8'(x) -I - UCLA

1 Chapter 3: Survival Distributions and Life Tables Distribution function of X: Force of mortality flea:): Fx(:r) = Pr(X S; :1;). /1(:1:). Survival function B(.1:): 8' (x). sex). Relations between survival functions and Probability of death between age :r and force of mortality: age y: Pr(.r < X S; z) ( (z) - Fx (:1:). - B(Z). exp ( -I "(Y)d ll ). Probability of death between age age y given survival to age :r:: and nPx exp (. - ! x+n p.(y)ely ). Pr(:1; < X S; zlX > Derivatives: d dt t(jx d Notations: dt tPx tPx . It (:r: + t). d tlJx PriT(.r) tl -T. dt '" prob. (3:) dies within t years d distribution function of T(a:) -L. dt x tPx Pr[T(:c) > t] d -1. attains age ;1; +t Mean and variance of T and ](: E[T(:r)] complete expectation of life Pr[t < Tel') t + 'Ill t+ul]x - t])a' t+u])x t(jx =./. o tP:B elt tPx' u(]x-t-t curtate expectation of life ex). Relations with survival functions: 00.

2 :T(:r:) ] . o ,J. t . tPx u,t , 2. - ex 00. Curtate future lifetime (K(:r) greatest Vo:r[K(.r}] 2)2k -1) kP:r 2. e". integer in T(x)): k=l Pr[K(.l') k] Pr[k T(:r) < k + 1] Total lifetime after age .r: Ta k]Jx k+lPx ex;. kP", . qx+k T-r: ./ lx+t dt klJx o Exam rv[ Life C;onting;;;ncicH - LGD@ 1. ~, Varying benefit insul'ances: Interest theory reminder 1 vn (IA)x = ./It + IJlIt, tPx !/'x(t)dt am 1 l,n i 0fll 0 n'fll 5 8../It +. 11. 1 1 1. (L4)~:m IJI/ ' t1 Jxlt x(t)dt /5 '. 00Cl - i d 0 - nvn (X) (Ia)fll (IA)", ./ t ' l,t , tP:r p'o,(t)dt 0. 11 1. (IO)OCl (IA);"fll ./ t ' 7,t , tPx It x (t)dt 52. (n + l)Ofll (Ia)fll + (Da)m 0. 11. - 5. CD"4.);':fll ./(n ItJ ' tPx fJ,x(t)dt i'IJ. 0 1 1 +i T1. id 12. (DA);':fll ./(n - t)vt , tPdl'x(t)dt 0. Doubling the constant force of interest 5. (IA)x Ax + VP:L,(1A)x+l lIqx + 1)1'rr' + 1 +i -4 (1 + i)2. v 1)2. (DA)~:fll nvqx + vpx(DA)x~l:n_ll -4 2i + i2.

3 (IA);:fll + (15A);:fll d --+ 2d d 2. (IA)~:fll + (D)l)~:fll = (n + l)A;:fll i 2i + i 2 (IA )~:fll + (DA)~:fll (n + l)A~;m -+. 5 25 Accumulated cost of insurance: Limit of interest rate i = 0: A o, 1. A~:fll nqx n!Ax 11}1X. Share of the survivor: Ax:fll 1 mlnqx accumulation factor (JA)x 1 +e:r: (IA)x eo, Exam l'vl - Life $ - LC;D'V 4. Chapter 5: Life Annuities Whole life annuity: ax Recursion relations J. 00. Elan] at! t]Jx + t)dt +. o + nl 00 ,x,). 1 +vpx Jvt'tPxdt o J. o tEx dt 2. 1 + v Px 1l or [an]. n-year temporary annuity: (Iii)x n n J. o v t . tllx dt = J. 0 Whole life annuity due: 0,,;. 1l oriY] 00. E[ii K+lll L 11k. kPx '..=0. n-year deferred annuity: Yor[oK+lll J JtE~,dt rAJ OC. 1,t . tPx dt n n n-yr temporary annuity due: '11-1. 2. Vor[Y] aX!n) E[Y] = Lv k . k]lx k=O. n-yr certain and life annuity: n-yr deferred annuity due: + na,x +. ex). E[Y] = L.

