Economics of Insurance
ECO College of Insurance
Economics of Insurance
(BA of Insurance Management)
By Ghadir Mahdavi
Associate Professor of Financial Economics and Insurance,
ECO College of Insurance, Allameh Tabataba’i University
mahdavi@atu.ac.ir
(2020-2021)
1
Economics of Insurance
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Economics of Insurance
Table of Contents
Preface: The economic contribution of insurance……………………..………..…3
Chapter 1. The Theory of Insurance and Rate Making
Introduction……………………………………………………………………….5
1-1-Rate-Making via the Principle of Equivalence……………………..…...….….9
1-2-Rate-Making in a Pure Monopoly Insurance Market…………………………13
1-3-The importance and significance of the Law of Large Numbers……………..24
1-4-Rate making in Perfect Competition Market………………………………….39
1-5-Rate making in an Oligopoly Insurance Market……………….……..……….54
1-6-Total Premium………………………………………………………………...58
1-7-Application of Model to Iranian Fire Insurance Market………………………69
1-8-Concluding Remarks…………………………………………………………..82
Chapter 2. The Effect of Risk Level and Risk Aversion Level ……………….87
References………………………………………………….…………………95
Appendix1. Distributions…………………………………..…………………97
Appendix 2. Definitions…………………………………..………………….109
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Economics of Insurance
Preface
The economic contribution of insurance
1. Insurance reduces or removes uncertainty and controls or manages
risks and brings peace of mind for all of the activities of human
being. In other words, it creates security for all economic activities.
2. Insurance increases investment and reduces unemployment by
utilizing the reserves of insurance companies in different investment
projects. Since there is time interval between premium payment and
loss coverage, insurance company usually have a lot of idle reserves
that can be invested in different areas. In the countries with higher
penetration rates, insurance companies (especially life insurance
companies) usually have large amount of reserves which can be
invested in economy and stimulate production markets.
3. Insurance can reduce the risks and control the probability of events
by imposing standards.
4. Insurance helps to fair distribution of risk in the society. If there is no
insurance in the society, the burden of events and risks should be
tolerated by a small fraction of the people who encountered the loss.
But if there is insurance in the society and the majority of the people
are insured against the losses, the burden of losses is tolerated by all
of the people who are insured. Consequently, insurance makes the
distribution of risk fair.
4
Economics of Insurance
Chapter one
The Theory of Insurance and Rate
Making
1
Introduction
The premium determined by insurance companies should be sufficient to cover for the
total expected losses, guarantee predetermined solvency margin and provide stability to the
market. Therefore, premium-determination is one of the most important functions of insurers. In
the absence of a precise and appropriate price for insurance products, the insurance company
may quickly collapse and face bankruptcy.
In this research, calculation of an appropriate premium for a monopolistic insurance market
based on the principle of equivalence, which brings stability for insurance industry, along with
satisfying sufficient financial solvency margin for the insurance company is rendered by a new
and innovative method that is called by the author as Potential Deviation Ratio (PDR) Method.
To develop the method, we assume the severity of claims (Average Loss: AL) is fixed and the
distribution of number of losses (frequency) is normal. Under these assumptions and obtaining
associated PDR, an appropriate pure premium is calculated. In real stochastic world, we have to
relax both of these assumptions and obtain the real distributions of severity and frequency.
This pamphlet is prepared step by step during the years I was teaching the course “Economic
Theory of Insurance” to undergraduate and graduate students at ECO College of Insurance,
Allameh Tabataba’i University. I would like to appreciate all students who participated the
discussions. It is actually a report on the Contributions I have made to “Economics of
Insurance”. The method presented for Price Determination in insurance industry in this report is
originally my own contribution. This method is comprehensive in considering predetermined
solvency margin and the number of customers together with the price-Determination. The
conventional methods of price determination lack this comprehensiveness.
1
5
Economics of Insurance
We also show the number of insureds (customers) is a critical factor for premium determination.
The larger the number of customers for any insurance service, the smaller the percentage that
should be added to actuarial fair premium to satisfy the predetermined solvency margin, and
hence, the closer the determined premium to actuarial fair premium.
Since, the financial solvency margin and the number of insureds for satisfying the law of large
numbers are held as the key policy factors of premium calculation; this method guarantees the
stability in insurance industry.
The Price-Determination under perfect competition insurance market is also discussed in detail.
Under perfect competition conditions, the price will be the actuarial fair premium level. The
company to guarantee its predetermined solvency margin should keep sufficient Required
Reserve (RR). The amount of required reserve is calculated for different levels of number of
customers. It is shown that the amount of required reserve in perfect competition insurance
market is actually equal to pure premium satisfying predetermined solvency margin in
monopolistic insurance market minus actuarial fair premium.
The price-determination in an oligopoly insurance market is also discussed. This market
is more realistic since just a percentage of the change required for satisfying
predetermined solvency margin can be accomplished by increasing the premium. This
percentage is referred to as Monopoly Power (µ). The remainder should be accomplished
by keeping Required Reserves. The percentage attributed to Required Reserve will be
equal to (1-µ).
In the last section of this research, we apply this method to Iranian Fire Insurance Market
to obtain the price of fire insurance associated with the solvency margin and the number
of customers.
The comparative advantage of this method is the fact that while calculating the pure and
total premium based on the actual distributions of frequency and severity of loss, the
financial solvency margin of the insurance company, which is the level of confidence at
which the insurance company will be able to cover all the future losses, will be obtained.
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Economics of Insurance
The difference between the method presented here and the conventional methods used by
actuaries is that the distributions of severity and frequency are used in our method to
calculate premiums together with considering the predetermined solvency margin defined
by the possibility of meeting all the future claims for the insurance company,
simultaneously. While, the conventional methods concentrate on historical data and the
means of severity and frequency, not on their distributions.
In conventional methods, different financial ratios of insurance companies are used as an
indicator for financial solvency, which sometimes do not fully correspond to the concept
of solvency. For example, according to Regulation 69 of the Central Insurance of Iran,
the financial solvency index is calculated according to the following ratio:
𝐹𝑖𝑛𝑎𝑛𝑐𝑖𝑎𝑙 𝑆𝑜𝑙𝑣𝑒𝑛𝑐𝑦 𝐼𝑛𝑑𝑒𝑥 =
𝐴𝑣𝑎𝑖𝑙𝑎𝑏𝑙𝑒 𝐶𝑎𝑝𝑖𝑡𝑎𝑙
× 100
𝑅𝑒𝑞𝑢𝑖𝑟𝑒𝑑 𝐶𝑎𝑝𝑖𝑡𝑎𝑙
This definition does not have full compliance with the concept of financial solvency of
insurance companies.
While in our method, solvency margin of any insurance company is defined as the
possibility that the company is able to afford for the claims.
The proposed method of calculating financial solvency can be used as a basic method in
calculating the real financial solvency of insurance companies. This method is a good
alternative to methods that are generally based on the financial ratios of companies in
which the basis of their calculations is the historical data of the financial statements of
companies.
Moreover, the conventional methods do not link directly between premium
determinations and solvency margins, while we determine the premium associated with
the predetermined solvency margin. In our method, the price of insurance products can
be determined in such a way that the real financial solvency margin of the insurance
company can also be satisfied. Therefore, this method can be of special importance for
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Economics of Insurance
the legislator of the insurance industry as the conventional pricing methods do not pay
attention to the issue of financial solvency at the same time.
This method can be easily practiced and applied by the insurance companies of ECO
member countries in their premium-making which satisfies their predetermined solvency
margin simultaneously.
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Economics of Insurance
1- Rate-Making via the Principle of Equivalence
Rate making or pricing is a process of allocating collected premiums to the claims. In
this section we calculate the insurance services prices and their rates under Ceteris
Paribus condition. We assume the only factor determining the price is the cost of losses
or the cost of claims of insureds. Thus, the price obtained will be Pure Premium since
pure premium is a fraction of total premium which pays just for the losses and claims.
To obtain pure premium we utilize the definition of insurance and the principle of
equivalence.
The definition of insurance: Insurance is an economic device whereby individual
substitutes a small certain cost (premium) for a large uncertain financial loss (the
event that will be insured).
Total Premium
Collected
(Small certain cost)
Total Expected Loss
(Large uncertain financial loss)
If we equalize two important elements of this economic device (insurance),
we obtain the amount of pure premium. [Pure premium is a fraction of total
premium which covers just the losses.]
Total premium collected =Total expected (uncertain) loss
This equation is called principle of equivalence.
For the time being we propose there is no other cost for insurance activity except the loss
itself. Later on, we will consider other costs such as administrative costs, commissions,
profits, taxes, moral hazard costs and etc. in determining total premium.
n ×
↓
Number of
insureds
Pu.Pr.
↓
Pure
Premium
=
x̅
↓
×
Average
Number of
Loss
AL
↓
Average
Loss
principle of equivalence
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Economics of Insurance
From the principle of equivalence, we obtain pure premium as following:
𝑃𝑢. 𝑃𝑟. =
𝑥̅ × 𝐴𝐿
𝑥̅
=
𝑛
𝑛
Pure Premium
=
×
𝐴𝐿
Loss Ratio × Average Loss
The basic formula in actuarial science for rate making
Numerical Examples:
Example1:
An insurance company insures houses against fire in a large city. If the past experience
shows out of each 10,000 houses 20 houses catch on fire and average loss per each
accident is 20,000,000 Toomans, find pure premium and show how the principle of
equivalence applies.
Answer:
Pu.P.r =
Pu.Pr. =
𝑥̅ × 𝐴.𝐿
𝑛
20
10,000
=
𝑥̅
𝑛
×
𝐴𝐿
× 20,000,000 = 40,000 Toomans
Principle of equivalence:
n × Pu.Pr.
= x̅
× AL
10,000 × 40,000 = 20 × 20,000,000
400,000,000
=
400,000,000
Example2:
A life insurance company provides whole-life policy in a large country. If the mortality
rate regardless the age and gender is equal to
2
1000
and the death benefit is equal to
300,000,000 Toomans, calculate pure premium for the contract and show how the
principle of equivalence applies.
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Economics of Insurance
Answer:
Pu.Pr. =
𝑥̅
×
𝑛
2
Pu.Pr. =
1000
𝐴𝐿
× 300,000,000 = 600,000
Principle of equivalence:
n × Pu.Pr.
= x̅
× AL
1000 × 600,000 = 2 × 300,000,000
600,000,000 = 600,000,000
Example3:
An insurance company sells car collision insurance in a country to one million drivers.
If past experiences show an average number of 50 accidents out of 1000 cars and the
average claim per each accident is equal to 20,000,000 Toomans, calculate pure
premium and show how the principle of equivalence applies. Discuss how much the
company is confident that can afford the claims (How much the solvency margin of the
company is).
Answer:
Pu.Pr. =
Pu.Pr. =
𝑥̅
𝑛
×
50,000
1,000,000
𝐴𝐿
× 20,000,000 = 1,000,000
Principle of equivalence:
n × Pu.Pr.
= x̅
× AL
1,000,000 × 1,000,000 = 50,000 × 20,000,000
1,000,000,000,000 = 1,000,000,000,000
1000B =1000B
As we can see the total amount of 1,000B Ts is collected from 1,000,000 policyholders
and exactly this amount of money is given as claims to 50,000 people who are
encountered with the loss and are reimbursed 20,000,000 Ts each in average.
To derive the solvency margin of insurance company we need to know the distribution
of number of loss. If the distribution of number of loss is assumed to be normal, the
solvency margin can be obtained as following:
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Economics of Insurance
The company to be able to afford for the claims and remain solvent the real number of
loss in the next year should be less than the average number of 50,000.
xi
x̅=50,000
Solvency margin= Pr (xi≤ x̅) = 50 %
Solvency margin= Pr (xi≤ 50,000) = 50 %
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Economics of Insurance
2- Rate-Making in a Pure Monopoly Insurance Market
The price determination discussed so far based on the definition of Insurance and
the Principle of Equivalence is general for all types of markets. As we got to know
the premium obtained brings low level of solvency margin or solvency ratio for
insurance companies. No one insurance company can operate under such low level
of solvency or saying in other way under such high Ruin Probability of 50%. The
companies need to increase their confidence level of affordability for paying for the
claims. The strategies for increasing the solvency ratio changes from market to
market. The strategy in monopoly market is completely different from perfect
competition market or oligopoly. In the next section of this section, we discuss about
price determination under monopoly insurance market which brings sufficient
solvency margin for the company to operate. First, we need to identify and
distinguish what a monopoly (or pure monopoly) insurance market is.
A market is referred to as monopoly which has the following characteristics:
1. The market structure is characterized by a single seller, or sole producer
selling a unique product in the market.
2. The seller faces no competition, as he is the sole seller of goods or services
with no close substitute.
3. There is only one seller in the market, meaning the company becomes the
same as the industry it serves.
4. The single seller becomes market controller as well as the price maker. He
enjoys the power of setting the price for his goods.
Shortly speaking, a market is called to be monopoly where there is only a sole
supplier (only one supplier) rendering a commodity or service. Since there is only
one producer, the producer determines the price. Thus the supplier is the price maker.
In the insurance industry, a pure monopoly insurance market is a market with only
one insurance company selling a unique insurance service in a large city or country.
The company determines the premium rates as it is price maker. In other words, there
exists only one insurer or insurance company that determines (makes) the prices.
Under this assumption, the company can increase its prices (premium) in order to
increase its solvency margin (The possibility that the company can cover the claim
of customers).
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Economics of Insurance
We use numerical examples to verify the monopolist insurance market strategies to
increase its solvency margin.
Example 4:
A life insurance company that has perfect monopoly power provides whole life
policy in a country. If the past data shows an average of 10 mortalities out of each
10,000 people yearly, and the company promises to pay 400,000,000Ts as death
benefit in the case of death.
a) Find pure premium and Fair premium and their rates.
b) If the number of deaths follows normal distribution. Calculate the solvency
margin or the confidence level that the company can afford the claims.
c) If the standard deviation for the distribution of death is 2. Find the percentage
that should be added to the pure premium by monopolist insurance company
in order to increase the solvency margin to %97.7.
d) Find pure premium and its rate and compare it with fair premium and its rate.
Answer:
Part a)
n= 10,000
𝑥̅ = 10
AL= 400,000,000
𝑥̅
10
𝑛
10,000
𝑝𝑢. 𝑝𝑟 = × 𝐴𝐿 =
×400,000,000= 400,000= 𝐹𝑎𝑖𝑟 𝑝𝑟𝑒𝑚𝑖𝑢𝑚
This pure premium of 400,000 is also the Fair Premium since it is obtained by using
the actual loss ratio of demanders. From the viewpoint of demanders this premium
level is Fair since it is obtained from the principle of equivalence, and total amount
of premium the insureds pay is equal to total expected loss. Thus they are paying a
fair amount of price for covering their risk. They are not overpaying for the losses.
Fairness of premium levels is from the point of view of customers.
𝑝𝑢𝑝𝑟 𝑟𝑎𝑡𝑒 =
400,000
400,000,000
=
10
10,000
= 𝐹𝑎𝑖𝑟 𝑝𝑟. 𝑟𝑎𝑡𝑒
The fair premium & its rate is equal to pure premium and its rate, respectively, since
Pu. Pr rate = loss ratio
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Economics of Insurance
Part b)
Solvency margin = pr (𝑥𝑖 ≤ 𝑥̅ ) =50%
Solvency margin = pr (𝑥𝑖 ≤ 10) =50%
Part c)
Solvency margin = pr (𝑥𝑖 ≤ 𝑥̅ +z 𝜎) = 97.7 %
Solvency margin = pr (𝑥𝑖 ≤ 10+2× 2) = pr (𝑥𝑖 ≤ 14) = 97.7 %
The company in order to reach to the confidence level of %97.7, should consider the
number of deaths to be equal to 14 (instead of 10). So the company should assume
the number of death equal to 14 instead of experienced average number of loss of 10
for calculation of pure premium in order to reach to the solvency margin of 97.7%.
Thus, the Percentage of increase in the number of loss and accordingly in pure
premium is equal to 40%:
14 − 10
× 100 = %40
10
The company should consider the number of deaths %40 more than the past data
average. Consequently, the monopolist insurance company, should increase the
premium by %40 as well. We will have:
Part d)
𝑃𝑢. 𝑃𝑟.%97.7 = 𝑃𝑢. 𝑃𝑟%50 + %40 × 𝑃𝑢. 𝑃𝑟%50
𝑃𝑢. 𝑃𝑟%97.7 = 400,000+ 0.4× 400,000 = 560,000
By this premium (560.000$) the insurance company reaches the confidence level of
%97.7. It can also be obtained directly:
𝑃𝑢. 𝑃𝑟%𝛼 =
(𝑥̅ +𝑧%𝛼 𝜎)
𝑃𝑢. 𝑃𝑟%97.7 =
𝑛
14
10,000
× AL
×400,000,000=560,000.
The monopolist insurance company to reach the solvency margin of 97.7% should
ask for the premium level of 560,000 (instead of 400,000).
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Economics of Insurance
To obtain the pure premium rate we have to devide the premium level into average
loss:
𝑃𝑢. Pr 𝑟𝑎𝑡𝑒 %97.7 =
𝑃𝑢. 𝑃𝑟%97.7
560,000
14
× 100 =
=
𝐴𝐿
400,000,000
10,000
The rate the insureds are charged (14/10,000) is greater than the Fair rate of
10/10,000 by 40%. Similarly, the premium the insureds are supposed to pay
(560,000) is greater than the fair premium level of 400,000.
Potential Deviation Ratio (PDR): Potential Deviation Ratio is the percentage that
should be added to the premium in order to satisfy and guarantee a pre-determined
solvency ratio. PDR can be obtained as following:
𝑷𝑫𝑹%𝜶 =
(𝒙
̅ + 𝒛 𝝈) − 𝒙
̅
𝒛%𝜶 𝝈
× 𝟏𝟎𝟎 =
× 𝟏𝟎𝟎
̅
̅
𝒙
𝒙
Example5:
An insurance company sells fire insurance policy in a big city. If average claim for
fire events is 100,000,000 Ts. and past experiences show the average number of the
fire equal to 10 per each 10,000 houses in a year,
a) calculate pure premium and actuarial fair premium and their rates.
b) Discuss about the solvency margin that this premium satisfies if the yearly
average number of loss follows normal distribution.
c) The company plans to increase its Solvency Margin to 99.8%, find PDR if
standard deviation of the distribution of number of loss is equal to 2?
d) Find pure premium and its rate for the pre-determined solvency margin of 99.8%.
What the Actuarial Fair Premium and its rate are?
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Economics of Insurance
Answer:
Part a)
Pu.Pr =
Pu.Pr =
𝑥̅
×
𝑛
𝐴. 𝐿
10
× 100,000,000 = 100,000
10,000
Pu.Pr Rate=
100,000
100,000,000,00
Actuarial Fair Premium=
=
10
10,000
10
× 100,000,000 = 100,000
10,000
Actuarial Fair premium satisfies the principle of equivalence.
Actuarial Fair Premium Rate=
10
10,000
In this part pure premium is equal to actuarial premium.
Part b)
Pr (xi≤ x̅) = Pr (xi≤ 10) = 50% so the solvency margin is 50%
Part c)
Pr (xi≤ x̅ + zδi) = 99.8 %
Pr (xi≤ 10+3×2) = 99.8 %
Pr (xi ≤ 16) = 99.8 %
The percentage that should be added to the original pure premium in order to
increase the solvency margin to a predetermined level is called PDR.
(𝑥̅ + 𝑧 𝜎) − 𝑥̅
𝑧%𝛼 𝜎
3×2
𝑃𝐷𝑅%𝛼 =
× 100 =
× 100 =
× 100 = 60%
𝑥̅
𝑥̅
10
Part d)
𝑃𝑢. 𝑃𝑟%𝛼 =
(𝑥̅ +𝑧%𝛼 𝜎)
𝑃𝑢. 𝑃𝑟%99.8 ==
𝑛
16
10,000
× AL
× 100,000,000=160,000
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Economics of Insurance
𝑃𝑢. Pr 𝑅𝑎𝑡𝑒 %98.8 =
𝑃𝑢. 𝑃𝑟%98.8
160,000
16
× 100 =
=
𝐴𝐿
100,000,000
10,000
The rate the insureds are charged (16/10,000) is greater than the Fair rate of
10/10,000 by 60%. Similarly, the premium the insureds are supposed to pay
(160,000) is greater than the fair premium level of 100,000. In other words, the
company to improve its solvency margin to 99.8% should assume or consider the
number of claims to be 16 instead of 10 and should ask for the premium of 160,000
instead of 100,000.
important notes:
1. The monopolistic market allows the insurance company to increase the premium to
160,000. If the market was not monopoly and there was some competition among the
companies, it was not possible for the company to increase the premium to 160,000 to
satisfy and guarantee the predetermined solvency margin. In such a case, other strategies
should be taken.
2. Actuarial fair premium and actuarial fair rate: the premium and its rate obtained from the
principle of equivalence are actually the actuarial fair premium and its rate. They are fair
from perspective of customers.
𝑥̅
Pu.Pr = 𝑛 × 𝐴. 𝐿 =Actuarial Fair premium
,
𝑥̅
𝑁
= Actuarial Fair Rate
3. Potential Deviation Ratio (PDR) is the percentage that should be added to the premium in
order to satisfy a pre-determined confidence level (Solvency Margin) for the suppliers
(Insurance companies) that can afford the losses.
PDRᵦ =
̅̅̅
(𝑥 +𝑧ᵦ𝛿)−𝑥̅
𝑥̅
× 100 =
𝑧ᵦ𝛿
𝑥̅
× 100
Example6:
A life insurance company sells whole-life policy in a country. The mortality table
suggests an average mortality rate of
10
10000
yearly regardless of age and gender. If the
death benefit is equal to 400,000,000 toman,
a) Calculate pure premium and pure premium rate.
b) What is the fair premium and fair premium rate from the view point of consumers?
c) If the number of deaths follows normal distribution find the confidence level
(solvency margin) that the company can afford the losses (can pay for the claims).
18
Economics of Insurance
d) If the standard deviation of the distribution is equal to 2, find the percentage that the
monopolistic insurance company should increase the premium in order to increase the
solvency margin to 99.8 %. What the premium level should be.
Answer:
x̅ = 10
n=10,000
A.L =400,000,000
Part a)
Pu.Pr =
𝑥̅
𝑛
10
× 𝐴𝐿 =
Pu.PrRate =
10000
𝑃𝑢𝑟𝑒 𝑃𝑟𝑒𝑚𝑖𝑢𝑚
𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝐿𝑜𝑠𝑠
=
× 400,000,000 = 400,000
400,000
400,000,000
=
10
10000
= Loss Ratio
Part b)
Fair Premium = 400,000
Fair Premium Rate =
10
10000
→ (Fair Rate)
Part c)
