《保险精算20140901》由会员分享，可在线阅读，更多相关《保险精算20140901（52页珍藏版）》请在金锄头文库上搜索。
1、 0 5 x 6 4 40 n 60 fe xgduan PPP FT t sT t oL eI A C T nd 5 x 6 4 40 n 60 fe xgduan PPP FT t sT t oL eI A C T nd f 1 l 1 20 u 4 5 30c b 5 XJl 12cc m zc zcI 30c c7 2 n370 10 000 5c x x73k cc G b d N l 0 6931 ln2 i 5 b x 3 g5 O KJ 29 K V 5 I G a y3d What is the value today of a random sum of money whi
2、ch will be paid at a random time in the future 5 p G0 n G contingent payment contingent dependent on an uncertain event or circumstance that may or may not occur fo f xn 1 7a k md 8U 1 a u5c 1 a m 7a 1 k 2 x7 G U G U G G m X uc 2 5c x G u n x 5cS k XJ n5cSk G m n k cc o x x L 5 5 5 U E K p 5 5 u E 5
3、 9 du g E 5 x x x LV O 7K En 9O E x 5 1 n n x 9 x x i G l S x J n SN do 13 1 n 1n o K 1 n 0 N 5 1 o y Vg x x t t 0 X L 0 Lt V 5 p P k s t L Y C k force of mortality Y C k C X Y C Ak P X t t 0 X t lim t 0 P t t t t 0 1 k X t A t Ne mk U5 N w X t fX t sX t t 0 fX t X t sX t t 0 d X t d dt log sX t t 0
4、 1 2 1 sX t exp Z t 0 X s ds fX t X t exp Z t 0 X s ds x A I b E X rX x d dx log Z x sX t dt 1 x 0 l C k l C k b X l C V a 1 a2 p1p2 0 a1 a2 t Y aj t sX aj sX aj 1 Y aj t 1 X aj 3 pj X aj sX aj 1 X aj j 1 Y k 1 1 X ak e Vg x N m 1 e0 E X 2 c e0 E K 0 1 2 2 e0 Z 0 sX t dt E X2 Z 0 2tsX t dt e0 X n 1
5、sX n E K 0 2 X n 1 2n 1 sX n y e0 Z 0 yfX y dy Z 0 Z y 0 1dtfX y dy Z 0 Z t fX y dydt E X2 Z 0 P X2 t dt Z 0 P X t dt Z 0 2tSX t dt y Ppk P K 0 k k 0 1 2 e0 X k 1 kpk X k 1 k X n 1 1 pk X n 1 P K 0 n X n 1 P X n X n 1 sX n E K 0 2 X k 1 k2pk 2 X k 1 1 2k k 1 pk e0 2 X k 1 k X n 1 n pk e0 X n 1 2nsX
6、n e0 S 1 X U 0 O k 9 C 2 X Exp O 9k A k 1 de Moivre 1729 k w 0 t 1 w t 0 t 0 C 1 t BCt t 0 A s t exp Z t 0 BCsds exp B logC Ct 1 3 Makeham 1860 k B 0 A B C 1 t A BCt t 0 A s t exp At B Ct 1 log C 4 Weibull 1939 k k 0 n 0 t ktn t 0 A s t exp Z t 0 ksnds exp k n 1t n 1 x N A P x L Lx b x N T x L x K x
7、 S x OL T x 9 FT x sT x T x L x T x A k 5 x A b L x X x dT x FT x t P T x t P X x t X x 1 sX x t sX x sT x t sX x t sX x fT x t fX x t sX x T x t fT x t sT x t X x t x 0 m X 1 3 1 FT x t 1 sX x t sX x sT x t sX x t sX x fT x t fX x t sX x T x t X x t E T x E X x X x Z 0 tfT x t dt R 0 tfX x t dt sX
8、x Z 0 sT x t dt Z 0 sX x t sX x dt R x sX t dt sX x S 1 X U 0 0 b x s T x t P T x t s s 0 y P T x t s T x t P X x t s X x t P X x t s X x t P T x t s y P T x t s T x t sT x s t sT x t exp n Rs t 0 X x u du o exp n Rt 0 X x u du o exp Z s t t X x u du exp Z s 0 X x t u du P T x t s L IS P L V Nk tpx
9、x Lc x t V x 2 tc V tqx x 3 5tcSk V u tqx x 3c x u x u t k V x Lx u 3 e5tcSk V t 1 Xe zL px 1px qx 1qx u qx u 1qx d w tpx sT x t P T x t tqx FT x t P T x t 1 3 3 1 tpx sX x t sX x 2 t 0 u 0 tqx 1 tpx u tqx upx tqx u u tqx upx u tpx 3 0 h x y sX x r 0 r x exp Z x 0 du r u y 5 r x Z 0 sX x t sX x dt R
10、 x sX t dt sX x 1 r x sX x R x sX t dt d dx log Z x sX t dt S d log Z x sX t dt log Z 0 sX t dt Z x 0 du r u C C 0 log Z x sX t dt logr 0 Z x 0 du r u Z x sX t dt r 0 exp Z x 0 du r u KKK JJJ b 4 Ndl0 N 9 K N 8Cz k X T N 5 Cz u b N k V b Nzc k k d u k b J u N b A n y 9k u 1 1eXi x K 0 l x L T N Lx K
11、l x l0sX x tDx tdx OL x x t mSk 9 tDx l0 X i 1 I x Xi x t tdx lx lx t d Pdx 1dx 1 4 1 1 tpx lx t lx tqx tdx lx 2 dlx dx lx X x lx t lxe Rx t x X u du ndx Z x n x ly X y dy 9 fff y Z t 0 slx s X x s ds tlx t Z t 0 lx sds R t 0 lx sds L l0 N3c x x t Oo m 1 Lx t Nklx t g3c x x t tc d N3dc o m tlx t 2 3
12、x s k dlx s lx s X x s ds 3 c x x t m slx s X x s ds d 3c x x t k c 0 05 f T1 0 T1600 0 OL 1600 5 m T1 10 T540 10 L 540 10 N 5 m K Y1 1600 X i 1 I Ti 70 Y2 1600 X i 1 I Ti 10 60 E Y1 160070p0 1600lx l0 160026 40 1040 E Y2 54060p10 1600 lx l10 54026 39 360 var Y1 1600var I T1 70 160070p0 1 70p0 364 v
13、ar Y2 540var I T1 10 60 54060p10 1 60p10 120 f P E Y1 E Y2 1400 2 var Y1 var Y2 484 N 2 CqY1 Y2 P Y1 Y2 c P Y 1 Y2 c P Y 1 Y2 c 1400 22 c 1400 22 0 05 IO dd c 1 645 22 1400 1436 2 5P d 5O N 5 c ex ex X n 1 npx X n 1 lx n lx L 3 x f O 7 J I x k V 3 xxx kkk VVV ddd LLL xxx L x N 5 5 L k P E 3z c k V L
14、 c N3 5k Cz L L A P kkk nmx nqx Rn 0 tpxdt P mx 1mx A P a x R1 0 ttqx T x t dt qx R1 0 tfT x t dt qx tLx Z t 0 lx sds Lx 1Lx Tx Lx Yx Z 0 Tx sds g P L A 1 5 1 a x E T x T x 1 nmx ndx nLx 1 5 2 nmx nLx lx lx n tLx lxE T x t Tx lx ex L fff y Lx a x lx 1 a x lx 1 a x lx lx 1 lx 1 a x dx lx 1 AO e a x 1 2 K Lx lx lx 1 2 g
