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Summary Introduction to Actuarial Science - All formulas and notations

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Introduction to Actuarial Science (EBB827A05)

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Studiejaar: 2011/2012

Boeken in lijstActuarial Mathematics Actuarial Mathematics for Life Contingent Risks

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Chapter 3: Survival Distributions and Life Tables

Distribution function ofX:

F

(x) = Pr(X≤x)

Survival functions(x):

s(x) = 1−F X

(x)

Probability of death between age x and

agey:

Pr(x &lt; X≤z) = F X

(z)−F X

(x)

= s(x)−s(z)

Probability of death between age x and

ageygiven survival to agex:

Pr(x &lt; X≤z|X &gt; x) =

FX(z)−FX(x)

1 −FX(x)

=

s(x)−s(z)

s(x)

Notations:

q x

= Pr[T(x)≤t]

= prob. (x) dies withintyears

= distribution function ofT(x)

p x

= Pr[T(x)&gt; t]

= prob. (x) attains agex+t

= 1−tqx

t|u

qx = Pr[t &lt; T(x)≤t+u]

=

t+u

q x

−

q x

= tpx−t+upx

=

p x

·

q x+t

Relations with survival functions:

p x

=

s(x+t)

s(x)

tqx = 1−

s(x+t)

s(x)

Curtate future lifetime (K(x)≡ greatest

integer inT(x)):

Pr[K(x) =k] = Pr[k≤T(x)&lt; k+ 1]

=

p x

−

p x

=

p x

·q x+k

=

Force of mortalityμ(x):

μ(x) =

fX(x)

1 −F

(x)

= −

′ (x)

s(x)

Relations between survival functions and

force of mortality:

s(x) = exp





−

x ∫

μ(y)dy





p x

= exp





−

x+n ∫

μ(y)dy





Derivatives:

q x

=

p x

·μ(x+t) =f T(x)

(t)

p x

= −

p x

·μ(x+t)

T

= −l x

L

= −d x

̊ex = μ(x) ̊ex− 1

Mean and variance ofT andK:

E[T(x)] ≡ complete expectation of life

≡ ̊e x

=

∞ ∫

p x

E[K(x)] ≡ curtate expectation of life

≡ e x

=

∞ ∑

p x

V ar[T(x)] = 2

∞ ∫

t· t

p x

dt− ̊e

V ar[K(x)] =

∞ ∑

(2k−1) k

p x

−e

Total lifetime after agex: T x

Tx=

∞ ∫

lx+tdt

©c

Total lifetime between agexandx+ 1: Lx

L

= T

−T

=

1 ∫

lx+tdt=

1 ∫

lx·tpxdt

Total lifetime from agextox+n: nLx

nLx = Tx−Tx+n=

n− 1 ∑

Lx+k

=

n ∫

lx+tdt

Average lifetime afterx: ̊ex

̊ex=

T

l x

Average lifetime fromxtox+ 1: ̊e x: 1

̊e x: 1

=

L

Median future lifetime of (x): m(x)

P r[T(x)&gt; m(x)] =

s(x+m(x))

s(x)

=

1

2

Central death rate: mx

mx =

l x

−l x+

L

nmx =

l x

−l x+n

L

Fraction of year lived between agexand

agex+ 1bydx: a(x)

a(x) =

1 ∫

t· t

p x

·μ(x+t)dt

1 ∫

tpx·μ(x+t)dt

=

L

−l x+

l x

−l x+

Recursion formulas:

E[K] =e x

= p x

(1 +e x+

)

E[T] = ̊e x

= p x

(1 + ̊e x+

) +q x

a(x)

ex = ex:n+npxex+n

̊e x

= ̊e x:n

+

p x

̊e x+n

E[K∧(m+n)] = e x:m+n

= e x:m

+

p x

e x+m:n

E[T∧(m+n)] = ̊e x:m+n

= ̊ex:m+mpx ̊ex+m:n

©c

Varying benefit insurances:

(I

̄

A)

=

∞ ∫

⌊t+ 1⌋v

t · t

p x

μ x

(t)dt

(I

̄

A)

x:n

=

n ∫

⌊t+ 1⌋v

t · t

p x

μ x

(t)dt

(IA)

