CHAPTER 8
INSURANCE PRICING
Mahdzan & Boey 2015
Learning Outcomes
Define insurance pricing and explain the components of insurance
cost
Discuss the implications of homogeneous and heterogeneous buyers
in insurance pricing.
Describe how investment returns, administrative cost and profit
loading determine fair insurance premium.
Compute fair premium based on the components of insurance cost
Compute life insurance pricing for term insurance and whole-life
insurance
Compute life insurance pricing under single premium and level
premiums.
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1.0 Insurance Pricing and
Components of Insurance Cost
Insurance pricing: The determination of rates (premiums)
that an insurer charges for insurance.
Pure premium: The minimum premium that is necessary
to cover the expected losses or expected claim cost.
Mahdzan & Boey 2015
1.0 Insurance Pricing and
Components of Insurance Cost
Fair insurance
premium
Expected
claim cost
Investment
Returns
Administrative
costs
Fair profit
loading
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1.0 Insurance Pricing and
Components of Insurance Cost
1.1 Expected claim cost (Pure premium)
Represents the expected loss that will be faced by the
insured, which then passes on to the insurer.
To illustrate, we classify buyers of insurance as:
(i) Homogenous buyers: Buyers of insurance who
are equal in terms of their loss distribution
(ii) Heterogeneous buyers: Buyers of insurance
that have different loss probabilities.
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Homogeneous buyers
Everyone in the risk pool has the same probabilities of loss &
expected claim cost.
Losses are independent of each other (uncorrelated).
Large number of participants in the pool
Table 1: Loss distribution of homogeneous buyers
Outcome (x)
Probability (p)
p*x
Expected
claim (RM)
Loss (RM1,000)
0.10
(0.10 x RM1,000)
100
No loss (RM0)
0.90
(0.90 x RM0)
0
Expected claim cost = RM100
100
By charging each buyer equally by RM100, the insurer can cover its
expected losses.
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Heterogeneous buyers
Everyone in the risk pool have different risk exposures depending on
their own unique lifestyle and circumstances.
Assume two groups of insurance buyers – low-risk buyers and high-risk
buyers.
Stay-at-home mums & Factory-operator mums
Table 1: Loss distribution of homogeneous buyers
Outcomes (x)
Probability (p)
p*x
Expected
claim
Stay home mums Loss (RM1,000)
0.05
0.05 x RM1,000 =
RM50
Factory-operator
50
No loss (RM0)
0.95
0.95 x RM0 = RM0
Loss (RM1,000)
0.20
0.20 x RM1,000 =
mums
RM200
No loss (RM0)
0.80
200
0.80 x RM0 = RM0
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Same Rate Insurance Co
Unable to differentiate the two groups
Charge same premium rates to both groups
Takes the average of the premiums, (RM50+200)/2 = RM125
Stay-home mums – disadvantaged
Factory-operator mums – advantaged
Stay-home mums would be subsidizing the premiums of the factoryoperator mums
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Selective Insurance Co
Excellent team who can correctly distinguish the two groups
Carefully select its policyholders to comprise only stay-home mums, and
deny coverage on factory-operator mums
Premiums for stay-home mums < RM125 - giving Selective a pricing
advantage over SameRate Co.
E.g. Selective charges RM100 to stay-home mums - still make a gain of
RM50 since the expected claim cost of stay-home mums are RM50.
Stay-home mums would choose buying from Selective rather than
SameRate.
SameRate would then be left with factory-operator mums as
policyholders (factory-operators cannot switch to Selective because
Selective only accepts stay-home mums as policyholders and deny
coverage on factory-operators).
The final outcome of this method would be that Selective would carry
all low risk buyers, and SameRate would carry all high risk buyers.
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1.0 Insurance Pricing and
Components of Insurance Cost
The preceding situation would result in a situation called
Adverse Selection.
Adverse selection: A high tendency for insurance to be
purchased or selected by individuals faced with high risk (in
adverse situations) as opposed to those with low risk, when
charged with the same premiums.
