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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
    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
    3
    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.
    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
    4
    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:
    f x
    x x x x
    ( )
    . . .
    . ,
    =
    = = = =
    R S
    ||
    T
    ||
    080 0
    010 500
    0 08 5000
    0 02 15 000
    The car owner’s expected loss is the mean of this distribution, E X :
    E X x f x [ ] ( ) . . . . , = ⋅ = ⋅ + ⋅ + ⋅ + ⋅ = ∑ 0 80 0 010 500 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:
    σ σ
    X
    X
    x E X f x 2 2
    2 2 2 2 0 80 750 010 250 0 08 4250 0 02 14 250 5 962 500
    5 962 500 2442
    = −
    = ⋅ − + ⋅ − + ⋅ + ⋅ =
    = =
    ∑ [ ] ( )
    . ( ) . ( ) . ( ) . ( , ) , ,
    , ,
    b g
    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 σ2 . Let n n X ... X X S + + + = 2 1 . Then:
    [ ] [ ] μ n X E n S E i n = ⋅ = , and
    [ ] [ ] 2 σ ⋅ = ⋅ = n X Var n S Var i n .
    The standard deviation of Sn is σ ⋅ n , which is less than n σ, 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 n
    n n
    =
    σ μ
    σ
    μ
    . This is smaller than
    σ μ
    , the coefficient of variation for each individual Xi .
    5
    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 750 326 = . 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.
    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.
    6
    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,
    and so on.
    7
    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
    savings.
    (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
    company.
    (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.
    8
    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:
    f y
    loss or y
    loss y
    loss y
    ( )
    . .
    . , ,
    =
    = =
    = =
    = =
    R
    S|
    T|
    0 90 0 500 0
    0 08 5000 4500
    0 02 15 000 14 500
    The expected claim payments and standard deviation for one policy would be:
    E Y
    Y Y
    [ ] . . . ,
    . ( ) . ( ) . ( , ) , ,
    , ,
    = ⋅ + ⋅ + ⋅ =
    = ⋅ − + ⋅ + ⋅ =
    = =
    0 90 0 0 08 4500 0 02 14 500 650
    0 90 650 0 08 3850 0 02 13 850 5 402 500
    5 402 500 2324
    2 2 2 2 σ
    σ
    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
    capacity.
    (2) In the context of risk, an upper bound to the benefit lessens the risk assumed by the
    insurer.
    (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
    9
    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:
    f y
    loss or y
    loss y
    loss y
    ( )
    . .
    . , ,
    =
    = =
    = =
    = =
    R
    S|
    T|
    0 90 0 500 0
    0 08 5000 4500
    0 02 15 000 12 500
    The expected claim payments and standard deviation for one policy would be:
    E Y
    Y Y
    [ ] . . . ,
    . ( ) . ( ) . ( , ) , ,
    , ,
    = ⋅ + ⋅ + ⋅ =
    = ⋅ − + ⋅ + ⋅ =
    = =
    090 0 008 4500 002 12 500 610
    090 610 008 3890 002 11890 4 372 900
    4 372 900 2091
    2 2 2 2 σ
    σ
    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 5000θ1.10=5500 in year 2; a loss of 5000θ(1.10)2=6050 in year 3; and a
    loss of 5000θ(1.10)3=6655 in year 4.
    10
    The claim payment distributions, expected losses, expected claim payments, and standard
    deviations for each policy are:
    Policy with a 500 Deductible
    f(y,t) 0.80 0.10 0.08 0.02
    Expected
    Amount
    Standard
    Deviation
    Year 1
    Loss 0 500 5000 15,000 750
    Claim 0 0 4500 14,500 650 2324
    Year 2
    Loss 0 550 5500 16,500 825
    Claim 0 50 5000 16,000 725 2568
    Year 3
    Loss 0 605 6050 18,150 908
    Claim 0 105 5550 17,650 808 2836
    Year 4
    Loss 0 666 6655 19,965 998
    Claim 0 166 6155 19,465 898 3131
    Year 5
    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.
    11
    Policy with a Deductible of 500 and Maximum Claim Payment of 12,500
    f(y,t) 0.80 0.10 0.08 0.02
    Expected
    Amount
    Standard
    Deviation
    Year 1
    Loss 0 500 5000 15,000 750
    Claim 0 0 4500 12,500 610 2091
    Year 2
    Loss 0 550 5500 16,500 825
    Claim 0 50 5000 12,500 655 2167
    Year 3
    Loss 0 605 6050 18,150 908
    Claim 0 105 5550 12,500 705 2257
    Year 4
    Loss 0 666 6655 19,965 998
    Claim 0 166 6155 12,500 759 2363
    Year 5
    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 ( = ) = . 1 015.
    Once an individual is hospitalized, the charges X have a probability density function (p.d.f.)
    f xH e X
    x = = − 1 01 0 1 c h . . 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.