=

∞ ∫

t·v

t · t

p x

μ x

(t)dt

(IA)

x:n

=

n ∫

t·v

t · t

p x

μ x

(t)dt

(D

̄

A)

x:n

=

n ∫

(n−⌊t⌋)v

t · t

p x

μ x

(t)dt

(DA)

x:n

=

n ∫

(n−t)v

t · t

p x

μ x

(t)dt

(IA)x = Ax+vpx(IA)x+

= vq x

+vp x

[(IA)

+A

]

(DA)

x:n

= nvqx+vpx(DA)

x+1:n− 1

(IA)

x:n

+ (DA)

x:n

̄

A

x:n

(I

̄

A)

x:n

+ (D

̄

A)

x:n

= (n+ 1)

̄

A

x:n

(IA)

x:n

+ (DA)

x:n

= (n+ 1)A

x:n

Accumulated cost of insurance:

̄

k x

=

̄

A

x:n

nEx

Share of the survivor:

accumulation factor =

1

nEx

=

(1 +i)

npx

Interest theory reminder

a n

=

1 −v

̄a n

=

1 −v

=

a n

̄a∞ =

1

, a∞=

1

, ̈a∞=

1

(Ia) n

=

̈a n

−nv

(Da) n

=

n− ̄an

(Ia) ∞

=

1

(n+ 1)an = (Ia)n+ (Da)n

s ̄ 1

=

d = iv

(Ia) ∞

=

1 =

1 +i

Doubling the constant force of interestδ

1 +i → (1 +i)

v → v

i → 2 i+i

d → 2 d−d

→

2 i+i

2 δ

Limit of interest ratei= 0:

−→ 1

A

x:n

−→ n

q x

−→ npx

A

x:n

−→ 1

m|n

A

−→ m|n

q x

(IA)

−→ 1 +e x

(IA)

−→ ̊e x

©c

Chapter 5: Life Annuities

Whole life annuity: ̄a x

̄a x

= E[ ̄a T

] =

∞ ∫

̄a t

·

p x

μ(x+t)dt

=

∞ ∫

t · t

p x

dt=

∞ ∫

E

V ar[ ̄a T

] =

2 ̄

A

−(

̄

A

)

n-year temporary annuity: ̄a x:n

̄ax:n =

n ∫

·tpxdt=

n ∫

tExdt

V ar[Y] =

2 ̄

A

x:n

−(

̄

A

x:n

)

n-year deferred annuity: n|

̄a x

̄ax =

∞ ∫

·tpxdt=

∞ ∫

tExdt

̄ax = v

·npx ̄ax+n= nEx·a ̄x+n

V ar[Y] =

2

2 n · n

p x

( ̄a x+n

−

2 ̄a x+n

)−(

̄a x

)

n-yr certain and life annuity: ̄a x:n

̄a x:n

= ̄ax+ ̄a n

− ̄a x:n

= ̄a n

+

a ̄ x

= ̄a n

+

E

·a ̄ x+n

Most important identity

1 = δ ̄a x

+

̄

A

̄a x

=

1 −

̄

Ax:n = 1−δa ̄x:n

2 ̄

A

x:n

= 1−(2δ)

2 a ̄ x:n

̈ax =

1 −A

̈a x:n

=

1 −Ax:n

1 = d ̈a x:n

+A

x:n

Recursion relations

̄a x

= ̄a x: 1

+vp x

̄a x+

̄a x

= ̄a x:n

+

a ̄ x

̈a x

= 1 +vp x

̈a x+

2 ̈a x

= 1 +v

2 p x

2 ̈a x+

̈ax:n = 1 +vpx ̈a x+1:n− 1

̈a

(m)

x:n

= ̈a

(m)

−

E

· ̈a

(m)

x+n

ax = vpx+vpxax+

a x:n

= vp x

+vp x

a x+1:n− 1

(I ̈a)x = 1 +vpx[(I ̈a)x+1+ ̈ax+1]

= ̈a x

+vp x

(I ̈a) x+

Whole life annuity due:a ̈x

̈ax = E[ ̈a K+

] =

∞ ∑

· k

V ar[ ̈a K+

] =

2 A x

−(A

)

n-yr temporary annuity due: ̈ax:n

̈ax:n = E[Y] =

n− 1 ∑

·kpx

V ar[Y] =

2 A x:n

−(A

x:n

)

n-yr deferred annuity due: n|

̈a x

̈ax = E[Y] =

∞ ∑

k=n

·kpx

= ̈a x

− ̈a x:n

= nEx· ̈ax+n

n-yr certain and life due: ̈a x:n

̈a x:n

= ̈a x

̈a n

−a ̈ x:n

= ̈a n

+

∞ ∑

k=n

k · k

p x

= ̈a n

+

̈a x

©c

Chapter 6: Benefit Premiums

Loss function:

