LIFE PRICING FUNDAMENTALS
Richard MacMinn
Understand the law of large numbers as it
relates to insurance.
 Describe insurers’ pricing objectives and explain
why they are of relevance to the life insurer and
consumer.
 Outline elements of life insurance rate making
including the assumptions made in the absence of
perfect information.
 Draw distinctions between participating and
guaranteed cost, nonparticipating life insurance.
 Explain how asset share analysis is used to test
the adequacy of life insurance rates.
28 June 2016
OBJECTIVES

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2

28 June 2016
LAW OF LARGE NUMBERS
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The Weak Law of Large Numbers: For each n =
1, 2, . . ., suppose that R1, R2, . . . , Rn are
independent random variables on a given
probability space, each having finite mean and
variance. Assume that the variances are
uniformly bounded; that is, assume that there is
2i  M for
some finite positive
number
M
such
that
n
all i. Let Sn   Ri Then,
i 1
 S  E  Sn 

P n
    0 as n  
n


3

Adequacy

Equity


The payments generated by a block of policies plus
any investment return on same must be sufficient to
cover the current and future benefits and costs
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
28 June 2016
PRICING OBJECTIVES
This equity refers to setting premiums commensurate
with the expected losses and expenses; it also
suggests no cross subsidization. The equity notion
sets a floor.
Not excessive
The excessive notion sets a ceiling
 Regulation
 Competition

4

Probability of insured event

Mortality and morbidity tables
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
28 June 2016
ELEMENTS OF RATE MAKING
Time value of money
Premiums paid now
 Interest on accumulated funds


Promised benefit
period of coverage
 level of coverage
 type of coverage


Loading or expenses, taxes, contingencies and
profit
5
LIFE INSURANCE RATE COMPUTATION

Yearly renewable term life
insurance




6
The YRT covers the life for one
year at a set premium and is
renewable
The YRT premium for a 30 year
old male would be $1.73 per
$1,000 of coverage while it would
be $1.38 for a female the same
age. If investment income is
included then the company
would set the premium at $1.65
and $1.31 for males and females
respectively
Table 2-5
Illustrative Net Level Premium Calculation
1
2
3
Policy Year
1
2
3
4
5
4
5
Net Level
Present Value
Number
Premium to be
of Total Net
Living at the Present Value
Paid Annually
Level
Beginning of Factor at 5%
by Each
Premiums [(2)
Each Year
Survivor
x (3) x (4)]
1
1
1
1
1
100,000
67,000
41,205
21,427
7,328
Richard:
This is the level premium or premium per
year.
1.0000
0.9524
0.9070
0.8638
0.8227
Total PV
premium
$100,000
$63,810
$37,374
$18,509
$6,029
$225,722
$395.40
The $225,722 is the present value per dollar
in premiums paid each year of the policy.
Hence, that times the premium per year must
equal the present value of the claims, i.e.,
the $89,251,339. By altering the interest
rate in table 2-3 cell C1 is possible to see how
the level premium changes in table 2-5 cell
E12
Single premium plan
Level premium plan
28 June 2016
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Richard D. MacMinn:
This allows us to calculate
the present value of a
one dollar premium flow
per customer.
SINGLE PREMIUM PLAN

This plan provides multiyear coverage for a single
premium now
This eliminates the rising
premiums associated with
the YRT.
 This gives the insurer the
ability to generate
compound interest and
reduce the rate for
coverage

