Chapter 3: Survival Distributions and Life Tables
Distribution function of X:
Fx(:r)
= Pr(X
S;
Force of mortality flea:):
:1;)
/1(:1:)
Survival function B(.1:):
8' (x)
sex)
Relations between survival functions and
force of mortality:
Probability of death between age :r and
age y:
Pr(.r < X S; z)
F.J( (z)
- Fx (:1:)
exp (
- B(Z)
Probability of death between age
age y given survival to age :r::
-I
"(Y)d ll )
!
and
x+n
exp
nPx
(
Pr(:1; < X S; zlX >
)
p.(y)ely
Derivatives:
d
dt t(jx d
Notations:
tlJx dt
-T
dt··'" prob. (3:) dies within t years
d
-L
dt x distribution function of T(a:)
tPx
+ t)
d
tl
PriT(.r)
tPx . It (:r:
tPx
d
Pr[T(:c) > t]
attains age
;1;
-1
+t
Mean and variance of T and ](:
Pr[t < Tel')
t+ul]x t])a'
t
E[T(:r)]
+ 'Ill
complete expectation of life
=./
t(jx
tP:B elt
o
t+u])x
curtate expectation of life
tPx' u(]x-t-t
ex)
Relations with survival functions:
00
Vo.7:T(:r:) ] 2./
.
,J
,·2
t . tPx u,t
- ex
o
00
Curtate future lifetime (K(:r)
integer in T(x)):
Pr[K(.l')
k]
Vo:r[K(.r}] greatest
Pr[k
T(:r) < k
k]Jx
k+lPx
k=l
+ 1]
Total lifetime after age .r: Ta
ex;
kP", . qx+k
T-r:
klJx
Exam rv[
Life C;onting;;;ncicH - LGD@
2)2k -1) kP:r
./ lx+t dt
o
1
2
e".
~,
Varying benefit insul'ances:
(IA)x
=
./It + IJlIt,
./It +
Interest theory reminder
am
tPx !/'x(t)dt
n'fll
0
1
l,n
1
/5 '
IJI/ ' t1Jxlt x(t)dt
0
i
0fll
8
1
00Cl
i
1
d
- nvn
(Ia)fll
(X)
(IA)", vn
5
11
(L4)~:m
1
./ t ' l,t , tP:r p'o,(t)dt
0
11
(IA);"fll
1
(IO)OCl
./ t ' 7,t , tPx It x (t)dt
(n
0
52
(Ia)fll + (Da)m
+ l)Ofll
11
CD"4.);':fll
-
./(n ItJ
5
' tPx fJ,x(t)dt
i'IJ
1
1 +i
id
12
0
T1
(DA);':fll
./(n - t)vt , tPdl'x(t)dt
0
(IA)x (DA)~:fll
Ax + VP:L,(1A)x+l
lIqx + 1)1'rr'
nvqx
Doubling the constant force of interest 5
1 +i
+
-4
+ (15A);:fll
(IA)~:fll + (D)l)~:fll = (n + l)A;:fll
(IA )~:fll + (DA)~:fll (n + l)A~;m
(IA);:fll
d
i
5
+ i)2
--+
-+
2i
+ i2
2d ­ d 2
2i + i 2 25 Limit of interest rate i = 0:
Accumulated cost of insurance:
A o,
Share of the survivor:
Exam l'vl - Life Cont.ingenciel$ - LC;D'V (1
1)2
v
+ vpx(DA)x~l:n_ll
accumulation factor
-4
1
A~:fll
nqx
n!Ax
11}1X
Ax:fll
mlnqx
1
4
(JA)x
1 +e:r:
(IA)x
eo,
Chapter 5: Life Annuities
ax
Whole life annuity:
Recursion relations
J
00
Elan]
at!·
t]Jx
+
+ nl
+ t)dt
o
1 +vpx
,x,)
00
Jvt'tPxdt
J
2
1 + v Px
tEx dt
o
o
1l or [an]
n-year temporary annuity:
n
J
v
(Iii)x
n
t
. tllx
dt
J
=
o
0
Whole life annuity due: 0,,;
1l oriY]
00
L
E[ii K+lll
'..=0
n-year deferred annuity:
J
Yor[oK+lll
JtE~,dt
OC.
