Formula Sheet for Actuarial Mathematics - Exam MLC - (ASM 2014) PDF

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1 Lesson 1 - Probability Review Lesson 2 – Survival Distributions: Probability Functions 1.1 𝜇2 = 𝜇2′ − 𝜇 2 2.1 𝑆𝑥+𝑡 (𝑢) = 𝑆𝑥 (𝑡+𝑢) 𝑆𝑥 (𝑡) 1.2 𝜇3 = 𝜇3′ − 3𝜇2′ 𝜇 − 2𝜇 3 𝑆0 (𝑥+𝑡) 1.3 𝑉𝑎𝑟(𝑋) = 𝐸[𝑋 2 ] − 𝐸[𝑋]2 2.2 𝑆𝑥 (𝑡) = 𝑆0 (𝑥) 1.4 𝑉𝑎𝑟(𝑎𝑋 + 𝑏𝑌) = 𝑎2 𝑉𝑎𝑟(𝑋) + 2𝑎𝑏𝐶𝑜𝑣(𝑋, 𝑌) + 𝑏 2 𝑉𝑎𝑟(𝑌) 𝐹0 (𝑥+𝑡)−𝐹0 (𝑥) 2...

Description

1 Lesson 2 – Survival Distributions: Probability Functions

Lesson 1 - Probability Review 1.1 1.2 1.3 1.4 1.5 1.6

� = �′ − � � = �′ − �′ � − � = [ ]− [ ] + = ∑= = ̅ =

1.7

Bayes Theorem

1.8

Pr

1.9 1.10 1.11 1.12 1.13

|

|

=

=

∑=

=

Pr( | )Pr⁡

2.1 2.2 +

�

=

,

�

+

2.4 2.5

Distribution Bernoulli Binomial Uniform Exponential

Mean

+

�

|

]+

Variance − −

�

=

|

+

=

+ −

=

−

=

|

+

−

−

+

+

Life Table Functions

Pr⁡ ( | )

[

=

2.3

=

Law of Total Probability (Discrete) Pr⁡ | = ∑ Pr Pr = ∑ Pr ∩ Law of Total Probability (Continuous) Pr = ∫ Pr | Conditional Mean Formula [ ] = [ [ | ]] Double Expectation Formula ]= [ [ [ | ]] Conditional Variance Formula =

+

=

| +

= =

+

= +

− +

=

+ −

+ +

+

[ | ]

−

Bernoulli Shortcut: If a random variable can only assume two values with prob and − , then = − −

Monica E. Revadulla EXAM MLC – Models for Life Contingencies

and

Formula Summary of ASM 2014

2 Lesson 4 – Survival Distribution: Mortality . Go pe tz’ La � = >

Lesson 3 – Survival Distributions: Force of Mortality 3.1 �

3.2 �

3.3 �

3.4 �

3.5 3.6 3.7 3.8 3.9

+ +

+

+

=

�

⁡

=�

=−

=−

4.2

)

n( n

.

= exp⁡ − ∫ �

= exp⁡ − ∫ � = exp⁡ − ∫

+

+

+

�

� + = = < < + =

3.10 =∫ � + ′ If � + = � + + for If � + = �̂ + + �̅ + for

If � ′ + = �

+

for

4.4 |

�

+

=∫

+

�

′

=(

Makeha ’s La

n⁡

= exp⁡ −

−

)

−

̂

. ̅

−

� = + > A is constant part of force of mortality *Adding A to � multiplies by e−μt

Weibull Distribution

+

′ = then then =

then

= exp⁡ −

n⁡

−

� =

Constant Force of Mortality 4.5 � = � 4.6 = e−μt 4.7 (BLANK) Uniform Distribution 4.8 � = ⁡⁡⁡⁡

4.9

4.10

=

−

=

−

= 4.11 | 4.12 (BLANK)

−

=

−

+

−

Beta Distribution � 4.13 � = 4.14

�

− −

− − −

�

�

= exp⁡ −

+

+

�−

�−

�−

�−

*The force of mortality is the sum of two uniform forces. is the product of uniform probabilities