Pr (xi≤ x̅) = Pr (xi≤ 10) = 50%
Xi
x̅ = 10
Solvency
Margin
Part d)
From the definition of solvency margin:
Pr ( xi≤ x̅ + zδi ) = 99.8 %
Pr ( xi≤ 10+3×2 ) = 99.8 %
Pr ( xi≤ 16 ) = 99.8 %
19
Economics of Insurance
The insurance company in order to reach to the confidence level of 99.8% (instead
of 50%), should consider the number of losses to be 16 (instead of 10). Thus the
percentage to increase the premium in order to reach to the solvency margin of
99.8% can be calculated as following:
16−10
10
× 100 =
6
10
× 100 = 60 %
So the premium should increase by 60%. Obviously, the pure premium offered by
monopolistic insurance company will be equal to:
𝑃𝑢. 𝑃𝑟%99.8 = 400,000 + 60% × 400,000 = 640,000
Equivalently by using the Formula:
𝑃𝑢. 𝑃𝑟%99.8 =
𝑥̅ + 𝑧𝛿
× 𝐴. 𝐿 =
𝑛
10+3×2
10,000
× 400,000,000 = 640,000
Example7:
A property insurance company which has monopoly power sells car collision
insurance. If the data indicates 100 accidents out of 10,000 cars per year and the
average claims are equal to 20,000,000 Ts. per each accident.
a) Calculate pure premium and fair premium and their rates.
a) If the number of accidents follows normal distribution, calculate the solvency
margin of the project.
a) if the standard deviation is equal to 20 and the company plans to increase the
solvency margin to 97.7%, find PDR, pure and Fair premiums.
d) Find pure premium rate and compare it to fair premium rate.
Answer:
x̅ = 100
Part a)
n=10,000
𝑥̅
100
Pu.Pr = × 𝐴. 𝐿 =
𝑛
Pu.Pr Rate =
A.L =20,000,000
10000
𝑃𝑢𝑟𝑒 𝑃𝑟𝑒𝑚𝑖𝑢𝑚
𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝐿𝑜𝑠𝑠
× 20,000,000 = 200,000
=
200,000
20,000,000
=
100
10000
= Loss Ratio
Fair Premium = 200,000
Fair Premium Rate =
100
10000
→ (Actuarial Fair Rate)
20
Economics of Insurance
= x̅
× A.L
= 100 × 20,000,000
= Total expected loss
n × Pu.Pr
10,000 × 200,000
Total premium collected
Fair Premium Rate = Loss Ratio=
100
10000
Note: Fair premium rate actually is equal to loss ratio by definition.
Part b)
Pr( xi≤ x̅ ) = Pr ( xi≤ 100 ) = 50%
Part c)
𝑃𝐷𝑅%97.7 % =
=
̅̅̅
(𝑥+𝑧97.7%𝛿)−𝑥̅
𝑥̅
100 + 2 × 20 − 100
100
𝑃𝑢. 𝑃𝑟%97.7 =
× 100 =
𝑧97.7%𝛿
𝑥̅
× 100
× 100 = 40 %
𝑥̅ + 𝑧𝛿
𝑛
× 𝐴. 𝐿 =
100+2×20
10,000
× 20,000,000 = 280,000
Pr( xi≤ x̅ + zδi ) = 97.7 %
Pr ( xi≤ 100+2×20 ) = 97.7 %
Pr ( xi≤ 140) = 97.7 %
xi
x̅ = 100
xi = x̅ + zδ
Fair Premium level will be 200,000 since this amount satisfies principle of
equivalence.
d)
𝑃𝑢. 𝑃𝑟%97.7 𝑅𝑎𝑡𝑒 =
𝑃𝑢.𝑃𝑟%97.7
𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝐿𝑜𝑠𝑠
200,000
Fair Premium Rate =
20,000,000
=
=
280,000
20,000,000
100
10,000
=
140
10,000
= 𝐿𝑜𝑠𝑠 𝑅𝑎𝑡𝑖𝑜
21
Economics of Insurance
Example8:
A life insurance company sells whole-life policy in a country. If the mortality rate
regardless the age and gender is equal to 10 out of 10,000 and death benefit is equal
to 200,000,000 Ts.
a) find fair premium and pure premium and their rates.
b) if the number of deaths follows normal distribution, find the solvency margin that
this pure premium satisfies.
c) if the insurance company plans to increase the solvency margin to 99.8%, find
PDR, Fair and pure premiums and their rates If the standard deviation of the
distribution is equal to 2 (Z = 3).
d) Discuss how the GAP between actuarial fair premium and the premium for the
confidence level of 99.8% is covered. (In other words, explain how the GAP between
willingness to pay in customers side and willingness to receive in suppliers’ side
does not deteriorate the market).
22
Economics of Insurance
Answer:
x̅ = 10
n=10,000
A.L =200,000,000
a)
𝑃𝑢. 𝑃𝑟%50 =
𝑥̅
×
𝑛
𝑃𝑢. 𝑃𝑟𝑅𝑎𝑡𝑒%50 =
𝐴. 𝐿 =
𝑃𝑢𝑟𝑒 𝑃𝑟𝑒𝑚𝑖𝑢𝑚
𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝐿𝑜𝑠𝑠
10
× 200,000,000 = 200,000
10000
200,000
=
200,000,000
=
10
10000
= Loss Ratio
Fair Premium = 200,000
Fair Premium Rate = Loss Ratio =
100
10000
→ (Fair Rate)
n × Pu.Pr
=
x̅
×
10,000 × 200,000 =
10 × 200,000,000
Fair Premium = Loss Ratio × A.L = 200,000
Pure Premium = Fair Premium = 200,000
A.L
b)
Pr( xi≤ x̅ ) = Pr ( xi≤ 10 ) = 50%
This is the definition of solvency margin because the claims are affordable if xi≤ x̅
, otherwise the amount of total premium collected will not be sufficient for covering
all the claims.
c)
Pr( xi≤ x̅ + zδi ) = 99.8%
Pr ( xi≤ 10+3×2 ) = 99.8%
Pr ( xi≤ 16) = 99.8%
In order to reach to the confidence level of 99.8 %, the insurance company should
consider the number of death equal to 16 (instead of 10).
𝑃𝐷𝑅%99.8 =
=
̅̅̅
(𝑥 +𝑧𝛼 𝛿)−𝑥̅
× 100 =
𝑥̅
10 + 3 × 2 − 10
10
𝑧𝛼 𝛿
𝑥̅
× 100
×100=60%
The company should increase the premium 60% in order to reach to the confidence
level of 99.8% that can afford the losses.
23
Economics of Insurance
𝑃𝑢. 𝑃𝑟%99.8 =
𝑥̅ + 𝑧𝛿
𝑛
× 𝐴. 𝐿 =
10+3×2
10,000
× 200,000,000 = 320,000
The company should ask for 320,000 (instead of 200,000) in order to increase the
solvency
margin
to
99.8
%
(instead
of
50
%).
This gap (320,000 and 200,000) may make the company collapse (people think they
are overpaying) and company remains with risky individuals and company can't pay
claims.
Pu.PrRate99.8% =
𝑃𝑢𝑟𝑒 𝑃𝑟𝑒𝑚𝑖𝑢𝑚
𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝐿𝑜𝑠𝑠
=
320,000
200,000,000
10
Fair Premium Rate = Loss Ratio =
10000
=
16
10000
→ (Fair Rate)
Fair Premium = 200,000
d) Part “d” is left to next section.
3-The importance and significance of the Law of Large Numbers
It is usually emphasized that insurance industry operates based on low of large
numbers. In this section we are going to discuss about the grounds of this statement
and identify how important the application of the law of large numbers is in
insurance industry. We will realize that when the number of insured increases further
and further the gap between pure premium rate and fair premium rate diminishes and
pure rate converges to fair rate.
Reminder: The law of large numbers indicates that when the number of samples increases and
increases further, sample’s mean converges towards population mean.
There is also another definition for the law of large numbers: whenever the number of samples
increases; the experimental mean converges towards mathematical mean.
In part “d” of numerical example 8 we were asked to explain how the GAP between
willingness to pay in customers side and willingness to receive in suppliers’ side
does not deteriorate the market. The demanders would like to pay just 200,000 Ts to
purchase the service as it is attributed to their real loss ratio of 10/10,000. While at
24
Economics of Insurance
the other side, the suppliers deliver the service with sufficient confidence for
affordability of 99.8% for the price of 320,000 Ts with the premium rate of
16/10,000. If the company asks for this premium level of 320,000, many low-risk
individuals who think this premium is excessive, will not purchase the policy and
leave the market. By exiting low-risk individuals, the company remains with just
high-risk individuals. In such a case, even the previous premium level of 320,000
will not bring sufficient solvency margin for the company to operate. If the company
continues the operation may go bankrupt and collapse.
According to the Law of large numbers, if the company could be able to absorb more
and more customers, the loss ratio of sample of customers will converge towards the
population loss ratio of 10/10,000. In such a case the company will be able to afford
the claims with the premium level of 200,000 and the rate of 10/10,000 even with
high confidence level of 99.8%. This is the secret of the law of large numbers. It is
usually said that the insurance industry stays stable at the wings of the law of large
numbers. Without the application of the law of large numbers, the insurer will remain
insolvent and go bankrupt.
In other words, If the company cannot absorb enough customers so as the law of
large numbers applies, should ask for high premium levels greater than actuarial fair
premium level. By doing so, the low-risk individuals realize they are overpaying the
policy and drop out of the market. By exiting low-risk individuals, the company
faces with higher loss ratios and inevitably should increase premium further. Again,
more low-risk customers leave the market and this follows up to the point the
company remains with very high-risk individuals with very high loss ratio whose
risk actually is not easily affordable.
follow the discussion by Numerical example.
25
Economics of Insurance
Example 9:
Suppose a life insurance company issues whole-life policy in a big city. The
mortality table shows the average mortality rate of
1
1000
for each individual
regardless the age and gender, and all individuals are exposed to the same risk.
The company sells the product to 10,000 customers with the death benefit of
200,000,000 Ts and the policy does not offer any surrender value. The number of
claims follows a normal distribution with standard deviation equal to 2.
A. Find pure premium, Fair Premium and their rates and loss ratio.
B. Discuss about the solvency margin that this premium brings for the insurance
company.
C. Suppose the company plans to increase its solvency margin to 99.8%, find
Pure Premium, Fair Premium and their rates, loss ratio, and PDR.
D. Now suppose the company can absorb more customers by efficient marketing
policy equal to 40,000 customers. Find pure premium, Fair Premium and their
rates and PDR for the same solvency margin of 99.8%.
E. What happens to pure premium, Fair Premium and their rates and PDR if the
number of customers increases to 1000,000?
F. What will be the pure premium, Fair Premium and their rates and PDR if the
number of customers increases to 25,000,000?
G. Based on the answer obtained discusses about the importance of the law of
large numbers in insurance market.
H. Specify how the market may collapse if the company cannot absorb enough
customers.
Answer:
A.
𝑥̅1 = 10 , 𝑛1 =10,000
Pu.Pr =
𝑥̅1
×
𝑛1
Pu.PrRate =
A.L =200,000,000
𝐴. 𝐿 =
𝑃𝑢𝑟𝑒 𝑃𝑟𝑒𝑚𝑖𝑢𝑚
𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝐿𝑜𝑠𝑠
10
Actuarial fair rate=
=
10
10000
200,000
× 200,000,000 = 200,000
200,000,000
=
10
10000
= Loss Ratio
10000
Customers consider the rate of 10/10,000 as he fair rate for them. Any rate
more than
10
10000
won’t be fair to them.
26
Economics of Insurance
B.
Pr( 𝑥𝑖1 ≤ 𝑥̅1 ) = Pr ( 𝑥𝑖1 ≤ 10) = 50%
C.
Pr( 𝑥𝑖1 ≤ 𝑥̅1 + 𝑧𝛿) = 99.8 %
Pr ( 𝑥𝑖1 ≤ 10+3×2) = 99.8 %
Pr (𝑥𝑖1 ≤ 16) = 99.8 %
PDRᵦ =
𝑧ᵦ𝛿
𝑥̅
× 100=60%
𝑃𝑢. 𝑃𝑟%98.8,𝑛1 =
̅𝑥̅̅1̅+𝑧%98.8 𝛿1
𝑁3
× 𝐴. 𝐿 =
10+3×2
10,000
× 200,000,000 = 320,000
The insurance company should consider the number of death equal to 16
instead of 10. Consequently, the insurance company should ask for the
premium equal to 320,000 instead of 200,000 in order to increase its
confidence level to 99.8%.
Since the market is assumed to be monopoly, the company is able to do that.
𝑃𝑢. 𝑃𝑟𝑅𝑎𝑡𝑒%98.8,𝑛1 =
Loss Ratio =
10
10000
𝑃𝑢𝑟𝑒 𝑃𝑟𝑒𝑚𝑖𝑢𝑚
𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝐿𝑜𝑠𝑠
=
320,000
200,000,000
=
16
10000
→ (Actuarial Fair Rate)
D.
x̅2 = 4 x̅1 = 40 (they are homogeneous)
𝛿2 = 2 𝛿1
n2 = 4 n1 = 40,000
A.L = 200,000,000
Proof:
Variance (xi2) =
Variance (xi2) =
𝛴 (𝑥𝑖2 −𝑥̅2 )2
𝑛2
𝛴 (𝛼𝑥𝑖1 −𝛼𝑥̅1 )2
Variance (xi2) = α
𝛼𝑛1
𝛴 (𝑥𝑖1 −𝑥̅1
𝑛1
)2
=
𝛼 2 𝛴 (𝑥𝑖1 −𝑥̅1 )2
𝛼𝑛1
= α variance (xi1)
𝛿2 2 = 𝛼 𝛿1 2
𝛿 2 = √𝛼 𝛿 1
27
Economics of Insurance
Solvency Margin= Pr(x i2≤ x̅2 + zδ2) = 99.8%
= Pr(x i2≤ 40 + 3×4) = 99.8%
𝑧99.8%𝛿2
PDR99.8%,𝑛2 =
PuPr99.8%,𝑛2 =
× 100=
𝑥̅2
𝑥̅2 + 𝑧𝛿2
PuPrRate99.8%,𝑛2 =
3×4
× 100 = 30 %
40
40+12
× 𝐴. 𝐿 =
× 200,000,000 = 260,000
𝑁2
40,000
𝑃𝑢𝑟𝑒 𝑃𝑟𝑒𝑚𝑖𝑢𝑚
260,000
=
𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝐿𝑜𝑠𝑠
200,000,000
=
Actuarial Fair Premium Rate = Loss Ratio =
13
10000
10
10000
E.
n3 = 25n2 = 1000,000
x̅3 = 25 x̅2 = 1000
A.L = 200,000,000
𝛿3 = 5 𝛿2
Pr(x i3≤ x̅3 + zδ3) = 99.8%
= Pr(x i3≤ 1000,000 + 3×20) = 99.8%
PDR99.8%, n3 =
PuPr99.8%, n3 =
𝑧99.8%𝛿3
× 100 =
𝑥̅3
𝑥̅3 + 𝑧𝛿3
3×20
× 100 = 6%
1000
1000+60
× 𝐴. 𝐿 =
× 200,000,000 = 212,000
𝑛3
1000,000
𝑃𝑢𝑟𝑒 𝑃𝑟𝑒𝑚𝑖𝑢𝑚
212,000
PuPrRate99.8%, n3 =
𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝐿𝑜𝑠𝑠
10
AF.P. Rate = Loss Ratio =
=
200,000,000
=
10.6
10000
10000
F.
n4 = 25n3 = 25000,000
x̅4 = 25 x̅3 = 25000
A.L = 200,000,000
𝛿4 = 5 𝛿3
Pr(x i4≤ x̅4 + zδ4) = 99.8%
= Pr(x i4≤ 25000,000 + 3×100) = 99.8%
PDR99.8%, n4 =
PuPr99.8%, n4 =
𝑧99.8%𝛿4
× 100 =
𝑥̅4
𝑥̅4 + 𝑧𝛿4
3×100
× 100 = 1.2%
25000
25000+300
× 𝐴. 𝐿 =
𝑛4
25000,000
𝑃𝑢𝑟𝑒 𝑃𝑟𝑒𝑚𝑖𝑢𝑚
212,000
PuPrRate99.8%, n4 =
𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝐿𝑜𝑠𝑠
10
AF.P. Rate = Loss Ratio =
=
× 200,000,000 = 202.400
200,000,000
=
10.12
10000
10000
28
Economics of Insurance
𝑛1
10,000
𝑛1
10,000
𝑛2 =4𝑛1
40,000
𝑛3 =25n2
1,000,000
𝑛4 =25n3
25,000,000
Solvency
Margin
50%
99.8%
99.8%
99.8%
99.8%
Loss Ratio
10
10,000
10
10,000
10
10,000
10
10,000
10
10,000
Fair
Premium
200,000
200,000
200,000
200,000
200,000
A.L
200,000,000
200,000,000
200,000,000
200,000,000
200,000,000
PDR
-
60%
30%
6%
1.2%
0
Pure
Premium
200,000
320,000
260,000
212,000
202,400
200,000
Pure
Premium
Rate
10
10,000
16
10,000
13
10,000
10.6
10,000
10.12
10,000
10
10,000
x̅
δ
10
10
40
1000
25,000
-
2
4
20
100
∞
G. and H.
As you can see by increasing the number of insureds from 10,000 to 25000,000 PDR
declines from 60% to 1.2% and in the case of infinity, PDR converges to zero.
Similarly, by increasing the number of insureds, the pure premium which satisfies
the solvency margin of 99.8% converges to its actuarial fair premium level of
200,000 and its rate converges to its actuarial fair premium rate of
10
10000
. This means
by increasing the number of insureds via efficient marketing policy, the company
reaches its predetermined solvency margin of 99.8% even by low premium level of
200,000 (equal to actuarial fair premium level). This means increasing the number
of insureds works as if the companies cost declines.
Now suppose the company is not able to increase its clients and plans to increase its
solvency margin to its predetermined level of 99.8%, the company should ask for the
premium rate of
60
10000
(or the premium level of 320,000). By doing so customers
realize they are overpaying the policy because they expect to pay actuarial fair
premium of 200,000. So, based on the theory of adverse selection, a part of low risk
individuals whose risk level is less than
10
10000
try to cancel their contract and exit
29
Economics of Insurance
from the market. Consequently, the company realizes that even the premium level of
320,000 is not enough for covering the claims and try set new premium levels which
is larger than the previous one. If this happens the next group of Low-Risk
individuals cancel their policies and again the company should increase the
premium and the next group of low-risks drop out of the market. If this happens the
company finally remains with very High-Risk individuals whose risks actually are
not insurable and the market collapses. Shortly speaking, we understand that the
stability of insurance market relies on the Application of the Law of large
numbers.
The law of large numbers states that if the number of samples increases further and
further the experimental means of samples converges to the mathematical mean.
Here, in insurance market if the number of insureds increases further and further,
their real actual claims ratio converges toward the population loss ratio.
Theorem: Prove that by increasing the number of insureds by “α” times the PDR
for satisfying the same solvency margin diminishes by √𝛼 times.
𝑃𝐷𝑅𝑎,𝑛1
𝑃𝐷𝑅𝑎,𝑛2 =
√𝛼
Proof:
𝑃𝐷𝑅𝑎,𝑛2 =
𝑧𝑎 𝛿2 𝑧𝑎 . √𝛼𝛿1 𝑧𝑎 . 𝛿1 𝑃𝐷𝑅𝑎,𝑛1
=
=
=
𝑥̅2
𝑎𝑥̅1
√𝛼𝑥̅1
√𝛼
Example 10:
A monopolist insurance company sells cars collision insurance. Based on the
historical data, 100 cars out of each 10,000 cars, face with accident. If average claim
for each accident is 40,000,000;
a) Calculate pure premium and fair premium and their rates.
b) If the number of accidents follows normal distribution with standard
deviation of 20, find PDR and pure premium and its rate for the confidence
level of %99.8.
30
Economics of Insurance
c) Now assume the company can increase the number of cars insureds to 40.000,
find PDR and pure premium rate for the same confidence level of %99.8.
Answer:
a)
𝑁1 = 10,000
𝑥̅1 = 100
𝛿1 = 20
AL= 40,000,000
𝑃𝑢. 𝑃𝑟%50 = loss ratio × A.L=
̅𝑥̅̅1̅
𝑁1
× 𝐴. 𝐿=
100
10,000
× 40,000 = 40,000
400,000
100
=
40,000.000 10.000
Fair premium and its rate are equal to pure premium and its rate, respectively.
𝑃𝑢. 𝑃𝑟 𝑟𝑎𝑡𝑒%50 =
b)
Solvency margin = pr (𝑥𝑖 ≤ 𝑥̅ +𝑧%99.8 𝜎) = %99.8
Solvency margin = pr (𝑥𝑖 ≤ 100+3× 20) = pr (𝑥𝑖 ≤ 160) = 99.8 %
𝑃𝐷𝑅%99.8 =
𝑧%99.8 𝛿1
3 × 20
× 100 =
× 100 = %60
𝑥̅1
100
𝑃𝑢. 𝑃𝑟%99.8 =
(𝑥̅ 1 +𝑧%99.8 𝛿1 )
𝑁1
𝑃𝑢. 𝑃𝑟 𝑟𝑎𝑡𝑒%99.8 =
100 ×60
× A.L =
10.000
= 640.000
640,000
160
=
40,000,000 10,000
c)
N2=4N1=40,000
𝑥̅2 = 4𝑥̅1 = 400
𝛿2 = 2𝛿1 = 40
Solvency margin = pr (𝑥𝑖 ≤ 𝑥̅ +𝑧%99.8 𝜎) = %99.8
Solvency margin = pr (𝑥𝑖 ≤ 400+3× 40) = pr (𝑥𝑖 ≤ 520) = 99.8 %
31
Economics of Insurance
𝑃𝐷𝑅%99.8 =
𝑧%99.8 𝛿2
3 × 40
× 100 =
× 100 = %30
𝑥̅2
400
𝑃𝑢. 𝑃𝑟%99.8 =
(𝑥̅ 2 +𝑧%99.8 𝛿2 )
𝑁2
400+120
×A.L=
40,000
× 40,000,000 = 520,000
520,000
130
=
40,000,000 10,000
If the number of insureds increases 4 times, PDR for the same confidence level
becomes half and the pure premium rate declines and get closer to fair premium
rate.
𝑃𝑢. 𝑃𝑟 𝑟𝑎𝑡𝑒%99.8 =
Example 11:
A non-life insurance company sells fire policy in a big city. If the number of
accidents in past 5 years and for each 1000 houses appear in the table below and
the average claim for each accident is equal to 50,000,000.
year
Number of accidents
1
10
2
11
3
13
4
7
5
9
a) Calculate pure premium and fair premium and their rates.
b) If the number of accidents follows normal distribution, calculate PDR and
pure premium rate for the confidence level of %97.7.
c) If the company can increase the number of insureds to 25,000, and all houses
are homogenous with respect to the risk. Calculate pure premium and its rate
and PDR.
d) Now assume the company can increase the number of insureds to 100,000
again. Find PDR, pure premium and pure premium rate for the same
confidence level.
e) If the company is successful in increasing the number of insureds to
2,500,000, calculate PDR, pure premium and pure premium rate for the same
confidence level.
f) Solve the same problem for the number of insureds equal to 10,000,000.
g) Based on the answer obtained discuss about the importance of the law of large
numbers in insurance industry and verify why the insurance company may
collapse if it could not absorb enough costumer.
32
Economics of Insurance
h) For which number of insureds, the PDR declines to 0.1%. What will be the
pure premium?
i) What is the relationship between number of insureds, PDRs and Pure
Premiums in successive changes?
Answer:
a)
𝑛1 = 1000
10 + 11 + 13 + 7 + 9
𝑥̅1 =
= 10
5
02 + 12 + 32 + (−3)2 + (−1)2
2
𝛿1 =
=4
5
AL= 50,000,000
𝑃𝑢. 𝑃𝑟%50,𝑁1 =
̅𝑥̅̅1̅
𝑛1
× 𝐴. 𝐿=
𝑃𝑢. 𝑃𝑟 𝑟𝑎𝑡𝑒%50,𝑛1 =
10
1000
,
𝛿1 = 2
× 50,000,000 = 500,000 = 𝐹. 𝑃𝑟
500,000
10
=
= 𝐹. 𝑃𝑟 𝑟𝑎𝑡𝑒
50,000,000 1000
b)
Solvency margin = pr (𝑥𝑖 ≤ ̅̅̅+𝑧
𝑥1 %97.7 𝛿1 ) = %97.7
Solvency margin = pr (𝑥𝑖 ≤ 10+2× 2) = pr (𝑥𝑖 ≤ 14) = 97.7 %
𝑃𝑢. 𝑃𝑟%97.7,𝑛1 =
̅𝑥̅̅1̅
𝑛1
× 𝐴. 𝐿=
𝑃𝑢. 𝑃𝑟 𝑟𝑎𝑡𝑒%97.7,𝑛1 =
𝑃𝐷𝑅%97.7,𝑛1 =
14
1000
× 50,000,000 = 700,000
700,000
14
=
50,000,000 1000
𝑧%97.7 𝛿1
2 ×2
× 100 =
× 100 = %40
𝑥̅1
10
c)
n2=25n1=25,000
𝑥̅2 = 25𝑥̅1 = 250
𝛿2 = 5𝛿1 = 10
33
Economics of Insurance
Solvency margin = pr (𝑥𝑖 ≤ ̅̅̅+𝑧
𝑥2 %97.7 𝛿2 ) = %97.7
Solvency margin = pr (𝑥𝑖 ≤ 250+2× 10) = pr (𝑥𝑖 ≤ 270) = 97.7 %
𝑃𝑢. 𝑃𝑟%97.7,𝑁1 =
̅𝑥̅̅2̅+𝑧%97.7 𝛿2
𝑛2
𝑃𝑢. 𝑃𝑟 𝑟𝑎𝑡𝑒%97.7,𝑛2 =
𝑃𝐷𝑅%97.7,𝑛2 =
× 𝐴. 𝐿=
250+20
25,000
× 50,000,000 = 540,000
540,000
10.8
=
50,000,000 1000
𝑧%97.7 𝛿2
2 × 10
× 100 =
× 100 = %8
𝑥̅2
250
By increasing the number of insureds to 25,000 (25 times as before) the company
reaches to the same confidence level of %97.7 inly by increasing the pure premium
from fair premium by %8.
d)
n3=4n2=100,000
𝑥̅3 = 4𝑥̅2 = 1000
𝛿3 = 2𝛿2 = 20
Solvency margin = pr (𝑥𝑖 ≤ ̅̅̅+𝑧
𝑥3 %97.7 𝛿3 ) = %97.7
Solvency margin = pr (𝑥𝑖 ≤ 1000+2× 20) = pr (𝑥𝑖 ≤ 1040) = 97.7 %
𝑃𝑢. 𝑃𝑟%97.7,𝑛3 =
̅𝑥̅̅3̅+𝑧%97.7 𝛿3
𝑁3
𝑃𝑢. 𝑃𝑟 𝑟𝑎𝑡𝑒%97.7,𝑛3 =
𝑃𝐷𝑅%97.7,𝑁3 =
× 𝐴. 𝐿=
1000+40
100,000
× 50,000,000 = 520,000
520,000
10.4
=
50,000,000 1000
𝑧%97.7 𝛿3
2 × 20
× 100 =
× 100 = %4
𝑥̅3
1000
e)
n4=25n3=2,500,000
𝑥̅4 = 25𝑥̅3 = 25,000
34
Economics of Insurance
𝛿4 = 5𝛿3 = 100
Solvency margin = pr (𝑥𝑖 ≤ ̅̅̅+𝑧
𝑥4 %97.7 𝛿4 ) = %97.7
Solvency margin = pr (𝑥𝑖 ≤ 25,000+2× 100) = pr (𝑥𝑖 ≤ 25,200) = 97.7 %
𝑃𝑢. 𝑃𝑟%97.7,𝑛4 =
̅𝑥̅̅4̅+𝑧%97.7 𝛿4
𝑛4
𝑃𝑢. 𝑃𝑟 𝑟𝑎𝑡𝑒%97.7,𝑛4 =
𝑃𝐷𝑅%97.7,𝑛4 =
× 𝐴. 𝐿=
25,000+200
2,500,000
× 50,000,000 = 504,000
504,000
10.08
=
50,000,000 1000
𝑧%97.7 𝛿4
2 × 100
× 100 =
× 100 = %0.8
𝑥̅4
25,000
f)
n5=4n4=10,000,000
𝑥̅5 = 45𝑥̅4 = 100,000
𝛿5 = 2𝛿4 = 200
Solvency margin = pr (𝑥𝑖 ≤ ̅̅̅+𝑧
𝑥5 %97.7 𝛿5 ) = %97.7
Solvency margin = pr (𝑥𝑖 ≤ 100,000+2× 200) = pr (𝑥𝑖 ≤ 100,400) = 97.7 %
𝑃𝑢. 𝑃𝑟%97.7,𝑛5 =
̅𝑥̅̅5̅+𝑧%97.7 𝛿5
𝑛5
𝑃𝑢. 𝑃𝑟 𝑟𝑎𝑡𝑒%97.7,𝑛5 =
𝑃𝐷𝑅%97.7,𝑛5 =
× 𝐴. 𝐿=
100,000+400
10,000,000
× 50,000,000 = 502,000
502,000
10.04
=
50,000,000 1000
𝑧%97.7 𝛿5
2 × 200
× 100 =
× 100 = %0.4
𝑥̅5
100,000
g)
As the answers indicate by increasing the number of insureds, the PDR becomes
smaller and smaller and finally converges to zero, which means by increasing the
number of insureds the company requires lower premiums for satisfying the same
confidence level. The answer also shows by increasing the number of insureds, pure
35
Economics of Insurance
premium converges to fair premium (500,000) and pure premium rate converges to
actuarial fair rate (10/1000).
Actuarial fair rate is the rate that consumers would like to pay for buying the
insurance services. If the rate suggested by insurance company is much larger than
the fair rate, some of the low- risk costumers cancel their contract and leave the
company with high-risk customers.
As a result, the prevailing premium will not be enough for covering the claim, and
insurance company to remain solvent should increase the premium further. By doing
so, the next group of low-risk customers drop out of the market and cancel their
contract, and consequently, the company remains with their high- risk neighbors.
This follows up to the point that the company remains with very high-risk
individuals, where their risk cannot be afforded and the company will not be able to
cover the claims. Obviously, the company collapses and goes bankrupt.
This is why we emphasize that insurance industry stands on the application of the
law of large numbers and the company should absorb enough customers so that the
law of large numbers applies.
n
ꝏ
n
𝒏𝟏 =1000
𝒏𝟏 =1000
Solvency
Margin
A.L.
%50
%97.7
%97.7
%97.7
%97.7
%97.7
%97.7
50,000,000
50,000,000
50,000,000
50,000,000
50,000,000
50,000,000
̅
𝒙
𝑥̅1 = 10
𝑥̅1 = 10
𝑥̅2 = 250
𝑥̅3 = 1000
𝑥̅4 = 25,000
𝑥̅5 = 100,000
50,000,0
00
−
Fair
Actuarial
rate
10
1000
10
1000
10
1000
10
1000
10
1000
10
1000
10
1000
𝜹
𝛿1 = 2
𝛿1 = 2
𝛿2 = 10
𝛿3 = 20
𝛿4 = 100
𝛿5 = 200
−
PDR
-
%40
%8
%4
%0.8
%0.4
0
Fair
Actuarial
Premium
Pu.Pr
500,000
500,000
500,000
500,000
500,000
500,000
n2=25,000 n3=100,000 n4=2,500,000 n5=10,000,000
500,000
500,000
700,000
540,000
520,000
504,000
502,000
500,000
36
Economics of Insurance
10
1000
Pu.Pr
rate
14
1000
10.8
1000
10.4
1000
10.08
1000
10.04
1000
10
1000
h)
𝑃𝐷𝑅%97.7,𝑛6 =
𝑃𝐷𝑅%97.7,𝑛5
0.1=
√𝛼
𝑧%97.7 𝛿6 𝑧%97.7 √𝛼𝛿5 𝑧%97.7 𝛿5 𝑃𝐷𝑅%97.7,𝑛5
=
=
=
𝑥̅6
𝛼𝑥̅5
√𝛼𝑥̅5
√𝛼
=
0.4
√𝛼
⇒ √𝛼 = 4 , 𝛼=16
n6=16 n5=160,000,000
𝑃𝑢. 𝑃𝑟%97.7,𝑛6 =500,000+%0.1×500,000=500,500
i)
When the number of insureds increases by "𝛼" times, the PDR declines by "√𝛼"
times. The percentages that should be added to Pure premium and its rate also
diminishes by "√𝛼" times. This is why the Pure premium declines to 500,500 from
502,000. By increasing the number of insureds by 16 times, the PDR diminishes 4
times, i.e. from 0.4% to 0.1% (= 0.4/4).
4-Rate making in Perfect Competition Market
A market is called perfect competition if it has the following characteristics:
37
Economics of Insurance
1. In a perfect competition market, there are an infinite number of insurers and
customers.
2. There is no barrier for entry to the market.
3. The suppliers (insurers) produce exactly the same services.
4. There are no information rents, and competition is complete so as the profit