Loss = PV of Benefits−PV of Premiums

Fully continuous equivalence premiums

(whole life and endowment only):

̄

P(

̄

A

) =

̄

̄a x

̄

P(

̄

A

) =

̄

A

1 −

̄

A

̄

P(

̄

A

) =

1

̄ax

−δ

V ar[L] =

(

1 +

̄

P

)

[ 2 ̄ Ax−(

̄

Ax)

]

V ar[L] =

2 ̄

A

−(

̄

A

)

(δ ̄a x

)

V ar[L] =

2 ̄

Ax−(

̄

Ax)

(1−

̄

A

)

Fully discrete equivalence premiums

(whole life and endowment only):

P(A

) =

̈a x

=P

P(A

) =

dAx

1 −Ax

P(Ax) =

1

̈a x

−d

V ar[L] =

(

1 +

P

)

2 [ 2 A x

−(A

)

]

V ar[L] =

2 A x

−(A

)

(d ̈ax)

V ar[L] =

2 Ax−(Ax)

(1−A

)

Semicontinuous equivalence premiums:

P(

̄

A

) =

̄

A

̈a x

m-thly equivalence premiums:

P

(m)

null

=

A

null

̈a

(m)

null

h-payment insurance premiums:

̄

P(

̄

A

) =

̄

̄a x:h

̄

P(

̄

A

x:n

) =

̄

A

x:n

̄a x:h

P

=

A

̈a x:h

hPx:n =

A

x:n

̈a x:h

Pure endowment annual premiumP

x:n

:

it is the reciprocal of the actuarial accumulated

value ̈s x:n

because the share of the survivor who

has depositedP

x:n

at the beginning of each year

for n years is the contractual $1 pure endow-

ment, i.

P

x:n

̈s x:n

= 1 (1)

P minusP overP problems:

The difference in magnitude of level benefit pre-

miums is solely attributable to the investment

feature of the contract. Hence, comparisons of

the policy values of survivors at agex+nmay

be done by analyzing future benefits:

(

P

−P

x:n

) ̈s x:n

= A

x+n

=

P

−P

x:n

P

x:n

(P

x:n

−

P

) ̈s x:n

= 1−A

x+n

=

P

x:n

−

P

x:n

(P

x:n

−P

x:n

) ̈s x:n

= 1 =

P

x:n

−P

x:n

P

x:n

Miscellaneous identities:

̄

A

x:n

=

̄

P(

̄

A

x:n

)

̄

P(

̄

A

x:n

) +δ

A

x:n

=

P

x:n

P

x:n

a ̄ x:n

=

1 ̄

P(

̄

Ax:n) +δ

a ̈ x:n

=

1 P

x:n

©c

Chapter 7: Benefit Reserves

Benefit reserve tV:

The expected value of the prospective loss at

timet.

Continuous reserve formulas:

Prospective: t

̄

V(

̄

Ax) =

̄

Ax+t−

̄

P(

̄

Ax) ̄ax+t

Retrospective: t

̄

V(

̄

A

) =

̄

P(

̄

A

) ̄s x:t

−

̄

A

x:n

E

Premium diff.: t

̄

V(

̄

Ax) =

[

̄

P(

̄

Ax+t)−

̄

P(

̄

Ax)

]

̄ax+t

Paid-up Ins.: t

̄

V(

̄

A

) =

[

1 −

̄

P(

̄

Ax)

̄

P(

̄

Ax+t)

]

̄

A

x+t

Annuity res.: t

̄

V(

̄

A

) = 1−

̄ax+t

̄ax

Death ben.: t

̄

V(

̄

A

) =

̄

A

x+t

−

̄

A

1 −

̄

A

Premium res.: t

̄

V(

̄

A

) =

̄

P(

̄

Ax+t)−

̄

P(

̄

Ax)

̄

P(

̄

A

x+t

) +δ

Discrete reserve formulas:

V

= A

x+k

−P

̈a x+k

V

x:n

=

[

P

x+k:n−k

−P

x:n

]

̈a x+k:n−k

V

x:n

=

[

1 −

P

x:n

P

x+k:n−k

]

A

x+k:n−k

tVx:n = Px:ns ̈x:n−tkx

kVx = 1−

a ̈ x+k

̈a x

kVx =

P

x+k

−P

P

x+k

V

=

A

x+k

−A

1 −A

h-payment reserves:

