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Mathematics Electives
David Kohler

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) Ch
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