    12
    The expected value of hospital charges is:
    E X P H E X H P H E X H
    x e dx x e e dx
    e
    x x x
    x
    = ≠ ≠ + = =
    = ⋅ + ⋅ = − ⋅ +
    = − ⋅ ⋅ =
    z z
    1 1 1 1
    085 0 015 01 015 015
    015 10 15
    0 1
    0
    0 1
    0
    0 1
    0
    0 1
    0
    b g b g
    . . . . .
    . .
    . . .
    .
    E X P H E X H P H E X H
    x e dx
    x e x e dx
    x
    x x
    2 2 2
    2 2 01
    0
    2 01
    0 0
    0 1
    1 1 1 1
    085 0 015 01
    015 015 10 01 2 30
    = ≠ ≠ + = =
    = ⋅ + ⋅
    = − ⋅ + ⋅ ⋅ ⋅ =
    ∞ ∞
    z
    z
    b g b g
    . . .
    . . .
    .
    . .
    The variance is: σX E X E X 2 2 2 2 30 15 27 75 = − = − = c h b g. .
    The standard deviation is: σX= = 27 75 527 . .
    The coefficient of variation is: σX E X / . /. . = = 527 15 351
    An alternative solution would recognize and use the fact that f XH X = 1 c his 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 Xi
    i
    =
    =
    ∑1
    200
    E S E X = = 200 300
    σ σ S X
    2 2 200 5550 = =
    σ σ S X = = 200 74 50 .
    13
    Coefficient of variation = = σS E S [ ] . 025
    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:
    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 yX H
    f y H
    P X H
    e
    P X H Y
    X
    y
    > = =
    + =
    > =
    =
    > =
    − +
    5 1
    5 1
    5 1
    01
    5 1
    0 1 5
    , . . ( ) c h c h
    c h c h
    Summing the three cases:
    E Y P H E Y H P X H E Y X H P X H E Y X H
    P H P H P X H P H P X H E Y X H
    y e dy e y e dy
    e
    y y
    = ≠ ≠ + ≤ = ≤ = + > = > =
    = ≠ ⋅ + = ⋅ ≤ = ⋅ + = ⋅ > = ⋅ > =
    = ⋅ = ⋅ ⋅
    = ⋅ ⋅ =
    − +
    − −
    z z
    1 1 5 1 5 1 5 1 5 1
    1 0 1 5 1 0 1 5 1 5 1
    015 01 015 01
    015 10 0 91
    0 1 5
    0
    0 5 0 1
    0
    0 5
    b g b g b g
    b g b gc h b gc h
    , [ , ] , ,
    ,
    . . . .
    . .
    . ( ) . .
    .
    Claim Payment with Deductible=5
    Y=max(0,X-5)
    0
    1
    2
    3
    4
    5
    0 2 4 6 8 10
    Hospital Charges (X)
    Claim Payment (
    Y)
    14
    ( ) ( ) ( ) ( ) 0.1 5 2 2 2 2
    0
    0.5 2 0.1
    0
    0.5
    1 0 1 5 1 0 0.15 0.1
    0.15 0.1
    30 18.20
    y
    y
    E Y P H P H P X H y e dy
    e y e dy
    e
    − +
    − −
    ⎡ ⎤ = ≠ ⋅ + = ⋅ ≤ = ⋅ + ⋅ ⎣ ⎦
    = ⋅ ⋅
    = =
    Y
    2 2 18 20 0 91 17 37 σ = − = . . . b g
    σY= = 17 37 417 . .
    For the pool of 200 individuals, let S Y Y i
    i
    =
    =
    ∑1
    200
    E S E Y Y = = 200 182
    σ σ S Y Y
    2 2 200 3474 = =
    σ σ S Y Y
    = = 200 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 S ES Y Y 80% 08 146 ⋅ = ⋅ = .
    σ σ 80%
    2 2 2 08 2223 S S Y Y
    = ⋅ = . b g
    σ σ 80% 08 4715 S S Y Y
    = ⋅ = . .
    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
    E
    15
    (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 σ then the mean of a large random sample of size n is
    approximately normally distributed with mean p and standard deviation
    σ n
    . 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
    payments.
    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.
    These are some of the common actuarial projects done for businesses facing risk. In addition,
    actuaries are involved in the design of new financial products, company management and strategic
    planning.
    X. CONCLUSION
    This study note is an introduction to the ideas and concepts behind actuarial work. The examples
    have been restricted to insurance, though many of the concepts can be applied to any situation
    where uncertain events create financial risks.
    Later Casualty Actuarial Society and Society of Actuaries exams cover topics including:
    adjustment for investment earnings; frequency models; severity models; aggregate loss models;
    survival models; fitting models to actual data; and the credibility that can be attributed to past data.
    In addition, both societies offer courses on the nature of particular perils and related business
    issues that need to be considered.

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