̄

V =

̄

A

x+t:n−t

−

̄

P(

̄

A

x:n

) ̄a x+t:h−t

Vx:n = A x+k:n−k

−hPx:n ̈a x+k:h−k

V(

̄

A

x:n

) =

̄

A

x+k:n−k

−

P(

̄

A

x:n

) ̈a x+k:h−k

V

1(m)

x:n

= A

x+k:n−k

−

P

(m)

x:n

̈a

(m)

x+k:h−k

Variance of the loss function

V ar[ t

L] =

(

1 +

̄

P

)

2 [

2 ̄ A x+t

−(

̄

A

x+t

)

]

V ar[ t

L] =

2 ̄

Ax+t−(

̄

Ax+t)

(1−

̄

Ax)

assuming EP

V ar[tL] =

(

1 +

̄

P

)

2 [

2 ̄ A x+t:n−t

−(

̄

A

x+t:n−t

)

]

V ar[ t

L] =

2 ̄

A

x+t:n−t

−(

̄

A

x+t:n−t

)

(1−

̄

A

x:n

)

assuming EP

Cost of insurance: funding of the accumu-

lated costs of the death claims incurred between

agexandx+tby the living att,e.

4 kx =

d x

(1 +i)

3 +d x+

(1 +i)

2 +d x+

(1 +i) +d x+

l x+

=

A

x: 4

4 Ex

k x

=

d x

lx+

=

q x

Accumulated differences of premiums:

(

P

−P

x:n

) ̈s x:n

=

V

−

V

x:n

)

= A

x+n

−0 =A

x+n

(

P

−P

) ̈s x:n

=

V

−

V

= Px ̈ax+n

(P

x:n

−P

) ̈s x:n

=

V

x:n

−

V

= 1−nVx

(

P

x:n

−

P

) ̈s x:m

=

V

x:n

−

V

= A

x+m:n−m

−Ax+m

Relation between various terminal re-

serves (whole life/endowment only):

m+n+pVx = 1−

(1−

V

)(1−

V

x+m

)(1−

V

x+m+n

)

©c

If the death benefit is equal to $1 plus the benefit reserve for the firstnpolicy years andqx+h≡q

constant

nV=Ps ̈n−vq ̈sn = (P−vq) ̈sn

Reserves at fractional durations:

(

V+π h

)(1 +i)

s = s

p x+h

·

h+s

V +

q x+h

1 −s

UDD ⇒ (hV+πh)(1 +i)

= (1−s·qx+h)h+sV +s·qx+hv

1 −s

=

h+s

V +s·q x+h

1 −s − h+s

V)

h+sV = v

1 −s

· 1 −sqx+h+s·bh+1+v

1 −s

· 1 −spx+h+s·h+1V

UDD ⇒

h+s

V = (1−s)( h

V +π h

) +s( h+

V)

i. h+s

V = (1−s)( h

V) + (s)( h+

V) + (1−s)(π h

)

︸︷︷︸

unearned premium

Next year losses:

Λ

≡ losses incurred from timehtoh+ 1

E[Λ

] = 0

V ar[Λh] = v

(bh+1−h+1V)

px+hqx+h

The Hattendorf theorem

V ar[hL] = V ar[Λh] +v

px+hV ar[h+1L]

= v

2 (b h+

−

V)

2 p x+h

·q x+h

2 p x+h

V ar[ h+

L]

V ar[hL] = v

(bh+1−h+1V)

px+h·qx+h

4 (b h+

−

V)

2 p x+h

·p x+h+

·q x+h+

(bh+3−h+3V)

px+h·px+h+1·px+h+2·qx+h+2+···

©c

Chapter 9: Multiple Life Functions

Joint survival function:

s T(x)T(y)

(s, t) = P r[T(x)&gt; s&amp;T(y)&gt; t]

p xy

= s T(x)T(y)

(t, t)

= P r[T(x)&gt; tandT(y)&gt; t]

Joint life status:

F

(t) = P r[min(T(x), T(y))≤t]

= tqxy

= 1−

p xy

Independant lives

p xy

=

p x

·

p y

tqxy = tqx+tqy−tqx·tqy

Complete expectation of the joint-life sta-

tus:

̊e xy

=

∞ ∫

p xy

PDF joint-life status:

f T(xy)

(t) = tpxy·μxy(t)