Monica E. Revadulla EXAM MLC – Models for Life Contingencies

Formula Summary of ASM 2014

3 Lesson 5 – Survival Distributions: Moments Complete Future Lifetime ∞ 5.1 =∫ � +

5.2

5.3

5.4 5.5 5.6 5.7

∞

=∫ [ ]=

∫

∞

∞

= ∫ − = [min , ] � + + :̅̅̅| = ∫

:̅̅̅|

:̅̅̅| = ∫ [min ,

]=

∫

Special Mortality Laws − 5.8 = [ ]= = [ ]=

5.9

= [ ]= =

=

5.10 5.11

:̅̅̅|

:̅̅̅|

: ̅|

=

= =

=

�

�+ −

�

− − −

−

e−μ

⁡⁡ (CFM) = ∑∞= e−μ = −e−μ = + .5 (UDD) (UDD) :̅̅̅| + .5 :̅̅̅| =

*For those surviving n years, min , = *For those not surviving n years, average future lifetime is , since future lifetime is uniform. * = [min , ] * = [ ] *If curtate,⁡ + :̅̅̅̅̅̅̅ + | , < , < , �Ƶ is the same as + :̅̅̅|

(Beta)

�+ − �

5.20 5.21 5.22

(UDD) (CFM) �+

(Beta) (UDD) (CFM)

+

+ .5

(UDD) +

(UDD)

−

(UDD)

Curtate Future Lifetime 5.12 = ∑∞= | − ∑ 5.13 + = ̅̅̅ : | = | ∞ 5.14 [ ] = ∑ = | ] = ∑ =− 5.15 [min , 5.16 = ∑∞= 5.17 :̅̅̅| = ∑ = − 5.18 [ ] = ∑∞= ]=∑ = 5.19 [min ,

|

−

+

Monica E. Revadulla EXAM MLC – Models for Life Contingencies

Formula Summary of ASM 2014

4 Lesson 6 – Survival Distributions: Percentiles and Recursions 6.1 = :̅̅̅| + + 6.2 = :̅̅̅| + + 6.3 = :̅̅̅̅̅̅̅ + + + − | + + ⁡⁡⁡ = 6.4 = + + = 6.5 = + < ̅̅̅ ̅̅̅̅ ̅̅̅̅̅̅̅̅ : | : | + : − | ⁡⁡⁡⁡ ( + + :̅̅̅̅̅̅̅̅ < 6.6 :̅̅̅̅̅̅̅̅ − |+ − | )⁡⁡⁡⁡ :̅̅̅| = 6.7 + + + :̅̅̅̅̅̅̅ :̅̅̅| = + :̅̅̅̅̅̅̅ − | = − |

Lesson 7 – Survival Distributions: Fractional Ages Uniform Distribution of Deaths 7.1 + = − + + = − 7.2 = 7.3

7.4 7.5 �

+

=

−

� + = = + =

=

−

Function



−

+ + +

,

+

−

7.6 = + (UDD) Recall: 5.11 ( : ̅| = + .5 7.7 :̅̅̅| + .5 :̅̅̅| = Constant Force of Mortality 7.8 = −� 7.9 � = −ln⁡ 7.10 = −� = ⁡⁡⁡ 7.11 + =

Monica E. Revadulla EXAM MLC – Models for Life Contingencies

=

)

−

Table 7.1: Summary of Formulas for Fractional Ages UDD CFM − + − − − + − � + − ln − − ln � + + .5 :̅̅̅| :̅̅̅| + .5 + .5 : ̅| : ̅|

=

:̅̅̅|

+

+ :̅̅̅̅̅̅̅ − |

Formula Summary of ASM 2014

5 Lesson 8 – Survival Distributions: Select Mortality  A man whose health was established 5 years ago will have better mortality than a randomly selected man.  A life selected at age can never become a life selected at any higher age. [ ] will never become [ + ].

Lesson 10 – Insurance: Annual and 1/mthly – Moments + + = ∑∞= 10.1 [ ] = ∑∞= + | ∞ + + ∑ = 10.2 [ ] = ∑∞= = |

Table 10.1: Actuarial notation for standard types of insurance Name Present Value of RV Symbol + Whole life + ⁡⁡⁡⁡ < Term life ́ :̅̅̅| { ⁡⁡⁡⁡⁡⁡ ⁡⁡⁡⁡⁡⁡ ⁡⁡⁡⁡⁡⁡ ⁡⁡⁡⁡⁡⁡ < Deferred life | { + ⁡⁡⁡ Deferred term ⁡⁡⁡⁡⁡⁡⁡⁡ ⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡ | ́ :̅̅̅̅| { + ⁡⁡⁡ < < + | ⁡⁡⁡⁡⁡⁡⁡ + ⁡⁡⁡ ⁡ ⁡⁡⁡⁡⁡⁡ < Pure Endowment ́| :̅̅̅ { ⁡⁡⁡ Endowment

