Actuarial Mathematics I
Actuarial Mathematics I MATH 3630
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This 12 page Class Notes was uploaded by Mary Veum on Thursday September 17, 2015. The Class Notes belongs to MATH 3630 at University of Connecticut taught by Emiliano Valdez in Fall. Since its upload, it has received 20 views. For similar materials see /class/205814/math-3630-university-of-connecticut in Mathematics (M) at University of Connecticut.
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Date Created: 09/17/15
Valuing Contingent Payments Lecture Weeks 4 6 l l An introduction a Central theme to quantify the value today of a random sum of money to be paid at a random time in the future a main application is in life insurance contracts but could be applied in other contexts 0 Generally computed in two steps 0 take the present value PV random variable bTvT and Q calculate the expected value EbTvT for the average value this value is referred to as the Actuarial Present Value APV o In general we want to understand the entire distribution of the PV random variable bTvT a it could be highly skewed in which case there is danger to use expectation a other ways of summarizing the distribution such as variances and percentilesquantiles would be use u I l l l A simple illustration Consider the simple illustration of valuing a two year term insurance policy issued to age 20 where if he dies within the first year a 100 benefit is payable at the end of his year of death If he dies within the second year a 200 benefit is payable at the end of his year of death Assume a constant discount rate of 5 and the following extract from a mortality table qm 20 0001 21 0002 22 0003 Calculate the APV of the benefits I l l l Chapter summary a Life insurance a benefits payable contingent upon death payment made to a designated beneficiary a actuarial present values APV a actuarial symbols and notation o Insurances payable at the moment of death a continuous a level benefits varying benefits eg increasing decreasing o Insurances payable at the end of year of death a discrete 0 level benefits varying benefits eg increasing decreasing a Chapter 5 Cunningham et al l l The present value random variable 0 Denote by Z the present value random variable 0 This gives the value at policy issue of the benefit payment a In the case where the benefit is payable at the moment of death Z clearly depends on the timeuntil death T o It is Z bTvT where o bT is called the benefit payment function a vT is the discount function l l Fixed term life insurance 0 An n year term life insurance provides payment if the insured dies within 71 years from issue 0 For a unit of benefit payment we have 7 17 TE 7 T ITi07 Tgtn andvTiv UT T lt n o The present value random IS therefore Z 7 i and EZ 0 T gt n 7 is called the actuarial present value APV of the insurance a Actuarial notation Aim EZ f0 Uttpmpmtdt l l Rule of moments o The j th moment of the distribution of Z can be expressed as TL 71 Elzj Utjtpz ztdt eil SltthImtdt 0 0 o This is actually equal to the APV but evaluated at the force of interest 36 o In general we have the following rule of moment E Z7 M Elz 712 o For example the variance can be expressed as Varlzl aim Aim I l l l Whole life insurance 0 For a Whole life insurance benefits are payable following death at any time in the future 0 Here we have bT 1 so that the present value random variable is Z UT 0 APV notation for whole life Am EZ f0 Uttpmpmtdt o Variance using rule of moments VarZ 214m 7 Amy a Whole life insurance is the limiting case of term life insurance as 71 H 00 l I Example wk41 For a whole life insurance of 1000 on with benefits payable at the moment of death you are given 6 7 0047 0ltt 10 quot 0tgtm and 7 00ltt m WH 0mtgtm Calculate the single benefit premium for this insurance l I Pure endowment insurance 0 For an n year pure endowment insurance a benefit is payable at the end of 71 years if the insured survives at least 71 years from issue lt 0 Here we have bT 0 Tin 17 Tgt7l ilt 7 n and UT u so that the PV rv U T gt n 39 o APV for pure endowment Am 7 nEm Unnpm o Variance using rule of moments VarZ v2 an n61m 2A l I Endowment insurance 0 For an n year endowment insurance a benefit is payable if death is within 71 years or if the insured survives at least 71 years from issue whichever occurs first T U T lt n o H h b 1 d 7 ere we ave T an UT Un7 Tgtn is Z i T7 T 3 n 7 U T gt n 39 so that the PV rv c 7 1 1 o APV endowment Amm 7 Amm AIR a Variance using rule of moments VarZ 214mm 7 Amm239