EDUCATION AND
EXAMINATION COMMITTEE
OF THE
SOCIETY OF ACTUARIES
Judy Feldman
Anderson, FSA and Robert L. Brown, FSA
Copyright 2005 by the
Society of Actuaries
The Education and Examination Committee provides
study notes to persons preparing for the examinations of the Society of
Actuaries. They are intended to acquaint candidates with some of the
theoretical and practical considerations involved in the various subjects.
While varying opinions are presented where appropriate, limits on the length of
the material and other considerations sometimes prevent the inclusion of all
possible opinions. These study notes do not, however, represent any official
opinion, interpretations or endorsement of the Society of Actuaries or its
Education and
Examination Committee. The Society is grateful to the
authors for their contributions in preparing the study notes.
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 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
080.

x =
0


R




x =
500


010.


f (x)
= S

x =
5000


_{}0.08


T0.02

x =15,000

The car owner’s expected loss is the mean of this
distribution, E X :
E [ X ] = ∑x ⋅
f ( x) = 080.
⋅ 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^{2}_{X}

= ∑b x − E [ X ]g^{2} f ( x)

= 0.80 ⋅ ( − 750) ^{2} + 010. ⋅ ( − 250) ^{2} + 0.08 ⋅ (4250) ^{2} + 0.02 ⋅ (14 ,250) ^{2} = 5,962 ,500


σ _{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 X_{1}
, X_{2}
,..., X_{n}
be independent random variables such that each X_{i} has an expected
value of μ and variance of σ^{2} . Let S_{n} = X_{1} + X
_{2} +...+
X _{n}
. Then:
E[S _{n} ] =
n ⋅E[ X _{i}
] = nμ
, and
Var[S _{n} ] =
n ⋅Var[ X _{i} ]= n ⋅σ ^{2}
.
The standard deviation of S_{n} is n ⋅σ
, which is less than nσ, the
sum of the standard deviations for each policy.
Furthermore, the coefficient of variation, which is
the ratio of the standard deviation to the mean,
is

n ⋅σ

=

σ

. This is smaller than

σ

, the coefficient of variation for each individual X_{i} .

n
⋅μ

n ⋅μ

μ

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
24,41875,000 =
0.326 , which is significantly less than the ratio of 2442750 =
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, zerovariance
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 welldefined 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.
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.
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


R




loss =
5000

y =
4500


f ( y) = S0.08




loss =
15,000

y =14,500


T0.02

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
σ_{Y}^{2} =
0.90 ⋅ ( − 650) ^{2} + 0.08 ⋅ (3850)
^{2} + 0.02 ⋅ (13,850)
^{2} = 5,402
,500
σ_{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
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
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:


R0.90

loss =
0 or 500

y =
0




loss =
5000

y =
4500


f ( y)
= S0.08




loss =
15,000

y =
12,500


T0.02

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
σ_{Y}^{2} = 090. ⋅ ( −610) ^{2} + 008. ⋅ (3890) ^{2} + 002. ⋅ (11890,) ^{2} =4,372,900
σ_{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 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.
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


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.
for x > 0.

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

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 _{X} cx H = 1h = 01.e^{−}^{0 .1x}
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


b

g

b

g


∞

∞


= 085.⋅ 0 + 015. z 01. x ⋅ e ^{−}^{0 .1x} dx

= − 015. x ⋅ e ^{−}^{0 .1x}

∞

+015. z e ^{−}^{0 .1x} dx


0


0

0


= − 015.⋅ 10 ⋅ e ^{−}^{0 .1x}

^{∞} =15.


0

E

X

2

= P

H ≠
1 E

X

2

H ≠ 1

+ P

H = 1
E

X ^{2}

H =1


b

g

b

g

∞


= 085.⋅ 0 ^{2} + 015. z 01. x ^{2} ⋅e ^{−}^{0 .1x} dx


0

∞


= − 015. x ^{2} ⋅ e ^{−}^{0 .1x}

^{∞} + 015.⋅ 10z 01.⋅ 2 x ⋅ e ^{−} ^{0 .1x} dx = 30


0

0

The variance is: σ^{2}_{X} = E
X ^{2 } − c E X h^{2} = 30 − b15.g^{2} = 27.75
The standard deviation is: σ _{X} = 27.75 = 527.
The coefficient of variation is: σ _{X} / E X = 527. / 15. = 351.
An alternative solution would recognize and use the fact
that f

X c

X

H =1
is an exponential


h


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.
200
Let S =
∑X_{i}
i=1
E S = 200 E
X = 300
σ ^{2}_{S} = 200
σ^{2}_{X} =
5550
σ _{S} = 200 σ _{X}
= 74.50
Coefficient of variation = ^{σ} ^{S}
_{E} _{[} _{S}_{]} = 0.25
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


5

Y=max(0,X5)


(Y)

4


Payment

3


2


Claim


1


0


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

−0.1( y+5)


f

_{Y} c

y

X > 5, H = 1

=

X c

h

=

01. e


h

P

c

X > 5

H =1

P

c

X > 5

H =1


h

h

Summing the three cases:

E

Y

= P

H ≠
1 E

Y

H ≠ 1

+ P

b

X ≤ 5,
H = 1
E[Y

X ≤ 5,
H = 1]
+ P

b

X > 5,
H = 1
E

Y

X > 5,
H =1


b

g

c

g

c

g


= 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


b

g

b

g

h

b

g

h


∞

∞


= 015.z 01. y ⋅ e ^{−}^{0.1(} ^{y} ^{+}^{5)} dy

= 015.⋅ e ^{−}^{0.5} z 01. y ⋅e^{−}^{0 .1y} dy


0

0

= 015.⋅ e ^{−}^{0 .5} ⋅ 10 = 0.91
13
= P ( H ≠ 1) ⋅ 0

+ P ( H = 1) ⋅ P ( X ≤ 5

H = 1) ⋅ 0

∞

−0
.1( y +5)


2

2

2

+ 0.15 _{∫}0.1y

2

⋅e

dy


E Y


0


= 0.15 ⋅ e ^{−}^{0 .5}

^{∞}_{∫}0.1y ^{2} ⋅e ^{−}^{0.1y} dy


0

= 30e^{−}^{0.5} =18.20
σ
_{Y}^{2} = 18.20 − b 0.91g^{2} =17.37
σ
_{Y} = 17.37 =
417.
200
For the pool of 200 individuals, let S_{Y} = ∑Y_{i}
i =1
E S_{Y
} = 200 E
Y =182
σ
^{2}_{S}_{Y} = 200 σ_{Y}^{2} = 3474
σ
_{S}_{Y} = 200 σ_{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% ⋅
S_{Y } = 08.⋅ E S_{Y
} =146
σ
_{80%}^{2} _{S} _{Y} = b 08.g^{2} ⋅ σ ^{2}_{S}_{Y} = 2223
σ
_{80%} _{S}_{Y} = 08.⋅ σ _{S}_{Y} = 4715.
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 70yearold
woman will die within one year. The actuary gathers data using a large random
sample of 70yearold 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 70yearold 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 longterm 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.
16
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