4 KP". Most important identity k=n 1 ba'T + )Ix 1 )Ix 1 ba'x:111. n-yr certain and life due: ii'x:111. 1 - (2b). d 1 Ax:111 0111 +L v k . kllx d k=n 1 (lii J ;:111 + +n,O'T. Exam f,/l - Life COlltingencieh 5. Whole life immediate: ax Accumulation function: =L . ~'P2: 11. 1. k=1. =/-1o m-thly annuities Limit of interest rate i 0: ;=0. ax ---+ ex [Y] ii,x 1 + c 2: II x ex 1. rn. :11I cx:rrl 1 ;=0. (ra) .. (m). -(I ---+ 1+. o'x:nl ax:-:m 'm ex:rrl 6. Chapter 6: Benefit Premiums Loss function: h-payment insurance premiums: Loss PV of Benefit,s - PV of Premiums Fully continuous equivalence premiums (whole life and endowment only): P(Ax). ii", A,,;. (L4 x P(A",) x:h\. 1. 1. P(A:r) =- -6. [. (l,;r \/ar[L] (1 + ~r . 2]. (A,,:). Pure endowment annual premium PJ::~: it is the reciprocal of the actuarial accumulated value because the share of the survivor who Var[L] has deposited P:r:4 at the beginning of each year for n years is the contractual $1 pure endow.)))

5 Var[L] ment, Fully discrete equivalence premiums (1). (whole life and endowment only): P minus P over P problems: The difference in magnitude of level benefit pre miums is solely attributable the investment P(A,,:) Px feature of the contract. Hence, comparisons of dAa: the policy values of survivors at age :/: + n lllay P(Ax). 1- Ax he done by future benefits: 1. P(Ax) d ( pr ax l'". ( n Px - P x:nl)8 x :m VadL] 1+ d [ (A,,:) 2]. 2Ax (Ax? \/ar[L]. (dii.".)2 <Ax - (Ax)2. \/ar[L] =. (1- A."Y lVIiscellaneous identities: Semicontinuous equivalence premiums: P(A x :nl ). P(Ax:m) +6. m-thly equivalence premiums: p(m) 1. #. +d tv! LIfe Contin)1;en-C'ies - LGD(':;: 7. /~. Chapter 7: Benefit Reserves Benefit reserve tV: Variance of the loss function The expected value of the loss at time t. Continuous reserve formulas: Vad tL] assuming EP. Prospective: t V(Ax) Ax+t - P(Ax) +t Retrospective: tti(Ax) = P(AT).)))

6 'lT;t]-- Premium diff.: Var[t L ] ass1lming EP. Paid-np Ins.: Cost of insurance: funding of the accumu . Annuity res.: tV(A;t.) = 1. lated costs of the death claims incurred between age ;1: and x + t by the living at t, Discrete reserve formulas: 4E x qT. lkx =. lX+1 ACCU ilmlated differences of premiums: ~Vx - n~::nl). Ax-;-n 0 = A~'+n .~Vx - nVT. - ~Vr AXTm:n-~ - Ax+m h-payment reserves: ~V Relation between various terminal re . serves (whole life/endowment only): hr7' A ). Ie' ~ '"ix:nl = 1 . (I - m~~)(l- nV.,,+m)(l- Exam ?vI - Contil1j2; l lcieb - LG D 8. Chapter 8: Benefit Reserves Notations: br death benefit payable at the end of year of death for the j-th policy year 71J~l: benefit premium paid at the beginning of the j-th policy year bt : death benefit payable at the moment of death 7ft: annual rate of benefit premiums payable continuously at t Benefit reserve: 00 00.

7 HI! = Lbh+j j]Jx+h qx+h-tj - L. j=O j=O V. U ul)x+tfJ~,(t + n)dl1 L 7ft+u V. li U P2'+t dv o Recursion relations: hI! + 7fh l' f]x--;-h . bh-'-l + 11 Px+h . h+1 If (" ~7 + 7fh)(l + i) qx+h . + 11x+h . h+1 If (hI! + (l+i) h+ll! + - h+1 V). Terminology: "policy year h+ 1" the policy year from time t = h to time t = h+1. "h V + == initial benefit reserve for policy year h + 1. terminal benefit reserve for polky year h terminal benefit reserve for policy year h + 1. Net amount at Risk for policy year h +1. ='let Amount Risk \Vhen the death benefit is defined as a function of the reserve: For each preminm P, the cost of providing the ensuing year's death benefit, based on the net amount at risk at age .T + h, is : - h-ti V). The leftover, P - vqx,h(/)h+l - h+IV) is the source of reserve creation. Accullmlated to age :r + 'TI, we have: 71-1. L. h=O. - htl it)] (1 +.