rAJ
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] =
Most important identity
1
ba'T
+ )Ix 1
)Ix k=n
1 ba'x:111
1 - (2b)
n-yr certain and life due: ii'x:111
d
1
Ax:111
0111
d
1
Exam
f,/l -
+L
k=n
(lii J ;:111 +
Life COlltingencieh L . kP"
+n,O'T
5
v
k
. kllx
11k.
kPx
Accumulation function:
Whole life immediate: ax
=L
. ~'P2:
11
=/-1
k=1
1
o
m-thly annuities
Limit of interest rate i
ax
Vo.r[Y]
1
(ra)
.. (m)
ax:-:m
1 + c 2:
II x
ex
cx:rrl
;=0
---+
1
-(I­
ex
ii,x
o.x:11I
rn.
o'x:nl
;=0
---+
1+
'm
ex:rrl
6
0:
Chapter 6: Benefit Premiums
h-payment insurance premiums:
Loss function: 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
=- -6
P(A:r)
(l,;r
(1 + ~r
\/ar[L]
[
. 2]
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 endow­
ment, i.e.
(A,,:)
Var[L]
Var[L]
(1)
Fully discrete equivalence premiums
(whole life and endowment only):
P(A,,:)
Px
dAa:
1- Ax
P(Ax)
1
P(Ax)
d
( pr
ax
1+ d
VadL]
\/ar[L]
\/ar[L]
P minus P over P problems: The difference in magnitude of level benefit pre­ miums 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: =
( n Px
[
-
l'"
P x:nl)8
x :m
(A,,:) 2]
2Ax (Ax?
(dii.".)2 <Ax - (Ax)2
(1-
A."Y
lVIiscellaneous identities:
P(A x :nl )
Semicontinuous equivalence premiums:
P(Ax:m) +6
m-thly equivalence premiums:
p(m)
#
Exa.m tv!
LIfe Contin)1;en-C'ies - LGD(':;:
1
+d
7
/~.
Chapter 7: Benefit Reserves
Benefit reserve tV: The expected value of the prospect.ive loss at time t. Variance of the loss function
Continuous reserve formulas:
Prospective: t V(Ax)
Vad tL]
assuming EP
Ax+t - P(Ax)a.x+t
Retrospective: tti(Ax) = P(AT).'lT;t]-Premium diff.:
Var[t L ]
ass1lming EP
Paid-np Ins.:
Cost of insurance: funding of the accumu­
lated costs of the death claims incurred between
age ;1: and x + t by the living at t, e.g.
Annuity res.: tV(A;t.) = 1
4E x
Discrete reserve formulas:
lkx
=
qT
P.T
lX+1
ACCUilmlated differences of premiums:
~Vx
-
Ax-;-n
n~::nl)
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:
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:
br
00 hI!
00
= Lbh+j
j]Jx+h qx+h-tj -
j=O
L
j=O V
U
L
ul)x+tfJ~,(t + n)dl1
7ft+u V
li
U
P2'+t dv
o
Recursion relations:
hI! + 7fh
(" ~7
l'
+ 7fh)(l + i)
(hI! +
+ 11 Px+h . h+1 If
+ 11x+h . h+1 If
f]x--;-h . bh-'-l
qx+h .
(l+i)
h+ll!
+
h+1 V)
-
Terminology:
"policy year h+ 1"
the policy year from time t = h to time t
"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
= h+1
+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
+
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
• If the death benefit is equal to
L
+
h=O
Exam
R
Life Contingencies - LGD(:) policy yean,
plus the benefit reserve for the fiTst n policy years
71-1
nV=V~m-
17
9
• If the death benefit is equal to $1 pIue; the benefit reserve for the first n policy years ane
COllfltant
nV =
Reserves at fractional durations:
(h1/
+ 7Th)(l +
+
+ 7T h)(l +
sPx+h' h~:.sV
UDD
'*
(hll
V +8'
Vl~8
h+.sV
UDD
i.E.
'*
.
h+8V
his 1/)
I-sqx+h+s . bh + 1
+
. l-sPx+h+s
(1-8)("V+7Th)+"("+lVr)
(1 .'\)(h1/)+
V)+
.
(1-.5)(7Th)
'-..r---"
unearned premium
Next year losses:
Ah
E[Ah]
losses incurred from time h
to
h
+1
o
V01'[A h l The Hattendorf theorem
~"
-
.
