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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