A(t) is the amount function. a(t) is the accumulation function. a(t) = A(t) k at time s ka(t) A(s)

Transcription

1 Interest Theory At) is the aount function at) is the accuulation function at) At) A) k at tie s kat) As) at tie t v t, t, is calle the iscount function iscount function at) k at tie s kv t v s at tie t Uner copoun interest v t + i) t i is the annual effective rate of interest + i is the one year interest factor ν is the one year iscount factor is the annual rate of iscount i ν, i ν i + ν, + i ν, iν, ) + i) i ) is the noinal rate of interest copoune ties a year ) is the noinal rate of iscount copoune ties a year ) ) + i + i) ) ) The force of interest is t t lnv t) t ln at) a t) at) t ln At) A t) At) v t e R t s s, at) e R t s s Annuities The cashflow, present an future values of an annuity ue with level payents of one are: Contributions Tie n n ä n i νn an s n i + i)n The cashflow, present an future values of an annuity ieiate with level payents of one: Contributions Tie n

2 a n i νn an s n i + i)n i i The cashflow an present value of an perpetuity ue with level payents of one are: Contributions Tie ä i The cashflow an present value of a perpetuity ieiate with level payents of one are: Contributions Tie a i i The cashflow an present value of a geoetric annuity ue with first payent of one are: Payents + r + r) + r) n Tie n an Gä) n i,r ä n i r +r The cashflow an present value of a geoetric annuity ieiate with first payent of one are: Payents + r + r) + r) n Tie 3 n Ga) n i,r + r a n i r +r The cashflow an present value of a geoetric perpetuity ue with first payent of one are: an an Payents + r + r) + r) n Tie n Gä) i,r { +i i r if i > r, if i r The cashflow an present value of a geoetric perpetuity ieiate with first payent of one are:

3 an Payents + r + r) + r) n Tie 3 n { Ga) i,r if i > r i r if i r The cashflow, present an future values of a ue increasing annuity with first payent of one are: Payents 3 n Tie n Iä) n i än i nν n an I s) n i s n i n The cashflow, present an future values of an ieiate increasing annuity with first payent of one are: Payents 3 n Tie 3 n Ia) n i än i nν n an Is) i n i s n i n i The cashflow an present value of an increasing ue perpetuity with first payent of one are: Payents 3 Tie an Iä) i The cashflow an present value of an increasing ieiate perpetuity with first payent of one are: Payents 3 Tie 3 an Ia) i i The cashflow, present an future values of a ecreasing ue annuity with first payent of one are: Payents n n n Tie n Dä) n i n a n i an D s) n i n + i)n s n i The cashflow, present an future values of a ecreasing ieiate annuity with first payent of one are: 3

4 Payents n n n Tie 3 n Da) n i n a n i an Ds) i n i n + i)n s n i i The cashflow, present an future values of a ue annuity pai ties a year are Contributions Tie in years) ä ) n i νn ) + n an s ) n i + i)n ) The cashflow, present an future values of an ieiate annuity pai ties a year are Contributions Tie in years) + n a ) n i νn an s ) i ) n i + i)n i ) The present value of a continuous annuity with rate Ct) is t Cs)v s s The present value of a continuous annuity with constant unit rate is a n i v t t vn The present value of an annually increasing continuous annuity is Iā) n i [t + ]v t t än i nv n The present value of a continuously increasing annuity is Īā n )n i tv t t ān i nv n The present value of an annually ecreasing continuous annuity is Da) n i [n + t]v t t n a n i The present value of a continuously ecreasing continuous annuity with is Da )n i n t)v t t n a n i n 4

5 Survival oels The cuulative istribution function of the rv X is F X ) P {X }, R The survival function of the nonnegative rv X is S ) s) Pr{X > }, If h an H) ht) t,, then E[HX)] st)ht) t In particular, E[X] If X is a iscrete rv st) t, E[X p ] E[HX)] st)pt p t, E[inX, a)] Pr{X k}hk) Hk )) In particular, for a positive integer a, E[X] Pr{X k}, E[X ] Pr{X k}k ), E[inX, a)] a st) t a Pr{X k} ) is calle a life age T ) T X is the future lifetie of ) The survival function of T ) is t p s+t), t The cf of T ) is s) tq s) s+t), s) t We have that tq t p, p p, q q, s t q Pr{s < T ) s + t} s p s+t p s p tq +s, +np p np +, n p p p + p +n, P k j n j p n p n p +n n3 p +n +n nk p + P k j n j The force of ortality is µ) µ ln S X) f X) S X ) ) S X ) ep µt) t, t p e R +t µy)y, f T ) t) t p µ + t) e E[X] e :n E[inT ), n)] tp t, e E[T )] tp t, tp t, e e :n + n p e+n t is the least integer greater than or equal to t, t k if k < t k K is the tie interval of eath of a life age K) is the curtate uration of eath of a life age, ie the nuber of coplete years live by this life K T ), K) K, K) T ), e E[K)] kp, E[K)) ] k ) kp, e p + e + ), e :n kp, e e :n + n p e +n, e :+n e : + p e+:n 5

