mlc formula sheet

= 1 - lx+t lx
q n m x
=
p q n x m x+n
=
p - p n x n+m x
=
l - l x+n
x+n+m
lx
=
dm x+n lx
·d dt
pt x
= - t m px x+t
( ) ( ) ò d ( ) dt
qt x
=
t
m px x+t
· mx+t
=
-S¢ x + t S x+t
( ) ( ) å( ) Õ for Var ex
use E éëK
X
2ù û
=
¥ k =1
n-1
2k -1
k
px
·
pn x
=
p p p K p x x+1 x+2
x +n -1
=
px+k
k=0
for
continuous
to
discrete
estimate
æçè
o
e
® ex
ö÷ø using
linear
LIFE TABLES
Memorize
t
¥
¥
for Uniform éë0,wùû Þ lx = w - x · qt x = 1 - t px · dt x = lx qt x = lx - lx+t = ò lx+wmx+wdw · Tx = ò lwdw = ò lx t pxdt
0
x
0
( ) ( ) o
k =1
o
o
= ex:n + n px ex+n
( ) ( ) ( ) ¥
å ex = E ëéK X ûù = k=1 k px = px + 2 px 1+ ex+2 = px + 2 px + 3 px 1 + ex+3 = px + 2 px + 3 px + 4 px + 5 px + K¥
SURVIVAL
Memorize
x
x
ò ò ( ) ( ) ( ) ( ) ( ) ( ) m
x
:m
x
¥
> 0 and ò m 0
x
dx = ¥ ·S
x
- l(u )du
- m (t )dt
= e-L(x) = e 0
=e 0
Ûm
x
= - d ln S dx
x
x+t
t
px
=
S (x + t) S (x)
A2 1 , Ax:n x:n
n Ax , n A x
A , Ax:n x:n
A 1 º n Ex
x:n
INSURANCE
De Moivre
Ex:w = 95, x = 25,n =15,i = .03
a
a
( ) w-xi = 70 .03 = 0.4160
w - x 70
aw-x (1+i)2 -1
d
Ax
=
iAxContinuous=
i d
Discrete
Û
d
( Continuous ) =i ( Discrete )
F:\SOA & Exams\MLC\Study Sheets\study sheet 19.docx
Symbols
Ax , Ax
2
2 Ax , Ax
1
A1 , Ax:n x: n 21
º m(x) =
-
d dx
lx
lx
· notation convention:
m ( x + t ) º mx+t
( ( ) ) t px + qt x
=1Û
pt x
= 1- t qx · t px
=
lx+t lx
=
l0Sx x + t l0Sx x
· t qx
= lx - lx+t lx
=
dt x lx
t px =
survival function for T ( X ) · f (t ) = t m px x+t = t px mx (t ) =
fx (x + t Sx (x)
)
,
x
is
constant
hazard
function
® l(x) =
m (x) =
f (x) S (x)
·
f
(x) =
o
·e
=
ex
+
1 2
· lx+t
=
1-t
lx +
t
lx+1
Û
qs x+t
=
sqx 1- tqx
,0 £
s+t
£1
( ) ( ) ( ) Constant Force Þ mx+t = m · t px =
px
t = e-mt · Hyperbolic Þ mx+t
= 1-
qx 1-t
qx
·
qs x+t
Makeham: m ( x) = A + Bcx for x > 0, B > 0, c > 1, A > -B « A = 0 Þ Gompertz
Weibull: m ( x) = kxnfor x ³ 0, k > 0, n > -1
x+n
x+n
n
ò Sx ( y ) mx ( y ) dy ( ) ò f y dy ò t px mx+tdt
®
lx+0.25
=
0.25
px lx
( ) P éëK
X
= k ùû =
p q k x x+k
=
p - p k x k+1 x
=t
qx
=
dx+k lx
·
central rate of failure ®
m n x
=
dn x Ln x
Principles
S (x) =
lx l0
Û lx
= l0S ( x) · mx
1
= Ax:n + A 1
0
0
n
x:n
x:n
( ) Exponential Û Constant Force Þ Ax =
m
2
· Ax =
m
1
·Ax:n =
m
m +d
m + 2d
m +d
1 - e-n(m+d )
· A 1 º E n x = e-n(m+d )
x:n
ò ( ) ( ) 2
Ax
=
¥
e-2d t
Ax
=
1 a
w - x w-x
Û
A x:n
=
1 a
w-x n
Generalized
De
Moivre
Þ
lx
=
(w
-
x )a
o
· ex
=
(w
a
- x)
+1
Percentiles
® tp
=
- ln m
p
®
Zp
=
e-d t p
n
· òte-dtdt 0
=
an
- ne-nd d
Principles
ò ( ) 2 1
a = 70 .0609
=
1
1- 1 1.03140
= 0.2308
w-x
70 70 1.032 -1
a
a
( ) ni = 15 .03 = 0.1705
n
n
n
ex:n = ò S t dt = ò t pxdt = ò t t pxmx+t dt + n n px · ex = px 1+ ex+1 · Constant Force, mx = m Þ t px = e-mt
0
0
0
Uniform Û UDD
Û
"Linear Interpolation" Þ mx+t
S (x)m (x)· mt
º
m (t)· L(x) =
x
ò l (u )du
0
=
x
ò m (t) dt
0
( ) P (T ( x) > t ) = S (t ) = P X > x + t½X > x = 1- F (t )
( ) ( ) S ( x + t )
qt x = 1 - S ( x) = P
·t
px
ò- m(w)dw
=e x
where m ( x) =
-S '(x) S (x)
=
f (x) o S (x) ·ex
¥
=ò 0
S
x(
Sx
x
(
+t
x)
)
dt
¥
=ò x
Sx Sx
(t ) (x)
dt
Uniform