Chapter 3: Survival Distributions and Life Tables
Distribution function of X:
Fx(:r) = Pr(X S; :1;)
Survival function B(.1:):
Probability of death between age :r and age y:
Pr(.r < X S; z) F.J( (z) - Fx (:1:)
- B(Z)
Probability of death between age and age y given survival to age :r::
Pr(:1; < X S; zlX >
Notations:
tlJx PriT(.r) tl prob. (3:) dies within t years
distribution function of T(a:)
tPx Pr[T(:c) > t] attains age ;1; + t
Pr[t < Tel') t + 'Ill t+ul]x - t(jx
t])a' t+u])x
tPx' u(]x-t-t
Relations with survival functions:
Curtate future lifetime (K(:r) greatest integer in T(x)):
Pr[K(.l') k] Pr[k T(:r) < k + 1]
k]Jx k+lPx
kP", . qx+k
klJx
Exam rv[ Life C;onting;;;ncicH - LGD@
Force of mortality flea:):
/1(:1:)
8' (x)
sex)
Relations between survival functions and force of mortality:
exp ( -I"(Y)dll)
x+n )
nPx exp - ! p.(y)ely (
Derivatives:
d dt t(jx
d dt tPx tPx . It (:r: + t)
d-Tdt··'" d-Ldt x
d -1
Mean and variance of T and ](:
E[T(:r)]
Vo.7:T(:r:) ]
Vo:r[K(.r}]
complete expectation of life
=./ tP:B elt o
curtate expectation of life ex)
,J ,·2 . t . tPx u,t - ex2./00
o 00
2)2k -1) kP:r e".2
k=l
Total lifetime after age .r: Ta
ex;
T-r: ./ lx+t dt
o
1
1
~, Varying benefit insul'ances:
(IA)x = ./It + IJlIt, tPx !/'x(t)dt
0 11
(L4)~:m ./It + IJI/ ' t1Jxltx(t)dt
0 (X)
(IA)", ./ t ' l,t , tP:r p'o,(t)dt
0 11
(IA);"fll ./ t ' 7,t , tPx Itx (t)dt
0 11
CD"4.);':fll ./(n ItJ ' tPx fJ,x(t)dt
0 T1
(DA);':fll ./(n - t)vt , tPdl'x(t)dt
0
(IA)x Ax + VP:L,(1A)x+l
lIqx + 1)1'rr' + (DA)~:fll nvqx + vpx(DA)x~l:n_ll
(IA);:fll + (15A);:fll
(IA)~:fll + (D)l)~:fll = (n + l)A;:fll
(IA )~:fll + (DA)~:fll (n + l)A~;m
Accumulated cost of insurance:
Share of the survivor:
accumulation factor 1
Interest theory reminder
1 vn
am
l,n1 i n'fll 80fll
5 1 1
-/5 ' 00Cl
i d - nvn
(Ia)fll
1 (IO)OCl 52
(n + l)Ofll (Ia)fll + (Da)m
-5 i'IJ 1 1 +i id 12
Doubling the constant force of interest 5
1 +i (1 + i)2-4
1)2v
-4 2i + i 2
d2d --+ 2d i 2i + i 2
5 -+
25
Limit of interest rate i = 0:
Ao, 1
A~:fll nqx
n!Ax 11}1X
Ax:fll
mlnqx
(JA)x 1 +e:r:
(IA)x eo,
Exam l'vl - Life Cont.ingenciel$ - LC;D'V 4
Chapter 5: Life Annuities
Whole life annuity: ax
J00
Elan] at!· t]Jx + t)dt
o 00
Jvt'tPxdt J,x,)
tExdt
o o
1lor [an]
n-year temporary annuity:
tJn
. = Jn
v tllx dt o 0
1l oriY]
n-year deferred annuity:
rAJ OC.J . tPx dt JtE~,dt1,t
n n
Vor[Y] 2aX!n)
n-yr certain and life annuity:
+ na,x +
Most important identity
1 ba'T + )Ix
1 )Ix
1 ba'x:111
1 - (2b)
d 1 Ax:111
d 1 (lii J;:111 +
Recursion relations
+ + nl
(Iii)x
1 +vpx
1 + v 2Px
Whole life annuity due: 0,,;
00
E[ii K+lll L 11k. kPx '..=0
Yor[oK+lll
n-yr temporary annuity due:
'11-1
E[Y] = Lvk. k]lx
k=O
n-yr deferred annuity due:
ex)
E[Y] = L . kP" k=n
n-yr certain and life due: ii'x:111
k0111 + L v . kllx
k=n
+n,O'T
Exam f,/l - Life COlltingencieh 5
