CHARACTERISTICS OF AN INSURABLE RISK

We have stated previously that individuals see the purchase of insurance as economically
advantageous. The insurer will agree to the arrangement if the risks can be pooled, but will need
some safeguards. With these principles in mind, what makes a risk insurable? What kinds of risk
would an insurer be willing to insure?
The potential loss must be significant and important enough that substituting a known insurance
premium for an unknown economic outcome (given no insurance) is desirable.
The loss and its economic value must be well-defined and out of the policyholder’s control. The
policyholder should not be allowed to cause or encourage a loss that will lead to a benefit or claim
payment. After the loss occurs, the policyholder should not be able to unfairly adjust the value of
the loss (for example, by lying) in order to increase the amount of the benefit or claim payment.
Covered losses should be reasonably independent. The fact that one policyholder experiences a
loss should not have a major effect on whether other policyholders do. For example, an insurer
would not insure all the stores in one area against fire, because a fire in one store could spread to
the others, resulting in many large claim payments to be made by the insurer.
These criteria, if fully satisfied, mean that the risk is insurable. The fact that a potential loss does
not fully satisfy the criteria does not necessarily mean that insurance will not be issued, but some
special care or additional risk sharing with other insurers may be necessary.

A MATHEMATICAL EXPLANATION

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

HOW INSURANCE WORKS

Insurance is an agreement where, for a stipulated payment called the premium, one party (the insurer) agrees to pay to the other (the policyholder or his designated beneficiary) a defined amount (the claim payment or benefit) upon the occurrence of a specific loss. This defined claim payment amount can be a fixed amount or can reimburse all or a part of the loss that occurred.
The insurer considers the losses expected for the insurance pool and the potential for variation in order to charge premiums that, in total, will be sufficient to cover all of the projected claim payments for the insurance pool. The premium charged to each of the pool participants is that participant’s share of the total premium for the pool. Each premium may be adjusted to reflect any special characteristics of the particular policy. As will be seen in the next section, the larger the policy pool, the more predictable its results.
Normally, only a small percentage of policyholders suffer losses. Their losses are paid out of the premiums collected from the pool of policyholders. Thus, the entire pool compensates the unfortunate few. Each policyholder exchanges an unknown loss for the payment of a known premium.
Under the formal arrangement, the party agreeing to make the claim payments is the insurance company or the insurer. The pool participant is the policyholder. The payments that the policyholder makes to the insurer are premiums. The insurance contract is the policy. The risk of any unanticipated losses is transferred from the policyholder to the insurer who has the right to specify the rules and conditions for participating in the insurance pool.
The insurer may restrict the particular kinds of losses covered. For example, a peril is a potential cause of a loss. Perils may include fires, hurricanes, theft, and heart attack. The insurance policy
may define specific perils that are covered, or it may cover all perils with certain named exclusions (for example, loss as a result of war or loss of life due to suicide).
Hazards are conditions that increase the probability or expected magnitude of a loss. Examples include smoking when considering potential healthcare losses, poor wiring in a house when
considering losses due to fires, or a California residence when considering earthquake damage.
In summary, an insurance contract covers a policyholder for economic loss caused by a peril named in the policy. The policyholder pays a known premium to have the insurer guarantee
payment for the unknown loss. In this manner, the policyholder transfers the economic risk to the insurance company. Risk, as discussed in Section I, is the variation in potential economic
outcomes. It is measured by the variation between possible outcomes and the expected outcome: the greater the standard deviation, the greater the risk.

RISK AND INSURANCE

People seek security. A sense of security may be the next basic goal after food, clothing, and
shelter. An individual with economic security is fairly certain that he can satisfy his needs (food,
shelter, medical care, and so on) in the present and in the future. Economic risk (which we will
refer to simply as risk) is the possibility of losing economic security. Most economic risk derives
from variation from the expected outcome.
One measure of risk, used in this study note, is the standard deviation of the possible outcomes.
As an example, consider the cost of a car accident for two different cars, a Porsche and a Toyota.
In the event of an accident the expected value of repairs for both cars is 2500. However, the
standard deviation for the Porsche is 1000 and the standard deviation for the Toyota is 400. If the
cost of repairs is normally distributed, then the probability that the repairs will cost more than
3000 is 31% for the Porsche but only 11% for the Toyota.
Modern society provides many examples of risk. A homeowner faces a large potential for
variation associated with the possibility of economic loss caused by a house fire. A driver faces a
potential economic loss if his car is damaged. A larger possible economic risk exists with respect
to potential damages a driver might have to pay if he injures a third party in a car accident for
which he is responsible.
Historically, economic risk was managed through informal agreements within a defined
community. If someone’s barn burned down and a herd of milking cows was destroyed, the
community would pitch in to rebuild the barn and to provide the farmer with enough cows to
replenish the milking stock. This cooperative (pooling) concept became formalized in the
insurance industry. Under a formal insurance arrangement, each insurance policy purchaser
(policyholder) still implicitly pools his risk with all other policyholders. However, it is no longer
necessary for any individual policyholder to know or have any direct connection with any other
policyholder.

Insurance

It is good business management to protect the assets of your business (including the owner) against unforeseen
events. This protection usually comes in the form of
insurance.
An insurance policy may be broadly defined as a contract
under which the insurer agrees, in return for a premium, to
indemnify the insured for loss suffered as a result of the
occurrence of specified events which cause the
destruction, loss or injury of something in which the
insured has an interest.
Here is a list of some of the types of business insurance
available.
Building
Generally only required if you own the premises in which
your business is located.
NOTE: It is common for a tenant to be required to insure
plate glass against breakage in leased premises.
If you are conducting a business from your home, the
insurer of your home should be notified so that the policy
can be noted and the premium adjusted if necessary.
Contents
This should cover all plant, equipment, fixtures and fittings
as well as business stock.
Cash
It is possible to insure against theft of cash held on the
business premises, in transit or held at home.
Loss of profit
If your business is unable to trade, due to the occurrence
of an insurable event, this cover can provide for loss of
profits incurred during the period of non-trading. Of course
you will have to provide evidence of actual profits lost.
Public liability
Provides insurance cover against accidental injury to
clients, customers and visitors to your business premises.
This cover is considered essential to all businesses and
cover of up to $5M is normal for a small retail operation.
Product liability
This indemnifies the manufacturer and/or distributor against
injury caused by their product, or the use of their product.
Professional indemnity
Provides insurance protection against negligence for
professionals and experts delivering services, advice and
information to clients.
Motor vehicle
Vehicles owned and used by the business should be
insured, as would your own personal vehicles. If you are
using your personal vehicle in the business, the insurer
should be notified so that the policy can be noted.
Key person
It is common in small business for the successful operation
of the business to hinge on the well being of one person.
Insurance cover can be taken against loss of income, in the
event of injury, disability or death to the key person.
Workers’ compensation
Accident and sickness insurance cover must be provided
for employees through an approved insurer. Self-employed
persons provide accident and sickness insurance through a
private insurer.
Superannuation
It is important for all people to provide for their retirement
years, including business owners. Superannuation is
generally the vehicle used to provide for a retirement plan.
For detailed advice and information on the type and cost of
insurance to best suit your business, you should consult
reputable insurance agents or brokers. Alternatively, speak
to your Industry Association for referrals to an agent who
can provide the right insurance.