How Insurance Companies Use Probability to Make Money

Every insurance policy is a bet. You pay a small certain amount today (the premium) in exchange for a large uncertain payment from the insurer if something bad happens. The insurer is taking the opposite side of that bet — collecting your premium today in exchange for a contingent liability that may or may not crystallise.

For the insurer to stay in business, that bet has to be priced so that — across millions of customers — premiums collected exceed claims paid. In expected value terms, the policy must have negative expected value for you and positive expected value for them. That is the entire industry in one sentence.

This is not a moral problem. It is the mechanism by which an insurer can credibly promise to pay you £200,000 if your house burns down. The negative EV is the price of that promise. What matters is understanding the maths underneath, so you can spot when the price is fair and when it is not.

Actuarial tables are the mathematical core of every insurance product. At their simplest, they are tables of empirical probabilities — for life insurance, the probability that a person of a given age, sex, and health profile will die within the next year. The UK Office for National Statistics publishes the national life tables that every UK life insurer uses as a starting point; the ONS National Life Tables are public data.

For a 40-year-old non-smoker in good health, the published one-year mortality probability is roughly 0.0015 — about 1.5 deaths per 1,000 people in that bracket per year. An insurer offering a £200,000 one-year term policy needs at minimum:

Expected payout = 0.0015 × £200,000 = £300

That is the pure premium — the actuarially fair price of the bet. The actual premium will be higher because the insurer also has to cover operating costs, regulatory capital, profit margin, and the cost of being wrong about the probability. The difference between the actuarially fair price and the price you actually pay is called the loading, and it is where insurer profit comes from.

The table above is illustrative — actual UK term-life pricing in 2026 varies by underwriter and product — but the structure is universal. Every personal-lines insurance product (motor, home, travel, pet) is built the same way. The insurer estimates the probability of a claim, multiplies by the expected claim size, and adds a loading.

The interesting questions are not about the formula. They are about the inputs. How does an insurer know the right probability for you specifically?

An insurer cannot predict whether you specifically will die next year, crash your car, or claim on your travel policy. The 0.0015 mortality rate above does not mean a 0.0015 chance for you — it is an average across a population of similar people. Some are healthier than the average and will not die; some are quietly sicker and will. The insurer does not know which.

What saves the insurer is the law of large numbers — a result from probability theory which says that as the number of independent trials increases, the average outcome converges to the expected value. With one customer, an insurer faces enormous variance: they either pay £200,000 or they pay £0. With 100,000 customers in the same risk class, the number of claims is extremely predictable.

If the true mortality rate is 0.0015, then with 100,000 policies sold the insurer expects 150 claims. The standard deviation of the number of claims is √(100,000 × 0.0015 × 0.9985) ≈ 12.2. So the actual number of claims will land somewhere between roughly 126 and 174 in 95% of years. Total payouts land between £25.2m and £34.8m on a pure-premium revenue of £30m. Profit is reliable; ruin is rare. This is the same reason a casino is happy to take many small bets but would never take one bet for its entire bankroll — see position sizing for the formal version of the argument.

For insurance to work, the risks must be:

1. Numerous — enough policies that the law of large numbers actually kicks in.
2. Independent — one person's claim should not make the next person's claim more likely.
3. Quantifiable — the underlying probability has to be estimable from data.
4. Not catastrophic in aggregate — the total possible payout in a worst-case year must be survivable.

When any of these break down, insurance breaks down with them. Earthquake insurance and pandemic-business-interruption cover are difficult to write because losses are correlated — when one customer claims, many do, and the law of large numbers fails. This is the risk vs uncertainty distinction in action: routine motor and home claims are risk (probabilities are known and stable); pandemic-induced commercial losses are uncertainty (probabilities cannot be reliably estimated from prior data).

Imagine an insurer selling identical £500/year health policies to everyone. Healthy 30-year-olds will look at the price, decide it is bad value, and decline. Chronically ill 60-year-olds will look at the same price, see a bargain, and rush to buy. The pool of people who actually buy is therefore worse than the average of the population the insurer based its price on. Claims come in higher than expected; the insurer raises premiums; even more healthy people leave; the cycle accelerates. This is adverse selection, and it has destroyed insurers that ignored it.

The academic foundation here is George Akerlof's 1970 paper The Market for Lemons, which won him a share of the 2001 Nobel prize in economics. Akerlof's insight: when buyers know more about themselves than sellers do, markets can collapse entirely. Insurers prevent this collapse with three tools:

Underwriting questions. A life insurer will ask about your medical history, smoking, weight, profession, and family history. A motor insurer will ask about driving record, claims history, garaging, and annual mileage. The questions are designed to extract the private information that you have and they do not.

Risk-class pricing. Once they have extracted information, insurers price you into a risk class with similar customers and charge the actuarial rate for that class. A 40-year-old smoker pays substantially more than a 40-year-old non-smoker because the data shows they should.

Mandates and pools. When adverse selection cannot be priced around, the only fix is to force the healthy to buy too. This is why every developed country mandates motor third-party insurance, and why countries with private health insurance (the United States most prominently) typically pair private cover with mandates or subsidised pools.

The screening process produces a base rate for each risk class. Customers often complain that insurer pricing 'unfair' on individual grounds — I have never made a claim, why am I paying so much? The answer is that insurers are pricing your risk class, not you, because that is the only thing the law of large numbers lets them price reliably.

Adverse selection is about who buys. Moral hazard is about how people behave once they have bought. It is the second great information problem in insurance, and it has its own Nobel-winning literature (Bengt Holmström, 2016 prize, on contract theory).