Whole life immediate: ax
= L . ~'P2: k=1
1
m-thly annuities
Vo.r[Y]
1
rn. .. (m) 1(ra) -(Io'x:nl ax:-:m
'm
Accumulation function:
11
=/-1 o
Limit of interest rate i 0:
;=0 ax ---+ ex
ii,x 1 + c2:
IIx ex
o.x:11I cx:rrl ;=0 ---+ 1 +
ex:rrl
6
Chapter 6: Benefit Premiums
Loss function: Loss PV of Benefit,s - PV of Premiums Fully continuous equivalence premiums (whole life and endowment only):
P(Ax) ii",
(L4xP(A",) 1 1
P(A:r) =- -6 (l,;r
.. 2]\/ar[L] (A,,:)(1 + ~r[ Var[L]
Var[L]
Fully discrete equivalence premiums (whole life and endowment only):
P(A,,:) Px
dAa:P(Ax)
1- Ax
P(Ax) d ax( 1
prVadL] 1 + d [ (A,,:) 2]
2Ax (Ax?\/ar[L]
(dii.".)2 <Ax - (Ax)2
\/ar[L] = (1- A."Y
Semicontinuous equivalence premiums:
m-thly equivalence premiums:
p(m) #
h-payment insurance premiums:
A,,;
°x:h\
Pure endowment annual premium PJ::~: it is the reciprocal of the actuarial accumulated value because the share of the survivor who has deposited P:r:4 at the beginning of each year for n years is the contractual $1 pure endowment, i.e.
(1)
P minus P over P problems: The difference in magnitude of level benefit premiums is solely attributable t.o the investment feature of the contract. Hence, comparisons of the policy values of survivors at age :/: + n lllay he done by ana.lyzing future benefits:
P l'"( n Px - x:nl)8x :m
lVIiscellaneous identities:
P(Ax :nl )
P(Ax:m) +6
1
+d
Exa.m tv! LIfe Contin)1;en-C'ies - LGD(':;: 7
lkx = lX+1 P.T
ACCUilmlated differences of premiums:
/~. Chapter 7: Benefit Reserves
Benefit reserve tV: The expected value of the prospect.ive loss at time t.
Continuous reserve formulas:
Prospective: t V(Ax) Ax+t - P(Ax)a.x+t
Retrospective:
Premium diff.:
Paid-np Ins.:
tti(Ax) = P(AT).'lT;t]--
Annuity res.: tV(A;t.) = 1
Discrete reserve formulas:
h-payment reserves:
~V
hr7' A )Ie' ~ ·'"ix:nl
Variance of the loss function
Vad tL] assuming EP
ass1lming EPVar[t L]
Cost of insurance: funding of the accumulated costs of the death claims incurred between age ;1: and x + t by the living at t, e.g.
4E x
qT
~Vx - n~::nl)
Ax-;-n 0 = A~'+n
.~Vx·- nVT
- ~Vr
AXTm:n-~ - Ax+m
Relation between various terminal reserves (whole life/endowment only):
= 1
(I - m~~)(l- nV.,,+m)(l-
Exam ?vI - Contil1j2;€llcieb - LGD© 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
hI! = Lbh+j j]Jx+h qx+h-tj - L j=O j=O
U li V ul)x+tfJ~,(t + n)dl1 L 7ft+u V U P2'+tdv
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
- htl it)] (1 +L h=O
n~1
- L1Hlx+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 Life Contingencies - LGD(:) 9R
• If the death benefit is equal to $1 pIue; the benefit reserve for the first n policy years ane qxlh == q
COllfltant
n V =
Reserves at fractional durations:
(h1/ + 7Th)(l + sPx+h' h~:.sV + UDD '* (hll + 7Th)(l +
V +8' his1/)
h+.sV Vl~8 . I-sqx+h+s . bh+ 1 + . l-sPx+h+s .
