List of Formulas for Actuarial Mathematics Courses PDF

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Summary

𝑗+1 1 AMAT 171 𝑢|𝑡 𝑞𝑥 = 𝑃[𝑢 < 𝑇𝑥 ≤ 𝑢 + 𝑡] ∫𝑗 𝑡𝑝𝑥 𝑑𝑡 ≈ ( 𝑗𝑝𝑥 + 𝑗+1𝑝𝑥 ) 2. 1−𝑡𝑞𝑥+𝑡 = 1 − (𝑝𝑥 )1−𝑡 2 CHAPTER 1 = 𝑆𝑥 (𝑢) − 𝑆𝑥 (𝑢 + 𝑡) 3. 𝑦𝑞𝑥+𝑡 = 1 − (𝑝𝑥 ) 𝑦 𝑆(𝑥) = 1 − 𝐹(𝑥) = 𝑃(𝑇𝑥 > 𝑡) Identities: Trapezoidal Rule 4. 𝑡 𝑡𝑝𝑥 µ𝑥+𝑡 = −(𝑝𝑥 ) 𝑙𝑛𝑝𝑥 𝑡𝑝𝑥 + 𝑡𝑞𝑥 = 1 𝑏 1 𝑃[𝑇𝑥 ≤ 𝑡] = 𝑃[𝑇0 ≤ 𝑥 + 𝑡|𝑇0 > 𝑥] 1....

Description

AMAT 171 CHAPTER 1 = − =� �[ ] = �[ � + −� = =

+ = + = = = lim →∞

> + |

+

> ]

1. 2.

3. 4.

+

5. 6.

=

lim

→∞

+ = | = = + µ =

�

�

=

CHAPTER 2 = ,

FORCE OF MORTALITY

�[

� → +

µ = µ

µ

µ

+ + +

−

µ =

=

=

{− ∫ µ + � }

+

+

�

=∫

=∫

≈µ

µ

µ

+

∞

∞

=∫ µ + ∞ [ ]=∫ µ ∞

= ∫ Var[ ] = [ ] − ∞ = ∫

= ] = �[ = |

−

+

−

≈

+

2. 3. 4.

+

+

< ={ < ∗ :̅̅̅ = [ ] = ∫

∗

+

µ

+

1.

3. 4.

+

=

+

=

+

=

−

+

=

− ∫ +

+ ]

+

=

−

= [ ]= = ∑∞= P[ = ] [ ] = ∑∞= � − ∞ ∑ − + = = Var[ ] = [ ] − [ ] = ∑∞= � − −

The complete and curtate expected future lifetimes + = ∑∞= ∫ ≈ +

UDD1: =s , UDD2: = + ]= �[

<

Some Results:

1. 2.

3. 4. 5.

�

�

µ

+

−

=

=

+

= +

Constant Force ∗

= −µ =

Some Results:

1.

=

−

µ

−

=

−

−

n

∫

+ −

−

=∫

=∫

=∫ =∫

=

∫

∫

+

=

∫

+

�

+

∫

+

+

+

�

µ

+

+

�

+

+

µ

µ

µ

µ

+

+

Some identities: 1. = 2.

3. 4. 5.

:̅

= ≈

=

+ −

−

−

−

+

+

+

+

+

+

+

+

+

�

�

=

=

−

− − −

+

µ

+

µ

−

=

=

�

+

−

−

= =

+

µ

→∞

∫

+

µ

∞

−

=

+

=∫ = lim ∫

=

+

+ + µ

∫ ∞

=∫

=

−µ∗ = − ∗ = −µ =

+

+

−

5.

Fractional Age Assumptions UDD ∞

+

µ

is a binomial random variable,

=

−

+

Some Identities:

Life Tables

=

−

=

2.

E[ ] = Let =

+

=

CHAPTER 3

If

−

+

−

=

+

−

= − − + = µ + =− +

−

Balducci

Temporary Complete Life Expectancy

+

∑∞= −

{− ∫ µ � }

+

+

= [ ]=∫

P[

Pareto Law of Mortality

µ =

+

�

+

Curtate Future Lifetime

ln

Some Results: =

≈∫ µ

+

Central Moments

+

+

=−

µ =−

> ]

=

=− =

+ |

−

+

Trapezoidal Rule ≈ ∫

µ + {− ∫ µ + � }

=∫

8.

∫

= −

=−

=

7.

µ = lim

+ ] +

= �[ < − = Identities: |

+

+

= [ |

< ]

+[ −

]

+

Monica Revadulla, BSAM - UPLB

Select life table = −

=

[ ]+

a.

− −

b.