 Due to information asymmetries
 Insurance pricing will not be optimal
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1.0 Insurance Pricing and
Components of Insurance Cost
Risk classification: The identification and grouping of people in
accordance to their level of risk and expected claims.
 Done by actuaries:- use historical data and statistical methods
to assess risk and estimate premiums for an insurance
company.
 Risk classification increases the incentive to lower down risks
by the policyholders.
Insurance companies must invest in additional underwriting
effort to classify their customers
Must do so at affordable cost in order to minimize premiums
charged to customers
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1.0 Insurance Pricing and
Components of Insurance Cost
Underwriting process: evaluating insurance applications from
buyers to determine the acceptability of the risk from the
perspective of the insurer and, if acceptable, whether special
conditions should be applied and the level of premium that is to
be charged.
Class rates: standard rates that apply to insurance applicants
within certain risk categories.
Experience rating: applies when class rates are adjusted
according to the loss experience of the insured.
Non-claim discount (NCD): discount given to the insured of a
motor vehicle insurance policy when no claims are made in a
given policy year.
Mahdzan & Boey 2015
1.0 Insurance Pricing and
Components of Insurance Cost
1.2 Investment returns
Reflect the ability of insurers to invest the premiums paid by its
customers.
The computation of fair premiums to include investment returns are
based on the time value of money
The resulting value: the discounted value of the expected claim cost.
The higher the interest rates or the longer the duration of the policy,
the lower the fair premiums, vice versa.
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Example 1
1)
Claims paid at the end of one year
Say that an insurer issues a policy that has an expected claim of RM1,000 (which is the
future value, FV) at the end of one year. Assuming that there are no administrative costs
and fair profit loading, we compute the premiums (P) the insurer should collect at the
beginning of the year. At an annual rate of r pear year for n years, the premium (which is
the present value, PV) that needs to be charged is as follows:
Beginning
Year 1
End
Year 1
Premiums
?
Exp. claims
RM1,000
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Example 1
Expected Claims (FV)
= PV (1 + r)n = 1000
If r = 5%,
Premiums (PV)
= FV / (1+r)n
= 1000 / (1.05)
= 952.38
If r = 7%,
Premiums (PV)
= FV / (1+r)n
= 1000 / (1.07)1
= 934.58
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Example 2
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Example 2
Expected claims (FV) = PV (1 + r)n = 1000
If r = 5%,
PV
= FV / (I + r)n
= 1000 / (1 + 0.05)2
= 1000 / 1.1025
= 907.03
If r = 7%,
PV
= FV / (I + r)n
= 1000 / (1 + 0.05)2
= 1000 / 1.1449
= 873.44
Mahdzan & Boey 2015
1.0 Insurance Pricing and
Components of Insurance Cost
1.3 Administrative costs
Include:
Underwriting costs: costs involved to determine the
acceptability of the risk, conducted by underwriters of the
insurer.
Agents’ renumeration: costs for the sales force that
market and sell insurance policies to customers.
Medical examination expenses
Marketing costs
Loss adjustment expenses: applies to general insurance
Other expenses: policy issuance, distribution costs, etc.
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1.0 Insurance Pricing and
Components of Insurance Cost
1.4 Profit loading
As with other types of business, insurance co’s are business entities in
pursuit of profits and long-term sustainability.
To compensate its investors for their capital, a certain level of profit
margin must be included, in excess of the discounted value of expected
claim cost plus administrative costs.
Claims could occur at any time from the inception of policy, investors of
the insurance company must provide capital that is sufficient to cover
claims and avoid insolvency.
A higher profit loading is required for claims that are less predictable.
Mahdzan & Boey 2015
1.0 Insurance Pricing and
Components of Insurance Cost
Example of Fair Premium Computation
Suppose a group of policyholders has the following loss
distribution as shown below. Assume a time period of two (2) years
after which claims will be made, and interest rate of 6%.
Administrative expenses are 15% of expected claim costs, and
profit loading is 5% of expected claim cost. Compute fair premiums
for the following loss distribution:
Outcome (x)
Probability
(p)
Loss (RM50,000)
0.10
No loss (RM0)
0.90
Loss distribution
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2.0 Life Insurance pricing
Mortality risk: The probability of dying for individuals based on
their age and gender, hence, the expected claim cost.