μxy(t) =

f T(xy)

(t)

1 −F

T(xy)

(t)

=

f T(xy)

(t)

p xy

Independant lives

μxy(t) = μ(x+t) +μ(y+t)

f T(xy)

(t) = t

p x

·

p y

[μ(x+t) +μ(y+t)]

Curtate joint-life functions:

p xy

=

p x

·

p y

[IL]

q xy

=

q x

+

q y

−

q x

·

q y

[IL]

P r[K=k] = k

p xy

−

p xy

=

p xy

·q x+k:y+k

= kpxy·qx+k:y+k= k|

qxy

q x+t:y+t

= q x+k

+q y+k

−q x+k

·q y+k

[IL]

exy = E[K(xy)] =

∞ ∑

kpxy

Last survivor statusT(xy):

T(xy) +T(xy) = T(x) +T(y)

T(xy)·T(xy) = T(x)·T(y)

f T(xy)

+f T(xy)

= f T(x)

+f T(y)

F

T(xy)

+F

T(xy)

= F

T(x)

+F

T(y)

p xy

+

p xy

=

p x

+

p y

̄

A

+

̄

A

=

̄

A

+

̄

A

̄axy+ ̄axy = ̄ax+ ̄ay

̊e xy

̊e xy

= ̊e x

̊e y

exy+exy = ex+ey

q xy

=

q x

+

q y

−

q xy

Complete expectation of the last-survivor

status:

̊exy =

∞ ∫

tpxydt

e xy

=

∞ ∑

p xy

Variances:

V ar[T(u)] = 2

∞ ∫

t·tpudt−( ̊eu)

V ar[T(xy)] = 2

∞ ∫

t·tpxydt−( ̊exy)

V ar[T(xy)] = 2

∞ ∫

t·tpxydt−( ̊exy)

Notes:

For joint-life status, work withp’s:

p xy

=

p x

·

p y

For last-survivor status, work withq’s:

q xy

=

q x

·

q y

“Exactly one” status:

[1]

= npxy−npxy

=

p x

+

p y

− 2

p x

·

p y

= nqx+nqy− 2 nqx·nqy

a ̄

[1]

= ̄ax+ ̄ay−2 ̄axy

©c

Chapter 10 &amp; 11: Multiple Decrement Models

Notations:

(j)

= probability of decrement in the next

tyears due to causej

(τ)

= probability of decrement in the next

tyears due to all causes

=

m ∑

(j)

= the force of decrement due only

to decrementj

(τ)

= the force of decrement due to all

causes simultaneously

=

m ∑

(j)

(τ)

= probability of survivingtyears

despite all decrements

= 1−

(τ)

= e

−

❘t

(τ)

x (s)ds

Derivative:

(

(τ)

)

=−

(

(τ)

)

=−

(τ)

Integral forms of t

q x

:

(j)

=

t ∫

(τ)

·μ

(j)

(s)ds

(τ)

=

t ∫

(τ)

·μ

(τ)

(s)ds

Probability density functions:

Joint PDF: f T,J

(t, j) = t

(τ)

·μ

(j)

(t)

Marginal PDF ofJ: f J

(j) = ∞

(j)

=

∞ ∫

fT,J(t, j)dt

Marginal PDF ofT: fT(t) =tp

(τ)

·μ

(τ)

(t)

=

m ∑

f T,J

(t, j)

Conditional PDF: f J|T

(j|t) =

(j)

x (t)

(τ)

x (t)

Survivorship group:

Group ofl

(τ)

a people at some ageaat timet= 0.

Each member of the group has a joint pdf for

time until decrement and cause of decrement.

(j)

= l

(τ)

·

x−a

(τ)

·

(j)

= l

(τ)

x−a+n ∫

x−a

(τ)

·μ

(j)

(t)dt

(τ)

=

m ∑

(j)

(τ)

=

m ∑

(j)

=

(j)

(τ)

Associated single decrement:

′ (j)

= probability of decrement from causejonly

′ (j)

= exp





−

t ∫

(j)

(s)ds





= 1−

′ (j)

©c

Basic relationships:

(τ)

= exp







−

t ∫

[

(1)

(s) +···+μ

(m)

(s)

]







(τ)

=

m ∏

′ (i)

′ (j)

≥

(j)

′ (j)

≥ tp

(τ)

UDD for multiple decrements:

(j)

= t·q

(j)

(τ)

= t·q

(τ)

(j)

