Life Premium - Life Contingencies - Lecture Notes, Study notes of Mathematical Statistics

Download Life Premium - Life Contingencies - Lecture Notes and more Study notes Mathematical Statistics in PDF only on Docsity! Chapter 6: Benefit Premiums 1 of 13 6.3 Fully Discrete Premiums Consider a unit of whole life policy with benefit payable at the end of the year of death (K+1). The loss function: 1 11 + + −⋅= Kx K aPvL && , where xP denotes a whole life premium paid annually, 1+K a&& is an annuity with payments due at the beginning of each year starting at policy issue and including the year of death. NOTE: There is no bar over the P. Let’s use the equivalent principle to determine the premium: [ ] [ ] [ ] 0Premiums of PVBenefits of PV =−= EELE [ ] [ ] 0 1 1 1 1 1 =−=    −=     −⋅= + + + + xxx Kx K Kx K aPA aEPvEaPvELE && &&&& Therefore, x x x a A P && = . Chapter 6: Benefit Premiums 2 of 13 The variance of the loss function is: [ ] [ ] ( )22 2 1 2 1 1 1 1 1 1 11 1 1 xx x KxxKx K x K Kx K AA d P vVar d P d P v d P Var d v PvVaraPvVarLVar −      +=       +=      −      +=             − −=     −⋅= ++ + + + + && NOTE: With x x x a A P && = , we have: [ ] ( ) ( )22 2 22 2 1 xx x xx xx x x AA ad Aad AA ad A LVar −      + =−      += && && && Recall: 1=+ xx Aad && Therefore, [ ] ( )2 22 x xx ad AA LVar && − = Docsity.com Chapter 6: Benefit Premiums 3 of 13 Chapter 6: Benefit Premiums 4 of 13 Example 6.3.5: Prove ( )nx nx xnnx APPP +−+= 11 : : . Example 6.11: Prove ( )x xx APPP 10|2020 20:30: 20 11 =− . Docsity.com Chapter 6: Benefit Premiums 9 of 13 Example: Given i= 0.06, x 90 91 92 93 lx 100 72 39 0 dx 28 33 39 0 Find premium if 1) A $1000 discrete whole-life insurance (payable at the end of the year of death) is purchased and is paid by annual premiums at the beginning of each policy year. Chapter 6: Benefit Premiums 10 of 13 2) A $1000 continuous whole-life insurance is purchased with continuous payments under UDD. 3) A $1000 continuous whole life insurance is purchased with annual payments starting age 90 under UDD. Docsity.com Chapter 6: Benefit Premiums 11 of 13 4) What is the p.m.f, mean, and variance of loss in the fully discrete case? Distribution of the loss: L K+1 Prob Chapter 6: Benefit Premiums 12 of 13 5) What is the probability of a loss (on the insurer)? 6) How much should each policyholder pay in order for the insurer to have a 5% chance of positive loss in the aggregate? Docsity.com Chapter 6: Benefit Premiums 13 of 13 Docsity.com