for each producer converges to zero.
5. The price in perfect competition market is determined by the market supply
and demand and no one company solely can affect the price i.e. that is the
companies are Price-Takers (not price makers).
6. The share of the market for each individual producer should be very small so
as no individual producer could have monopoly power.
7. The profit for each individual firm converges toward zero.
The actuarial fair premium satisfies zero profit since at this price total premium
collected is equal to total expected loss (Actuarial fair premium is obtained from the
principle of equivalence). Thus, the prices prevailing in the perfect competition
insurance market is actuarial fair premium and actuarial fair premium rate. Since any
company is price taker and cannot increase premium, the company should keep
required reserves in order to guarantee its predetermined solvency margin.
Since fair premium rate satisfies just the solvency margin of 50% and this level of
confidence is not sufficient for any company to operate in the market, the insurance
company is required to bring financial reserves to guarantee and fulfill a
predetermined confidence level (Let’s say 99.8%). This financial reserve is referred
to as Required Reserves (RR).
For the rest of the discussion, we try to find the amount of required reserve per each
contract and for any policy in general. It is obvious that the amount of required
reserve is actually equal to Pure premium satisfying predetermined solvency margin
Minus Fair Premium.
38
Economics of Insurance
RR1 =
RR1 =
𝑥̅ + 𝑧𝛿
𝑛
× 𝐴. 𝐿 -
𝑥̅
𝑛
× 𝐴. 𝐿
𝒛𝜹×𝑨.𝑳
𝒏
In other words, required reserves for each unit contract should be equal to the amount
suggested by PDR in the case of monopoly market.
RR1 = PDR ×
𝑥̅
𝑛
× 𝐴. 𝐿 =
𝑧𝛿
𝑥̅
𝑥̅
𝑧𝛿×𝐴.𝐿
𝑛
𝑛
× × 𝐴. 𝐿=
.
The total Required Reserve for a policy soled to “n” exposures can be obtained as
RRn = n ×
𝒛𝜹×𝑨.𝑳
𝒏
= 𝒛𝜹 × 𝑨. 𝑳
Example12:
A company doing its business under perfect competition market sells fire insurance.
If the past data shows an average number of accidents of 5 out of each 10,000 houses
and the average claim against the company for each accident in past years is equal
to 100,000,000 Tomans;
a) calculate market price (Actuarial Fair Premium) and its rate.
b) If the company plans to reach to the confidence level of 97.7% find the amount
of required reserves for each unit contract and for the policy. (The standard deviation
of the distribution of number of loss is assumed to be equal to 1).
Answer:
a)
n =10,000
x̅=5
A.L = 100,000,000
𝛿=1
Since market price in perfect competition insurance market is equal to fair
premium the price can be obtained from the principle of equivalence:
Fair Premium = Loss Ratio × A.L =
5
10,000
× 100,000,000 = 50,000
b)
39
Economics of Insurance
𝑧𝛿×𝐴.𝐿
RR1 =
𝑛
=
2×1×100,000,000
10,000
= 20,000
RR1 also can be found by multiplication of PDR and Actuarial Fair Premium:
zδ
2
x
5
RR1=PDR× 𝐴𝐹𝑃= ̅ × 𝐴𝐹𝑃 = × 50,000 = 20,000
RRn = 𝑧𝛿 × 𝐴. 𝐿 = 2×1×100,000,000= 200,000,000
RRn also can be found by multiplication of “n” number of insureds to RR1
RRn =10,000×20,000=200,000,000
Fair Premium rate =
5
10,000
In a perfect competition insurance market, the company should bring the Required
Reserve equal to 200,000,000 Ts in order to be confident by 97.7% that can afford
the claims.
Example13:
Suppose a Life Insurance Company operates under perfect competition market. If
the mortality rate regardless the age and gender is 1/1000 and the death benefit is
100,000,000Ts.
a) calculate market price and its rate. What is the solvency margin of the company?
b) If the so-called company sells 10,000 policies and plans to increase its solvency
margin to 99.8%, calculate the required reserves for each contract and the total
required reserves (number of loss follows normal distribution with a standard
deviation of 2).
Answer:
a)
n=10,000 , x̅=10 , A.L = 100,000,000 , 𝛿=2
As mentioned, market price in perfect competition insurance market is equal to fair
premium.
Fair Premium = Loss Ratio × A.L =
10
10,000
× 100,000,000 = 100,000
40
Economics of Insurance
Solvency margin= Pr( xi≤ x̅ ) = Pr ( xi≤ 10 ) = 50%
b)
𝑧𝛿×𝐴.𝐿
RR1 =
𝑛
=
3×2×100,000,000
10,000
= 60,000
RRN = 𝑧𝛿 × 𝐴. 𝐿 = 3×2×100,000,000= 600,000,000
The company should bring the Required Reserve equal to 600,000,000Ts in order to
be confident by 98.8% that can afford the claims.
Example 14:
Now suppose the company can increase its number of customers to 40,000 and all
customers are exposed to the same risk. Calculate RR1 and RRn for the same
confidence level of 98.8%.
Answer:
n2 = 4 n1=40,000
A.L = 100,000,000
𝑧𝛿2×𝐴.𝐿
RR1 =
𝑛2
=
x̅2 = 4 x̅1 =40
𝛿 2 = 2 𝛿 1=4
3×4×100,000,000
40,000
= 30,000
Or equivalently:
𝑧𝛿2
12
𝑥2
40
RR1=PDR× 𝐴𝐹𝑃= ̅̅̅̅ × 𝐴𝐹𝑃 =
× 100,000 = 30,000
The Required Reserve for one unit of contract diminishes from 60,000Ts to
30,000Ts. This is because the number of insureds is increased by four times and the
PDR declines by √4 times and RR1 diminishes respectively.
RRn = 𝑧𝛿 × 𝐴. 𝐿 = 3×4×100,000,000= 1,200,000,000
Or equivalently:
RRn = n2 × RR1 = 40,000 × 30,000= 1,200,000,000
The Law of Large Numbers under Perfect Competition Insurance Market
As we learned in monopoly market by increasing the number of insureds PDR
diminishes and finally converges to zero. In perfect competition market RR1
41
Economics of Insurance
(required reserves for each contract) diminishes by increasing the number of
insureds respectively but RRn (required reserves for the policy) increases.
Corollary: If the number of insureds increases by ‘α’ times, the RR1 declines by
√𝛼 times, but the RRn increases by √𝛼 times:
Proof:
n2 = 𝛼 n1
𝑧𝛿2×𝐴.𝐿
RR1n2 =
𝑛2
=
𝑧√𝛼𝛿1×𝐴.𝐿 𝑧𝛿1×𝐴.𝐿
α𝑛1
RRnn1 = n1 × RR1 = n1 ×
=
√𝛼𝑛1
𝑧𝛿1 ×𝐴.𝐿
𝑛1
= RR1n1/√𝛼
= 𝑧𝛿1 × 𝐴. 𝐿
RRnn2 = 𝑧𝛿2 × 𝐴. 𝐿 = z × √𝛼 𝛿1 × A.L = √𝛼 × z𝛿1 × A.L = √𝛼 RRnn1
In Short:
RR1𝒏𝟐 = RR1𝒏𝟏 /√𝜶
RRn𝒏𝟐 = √𝜶 RRn𝒏𝟏
Example 15:
A company does its business under perfect competition insurance market. The
company absorbs 1,000 clients and the past experience shows the average claim of
20 out of each 1,000 customers with standard deviation of 4 and average loss of
50,000,000Ts. .
a) Find the market price and its rate.
b) If the company plans to increase its solvency margin to 99.8% find RR1 and RRn.
c) If the company can sell the product to 100,000 customers find RR1 and RRn and
market price and its rate.
d) If the company can sell the product to 2,500,000 customers find RR1 and RRn and
market price and its rate.
e) If the company is successful in selling the product to 10,000,000 clients, solve the
same problem for the same confidence level.
f) By answers obtained discuss about the importance of the law of large numbers in
insurance industry. (𝛿1 =4)
42
Economics of Insurance
Answer:
n1 =1,000
x̅1=20
𝛿1 =4
A.L = 50,000,000
a)
Fair Premium =
𝑥̅1
𝑛1
× A.L =
20
1,000
× 50,000,000 = 1,000,000
Fair Premium rate = Loss Ratio =
20
1,000
b)
𝑧𝛿1 ×𝐴.𝐿
RR1n1 =
𝑛1
=
3×4×50,000,000
1,000
= 600,000
RRNn1 = 𝑧𝛿1 × 𝐴. 𝐿= 3×4×50,000,000= 600,000,000
c)
n2 =100 n1= 100,000
𝑧𝛿2 ×𝐴.𝐿
RR1n2 =
𝑛2
=
x̅2=100 x̅1 = 2,000,
3×40×50,000,000
100,000
𝛿2 =10𝛿1 = 40
= 60,000
RRnn2 = 𝑧𝛿2 × 𝐴. 𝐿= 3×40×50,000,000= 6,000,000,000
Fair Premium =
𝑥̅2
𝑛2
× A.L =
Fair Premium rate =
𝑥̅2
𝑛2
2000
100,000
= Loss Ratio =
d)
n3 =25 n2= 2,500,000
𝑧𝛿3 ×𝐴.𝐿
RR1n3 =
𝑛3
=
× 50,000,000 = 1,000,000
20
1,000
x̅3=25 x̅2 = 50,000,
3×200×50,000,000
2,500,000
𝛿3 =5𝛿2 = 200
= 12,000
RRnn3 = 𝑧𝛿3 × 𝐴. 𝐿= 3×200×50,000,000= 30,000,000,000
Fair Premium =
𝑥̅2
𝑛2
× A.L =
Fair Premium rate =
𝑥̅2
𝑛2
e)
n4 =10,000,000 = 4 n3
𝑧𝛿4 ×𝐴.𝐿
RR1n4 =
𝑛4
=
2000
100,000
× 50,000,000 = 1,000,000
= Loss Ratio =
20
1,000
x̅4=4 x̅3 = 200,000,
3×400×50,000,000
10,000,000
𝛿4 =2𝛿3 = 400
= 6,000
RRNn4 = 𝑧𝛿4 × 𝐴. 𝐿= 3×400×50,000,000= 60,000,000,000
43
Economics of Insurance
Fair Premium =
𝑥̅2
𝑛2
× A.L =
Fair Premium rate =
𝑥̅2
𝑛2
2000
100,000
× 50,000,000 = 1,000,000
= Loss Ratio =
20
1,000
f)
By increasing the number of insureds RR1 declines from 600,000 to 60,000, 12,000
and finally to 6,000 and in infinite it converges to zero. This means by increasing
the number of customers the required reserves for each contract becomes smaller
and smaller. In other words, by increasing the number of insureds further and
further, the company needs to keep less and less reserves in order to satisfy the
confidence level of 99.8%. This is because of the grace of the application of law of
large numbers. As the corollary suggests if the number of insureds increases by ‘α’
times, the RR1 declines by √𝛼 times, but the RRn increases by √𝛼 times.
Example16:
Suppose an insurance company sells fire insurance policy. If the past data shows an
average of 10 accidents per each 10,000 houses insured and the average claim per
each accident is equal to 10,000,000 Ts. (𝛿1 =2)
a) Calculate fair premium and its rate and verify the solvency margin this premium
level fulfills.
b) If the company plans to increase its solvency margin to 97.7%, calculate PDR,
pure premium, fair premium and their rates in the case the insurance market is
monopoly.
c) Answer the same problem for the same confidence level (solvency margin) if the
company can absorb 40,000 clients, assuming all customers are subject to the same
risk.
d) what happens to PDR, pure premium, fair premium and their rates if the company
can increase its number of customers to 1,000,000.
e) Solve the same problem for the case the number of houses insureds increases to
100,000,000.
f) Now assume the insurance market is perfect competition; find pure premium for
the perfect competition market and the amount of required reserves for one contract
and for the policy in general in all sections mentioned before. Tabulate your answers
to show the convergence.
44
Economics of Insurance
g) Discuss about the importance of the law of large numbers using the answers
obtained in perfect competition and monopoly markets.
Answer:
n1 =10,000
x̅1=10
A.L = 10,000,000
𝛿1 =2
𝑥̅
10
a) Fair Premium = 1 × A.L =
× 10,000,000 = 10,000
𝑛1
10,000
10
Fair Premium rate = Loss Ratio =
10,000
Solvency Margin: Pr( xi≤ x̅ ) = Pr ( xi≤ 10 ) = 50%
b)
Pr(xi1≤x̅1+zδ1)=Pr(xi≤14)=97.7%
PDR99.7%,n1 =
𝑧𝛿1
× 100=
𝑥̅1
𝑥̅1 + 𝑧𝛿1
PuPr99.7%,n1=
𝑛1
Pu.PrRate99.7%,n1 =
2×2
10
× 𝐴. 𝐿 =
× 100 = 40 %
10 + 2×2
× 10,000,000=14,000
10,000
𝑃𝑢𝑟𝑒 𝑃𝑟𝑒𝑚𝑖𝑢𝑚
14,000
𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝐿𝑜𝑠𝑠
=
10,000,000
=
14
10,000
AFP=Loss.Ratio×A.L=10,000
Fair Premium Rate = Loss Ratio =
c)
n2 =4 n1= 40,000
Fair
Premium=
10
10,000
x̅2=4 x̅1 = 40 ,
𝛿2 =2𝛿1 = 4
Loss
Ratio
×
A.L
Fair Premium Rate = Loss Ratio =
PDR99.7%,n2 =
𝑧𝛿2
× 100=
𝑥̅2
𝑥̅ 2 + 𝑧𝛿2
PuPr99.7%,n2=
𝑛2
Pu.PrRate99.7%,n2 =
2×4
10,000
10
10,000
× 100 = 20 %
40
40 + 2×4
× 𝐴. 𝐿=
× 10,000,000=12,000
40,000
𝑃𝑢𝑟𝑒 𝑃𝑟𝑒𝑚𝑖𝑢𝑚
12,000
𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝐿𝑜𝑠𝑠
d)
n3 =25n2= 1,000,000,
=
=
10,000,000
x̅3=25 x̅2 = 1000,
=
12
10,000
𝛿3 =5𝛿2 = 20
45
Economics of Insurance
AFP=Loss.Ratio×A.L=10,000
10
Fair Premium Rate = Loss Ratio =
PDR99.7%,n3 =
𝑧97.7%𝛿3
𝑥̅3
𝑥̅ 3 + 𝑧𝛿3
PuPr99.7%,n3=
𝑛3
Pu.PrRate99.7%,n3 =
10,000
2×20
× 100=
× 𝐴. 𝐿=
× 100 = 4 %
1000
1000 + 2×20
× 10,000,000=10,400
1,000,000
𝑃𝑢𝑟𝑒 𝑃𝑟𝑒𝑚𝑖𝑢𝑚
10,400
𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝐿𝑜𝑠𝑠
=
10,000,000
=
10.4
10,000
e)
n4 =100n3= 100,000,000 ,
x̅4=100 x̅3 = 100,000,
Fair.Premium=Loss.Ratio×A.L=10,000
10
Fair Premium Rate = Loss Ratio =
PDR99.7%,n4 =
𝑧97.7%𝛿4
𝑥̅4
𝑥̅4 + 𝑧𝛿4
PuPr99.7%,n4=
𝑛4
Pu.PrRate99.7%,n4 =
× 100=
× 𝐴. 𝐿=
𝛿4 =10𝛿3 = 200
10,000
2×200
× 100 = 0.4 %
100,000
100,000 + 2×200
100,000,000
𝑃𝑢𝑟𝑒 𝑃𝑟𝑒𝑚𝑖𝑢𝑚
10,040
𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝐿𝑜𝑠𝑠
=
10,000,000=10,040
10,000,000
=
10.04
10,000
f)
𝑧𝛿1 ×𝐴.𝐿
RR1n1,97.7% =
𝑛1
=
2×2×10,000,000
10,000
= 4,000
RRnn1,97.7% = 𝑧𝛿1 × 𝐴. 𝐿= 2×2×10,000,000= 40,000,000
𝑧𝛿2 ×𝐴.𝐿
RR1n2,97.7% =
𝑛2
=
2×4×10,000,000
40,000
= 2,000
RRnn2,99.7% = 𝑧𝛿2 × 𝐴. 𝐿= 2×4×10,000,000= 80,000,000
𝑧𝛿3 ×𝐴.𝐿
RR1n3,99.7% =
𝑛3
=
2×20×10,000,000
1,000,000
=400
RRnn3,99.7% = 𝑧𝛿3 × 𝐴. 𝐿= 2×20×10,000,000=400,000,000
𝑧𝛿4 ×𝐴.𝐿
RR1n4,99.7% =
𝑛4
=
2×200×10,000,000
100,000,000
=40
RRnn4,99.7% = 𝑧𝛿4 × 𝐴. 𝐿= 2×200×10,000,000= 4,000,000,000
Note:
RR1n2 = RR1n1/√𝜶
RRnn2 = √𝜶 RRnn1
46
Economics of Insurance
𝒏𝟏
10,000
n
Solvency
Margin
Loss
Ratio
𝒏𝟏
10,000
𝒏𝟐 =4𝒏𝟏
40,000
𝒏𝟑 =25n2
1,000,000
𝒏𝟒 =100n3
100,000,000
∞
50%
97.7%
97.7%
97.7%
97.7%
97.7%
10
10,000
10
10,000
10
10,000
10
10,000
10
10,000
10
10,000
Fair
Premium
10,000
10,000
10,000
10,000
10,000
10,000
Fair
Premium
Rate
10
10,000
10
10,000
10
10,000
10
10,000
10
10,000
10
10,000
10,000,000
10,000,000
10,000,000
A.L
10,000,000 10,000,000 10,000,000
PDR
-
40%
20%
4%
0.4%
0
Pure
Premium
10,000
14,000
12,000
10,400
10,040
10,000
10
10,000
14
10,000
12
10,000
10.4
10,000
10.04
10,000
10
10,000
Pure
Premium
Rate
x̅
𝜹
RR1
RRn
10
-
10
40
1000
100,000
2
4
20
200
4,000
2,000
400
40
40,000,000 80,000,000 400,000,000 4,000,000,000
0
∞
g)
As the answers indicate, by increasing the number of insureds the PDR (the
percentage increase in pure premium which is needed in order to satisfy the
predetermined solvency margin) converges to zero and the pure premium and its
rate converges to fair premium and its rate. (10,000,
10
10,000
).
This means that by increasing the number of insureds the company in the case of
monopoly market does not need to increase the premium for satisfying a high level
of solvency margin of 97.7%;
If the company is efficient in its marketing and can absorb enough customers, does
not need to increase the premium further so as to discourage low-risk individuals
from purchasing the policy. Consequently, the insurance market will become stable.
47
Economics of Insurance
But for any reason if the company cannot absorb enough customers, should increase
the premium considerably in order to reach to the predetermined solvency margin
of 97.7%. By doing so, low-risk customers cancel their contracts since they realize
the premium is not fair to them and they are overpaying the policy. By dropping out
these low-risk clients, the company remains with high-risk individuals. This requires
even higher premium rates in order to satisfy the same predetermined solvency
margin. By increasing the premium in the next step again the second group of lowrisk individuals exit from the market and cancel their contracts. This follows up to
the point the company collapses.
In the case of perfect competition market, when the number of insureds increases
further and further, the required reserve for each contract declines and finally
converges to zero. Thus the company requires less reserve for each contract in order
to guarantee a predetermined confidence level. This means the market becomes
more stable.
Example 17:
An insurance company offers car insurance. The past data experience shows an
average of 20 crashes out of each 10,000 exposure with the average claim of
50,000,000Ts. If the company operates under perfect competition and the standard
deviation of the number of accidents, which follows normal distribution is equal to
4;
a) Calculate market price and its rate.
b) If the company plans to increase its solvency margin to %99.8, find required
reserves for one and all policy, and market price and its rate.
c) If the company can increase its costumers to 250,000 clients and assuming all
clients are subject to the same risk, find required reserves for each contract
and the policy in general, and also calculate the price and its rate.
d) If the company is efficient in marketing and can sell the product to 1,000,000
costumers, solve the same problem for the same confidence level.
e) For which number of costumers, the RR for each contract diminishes to 1000.
f) For which number of consumers, the required reserve for the policy becomes
72,000,000,000.
48
Economics of Insurance
g) By the answers obtained discuss about the importance of law of large number
in perfect competition insurance market. Tabulate your answers to notify the
importance of the Law of Large Numbers.
Answer:
a)
𝑛1 = 10,000
𝑥̅1 = 20
AL= 50,000,000
𝛿1 = 4
𝐴𝐹𝑃 =
̅𝑥̅̅1̅
𝑛1
𝐴𝐹𝑃𝑅 =
× 𝐴𝐿 =
̅𝑥̅̅1̅
𝑁
=
20
10,000
20
10,000
×50,000,000= 100,000
= loss ratio
b)
𝑅𝑅¹ =
𝑍. 𝛿1 . 𝐴𝐿 3 × 4 × 50,000,000
=
= 600,000
𝑛1
10,000
RRn1= 𝑍. 𝛿1 . 𝐴𝐿 = 3 × 4 × 50,000,000= 600,000,000
Market price =AFP = 100,000
Market rate= AFPR =
20
10,000
c)
𝑛2 = 250,000
𝑥̅2 = 500
𝛿2 = 20
49
Economics of Insurance
𝑅𝑅¹ =
𝑍. 𝛿2 . 𝐴𝐿 3 × 20 × 50,000,000
=
= 12,000
𝑛2
250,000
RRn2= 𝑍. 𝛿2 . 𝐴𝐿 = 3 × 20 × 50,000,000= 3,000,000,000
Market price = AFP = 100,000
Market rate=AFPR =
20
10,000
d)
𝑛3 = 1,000,000
𝑥̅3 = 2,000
𝛿3 = 40
𝑍. 𝛿3 . 𝐴𝐿 3 × 40 × 50,000,000
𝑅𝑅¹ =
=
= 6,000
𝑛3
1,000,000
RRn3= 𝑍. 𝛿3 . 𝐴𝐿 = 3× 40 × 50,000,000= 6,000,000,000
Market price = AFP = 100,000
Market rate= AFPR =
20
10,000
e)
Note: if the number of insureds increases by “α” time, then RR¹ diminishes by "√𝛼"
times and RRn increases by √α𝑡𝑖𝑚𝑒𝑠.
𝑅𝑅¹𝑚2 =
𝑍𝛿2 . 𝐴𝐿 𝑍√𝛼 𝛿1 . 𝐴𝐿
𝑍𝛿1 . 𝐴𝐿 𝑅𝑅¹𝑚1
=
=
=
𝑚2
𝛼𝑚1
√𝛼𝑚1
√𝛼
𝑅𝑅¹𝑚2 = 1000
1000 = 6000/√𝛼
√𝛼 = 6 , 𝛼 = 36
50
Economics of Insurance
𝑛4 = 36 × 𝑛3 = 36 ×1,000,000 = 36,000,000
f)
RRn4= 𝑛4 ×RR¹
RRn4= 36,000,000 × 1000 = 36,000,000,000
72,000,000,000 =√𝛽 × 36,000,000,000
𝛽 =4
n5 = 4n4 = 4× 36,000,000 = 144,000,000,000
g)
As the answers show, by increasing the number of insureds the required reserves for
each contract diminishes and finally converges to zero. This means by increasing
the number of consumers the company needs to keep lower required reserves for
each unit contract in order to reach to the same confidence level. This indicates that
the application of the law of large number is also very important in perfect
competition insurance market. However, the required reserves for policy in general
increases by any increase in the number of insureds, this increase is less than the
increase in the number of insureds. Specifically, if the number of insureds increases
by 𝛼 times, required reserves for the policy increases by √𝛼 times. This is because
of the grace of law of large number.
n
∞
n1=10,000
n1=10,000
n2=25n1
n2=250,000
n3=4n2
n3=1,000,000
n4=36n3
n4=36,000,000
n5=4n4
n5=144,000,000
S.M.
%50
%99.8
%99.8
%99.8
%99.8
%99.8
%99.8
AFP
100,000
100,000
100,000
100,000
100,000
100,000
100,000
𝟐𝟎
𝟏𝟎𝟎𝟎
𝟐𝟎
𝟏𝟎𝟎𝟎
𝟐𝟎
𝟏𝟎𝟎𝟎
𝟐𝟎
𝟏𝟎𝟎𝟎
𝟐𝟎
𝟏𝟎𝟎𝟎
𝟐𝟎
𝟏𝟎𝟎𝟎
𝟐𝟎
𝟏𝟎𝟎𝟎
n
AFPR
51
Economics of Insurance
𝛿
𝜹𝟏 = 𝟒
𝜹𝟏 = 𝟒
𝜹𝟐 = 𝟐𝟎
𝜹𝟑 = 𝟒𝟎
𝜹𝟒 = 𝟐𝟒𝟎
𝜹𝟓 = 𝟒𝟖𝟎
−
𝑥̅
̅𝟏 = 𝟐𝟎
𝒙
̅𝟏 = 𝟐𝟎
𝒙
̅𝟐 = 𝟓𝟎𝟎
𝒙
̅𝟑 = 𝟐𝟎𝟎𝟎
𝒙
̅𝟒 = 𝟕𝟐, 𝟎𝟎𝟎
𝒙
̅𝟓 = 𝟐𝟖𝟖, 𝟎𝟎𝟎
𝒙
−
RR¹
−
𝟔𝟎. 𝟎𝟎𝟎
𝟏𝟐. 𝟎𝟎𝟎
𝟔. 𝟎𝟎𝟎
𝟏. 𝟎𝟎𝟎
𝟓𝟎𝟎
𝟎
RRn
−
𝟔𝟎𝟎, 𝟎𝟎𝟎, 𝟎𝟎𝟎
𝟑, 𝟎𝟎𝟎, 𝟎𝟎𝟎, 𝟎𝟎𝟎
𝟔, 𝟎𝟎𝟎, 𝟎𝟎𝟎, 𝟎𝟎𝟎
𝟑𝟔, 𝟎𝟎𝟎, 𝟎𝟎𝟎, 𝟎𝟎𝟎
𝟕𝟐, 𝟎𝟎𝟎, 𝟎𝟎𝟎, 𝟎𝟎𝟎
∞
52
Economics of Insurance
5-Rate making in an Oligopoly Insurance Market
A market is referred to as oligopoly if the number of suppliers is limited and each
supplier has monopoly power equal to a percentage of PDR to increase the premium
and has to keep required reserve (RR) for the remainder of PDR in order to fulfill
the predetermined solvency margin.
In the case of pure monopoly, the sole producer had perfect power to increase the
premium by 100% of PDR in order to fulfill the predetermined solvency margin and
did not need to keep any reserve for that purpose. While in the perfect competition
insurance market, any individual producer or supplier should take the market price
of fair premium level and cannot increase the premium for the purpose of fulfilling
the predetermined solvency margin. Instead of increasing the premium, the
individual supplier should keep reserve equal to 100% of PDR in order to reach to
predetermined confidence level.
As mentioned in the case of oligopoly, just a percentage of the change required for
satisfying predetermined solvency margin can be accomplished by increasing the
premium. This percentage is referred to as Monopoly Power ( µ ). The remainder
can be accomplished by keeping Required Reserves. The percentage attributed to
Required Reserve will be equal to (1-µ). It is obvious that for the case of monopoly,
µ=1, and for the case of perfect competition, µ=0.
We will have:
Pu.PrOli β% =AFP + µ ×PDRβ% × AFP = (1+ µ.PDR) AFP
RR1,Oliβ% = (1-µ)PDR × AFP
RRn,Oliβ% = n(1-µ)PDR × AFP
Example18:
Suppose the actuarial fair premium is equal to 100,000 and the PDR for solvency
margin of 99% is equal to 40%. Find pure premium and RR for each contract in the
case of monopoly, perfect competition and oligopoly market with the monopoly
power for the individual company doing business under oligopoly equal to 60%
(µ=60%=0.6)
Answer:
53
Economics of Insurance
Fair premium= 100,000
PDR= 40%
µ=0.6
a) Monopoly:
Pu.PrMO 99% = AFP + PDR99% × AFP = (1+PDR) AFP
= 100,000 + 40% × 100,000 = (1.4) 100,000 = 140,000
MO
RR = 0
b) Perfect competition:
Pu.PrP.C 99% = AFP = 100,000
RRP.C = PDR99% × AFP = 0.4 × 100,000 = 40,000
c) Oligopoly:
PuPrOli 99% = AFP + µ PDR99% × AFP = (1+ µ. PDR) AFP
= (1+0.6×0.4) ×100,000 = 1.24 ×100,000 = 124,000
1, Oli
RR 99% = (1-µ) PDR × AFP
= 0.4 ×0.4×100,000 = 16,000
Example 19:
A company is selling car collision insurance in a country. If the past records show
an average of 100 accidents out of each 10,000 cars with the average claims of
20,000,000Ts, find pure premium for the cases of monopoly, perfect competition
and oligopoly market with monopoly power of 20% if the standard deviation of the
number of accidents is equal to 20 and the company plans to reach to the solvency
margin of 99.8%. Find the amount of required reserve for all three types of markets.
What will happen to the answers if the company can sell the product to 1,000,000
clients?
Answer:
n1 = 10,000
x̅1 = 100
A.L = 20,000,000
𝛿1 =20
µ=20%
100
Fair Premium = Loss Ratio × A.L =
× 20,000,000 = 200,000
10,000
a) Monopoly:
PDR 99.8%, n1 =
𝑧99.8%𝛿1
𝑥̅1
× 100=
3×20
100
× 100 = 60 %
54
Economics of Insurance
Pu.PrMO 99.8%, n1 = AFP + PDR99.8% × AFP = (1+PDR) AFP
= 200,000 + 60% × 200,000 = (1.6) 200,000 = 320,000
RR1, MOn1,99.8% = 0
b) Perfect Competition:
Pu.PrP.C 99.8%, n1 = AFP = 200,000
RR1,P.C99.8%, n1 = PDR99.8%,n1 × AFP = 0.6 × 200,000 = 120,000
RRn,P.C99.8%, n1 = n1× RR1,P.C99.8%, n1=10,000×120,000=1,200,000,000
c) Oligopoly:
Pu.PrOli 99.8%, n1 = AFP + µ PDR99.8% × AFP = (1+ µ.PDR) AFP
= (1+0.2×0.6)×200,000 = 1.12 ×200,000 = 224,000
1,Oli
RR 99.8%, n1 = (1-µ)PDR × AFP= 0.8 ×0.6×200,000 = 96,000
RRn,Oli99.8%, n1 = 10,000 × 96,000 = 960,000,000
n2 = 100 n1 =1,000,000 x̅2 = 100 x̅1 = 10,000
A.L = 20,000,000
𝛿2 =10𝛿1 =200
µ=20%
Fair Premium = 200,000
a) Monopoly:
PDR 99.8%, n2 =
𝑧99.8%𝛿2
𝑥̅2
× 100=
3×200
10,000
× 100 = 6 %
Pu.PrMO 99.8%, n2 = AFP + PDR99.8% × AFP = (1+PDR) AFP
= 200,000 + 6% × 200,000 = (1.06) 200,000 = 212,000
RR1, MOn2,99.8% = 0
b) Perfect Competition:
PuPrP.C 99.8%, n2 = AFP = 200,000
RR1,P.C99.8%, n2 = PDR99.8% × AFP = 0.06 × 200,000 = 12,000
RRn,P.C99.8%, n2 = 1,000,000 × 12,000 = 12,000,000,000
c) Oligopoly:
PuPrOli 99.8%, n2 = AFP + µ PDR99.8% × AFP = (1+ µPDR) AFP
= (1+0.2×0.06)×200,000 = 1.012 ×200,000 = 202,400
1,Oli
RR 99.8%, n2 = (1-µ)PDR × AFP= 0.8 ×0.06×200,000 = 9,600
55
Economics of Insurance
RRn,Oli99.8%, n2 = 1,000,000 × 9,600 = 9,600,000,000
The total amount of money that should be sacrificed in order to increase the solvency
margin is equal in all markets. But in the case of pure monopoly the total amount of
burden is imposed to demanders, in perfect competition the companies should
sacrifice the amount of money required and in an oligopoly insurance market the
burden is imposed to both of demanders and suppliers.
In example 20, the total amount sacrificed for this purpose in the first case when n1
= 10,000, was equal to 1,200,000,000Ts, and became 12,000,000,000Ts when
n2=100 n1=1,000,000.
In the first case of n1=10,000, in Oligopoly Market, out of the total cost of
1,200,000,000 that should be sacrificed, an amount of 960,000,000Ts should be kept
as required reserve by the company, and the remainder of 240,000,000
(=10,000×24,000) is imposed to demanders. In the second case of n2=100
n1=1,000,000, out of the total cost of 12,000,000,000 that should be sacrificed, an
amount of 9,600,000,000Ts should be kept as required reserve by the company, and