Exam Ivi - Life ContingEl1cies
LGDZ: 10
qxlh
==
q
Chapter 9: Multiple Life Functions
Joint survival function:
(,~,
t)
Last survivor status T(xy):
Pr[T(.1:)
T(J::Y) + T(.TY)
T(:ry) . T(XIJ)
> 8&T(y) > *1
(t,t)
Pr[T(:r.)
> t and T(y) > tj
+ h(xy)
Fy(xy) +
tP:1'1I + tP.TY
J oint life status:
FT(t)
=
Axy+
Prlmin(T(:r),T(y)).s; t]
i'i. xy
T(:r.)
+ T(y) T(:r.) . T(y) +
FT(x) +
tPx + tl)y
== Ax + -,,4y
+
+
+
+
ex
+ ey
Independant lives
Complete expectation of the last-survivor
status:
tjJ2' . tPy
t!J"
+ tqy
tq," . tqll
J
tl'Tydt
Complete expectation of the joint-life sta­
tus:
o
(Xl
J
=
t1i xy dt
o
Variances:
00
PDF joint-life status:
FarfT(l1)]
J
t·
2
(t)
tl)u dt
o
J
J
00
Va·r[T(.I:Y)]
2
t . tPxy dt ­
o
Independant lives
(t)
\/ar[T(;ry)]
+ t) + I-L(Y + t)
tJ)ylf.L(:r. + t) + f.1(Y + t)]
2
t· t'Pxy dt ­
o
t])x .
Notes:
For joint-life Htat1lH, work with p's:
Curtate joint-life functions:
/,])xy
/,P2' . kPy
k(jxy
kqx
Pr[K = k] +
k(jy -
k])xy -
nPxll =
[IL]
For last-survivor status, work with q's:
kqx . Ic(jy
[IL]
k+1Pxy kPxy . qx+k,y+" "Exactly one" status:
=
kPxy'
IJx+k
nPxy -
+
nPx
00
E[K(;ry)]
2.:: kP,ry
Life Com.inl'?>~ncies - LGDCS
nPxy
+ nPy
2
11
px' nPy
+ nqy - 211.I]x
+ o.y - 20 xy
n!]x
1
Exam
n])2' ' nPy
11
. n(jy
Common shock model:
Insurances:
(t) (t) . 8 z (t)
1-
ST*(x)(t) .
(t)
1 - Oo.xy
(t) .
1
(t) . C- At
(t)··
(t)
'T*(y)
=
j1(;r
(t) . 8.,(t)
Premiums:
(t) . C- At
8Y*(X)(t) .
J1xy(t)
+ t) + J1(Y + t) + A
d
1 _ d
Insurance functions:
A"
=
Ofj,~.
L
1
. "p." . qu+k
-d
k=Q
L
k=O
k]
Pl'[E(
Annuity functions:
00
00
. kPxy'
)' v
t
.
tPu dt
o
00
k=O
Reversionary annuities:
A reversioanry annuity is payable during the ex­
istence 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).
Variance of insurance functions:
- (An)2
Vor[Z]
Vor[Z]
2Axy
,VY(x y )]
(A~:
(A 2 •y)2
i1xy)( Ay ­
Covariance of T(:ry) and T(x!7): Call [T(:ry), T(.TY)]
Call [T(;r) ,T(y)]
C01l
T(y)]
(ex
Exa,m 1..,{
Life Contingencie~ - LCD@ (e y
+ {E fT(;r)] +
[IL]
12
E [T(:J:Y)]}' {(E [T(y)]
E :Tery)]}
Chapter 10 & 11: Multiple Decrement Models
Notations:
Probability density functions:
hAt, j)
.Joint PDF:
probability of decrement in the next
t yearH due to caUHe .7
fA.j)
Marginal PDF of J:
=
(t)
00
q~j)
iX'
=
probability of decrement in the next
f:r,J(tLi)dt
= /
t yearH due to all caUHes
o
m
L
Marginal PDF of T:
f:r(t)
(t)
j=l
171
=
the force of decrement d He only
LhAt,
to decrement j
hlrUlt) = ---,.---
Conditional PDF:
Il~T) the force of decrement due to all causeH simultaneously rn
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. L
j=l
probability of surviving t yearH
despite all decrementH
1
t
e
-/
T.~a+n
(B)ds
(l
/
tP~~), fL!!) (t)dt
:r--a
m
L
j=1
Derivative:
171
Ll~)
j
) _!idt (
cce
1
Integral forms of tqx :
Associated single decrement:
t
/
S
probability of decrement from caUHe j only
p~;T) , p,~)
e~p -I"~) (,' )d'1
o
[
1 - tq~(j)
Exam IvI - Life Contingencie~
LGD© 13
Basic relationships:
Actuarial present values
rn
II
I(;)
t Px
Irh<;tead of summing the benefit8 for each pos­
sible 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.
