1. education

HOW TO USE THESE FLASHCARDS
These flashcards are designed to help you to prepare efficiently in the
run-up to the Course M exam of the Society of Actuaries. They include
conceptual ideas, key formulas and techniques for efficient problem
solving. The Course M Examination has a number of problems that
require first principles reasoning as well as a fair amount of computation.
So don’t look at the lists of formulas as simply being memorization work.
There are often simple intuitive ideas that underlie the formulas as well as
basic mathematical reasons why they are correct. Strive to understand
and learn the key relations from these points of view and your knowledge
will not be the superficial type that may collapse under the stress of
taking the examination. The more that you understand, the easier it
becomes to retain the key ideas and write them down quickly and
accurately.
We have designed the flashcards so that they can be carried conveniently
and read frequently in the final run-up to the exam, eg when commuting
to work. We hope that you will personalize them by adding your own
comments and notes, and checking each section when you feel confident
with the material covered.
You will probably also find these summaries useful when you are at the
stage of working through the past exams. If you see a particular point
being examined that is not summarized here add it to these flashcards.
Let us know if you find some key ideas that are missing.
Good luck with your studying.
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© BPP Professional Education: 2005 exams
SURVIVAL MODELS
X is the random lifetime of a newborn. There are a number of
equivalent ways of specifying the distribution:
1.
The probability density function: f X ( x ) . It is not a probability,
but f X ( x ) dx ≈ Pr ( x ≤ X ≤ x + dx ) .
2.
The distribution function: FX ( x ) = Pr ( X ≤ x ) is the probability
that the death of a newborn life occurs by age x.
3.
The survival function: sX ( x ) = Pr ( X > x ) is the probability that
a newborn life is surviving at age x.
4.
The life table: lx is the expected number of survivors at age x
from a group of l0 newborn lives.
5.
The force of mortality: µ ( x ) is an instantaneous measure of
mortality at age x.
Key properties and relations for the pdf and cdf
1.
FX ( x ) is continuous and non-decreasing with FX ( 0 ) = 0 and
FX (ω ) = 1 .
2.
f X ( x ) is non-negative and continuous (or possibly piecewise
continuous) on  0 , ω  .
3.
Relations:
FX ( x ) =
x
∫0
f X ( y ) dy
f X ( x ) = FX′ ( x ) (where it exists)
b
Pr ( a ≤ X ≤ b ) = FX ( b ) − FX ( b ) =
f X ( x ) dx
a
∫
© BPP Professional Education: 2005 exams
Page 3
SURVIVAL MODELS
Key properties and relations for the survival function
1.
sX ( x ) is continuous and non-increasing with sX ( 0 ) = 1 and
s X (ω ) = 0 .
2.
Relations:
′ (x)
sX ( x ) = 1 − FX ( x ) and f X ( x ) = − sX
sX ( x ) =
ω
∫x
f X ( y ) dy
Pr ( a ≤ X ≤ b ) =
b
∫ a f X ( x ) dx = sX ( a ) − sX ( b )
Key properties and relations for the life table function
1.
lx is the expected number of survivors at age x from a group
of l0 newborn lives.
2.
lx = l0 sX ( x ) is continuous and non-increasing with lω = 0 .
3.
sX ( x ) =
lx
l0
Key properties and relations for the force of mortality
1.
µ ( x ) is non-negative and piece-wise continuous.
2.
µ (x) =
3.
fX ( x )
sX ( x )
=−
′ (x)
sX
l′
= − ln ( sX ( x ) ) ′ = − x
sX ( x )
lx
(
 x

sX ( x ) = exp  − ∫ µ ( y ) dy  and
 0

sX (ω ) = 0
4.
µ ( x ) ⋅ ∆x ≈ Pr ( X ≤ x + ∆x | X ≥ x )
5.
Shortcut: µ ( x ) = c
Page 4
p′ ( x )
p (x)
)
ω
∫ 0 µ ( y ) dy = ∞
 p (0) 
⇒ sX ( x ) = 