the remainder of 2,400,000,000 (=1000,000×2,400) is imposed to demanders.
56
Economics of Insurance
6-Total Premium
The premium discussed so far is called pure premium which covers just for the
claims. Insurance business has other costs and expenses that can be outlined as
follows:
1. Administrative cost
2. Commission for agents and brokers
3. Taxes
4. Profit for insurers
5. Moral hazard costs (the costs of fraudulent activities)
6. Adverse selection cost
7. Cost of contingencies
8. Etc.
We refer to these costs as Loading Costs. Consequently, total premium of a policy
can be obtained by Using this relationship:
Total Premium= Pure Premium + Loading Costs
T.Pr = Pu.Pr + L.C (Loading Costs)
T.Pr = Pu.Pr + 𝜾 T.Pr
α: Loading Ratio
T.Pr - 𝜾 T.Pr = Pu.Pr
(1- 𝜾) T.Pr = Pu.Pr
T.Pr =
𝑷𝒖 .𝑷𝒓
(𝟏−𝜾)
Example 20:
If the pure premium for a contract is 1 million and the loading ratio is 20% find
total premium and the amount of loading costs.
Answer:
T.Pr =
𝑃𝑢 𝑃𝑟
(1−𝜾)
=
1000,000
1− 0.2
=
1000,000
0.8
= 1,250,000
T.Pr = Pu.Pr + L.C
L.C= T.Pr- Pu.Pr =1,250,000-1,000,000=250,000
57
Economics of Insurance
Example 21:
If the pure premium is equal to 200, and loading ratio is equal to 20%, find total
premium and show how the equality of “T.Pr = PuPr + 𝜾 T.Pr” applies.
Answer:
T.Pr =
𝑃𝑢 𝑃𝑟
(1−𝜾)
=
200
1− 0.2
=
200
0.8
= 250
T.Pr = PuPr + 𝜾 T.Pr
250 = 200 + 0.2× 250
250= 200+50 = 250
Example22:
A company sells fire insurance. If the past data shows an average number of 10
accidents out of each 10,000 houses and the average claim per each accident is equal
to 10,000,000Ts
a) find fair premium and its rate.
b) If the market is pure monopoly and the company wishes to increase its solvency
margin to 99.8%, find its PDR, pure premium and its rate and compare it with fair
premium. (𝛿=2).
c) Suppose the market is perfect competition, find pure premium and its rate and the
required reserve for the same solvency margin.
d) If the market is oligopoly with the monopoly power of 50%, find pure premium
and required reserve.
e) Find total premium for all cases if the loading ratio is equal to 20%.
Answer:
n = 10,000
Z99.8% = 3
x̅ = 10
A.L = 10,000,000 𝛿=2
µ=50%
a)
Fair Premium = Loss Ratio × A.L =
10
10,000
× 10,000,000 = 10,000
58
Economics of Insurance
Fair Premium Rate = Loss Ratio=
10
10,000
b)
PDR 99.8%=
𝑧99.8% 𝛿
𝑥̅
× 100=
𝑥̅+ 𝑧𝛿
Pu.PrMO 99.8%=
𝑛
Pu.PrRateMO 99.8%=
3×2
× 100 = 60 %
10
10+3×2
× 𝐴. 𝐿 =
× 10,000,000 = 16,000
10,000
𝑃𝑢𝑟𝑒 𝑃𝑟𝑒𝑚𝑖𝑢𝑚
16,000
𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝐿𝑜𝑠𝑠
=
10,000,000
=
16
10,000
c)
Pure premium in perfect competition is equal to Actuarial Fair Premium.
PuPrP.C 99.8%=AFP = 10,000
RR1,P.C99.8% = PDR99.8% × AFP = 0.6 × 10,000 = 6,000
RRn,P.C99.8% = 10,000 × 6,000 =60,000,000
d)
µ=50%
Pu.PrOli 99.8% = AFP + µ PDR99.8% × AFP = (1+ µPDR) AFP
= (1+0.5×0.6)×10,000 = 1.3 ×10,000 = 13,000
RR1,Oli99.8% = (1-µ)PDR × AFP= 0.5 ×0.6×10,000 = 3,000
RRn,Oli99.8% = 10,000 × 3,000 = 30,000,000
e)
T.Pr =
T.Pr
𝐴𝐹𝑃
=
10,000
=
(1−𝛼) 1− 0.2
𝑃𝑢 .𝑃𝑟
16,000
MO
=
T.Pr PC =
T.Pr Oli=
(1−𝛼)
𝑃𝑢. 𝑃𝑟
=
=
=
1− 0.2
10,000
=
1− 0.2
= 12,500
0.8
16,000
=
(1−𝛼) 1− 0.2
𝑃𝑢. 𝑃𝑟
13,000
(1−𝛼)
10,000
=
0.8
10,000
0.8
13,000
0.8
= 20,000
= 12,500
= 16,250
Example 23:
A company provides fire insurance contract in an oligopoly insurance market with
the monopoly power of %40. Suppose the number of loss, the number of insureds
and average loss equal to 10, 1000, 100.000.000, respectively; and standard
59
Economics of Insurance
deviation of the number of loss is equal to 2; find pure premium and its rate and
required reserves per each contract and for the policy, if the company determines
confidence level of %99.8.
Answer:
𝑥̅ = 10,
n=1,000,
𝛿 = 2,
𝑥̅
10
𝑛
1000
𝑃𝑢. 𝑃𝑟%50 = × 𝐴. 𝐿=
𝑃𝑢. 𝑃𝑟 𝑅𝑎𝑡𝑒 =
𝑃𝐷𝑅%99.8 =
1,000,000
100,000,000
𝜇 = 0.4,
A.L = 100,000,000
× 100,000,000 = 1,000,000 = 𝐴𝐹𝑃
=
10
1000
=AFPR
𝑧%99.8 𝛿
3 ×2
× 100 =
× 100 = %60
𝑥̅
1000
Pu.PrOli%99.8 = 1,000,000+ 0.4 × 0.6 × 1,000,000 = 1,240,000
The company can increase the price to 1,240,000$ , and the amount of required
reserves is as follows:
RRPC,¹ =
𝑍𝛿.𝐴𝐿
𝑛
=
3 ×2×100,000,000
1000
= 600,000
RR for one contract in perfect
competition insurance market.
RR1,Oli =600,000 – 240,000 = 360,000
The direct formulas for calculation of the pure premium in oligopoly market is as
follows:
𝑥̅
𝑥̅
𝑥̅
𝑛
𝑛
𝑁
Pu.Pr Oli= × 𝐴. 𝐿 + 𝜇 × 𝑃𝐷𝑅 × × 𝐴. 𝐿 =
× 𝐴. 𝐿 (1 + 𝜇 × 𝑃𝐷𝑅)
Therefore, we have:
Pu.Pr Oli= 1,000,000 (1 + 0.4 × 0.6) = 1,240,000
RRn,Oli = n × RR1,Oli= 1,000× 360,000 = 360,000,000
60
Economics of Insurance
Example 24:
A company sells fire insurance. If the past data shows 10 accidents out of 10,000
houses with the average loss of 100,000,000Ts. and standard deviation of 2 for the
number of losses:
a) Calculate pure premium and fair premium, and their rates. Verify the solvency
margin that this premium satisfies.
b) Assume the market is monopoly and the company plans to reach the solvency
of %97.7, calculate PDR, pure premium and fair premium and their rates.
c) Assume all exposures are subject to the same risk and the company can
increase the number of insureds to 40,000. Calculate PDR, pure premium, fair
premium and their rates.
d) Solve the same problem if the number of houses insured increases to
1,000,000 for the same solvency margin.
e) What happens to the answers obtained if the company is very efficient in
marketing and can sell the products to 100,000,000 clients.
f) Find the number of insureds if the pure premium of 100,200 guarantees the
same confidence level.
g) Find required reserves for one contract and for the policy in general if the
market is perfect competition for the same solvency margin.
h) If the market is oligopoly and the mentioned company has monopoly power
of %40, fine pure premium and required reserves for previous sections.
i) It the loading ratio is equal to %20, find total premium for parts ‘a’, ‘b’ and
‘c’.
Answer:
a)
𝑛1 = 10,000
𝑥̅1 = 10
𝛿1 = 2
61
Economics of Insurance
AL= 100,000,000
𝑝𝑢. 𝑝𝑟 =
̅𝑥̅̅1̅
𝑛1
× 𝐴𝐿 =
𝑝𝑢. 𝑝𝑟 𝑅𝑎𝑡𝑒 =
10
10,000
100,000
×100,000,000= 100,000= AFP
=
10
100,000,000 10,000
= AFPR
Solvency margin = pr (𝑥𝑖 ≤ ̅̅̅)
𝑥1 = %50
Solvency margin = pr (𝑥𝑖 ≤ 10) = %50
b)
Solvency margin = pr (𝑥𝑖 ≤ ̅̅̅+𝑧
𝑥1 %97.7 𝛿1 ) = %97.7
Solvency margin = pr (𝑥𝑖 ≤ 10+2× 2) = pr (𝑥𝑖 ≤ 14) = 97.7 %
𝑃𝐷𝑅%97.7,𝑛1 =
𝑧%97.7 𝛿1
2 ×2
× 100 =
× 100 = %40
𝑥̅1
10
𝑃𝑢. 𝑃𝑟%97.7,𝑛1 =
̅𝑥̅̅1̅
𝑛1
× 𝐴. 𝐿=
𝑃𝑢. 𝑃𝑟 𝑅𝑎𝑡𝑒%97.7,𝑛1 =
14
10,000
× 10,000,000 = 140,000
140,000
14
=
100,000,000 10,000
Fair premium and its rate are 10,000 and
10
10,000
, respectively.
c)
n2=4n1=40,000
𝑥̅2 = 4𝑥̅1 = 40
𝛿2 = 2𝛿1 = 4
Solvency margin = pr (𝑥𝑖 ≤ ̅̅̅+𝑧
𝑥2 %97.7 𝛿2 ) = %97.7
Solvency margin = pr (𝑥𝑖 ≤ 40+2× 4) = pr (𝑥𝑖 ≤ 48) = 97.7 %
62
Economics of Insurance
𝑃𝐷𝑅%97.7,𝑛2 =
𝑧%97.7 𝛿2
2 ×4
× 100 =
× 100 = %20
𝑥̅2
40
𝑃𝑢. 𝑃𝑟%97.7,𝑛2 =
̅𝑥̅̅2̅+𝑧%97.7 𝛿2
𝑛2
× 𝐴. 𝐿=
40+8
40,000
× 100,000,000 = 120,000
120,000
12
=
100,000,000 10,000
10
Fair premium and its rate are 100,000 and
, respectively.
𝑃𝑢. 𝑃𝑟 𝑅𝑎𝑡𝑒%97.7,𝑛2 =
10,000
d)
n3=25n2=1,000,000
𝑥̅3 = 25𝑥̅2 = 1,000
𝛿3 = 5𝛿2 = 20
Solvency margin = pr (𝑥𝑖 ≤ ̅̅̅+𝑧
𝑥3 %97.7 𝛿3 ) = %97.7
𝑃𝐷𝑅%97.7,𝑛3 = %4
𝑃𝑢. 𝑃𝑟%97.7,𝑛3 = 104,000
10.4
𝑃𝑢. 𝑃𝑟 𝑅𝑎𝑡𝑒%97.7,𝑁3 =
10,000
Fair premium and its rate are 10,000 and
10
10,000
, respectively.
e)
n4=100n3=100,000,000
𝑥̅4 = 100𝑥̅3 = 100,000
𝛿4 = 10𝛿3 = 200
𝑃𝐷𝑅%97.7,𝑛4 = %0.4
𝑃𝑢. 𝑃𝑟%97.7,𝑛4 = 100,400
10.04
𝑃𝑢. 𝑃𝑟 𝑅𝑎𝑡𝑒%97.7,𝑛4 =
10,000
Fair premium and its rate are 10,000 and
10
10,000
, respectively.
63
Economics of Insurance
f)
𝑃𝑢. 𝑃𝑟%97.7,𝑛5 = 100,200
When the number of insureds is increased 𝛼 times, the relation between 𝑅𝑅%97.7,𝑛4
and 𝑅𝑅%97.7,𝑛5 can be described as follows:
𝑅𝑅%97.7,𝑛5 =
𝑅𝑅%97.7,𝑛4
√𝛼
400
200=
√𝛼
𝛼=4
𝑛5 = 4 × 100,000,000 = 400,000,000
g)
RR1n1=
𝑍.𝛿1 .𝐴𝐿
𝑛1
= 40,000
RRnn1= 𝑍. 𝛿1 . 𝐴𝐿 = 400,000,000
RR1n2=
𝑍.𝛿2 .𝐴𝐿
𝑛2
= 20,000
RRnn2= 𝑍. 𝛿2 . 𝐴𝐿 = 800,000,000
RR1n3=4000
RRnn3= 4,000,000,000
RR1n4=400
RRnn4= 40,000,000,000
64
Economics of Insurance
RR1n5= 200
RRnn5= 80,000,000,000
h)
𝜇 = 0.4
𝑃𝑢. 𝑃𝑟 𝑂𝑙𝑖 %97.7,𝑛1 = 𝐴𝐹𝑃 + ( 𝜇 × 𝑃𝐷𝑅 × 𝐴𝐹𝑃) = (1 + 𝜇 × 𝑃𝐷𝑅) × 𝐴𝐹𝑃
𝑃𝑢. 𝑃𝑟 𝑂𝑙𝑖 %97.7,𝑛1 = 1.16 × 10,000 = 116,000
RR1,Oli%97.7, n1= 4,000 − 1,600 = 24,000
RRn,Oli%97.7,n1= 24,000 × 10,000 = 240,000,000
𝑃𝑢. 𝑃𝑟 𝑂𝑙𝑖 %97.7,𝑛2 = 108,000
RR1,Oli%97.7, n2= 20,000 − 8,000 = 12,000
RRn,Oli%97.7,n= 12,000 × 40,000 = 480,000,000
The calculation for n3 to n5 are left to the students.
i)
𝑃𝑢.𝑃𝑟
TP=
1−𝛼
𝑇𝑃𝑛1,%50 =
100,000
1−0.20
𝑇𝑃𝑛1,%97.7 =
𝑇𝑃𝑛2,%97.7 =
= 125,000
140,000
1−0.25
= 175,000
120,000
= 150,000
1 − 0.2
The calculation for n3 to n5 are left to the students
65
Economics of Insurance
7-Application of Model to Iranian Fire Insurance Market2
In this part we try to apply the model to Iranian Fire Insurance market. First the
primary input data is selected from an Iranian Insurance Company (Mellat Insurance
Company) fire insurance claim and after statistical modeling of frequency of loss the
pure premium for different distributions of frequency are calculated. In the next
section, the potential deviation ratios required to increase the financial solvency
margin of the insurance company according to different distributions are calculated.
Finally, based on the Law of Large Numbers, the effect of increasing the number of
clients on the amount of potential deviation ratio has been examined and finally, by
considering loading factors, total premiums are calculated.
7-1-Description of data
The monthly data of fire claims of Mellat insurance company from the year 1395 to
1398 Solar Hijri (2016-2019) is used in the research.
Table 7-1- Monthly data of frequency and severity of fire loss
Number
Year
Month
Number of
Contracts
Frequency
(No. of
claims)
Severity per
Contract
(Average Loss)
(Rial)
1
01
3,368
25
14,650,201
2
02
6,502
49
13,029,348
3
1395
03
5,233
14
53,702,071
4
(2016)
04
8,445
22
25,749,984
5
05
7,014
14
18,947,693
6
06
6,301
17
6,113,502
2
This practical work is the main part of the Master dissertation of Hamideh Heidari supervised by me in the field of
actuarial science at ECO college of insurance. I earnestly acknowledge the sincere efforts given by her.
66
Economics of Insurance
Number
Year
Month
Number of
Contracts
Frequency
(No. of
claims)
Severity per
Contract
(Average Loss)
(Rial)
7
07
7,430
30
33,022,267
8
08
9,572
19
128,563,804
9
09
6,070
17
20,890,187
10
10
6,736
26
59,359,619
11
11
8,859
21
111,232,387
12
12
7,530
21
17,440,429
13
01
3,154
9
13,757,130
14
02
6,161
15
28,689,078
15
03
10,489
19
48,233,281
16
04
12,168
28
24,365,821
17
05
10,979
36
125,739,441
18
1396
06
8,022
21
22,360,984
19
(2017)
07
7,064
27
63,237,380
20
08
9,329
21
28,116,850
21
09
8,217
19
29,844,935
22
10
12,728
60
24,787,090
23
11
18,115
75
26,346,756
24
12
10,394
70
29,473,199
25
01
3,535
10
55,353,020
26
02
6,320
69
58,695,468
27
03
7,285
44
80,501,205
28
04
9,363
39
25,749,479
05
8,886
80
49,402,235
06
8,841
63
58,909,076
31
07
8,994
62
38,094,064
32
08
9,578
86
49,907,931
33
09
9,285
36
55,921,165
34
10
10,168
40
43,637,436
35
11
12,220
66
66,211,648
29
30
1397
(2018)
67
Economics of Insurance
Number
Year
Month
Number of
Contracts
Frequency
(No. of
claims)
Severity per
Contract
(Average Loss)
(Rial)
36
12
12,548
64
43,054,227
37
01
4,813
24
38,918,940
38
02
8,167
29
40,934,338
39
03
8,856
29
102,367,876
40
04
11,022
41
66,614,756
41
05
27,665
40
209,512,776
42
1398
06
14,876
21
212,270,071
43
(2019)
07
11,053
16
537,097,138
44
08
10,029
23
233,825,845
45
09
7,951
16
192,994,284
46
10
11,345
36
259,187,139
47
11
13,376
30
159,154,459
48
12
5,105
24
95,926,772
Source: Mellat Insurance Data Base
Trends of Severity and Frequency and their Histograms are drawn at the graphs
below.
68
Economics of Insurance
8
80
5
11
12
2
11
12
6 7
2
3
10
4
40
Frequency
60
10
4
9
5
10
5
7
4
10
1
4
20
8
6
3
11
3
1
5
9
7
2
12
8
6
8 9
3
9
2
7
6
11 12
1
1
0
10
20
30
40
Time
Graph 7-1- Frequency Trend
1395
1396
1397
7
300
10
8
200
5
6
9
11
8
5
100
11
3
3
3
1
2
7
10
4
5
7
9
12
6
2
1
3
4
6
8
1
9
10 11
12
2
5
6
7
4
8
9
10
12
4
11
12
1 2
0
Severity
400
500
1398
0
10
20
30
40
Time
Graph 7-2- Severity Trend
69
Economics of Insurance
Graph 7-3-Histograms: Severity and Frequency
As the trends indicate, both frequency and severity of the fire claims (losses) do not
follow any sort of specific trends. Yet, the Histogram of both frequency and severity
are right-skewed.
7-2-Claim Frequency Models
To calculate the pure premium as the multiplication of severity and frequency,
different distributions are fit to the frequency data set and pure premium is obtained
based on different distributions introduced. It should be noted that loss severity is
considered constant at the average rate of damage.
In order to fit the statistical distributions on the frequency variables, discrete and
continuous statistical distributions are used. Among the statistical distributions,
Gamma, Lognormal, Weibull and Negative Binominal are well known to have good
data fit on loss frequency. The Kolmogorov-Smirnov, Anderson-Darling and ChiSquare test are used to investigate the goodness of fit.
In order to compare the amount of pure premium and potential deviation from the
calculated average, the normal distribution is also fit on the data. Moreover,
70
Economics of Insurance
Maximum Likelihood Estimates (MLE) method is used to estimate the parameters
of the distributions, which leads to the amount of mean, standard deviation and
skewness of each distribution.
The results of fitting each distribution are presented in the form of graphs and tables
in Appendix 1. Based on the results of goodness of fit tests, Gamma, Lognormal,
Weibull and Negative Binomial distributions have good fit on the data and Normal
distribution does not have a suitable fit on the data. In addition, Poisson distribution
does not fit on the data. It should be noted that it is not possible to determine which
distribution is superior to the other. It is only the results of the fit that determine
which distribution is best for the data.
7-3- Actuarial Fair Premium
Based on the principle of equivalence the total premium collected is equal to the
total expected losses. Since, the total premium for the losses is the multiplication of
pure premium and the number of exposures, and the total expected losses is the
multiplication of the average number of losses and the average severity, the pure
premium is calculated as below:
𝑃𝑢𝑟𝑒 𝑃𝑟𝑒𝑚𝑖𝑢𝑚 =
𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑙𝑜𝑠𝑠𝑒𝑠 × 𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑠𝑒𝑣𝑒𝑟𝑖𝑡𝑦
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑒𝑥𝑝𝑜𝑠𝑢𝑟𝑒𝑠
Due to the frequency distribution fit, pure premium is calculated by multiplying the
mean of distributions divided by the number of contracts (exposures) by severity of
the loss (constant average loss), which is the mean of severities (77,956,141 Rials).
The result is shown in the table below:
71
Economics of Insurance
Table 7-2- Fair premium
The Frequency Distribution
Fair premium
Gamma (3P)
Lognormal (3P)
Weibull (3P)
Neg. Binomial
Normal
293,865
296,265
293,822
274,382
293,865
Table 7-2 shows the amount of actuarial fair premium for different frequency
distributions.
7-4- Potential Deviation Ratio (PDR)
At this stage, the calculated values of fair insurance premiums are analyzed based
on the frequency distribution and average of damages, and the potential deviation
ratio is calculated. In fact, for each distribution, the potential deviation ratio at a
certain level of confidence (e.g. 90%) is equal to the percentage deviation of the
desired value at the relevant confidence level.
𝑃𝐷𝑅 =
𝑣𝑎𝑙𝑢𝑒𝑝(1−𝛼)− 𝑣𝑎𝑙𝑢𝑒𝑝𝑚𝑒𝑎𝑛
𝑣𝑎𝑙𝑢𝑒𝑝𝑚𝑒𝑎𝑛
× 100
Assuming the Gamma distribution on the frequency of claims, PDR is calculated
as below:
Pr(𝑓𝑖 ≤ 𝑓 ̅ + 𝑍𝜎) = (1 − 𝛼)%
𝑖𝑓 𝑍 = 0 ,
𝑓 ̅ = 34.646
Pr(𝑓𝑖 ≤ 𝑓 )̅ = (1 − 𝛼)%
Pr(𝑓𝑖 ≤ 34.646) = 60%
72
Economics of Insurance
Table 7-3- Frequency for Gamma cumulative distribution function
Confidence level
60%
90%
95%
99%
f
35
62
76
105
The values in the table above are derived from Gamma cumulative distribution
function fit on the data. This means that the Gamma distribution suggests that if the
insurance company calculates the premium based on the average number of losses
of 35, the premium level satisfies the solvency margin level of 60%.
If the company plans to increase its solvency margin to %90, %95, and %99 , the
company should consider the number of claims as 62, 76, and 105 respectively.
Also, by using the concept of PDR the required change in frequency can be obtained
as:
𝑃𝐷𝑅 =
62 − 35
× 100 = 77%
35
If the company adds as much as this percentage to the amount of fair premium
calculated in the previous step, the solvency margin of the insurance company
increases from 60% to 90%.
In the table below the potential deviation ratio or the percentage required to be added
to fair premiums to increase the solvency margin are shown. Obviously, the potential
deviation ratio at the confidence level corresponds to the mean of the distribution is
zero.
It should be noted that the actuarial fair premium levels calculated in the table 7-3
above are obtained by the average amount (mean) of the corresponding distributions.
These averages occur at different levels of confidence.
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For other distributions, PDR is calculated and the results are shown in the tables
below:
Table 7-4- PDR based on Gamma distribution
Confidence level
60%
90%
95%
99%
N
35
62
76
105
PDR
0%
80%
118%
203%
The considered number of claim (number of loss) and associated PDR for lognormal
distribution of frequency appears in table below:
Table 7-5- PDR based on Lognormal distribution
Confidence level
63%
90%
95%
99%
N
35
62
78
122
PDR
0%
78%
125%
249%
The considered number of claim (number of loss) and associated PDR for Weibull
distribution of frequency appears in table below:
Table 7-6- PDR based on Weibull distribution
Confidence level
59%
90%
95%
99%
N
35
62
75
101
PDR
0%
80%
116%
192%
Similarly, the considered number of claim (number of loss) and associated PDR for
Binomial distribution of frequency appears in table below:
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Economics of Insurance
Table 7-7- PDR based on negative Binomial distribution
Confidence level
57%
90%
95%
99%
N
35
59
69
94
PDR
0%
82%
113%
191%
Finally, the considered number of claim (number of loss) and associated PDR for
normal distribution of frequency appears in table below:
Table 7-8- PDR based on Normal distribution
Confidence level
50%
90%
95%
99%
N
35
61
68
82
PDR
0%
76%
97%
137%
Table 7-9 compares the PDR obtained from different distributions with each other.
Table 7-9- PDR comparison of different distributions
Confidence level
90%
95%
99%
Gamma (3P)
79.99%
117.93%
203.27%
Lognormal (3P)
78.31%
124.53%
248.71%
Weibull (3P)
80.17%
115.57%
191.65%
Neg. Binomial
82.39%
113.30%
190.58%
Normal
75.53%
96.94%
137.10%
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Economics of Insurance
7-5- Premium Calculation
Based on the amount of potential deviation ratio for different distributions of claim
frequency, the amount of pure premium to achieve the required solvency margin is
calculated. The table 7-10 summaries the results.
Table 7-10- Pure Premium for different solvency margin (Rials)
Frequency
Confidence
level
Gamma (3P)
Lognormal (3P)
Weibull (3P)
Neg. Binomial
Normal
90%
528,941
528,271
529,382
500,434
515,811
95%
640,419
665,195
633,405
585,253
578,730
99%
891,196
1,033,099
856,929
797,301
696,748
Premiums calculated in the table 7-10 are in the direct relationship with the PDR
calculated in the table 7-9. This table gives the premiums that insurance company
should ask for from clients in order to remain confident that can afford the claims by
the attributed confidence level.
7-6- The importance of the Law of Large Numbers
In this section, we identify how important the application of the law of large numbers
is in insurance industry. According to the law of large numbers and the Central Limit
theorem, if the number of samples increases 𝛼 times, mean and variance increase 𝛼
times, but SD increases √𝛼 times. Therefore, if the number of clients increases 𝛼
times, the formula for PDR changes as below:
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Economics of Insurance
𝑃𝐷𝑅 𝑛2 =
𝑧𝛽 𝛿2 𝑧𝛽 √𝛼𝛿1
𝑧𝛽 𝛿1
𝑃𝐷𝑅 𝑛1
=
=
=
𝑥̅2
𝛼𝑥̅1
√𝛼𝑥̅1
√𝛼
This relationship is for normal distribution. For other distributions the relationships
should be obtained through separate calculations. Tables 7-11 to 7-15 gives the
results of calculations of declined PDRs for different distributions for the cases that
the number of clients increases 4, 9, 16, 25 and 100 times from its initial numbers.
Table 7-11-The effect of law of large number on PDR for Gamma distribution.
Confidence level
90%
95%
99%
PDR
79.99%
117.93%
203.27%
PDR(4)
40.00%
58.96%
101.63%
PDR(9)
26.66%
39.31%
67.76%
PDR(16)
20.00%
29.48%
50.82%
PDR(25)
16.00%
23.59%
40.65%
PDR(100)
8.00%
11.79%
20.33%
The shrink of PDR when the number of clients increases 4,9,16,25 and 100 times
As the answers indicate, when the number of customers increases further and further,
the gap between pure premium and actuarial fair premium diminishes and pure
premiums tend to converges to actuarial fair premiums. In other words, by increasing
the size of portfolio by the number of contracts, PDR gets smaller and smaller and
finally converges to zero. This means by increasing the number of contracts, the
insurance company requires lower premium for satisfying the same confidence level.
Table 7-12 shows the same effect for lognormal distribution.
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Economics of Insurance
Table 7-12-The effect of law of large number on PDR for lognormal distribution.
Confidence level
90%
95%
99%
PDR
78.31%
124.53%
248.71%
PDR(4)
39.16%
62.26%
124.35%
PDR(9)
26.10%
41.51%
82.90%
PDR(16)
19.58%
31.13%
62.18%
PDR(25)
15.66%
24.91%
49.74%
PDR(100)
7.83%
12.45%
24.87%
The shrink of PDR when the number of clients increases 4,9,16,25 and 100 times
Table 7-13 shows the same effect for Weibull distribution.
Table 7-13-The effect of law of large number on PDR for Weibull distribution.
Confidence level
90%
95%
99%
PDR
80.14%
115.54%
191.61%
PDR(4)
40.07%
57.77%
95.80%
PDR(9)
26.71%
38.51%
63.87%
PDR(16)
20.04%
28.89%
47.90%
PDR(25)
16.03%
23.11%
38.32%
PDR(100)
8.01%
11.55%
19.16%
The shrink of PDR when the number of clients increases 4,9,16,25 and 100 times
Table 7-14 gives the effect of increase in number of clients on PDR for Binomial
distribution.
Table 7-14-The effect of law of large number on PDR for negative binomial distribution
Confidence level
90%
95%
99%
PDR
82.39%
113.30%
190.58%
PDR(4)
41.19%
56.65%
95.29%
PDR(9)
27.46%
37.77%
63.53%
PDR(16)
20.60%
28.32%
47.65%
PDR(25)
16.48%
22.66%
38.12%
PDR(100)
8.24%
11.33%
19.06%
The shrink of PDR when the number of clients increases 4,9,16,25 and 100 times
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Economics of Insurance
Finally, table 7-15 shows the relationship between PDRs for normal distribution
when the number of customers increases 4, 9, 16, 25, and 100 times. As we expecte
when the number of insureds increases 𝛼 times, the PDRs decline by √𝛼 times.
Table 7-15-The effect of law of large number on PDR for Normal distribution
Confidence level
90%
95%
99%
PDR
75.53%
96.94%
137.10%
PDR(4)
37.76%
48.47%
68.55%
PDR(9)
25.18%
32.31%
45.70%
PDR(16)
18.88%
24.23%
34.27%
PDR(25)
15.11%
19.39%
27.42%
PDR(100)
7.55%
9.69%
13.71%
The shrink of PDR when the number of clients increases 4,9,16,25 and 100 times
7-7- Total premium calculation
The premium discussed so far is a fraction of total premium necessary for paying for
the claims and is called “Pure Premium”. As discussed earlier, different loading
factors should be considered. The relationship between total premium and pure
premium was discussed to be as following. In this relationship, “𝜄" is loading ratio.
𝑇𝑜𝑡𝑎𝑙 𝑃𝑟𝑒𝑚𝑖𝑢𝑚 =
𝑃𝑢𝑟𝑒 𝑃𝑟𝑒𝑚𝑖𝑢𝑚
1−𝜄
Assuming loading ratio to be equal to 15%, the total premiums calculated based on
the pure premiums for different distributions for the frequency of claims for the
