David K.H. Begg, Gianluigi Vernasca-Economics-McGraw Hill Higher Education (2011)

12.2 Insurance and risk
Life assurance companies take in premium payments in exchange for a promise to pay a large amount to
the family if the insured person dies. The company can make this promise with great certainty because it
pools risks over many clients. Since the company cannot guarantee that exactly 1 per cent of its many
55-year-olds will die in any one year, there is a small element of residual risk for the company to bear, and
it makes a small charge for this in calculating its premiums. However, the company's ability to pool the risk
means that it will make only a small charge. If life assurance companies try to charge more, new entrants
join the industry knowing that the profits more than compensate for the small residual risk to be borne.
Risk pooling does not work when all individuals face the same risk. Suppose there
is a 10 per cent risk of a nuclear war in Europe in the next ten years. If it happens,
everyone in Europe dies, leaving money to their nearest surviving relative in the
rest of the world. Ten million people in Western Europe offer to buy insurance
from an American company.
Risk pooling aggregates
independent risks to make
the aggregate more certain.
Despite the number of people, the risk cannot be pooled. If everybody in Europe dies, if anybody dies, the
insurance company either pays out to everybody's relatives or it pays nothing. In the aggregate there is still
a 10 per cent chance of having to pay out, just as individual Europeans face a 10 per cent chance of disaster.
When the same thing happens to everybody, if it happens at all, the aggregate behaves like the individual.
There is no risk reduction from pooling.
Many insurance companies do not insure against what they call 'acts of God' - floods, earthquakes,
epidemics. Such disasters are no more natural or unnatural than a heart attack. But they affect large
numbers of the insurance company's clients if they happen at all. The risk cannot be reduced by pooling.
Companies cannot quote the low premium rates that apply for heart attacks, where risks are independent
and the aggregate outcome is fairly certain.
There is another way to reduce the cost of risk bearing. This is known as risk
sharing, and the most famous example is the Lloyd's insurance market in London.
Risk sharing is necessary when it has proved impossible to reduce the risk by
pooling. Lloyd's offers insurance on earthquakes in California, and insurance of a
film star's legs.
Risk sharing works by
reducing the stake.
To understand risk sharing we return to diminishing marginal utility. We argued that the utility benefit
from an extra £10 000 is less than the utility sacrificed when £10 000 is given up. However, this difference
in marginal utility for equivalent monetary gains and losses is tiny if the size of the stake is tiny. The
marginal utility from an extra £1 is only fractionally less than the utility lost by sacrificing £1. For small
stakes, people are almost risk-neutral. You would probably toss a coin with us to win or lose £0.10, but not
to win or lose £10 000. The larger the stake, the more diminishing marginal utility bites.
You go to Lloyd's to insure the US space shuttle launch for £20 billion - a big risk. Only part of this risk
can be pooled as part of a larger portfolio of risks. It is too big for anyone to take on at a reasonable
premium.
The Lloyd's market in London has hundreds of 'syndicates: each a group of 20 or so individuals who have
each put up £100 000. Each syndicate takes perhaps 1 per cent of the £20 billion deal and then resells the
risk to yet other people in the insurance industry. By the time the deal has been subdivided and subdivided
again, each syndicate or insurance company holds a tiny share of the total. And each syndicate risk is
further subdivided among its 20 members. The risk is shared out until each individual's stake is so small
that there is a tiny difference between the marginal utility from a gain and the marginal loss of utility in the
event of a disaster. It now takes only a small premium to cover this risk. The package can be sold to the
client at a premium low enough to attract the business.
By pooling and sharing risks, insurance allows individuals to deal with many risks at affordable premiums.
But two things inhibit the operation of insurance markets, reducing the extent to which individuals can
use insurance to buy their way out of risky situations.
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