnity policies pay the loss; benefit policies pay a fixed sum and are allowed only where the loss is hard to measure but genuinely proportional, as with a sum assured on death.

4. The event must have a small probability. Not because large probabilities cannot be priced, but because a near-certain event is better paid for out of income than insured: the premium would be most of the loss, and the insurer's expenses and margin would be added on top. Insurance is a device for rare, large losses, and routine, predictable costs belong in a budget.

5. Risks must be independent, or close to it, and there must be enough of them. This is the condition that actually makes the business work, and it is the law of large numbers: pooling many independent risks makes the average claim per policy predictable even though each policy is not. The coefficient of variation of the average falls like one over the square root of the number of policies, so a pool of ten thousand similar risks has a hundredth of the relative uncertainty of a single one.

6. There must be an ultimate limit on the liability. An insurer cannot accept an unbounded loss, because no amount of capital covers it. This is why policies carry sums insured, limits and exclusions, and why unlimited cover — motor third-party liability in some jurisdictions — is unusual and closely regulated.

7. Moral hazard must be eliminated as far as possible.

8. There must be sufficient existing statistical data.

The last two are the ones worth real attention, and they are where the interesting failures live.

Moral hazard, and why a deductible is the answer

Moral hazard is the change in behaviour that insurance itself causes. A fully insured person takes less care, because the cost of carelessness has been transferred. The risk the insurer actually carries is therefore not the risk it measured before the policy existed.

Note what this is not. It is not dishonesty, although fraud is an extreme form of it. It is the ordinary and entirely rational response of someone who no longer bears the consequence — parking in a worse street, replacing a worn roof later, driving a little faster.

The standard answer is to leave the policyholder with some of the loss, which restores some of the incentive:

a deductible or excess: the policyholder pays the first part of every claim;
co-insurance: the policyholder pays a share of every claim;
a no-claims discount, which makes a claim expensive in future premiums rather than now;
policy conditions requiring specific care, such as approved locks or an alarm.

The arithmetic below is the point of this page, because it shows that a deductible is worth considerably more than the claims it removes. It cuts what the insurer pays on each claim, and it also cuts how often claims happen — and the second effect is usually the larger one.

Data, and adverse selection

Condition 8 looks dull and is the one that stops new products. Pricing needs an estimate of frequency and severity, and for a genuinely new risk there is none — which is why cover for emerging risks is written narrowly, with low limits, until experience exists.

The related failure is adverse selection: the policyholder knows more about their own risk than the insurer does, so at any single premium the worse risks find the cover better value and buy more of it. The pool that turns up is then worse than the pool that was priced.

The worked example prices this, and the striking part is how fast it moves. A premium set on the average of two groups is more than three times what the better group's own risk costs, so they decline it; and once they have gone the remaining pool costs far more than the premium collects. Raise the premium to cover that and the next-best group leaves. This is the adverse selection spiral, and it is the mechanism by which a market for a product can disappear entirely even though every individual risk in it was insurable.

The defences are underwriting, which is the insurer buying information; compulsory cover, which removes the choice to leave; and pooling at the level of an employer or a state, which does the same thing. The economics of selection are treated more fully on the CB2 page on adverse selection and moral hazard; what matters here is which insurability condition each one threatens.

The risks that fail, and which condition they fail

A question will give a list of proposed covers and ask which are insurable. The answer is always to name the condition.

A business's loss of profit from a pandemic. Fails independence — every policy claims at once, so pooling does nothing. This is why pandemic business-interruption cover was largely withdrawn, and why such risks end up with governments, which can pool across time instead of across policyholders.
Cover against a fall in the stock market. Fails independence for the same reason, and it is anyway available more cheaply as a traded put option. Where a risk is already traded, insurance is a worse wrapper for it.
Insuring a competitor's factory burning down. Fails insurable interest: the buyer gains from the event.
Cover for a new technology with no loss history. Fails the data condition. Insurable in principle, unpriceable in practice, so it is written small while data accumulates.
Insuring a routine annual maintenance cost. Fails the small-probability condition. The event is near certain, so the premium is the cost plus expenses, and the policyholder is worse off than if they had simply paid it.
Unlimited liability for an environmental clean-up. Fails the limit condition, which is why such cover carries caps and exclusions and why the uncovered part lands on the state.

Where marks are lost

Listing the conditions without applying them. A question that gives four proposed products wants four verdicts, each naming the condition at issue. The list on its own earns little.

Confusing moral hazard with adverse selection. Moral hazard is about behaviour after the policy is written; adverse selection is about who buys it in the first place. Deductibles address the first; underwriting addresses the second. A deductible does help a little against selection, because the worst risks dislike it most, and saying so is a good answer rather than a confused one.

Saying a correlated risk is uninsurable. It is harder and more expensive to insure, not impossible. Catastrophe reinsurance, catastrophe bonds and government backstops all exist to carry exactly these risks. What correlation destroys is the law-of-large-numbers argument, so the capital needed rises enormously — and that is the right thing to say.

Treating the conditions as all-or-nothing. Almost every real risk fails one of them slightly. Insurance is written anyway, with the policy designed around the failure — a deductible for moral hazard, a limit for unbounded liability, underwriting for selection. The design is the answer to the question.

Questions students ask

Is life insurance really an indemnity?

No, and it is the standard exception. A death benefit is a fixed sum rather than a measured loss, because the financial loss from a death cannot be measured in the way a burnt building can. It survives condition 3 because the sum assured is chosen to bear a sensible relationship to the dependants' loss, and it survives condition 1 because the law requires an insurable interest — which is why you cannot insure the life of a stranger.

Why is a deductible worth more than the claims it removes?

