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MTHEL 131
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Mathematics Electives

MTHEL 131

David Kohler

Fall

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MTHEL131 Textbook notes – Life and Health Insurance 13e (Black and Skipper)
Chapter 28
Erin Edward
Chapter 28-Net Premiums
Net single Premiums
Calculation Process
Requires:
1. age and sex of insured
2. benefits to be provided
3. mortality rates used
4. rate of interest consumed
Term Insurance
the insurer wants to make sure they have enough money to pay the claims of the year
o wants to know probability of paying death claim (insured dying within the term) – uses
the mortality table
to calculate the premium needed from individuals in term insurance (net single premium):
o The aggregate approach – emphasizes total funds necessary to meet death claims as
they occur
EX: dollar discounted from the projected interest earned x projected value of claims,
divided total number of insured (refer to Appendix 27.4 on page 722)
$1 discounted at 5% is $0.952 at the beginning of the year
0.952 x 1000 (policy value) x 41,907 (estimated death claims)
9,210,289 (total insured)
=$4.33 (pg. 274)
o The expected value approach
EX: estimated death claims divided by the total number of insured, multiplied by the
value of the policy, discounted from the projected interest earned (refer to Appendix
27.4 on page 722)
41,907 (estimated death claims)
9,210,289 (total insured) x $1000 (policy value)
=$4.55
Discounted to $4.33 by 5% annual interest
o Actuarial notation: (1,000)xq )(v)=(1,000)()(v)
l45
Insurance companies can’t insure a single person, must deal with a large group to apply law of
large number
o Does not need to insure this whole group with the same policy at the same age.
These processes give the present expected value
Term insurance can be purchased in a single sum for a specific period (ie. 5 years)
o Premium is paid once, at time of policy inception
o Death claims are paid at end of year in which claim occurs (not at the end of 5 year
period). This is significant to the amount of interest earned on the premium (significant
to computing the cost) MTHEL131 Textbook notes – Life and Health Insurance 13e (Black and Skipper)
Chapter 28
Erin Edward
o Must compute every year’s mortality costs separately because not all deaths will occur
in the same year (ie. Within 5 years)
o Actuarial notation: (45 year old man)
d45 41,907
EX: l45 =9,210,2890.00455
d46 45,108
l45=9,210,2890.00490
etc.
These computations show (ignoring tax, expenses) the amount required to be paid by each
policy owner to ensure each claim is paid
Whole life insurance
Chance of dying in each separate year:
𝑑𝑥+𝑡−1
𝑙
𝑥
x=age
t=policy year
This is multiplied by the amount of policy and discounted for the number of years between issue
of the policy (ie. The payment of the single premium) – see table 28-2 pg. 726
The calculated value represents the present value of the policy’s share of all the expected death
claims payable from a certain age, based on the stated assumptions
EX: this calculation will show that it is highly unlikely for a man to die in his 100 year, because it
is unlikely for him to attain such an age in the first place.
Net single-premium calculation for a whole life policy apportions the probability of dying
Endowments
1. Pure Endowments
Paid if and only if insured survives a certain period (ex: 5 years)
Probability of the event insured against occurring is the probability of the insured
surviving the next 5 years
Pure endowment cannot obtain possession of the money invested in a pure endowment
until the end of the endowment period. Nothing is returned if the insured should die
within this period.
o In comparison, a bank savings account is not lost on the death of the saver.
o This contrast makes it possible to divide the policy ($1000) that will be paid in
case of survival through endowment period into two funds (investment fund
and benefit of survivorship fund)
Some of the policy will be paid for with the survivor’s share of the amounts left by those
insureds who died before their policy matured – the benefit of survivorship contribution
A life annuity is a series of pure endowments
Some pure endowments are prohibited by law
2. Endowment Insurance
Includes pure endowment feature, as well as insurance against death during the term of
endowment MTHEL131 Textbook notes – Life and Health Insurance 13e (Black and Skipper)
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