UDD '* h+8V (1-8)("V+7Th)+"("+lVr)
i.E. (1 .'\)(h1/)+ V)+ (1-.5)(7Th) '-..r---"
unearned premium
Next year losses:
Ah losses incurred from time h to h + 1
E[Ah] o V01'[Ah l
The Hattendorf theorem
~"
-. Exam Ivi - Life ContingEl1cies LGDZ: 10
Chapter 9: Multiple Life Functions
Joint survival function:
(,~, t) Pr[T(.1:) > 8&T(y) > *1 (t,t)
Pr[T(:r.) > t and T(y) > tj
Joint life status:
FT(t) = Prlmin(T(:r),T(y)).s; t]
Independant lives
tjJ2' . tPy
t!J" + tqy tq," . tqll
Complete expectation of the joint-life status:
= J(Xl
t1ixy dt
o
PDF joint-life status:
(t)
Independant lives
+ t) + I-L(Y + t)
(t) t])x . tJ)ylf.L(:r. + t) + f.1(Y + t)]
Curtate joint-life functions:
/,])xy /,P2' . kPy [IL] k(jxy kqx + k(jy - kqx . Ic(jy [IL]
Pr[K = k] k])xy - k+1Pxy
kPxy . qx+k,y+"
kPxy' =
IJx+k + 00
E[K(;ry)] 2.:: kP,ry 1
Exam Life Com.inl'?>~ncies - LGDCS
Last survivor status T(xy):
T(J::Y) + T(.TY)
T(:ry) .T(XIJ)
+ h(xy)
Fy(xy) + tP:1'1I + tP.TY
Axy+
i'i.xy + + +
Variances:
FarfT(l1)]
Va·r[T(.I:Y)]
\/ar[T(;ry)]
Notes:
T(:r.) +T(y)
T(:r.) . T(y)
+ FT(x) + tPx + tl)y
== Ax + -,,4y
+ ex + ey
Complete expectation of the last-survivor status:
Jtl'Tydt
o
2 J00
t· tl)udt
o 00
2 Jt . tPxy dt
o
2 Jt· t'Pxy dt
o
For joint-life Htat1lH, work with p's:
nPxll = n])2' ' nPy
For last-survivor status, work with q's:
"Exactly one" status:
nPxy - nPxy
nPx + nPy 2 11 px' nPy
n!]x + nqy - 211.I]x . n(jy
+ o.y - 20xy
11
Common shock model:
(t) (t) . 8 z (t)
ST*(x)(t) .
(t) (t) .
(t) . C- At
(t) (t)·· 'T*(y) (t) . 8.,(t)
8Y*(X)(t) . (t) . C-At
J1xy(t) = j1(;r + t) + J1(Y + t) + A
Insurance functions:
A" = L k=Q
. "p." . qu+k
L k=O 00
Pl'[E( k]
. kPxy'
00
k=O
Variance of insurance functions:
Vor[Z] - (An)2
Vor[Z] 2Axy (A2•y)2
,VY(xy )] (A~: i1xy)( Ay
Covariance of T(:ry) and T(x!7):
Call [T(:ry), T(.TY)] Call [T(;r) ,T(y)] + {E fT(;r)]
C01l T(y)] + (ex (ey [IL]
Insurances:
1 - Ofj,~.
1 - Oo.xy
1
Premiums:
d
1 _ d
1 -d
Annuity functions:
00
)' v t . tPu dt
o
Reversionary annuities: A reversioanry annuity is payable during the existence of one status n only if another status v has failed. E.g. an annuity of 1 per year payable continnollsly to (y) after the death of (x).
E [T(:J:Y)]}' {(E [T(y)] E :Tery)]}
Exa,m 1..,{ Life Contingencie~ - LCD@ 12
Chapter 10 & 11: Multiple Decrement Models
Notations:
probability of decrement in the next
t yearH due to caUHe .7
= probability of decrement in the next
t yearH due to all caUHes m
L j=l
the force of decrement dHe only
to decrement j
Il~T) the force of decrement due to all
causeH simultaneously rn
L j=l
probability of surviving t yearH
despite all decrementH
1 t
-/ (B)ds e (l
Derivative:
_!i () dt
Integral forms of tqx :
t
/ S p~;T) , p,~) o
Exam IvI - Life Contingencie~ LGD© 13
Probability density functions:
.Joint PDF: hAt, j) (t)
Marginal PDF of J: fA.j) = 00 q~j) iX'
= / f:r,J(tLi)dt o
Marginal PDF of T: f:r(t) (t) 171
= LhAt,
Conditional PDF: hlrUlt) = ---,.---
Survivorship group: Group of l~T) people at some age a at time t o. Each member of the group has a joint pdf for time until decrement and cause of decrement.