The basic mechanism is simple: insurance reduces the cost of bad outcomes, which reduces the incentive to avoid them. Drivers with comprehensive cover may park more carelessly. Skiers with travel insurance may take a riskier slope. Restaurant owners with business-interruption cover may invest less in fire suppression than if every pound of loss came out of their own pocket. None of this is fraud — it is just the rational response of a person whose private cost of bad outcomes has been partly externalised to the insurer.

Moral hazard is invisible at the individual level but visible in the aggregate. Insurers manage it with three primary tools:

Excess (deductible) and co-insurance. If you pay the first £500 of every claim, you still have skin in the game. The bigger the excess, the more residual risk you carry, and the cheaper the policy.

Limits and exclusions. Most travel policies cap medical evacuation at a defined limit. Most home policies exclude wear and tear, gradual damage, and 'unoccupied' losses. These narrow what counts as a covered loss and reduce the insurer's exposure to behaviours they cannot easily monitor.

Bonus-malus and experience rating. Motor no-claims discounts are the textbook example: every claim-free year reduces your premium, every claim raises it. Customers face direct future-cost consequences of claiming, partially restoring the incentives that insurance otherwise dilutes.

The deeper point is that insurance is not just a money transfer — it is a contract that has to be designed to keep the underlying probabilities stable. If insurance changes behaviour enough that the probability of a claim doubles, the actuarial pricing collapses. The whole machine has to be engineered to prevent that.

There is one more piece of the picture, and it is a big one. Premiums are paid up front. Claims are paid later — sometimes years later for liability and life lines. In the gap, the insurer holds your money. That money is called the float, and it can be invested.

For short-tail business (motor, home), the float is held for months. For long-tail business (life, annuities, employer's liability), it is held for decades. Warren Buffett's letters to Berkshire Hathaway shareholders are the most accessible exposition of the economics of float. Berkshire's insurance subsidiaries have, in many years, generated negative underwriting cost — meaning Berkshire was effectively paid to hold investable float — while also earning the investment return on that float. That double layer is why insurance has historically been a structural source of compounded capital for the patient.

For the customer, the existence of investment float is mostly invisible. But it explains why insurers are willing to write some lines of business at very thin underwriting margins (or even small underwriting losses): the float economics make the deal work even when the policy itself does not.

Once you understand the maths, the natural question is: why play the game? If insurance has negative expected value by construction, why does any rational person buy it?

The answer is that expected value is not the only thing that matters when stakes are large relative to wealth. The mathematical concept here is utility — the subjective value you assign to outcomes — and the key empirical observation is that for most people utility is concave in wealth. The loss of £200,000 hurts roughly fifty times as much as the gain of £200,000 helps, because losing your house ruins your life in a way that a windfall does not symmetrically improve it.

This interacts with loss aversion — the well-documented behavioural pattern where losses loom about twice as large as equivalent gains in subjective experience. But even setting loss aversion aside, the objective concavity of utility justifies paying a premium to eliminate catastrophic outcomes.

Formally: a rational decision-maker should buy insurance whenever the utility-adjusted expected value of insuring exceeds the utility-adjusted expected value of self-insuring. For high-frequency low-severity risks (lost phone, dented bumper), self-insurance usually wins — the loading on small policies is high in percentage terms, and the loss is easily absorbed. For low-frequency high-severity risks (house fire, premature death with dependants, long-term disability), insurance usually wins — the loading is small relative to the catastrophic downside it removes.

The rule of thumb is straightforward: insure against losses you cannot absorb; self-insure against losses you can. The smaller the loading and the bigger the tail, the better the buy.

The pricing machine described above is impressive, but it is not perfect. Three failure modes recur often enough to be worth knowing about.

1. Stale historical data. Actuarial pricing assumes the future will look like the past. When the underlying probability is shifting — climate-driven flood frequency, motor accident rates as cars get safer, mortality during a novel pandemic — published rate tables lag. Customers in shifting risk classes can be over- or under-priced for years before the tables catch up.

2. Crude risk classification. Insurers price the class, not the individual. Within any class, some customers are genuinely lower-risk than the class average; others are higher. If you have private information that you are a lower-than-average risk in your class (no claims history, advanced driving certificate, garaged car), you are subsidising the worse risks in your class. Telematics motor policies are an attempt to break the class down further and price closer to the individual.

3. Anchoring on renewal premiums. UK motor and home insurers have historically loaded renewal premiums on existing customers — what the FCA called the 'loyalty penalty'. The 2022 FCA rules now prohibit charging existing customers more than equivalent new ones, but anchoring bias still does work for the insurer: customers who do not shop around accept the renewal quote and overpay. The empirical advice is universal: get three quotes at every renewal, and treat the renewal letter as a starting bid, not a final price.

A fourth (less common) failure mode is screening that misses a key signal. A medical underwriter who does not ask about a relevant family-history factor is effectively pricing without the information that would put you in a higher-risk class — a false-positive paradox-style mismatch between what is being measured and what actually matters.

Insurance is a probability-pricing business. Insurers estimate the probability of a covered loss for each class of customer, multiply by the expected loss size, and add a loading for costs, capital, and profit. The law of large numbers makes the aggregate cost of claims predictable even though no individual claim is. Adverse selection and moral hazard are managed with underwriting, pricing tiers, excesses, exclusions, and experience rating. Investment float is a second profit engine on top of underwriting margin. For you as a buyer, the only number that matters is whether the loss being insured against is one you cannot absorb. If it is, the negative expected value is the price of removing variance you cannot live with — and that is a rational thing to pay for.