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RISK AND INSURANCE Member SOA

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RISK AND INSURANCE I. INTRODUCTION People seek security. A sense of security may be the next basic goal after food, clothing, and ... a Porsche and a Toyota.



RISK AND INSURANCE
I INTRODUCTION
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
II 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
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
III 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
0 10 x 500
0 08 x 5000
0 02 x 15 000
The car owner s expected loss is the mean of this distribution E X
E X x 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
X2 x E X f x
0 80 750 2 0 10 250 2 0 08 4250 2 0 02 14 250 2 5 962 500
X 5 962 500 2442
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 X1 X 2 X n be independent random variables such that each X i has an expected
value of and variance of 2 Let Sn X1 X 2 X n Then
E Sn n E X i n and
Var Sn n Var X i n 2
The standard deviation of S n is n which is less than n 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
is This is smaller than the coefficient of variation for each individual X i
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 24 418 which 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 accident
related 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 750 130 975 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
IV 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
V EXAMPLES OF INSURANCE
Some readers of this note may already have used insurance to reduce economic risk In many
places to drive a car legally you must have liability insurance which will pay benefits to a person
that you might injure or for property damage from a car accident You may purchase collision
insurance for your car which will pay toward having your car repaired or replaced in case of an
accident You can also buy coverage that will pay for damage to your car from causes other than
collision for example damage from hailstones or vandalism
Insurance on your residence will pay toward repairing or replacing your home in case of damage
from a covered peril The contents of your house will also be covered in case of damage or theft
However some perils may not be covered For example flood damage may not be covered if
your house is in a floodplain
At some point you will probably consider the purchase of life insurance to provide your family
with additional economic security should you die unexpectedly Generally life insurance provides
for a fixed benefit at death However the benefit may vary over time In addition the length of
the premium payment period and the period during which a death is eligible for a benefit may each
vary Many combinations and variations exist
When it is time to retire you may wish to purchase an annuity that will provide regular income to
meet your expenses A basic form of an annuity is called a life annuity which pays a regular
amount for as long as you live Annuities are the complement of life insurance Since payments
are made until death the peril is survival and the risk you have shifted to the insurer is the risk of
living longer than your savings would last There are also annuities that combine the basic life
annuity with a benefit payable upon death There are many different forms of death benefits that
can be combined with annuities
Disability income insurance replaces all or a portion of your income should you become disabled
Health insurance pays benefits to help offset the costs of medical care hospitalization dental care
Employers may provide many of the insurance coverages listed above to their employees
VI LIMITS ON POLICY BENEFITS
In all types of insurance there may be limits on benefits or claim payments More specifically
there may be a maximum limit on the total reimbursed there may be a minimum limit on losses
that will be reimbursed only a certain percentage of each loss may be reimbursed or there may be
different limits applied to particular types of losses
In each of these situations the insurer does not reimburse the entire loss Rather the policyholder
must cover part of the loss himself This is often referred to as coinsurance
The next two sections discuss specific types of limits on policy benefits
DEDUCTIBLES
A policy may stipulate that losses are to be reimbursed only in excess of a stated threshold
amount called a deductible For example consider insurance that covers a loss resulting from an
accident but includes a 500 deductible If the loss is less than 500 the insurer will not pay
anything to the policyholder On the other hand if the loss is more than 500 the insurer will pay
for the loss in excess of the deductible In other words if the loss is 2000 the insurer will pay
1500 Reasons for deductibles include the following
1 Small losses do not create a claim payment thus saving the expenses of processing the claim
