Continuous Life Annuity - Life Contingencies - Lecture Notes, Study notes of Mathematical Statistics

Download Continuous Life Annuity - Life Contingencies - Lecture Notes and more Study notes Mathematical Statistics in PDF only on Docsity! Chapter 5: Life Annuity Summary 1 of 5 Continuous Life Annuity Cumulative Distribution of Y 1 ln(1 ) 1 ( ) [ ] 0 T Y T e y F y P Y y P y F y δ δ δ δ δ − − − −  = ≤ = ≤ = < <      Density Function of Y 1 ln(1 ) 1 ( ) 0 1 Y T y f y f y y δ δ δ δ   − −  = < <   −    Whole Life Annuity 0 0 ( ) ( ) x t xT t t t x a E a a p x t dt v p dt µ ∞ ∞ = = + = ∫ ∫ ( )22 2 2 2 2 [ ] ( ) x xx x xT a aA A Var Y Var a a δ δ −− = = = − Constant force of mortality xA µ µ δ = + 1 xa µ δ = + 2 2 2 22 2 )( δ δµ µ δµ µ δ       + − + = − = xx T AA aVar DeMoivre’s law xw xw x a xw e xw A − −− − = − ⋅ − = 111 )( δ δ 1 1 1 1 w x x x w x a A w xa Da w xδ δ − − − − −= = = − Docsity.com Chapter 5: Life Annuity Summary 2 of 5 2 2 2 2 22 11 )( δδ       − − − = − = −− xwxw xx T a xw a xwAA aVar n-year Temporary Life Annuity ( ) : 0 0 0 ( ) ( ) n t x n xx n t n n n t t x t x a E Y a p x t dt a p v p dt E dt µ= = + + = = ∫ ∫ ∫ ( )22 2 : :: : 2 2 : 2 [ ] x n x nx n x n x n a aA A Var Y a δ δ −− = = − Recursion relationship: 1|1: ++= xxxx avpaa In general: nxxn n nxx apvaa ++= |: n-year Deferred Whole Life Annuity | ( ) ( ) n n x t xt nn a E Y v a p x t dtµ ∞ − = = +∫ | : | t n n x x x n t x n x x n n a a a v p dt v p a ∞ += − = =∫ ( ) 2 22 2( )( ) n x n x nn x n x a a Var Y v p a δ + +  − = −    n-year Certain and Life Annuity ∫ ∞ ++== n xttnnx dttxpaaYEa )()( |: µ xnnnx aaa |||: += ( )2 2 2 )(2][ xn nxnx xn n a aa pvYVar −      − = ++ δ Discrete Life Annuities Docsity.com