6 For e Moivre s law: f X ) ω, S X) ω ω, µ), for < ω, ω tp ω t ω, tq t ω, t ω, e ω, VarT )) ω ) Uner constant force of ortality µ:, e ω, VarK)) ω ) S X ) e µ, F X ) e µ, f X ) µe µ, µ) µ, for >, tp Pr{T ) > t} e µ, e :n e µn µ 3 Life tables s + t) s) e µt,, VarT )) µ, e p, e :n p p n ) q q, VarK)) p q l enote the nuber of iniviuals alive at age The nuber of iniviuals which ie between ages an + t is t l l +t The nuber of iniviuals which ie between ages an + is l l + We have that s) l, F X ) l l, µ) l logl ), l tp l +t, t q l l +t l l t l, p l + l, q l l + l l e, e l n +t t, l +t e :n t, e l l l l, n q l +n l +n+ l l +k l, e :n l +k l The epecte nuber of years live between age an age + n by the l survivors at age is n L nl l e:n l +t t, L L l e:, e k L k l, e :n +n k L k l Interpolation l +t tp L unifor istribution of eaths l + tl + l ) tq l +l + eponential interpolation l p t p t log p Balucci assuption t) l +t l + 6 p t+ t)p l + log p q

7 Uner unifor istribution of eaths: l +t l + tl + l ), t p tq, f T ) t) q, µ +t L l + l +, e e + Uner eponential interpolation: q tq, t, l +t l p t, t p p t, f T t) p t log p, µ +t log p, t Uner Balucci assuption) haronic interpolation: 4 Life insurance t) + t, t p l +t l l + p t + t)p type of insurance payent whole life insurance Z v K n year ter life insurance Z:n vk IK n) n year eferre life insurance n Z v K In < K ) n year pure enowent life insurance Z:n In < K ) n year enowent life insurance Z :n v ink,n) year eferre n year ter life insurance n Z v K I < K + n) Whole life insurance pai at the en of the year: Z v K, A E[Z ] v k k p q +k, A VarZ ) A A, A vq + vp A + v k k p q +k, n year ter life insurance pai at the en of the year: Z:n v K IK n), A :n E[Z:n ] v k k q, A :n VarZ :n ) A :n A :n, A :n vq + vp A +:n v k k q, n year eferre life insurance pai at the en of the year: n Z v K In < K ), n A E[ n Z ] v k k q, n A v k k q, kn+ kn+ Var n Z ) n A n A, n A v n n p q +n + n+ A 7

8 n year pure enowent life insurance pai at the en of the year: Z :n v n In < K ), A :n E[Z :n ] n E v n np, A :n v n np, VarZ :n ) A :n A :n n year enowent life insurance pai at the en of the year: Z :n v ink,n), A :n n E E[Z :n ] v k k q + v n np, A :n v k k q + v n np, VarZ :n ) A :n A :n n A n E A +n, A A :n + n A A :n + n E A +n, A A :n + n A, A :n A :n + A :n, A :n A :n + A :n, Increasing/ecreasing life insurance pai at the en of the year: IA) kv k k q, IA) :n kv k k q, DA) :n n + k)v k k q Uner e Moivre s oel with terinal age ω, if ω,, n are a positive integers, A a ω i ω, A :n a n ω, A :n v n ω n ω, n A v n a ω n i ω Uner constant force of ortality: A q q + i, A :n e nµ+), n A e nµ+) q q + i, A :n e nµ+) q ) q + i type of insurance whole life insurance payent Z v K n year ter life insurance Z :n v T IT n) n year eferre life insurance n Z v T In < T ) n year pure enowent life insurance Z :n v n In < T ) n year enowent life insurance Z :n v int,n) year eferre n-year ter life insurance n Z v T I T + n) Whole life insurance pai at the tie of eath: Z v T A E[Z ] A E[Z ) ] v t f T t) t, v t f T t) t, VarZ ) A A 8