T.~a+n
/ tP~~), fL!!) (t)dt :r--a
L m
j=1
171
Ll~) j cce 1
Associated single decrement:
probability of decrement from caUHe j only
e~p [-I"~) (,')d'1 1 - tq~(j)
Basic relationships:
II rn
PI(;)
t x ;=1
UDD for multiple decrements:
t· t. q~T)
Decrements uniformly distributed in the associated single decrement table:
t·
1 .1(2) \= q~(l) (1 -q I2 x )
~ Qx1(1))(1
2
l(1) ( 1 1(2) 1 ...1. ~ql(2) . ql(3))
qx \ 1 - "2 Qx . - 2 . 3 x x
Actuarial present values
Irh<;tead of summing the benefit8 for each possible cause of death, it is often easier to write the benefit as one benefit given regardless of the cause of death and add/8ubtract other henefit8 according to the ca,lse of death.
Premiums:
p(T) x
EXtn11 rvI Life Contingl;'ncies - LGDe~ 14
Chapter 15: lVlodels Including Expenses
Notations:
G expense loaded ( or gross) premium
b face amount of the policy
G /b per unit gross premium
Expense policy fee: The portion of G that is independent of b.
Asset shares notations:
G level ann11al contract premimn
kAS asset share assigned to the policy at time t = k
Ck fraction of premium paid for expenses at k cG is the expel1xe premium)
Ck expenseH paid per policy at time t = k
probability of decrement by death
probability of decrement by withdrawal
"CiI cash amount due to the policy holder as a withdrawal benefit
b" death benefit due at time t = k
Recursion formula:
(1 + i)
Direct formula:
f G(l Ch) - eh h+1 Cl1
h=O
EXalTI l\j - Life COTIringencies - LGD@ 15
Constant Force of Mortality
• Chapter 3
S CJ:)
Z" nPJ:
1 E[T] E'X]
nPx) 1
1/or[T] Var[.1:] =
p.
ln2 . ]Yledian[T] = Mechan,X p.
ex Px = E[K] qar
• Chapter 4
/-1. + (j f."
f.."+26
Ax (1 nEx)
q+i
• Chapter 5
1
p.+ 1
ax
• Chapter 6
llqx = P;':11i P(A~,) p
For fully discrete whole wi EP,
For fully continuous whole life, w IEP.
1/m'[Loss] =
• Chapter 7
t V(A)x O. t 2: 0
O. k=O,l,2 ....
For discrete whole life, assuming
Vo1'[ "Los.s] p. If = 0, 1,2, ...
For fully continuous whole as:mming EP,
1/a/' [ tLo.~81 1,4x , t 2: 0
• Chapter 9 For two COllstant forces, i. e. acting on (.1:) and Il.F acting on (y), we have:
(hy + i 1 +i
qJ,y + i 1Jxy
exy qJ'Y
Exam fvI Lift Contingencies - LGD<9 16
De Moivre's Law
• Chapter 3
8(.1; )
,,1','
t1Jx p(.r + t)
Vor[T]
0(.1;)
• Chapter 4
• Chapter 3
8(.r)
ft·(;r)
"Px
a 2(w-x) I2AX :r 2(w - x)1- W ow-xl
AxW -;r W -.rlo-- ex: (w - .1;)
W 07il A~:7il1 W -.r!J.('r) =-
W -.r (Io)w_xl(IA)x
W -.1: W -.T
(Io)w-xlw-.r-T/ (IA)x w-.rw-.r
qx = !J.('r) = h(.r) , O.:s; t < W -.T (Io)7il(IAX:7il w-.r
"2(lx + Ix+d 1
(Io)7il(IAX:7ilw-;r: W -.1:-2- = E[T] = Median[T]
W - .r _ ~ = E[K] 2 2 • Chapter 5
(w-.r)2 1-Ax d heNo useful formulas: use ii. x -d- an. t 12 chapter 4 formulas.
(w _.r)2 -1
12 • Chapter 9 qx 2dx
1 - ~qx lx + l1:+1 W-;]:1 --(= MDML with p. = 2/(w - .1:))E[8]. =-2 3
n 2(w - .1:)nnPx + '2 "qx
3 T/
y-x]Jx Cyy + y-1,qx cye1':7il + '2 "l]x
For two lives with different w's, simply translate a'w_xl one of the age by the difference in w's. E.g. w-:r:
a'7il w-.r Age 30, w = 100 {o} Age 15, w = 85
Modified De Moivre's Law
V or[T]
(1- ~r • Chapter 9
(w-.1:)C c10 -----:;- ex: (w - .r)
W -.1:
c 2c+ 1 w -.r 2c
ex o.
wIth j.t = - w -.r
w -.r
w - .1: = E[T]
(w-.T-n)C
c+1
Exam rvI - Life Contingenciet> - LGD© 17
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