2 Claim payments are reduced by the amount of the deductible which is translated into premium
3 The deductible puts the policyholder at risk and therefore provides an economic incentive for
the policyholder to prevent losses that would lead to claim payments
Problems associated with deductibles include the following
1 The policyholder may be disappointed that losses are not paid in full Certainly deductibles
increase the risk for which the policyholder remains responsible
2 Deductibles can lead to misunderstandings and bad public relations for the insurance company
3 Deductibles may make the marketing of the coverage more difficult for the insurance
4 The policyholder may overstate the loss to recover the deductible
Note that if there is a deductible there is a difference between the value of a loss and the
associated claim payment In fact for a very small loss there will be no claim payment Thus it is
essential to differentiate between losses and claim payments as to both frequency and severity
Example Consider the group of 100 car owners that was discussed earlier If the policy provides
for a 500 deductible what would the expected claim payments and the insurer s risk be
The claim payment distribution for each policy would now be
0 90 loss 0 or 500 y 0
f y 0 08 loss 5000 y 4500
0 02 loss 15 000 y 14 500
The expected claim payments and standard deviation for one policy would be
E Y 0 90 0 0 08 4500 0 02 14 500 650
Y2 0 90 650 2 0 08 3850 2 0 02 13 850 2 5 402 500
Y 5 402 500 2324
The expected claim payments for the hundred policies would be 65 000 the variance would be
540 250 000 and the standard deviation would be 23 243
As shown in this example the presence of the deductible will save the insurer from having to
process the relatively small claim payments of 500 The probability of a claim occurring drops
from 20 to 10 per policy The deductible lowers the expected claim payments for the hundred
policies from 75 000 to 65 000 and the standard deviation will fall from 24 418 to 23 243
BENEFIT LIMITS
A benefit limit sets an upper bound on how much the insurer will pay for any loss Reasons for
placing a limit on the benefits include the following
1 The limit prevents total claim payments from exceeding the insurer s financial
2 In the context of risk an upper bound to the benefit lessens the risk assumed by the
3 Having different benefit limits allows the policyholder to choose appropriate coverage
at an appropriate price since the premium will be lower for lower benefit limits
In general the lower the benefit limit the lower the premium However in some instances the
premium differences are relatively small For example an increase from 1 million to 2 million
liability coverage in an auto policy would result in a very small increase in premium This is
because losses in excess of 1 million are rare events and the premium determined by the insurer is
based primarily on the expected value of the claim payments
As has been implied previously a policy may have more than one limit and overall there is more
than one way to provide limits on benefits Different limits may be set for different perils Limits
might also be set as a percentage of total loss For example a health insurance policy may pay
healthcare costs up to 5000 and it may only reimburse for 80 of these costs In this case if
costs were 6000 the insurance would reimburse 4000 which is 80 of the lesser of 5000 and the
actual cost
Example Looking again at the 100 insured car owners assume that the insurer has not only
included a 500 deductible but has also placed a maximum on a claim payment of 12 500 What
would the expected claim payments and the insurer s risk be
The claim payment distribution for each policy would now be
0 90 loss 0 or 500 y 0
f y 0 08 loss 5000 y 4500
0 02 loss 15 000 y 12 500
The expected claim payments and standard deviation for one policy would be
E Y 0 90 0 0 08 4500 0 02 12 500 610
Y2 0 90 610 2 0 08 3890 2 0 02 11 890 2 4 372 900
Y 4 372 900 2091
The expected claim payments for the hundred policies would be 61 000 the variance would be
437 290 000 and the standard deviation would be 20 911
In this case the presence of the deductible and the benefit limit lowers the insurer s expected
claim payments for the hundred policies from 75 000 to 61 000 and the standard deviation will fall
from 24 418 to 20 911
VII INFLATION
Many insurance policies pay benefits based on the amount of loss at existing price levels When
there is price inflation the claim payments increase accordingly However many deductibles and
benefit limits are expressed in fixed amounts that do not increase automatically as inflation
increases claim payments Thus the impact of inflation is altered when deductibles and other
limits are not adjusted
Example Looking again at the 100 insured car owners with a 500 deductible and no benefit
