Class Notes (839,116)
Canada (511,194)
ACTSC 232 (47)
Jun Cai (4)
Lecture

Summary of Part IV (formulas) All the formulas you need for the fourth part of the course organized in a pdf file

6 Pages
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Department
Actuarial Science
Course Code
ACTSC 232
Professor
Jun Cai

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Summary of Lecture Notes { ACTSC 232, Winter 2010 Part 4 { Bene▯t Premiums ▯ The loss random variable L for an insurance or annuity is de▯ned as L = The PV of bene▯ts ▯ The PV of premiums. ▯ The expectation of the loss random variable is E[L] = The APV of bene▯ts ▯ The APV of premiums. ▯ Equivalence Principle (EP): Set premiums such that E[L] = 0 or The APV of bene▯ts = The APV of premiums. ▯ Percentile Principle (PP): Set premiums such that PrfL > 0g = ▯. Note that we assume the equivalence principle throughout this courses unless otherwise stated. 4.1 Fully continuous bene▯t premiums { In an insurance, the bene▯ts of the insurance form a continuous life insurance while the premiums of the insurance form a continuous life annuity. (a) A fully continuous whole life insurance of 1 on (x) with an annual premium rate of P: L = v Tx▯ P ▯ ; Tx A▯x ▯ Ax 1 ▯ ▯▯ax P = = = ; ▯x 1 ▯ Ax ▯x ▯!2 ▯! 2h i V ar[L] = 1 + P V ar(v ) = 1 + P 2A ▯ (A ) 2 ▯ ▯ x x 2Ax▯ (A x 2 = 2 : (▯▯x) (b) A fully continuous h-Payment whole life insurance of 1 on (x) with an annual pre- mium rate of P: L = v x ▯ P ▯T ^h; x A▯x P▯ = : ▯x:h 1 (c) A fully continuous n-year term insurance of 1 on (x) with an annual premium rate of P: 8 < v x ▯ P ▯ ; T ▯ n L = Tx x : 0 ▯ P ▯n; T x n A▯x: P▯ = : ▯ x:n (d) A fully continuous n-year endowment insurance of 1 on (x) with an annual premium rate of P: Tx^n ▯ L = v ▯ P ▯Tx^n; ▯ ▯ ▯ Ax:n ▯ Ax:n 1 ▯ ▯▯x:n P = ▯ = 1 ▯ A = ▯ ; x:n ! x:n x:n ! P 2 P 2h i V ar[L] = 1 + V ar(vTx^n) = 1 + 2A▯x:n (A▯x:n2 ▯ ▯ 2 2 Ax:n▯ (A▯x:n = 2 : (▯▯x:n) (e) A fully continuous n-year pure endowment of 1 on (x) with an annual premium rate of P: 8 < 0 ▯ P ▯ ; Tx▯ n L = Tx : v ▯ P a▯n; T x n A x:n nE x P = = : ▯x:n ▯x:n (f) A fully continuous n-year deferred whole life annuity of 1 on (x) with an annual premium rate of P: 8 < 0 ▯ P ▯ ; T ▯ n L = Tx x : v a▯ ▯ P a▯n; Tx> n Tx▯n nj▯x nE x▯x+n ▯x▯ a▯x:n P▯ = = = : ▯x:n ▯x:n ▯x:n (g) A fully continuous h-Payment, n-year endowment insurance of 1 on (x)(h < n) with an annual premium rate of P: Tx^n ▯ L = v ▯ P ▯ x ^h ▯ ▯ A x:n P = ▯ : x:h 4.2 Fully discrete bene▯t premiums { In an insurance, the bene▯ts of the insurance form a discrete life insurance while the premiums of the insurance form a discrete life annuity- due. 2 (a) A fully discrete whole life insurance of 1 on (x) with an annual premium of P: K x1 L = v ▯ P Kx+1; A dA 1 ▯ d P = x = x = x; x 1 ▯ Ax x ▯ ▯ 2 ▯ ▯ 2 P K x1 P h2 2i V ar[L] = 1 + V ar(v ) = 1 + Ax▯ (A x d d 2A x (A )x 2 = : (dx)2 (b) A fully discrete h-Payment whole life insurance of 1 on (x) with an annual premium of P: L = v Kx+1 ▯ P (K +1)^h x Ax P = : x:h (c) A fully discrete n-year term insurance of 1 on (x) with an annual premium of P: 8 < vKx+1 ▯ P Kx+1; K x n ▯ 1 L = : 0 ▯ P n; Kx> n ▯ 1 1 Ax:n P = : x:n (d) A fully discrete n-year endowment insurance of 1 on (x) with an annual premium of P: (Kx+1)^n L = v ▯ P (Kx+1)^n; A x:n dA xn 1 ▯ daxn
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