actuarial fair premiums (table 7-2) are indicated in table 7-16.
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Economics of Insurance
Table 7-16-Total Premium for the mean of distribution indicating confidence level for 𝜄 = 0.15
Frequency Distribution
Pure Premium
(Rials)
Gamma (3P)
Lognormal (3P)
Weibull (3P)
Neg. Binomial
Normal
345,723
348,547
345,673
322,802
345,723
Total premium for other confidence levels (solvency margins) are indicated in table
below.
Table 7-17- Total Premium for different confidence levels for 𝜄 = 0.15
Frequency Distribution
Confidence
level
Gamma (3P)
Lognormal (3P)
Weibull (3P)
Neg. Binomial
Normal
90%
622,284
621,496
622,803
588,745
606,837
95%
753,434
782,582
745,182
688,533
680,859
99%
1,048,466
1,215,410
1,008,152
938,001
819,703
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Economics of Insurance
8- Concluding Remarks
Assuming the severity as fixed for simplicity, it is advised to relax this assumption
and assume both of severity and frequency are stochastic and follow different
distributions. While calculating the pure and total premium based on the actual
distributions of frequency and severity of loss, we simultaneously calculate the
financial solvency margin of the insurance company, which is the level of
confidence at which the insurance company will be able to cover all the future losses.
These premiums have inverse relationship with the number of clients.
The difference between the method presented here and the conventional methods is
that the distributions of severity and frequency are used in our method to calculate
premiums together with considering the predetermined solvency margin defined by
the possibility of meeting all the future claims for the insurance company,
simultaneously. But the conventional methods concentrate on historical data and the
means of severity and frequency, not on their distributions. In our setting, the
financial solvency margin is a probability, which indicates that the insurance
company will be able to fulfill all the possible claims it has accepted by that
probability. This index is completely in line with the real concept of financial
solvency.
In conventional methods, different financial ratios of insurance companies are used
as an indicator for financial solvency, which sometimes do not fully correspond to
the concept of solvency. For example, according to Regulation 69 of the Central
Insurance of Iran, the financial solvency index is calculated according to the
following ratio:
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Economics of Insurance
𝐹𝑖𝑛𝑎𝑛𝑐𝑖𝑎𝑙 𝑆𝑜𝑙𝑣𝑒𝑛𝑐𝑦 𝐼𝑛𝑑𝑒𝑥 =
𝐴𝑣𝑎𝑖𝑙𝑎𝑏𝑙𝑒 𝐶𝑎𝑝𝑖𝑡𝑎𝑙
𝑅𝑒𝑞𝑢𝑖𝑟𝑒𝑑 𝐶𝑎𝑝𝑖𝑡𝑎𝑙
×100
This index does not have full compliance with the concept of financial solvency of
insurance companies. In this conventional method, the financial solvency index is a
number that can be greater or less than 100. For example, the value of this index
announced by the Central Insurance of Iran for Mellat insurance company is 194 in
year 1398 (2019) .
While in our method, solvency margin of any insurance company is defined as the
possibility that the company is able to afford for the claims and will be between zero
and one.
The proposed method of calculating financial solvency can be used as a basic method
in calculating the real financial solvency of insurance companies. This method is a
good alternative to methods that are generally based on the financial ratios of
companies in which the basis of their calculations is the historical information of the
financial statements of companies.
Moreover, the conventional methods do not link directly between premium
determinations and solvency margins, while our method determine the premium
associated with the predetermined solvency margin.
In our method, the price of insurance products can be determined in such a way that
the real financial solvency margin of the insurance companies can also be satisfied.
The model introduced in this research also considers the importance of application
of Law of Large Numbers. We show that by increasing the number of clients, the
total premium for satisfying the predetermined solvency margin declines.
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Economics of Insurance
In this research, Frequency was modeled and severity was considered to be constant.
It is suggested for further researches to model both frequency and severity at the
same time to obtain more accurate price for insurance products.
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Economics of Insurance
Chapter 2
The Effect of Risk Level and Risk Aversion Level on the
Demand for Insurance and Rate Making
1-1-Introduction
Before we start the discussion on the effect of Risk Level and Risk Aversion Level
on the Demand for Insurance, we need to introduce the terminologies mostly used
in this chapter.
a. Risk Level
The experience of risk; risk exposure which shows the history of involvement
with risk. There are different risk level groups in society but for simplicity we
assume there are two groups of High-Risk Individuals and Low-RiskIndividuals.
b. Risk Aversion Level
Risk aversion is the behavior of individuals regarding risk and uncertainty. It
is actually the sensitivity towards risk and is affected by the character of
individuals. The risk aversion coefficient is utility-based. In economics and
finance, risk aversion level of a person is related to the type of Utility Function
of individuals that will be discussed in detail in the next chapter. Here we just
introduce two well-known criteria for the degree of risk aversion level.
• Arrow-Pratt Absolute Risk Aversion Index:
𝐴𝑅𝐴𝐼 = −
𝑈′′(𝑋)
𝑈′(𝑋)
• Arrow-Pratt Relative Risk Aversion Index:
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Economics of Insurance
𝑅𝑅𝐴𝐼 = −𝑋.
𝑈′′(𝑋)
𝑈′(𝑋)
c. Asymmetric Information
Asymmetric information in insurance market conventionally indicates that
demanders (policy holders) know more about their risk level compared to the
suppliers (insurance companies). The policy holders know to which group of
risk level they belong while the company cannot distinguish the risk level of
policy holders.
1-2- Conventional Theory of Demand for Insurance: Risk Level
matters
This theory originates from the research of Rothschild and Stiglitz (1976) where they
stated whenever an insurance service is rendered by an insurance company, the highrisk-individuals demand more since the most important factor behind the decision
to purchase insurance services is the risk history (Risk Level) of demanders. As the
price is set based on the population risk level or population Loss Ratio (Actuarial
Fair Rate), the company will not be able to afford the claims and remains insolvent
in this rate. This phenomenon is called Adverse Selection problem. Adverse
Selection problem refers to a situation where High-Risk individuals are absorbed by
insurance companies while the preference of insurance companies is to collect LowRisk customers. This phenomenon occurs because of asymmetric information the
insurance market inherits. If the market is not evolved in a way to create asymmetric
information between demanders and suppliers of the market and the company can
classify the customers, the adverse selection problem never occurs.
Adverse selection is originally defined in insurance theory to describe a situation
where the information asymmetry between policyholders and insurers leads the
market to a situation that the policyholders claim losses that are higher than the
average rate of loss of population used by the insurers to set their premiums.
According to the conventional theory of demand for insurance under asymmetric
information, insurers consider the perceived loss rates of population to set the
premium, while the individuals can be divided into two groups of risk level, let’s
say, low- and high-risk groups, and the insurance companies can't distinguish
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Economics of Insurance
between them but the individuals know what group they belong to. Low-risk
individuals realize that their loss rate is low and they are subsidizing high-risk
individuals so will be reluctant to insure, while high-risk individuals will have
motivation for purchasing more insurance as they are paying less than their real rate
and are actually receiving subsidy from low-risk individuals. Consequently, the
average loss rate of purchasers of insurance services will be higher than the
perceived loss rate by insurance companies and thus the companies end up with
policyholders who are of higher-than-average risk rates.
The conventional theory of demand for insurance which leads to adverse selection
is based on the following assumptions: (1) The difference in exposure to risk:
People differ in the level of exogenously determined risk exposures. For simplicity,
we consider that people are divided into two groups of risk levels, high- and lowrisk groups. (2) The most important factor behind the decision to purchase insurance
service is the Risk Level of customers. (3) Positive correlation between selfperceived risk level and real risk level: Adverse selection occurs when the
individuals’ beliefs about their risk level and their actual rates are positively
correlated. If not, there will not be a systematic difference between policyholders’
risk levels and hence no adverse selection occurs. (4) No relationship between the
level of risk aversion and riskiness: In other words, there’s no way to claim whether
high-risk individuals are less risk averse than low-risk individuals and vice versa.
(5) Customers know more about their riskiness than the insurers and efficiently use
their information against the insurers.
The consequences and implications of conventional theory of demand can be
outlined as below:
1. Insufficient provision of insurance services: because of adverse selection
problem, the companies should consider positive loading and increase the
premium, thus low-risk customers will surrender their policies and drop out
of the market and the market settles will lower than equilibrium level of
demand.
2. The premium rates will be higher than the actuarial fair premiums: the
companies to be able to afford the claims, should increase the premium rates.
3. Instability in the market: Since the company considers positive loading in
order to be able to afford the claims, low-risk customers will have motivation
to drop out of market. If low-risk individuals surrender their contracts, the
company remains with high-risk customers. Thus the company needs to
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Economics of Insurance
increase the premium further. Consequently, the insurance market will be
instable.
4. Contradiction with real world: Despite the straightforward understanding
from the conventional theory of insurance demand under asymmetric
information, this theory is not supported by most of the empirical works.
There are many empirical evidences that appear to conflict with the standard
theory of adverse selection in insurance market. Mahdavi (2003) finds that the
risk level of demanders of comprehensive life insurance policies in Kyoto city
is considerably smaller than the risk level of those who didn’t purchase
comprehensive policies. Hemenway (1990) finds that at a hospital in Texas,
the percentage of insured individuals amongst helmeted and unhelmeted
motorcyclists is 73% and 59%, respectively. He also found that amongst
drivers, 40 percent of those who wore their seat belt bought insurance while
only 33 percent of those not wearing the belt purchased the coverage. Both
examples show that high-risk individuals (unhelmeted and not wearing the
belts) purchase less coverage.
McCarthy and Mitchell (2003) found that the mortality rate of UK and US
males and females purchasing term- and whole-life insurance is below that of
the uninsureds. For example, they found that mortality rates for male and
female purchasers of whole-life insurance are only 77.5 and 68.5 percent of
the total population mortality rate for the UK, and 78.6 and 90.9, for the US,
respectively.
Meza and Webb (2001) state that in addition to precautionary effort that
explains the negative correlation between insurance demand and risk level,
heterogeneous optimism also supports this negative correlation: High risks are
more optimistic about the events to be improbable, so they purchase less
insurance.
1-3- Alternative (Modern) Theory of Demand for Insurance: Risk
Aversion Level Matters
The alternative theory of demand for insurance focusses on Risk-Aversion level and
emphasizes that whenever an insurance policy is issued by an insurance company,
the individuals who are more risk averse demand the services more since the most
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Economics of Insurance
important factor behind the decision to purchase insurance services is the RiskAversion Level of demanders.
More risk-averse individuals have lower risk level because there is a negative
correlation between risk aversion level and risk level naturally. Moreover, more riskaverse individuals undertake more precautionary efforts, and consequently their risk
level declines further.
The result of this behavior will be the insurers end up (face with) low-risk customers
and the loss ratio the company faces with will be lower than the actuarial fair rates
that the premiums are determined based on those rates. This situation will be
favorable to insurers. This is why we call the alternative theory of demand the
Advantageous Selection theory. The alternative advantageous selection theory
assumes a negative correlation between risk aversion and risk exposure and
considers the effect of precautionary activity on the risk exposure. Under these
assumptions, insurers end up with relatively low-risk individuals, the market offers
sufficient provision of policies and, the selection effect will be propitious to insurers
as more risk-averse low-risk individuals are not only willing to pay more for
precautionary efforts but also are more inclined to insure.
The modern theory of demand for insurance which leads to advantageous selection
is based on the following assumptions: (1) The difference in risk-aversion levels:
People differ in their risk-aversion levels. For simplicity, we consider that people are
divided into two groups of risk-aversion levels, high- and low-risk aversion groups.
(2) The most important factor behind the decision to purchase insurance service is
the Risk-Aversion Level of customers. (3) Negative relationship between the level
of risk aversion and riskiness: Those who are more risk-averse engage less in risky
activities. (4) Effectiveness of precautionary efforts: more risk-averse individuals
undertake more precautionary efforts that contribute in improving their risk levels.
The consequences and implications of alternative theory of demand for insurance
can be outlined as below:
1. Sufficient provision of insurance services: the prices will be favorable to high
risk group and consequently they remain in the market. Low risk group also
remains in the market since they are assumed to be more risk-averse and
valuing insurance so highly that they can tolerate the rates which are actually
higher than fair prices for them. Even though they realize they are subsidizing
high-risk individuals, they continue purchasing the insurance services as they
are more risk-averse cautious and prudent customers.
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Economics of Insurance
2. The premium rates will be lower than the actuarial fair premiums: the
companies to absorb more customers, are able even to consider negative
loading and decline the premium levels. Since the premium rates the
companies face with are lower than the actuarial fair rates, they are able to
afford the claims even in lower prices.
3. Stability in the market: Since the companies consider negative loading, highrisk customers will have motivation to stay in the market. Furthermore, lowrisks are also satisfied since they are more risk-averse individuals who
overvalue insurance services and continue purchasing the policies.
4. No Contradiction with real world. As mentioned many researches lack to
support the conventional theory of demand for insurance. These researches
are in accordance with the modern theory of demand for insurance which
leads to advantageous selection.
1-4-Ignorances of Conventional Theory of Demand
The conventional theory of demand for insurance does not consider the role of risk
aversion level on the decision to purchase insurance services while the sensitivity
towards risk which is the risk-aversion level of customers can play important role in
demanding insurance. Researches on the effect of risk aversion level on the demand
for life and car collision insurance (Mahdavi, 2013, 2016) indicate that the customers
are more motivated by risk aversion level than the risk level in order to decide to
purchase insurance policies.
The conventional theory of demand for insurance also ignores the negative
correlation between risk-aversion level and riskiness (risk level). While more riskaversion implies more cautiousness and prudence regarding exposing risk and
consequently facing with less risks, the adverse selection theory doesn’t consider
this important relationship.
Another ignorance of the conventional theory of demand for insurance is the fact
that the theory ignores the effect of precautionary efforts which lowers the risk level
considerably.
1-5- The Effect of the Models on Rate-Making
It is easily understandable if risk level is the main factor behind the demand for
insurance, high-risk-individuals demand more. Thus, the company faces with the
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Economics of Insurance
loss ratio which is higher than the population loss ratio(=actuarial fair rate) which is
the basis for rate-making. In such a case the company should consider positive
adverse selection cost and positive loading.
But if the most effective factor behind the demand for insurance is Risk Aversion
Level, the company expects low-risk-individuals who are more risk-averse,
purchase more insurance services. This results in the fact the company faces with
the loss ratio which is less than population loss ratio (=actuarial fair rate) and benefits
from the behavior of its clients. In such a case the company realizes that it is rational
to decrease the premium rate by considering negative loading for marketing
justifications.
Assignments:
1. Distribute 10 questionnaires covering at least 5 questions about different
insurable risks in order to understand which one matters more, Risk Level or
Risk aversion Level.
2. Distribute 10 questionnaires covering at least 10 questions about a specific
risk (e.g. the risk of COVID19, the risk of car accident, financial risk, the risk
of fire, etc.) to find the risk aversion level of those people with respect to
selected specific risk. The respondents may include your family, your
classmates, and even yourself.
(Notes: Choose 4-choice or 5-choice questions. The risk aversion level should
be between zero and one)
Suggestions for more study:
Paper 1
When Effort Rimes with Advantageous Selection: A New Approach to Life
Insurance Pricing, Mahdavi and Rinaz, The Kyoto Economic Review, No.
158, June 2006.
Paper 2
Advantageous Selection Versus Adverse Selection in Life Insurance Market,
Mahdavi, Paper Presented at University of Athens, Greece, 2005.
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Economics of Insurance
References
1- Antonio, K. and Valdez, E.A., 2012. Statistical concepts of a priori and a posteriori risk
classification in insurance. AStA Advances in Statistical Analysis, 96(2), pp.187-224.
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Economics of Insurance
2- Bardey, D. and Buitrago, G., 2017. Supplemental health insurance in the Colombian
managed care system: Adverse or advantageous selection? Journal of health economics,
56, pp.317-329.
3- Bühlmann, H., 2007. Mathematical methods in risk theory (Vol. 172). Springer Science &
Business Media.
4- Bahnemann, D., 2015. Distributions for actuaries. CAS Monograph Series, (2).
5- Chiappori, P.A., Jullien, B., Salanié, B. and Salanie, F., 2006. Asymmetric information in
insurance: General testable implications. The RAND Journal of Economics, 37(4), pp.783798.
6- David, M., 2015. A review of theoretical concepts and empirical literature of non-life
insurance pricing. Procedia Economics and Finance, 20, pp.157-162.
7- Denuit, M., Maréchal, X., Pitrebois, S. and Walhin, J.F., 2007. Actuarial modelling of
claim counts: Risk classification, credibility and bonus-malus systems. John Wiley & Sons.
8- Dionne, G., Michaud, P.C. and Pinquet, J., 2013. A review of recent theoretical and
empirical analyses of asymmetric information in road safety and automobile insurance.
Research in transportation economics, 43(1), pp.85-97.
9- De Meza, D. and Webb, D.C., 2001. Advantageous selection in insurance markets. RAND
Journal of Economics, pp.249-262.
10- Dudewicz, E.J. and Mishra, S.N., 1988. Modern Mathematical Statistics. (John W & Sons,
Ltd. Inc)
11- Ezenekwe, R.U. and Uzonwanne, M.C., 2018. Economics Study Material, pp.113-124
12- Finkelstein, A. and McGarry, K., 2006. Multiple dimensions of private information:
evidence from the long-term care insurance market. American Economic Review, 96(4),
pp.938-958.
13- Hemenway, D., 1990. Propitious selection. Quarterly Journal of Economics, 105(4),
pp.1063-1069.
14- Mahdavi, Gh. and Nasiri, F., 2013, theoretical principles and foundations of insurance. In
Insurance Research Center (Vol. 1).
15- Mahdavi, G. and Izadi, Z., 2012. Evidence of Adverse Selection in Iranian Supplementary
Health Insurance Market. Iranian journal of public health, 41(7), p.44.
92
Economics of Insurance
16- Mahdavi, G., 2005. Advantageous selection versus adverse selection in life insurance
market. Japanese Society for the Promotion of Science.
17- Rouaud, M., 2013. Probability, statistics and estimation. Propagation of uncertainties.
18- Siegelman, P., 2003. Adverse selection in insurance markets: an exaggerated threat. Yale
LJ, 113, p.1223.
19- Tse, Y.K., 2009. Nonlife actuarial models: theory, methods and evaluation. Cambridge
University Press.
20- Tinungki, G.M., 2018, March. The Application Law of Large Numbers That Predicts the
Amount of Actual Loss in Insurance of Life. In Journal of Physics: Conference Series (Vol.
979, No. 1, p. 012088). IOP Publishing.
21- Werner, G. and Modlin, C., 2016, May. Basic ratemaking. In Casualty Actuarial Society
(Vol. 5).
22- Zhao, Y.F., Chai, Z.H., Delgado, M.S. and Preckel, P.V., 2017. A test on adverse selection
of farmers in crop insurance: Results from Inner Mongolia, China. Journal of integrative
agriculture, 16(2), pp.478-485.
Appendix 1- Distributions
93
Economics of Insurance
1.Gamma distribution: Probability density function (top left), cumulative
distribution function (top right), Q-Q plot (down left), P-P plot (down right),
probability difference (down middle) of Gamma distribution fit on data of frequency.
Graph 1- Gamma distribution
Probability Density Function
Cumulative Distribution Function
1
0.36
0.9
0.32
0.8
0.28
0.7
F(x)
f(x)
0.24
0.2
0.6
0.5
0.16
0.4
0.12
0.3
0.08
0.2
0.04
0.1
0
0
16
24
32
40
48
56
64
72
80
16
24
32
40
x
Histogram
48
56
64
72
80
x
Gamma (3P)
Sample
Gamma (3P)
P-P Plot
Q-Q Plot
80
0.9
72
0.8
64
0.7
P (Model)
Quantile (Model)
1
56
48
40
0.6
0.5
0.4
32
0.3
24
0.2
16
0.1
8
0
8
16
24
32
40
48
x
Gamma (3P)
56
64
72
80
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
P (Empirical)
Gamma (3P)
94
Economics of Insurance
Probability Difference
0.2
Probability Difference
0.16
0.12
0.08
0.04
0
-0.04
-0.08
-0.12
-0.16
-0.2
16
24
32
40
48
56
64
72
80
x
Gamma (3P)
Table 1- Gamma distribution goodness of fit test results
Gamma (3P)
Kolmogorov-Smirnov
Sample Size
48
Statistic
0.09532
P-Value
0.73984
Rank
7

