cix
I
=
+
i
g. + i
benefit
,
-
-
I
Formulas
which
not
are
in
previous
sheet
.
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:
F.J( (z)
Pr(.r < X S; z)
- Fx (:1:)
exp (
- B(Z)
Probability of death between age
age y given survival to age :r::
and
-I
"(Y)d ll )
!
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
Pr[T(:c) > t]
d
attains age
;1;
-1
+t
Mean and variance of T and ](:
Pr[t < Tel')
t+ul]x t])a'
t
complete expectation of life
E[T(:r)]
+ 'Ill
=./
t(jx
tP:B elt
o
t+u])x
curtate expectation of life
tPx' u(]x-t-t
ex)
Relations with survival functions:
2./
00
Vo.7:T(:r:) ]
.
,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:
Life C;onting;;;ncicH - LGD@
./ lx+t dt
o
klJx
Exam rv[
2)2k -1) kP:r
1
2
e".
Varying benefit insul'ances:
(IA)x
=
./It + IJlIt,
./It +
Interest theory reminder
am
tPx !/'x(t)dt
n'fll
0
11
vn
1
l,n
1
/5 '
IJI/ ' t1Jxlt x(t)dt
0
11
./ t ' 7,t , tPx It x (t)dt
(n
+ l)Ofll
-
./(n ItJ
5
' tPx fJ,x(t)dt
i'IJ
T1
./(n - t)vt , tPdl'x(t)dt
0
Ax + VP:L,(1A)x+l
lIqx + 1)1'rr'
nvqx
(IA);:fll
(IA
1
d
(Ia)fll + (Da)m
0
(IA)x
- nvn
1
i
52
11
(DA);':fll
00Cl
1
(IO)OCl
0
CD"4.);':fll
0fll
8
./ t ' l,t , tP:r p'o,(t)dt
0
(IA);"fll
i
5
(Ia)fll
(X)
(IA)",
1
1
1 +i
id
12
Doubling the constant force of interest 5
1 +i
+
-4
v
+
-4
+ (15A);:fll
= (n + l)A;:fll
+
(n +
+
d
i
5
(1
+ i)2
1)2
--+
-+
2i
+ i2
2d - d 2
2i + i 2
25
Limit of interest rate i = 0:
Accumulated cost of insurance:
A o,
1
nqx
n!Ax
Share of the survivor:
accumulation factor
Exam l'vl - Life Cont.ingenciel$ - LC;D'V
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
00
1 +vpx
J
,x,)
2
tEx dt
Jvt'tPxdt
1 + v Px
o
o
1l or [an]
n-year temporary annuity:
J
v
(Iii)x
J
n
n
t
. tllx
dt
=
o
0
Whole life annuity due: 0,,;
1l oriY]
00
L
E[ii K+lll
'..=0
n-year deferred annuity:
J
Yor[oK+lll
J
OC.
rAJ
1,t .
tPx dt
n
n-yr temporary annuity due:
n
'11-1
2
aX!n)
Vor[Y]
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
.
=/-1
11
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'm
ex
ii,x
o.x:11I
rn.
o'x:nl
;=0
---+
1+
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 +
\/ar[L]
=
if
µ
constant
force of
moot
. 2]
Pure endowment annual premium
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.
(A,,:)
Var[L]
M
Var[L]
=
L
µ +28
(1)
Fully discrete equivalence premiums
(whole life and endowment only):
P(A,,:)
Px
dAa:
1- Ax
1
d
P(Ax)
( pr
P(Ax)
ax
1+ d
VadL]
\/ar[L]
\/ar[L]
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:
=
( n Px
[
-
l'"
P x:nl)8
x :m
(A,,:) 2]
2Ax (Ax?
(dii.".)2
<Ax - (Ax)2
(1-
lVIiscellaneous identities:
A."Y
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
É
É
É
¥