() t ( ) ( ) ( ) ( ) ( ) ( ) ω ω. SURVIVAL Memorize + + x x. m = = =

SURVIVL ' uu λ -Λ : > l + S e e e S ω ο ω ω Ufrm DeMvre S X e Vr X ω λ Eel S X e e λ ω ω ww S + S f ο S + S e where e S S S S S Prcles T X s rm vrble fr remg me ul eh f sus ge f + survvl fuc fr T X f, s cs + S f hzr fuc λ f S Λ λ u u S P T > S P X > + X > F S + q P T F P X + X > S S ο ο e E T X e E X f X S X P K X q P K X k q q k k + k Ufrm DeMvre f F X E X s T U, ω ω ω ω λ Eel λ λ f λe E X Vr X m λ λ Eel Memryless Prery P X > s+ > s P X > Mkehm: + Bc fr >, B >, c >, > B Gmerz Webull: k fr, k >, > + + S y y y f y y + Cerl Re f Flure, + m S y y S y y + + C:\Users\JMLJR\Dcumes\Hme Offce\JMLJRPersl\SO & Ems\MLC\Suy Shees\suy shee 9.c

LIFE TBLES fr Ufrm, ω l ω q l q l l l w T l w l e : S + + w + w w +.5 + e + e Cs Frce, e + + q q + Ufrm UDD "Ler Ierl" q l l + + q + UDD l.5.75 l +.5 l Hyerblc.75 +.5 + + l l l + P K X k q q cerl re f flure m Prcles + k k + k k k+ l L l l S l l S cve: + l l + l ls + l l l + q q q l l S l l l m + + + + + + l l q q l l + + + m m + m + + m + S + Cs l q f L l T L L T T + + + + + S T l T s "Esure" "Lfe yers Lve fer ge " T l T e l l w e w E X + l l l l l L T T l e e e e e + e l l l + + e E K X : : k : k + k k + + k 3 3 3 4 5 k fr Vr e use E K X k K k + + + + k k k.5 +.5 + + e + + + e + + + + + K fr cuus scree esme e e usg ler er l/udd use uuur e e + e e Frcl ge Ierl < < Ufrm UDD "Ler Ierl" sq q q q e e + l l + l q, s+ q + + s + Cs Frce + C:\Users\JMLJR\Dcumes\Hme Offce\JMLJRPersl\SO & Ems\MLC\Suy Shees\suy shee 9.c q sq e Hyerblc q + s + q s + q l

INSURNCE : + + + : : + : : e e e E e + + Eel Cs Frce : e E e + + + : m e Vr Z b m υ m + m : + m + υ + υ υ + υ + υ υ + : + : + : q q I I D q ω D ω + : UDD ssume fr ll I I X Berull, b, Vr X b Ufrm / DeMvre ω : ω ω Geerlze De Mvre l e α + l Perceles Z e e Prcles : e : e υ VR Z e q + Relshs Bewee Isurces: m α E + m + m m : υ e k+ + + E q : : + k + k k E E E E E E E m m + m 3 35 35 45 Recurss: υ Eel + + υ υ q + υ υ q + υ I + υ I + : + : : : + : Cuus I I Cuus Dscree Relshs: UDD ssume fr ll Cuus Dscree Cuus Dscree C:\Users\JMLJR\Dcumes\Hme Offce\JMLJRPersl\SO & Ems\MLC\Suy Shees\suy shee 9.c

INSURNCE Symbls De Mvre E : ω 95, 5, 5,.3 Cs Frce / Eel E :.7,., 5 ω, ω, 7.3.46 7.3.38 7 7.3 ω + 7.69 4 ω, : 5.3.75 : ω 7.7.3684 +.7 +..7.58 +.7 +.4 +.7 5.7 +. e e.347 +.7 +., : :, υ υ + 5.69.379 ω ω 7 5 7.3.3.455 ω.3 7 +.7 5.7 +.4 e e.36 +.7 +.4 +.7 5.7 +. e e.347 +.7 +. :, : : : 55 + +.6748 : +.347+.578.449 5.3 5 : 7.3 7 E ω + υ ω 55.543 5.3 7 e : 5.7. e.578 + + C:\Users\JMLJR\Dcumes\Hme Offce\JMLJRPersl\SO & Ems\MLC\Suy Shees\suy shee 9.c

NNUITIES T υ + + T Vr + e lm ω Eel : De Mvre + + ω : : : υ + + : && && υ && && && E && && && E BUT Recurs: + + : + : + : : && && υ + BUT! Beer υ && chge mmee ue Prcles m m && + υ && + Mhly && && + m T υ T E T f f Symbls De Mvre E : ω 95, 5, 5,.3.9 && && ω,, : : ω 7.3 7.5.9 +.75 +.543.7.9 : : : Cs Frce / Eel E :.7,., 5 5.63 +.7 +. 5.7. + + e e 4.9587 +.7 +. C:\Users\JMLJR\Dcumes\Hme Offce\JMLJRPersl\SO & Ems\MLC\Suy Shees\suy shee 9.c