=

(τ)

·μ

(j)

(t)

(j)

(t) =

(j)

(τ)

=

(j)

1 −t·q

(τ)

′ (j)

=

(

(τ)

(j)

qxt

Decrements uniformly distributed in the

associated single decrement table:

′ (j)

= t·q

′ (j)

(1)

= q

′ (1)

(

1 −

1

2

′ (2)

)

(2)

= q

′ (2)

(

1 −

1

2

′ (1)

)

(1)

= q

′ (1)

(

1 −

1

2

′ (2)

−

1

2

′ (3)

+

1

3

′ (2)

·q

′ (3)

)

Actuarial present values

̄

A=

m ∑





∞ ∫

B

(j)

x+t

· t

(τ)

(j)

(t)dt





Instead of summing the benefits 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/subtract other benefits

according to the cause of death.

Premiums:

P

(τ)

=

∞ ∑

B

(τ)

k+ · k

(τ)

x ·q

(τ)

x+k

∞ ∑

k · k

(τ)

P

(j)

=

∞ ∑

B

(j)

k+ · k

(τ)

·q

(j)

x+k

∞ ∑

k · k

(τ)

©c

Constant Force of Mortality

Chapter 3

μ(x) = μ &gt; 0 , ∀x

s(x) = e

−μx

l x

= l 0

−μx

npx = e

−nμ

= (px)

̊ex =

1 =E[T] =E[X]

̊ex:n = ̊ex(1−npx)

V ar[T] = V ar[x] =

1

m x

= μ

Median[T] =

ln 2

= Median[X]

ex =

p x

q x

=E[K]

V ar[K] =

p x

(q x

)

Chapter 4

̄

A

=

μ+δ

2 ̄

A

=

μ+ 2δ

̄

A

x:n

=

̄

Ax(1−nEx)

E

= e

−n(μ+δ)

(IA)

=

(μ+δ)

A

=

q+i

A x

=

q+ 2i+i

A

x:n

= A

(1−

E

)

Chapter 5

̄a x

=

1

μ+δ

2 ̄a x

=

1

μ+ 2δ

̈a x

=

1 +i

q+i

2 ̈a x

=

(1 +i)

q+ 2i+i

̄a x:n

= ̄a x

(1−

E

)

̈ax:n = ̈ax(1−nEx)

(IA)

=

̄

A

̄a x

=

(μ+δ)

(I

̄

A)

=

̄

A

̈a x

=

μ(1 +i)

(μ+δ)(q+i)

(IA)x = Ax ̈ax=

q(1 +i)

(q+i)

(Ia) x

= ( ̄a x

)

2 =

1

(μ+δ)

(Ia ̈) x

= ( ̈a x

)

2 =

(

1 +i

q+i

)

Chapter 6

Px = vqx=P

x:n

̄

P(

̄

A

) = μ=

̄

P(

̄

A

x:n

)

For fully discrete whole life, w/ EP,

V ar[Loss] =p·

For fully continuous whole life, w/EP,

V ar[Loss] =

2 ̄

A

Chapter 7

̄

V(

̄

Ax) = 0, t≥ 0

V

= 0, k= 0, 1 , 2 ,...

For fully discrete whole life, assuming EP,

V ar[ k

Loss] =p·

Ax, k= 0, 1 , 2 ,...

For fully continuous whole life, assuming EP,

V ar[ t

Loss] =

2 ̄

A

, t≥ 0

Chapter 9

For two constant forces,i. μ

M acting on (x)

andμ

F acting on (y), we have:

̄

Axy =

M +μ

F +δ

a ̄xy =

1

M +μ

F +δ

̊exy =

1

M +μ

A

=

q xy

qxy+i

a ̈xy =

1 +i

q xy

e xy

=

p xy

q xy

©c

De Moivre’s Law

Chapter 3

s(x) = 1−

l x

= l 0

ω−x

∝(ω−x)

q x

= μ(x) =

1

ω−x

n|m

qx =

ω−x

npx =

ω−x−n

ω−x

p x

μ(x+t) = q x

=μ(x) =f T

(x), 0 ≤t &lt; ω−x

L

=

1

2

(l x

+l x+

)

̊ex =

ω−x

2

=E[T] = Median[T]

ex =

ω−x

2 −

1

2 =E[K]

V ar[T] =

(ω−x)

12

V ar[K] =

(ω−x)