Critical Value
Reject?
0.2
0.1
0.05
0.02
0.01
0.1513 0.17302 0.19221 0.21493 0.23059
No
No
No
No
No
0.2
0.1
0.05
0.02
0.01
1.3749
1.9286
2.5018
3.2892
3.9074
Anderson-Darling
Sample Size
48
Statistic
0.50514
Rank
13

Critical Value
95
Economics of Insurance
Reject?
No
No
No
No
No
0.2
0.1
0.05
0.02
0.01
7.2893
9.2364
11.07
13.388
15.086
No
No
No
No
No
Chi-Square
Deg. of freedom
5
Statistic
2.1051
P-Value
0.83442
Rank
12

Critical Value
Reject?
2. Lognormal Distribution: Probability density function (top left), cumulative
distribution function (top right), Q-Q plot (down left), P-P plot (down right),
probability difference (down middle) of lognormal distribution fit on frequency data
Graph 2- Lognormal distribution
Cumulative Distribution Function
Probability Density Function
1
0.36
0.9
0.32
0.8
0.28
0.7
F(x)
f(x)
0.24
0.2
0.6
0.5
0.16
0.4
0.12
0.3
0.08
0.2
0.04
0.1
0
0
16
24
32
40
48
56
64
72
80
10
20
30
x
Histogram
Lognormal (3P)
40
50
60
70
80
x
Sample
Lognormal (3P)
96
Economics of Insurance
P-P Plot
Q-Q Plot
0.9
72
0.8
64
0.7
P (Model)
80
56
48
40
0.6
0.5
0.4
32
0.3
24
0.2
16
0.1
8
0
8
16
24
32
40
48
56
64
72
0.1
80
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
P (Empirical)
x
Lognormal (3P)
Lognormal (3P)
Probability Difference
0.16
Probability Difference
Quantile (Model)
1
0.12
0.08
0.04
0
-0.04
-0.08
-0.12
-0.16
16
24
32
40
48
56
64
72
80
x
Lognormal (3P)
Table 2- lognormal distribution goodness of fit test results
Lognormal (3P)
Kolmogorov-Smirnov
Sample Size
48
Statistic
0.0975
P-Value
0.71487
Rank
11