Because it changes two things at once. It removes the first part of every claim, which is a straightforward saving, and it gives the policyholder a reason to take care, which reduces how often claims happen at all. On the example below the frequency effect is two and a half times the size of the severity effect, and it is the part the policyholder's own behaviour produces.

Does genetic testing break insurability?

It threatens the data condition in an unusual direction: it is adverse selection with the information asymmetry made sharper. If applicants know their results and insurers may not use them, the worse risks buy more cover at a premium set for the average, and the spiral in the example below is the consequence. This is why the use of genetic results in underwriting is restricted by agreement or law in many countries, and why those restrictions are usually paired with limits on the sum assured — capping the amount caps the selection.

How many risks are "enough"?

It depends on how variable each one is, and the square root is the guide. The relative uncertainty of the average claim falls like one over the root of the number of policies, so a hundred policies have a tenth of the relative uncertainty of one and ten thousand have a hundredth. For a class with a few large claims, "enough" is many thousands; for a homogeneous class of small claims it is far fewer. That is the law of large numbers doing the work that the capital would otherwise have to do.

A worked example

Two pieces of arithmetic, one for adverse selection and one for moral hazard. Every figure is invented for this example — there is no real portfolio or product here.

Part 1: what adverse selection costs

A risk of a ₹5,00,000 loss. The population splits in equal numbers into two groups, and each person knows which they are in while the insurer does not:

Group	Probability of a loss	Their own expected claim
Low risk	0.02	₹10,000
High risk	0.10	₹50,000

The insurer cannot tell them apart, so it prices the average with a 20% loading:

pooled expected claim = 0.5 × 10,000 + 0.5 × 50,000 = ₹30,000
pooled premium = 1.20 × 30,000 = ₹36,000

How that premium looks to each group

to a low risk: 36,000 / 10,000 = 3.60 times their own expected cost
to a high risk: 36,000 / 50,000 = 0.72 times theirs

A low risk is being asked to pay three and a half times what their risk costs. Unless they are very risk averse indeed they decline. A high risk is offered cover at 72 paise in the rupee and buys as much as is available. The insurer has not made a pricing error; it has made an information error, and the market has sorted itself in response.

What happens when the low risks have gone

true expected claim of the remaining pool = ₹50,000
premium being collected = ₹36,000
loss per policy = ₹14,000

The insurer is now losing ₹14,000 on every policy it writes, and nothing about any individual risk has changed. Raise the premium to 1.20 × 50,000 = ₹60,000 and the book is sound again — but if there had been a third, middling group, they would now be the ones facing a premium far above their own cost, and they would leave in turn. That is the spiral, and each turn of it is the same arithmetic.

What underwriting is worth

If the insurer can tell the groups apart, it charges each its own price:

low risk: 1.20 × 10,000 = ₹12,000
high risk: 1.20 × 50,000 = ₹60,000

Both groups are now insurable and both buy. The value of the underwriting is not that it raises revenue — it is that it keeps the low risks in the market at all. Everything an insurer spends on proposal forms, medical evidence and risk questions is buying its way out of the ₹14,000 above, and that figure is what tells you how much it is worth spending.

Part 2: what a deductible is worth

Now take the high-risk group, with a 0.10 chance of a ₹5,00,000 loss, and add a deductible of ₹50,000. Suppose that having ₹50,000 of their own money at stake makes policyholders more careful, so the probability of a loss falls from 0.10 to 0.08. That behavioural response is an assumption, invented for this example; the whole point of the calculation is that its size matters.

	Full cover	With a ₹50,000 deductible
Probability of a loss	0.10	0.08
Insurer pays per claim	₹5,00,000	₹4,50,000
Insurer expected cost	₹50,000	₹36,000
Policyholder expected cost	nil	₹4,000
Total expected cost of the risk	₹50,000	₹40,000

Splitting the insurer's ₹14,000 saving

The saving comes from two separate effects and it is worth decomposing, because they are not the same size and only one of them is the deductible itself.

frequency effect: (0.10 − 0.08) × 5,00,000 = ₹10,000
severity effect: 0.08 × 50,000 = ₹4,000
total = ₹14,000

And the direct calculation agrees: 50,000 − 36,000 = ₹14,000. Two routes, one answer.

Read the split. Only ₹4,000 of the ₹14,000 is the deductible mechanically removing part of each claim. ₹10,000 — nearly three-quarters of it — is the change in behaviour. That is moral hazard being undone, and it is the reason deductibles exist. If care had not improved at all, the saving would have been ₹5,000 rather than ₹14,000.

Is the policyholder better off?

At a 20% loading the premium falls from 1.20 × 50,000 = ₹60,000 to 1.20 × 36,000 = ₹43,200.

premium saved = ₹16,800
expected out-of-pocket on the deductible = ₹4,000
net expected gain to the policyholder = ₹12,800

So the policyholder gives up ₹4,000 of expected cover and is paid ₹16,800 for it. They are ₹12,800 a year better off in expectation and they have taken on a ₹50,000 exposure they did not have before — which is precisely the trade the utility theory at the start of this paper is about, and whether they should accept it depends on their risk aversion and on whether they could find ₹50,000 at short notice.

Notice also where the extra value came from. The total expected cost of the risk fell from ₹50,000 to ₹40,000 — ₹10,000 of real resources that are no longer destroyed, because the losses did not happen. Insurance did not move that ₹10,000 from one party to the other; the policy design created it, by giving somebody a reason to be careful. That is the strongest argument there is for deductibles, and it is why a policy with no deductible at all is usually a badly designed policy rather than a generous one.

Where this fits in CM2

This page explains one idea from CM2 Financial Engineering & Loss Reserving. The full CM2 syllabus, with what each part is examined on, sits on the course page, and past-paper style mock exams with full solutions are where you find out whether the idea has stuck under time pressure.

What makes a risk insurable, and what to do when it almost is