To compute life insurance pricing, we will differentiate between:
(i) Term insurance
(ii) Whole life insurance.
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2.0 Life Insurance pricing
One-year term
For a one year term insurance, an insurer needs to consider the
probability of a person dying at his current age.
E.g.:
Mr. Isaac, 35 year old, purchases a one year term life insurance
policy with a coverage of RM200,000. Should death occur during
the tenure of the policy, Mr. Isaac’s family will receive
RM200,000. The probability of dying is 0.001592. What this
means is that the probability of a 35 year old male surviving for
another year is 0.998408 (1-0.001592). The loss distribution is
summarized in the following table. For simplicity, we ignore
administrative expense and profit loading.
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Loss distribution for Mr. Isaac, age 35
Outcome (x)
Step 1:
Probability (p)
Loss (RM200,000)
0.001592
No loss (RM0)
0.998408
Compute expected claim cost
Exp. claim cost
= (0.001592 x RM200,000) + (0.998408 x RM0)
= RM318.40
Step 2:
Compute discounted value of expected claim cost
Discounted value of exp. claim cost (PV)
= FV / (1+r)n
= RM318.40 / (1.05)
= RM303.24
Hence, the premium of a one year term policy is RM303.24.
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If Mr. Isaac were to purchase another one-year policy the following year at
age 36, the loss distribution would be:
Outcome (x)
Probability (p)
Loss (RM200,000)
0.001660
No loss (RM0)
0.998340
Loss distribution for Mr. Isaac, age 36
Step 1:
Compute expected claim cost
Exp. claim cost
= (0.001660 x RM200,000) + (0.998340 x RM0)
= RM332.00
Step 2:
Compute discounted value of expected claim cost
Discounted value of exp. claim cost (PV)
= FV / (1+r)n
= RM332.00 / (1.05)
= RM316.19
Hence, the present value of premiums of a one-year term policy at Mr. Isaac’s age of
36 is RM316.19.
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Two year term
To show how premiums differ according to the duration of a policy, we now
assume that Mr. Isaac, age 35, purchases a two-year term insurance policy.
The investment return is assumed to be 5%, and other loadings
(administrative and profit) are ignored. There are two types of payment
methods that can be charged by the insurer, single premium and level
premium. Computation of the two payment methods will be demonstrated
below.
(i) Single premium- Policyholders pay a one-time premium at the inception of
the policy.
Since we are ignoring administrative and profit loading, the inflow (premium)
must equal the outflow (which is the pure premium or discounted expected
claim cost), as shown in Column v in Table 4.
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i) Single premium- Policyholders pay a one-time premium at the inception
of the policy.
Since we are ignoring administrative and profit loading, the inflow
(premium) must equal the outflow (which is the pure premium or
discounted expected claim cost), as shown in Column v in Table 4.
Mahdzan & Boey 2015
Table 8.7: Summary of computation for a single premium two-year term policy
i
ii
Time
Claim cost
End of Year 1
End of Year 2
RM200,000
RM200,000
iii
Probability of
dying (subject
to survival at
start of policy)
0.001592
(160/96736)
= 0.001657
iv
v
Exp.claim cost
PV of Exp.claim
cost
318.40
331.40
303.24
300.60
603.84
Total
Step 1:
Compute expected claim cost
Year 1: (0.001592 x RM200,000) + (0.998408 x RM0) = RM318.40
Year 2: (0.001657 x RM200,000) + (0.998343 x RM0) = RM331.40
Step 2:
Compute discounted value of expected claim cost (PV of exp. claim cost)
Discounted value of exp. claim cost (PV)
= FV / (1+r)n
Year 1: RM318.40 / (1.05) = RM303.24
Year 2: RM331.40 / (1.05)2 = RM300.60
Step 3:
Total up the discounted exp.claim cost for Year 1 and year 2
Total discounted exp.claim cost = RM603.84
Hence, the single premium to be charged for a two-year term policy is RM603.84.