Critical Value
0.2
0.1
0.05
0.02
0.01
0.1513 0.17302 0.19221 0.21493 0.23059
97
Economics of Insurance
Reject?
No
No
No
No
No
0.2
0.1
0.05
0.02
0.01
1.3749
1.9286
2.5018
3.2892
3.9074
No
No
No
No
No
0.2
0.1
0.05
0.02
0.01
5.9886
7.7794
9.4877
11.668
13.277
No
No
No
No
No
Anderson-Darling
Sample Size
48
Statistic
0.41847
Rank
6

Critical Value
Reject?
Chi-Squared
Deg. of freedom 4
Statistic
2.63
P-Value
0.62152
Rank
15

Critical Value
Reject?
98
Economics of Insurance
3.Poisson distribution: Probability density function (top left), cumulative
distribution function (top right), P-P plot (down left), probability difference (down
right) of Poisson distribution fit on frequency data
Graph 3- Poisson distribution
Cumulative Distribution Function
1
0.1
0.09
0.9
0.08
0.7
0.07
0.6
0.8
F(x)
f(x)
Probability Density Function
0.11
0.06
0.05
0.5
0.4
0.04
0.3
0.03
0.2
0.02
0.01
0.1
0
0
8
16
24
32
40
48
56
64
72
8
80
16
24
32
40
P-P Plot
1
0.9
0.8
Probability Difference
1
P (Model)
0.7
0.6
0.5
0.4
0.3
0.2
56
64
72
80
Sample Poisson
Sample Poisson
0.8
48
x
x
Probability Difference
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
0.1
-1
0
0.1
0.2
0.3
0.4
0.5
0.6
P (Empirical)
Poisson
0.7
0.8
0.9
1
16
24
32
40
48
56
64
72
80
x
Poisson
99
Economics of Insurance
Table 3- Poisson distribution goodness of fit test results
Poisson
Kolmogorov-Smirnov
Sample Size
48
Statistic
0.40366
P-Value
1.7595E-7
Rank
4