−

[ ]+

+

=

=

Term Insurance

+�

�−

[ ]+

[ ]+

Z={ ̅

[ ]+

[ ]+

3.

̅

={

>

−�

[ ]=∫

=∫

=E[ [ ]=

]= ∫ ̅ ́: −

́ :̅

̅

Discrete Case:

+

+

=

+

�

[ ]=∫ =∫

∞

∞

−

−�

µ

+

={ ́ :̅

Whole Life Insurance (Continuous) Z= = −δTx ̅ =

=

́ :̅

CHAPTER 4 Assumptions 1. = 2. � =

−

[ ]=

µ

+

= [ ]=∑ = ∑ =− − ́ :̅ = ∑ =

+

−

+

+

+

=∑ =

[ ]= ̅ =E[ ] = ∫∞ − � µ + Mth-ly ̅ − ̅ + [ ]= For constant force of mortality µ and force of Z={ interest δ, ̅ = [ ] = ∫∞ −� −µ µ = µ δ+µ [ ] = ∑ =− ́: ̅ = Discrete Case: + + = ∑ =− ́: = ∞ + = [ ]=∑ = | [ ]= + ́: − = ∑∞ =

+ = ∑∞= ∞ =∑ = [ ]= −

| +

+

P[

=

= ] = �[ ∑∞=

= ∑∞=

=

+

|

|

+

= |

µ

|

=

|

́:

|

+

> : ́ = [ ]= Continuous

̅ = ∫∞ = µ + Some identities: ̅ +́ : 1. | ̅ ́ : = ̅ 2. | ̅ ́ : = ̅ ́ :̅̅̅̅̅̅ + - ́:

́:

̅

=∑ ́: ̅ = ∑∞=

3. 4. 5. 6.

= = ́:

́:

− =

|

+ −

|

̅

|

Z={ ̅

:

=∫

Var[Z]= ={

̅

> :

−

µ

>

̅

+

:

+

́:

̅

̅

: ́

|

≈

:

́:

�

=

+

Z=

+

̅ )x=∫∞ ( Z= ∞ ( ̅ ̅)x=∫

+

µ

+

µ

Annually decreasing n-year insurance + = , ,…, − − Z={ = , + ,… − + − ́ : =∑ = |

Some identities: − 1. ́ : =∑ = = ∑ =− 2. ́ :̅̅̅̅̅̅̅ − = [v 3.

∞

+

́:

|

́ :̅̅̅̅̅̅ −�

́ :̅̅̅̅̅̅̅̅̅̅̅̅̅ + − + ⌉]

x=0,1,2,…,y-1 − ́ :̅̅̅̅̅̅̅ − = v

+

+

= [v

=

́:̅

́ :̅̅̅̅̅̅̅̅̅̅̅̅̅ + − + ⌉

̅

+

]+

+

= − ̅̅̅

CONTINUOUS LIFE ANNUITIES =̅ ̅

=

+

+

=∫

−

́ :̅̅̅̅̅̅̅̅̅̅̅̅̅ + − + ⌉,

+

CHAPTER 5

�

Varying Benefit Insurance Z= + ∞ APV=(I ̅)x=∫ + µ �

Annually increasing whole life insurance + Z=( + + = ∑∞= + |

5.

+

̅ =

+

]

Annually decreasing n-year term insurance Z= − ) − µ + (D ̅ ́ : = ∫

4.

́:

̅ � , �� � � �� Relationship of � Under UDD

̅

=

+

Deferred Insurance > Z={

+

|

+

[ ,

Standard Ultimate Survival Model Makeham’s Law: µ = + A=0.00022 B=2.7x10-6 C=1.124

+

−

+

=

2.

+

=

−

́:

Z={

+ ]

−

+

́:

+

+

ENDOWMENT INSURANCE Pure Endowment

+

M-thly

=

µ

−

− =

µ

> = + [ ] = ̅ ́: + : ́ [ ]+ [ ]+ [ ]= [ , ]=− [ ] [ ] Recursions 1. = + +

|

=

− �

̅ =

ln −� �

�� −

−�

∞

=∫

∞

−�̅ �

For constant δ and µ, ̅ = ̅

=

<

[ ]=∫ ̅ =

̅

,

�̅ − �̅ �

<

�

µ

+

µ+�

Monica Revadulla, BSAM - UPLB

=∫

−�̅

=

�

:

̅

=

̅

− �

:

1. 2. 3. 4. :

= ̅ + =̅ −̅ : ∞ [ ]= ∫ ̅̅̅̅̅̅̅̅ − =∫

|̅

=

∞

̅̅̅̅̅̅̅̅ −

�

̅

µ

+

+

−

µ ̅

+

+

�

�

µ ∫ ̅ ∞ = ̅ +∫

̅

:

= µ +� ̅

|̅ = −

:

−

−

̅̅̅̅̅̅̅ + =∑∞=

If g(k)= ̅̅̅̅̅̅̅̅̅ + ,� �

=

−

+

=

− = − =

̅̅̅̅̅̅ +

Some identities: + 1. :̅̅̅̅̅̅̅ − = :̅̅̅

=

=

−

−

:̅̅̅

= ∑∞=

=

|

={

̅|

̅̅̅̅̅̅ : ̅|

+

−

=

̅̅̅̅̅̅̅̅̅̅ + | ̅|

+

+

=

−

:̅̅̅

+ ∑∞=

∑∞=

̅̅̅̅̅̅ : ̅|

=

̅̅̅̅̅̅̅̅ + |

̅|

+

−

+

:̅̅̅

Whole Life Annuity Immediate = ̅̅̅̅̅ ∞ = ∑ = ̅̅̅ = ∑∞= +

Note:

́ :̅|

−

=

: ̅|

=

= [

−

]=

−

+ �

=

=

Var(Y)=

: ̅|

−(�

=

́ :̅| )

+

−

́ : ̅|

]+

−

́ : ̅|

−

:̅

−

+

=

∑∞ ℎ=

�

=

− ℎ

−

−

=

̅|

−

+

+ �

: ̅́

́ :̅| � :̅́

̅|

�

+ /

�

=

ℎ

Some identities: 1. = + − ∞ 2. = ̅| ̅| 3. 4.

: ̅|

CHAPTER 6

̅̅̅

[ �

=∑

̅̅̅

n-year certain and life annuity-due

Summation by parts: ∑ = = [g n + f n + � g m f m ] − ∑ = f k + Δg k

+

̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅ + : − +

−� :̅̅̅ � :̅̅̅

+

+

| ̅̅̅̅̅̅̅̅̅̅̅̅̅ + − |

̅̅̅̅̅̅ : ̅| = ̅̅̅|

+

�

̅̅̅̅̅

Life Annuities with m-thly payments

̅̅̅

=∑

:̅̅̅

={

̅̅̅̅̅̅̅̅̅ +

̅̅̅̅̅̅̅̅̅ + ]

−

n-year deferred whole life annuity-due

Whole life Annuity-Due = [ = ∑∞=

={

=

= ̅̅

Temporary life annuity-immediate

̅̅̅̅̅̅̅̅̅̅ +

=∑

:̅̅̅

2.

Discrete Life Annuities Y=

Some Identities: = + + −� = � = + = ∞ ̅̅̅̅ − ̅̅̅̅ ∞

={

+

̅ = [µ + �] ̅ −

+

N-year temporary life annuity-due

n-year certain and life annuity ̅ Y={ > ̅ ̅̅̅̅̅̅̅ µ + =∫ ̅ + :̅ ∞

+

̅̅̅̅̅̅̅̅̅ +

n-year deferred whole life annuity < Y={ ̅̅̅̅̅̅̅̅̅ − |̅

+ ∑∞=

=

n-year temporary life annuity < ̅ Y={ ̅ µ + +̅ ̅ : =∫ ̅

−� �

=

+

=

+� = [

−� +

: ̅|

̅̅̅̅̅̅̅̅̅ +

−� − ]

Equivalence Principle: [ ] = L= PV Benefit – PV Premiums Paid

If

=

− �̅ ̅̅̅̅̅ �̅( ̅

=

�̅ ̅

�̅ � For constant force of mortality and constant force of interest, ̅ −( ̅ ) ]

=[

+

�̅ = µ,

Premium Formulae: 1. Whole life insurance −

−

Increasing Annuities Annuity-due where payment increases with time = ∑∞= +

Annuity is payable for a maximum of n payments − + : ̅| = ∑ =

Annuity is payable continuously, payments increasing by 1 at the end of each year ̅ : ̅| = ∑ =− + ̅ :̅| |

Whole life continuous annuity where payment is t at exact time t

̅ ̅ = ∫∞

Benefit Premiums

=̅

�̅( ̅

2. n-year term insurance

�̅( ̅

=

�̅ ̅

=

�̅ :̅| ̅ :̅|

=

́ : ̅|

3. n-year endowment

�̅( ̅

: ̅|

4. h-payment whole life ̅ ̅ = ℎ� ( 5. h-payment n-year term

̅ ̅

ℎ� (

=

́ : ̅|

�̅ ́ :̅| ̅ :̅|

�̅ ̅ :ℎ̅|

�̅ ́ :̅| ̅ :ℎ̅|

6. n-year pure endowment �̅ ́ �̅( ̅ ̅|́ = :̅| =

:

�̅

Some identities:

1. �̅( ̅

:̅| −

(� ̅

=

�̅

:̅| )

̅ :̅|

:̅|

��̅ −�̅

(n-year endowment)

Monica Revadulla, BSAM - UPLB

2. �̅( ̅ ̅

3.