limit assume that there is 10 annual inflation Over the next 5 years what would the expected
claim payments and the insurer s risk be
Because of the 10 annual inflation in new car and repair costs a 5000 loss in year 1 will be
equivalent to a loss of 5000 1 10 5500 in year 2 a loss of 5000 1 10 2 6050 in year 3 and a
loss of 5000 1 10 3 6655 in year 4
The claim payment distributions expected losses expected claim payments and standard
deviations for each policy are
Policy with a 500 Deductible
Expected Standard
f y t 0 80 0 10 0 08 0 02 Amount Deviation
Loss 0 500 5000 15 000 750
Claim 0 0 4500 14 500 650 2324
Loss 0 550 5500 16 500 825
Claim 0 50 5000 16 000 725 2568
Loss 0 605 6050 18 150 908
Claim 0 105 5550 17 650 808 2836
Loss 0 666 6655 19 965 998
Claim 0 166 6155 19 465 898 3131
Loss 0 732 7321 21 962 1098
Claim 0 232 6821 21 462 998 3456
Looking at the increases from one year to the next the expected losses increase by 10 each year
but the expected claim payments increase by more than 10 annually For example expected
losses grow from 750 in year 1 to 1098 in year 5 an increase of 46 However expected claim
payments grow from 650 in year 1 to 998 in year 5 an increase of 54 Similarly the standard
deviation of claim payments also increases by more than 10 annually Both phenomena are
caused by a deductible that does not increase with inflation
Next consider the effect of inflation if the policy also has a limit setting the maximum claim
payment at 12 500
Policy with a Deductible of 500 and Maximum Claim Payment of 12 500
Expected Standard
f y t 0 80 0 10 0 08 0 02 Amount Deviation
Loss 0 500 5000 15 000 750
Claim 0 0 4500 12 500 610 2091
Loss 0 550 5500 16 500 825
Claim 0 50 5000 12 500 655 2167
Loss 0 605 6050 18 150 908
Claim 0 105 5550 12 500 705 2257
Loss 0 666 6655 19 965 998
Claim 0 166 6155 12 500 759 2363
Loss 0 732 7321 21 962 1098
Claim 0 232 6821 12 500 819 2486
A fixed deductible with no maximum limit exaggerates the effect of inflation Adding a fixed
maximum on claim payments limits the effect of inflation Expected claim payments grow from
610 in year 1 to 819 in year 5 an increase of 34 which is less than the 46 increase in
expected losses Similarly the standard deviation of claim payments increases by less than the
10 annual increase in the standard deviation of losses Both phenomena occur because the
benefit limit does not increase with inflation
VIII A CONTINUOUS SEVERITY EXAMPLE
In the car insurance example we assumed that repair or replacement costs could take only a fixed
number of values In this section we repeat some of the concepts and calculations introduced in
prior sections but in the context of a continuous severity distribution
Consider an insurance policy that reimburses annual hospital charges for an insured individual
The probability of any individual being hospitalized in a year is 15 That is P H 1 0 15
Once an individual is hospitalized the charges X have a probability density function p d f
f X x H 1 0 1e 0 1x for x 0
Determine the expected value the standard deviation and the ratio of the standard deviation to
the mean coefficient of variation of hospital charges for an insured individual
The expected value of hospital charges is
E X P H 1 E X H 1 P H 1 E X H 1
0 85 0 0 15 0 1 x e 0 1x
dx 0 15 x e 0 1x
0 15 e 0 1x dx
0 15 10 e 0 1x 1 5
E X 2 P H 1 E X 2 H 1 P H 1 E X 2 H 1
0 85 02 0 15 0 1 x 2 e 0 1x dx
0 15 10 0 1 2 x e 0 1x dx 30
The variance is X2 E X 2 E X 30 1 5 2 27 75
The standard deviation is X 27 75 5 27
The coefficient of variation is X E X 5 27 1 5 3 51
An alternative solution would recognize and use the fact that f X x H 1 is an exponential
distribution to simplify the calculations
Determine the expected claim payments standard deviation and coefficient of variation for an
insurance pool that reimburses hospital charges for 200 individuals Assume that claims for each
individual are independent of the other individuals
Let S X i Then
E S 200 E X 300 S2 200 X2 5550 and
S 200 X 74 50
The coefficient of variation is
If the insurer includes a deductible of 5 on annual claim payments for each individual what would
the expected claim payments and the standard deviation be for the pool
The relationship of claim payments to hospital charges is shown in the graph below
Claim Payment with Deductible 5
Y max 0 X 5
Claim Payment Y
0 2 4 6 8 10
Hospital Charges X
There are three different cases to consider for an individual
1 There is no hospitalization and thus no claim payments
2 There is hospitalization but the charges are less than the deductible
3 There is hospitalization and the charges are greater than the deductible
In the third case the p d f of claim payments is
f y 5 H 1 0 1e 0 1 y 5
fY y X 5 H 1 X
P X 5 H 1 P X 5 H 1
Summing the three cases
E Y P H 1 E Y H 1 P X 5 H 1 E Y X 5 H 1 P X 5 H 1 E Y X 5 H 1
P H 1 0 P H 1 P X 5 H 1 0 P H 1 P X 5 H 1 E Y X 5 H 1
0 15 0 1 y e 0 1 y 5
dy 0 15 e 0 5
0 15 e 0 5 10 0 91
