Chapter 3. Life Insurance
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1 Cher 3 Life Insurnce
2 Life Tble = Morliy ble Noion l =number of eole live ge d =number of eole die b/w ge nd + Bsic Relionshi l + =l -d Life/Morliy ble X l 00,000 98,000 96,500 95,500 d
3 Life Tble Probbiliies n =Psurvive o +n survived o )= l n q =Pdies in n yrs survived o )= l =Psurvive o + survived o )= q =Pdies wihin yr survived o )= Emles 2 0 =96,500/00,000 2 q =98,000-95,500)/98,000 =96,500/98,000 q 0 =2,000/00,000 l n l l l l n l l X l l 00,000 98,000 96,500 95,500 d
4 Life eecncy How longer cn n ged erson live? Ide: e Y=number of yers n ged erson lives Life eecncy = EY) = kpy=k) k 0 k 0 kp Y k k Emles l k) l l k k 0 d k l k e 97 = )/800= e 99 =6/46=0.096 l k l k X l d
5 Life insurnce 壽險 ) Procedure Py $P o he insurer eriodiclly Receive $ c he ime of deh Why buy i? Wn roecion gins hving shor life
6 Life nnuiies 年金 ) Procedure Py $P o he insurer now Receive $ c he beginning of ech yer, unil deh Why buy i? Wn roecion gins hving long life
7 Versions of Life insurnce Whole life insurnce Deh benefi id he ime of deh Term life insurnce Deh benefi is rovided for only n yers Afer n yers, no benefi is rovided Endowmen insurnce Deh benefi is rovided for only n yers If olicyholder is live, ymen is mde ime n Mrkeed s sving ln Pure endowmen: If live, ge benefi ime n.
8 Versions of Life nnuiy Whole life nnuiy Benefis id unil he ime of deh Temorry life nnuiy Benefis is id for only n yers Deferred nnuiy The s ymen begins he k-h yer Gurneed ymen For gurneed eriod, benefis will be id regrdless of wheher he nnuin is live or no Lower bounded he los of nnuin
9 Remrk on remium nd benefi If no riculrly secified Premiums re id he beginning of he yer Benefis re id he end of he yer
10 Pricing Insurnce nd Annuiy The only KEY ide of ricing nnuiy: Find he Presen Vlue of ll ossible ymens, weighed by he robbiliy of mking ymens Probmke ymen) Csh flow of conrc Annuiy Price 0 ) c )
11 Pricing Insurnce nd Annuiy The only KEY ide of ricing insurnce: Find he Presen Vlue of ll ossible ymens, weighed by he robbiliy of mking ymens Probmke ymen) Csh flow of conrc Insurnce Price 0 ) c ) q
12 Emles of Pricing insurnce conrc Emle : emorry life nnuiy An ged 50 erson is rovided for $ yble ge 50, $2 yble ge 5, $3 52 nd $4 53. Suose q 50 =0., q 5 =0.2, q 52 =0.25. The ineres re i=50% for he s yr nd i=00% herefer. Wh is he rice?
13 Emles of Pricing insurnce conrc Emle : emorry life nnuiy An ged 50 erson is rovided for $ yble ge 50, $2 yble ge 5, $3 52 nd $4 53. Suose q 50 =0., q 5 =0.2, q 52 =0.25. The ineres re i=50% for he s yr nd i=00% herefer. Wh is he rice? Price [0.90.8] 4[ ]
14 Emles of Pricing insurnce conrc Emle 2: Gurneed nnuiy A erson ge 40 urchses life nnuiy h rovides 0000 ech yer for life, wih he firs ymen sring 4. The firs 0 ymens will be id regrdless of wheher he nnuin is live or no. Find formul for his remium.
15 Emles of Pricing insurnce conrc Emle 2: Gurneed nnuiy A erson ge 40 urchses life nnuiy h rovides 0000 ech yer for life, wih he firs ymen sring 4. The firs 0 ymens will be id regrdless of wheher he nnuin is live or no. Find formul for his remium. Price ) ) 40
16 Emles of Pricing insurnce conrc Emle 3: Deferred + gurneed Pension fund) A erson ge 40 urchses life nnuiy h rovides 0000 ech yer for life, wih he firs ymen sring 65. If he reches ge 65, he will receive les 0 ymens regrdless of wheher he is live or no. Find formul for his remium.
17 Emles of Pricing insurnce conrc Emle 3: Deferred + gurneed Pension fund) A erson ge 40 urchses life nnuiy h rovides 0000 ech yer for life, wih he firs ymen sring 65. If he reches ge 65, he will receive les 0 ymens regrdless of wheher he is live or no. Find formul for his remium. Price ) )
18 Emles of Pricing insurnce conrc Emle 4: Term life Insurnce An ged 60 erson is rovided for $80 yble he end of he yer of deh of he s yer, $75 for he 2 nd yer, $00 for he 3 rd yer. Nohing is id if he survives fer 63. Suose q 60 =0.2, q 6 =0.4, q 62 =0.5. The ineres re i=00%. Wh is he rice?