=

ℎ

−

=

:̅|

:̅| + � ̅

−

̅ � �

��̅ :̅| −�̅ :̅|

:̅|

=

� + �̅ − ln [ ] � � + �̅

<

Discrete Case

+

=

−�

� =

�

=

Premium Formulae:

̅̅̅̅̅̅̅̅̅ +

−

�

=

́ : ̅|

3. n-year endowment

�

=

: ̅|

4. h-payment whole life

�

: ̅|

́ : ̅|

=

:̅|

̅| :ℎ

:̅|

1. �

̅

=

=�

� ́ :̅|

ℎ�

=

ℎ� : ̅|

=

�

= �̅ { ̅ ̅|

:̅|

:̅|

�

( ̅

3. n-year endowment

�

( ̅

́ : ̅| : ̅|

4. h-payment whole life ℎ�

( ̅

= � �

2. �( ̅ ́ : ̅| ) = � � ́ :̅ (Term)

: ̅|

+

=

=

�

�

ℎ�

+

:̅̅̅|

( ̅

=

̅̅̅̅|

: ̅|

( ̅

�̅ ́ :̅|

=

Loss Formula 1. Whole Life: =

2. N-year term:

=

<

−

=

�

�̅

= −

�̅

�̅ :̅|

=

−

=

:̅|

=

:̅|

<

< ℎ + ̅ℎ| ̅̅̅

< + < ℎ + ̅̅̅̅ ℎ|

ℎ }

− �̅ { ̅ ̅|

}

−

<

}

Var[

ℎ }

De Moivre: Gompertz’: Makeham’s: Weibull: Pareto’s



De Moivre:



De Moivre:

µ

−

−

x

+ −

−

−

<

̅

}

+

−

−

−

=

̅

̅

−��

δ

δ

]

̅

[

] [

�̅ �̅ δ

] [

= P�[

−

�

−��

̅(̅ ) � +� ̅(̅ ) �+�

̅

̅ − |

− |

̅

> ] ]

+

̅ = ̅

> ]

]

� + �̅ ̅ − ln � � + �̅ ̅

•

̅

−

+

= ̅ − �̅

) � + �̅

( −

µ+δ Benefit Reserve = (APV of whole life insurance from age x+t)-(APV of future benefit premium payable after x+t at an annual rate of �̅ ̅ Trivial case: ̅ ̅ = Variance:

−

̅ ̅ � +� − ln ̅ ̅ � � +�

+

• � + �̅ ̅ Prospective Method: 1. Whole Life Insurance =

2.

̅ ( ̅[ ] ) = [ | > ] ̅ ̅ = + − �̅ ̅ ̅ + For constant μ and δ, ̅ ̅ = + − �̅ ̅ ̅ + = µ ̅ + = µ+δ �̅ ̅ = μ

̅| :ℎ

> ]= [ +

=

3.

+

|

�̅ �̅

Aggregate Mortality Assumption +

AMAT 172

�̅ :̅|

̅ ̅ ̅| − �

    

> ]=[ +

= =

Fully Continuous Benefit Reserve − = − �̅ ̅[ ] ̅̅̅̅̅̅̅̅̅ − |

̅| :ℎ

�

− �̅ { ̅ ̅| + ̅̅̅̅|

− �̅ { ̅ ̅|

− �̅ { ̅ ̅| + ̅̅̅̅|

�̅ �̅

|

=[ + Independent age:

Mortality and Survival Functions

̅| :ℎ

: ̅|

+

6. n-year pure endowment

̅| :ℎ

1. Whole life insurance: �

<

Var[

5. h-payment n-year endowment:

:̅|

5. h-payment n-year term

UDD Assumptions �̅

=

Benefit is paid at the moment of death

̅| :ℎ

� :̅| ́

=

4. h-payment whole life:

4. h-payment years, whole life

� :̅|

Write expression for L � = �, Set = in L, set L=0 and solve for P �̅

�

3. n-year endowment: � : ̅| =

Percentile Premiums 1. 2. 3.

=

� ́ : ̅| =

2. n-year term:

6. n-year pure endowment

�

�

1. Whole life:

2. n-year term insurance

:̅|

� :̅|

3. N-year endowment:

́ : ̅|

Benefit is paid at the end of year of death

Premium Formulae

ℎ� =