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ii) Level premium: Policyholders pay periodic, level premiums (P) over a fixed
number of years.
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Table 8.7: Computation of expected premium for a level premium two-year term policy
Premium 1 + PV of Premium 2 = PV of Exp.claim1 + PV of Exp. Claim2
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Step 1:
Compute expected premium for Year 1 and Year 2.
Year 1: P
Year 2: 0.998408P
Step 2:
Compute discounted value (PV) of expected premium and total for Year 1
and 2.
Discounted value of exp. premium (PV)
= FV / (1+r)n
Year 1: P
Year 2: 0.998408P/1.05 = 0.95086P
Total (Year 1 & 2): P + 0.95086P = 1.95086P
Step 3:
Compute the discounted expected claim cost.
Computation is the same as in the single premium method, hence it is
RM603.84.
Step 4:
Equate the discounted expected claim cost and discounted expected premium
1.95086P
= RM603.84
P
= RM603.84 / 1.95086
P
= RM309.53
Hence, the level premium that is charged at the beginning of Year 1 and 2 is RM309.53
per annum.
Mahdzan & Boey 2015
Table 8.9: Comparison of premiums paid for different term policies
Type of policy
Purchase
one-year
term policies,
renewed
yearly
Purchase a
two-year
policy, single
premium
Purchase a
two-year
policy, level
premium
Premiums
(Year 1)
Premiums
(Year 2)
Premiums
(Total Yr 1&2)
RM303.24
RM316.19
RM619.43
RM603.84
RM309.53
RM603.84
RM309.53
RM619.06
Observations
Premium increases from Year 1 to Year 2,
indicating higher mortality risks as one
increase in age. The disadvantage of
purchasing renewable term policies is that
the insurer may deny coverage if the
individual is no longer insurable (e.g. due
to health conditions).
By paying a single premium, premiums
appear to be the cheapest, although a
higher lumpsum amount is required at the
beginning of the two years.
Premiums are the same for both years. In a
way, Year 1 premium subsidizes Year 2
premium. The total is almost the same as
the one-year renewable term policy
premiums.
Mahdzan & Boey 2015
2.3 Pricing of Whole-life insurance
Single Premium
E.g.: Mr. Isaac, age 35, who wants to purchase a whole life policy with a face
amount of RM200,000. The computation of expected claim cost is
summarized in the following table.
I
ii
iii
iv
v
vi
vii
Age
Death
probability
No.of
living
No.of
deaths
Prob.of
death
Exp.
Claim Cost
PV of
Exp.Claim
Cost
35
0.001592
96,736
154
0.001592
318.40
303.24
36
0.001660
96,582
160
0.001657
331.40
300.65
37
0.001741
96,422
168
0.001735
347.00
299.81
38
0.001837
96,254
177
0.001828
365.60
300.75
39
0.001953
96,077
188
0.001940
388.00
303.96
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
95
0.266454
5,387
1,435
0.014838
2,967.60
151.31
96
0.286625
3,952
1,133
0.011710
2,342.00
113.72
97
0.305869
2,819
862
0.008913
1,782.60
82.44
98
0.323783
1,957
634
0.006550
1,310.00
57.70
99
0.339972
1,323
450
0.004650
930.00
39.01
198,839
30,447.52
Computation summary of a
single premium whole-life
policy
Mahdzan & Boey 2015
Step 1:
Compute the probability of death of a 35 year old occurring in the
respective subsequent years
Age 36:
160/96,736 = 0.001657
Age 37:
168/96,736 = 0.001735
.
Step 2:
Age 98:
634/96,736 = 0.006550
Age 99:
450/96,736 = 0.004650
Compute expected claim cost for each year (age 35-100)
Age 35:
0.001592 x 200,000 = 318.40
Age 36:
0.001657 x 200,000 = 331.40
.
.
Age 98:
0.006550 x 200,000 = 1310.00
Age 99:
0.004650 x 200,000 = 930.00
Mahdzan & Boey 2015
Step 3:
Compute discounted value (PV) of expected premium for each year and
total up the amounts.
