526 Chapter 13 Insurance and Risk Management
Now suppose you flip the coin 10 times. All heads or all tails is still within the realm of
possibility, but far less likely to happen than with only two flips. Still, though, you would
not be surprised to get four heads and six tails, or seven heads and three tails, and so on.
With 10 flips we still don’t necessarily expect the proportion to necessarily be 50%; but
we do have a much greater expectation of it being closer to 50% with 10 flips than we do
with two.
Suppose now that we flip the coin 1,000,000 times. All heads or all tails is again still
possible, but the likelihood of that happening is so small that it’s not worth considering.
It is still possible to get 30% or 40% heads or tails, but the chance of that happening
seems pretty remote. We still don’t necessarily expect an exactly 50/50 split—499,563
heads/500,437 tails certainly would not be a surprising outcome, for example. But with so
many flips, even though we can’t expect exactly 50%, we most certainly can expect the
proportion of heads or tails to be very close to 50%. This is the idea of the law of large
numbers—the greater the number of coin flips, the more predictable the proportions of
heads and tails.
Of course, we don’t buy insurance for coin flips. But the law of large numbers works
just as well with the things we do buy insurance for. For example, it is unlikely that any
one specific person will become disabled in a given year. But people do become disabled
all the time. Though the likelihood that you will be one of the people who does face dis-
ability in the next year may be small, the financial consequences of being unable to work
for a living could be disastrous to you. If you buy a disability insurance policy, neither
you nor the insurer can know in advance whether or not you will be one of the people who
does become disabled. But if the company sells lots of disability policies, it can reasonably
predict how many of its policyholders will face disability, even though it can’t predict the
specific ones. So the company can make a reasonable projection of how much it will need
to pay out in claims overall and can use this projection to determine how much premium it
will need to charge to be able to pay those claims. By buying a disability insurance policy,
you can take advantage of the insurance company’s ability to take advantage of the law of
large numbers to protect yourself financially.
As a simple illustration of how this works, imagine that an insurance company offers
a policy that will pay the policyholder a benefit of $200,000 if he becomes permanently
disabled within the next year.^2 From past experience, the company expects that 1 out of
every 5,000 policyholders will be entitled to make a claim against their policies. This
works out to $200,000(1/5,000) $200,000/5,000 $40 per policy. Thus, if the insurer
collects $40 from each policyholder, and if the number of policies sold is large enough that
we can expect the proportion of actual claims to match this 1 in 5,000 frequency, the premi-
ums collected should be enough to provide for the predicted claims expense. Even though
on any one individual policy the insurer would either pocket the entire $40 or pay out a
$200,000 claim, overall the premiums collected and claims paid would be expected to bal-
ance out. This $40 per policy is sometimes referred to as the pure premium^3 for the policy.
The actual premium charged to the policyholders would of course be higher, though, since
the company must also add on charges for its operating expenses, agents’ commissions,
taxes, profit, and other costs.
Even though the company expected the rate of claims to be 1 claim per 5,000 policies,
it is possible that more or fewer than the predicted number of claims will occur. The pure
premium does not guarantee that premiums will balance out with claims. The actual cost of
claims divided by the number of policies sold could work out to be more, or less, than $40.
Yet, the more of these policies they sell, the closer to the predicted $40 we would expect(^2) This is not entirely realistic. A disability policy would be much more likely to pay a certain income per week or
month while the person is disabled, not a single lump sum payment. The insurer would then need to take into
account not only the likelihood of someone fi ling a claim but also the likely duration of the disability. However,
this simple example will serve to illustrate the concept just fi ne without getting bogged down.
(^3) This is also referred to as the expected claims cost. This can be confusing at fi rst, though, because the word
expected is being used in a special technical sense that does not mean quite the same thing as it does in ordinary
speech. This use of the term expected is discussed in more detail in Chapter 16.