Losses depend on two random variables. The first is the number of losses that will occur in a
specified period. For example, a healthy policyholder with hospital insurance will have no losses
in most years, but in some years he could have one or more accidents or illnesses requiring
hospitalization. This random variable for the number of losses is commonly referred to as the
frequency of loss and its probability distribution is called the frequency distribution. The second
random variable is the amount of the loss, given that a loss has occurred. For example, the
hospital charges for an overnight hospital stay would be much lower than the charges for an
extended hospitalization. The amount of loss is often referred to as the severity and the probability
distribution for the amount of loss is called the severity distribution. By combining the frequency
distribution with the severity distribution we can determine the overall loss distribution.
Example: Consider a car owner who has an 80% chance of no accidents in a year, a 20%
chance of being in a single accident in a year, and no chance of being in more than one accident
in a year. For simplicity, assume that there is a 50% probability that after the accident the car
will need repairs costing 500, a 40% probability that the repairs will cost 5000, and a 10%
probability that the car will need to be replaced, which will cost 15,000. Combining the frequency
and severity distributions forms the following distribution of the random variable X, loss due to
accident:
The car owner’s expected loss is the mean of this distribution, E X :
E[X ] =Σx ⋅ f (x) = 0.80⋅0 + 0.10⋅500 + 0.08⋅5000 + 0.02⋅15,000 =750
On average, the car owner spends 750 on repairs due to car accidents. A 750 loss may not seem
like much to the car owner, but the possibility of a 5000 or 15,000 loss could create real concern.
To measure the potential variability of the car owner’s loss, consider the standard deviation of the
loss distribution:
If we look at a particular individual, we see that there can be an extremely large variation in
possible outcomes, each with a specific economic consequence. By purchasing an insurance
policy, the individual transfers this risk to an insurance company in exchange for a fixed premium.
We might conclude, therefore, that if an insurer sells n policies to n individuals, it assumes the
total risk of the n individuals. In reality, the risk assumed by the insurer is smaller in total than the
sum of the risks associated with each individual policyholder. These results are shown in the
following theorem.
Theorem: Let X X Xn
1 2 , ,..., be independent random variables such that each Xi has an
expected value of μ and variance of σ 2 . Let n n S = X + X + ...+ X 1 2 . Then:
E[S ] n E[X ] nμ n i = ⋅ = , and
Var[S ] = n ⋅Var[X ] = n ⋅σ 2 n i .
The standard deviation of Sn is n ⋅σ , which is less than nσ, the sum of the standard deviations
for each policy.
Furthermore, the coefficient of variation, which is the ratio of the standard deviation to the mean,
, the coefficient of variation for each individual Xi.
The coefficient of variation is useful for comparing variability between positive distributions with
different expected values. So, given n independent policyholders, as n becomes very large, the
insurer’s risk, as measured by the coefficient of variation, tends to zero.
Example: Going back to our example of the car owner, consider an insurance company that will
reimburse repair costs resulting from accidents for 100 car owners, each with the same risks as in
our earlier example. Each car owner has an expected loss of 750 and a standard deviation of
2442. As a group the expected loss is 75,000 and the variance is 596,250,000. The standard
deviation is 596,250,000 = 24,418which is significantly less than the sum of the standard
deviations, 244,182. The ratio of the standard deviation to the expected loss is
24,418 75,000 = 0.326 , which is significantly less than the ratio of 2442 750 = 3.26 for one car
owner.
It should be clear that the existence of a private insurance industry in and of itself does not
decrease the frequency or severity of loss. Viewed another way, merely entering into an insurance
contract does not change the policyholder’s expectation of loss. Thus, given perfect information,
the amount that any policyholder should have to pay an insurer equals the expected claim
payments plus an amount to cover the insurer’s expenses for selling and servicing the policy,
including some profit. The expected amount of claim payments is called the net premium or
benefit premium. The term gross premium refers to the total of the net premium and the amount to
cover the insurer’s expenses and a margin for unanticipated claim payments.
Example: Again considering the 100 car owners, if the insurer will pay for all of the accidentrelated
car repair losses, the insurer should collect a premium of at least 75,000 because that is
the expected amount of claim payments to policyholders. The net premium or benefit premium
would amount to 750 per policy. The insurer might charge the policyholders an additional 30%
so that there would be 22,500 to help the insurer pay expenses related to the insurance policies
and cover any unanticipated claim payments. In this case 750θ130%=975 would be the gross
premium for a policy.
Policyholders are willing to pay a gross premium for an insurance contract, which exceeds the
expected value of their losses, in order to substitute the fixed, zero-variance premium payment for
an unmanageable amount of risk inherent in not insuring
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