Discounted value of exp. claim cost (PV)
Age 35:
318.40/1.05 = 303.24
Age 36:
331.40/1.052 = 300.65
= FV / (1+r)n
.
.
Age 98:
1310/1.0564 = 57.70
Age 99:
930.00/1.0565 = 39.01
Total of discounted values of exp. claim cost = RM30,447.52
Hence, the single premium for a whole-life policy (with coverage of
RM200,000) that must be paid by a 35 year old individual at the inception of
the policy is RM30,447.52.
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Continuous Level premium: The insured pays a level premium from the
inception of the policy to its maturity (age:100).
i
ii
iii
iv
v
vii
Age
35
Prob.of
death
0.001592
No.of
living
96,736
Prob of 35
surviving till
specified age
1.000000
Exp.premium
payment
P
PV of
exp.premium
36
0.001660
96,582
0.998408
0.998408P
0.950865P
37
0.001741
96,422
0.996754
0.996754P
0.904085P
38
0.001837
96,254
0.995017
0.995017P
0.859533P
39
0.001953
96,077
0.993188
0.993188P
0.817098P
.
.
.
.
.
.
.
.
.
.
.
.
95
0.266454
5,387
0.055688
0.055688P
0.002981P
96
0.286625
3,952
0.040853
0.040853P
0.002083P
97
0.305869
2,819
0.029141
0.029141P
0.001415P
98
0.323783
1,957
0.020230
0.020230P
0.000936P
99
0.339972
1,323
0.013676
0.013676P
0.000602P
TOTAL
17.19506P
P
Table 8: Computation summary of a continuous level premiums whole-life policy
Mahdzan & Boey 2015
Step 1:
Compute expected premium for all years (from age 35-99), which is the
number of living at each respective year divided by the number of living at
age 35.
Age 35: 96736/96736 = 1
Age 36: 96582/96736 = 0.998408
.
.
Age 98: 1957/96736 = 0.020230
Age 99: 1323/96736 = 0.013676
Step 2:
Denote the expected premium payment as probability of surviving (column
iv) multiplied by the unknown level premiums, P.
Age 35: P
Age 36:0.998408P
.
.
Age 98: 0.020230P
Age 99: 0.013676P
Mahdzan & Boey 2015
Step 3:
Compute discounted value (PV) of expected premium (age 35-99) and
total up.
Discounted value of exp. premium (PV)
= FV / (1+r)n
Age 35: P
Age 36: 0.998408P/1.05 = 0.95086P
.
.
Age 98: 0.02030/1.0564 = 0.000936P
Age 99: 0.013676/1.0565 = 0.000602P
Total is 17.9506P.
Step 3:
Compute the discounted expected claim cost.
Computation is the same as in the single premium method, hence it is
RM30,447.51.
Step 4:
Equate the discounted expected claim cost and discounted expected
premium
17.9506P
= RM30,447.51
P
= RM30,447.51/17.9506
P
= RM1,696.18
Hence, the continuous level premium that is must be paid at the beginning of each year is
RM1,696.18 per annum.
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Limited payment premiums: Only take into consideration the
expected premium payment and its discounted value
E.g.: Say Mr. Isaac wants to pay premiums for 20 years only
instead of until his age of 99. We only take into
consideration the expected premium payment and its
discounted value till Mr. Isaac’s age of 54. The total
discounted value of the expected premium is 12.836409P
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We then equate the discounted premium to the discounted expected claim cost.
12.836409P = RM30,447.51
P
= RM2371.96
Hence, the limited payment premiums that is to be paid us RM2,371.96 per annum for
20 years.
Table 9: Comparison of premiums
(single, continuous level and limited payment premium)
Type of premium paid
Amount
of Duration of payment
Total premiums paid
premium
Single premium
RM30,447.51
Once
RM30,447.51
Continuous level premium
RM1,696.18
Age 35-99 = 65 years
RM1696.18 x 65 =
RM110,251.70
Limited payment premium
RM2,371.96
Age 35-54 = 20 years
RM2371.96 x 20
RM47,439.30
Mahdzan & Boey 2015