 p(x) 
in order that
c
© BPP Professional Education: 2005 exams
SURVIVAL MODELS
Standard actuarial notation and relations
1.
( x ) denotes a life currently age x
2.
n px
is the probability that a life age x will survive to age x + n
n px
3.
is the probability that a life age x will die by age x + n
n qx
n qx
4.
s ( x + n ) lx + n
= X
=
= Pr ( X > x + n|X > x )
sX ( x )
lx
n|m q x
l −l
d
= x x + n = n x = 1 − n px = Pr ( X ≤ x + n |X > x )
lx
lx
l
−l
= x + n x + n+ m = Pr ( x + n < X ≤ x + n + m|X > x )
lx
Key relations
n px
+ n qx = 1
= n px m px + n
px + n −1 (n integer)
n px = px px + 1
m + n px
n qx
= qx + 1|qx +
n|q x
= n px qx + n = n px − n+ 1 px = n + 1 qx − n qx
n|m qx
+ n−1|qx
= n px m q x + n = n px − n + m px = n + m q x − n q x
d
qx = x
lx
 l
−l 
−  x + 1 x 
( x + 1) − x 
chord slope
= 
=−
= Pr ( x < X ≤ x + 1|X > x )
number living
lx
µ (x) = −
tangent slope
lx′
=−
lx
number living
µ ( x ) ∆x ≈ Pr ( x < X ≤ x + ∆x|X > x ) = ∆x qx
© BPP Professional Education: 2005 exams
Page 5
SURVIVAL MODELS
Future lifetime after age x
T = T ( x ) = X − x|X > x is the random future lifetime after age x for
a survivor to age x :
s (x + t)
l
 t

s T ( t ) = Pr (T > t ) = t px = x +t = X
= exp  − ∫ µ ( x + s ) ds 
lx
sX ( x )
 0

FT ( t ) = Pr (T ≤ t ) = t qx = 1 − t px
Contrast the following relations:
f T ( t ) = t px µ ( x + t )
f T ( t ) dt ≈ Pr ( t < T ≤ t + dt )
f X ( x ) = x p0 µ ( x )
f X ( x ) dx ≈ Pr ( x < X ≤ x + dx )
µ ( x ) dx ≈ Pr ( x < X ≤ x + dx|X > x )
Curtate future lifetime after age x
K = K ( x ) = T ( x )  (the greatest integer function) is a discrete
random variable:
sK ( k ) = Pr ( K > k ) = Pr ( K ≥ k + 1 ) = k + 1 px
FK ( k ) = Pr ( K ≤ k ) = Pr ( K < k + 1 ) = k + 1qx
f K ( k ) = Pr ( K = k ) = k|qx = k px qx + k
Complete life expectancy
e x denotes E T ( x )  , the expected number of future years of life:
e x = E T ( x )  =
ω −x
=
∫0
ω −x
∫0
lx +t dt
lx
=
t fT ( t ) dt =
ω −x
∫0
t px
dt
Tx
, Tx is total future years lived by lx lives age x
lx
ω −x
2
E T ( x )  =
2t ⋅ t px dt


0
∫
Page 6
© BPP Professional Education: 2005 exams
SURVIVAL MODELS
Temporary life expectancy
e
is the expected number of years lived by a life currently age
x: n
x in the next n years.
T ( x ) ∧ n = min {T ( x ) , n} = the number of years lived in the next n years
e
=
ex =
n
∫0
e
t px
ω
∫x
t fT ( t ) dt +
∞
∫n
n fT ( t ) dt
Tx − Tx + n
lx
dt =
+ n px e x + n
x: n
after age x+n
next n years
Tx =
n
∫0
= E T ( x ) ∧ n  =
x: n
ly dy
Curtate life expectancy
The expected number of complete years lived after age x is ex
(fractional part of final year is ignored)
ex = E  K ( x )  =
ω − x −1
∑
k =0
k Pr ( K = k ) = 1 px + 2 px +
l
+ lx + 2 +
= x +1
lx
+ ω − x − 1 px
+ lω −1
l
+ 3 lx + 2 + 5 lx + 3 +
2
E K ( x )  = x +1


lx
ex: n = E  K ( x ) ∧ n  = 1 px + 2 px +
+ n px
e x = ex : n + n p x e x + n
UDD ⇒ e x: n = ex: n + .5 n qx (see the UDD below)
© BPP Professional Education: 2005 exams
Page 7
SURVIVAL MODELS
Other life table functions
1.
2.
For life expectancy calculations
x +1
Lx =
∫x
Tx =
∫x ly dy
ly dy
ω
= Lx + Lx + 1 +
+ Lω −1
Central rate of mortality
n
n mx
=
=
3.
∫ 0 t px µ ( x + t ) dt
n
∫ 0 t px dt
n dx
Tx − Tx + n
=
n qx
=
ex:n
=
n dx
Lx + Lx + n +
n dx
/ lx
(Tx − Tx +n ) / lx
+ Lx + n−1
=
deaths
exposure
Average part of the final year of life
a ( x ) = E T ( x )| T ( x ) ≤ 1 =
ex:1 =
dx ⋅ a ( x ) + lx + 1 ⋅ 1
1
∫0 t⋅
t px
µ (x + t)
t qx
dt
(weighted average over deaths
dx + lx + 1
and survivors)
=
dx ⋅ a ( x ) + lx + 1 ⋅ 1
⇒ a(x) =
Page 8
lx
ex:1 − px
qx
=
= q x a ( x ) + px
1
Lx − lx + 1
where ex:1 = ∫ t px dt
0
dx
© BPP Professional Education: 2005 exams