;=1
UDD for multiple decrements:
t·
t.
Premiums:
q~T)
p(T)
x
Decrements uniformly distributed in the
associated single decrement table:
t·
= q~(l)
(1
(1
l(1) (
qx
EXtn11 rvI
\
1-
1 .1(2) \
-q
I
2
~
2
1
x
)
1(1))
Qx
1(2)
. -
"2 Qx
Life Contingl;'ncies - LGDe~
1
2
...1.
~ql(2)
x
. 3
.
ql(3))
x
14
Chapter 15: lVlodels Including Expenses
Notations:
G
b
G /b
expense loaded ( or gross) premium
face amount of the policy
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
Ck
fraction of premium paid for expenses at k
Ck
expenseH paid per policy at time t
=
=
k
cG is the expel1xe premium)
k
probability of decrement by death
probability of decrement by withdrawal
"CiI
cash amount due to the policy holder as a withdrawal benefit
death benefit due at time t
b"
=
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
Var[.1:] =
1/or[T]
• Chapter 6
p.
llqx =
ln2
.
]
= Mechan,X
p.
Yledian[T]
P(A~,)
For fully discrete whole
Px = E[K]
ex
P;':11i p
wi EP,
qar
For fully continuous whole life, w IEP.
• Chapter 4
1/ m'[Loss] =
• Chapter 7
/-1.
+ (j
O. t 2: 0
t V(A)
x
f."
O. k=O,l,2 ....
f.."+26
Ax (1
nEx)
For
discrete whole life, assuming
Vo1'[ "Los.s]
p.
If = 0, 1,2, ...
For fully continuous whole
1/a/' [ tLo.~81
q+i
as:mming EP,
1,4 x , t 2: 0
• Chapter 9 For two COllstant forces, i. e.
and Il.F acting on (y), we have: • Chapter 5
1
p.+
1
ax
(hy
+i
1 +i
qJ,y
e xy
1Jxy
qJ'Y
Exam fvI
Lift Contingencies - LGD<9 16
+i
acting on (.1:) De Moivre's Law
• Chapter 3
1- ­
8(.1; )
W
W -;r
Ax
l o - - ex: (w -
.1;)
W
A~:7il
1
W -.r
!J.('r)
a 2(w-x) I
2(w - x)
ow-xl
W -.r
07il
W -.r
(Io)w_xl
2AX
:r
=-­
(IA)x
W -.1: W -.T w-.r-T/
w-.r qx = !J.('r) = h(.r) ,
1
,,1','
t1J x
p(.r
+ t)
"2(lx + Ix+d w-;r:
- 2 - = E[T]
(Io)w-xl
w-.r
(Io)7il
w-.r (Io)7il
(IA)x
O.:s; t < W
-.T (IAX:7il
(IAX:7il
W -.1:
= Median[T]
.r _ ~ = E[K] 2
2
(w-.r)2
W-
Vor[T]
• Chapter 5
No useful formulas: use ii. x
chapter 4 formulas.
12
(w _.r)2 -1
12
qx
2dx 1 - ~qx
lx + l1:+1 E[8].
0(.1;)
W-;]:
1
2
--
(= MDML with p.
3
2(w - .1:)
3
y-x]Jx Cyy + y-1,qx cy
=-
+ '2 "qx
e1 ':7il +
T/
'2 "l]x
• Chapter 4
w-:r:
a'7il
w-.r
Age 30, w = 100
{o}
Age 15, w = 85
Modified De Moivre's Law
• Chapter 3
ft·(;r)
"Px
V or[T]
(1- ~r
10
(w-.1:)C ex: (w -----:;-
.r)
• Chapter 9
c
W -.1:
c
w
(
2c+ 1
-.r
o.
w -
.1:
-.r
= E[T]
c+1
Exam rvI - Life Contingenciet> - LGD© 2c
ex wIth j.t = - ­
w -.r
w-.T-n)C
w
= 2/(w - .1:))
For two lives with different w's, simply translate
one of the age by the difference in w's. E.g.
a'w_xl
8(.r)
an.d t h e
• Chapter 9
n
nnPx
1-Ax
d-
17