2 − 1

12

m x

=

q x

1 −

q x

=

2 d x

l x

+l x+

a(x) = E[S] =

1

2

̊e x:n

= nnpx+

2

nqx

̊e x:n

= e x:n

+

2

q x

Chapter 4

̄

A

=

̄a ω−x

ω−x

̄

A

x:n

=

a ̄ n

ω−x

2 ̄ A x

=

̄a

2(ω−x)

A

=

a ω−x

ω−x

A

x:n

=

a n

ω−x

(IA)

=

(Ia) ω−x

ω−x

(IA)x =

(Ia) ω−x

ω−x

(IA)

x:n

=

(Ia)n

ω−x

(IA)

x:n

=

(Ia)n

ω−x

Chapter 5

No useful formulas: use ̈ax =

1 −Ax

and the

chapter 4 formulas.

Chapter 9

̊exx =

ω−x

3

(≡MDML withμ= 2/(ω−x))

̊exx =

2(ω−x)

3

̊exy = y−xpx ̊eyy+y−xqx ̊ey

For two lives with differentω’s, simply translate

one of the age by the difference inω’s.E.

Age 30, ω= 100⇔Age 15, ω= 85

Modified De Moivre’s Law

Chapter 3

s(x) =

(

1 −

)

l x

= l 0

(

ω−x

)

∝(ω−x)

μ(x) =

ω−x

p x

=

(

ω−x−n

ω−x

)

̊e x

=

ω−x

c+ 1

=E[T]

V ar[T] =

(ω−x)

2 c

(c+ 1)

2 (c+ 2)

Chapter 9

̊exx =

ω−x

2 c+ 1

≡ ̊exwithμ=

2 c

ω−x

©c

Chapter 1: The Poisson Process

Poisson process with rateλ:

P r[N(s+t)−N(s) =k] = e

−λt

(λt)

E[N(t)] = λt

V ar[N(t)] = λt

Interarrival time distribution:

The waiting time between events. Let T n

de-

note the time since occurence of the eventn−1.

Then theT n

are independent random variables

following an exponential distribution with mean

1 /λ.

P r[Ti≤t] = 1−e

−λt

f T

(t) = λe

−λt

E[T] =

1

V ar[T] =

1

Waiting time distribution:

LetSn be the time of then-occurence of the

event,i.e n

=

∑

T

.

S

has a gamma distribution with parametersn

andθ= 1/λ

S

≡ GammaRV[α=n, θ=

1 ]

P r[S n

≤t] =

∞ ∑

j=n

−λt

(λt)

f Sn

(t) = λe

−λt

(λt)

n− 1

(n−i)!

E[S

] =

V ar[S n

] =

Sum of Poisson processes:

IfN 1

,···, N

are independent Poisson processes

with ratesλ 1

,···, λ k

then,N=N 1

+···+N

is a Poisson process with rateλ=λ 1

+···+λ k

.

Special events in a Poisson process:

Let N be a Poisson process with rate λ.

Some events i are special with a probability

P r[event is special] =πiand

̃

Nicounts the spe-

cial events of kind i. Then,

̃

N

is a Poisson

process with rate

̃

λi=πiλand the

̃

Niare inde-

pendent of one another.

If the probabilityπ i

(t) changes with time, then

E[

̃

N

(t)] =λ

t ∫

π(s)ds

Non-homogeneous Poisson process:

λ(t) ≡ intensity function

m(t) ≡ mean value function

=

t ∫

λ(y)dy

P r[N(t) =k] = e

−m(t)

[m(t)]

Compound Poisson process:

X(t) =

N(t)

∑

Y

N(t) PoissonRV w/ rateλ

E[X(t)] = λt·E[Y]

V ar[X(t)] = λt·E[Y

]

Two competing Poisson processes:

Probability thatnevents in the Poisson process (N 1 ,λ 1 ) occur beforemevents in the Poisson process

(N

,λ 2

):

P r[S

&lt; S

] =

n+m− 1 ∑

k=n

(

n+m− 1

) (

λ 1

λ 1 +λ 2

)

(

λ 2

λ 1 +λ 2

)

n+m− 1 −k

P r[S

&lt; S

] =

(

λ 1

+λ 2

)

P r[S

&lt; S

] =

λ 1

+λ 2

Exam M - Loss Models - LGD

©c 1

Chapter 2&amp;3: Random Variables

k-th raw moment:

′

=E[X

]

k-th central moment:

μ k

=E[(X−μ)

k ]

Variance:

