Why do financial institutions buy insurance? There is something counter-intuitive about one financial institution agreeing to guarantee the risk of another financial institution.

To understand why any corporation buys insurance it is important to understand the economics and motivations behind a typical insurance transaction.

## Expected values and risk aversion

### Expected value

Understanding the economic drivers behind insurance requires knowledge of two principles – *expected value* and *risk aversion*.

Expected values have to do with prediction and probability[i]. A $1 million policy protecting against a risk that has a 5% probability of occurring – has an expected value of $50,000.

As an investor you would never buy an asset for more than its expected value, yet institutions often pay more than the expected value for insurance. The difference is risk aversion.

### Risk aversion

Many institutions are unwilling to risk losing a million dollars in a single year and would rather pay $50,000 (and some margin of profit for the insurer) than risk the volatility in their financials that would be caused by a million-dollar loss.

A loss in the first few years means the insured benefits from having paid less premium than their loss; if the event occurs later, or not at all, the insurer benefits from the collective premiums.

Simply put: buying insurance is hard to justify using the theory of expected value. Insurance is a transaction with a negative expected value, in dollar terms. It is only when we factor in the theory of risk aversion that we can explain why institutions ignore the principle of expected values when buying insurance.

### Roulette

Consider a simple risk aversion analogy. A player must play two spins on the roulette table. The first spin, he bets $100 on a single number – the payoff is 35:1 and he wins! The player is now ahead $3,500. This was a good bet using expected value rationale because he risked some amount and the return was proportional to his risk.

For the second spin, the player must bet his entire $3,500 with no potential to win more—but if the same number comes up again, he loses it all. The value of protecting his assets (his recently won $3500) is really the issue. What would it be worth to the player if the roulette dealer offers to ‘insure’ him and pay him back any losses?

Expected value theory tells us that the risk is 1:35 so the player should expect to pay $100. However, the $3,500 may be a large sum to our player and he may be averse to losing his hard-won gains. Would he pay $200 to be assured of keeping $3,300 ($3,500 less the $200 premium)? Perhaps he would pay much more depending on his need and potential utility[ii] for the proceeds.

The key difference between insurance and gambling is that with insurance you pay the insurer to assume your risk, with gambling you pay the casino for the right to assume risk – and hopefully a potential jackpot.

## Insurance companies are also risk averse

An insurance company is able to assume risk because insurance spreads risk amongst a large pool of counterparties.

The *law of large numbers* is a principle of probability. It states that as the number of events increase, the *actual ratio* of outcomes will converge on the *expected ratio* of outcomes. Simply put, the larger the sample size (in this case the number of insured parties) the more likely it is that the results will match the forecasted outcome. For example, if we forecast that half of the time the roulette ball will land on black, we have a better chance of actually achieving that result if we spin the wheel 1000 times versus 10 times. With millions of customers and comprehensive actuarial data, insurers have a very large sample size and are able to accurately forecast the expected outcome. They then use this forecast to develop their models and pricing for the policies they offer.

This is mathematics’ way of saying that an insurance company can do what an individual cannot – namely take on large amounts of diversified risk with intent of paying out on only a manageable number of claims. By insuring many, underwriters get the benefit of the law of large numbers; the larger an insurance company is the more likely they are to achieve the expected results.

### Premium-to-surplus ratio

Insurers are required to demonstrate that they have the financial strength and solvency to pay claims that may come due.

One measure of an insurance company’s solvency is its *premium-to-surplus ratio.* This ratio is computed by dividing net premiums written by surplus. An insurance company’s surplus is the amount by which assets exceed liabilities (similar to a corporation’s retained earnings).

The amount of premium[iii] written is a better measure than the total amount insured because the level of premiums is linked to the likelihood of claims. The lower the ratio, the greater the company’s financial strength.

Insurers may in turn purchase insurance to reduce specific exposures[iv] as a means of risk management.

## Understanding the value in insurance

It is not uncommon to hear people complain about paying insurance premiums. They argue that the risk is small or the premium is not proportional to the risk. No one likes paying for protection, which they hope will go unused. Understanding the reasoning behind risk aversion and your institution’s specific risk appetite are crucial to making sense of your firm’s risk management program and subsequent insurance purchases.

[i] Insured and insurer may have different estimates of probability; however it is generally presumed that the insurer will have more complete data regarding potential losses.

[ii] *Utility* is another term that is central to understanding the value of insurance. *Expected utility theorem* tries to unify the idea that different individuals or entities may place different values on the same results and the mathematical notion of expected value. This field of game theory or decision theory is interesting, but beyond the scope of this article.

[iii] State regulators have generally established a premium-to-surplus ratio of no higher than 3-to-1 as a guideline.

[iv] Such insurance is called reinsurance and can be purchased on a transactional basis or on a broader program basis, called facultative or treaty basis respectively.