7
Table 2-2
Modified Version of 1980 CSO Mortality Table
1
2
3
4
Age
Number Living
(Beginning of
Year)
Probability
of Death
(During
the Year)
Number
Dying
(During
the year)
95
100,000
0.330
33000
96
67,000
0.385
25795
97
41,205
0.480
19778
98
21,427
0.658
14099
99
7,328
1.000
7328
100
0
28 June 2016
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Table 2-5
Illustrative Net Level Premium Calculation
1
2
3
1
2
3
4
5
5
Net Level
Present Value
Number
Premium to be
of Total Net
Living at the Present Value
Paid Annually
Level
Beginning of Factor at 5%
by Each
Premiums [(2)
Each Year
Survivor
x (3) x (4)]
1
1
1
1
1
100,000
67,000
41,205
21,427
7,328
Richard:
This is the level premium or premium per
year.
The $225,722 is the present value per dollar
in premiums paid each year of the policy.
Hence, that times the premium per year must
equal the present value of the claims, i.e.,
the $89,251,339. By altering the interest
rate in table 2-3 cell C1 is possible to see how
the level premium changes in table 2-5 cell
E12
1.0000
0.9524
0.9070
0.8638
0.8227
Total PV
premium
Richard D. MacMinn:
This allows us to calculate
the present value of a
one dollar premium flow
per customer.
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Policy Year
4
28 June 2016
MODIFIED VERSION OF 1980 CSO
MORTALITY TABLE
$100,000
$63,810
$37,374
$18,509
$6,029
$225,722
$395.40
8
Table 2-5
Illustrative Net Level Premium Calculation
1
2
3
1
2
3
4
5
5
Net Level
Present Value
Number
Premium to be
of Total Net
Living at the Present Value
Paid Annually
Level
Beginning of Factor at 5%
by Each
Premiums [(2)
Each Year
Survivor
x (3) x (4)]
1
1
1
1
1
100,000
67,000
41,205
21,427
7,328
Richard:
This is the level premium or premium per
year.
The $225,722 is the present value per dollar
in premiums paid each year of the policy.
Hence, that times the premium per year must
equal the present value of the claims, i.e.,
the $89,251,339. By altering the interest
rate in table 2-3 cell C1 is possible to see how
the level premium changes in table 2-5 cell
E12
1.0000
0.9524
0.9070
0.8638
0.8227
Total PV
premium
Richard D. MacMinn:
This allows us to calculate
the present value of a
one dollar premium flow
per customer.
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Policy Year
4
28 June 2016
PRESENT VALUE OF CLAIMS FOR 95-YEAROLD MALES
$100,000
$63,810
$37,374
$18,509
$6,029
$225,722
$395.40
9
Table 2-5
Illustrative Net Level Premium Calculation
1
2
3
1
2
3
4
5
5
Net Level
Present Value
Number
Premium to be
of Total Net
Living at the Present Value
Paid Annually
Level
Beginning of Factor at 5%
by Each
Premiums [(2)
Each Year
Survivor
x (3) x (4)]
1
1
1
1
1
100,000
67,000
41,205
21,427
7,328
Richard:
This is the level premium or premium per
year.
The $225,722 is the present value per dollar
in premiums paid each year of the policy.
Hence, that times the premium per year must
equal the present value of the claims, i.e.,
the $89,251,339. By altering the interest
rate in table 2-3 cell C1 is possible to see how
the level premium changes in table 2-5 cell
E12
1.0000
0.9524
0.9070
0.8638
0.8227
Total PV
premium
$100,000
$63,810
$37,374
$18,509
$6,029
$225,722
Richard D. MacMinn:
This allows us to calculate
the present value of a
one dollar premium flow
per customer.
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Policy Year
4
28 June 2016
POLICY RESERVES FOR NET SINGLEPREMIUM WHOLE LIFE INSURANCE
$395.40
10
LEVEL PREMIUM PLAN



11
If some of the 100,000
policyholders prefer to pay
premiums on an annual basis
then how much must be charged
per year to make the insurer
indifferent between the single
premium and the annual level
premium?
Let pt be the proportion of the
insured population alive at the
beginning of policy year t. Let at
be the annuity factor for the
premium payment stream.
Let x be the level premium.
Then x must satisfy the last
equation on the RHS.
28 June 2016
T
pt
t1
t  1 (1  r)
aT  
 p1  p2
p4
1
1
 p3
2
1r
1  r 
1
1

p

5
3
4
1  r 
1  r 
aT x pv T (L)
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Table 2-5
Illustrative Net Level Premium Calculation
1
2
3
1
2
3
4
5
5
Net Level
Present Value
Number
Premium to be
of Total Net
Living at the Present Value
Paid Annually
Level
Beginning of Factor at 5%
by Each
Premiums [(2)
Each Year
Survivor
x (3) x (4)]
1
1
1
1
1
100,000
67,000
41,205
21,427
7,328
Richard:
This is the level premium or premium per
year.
The $225,722 is the present value per dollar
in premiums paid each year of the policy.
Hence, that times the premium per year must
equal the present value of the claims, i.e.,
the $89,251,339. By altering the interest
rate in table 2-3 cell C1 is possible to see how
the level premium changes in table 2-5 cell
E12
1.0000
0.9524
0.9070
0.8638
0.8227
Total PV
premium
$100,000
$63,810
$37,374
$18,509
$6,029
$225,722
Richard D. MacMinn:
This allows us to calculate
the present value of a
one dollar premium flow
per customer.
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Policy Year
4
28 June 2016
NET LEVEL PREMIUM CALCULATION
$395.40
12
EXPERIENCE PARTICIPATION IN

Guaranteed-cost, non-participating insurance
(without profits policies)

Policy elements fixed at inception
They offer no way of passing changes in mortality
(morbidity), interest or loading to policyholders
Participating insurance (with profits policies)
Policy gives its owner the right to share in surplus
accumulated due to experience
 Surplus is distributed as dividends


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

28 June 2016
INSURANCE
Current assumption insurance
Policy allows values to deviate from those at policy
inception on the upside and downside
 Unlike participating policies that adjust ex post the
current assumption policy adjusts ex ante; for example, if
the insurer expects a 7% return on investments backing
policy reserves then the policyholders may get a promised
6.5% credited to their cash values.

13
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The asset share calculation is a simulation of the
anticipated operating experience of a block of
policies
 An example

28 June 2016
ASSET SHARE CALCULATION
14