E Y P H 1 0 P H 1 P X 5 H 1 0 0 15 0 1 y 2 e 0 1 y 5 dy
0 15 e 0 5 0 1y 2 e 0 1 y dy
Y2 18 20 0 91 17 37 and Y 17 37 4 17
For the pool of 200 individuals let SY Yi
Then E SY 200E Y 182 2
SY 200 Y2 3474 and SY 200 Y 58 94
Assume further that the insurer only reimburses 80 of the charges in excess of the 5 deductible
What would the expected claim payments and the standard deviation be for the pool
E 80 SY 0 8 E SY 146 80 SY 0 8 SY 2223 and 80 SY 0 8 SY 47 15
IX THE ROLE OF THE ACTUARY
This study note has outlined some of the fundamentals of insurance Now the question is what is
the role of the actuary
At the most basic level actuaries have the mathematical statistical and business skills needed to
determine the expected costs and risks in any situation where there is financial uncertainty and
data for creating a model of those risks For insurance this includes developing net premiums
benefit premiums gross premiums and the amount of assets the insurer should have on hand to
assure that benefits and expenses can be paid as they arise
The actuary would begin by trying to estimate the frequency and severity distribution for a
particular insurance pool This process usually begins with an analysis of past experience The
actuary will try to use data gathered from the insurance pool or from a group as similar to the
insurance pool as possible For instance if a group of active workers were being insured for
healthcare expenditures the actuary would not want to use data that included disabled or retired
individuals
In analyzing past experience the actuary must also consider how reliable the past experience is as
a predictor of the future Assuming that the experience collected is representative of the insurance
pool the more data the more assurance that it will be a good predictor of the true underlying
probability distributions This is illustrated in the following example
An actuary is trying to determine the underlying probability that a 70 year old woman will die
within one year The actuary gathers data using a large random sample of 70 year old women
from previous years and identifies how many of them died within one year The probability is
estimated by the ratio of the number of deaths in the sample to the total number of 70 year old
women in the sample The Central Limit Theorem tells us that if the underlying distribution has a
mean of p and standard deviation of then the mean of a large random sample of size n is
approximately normally distributed with mean p and standard deviation The larger the size
of the sample the smaller the variation between the sample mean and the underlying value of p
When evaluating past experience the actuary must also watch for fundamental changes that will
alter the underlying probability distributions For example when estimating healthcare costs if
new but expensive techniques for treatment are discovered and implemented then the distribution
of healthcare costs will shift up to reflect the use of the new techniques
The frequency and severity distributions are developed from the analysis of the past experience
and combined to develop the loss distribution The claim payment distribution can then be
derived by adjusting the loss distribution to reflect the provisions in the policies such as
deductibles and benefit limits
If the claim payments could be affected by inflation the actuary will need to estimate future
inflation based on past experience and information about the current state of the economy In the
case of insurance coverages where today s premiums are invested to cover claim payments in the
years to come the actuary will also need to estimate expected investment returns
At this point the actuary has the tools to determine the net premium
The actuary can use similar techniques to estimate a sufficient margin to build into the gross
premium in order to cover both the insurer s expenses and a reasonable level of unanticipated
claim payments
Aside from establishing sufficient premium levels for future risks actuaries also use their skills to
determine whether the insurer s assets on hand are sufficient for the risks that the insurer has
already committed to cover Typically this involves at least two steps The first is to estimate the
current amount of assets necessary for the particular insurance pool The second is to estimate the
flow of claim payments premiums collected expenses and other income to assure that at each
point in time the insurer has enough cash as opposed to long term investments to make the
Actuaries will also do a variety of other projections of the insurer s future financial situation under
given circumstances For instance if an insurer is considering offering a new kind of policy the
actuary will project potential profit or loss The actuary will also use projections to assess
potential difficulties before they become significant


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