8 N~1. - L1 Hlx+h(bhl-l - 11-'-1 V)(l +. h=O If the death benefit is equaJ to the benefit reserve for the first 17 policy yean, If the death benefit is equal to plus the benefit reserve for the fiTst n policy years 71-1. nV=V~m- L +. h=O. Exam R Life Contingencies - LGD(:) 9. If the death benefit is equal to $1 pIue; the benefit reserve for the first n policy years ane qxlh == q COllfltant nV =. Reserves at fractional durations: (h1/ + 7Th)(l + sPx+h' h~:.sV +. UDD '* (hll + 7T h)(l +. V +8' his 1/). h+.sV Vl~8 . I-sqx+h+s . bh + 1 + . l-sPx+h+s . UDD '* h+8V (1-8)("V+7Th)+"("+lVr). (1 .'\)(h1/)+ V)+ ( )(7Th). ' ". unearned premium Next year losses: Ah losses incurred from time h to h +1. E[Ah] o V01'[A h l The Hattendorf theorem ~". - . Exam Ivi - Life ContingEl1cies LGDZ: 10. Chapter 9: Multiple Life Functions Joint survival function: Last survivor status T(xy): (,~, t) Pr[T(.))]]

9 1:) > 8&T(y) > *1 T(J::Y) + T(.TY) T(:r.) + T(y) (t,t) T(:ry) . T(XIJ) T(:r.) . T(y) Pr[T(:r.) > t and T(y) > tj + h(xy) +. Fy(xy) + FT(x) +. J oint life status: tP:1'1I + tPx + tl)y Axy+ == Ax + -,,4y FT(t) = Prlmin(T(:r),T(y)).s; t]. i'i. xy +. + +. + ex + ey Independant lives tjJ2' . tPy Complete expectation of the last-survivor status: t!J" + tqy tq," . tqll Complete expectation of the joint-life sta . J. o tl'Tydt tus: (Xl = J. o t1i xy dt Variances: 00. PDF joint-life status: FarfT(l1)] 2 J t tl)u dt (t) o J. 00. Va r[T(.I:Y)] 2 t . tPxy dt . o Independant lives + t) + I-L(Y + t). \/ar[T(;ry)] 2 J. o t t'Pxy dt . (t) t])x . tJ) (:r. + t) + (Y + t)]. Notes: For joint-life Htat1lH, work with p's: Curtate joint-life functions: nPxll = n])2' ' nPy /,])xy /,P2' . kPy [IL] For last-survivor status, work with q's: k(jxy kqx + k(jy - kqx . Ic(jy [IL]. Pr[K = k] k])xy - k+1 Pxy kPxy.

10 Qx+k,y+" "Exactly one" status: kPxy' =. nPxy - nPxy IJx+k +. 00 nPx + nPy 2 11 px' nPy E[K(;ry)] 2.:: kP,ry n!]x+ nqy - ]x . n(jy 1. + - 20 xy Exam Life '?>~ncies - LGDCS 11. Common shock model: Insurances: (t) (t) . 8 z (t). 1- Ofj,~. ST*(x)(t) . 1 - (t) (t) . 1. (t) . C- At (t) (t) 'T*(y) (t) . 8.,(t). Premiums: 8Y*(X)(t) . (t) . C- At J1xy(t) = j1(;r + t) + J1(Y + t) + A d 1 _ d Insurance functions: A" = L. k=Q.. "p." . qu+k 1. -d L. k=O. Pl'[E( k]. Annuity functions: 00. 00.. kPxy' t )' v . tPu dt 00 o k=O. Reversionary annuities: Variance of insurance functions: A reversioanry annuity is payable during the ex . Vor[Z] - (An)2 istence of one status n only if another status v 2 Axy (A 2 y)2 has failed. an annuity of 1 per year payable Vor[Z]. continnollsly to (y) after the death of (x). ,VY(x y )] (A~: i1xy)( Ay . Covariance of T(:ry) and T(x!7): Call [T(:ry), T(.)))