V ar[x] = σ

2 =E[X

2 ]−E[X]

= μ 2

=μ

′

−μ

Standard deviation:

σ=

√

V ar[X]

Coefficient of variation:

σ/μ

Skewness:

γ 1

=

E[(X−μ)

3 ]

=

μ 3

Kurtosis:

γ 1

=

E[(X−μ)

4 ]

=

μ 4

Left truncated and shifted variable (aka

excess loss variable):

Y

P =X−d|X &gt; d

Mean excess loss function:

e X

(d) ≡ e(d) =E[Y

P ]

= E[X−d|X &gt; d]

=

∫

∞

S(x)dx

1 −F(d)

=

E[X]−E[X∧d]

1 −F(d)

Higher moments of the excess loss vari-

able:

(d) = E[(X−d)

|X &gt; d]

=

∞ ∫

(x−d)

k f(x)dx

1 −F(d)

=

∑

x&gt;d

(x−d)

k p(x)

1 −F(d)

Left censored and shifted variable:

Y

= (X−d)+=

{

0 X &lt; d

X−d X≥d

Moments of the left censored and shifted

variable:

E[(X−d)

] =

∞ ∫

(x−d)

f(x)dx

=

∑

x&gt;d

(x−d)

k p(x)

= e

k (d)[1−F(d)]

Limited loss:

Y = (X∧u) =

{

X X &lt; u

u X≥u

Limited expected value:

E[X∧u] = −

0 ∫

−∞

F(x)dx+

u ∫

S(x)dx

=

u ∫

[1−F(x)]dx if X is always positive

Moments of the limited loss variable:

E[(X∧u)

k ] =

u ∫

−∞

k f(x)dx+u

k [1−F(u)]

=

∑

x≤u

p(x) +u

[1−F(u)]

Moment generating functionsm X

(t):

m X

(t) =E[e

tX ]

Sum of random variablesS k

=X 1 +···+X

:

m Sk

(t) =

k ∏

m Xj

(t)

Exam M - Loss Models - LGD

©c 2

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Summary Introduction to Actuarial Science - All formulas and notations

Vak: Introduction to Actuarial Science (EBB827A05)

54 Documenten

Studenten deelden 54 documenten in dit vak

Universiteit: Rijksuniversiteit Groningen

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Chapter 3: Survival Distributions and Life Tables

Distribution function of X:

FX(x) = Pr(X≤x)

Survival function s(x):

s(x) = 1 −FX(x)

Probability of death between age xand

age y:

Pr(x < X ≤z) = FX(z)−FX(x)

=s(x)−s(z)

Probability of death between age xand

age ygiven survival to age x:

Pr(x < X ≤z|X > x) = FX(z)−FX(x)

1−FX(x)

=s(x)−s(z)

s(x)

Notations:

tqx= Pr[T(x)≤t]

= prob. (x) dies within tyears

= distribution function of T(x)

tpx= Pr[T(x)> t]

= prob. (x) attains age x+t

= 1 −tqx

t|uqx= Pr[t < T (x)≤t+u]

=t+uqx−tqx

=tpx−t+upx

=tpx·uqx+t

Relations with survival functions:

tpx=s(x+t)

s(x)

tqx= 1 −s(x+t)

s(x)

Curtate future lifetime (K(x)≡greatest

integer in T(x)):

Pr[K(x) = k] = Pr[k≤T(x)< k + 1]

=kpx−k+1px

=kpx·qx+k

=k|qx

Force of mortality µ(x):

µ(x) = fX(x)

1−FX(x)

=−s′(x)

s(x)

Relations between survival functions and

force of mortality:

s(x) = exp 

−

µ(y)dy



npx= exp 

−

x+n

µ(y)dy



Derivatives:

dt tqx=tpx·µ(x+t) = fT(x)(t)

dt tpx=−tpx·µ(x+t)

dtTx=−lx

dtLx=−dx

dt˚ex=µ(x)˚ex−1

Mean and variance of Tand K:

E[T(x)] ≡complete expectation of life

≡˚ex=

∞

tpxdt

E[K(x)] ≡curtate expectation of life

≡ex=

∞

k=1

kpx

V ar[T(x)] = 2

∞

t·tpxdt −˚e2

V ar[K(x)] =

∞

k=1

(2k−1) kpx−e2

Total lifetime after age x: Tx

Tx=

∞

lx+tdt

Exam M - Life Contingencies - LGD c

1

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