9 n year ter life insurance pai at the tie of eath: Z :n v T IT n), A :n E[Z :n ] v t f T t) t, A :n E[Z n :n ] v t f T t) t, VarZ :n ) A :n A :n n year eferre life insurance pai at the tie of eath: n Z v T In < T ), n A E[ n Z ] n v t f T t) t, n A E[ n Z ] v t f T t) t, Var n Z ) n A n A n year enowent life insurance: Z :n v int,n), A :n E[Z :n ] A :n E[Z :n ) ] n v t f T t) t + v n Pr{T > n}, v t f T t) t + v n Pr{T > n}, VarZ :n ) A :n A :n Z Z :n + n Z, A A :n + n A, A A :n + n A, Z :n Z :n + E :n, A :n A :n + A :n, A :n A :n + A :n, n A n E A +n Uner e Moivre s oel with terinal age ω, A a ω i ω, A :n a n i ω, A Uner constant force of ortality: :n e n ω n ω, n A e n a ω n i ω A µ µ +, A :n e nµ+), n A e nµ+) µ µ +, A :n e nµ+) µ ) µ + Continuously increasing whole life insurance: b t t, t, I A ) tv t tp µ +t t by Annually increasing whole life insurance: b t t, t, present value is enote I A ) k k kv t tp µ +t t 9

10 n year ter continuously increasing whole life insurance: b t t, t n, ) n I A tv t :n tp µ +t t n year ter annually increasing whole life insurance: b t t, t n, I A ) :n k k kv t tp µ +t t Continuously ecreasing life insurance: b t n t, t n, ) n D A n t)v t :n tp µ +t t Annually ecreasing life insurance: b t n t, t n, D A ) :n k k Assuing a unifor istribution of eaths: n + k)v t tp µ +t t A i A, A :n i A :n, n A i n A, A :n i A :n + A :n, A ) i i A, A ) ) :n 5 Life annuities i i ) A :n, n A ) i i ) n A, A ) :n i i ) A :n + A :n ue annuities present value APV whole life Ÿ ä K Z ä A n year eferre life insurance n Ÿ v n ä IK K n > n) n ä n E ä +n n year ter Ÿ :n ä inkn,n) Z :n ä :n A :n ieiate annuities present value APV whole life Y a v Z K a v A n year eferre life insurance n Y v n a K n IK > n + ) n a n E a +n n year ter Y :n a ink,n) v Z :n+ a :n v A :n+ continuous annuities present value APV whole life Y a T Z a A n year eferre life insurance n Y v n a IT T n > n) n a n E a +n n year ter Y :n a int,n) vint,n) a :n A :n

11 Discrete whole life ue annuity: Ÿ ä K Z, ä A Whole life ieiate annuity: v k kp, VarŸ) k Y a K Ÿ v Z, a v A A A VarY ), a vp ä + vp + a + ) Whole life continuous annuity: Y a Z T, a A n year eferre iscrete ue annuity: n Ÿ v n ä K n IK > n), n ä n year eferre iscrete ieiate annuity: A A, ä + vp ä + v k kp, v t tp t, VarY ) v k kp n E ä +n kn n Y n+ Ÿ, n a n+ ä vp n a + n year eferre continuous annuity: n Y v n a T n IT > n), n a n year ter ue iscrete annuity: n v t tp t n E a +n Ÿ :n ä inkn,n) Z n :n, ä :n v k kp A :n, A :n A :n ) VarŸ:n ), ä :n+ ä :n + n E ä +n:, ä ä :n + n ä ä :n + n E ä +n n year ter iscrete ieiate annuity: k Y :n a ink,n) Ÿ:n+ v Z :n+, a :n ä :n+ v k kp v A :n+ A :n+ A :n+ ) VarY :n ), a n a + a :n n a + n E a +n, A A