19 Emles of Pricing insurnce conrc Emle 4: Term life Insurnce An ged 60 erson is rovided for $80 yble he end of he yer of deh of he s yer, $75 for he 2 nd yer, $00 for he 3 rd yer. Nohing is id if he survives fer 63. Suose q 60 =0.2, q 6 =0.4, q 62 =0.5. The ineres re i=00%. Wh is he rice? Price
20 Emles of Pricing insurnce conrc Emle 5: Term life Insurnce An ged 60 erson is rovided for $80 yble he end of he yer of deh of he s yer, $75 for he 2 nd yer, $00 for he 3 rd yer. Nohing is id if he survives fer 63. Suose q 60 =0.2, q 6 =0.4, q 62 =0.5. The ineres re i=00%. If his insurnce is urchsed by ying P, 2P nd 0.75P he beginning of ech yr, Wh is P?
21 Emles of Pricing insurnce conrc Emle 5: Term life Insurnce An ged 60 erson is rovided for $80 yble he end of he yer of deh of he s yer, $75 for he 2 nd yer, $00 for he 3 rd yer. Nohing is id if he survives fer 63. Suose q 60 =0.2, q 6 =0.4, q 62 =0.5. The ineres re i=00%. If his insurnce is urchsed by ying P, 2P nd 0.75P he beginning of ech yr, Wh is P? P 2 2 P P P 8.99
22 Emles of Pricing insurnce conrc Emle 6: Endowmen Insurnce Consider 20-yer endowmen insurnce wih he deh benefi equl o uni for he firs 0 yers, nd 2 unis for he second 0 yers. The moun of he ure endowmen 3 unis. Level remium of P re yble for 5 yers. Wh is P?
23 Emles of Pricing insurnce conrc Emle 6: Endowmen Insurnce Consider 20-yer endowmen insurnce wih he deh benefi equl o uni for he firs 0 yers, nd 2 unis for he second 0 yers. The moun of he ure endowmen 3 unis. Level remium of P re yble for 5 yers. Wh is P? ) 20) 3 ) ) 20) 3 ) ) ) q q P q q P
24 Emles of Pricing insurnce conrc Emle 7: Combined olicy A conrc on n ged 40 erson rovides for nnuiy benefis of er yer for life, beginning ge 65. If he dies before ge 65, deh benefi of 0 will be id. Level nnul remium of P re yble for 25 yers. Find P.
25 Emles of Pricing insurnce conrc Emle 7: Combined olicy A conrc on n ged 40 erson rovides for nnuiy benefis of er yer for life, beginning ge 65. If he dies before ge 65, deh benefi of 0 will be id. Level nnul remium of P re yble for 25 yers. Find P ) ) ) 0 ) ) 0 ) q P q P
26 Sndrd noions Annuiies Annuiy-immedie Temorry life Annuiy-immedie ged ) Annuiy-due n n ) n : n ) n n ) 0 n n Temorry life Annuiy-due ged ) s n s n n : n ) 0
27 Sndrd noions Pereuiies Pereuiy-immedie Whole life Annuiy-immedie ged ) Pereuiy-due ) ) Whole life Annuiy-due ged ) ) 0 0 )
28 Insurnce Life Insurnce Term life Insurnce ged ) Endowmen insurnce ged ) Pure endowmen insurnce ged ) Sndrd noions 0 ) q A 0 : ) n n q A n n n n q A ) ) 0 : n n n E )
29 Reserve +ve ci) inflow -ve ci) ouflow As we wn o mnge he risk in he fuure, we re more ineresed he Reserve hn he blnce Reserve 儲備 ) moun needed jus before) k for fuure obligion Reserve N jk c j) j k) POccurnceof cj) )
30 Emles of Blnce nd Reserve An ged 60 erson is rovided for $80 yble he end of he yer of deh of he s yer, $75 for he 2 nd yer, $70 if he survives fer 62. Suose q 60 =0.2, q 6 =0.4. The ineres re i=00%. This is urchsed by wo level nnul remiums. Wh is he fir remium? Find he reserves ime nd 2.
31 Emles of Reserve An ged 60 erson is rovided for $80 yble he end of he yer of deh of he s yer, $75 for he 2 nd yer, $70 if he survives fer 62. Suose q 60 =0.2, q 6 =0.4. The ineres re i=00%. This is urchsed by wo level nnul remiums. Wh is he fir remium? Find he reserves ime nd 2. P 2 Reserve Reserve ) ) 2 2 P 6
32 Summry Life Tble Life Insurnce Whole life insurnce Term life insurnce Endowmen insurnce Life Annuiy Whole life nnuiy Temorry life nnuiy Deferred nnuiy Gurneed ymen Key Ide = Creful discouning wih robbiliy
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