0.2
0.1
0.05
0.02
0.01
Critical Value 0.1513 0.17302 0.19221 0.21493 0.23059
Reject?
Yes
Yes
Yes
Yes
Yes
0.1
0.05
0.02
0.01
1.9286
2.5018
3.2892
3.9074
Yes
Yes
Yes
Yes
Anderson-Darling
Sample Size
48
Statistic
58.465
Rank
5

0.2
Critical Value 1.3749
Reject?
Yes
4.Weibull Distribution: Probability density function (top left), cumulative
distribution function (top right), Q-Q plot (down left), P-P plot (down right),
probability difference (down middle) of Weibull distribution fit on frequency data
100
Economics of Insurance
Graph 4- Weibull distribution
Probability Density Function
Cumulative Distribution Function
1
0.36
0.9
0.32
0.8
0.28
0.7
0.6
F(x)
f(x)
0.24
0.2
0.5
0.16
0.4
0.12
0.3
0.08
0.2
0.04
0.1
0
0
16
24
32
40
48
56
64
72
80
16
24
32
40
x
Histogram
48
56
64
72
80
x
Weibull (3P)
Sample
Q-Q Plot
Weibull (3P)
P-P Plot
0.9
72
0.8
64
0.7
P (Model)
80
56
48
40
0.6
0.5
0.4
32
0.3
24
0.2
16
0.1
8
8
16
24
32
40
48
56
64
72
0
80
0.1
x
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
P (Empirical)
Weibull (3P)
Weibull (3P)
Probability Difference
Probability Difference
Quantile (Model)
1
0.24
0.2
0.16
0.12
0.08
0.04
0
-0.04
-0.08
-0.12
-0.16
-0.2
-0.24
16
24
32
40
48
56
64
72
80
x
Weibull (3P)
101
Economics of Insurance
Table 4- Weibull distribution goodness of fit test results
Weibull (3P)
Kolmogorov-Smirnov
Sample Size
48
Statistic
0.09835
P-Value
0.70495
Rank
13

Critical Value
0.2
0.1
0.05
0.02
0.01
0.1513 0.17302 0.19221 0.21493 0.23059
Reject?
No
No
No
No
No
0.2
0.1
0.05
0.02
0.01
1.3749
1.9286
2.5018
3.2892
3.9074
No
No
No
No
No
0.2
0.1
0.05
0.02
0.01
7.2893
9.2364
11.07
13.388
15.086
No
No
No
No
No
Anderson-Darling
Sample Size
48
Statistic
0.56831
Rank
18

Critical Value
Reject?
Chi-Squared
Deg. of freedom
5
Statistic
3.1432
P-Value
0.67792
Rank
22

Critical Value
Reject?
102
Economics of Insurance
5.Negative Binominal Distribution: Probability density function (top left),
cumulative distribution function (top right), P-P plot (down left), probability
difference (down right) of negative binomial distribution fit on frequency data
Graph 5- negative binominal distribution
Cumulative Distribution Function
1
0.1
0.09
0.9
0.08
0.7
0.07
0.6
0.8
F(x)
f(x)
Probability Density Function
0.11
0.06
0.05
0.4
0.04
0.3
0.03
0.2
0.02
0.01
0
0.5
0.1
0
10
20
30
40
50
60
70
80
8
16
24
32
40
x
Sample
Neg. Binomial
Sample
P-P Plot
Probability Difference
P (Model)
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.4
0.5
0.6
P (Empirical)
Neg. Binomial
72
80
Neg. Binomial
Probability Difference
0.8
0.3
64
0.24
0.2
0.16
0.12
0.08
0.04
0
-0.04
-0.08
-0.12
-0.16
-0.2
-0.24
0.9
0.2
56
x
1
0.1
48
0.7
0.8
0.9
1
16
24
32
40
48
56
64
72
80
x
Neg. Binomial
103
Economics of Insurance
Table 5- negative binomial distribution goodness of fit test results
Neg. Binomial
Kolmogorov-Smirnov
Sample Size
48
Statistic
0.12884
P-Value
0.37118
Rank
1

Critical Value
0.2
0.1
0.05
0.02
0.01
0.1513 0.17302 0.19221 0.21493 0.23059
Reject?
No
No
No
No
No
0.2
0.1
0.05
0.02
0.01
1.3749
1.9286
2.5018
3.2892
3.9074
No
No
No
No
No
Anderson-Darling
Sample Size
48
Statistic
1.2571
Rank
1

Critical Value
Reject?
104
Economics of Insurance
6.Normal Distribution: Probability density function (top left), cumulative
distribution function (top right), Q-Q plot (down left), P-P plot (down right),
probability difference (down middle) of Normal distribution fit on frequency data
Graph 6- Normal distribution
Probability Density Function
Cumulative Distribution Function
1
0.36
0.9
0.32
0.8
0.28
0.7
F(x)
f(x)
0.24
0.2
0.16
0.6
0.5
0.4
0.12
0.3
0.08
0.2
0.04
0.1
0
0
16
24
32
40
48
56
64
72
80
16
24
32
40
x
48
56
64
0.6
0.7
72
80
x
Histogram Normal
Sample Normal
Q-Q Plot
P-P Plot
80
0.9
72
0.8
64
0.7
P (Model)
Quantile (Model)
1
56
48
40
0.6
0.5
0.4
32
0.3
24
0.2
16
0.1
8
8
16
24
32
40
48
x
Normal
56
64
72
80
0.1
0.2
0.3
0.4
0.5
0.8
0.9
1
P (Empirical)
Normal
105
Economics of Insurance
Probability Difference
0.48
0.4
0.32
0.24
0.16
0.08
0
-0.08
-0.16
-0.24
-0.32
-0.4
-0.48
Probability Difference
16
24
32
40
48
56
64
72
80
x
Normal
Table 6- Normal Distribution goodness of fit test results
Normal
Kolmogorov-Smirnov
Sample Size
48
Statistic
0.19416
P-Value
0.0464
Rank
47

0.2
Critical Value
0.1
0.05
0.02
0.01
0.1513 0.17302 0.19221 0.21493 0.23059
Reject?
Yes
Yes
Yes
No
No
0.1
0.05
0.02
0.01
Anderson-Darling
Sample Size
48
Statistic
2.3432
Rank
38

0.2
106
Economics of Insurance
Critical Value
1.3749
1.9286
2.5018
3.2892
3.9074
Yes
Yes
No
No
No
0.2
0.1
0.05
0.02
0.01
5.9886
7.7794
9.4877
11.668
13.277
No
No
No
No
No
Reject?
Chi-Squared
Deg. of freedom 4
Statistic
5.2303
P-Value
0.26447
Rank
31

Critical Value
Reject?
Table 7-The parameters of distribution of frequency
Distribution
Parameters
Gamma (3P)
=1.5945 =16.505 =8.3294
Lognormal (3P)
=0.67571 =3.1948 =4.2639
Weibull (3P)
=1.2793 =27.979 =8.7142
Neg. Binomial
n=3 p=0.08487
Normal
=20.418 =34.646
Estimated parameters for distributions on loss frequency data.
Table 8- Specifications of frequency distribution
Title
Gamma (3P)
Lognormal (3P)
Weibull (3P)
Neg. Binomial
Normal
Mean
34.646
34.929
34.641
32.349
34.646
Variance
434.34
544.18
416.9
381.16
416.91
SD
20.841
23.328
20.418
19.523
20.418
SK
1.5839
2.7223
1.3796
1.1558
0
Mean, variance, standard deviation of the frequency distribution are shown
107
Economics of Insurance
Appendix 2- Definitions
1)
Pure Premium (Pu.Pr): Pure premium for a contract is a fraction of total premium
that covers just for claims and losses. Consequently, pure premium does not cover
for other costs and expenses such as administrative costs, profits, commissions,
moral hazard costs, contingencies and etc.In other words the premium calculated by
using the principle of equivalence is called pure premium.
2)
Loading Factors (Loading Costs): the expenses of insurer such as administrative
costs, profit, commissions to agents or brokers, taxes, moral hazards and adverse
selection costs and other contingencies which are not considered in calculating pure
premium.
3)
Total Premium (Premium) (T.Pr) : premium is the price of a contract. The cost
of claims and loading costs both are considered in calculating premium. The
premium is referred to as total premium since includes pure premium and loading
factors.
Premium= Premium Rate × Average Loss
4) Premium Rate =
Premium
Average Loss
is the price for one unit of coverage.
5)Fair Premium (FP): From the view point of demanders of insurance services, a
premium is called fair premium where the premium rates are equal to loss ratio. The
rate they expect to pay is exactly equal their expected loss ratio. Fair Premium is also
called Actuarial Fair Premium (AFP). Fair premium is derived from principle of
equivalence.
6) Actuarial Fair Premium Rate(AFPR): The rate equal to loss ratio which makes
the demanders pleased for purchasing the policy. Any rate grater than loss ratio will
be unfair in viewpoint of customers.
7) Loss Ratio =
Number of losses
Number of insureds
=
x̅
n
108
Economics of Insurance
7) Solvency Margin (S.M): Solvency Margin is the confidence level for the insurer
that can afford for the claims. Or the possibility that the insurance company can pay
for the claims and cover the losses totally.
Solvency Margin = 1 - Ruin Probability
9) Ceteris Paribus condition: Ceteris paribus or caeteris paribus is a Latin phrase
meaning " other things held constant " ; English translations of the phrase include "all
other things being equal" or "all else unchanged”. The condition which assumes only
one factor affects the dependent variable and other factors are assumed to be fixed and
given and do not affect the dependent variable. This assumption is usually used in order
to focus on the effect of one independent factor on a dependent factor. The dictionary
of Investopedia explains Ceteris paribus, literally "holding other things constant," is a
Latin phrase that is commonly translated into English as "all else being equal." A
dominant assumption in mainstream economic thinking, it acts as a shorthand indication
of the effect of one economic variable on another, provided all other variables remain
the same.
10) Potential Deviation Ratio (PDR): Potential Deviation Ratio is the percentage that
should be added to the premium in order to satisfy and guarantee a pre-determined
solvency ratio. PDR can be obtained as following:
𝑷𝑫𝑹%𝜶 =
(𝒙
̅ + 𝒛 𝝈) − 𝒙
̅
𝒛%𝜶 𝝈
× 𝟏𝟎𝟎 =
× 𝟏𝟎𝟎
̅
̅
𝒙
𝒙
109
Economics of Insurance
110