PREMIUMS && υ P E L E L Vr L b+ Vr L P P + V P : Prcles P : OR OR + Equvlece Prcle PV Premums PV Beefs E L Equvlece Prcle PV Premums PV Beefs P P T Lss L bυ P E L Vr L E L P Vr L Vr L P : : P + k+ l l l k l k && + + + k+ chge b P Eel P && Vr L + P + P + + + P P + P : Lss υ π slve fr π Sem-Cuus && T jus Bewee Dscree Sem-Cuus uer UDD P P P P P P P use jus P C:\Users\JMLJR\Dcumes\Hme Offce\JMLJRPersl\SO & Ems\MLC\Suy Shees\suy shee 9.c

V, E V : Reserve surce remum remum uy V P + + && + k + V V k && + + RESERVES k Vr L T > b V 'e mu rsk' P P P V : : Prsecve Reserve me V PV Beefs PV Pr emums V P V P&& + + k + k + k Rersecve reverse beef/remum subrc rer ve by && P k : k : : V V E P&& s k k k : k : k : E E k π + V + + bq Recurs + + + V V π q υb υ V + + + Herf Vr Λ K υ b V 3 3 q + + Vr L K Vr Λ K + υ Vr L K 3 3 + V + π + bq V V + π + V b V q + + + V + π + b q 3 E: V 3 V + π q υb + υ V 3 3 V + π + V 3 beefs ssue remums ssue + + 3 3 + Prcles Reserves re efe rsecvely Prsecve Meh lwys Wrks Rersecve Wrks IF Premums eerme by Equvlece Prcle : PV : V Ps : P E PV E Summry f Vrces Isurce uy Lss E b V q INS INS INS P b INS INS b+ C:\Users\JMLJR\Dcumes\Hme Offce\JMLJRPersl\SO & Ems\MLC\Suy Shees\suy shee 9.c

MULTIPLE LIVES + y y q q + E e e e e : : y y y y y y y y + + q + q q + q y y y y y y y y y y y y y y y BUT!!! slve + + + + P + P P + P chge P eher r y y y y y y y y y + y E E E v v E, y + + y y y y y y y y q q q e + e e + e e De Mvre wh ervl leghs < b ey b K b 6b * * * * y y y y + λ + λ + λ + + λ Cmm Shck + e + e e e + e e y y y Cmm Shck em quess CS λ lwys Eel, usully ls survvr, clue CS whle, eclue CS + λ + λ * * * * y y y y * * * * y y + λ + λ + λ + + λ e e e e y y y y * * + y y λ + y+ λ e e e y y * * * * y y y y + λ + λ + λ + + λ + e + e e e + e e y y τ τ && && E&& && && + E&& : + : + τ τ q esy fr sgle ecreme + + Prcles y y BOTH ND y survve MIN T, T y "J Sus" EITHER OR y survves MX T, T y "Ls Survvr" y + + y y C:\Users\JMLJR\Dcumes\Hme Offce\JMLJRPersl\SO & Ems\MLC\Suy Shees\suy shee 9.c

MULTIPLE DECREMENTS ll emles wh ecremes; hree r mre, sme rcles + ww + ww + τ q q e e + + + τ τ τ + ww + w+ + ww + ww + ww τ τ q q q + q q e e e e q q q τ τ q τ l q τ τ τ τ + + + + l l q + + q E 3 3 q q q q 3 fr q UDD Sgle Decreme ble q q q q + 3 q.3 +. q τ τ q τ UDD Mulle ecreme ble q q Mul Decr Frc Yr E Prcles q q.3q.q τ l l τ q C:\Users\JMLJR\Dcumes\Hme Offce\JMLJRPersl\SO & Ems\MLC\Suy Shees\suy shee 9.c

INSURNCE WITH EXPENSES r r G&& F + c G + F + e c G F e + + + fuure beefs + fuure beefs eese le remums PV PV PV Prcles G Eese Le Premum Equvlece Prcle Eese Le PremumIsurce+Eeses G e me eeses + frs yer eeses & beefs + gg eeses & beefs T Res Be Res + E Res E Lg Eese Le Premum - Beef Premum G P e sse Shres, E: S e V + e Gc e + w S + c G e + bq CVq w q q w w sse Shres, E: S + c G e + bq + CVq + q q S Eese Reserve: e + V Eeses Yer % f Premum Per, Per Plcy ly c r e + c r e Geese le remum; remums; surce; Ffce mu POISSON PROCESSES λ e λ λ e P X E X Vr X λ M X e! E N E X b λ λ,, b Pss Prcess m, m λ u u E S + Vr S E N Vr X Vr N E X E N E X Prcles Pss Cug Prcess N λe X λ Number f Clms PssPrcess Severy f Clms X S X N C:\Users\JMLJR\Dcumes\Hme Offce\JMLJRPersl\SO & Ems\MLC\Suy Shees\suy shee 9.c

Summry Isurces, ues, Premums, Reserves Premums P P P Isurces ues P P+ P+ Reserves Isurce - ues + + V V V P OR OR P Vr L T > + + b + C:\Users\JMLJR\Dcumes\Hme Offce\JMLJRPersl\SO & Ems\MLC\Suy Shees\suy shee 9.c