12 n year ter continuous annuity: Y :n a int,n) vint,n) VarY :n ) A :n A :n ) Uner constant force of ortality: ä Z :n, a :n v s sp s A :n,, a :n+ a :n + n E a +n:, a a :n + n a + i vp q + i e, a +µ) vp q vp q + i Annuities pai ties a year For a whole life unity annuity ue to ) pai ties a year: e +µ) e, a +µ) µ + Ÿ ) Z), ä ) ) A) ) k v k k p, VarŸ ) ) A ) A ) ) ) ) For a whole life unity annuity ieiate to ) pai ties a year: Y ) a ) Ÿ ) ä ) VarY ) ) v/ Z ), ) v/ A ) ) A ) A ) ) ) ) v k k p, For a n year unity annuity ue to ) pai ties a year: :n Z) :n, ä ) :n A) :n Ÿ ) ) :n ) VarŸ ) :n ) VarZ) ) ) ) n k v k p, For a n year unity annuity ue to ) pai ties a year: :n Z) :n, ä ) :n A) :n Ÿ ) ) ) n k v k p, VarŸ ) :n ) :n ) VarZ) ) ) For a n year unity annuity ieiate to ) pai ties a year: Y ) :n Ÿ ) :n + Z :n, a ) :n ä) :n + ne

13 For a n year eferre unity annuity ue to ) pai ties a year: n Ÿ ) Z ä ) :n n Z ) ), n ä ) A :n n A ) ) ä ) :n + n ä ) ä ) :n + ne ä ) +n kn v k k p n E ä ) +n, For a n year eferre unity annuity ieiate to ) pai ties a year: n Y ) a ) n Ÿ ) Z :n, n a ) n E a ) +n n ä ) n E, a ) :n + n a ) a ) :n + ne a ) +n Uner an unifor istribution of eaths within each year: ä ) i A i ) ), a ) 6 Benefit Preius ä ) v/ i Fully iscrete insurance A i ) ), a i A Whole life insurance: L v K P ä K Z P Ÿ Z P Ÿ Z + P ) P, E[L ] A P ä A + P ) P, VarL ) + P ) VarZ ) + P ) A A ) Uner the equivalence principle: P A ä A A ä, VarL ) n year ter insurance: A A A ) A A ä ) L :n Z:n P Ÿ:n Z:n P Z :n,, t P A ä :t P :n P A :n ) A :n ä :n, t P :n P t A :n ) A :n ä :t 3

14 n year pure enowent: L:n Z:n P Ÿ:n Z:n P Z :n, P:n P A:n ) A :n, t P:n P t A ä :n ) A :n :n ä :t n year enowent: L :n v inn,k) P ä ink,n) Z :n P Ÿ:n Z :n P Z :n VarL :n ) + P ) VarZ :n ) + P ) A :n A :n ) ), + P ) Z :n P, P :n P A :n ) A :n, t P :n P t A :n ) A :n, ä :n ä :t VarL :n ) + P ) :n A :n A ) A :n A :n :n ) A:n A :n A :n ä:n ) n year eferre insurance: Properties: Whole life insurance: n year ter insurance: n Z P Ÿ, P n A ) n A, t P n A ) n A ä ä :t P :n P :n + P :n, n P P :n + P :n A +n Seicontinuous annual benefit preius P P A ) A a, t P t P A ) A a :t n year pure enowent: P :n A :n a :n, t P :n t P A :n ) A :n a :t P :n P A:n ) A :n, t P :n t P A a :n ) A :n :n a :t 4

15 n year enowent: n year eferre insurance: Whole life insurance: P :n P A :n ) A :n a :n, t P :n t P A :n ) A :n a :t P n A ) n A a :n, t P n A ) n A a :n Fully continuous insurance LA ) v T P a T Z P Y Z Z VarLA )) Z + P ) P, + P ) VarZ ) + P ) A A ), P A ) A A, t P A ) A a A a a :t VarLA )) + P A ) ) A A ) A A A ) A A a ) n year ter insurance: n year pure enowent: L Z :n P Y :n, P A :n ) A :n, t P A a :n ) A :n :n a :t L Z :n P Y :n, P A:n ) A :n, t P A a :n :n ) A :n a :t n year enowent: L Z :n P Y :n Z :n P Z :n VarL) + P ) A :n ) ) A :n, + P ) P Z :n, P A :n ) A :n a :n a :n a :n A :n A :n, VarL) A :n A :n A:n ), t P A :n ) A :n a :t n year eferre insurance: L n Z P Y :n, P n A ) n A a :n 5

16 n year eferre ue annuity: n year eferre ieiate annuity: n year eferre annuities L n Ÿ P Ÿ:n, P n ä ) n ä ä :n L n Y P Ÿ:n, P n a ) n a ä :n n year eferre continuous annuity fune iscretely: L n Y P Ÿ:n, P n a ) n a ä :n n year eferre continuous annuity fune continuously: L n Y P Y :n, P n a ) n a ä :n 6