= :̅̅̅| ⁡ − ( = + = − = 10.5 ⁡ | = + 10.6 −⁡ | = ́ :̅̅̅| = 10.7 = = ̅̅̅ ̅̅̅ ́: | + : | ⁡⁡⁡⁡ 10.8 =

10.3 10.4

+

Monica E. Revadulla EXAM MLC – Models for Life Contingencies

+

=

+ +

−

{

⁡

+

:̅̅̅| )

− − +

⁡⁡⁡⁡⁡⁡ ⁡⁡⁡

<

+

+

:̅̅̅|

+

(CFM)

Formula Summary of ASM 2014

6 Table 10.2: EPV for insurance payable at EOY of death Type of Insurance CFM UDD ̅̅̅̅̅̅̅̅ − | Whole Life + ̅̅̅| n-year term − + �− ̅̅̅̅̅̅̅̅̅̅̅̅̅̅ n-year deferred life − + | + �− n-year pure endowment (� − + ) �−

Lesson 11 – Insurance: Continuous – Moments – Part I ∞ 11.1 = ̅ =∫ ∞ 11.2 ̅ = ∫ −� � + 11.3 11.4

11.5 11.6 11.7 11.8

Monica E. Revadulla EXAM MLC – Models for Life Contingencies

̅ =

|

=

−

�+�

̅ = � ⁡⁡⁡⁡ �+� ̅ + = ̅ ⁡⁡⁡ ̅ ́ :̅̅̅| = ̅ ( − |

̅

|

̅

́ :̅̅̅̅|

:̅̅̅|

=

̅ = ́| = :̅̅̅

= ̅ ( �

�+� −

̅

)=

−

−

̅ −

) =

�+�

( − + �+�

−( � ⁡⁡

+

�

�+�

�+�

−

)=

)+

−

̅

�+�

� − �+�

−

�+�

�+�

−

−

�+�

Formula Summary of ASM 2014

7 Lesson 12 – Continuous – Moments – Part II

Lesson 13 – Insurance: Probabilities and Percentiles

Table 12.1 EPV for insurance payable at moment of death Type of Insurance CFM UDD � ̅̅̅̅̅̅̅̅̅ Whole Life − | �+ �− � ̅̅̅̅| n-year term − − �+� �+ �− � −� n-year deferred life − �+� ̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅ − + | �+ �− − �+� −� n-year pure endowment (� − + ) �−

To calculate Pr⁡ for continuous , draw a graph of as a function of . Identify the parts of the graph that are below the horizontal line = , and the corresponding ’s. The al ulate the p o a ilit of being in the range of those ’s.

=∫

12.1 Gamma

=∫

12.2 If n=1,

=∫

12.3 If n=2, 12.4 ∫

12.5

�̅̅

− ̅̅̅|

Variance If = + =

=

∞

∞

̅̅̅̅| −

=

⁡⁡

�

,

&⁡

∞

+

−

−

−

−

+

=

!

=

=

+

−

For CFM, Pr

=

For discrete , identify .

� �

and then identify

+

corresponding to that

To calculate percentiles of continuous , draw a graph of as a function of . Identify where the lower parts of the graph are, and how they vary as a function of . For example, for whole life, higher leads to lower . For year deferred whole life, both < and higher lead to lower . Write an equation for the probability is less than in terms of mortality probabilities expressed in terms of . Set it equal to the desired percentile, and solve for or for for any . Then solve for (which is often )

are mutually exclusive, −

Monica E. Revadulla EXAM MLC – Models for Life Contingencies

Formula Summary of ASM 2014

8 Lesson 14 – Insurance: Recursive Formulas, Varying Insurance Recursive Formulas 14.1 = + = + 14.2 ̅̅̅ : | 14.3 = + ́ :̅̅̅| 14.4 | = −

+ |

+

15.2

+ :̅̅̅̅̅̅̅ − | +́ :̅̅̅̅̅̅̅ − |

15.3 15.4

Applying whole life recursive equation twice: = + + + � 14.5 � ̅ ̅ = �+�

15.5 +

14.6 Continuously whole life insurance (CFM) � = �+ ̅ ́ :̅̅̅| 14.7 � ̅ ̅ ́ :̅̅̅| + ̅ ̅ ́ :̅̅̅| = ̅ ̅ ́ :̅̅̅| ̅ + 14.8 � ́ :̅̅̅| + ́ :̅̅̅| = 14.9

� �

+ ́ :̅̅̅| = ∑

́ :̅̅̅|

=

́ :̅̅̅|

=

− |

̅

+

́ : ̅|

Lesson 15 – Insurance: Relationships ( Uniform Distribution of Deaths 15.1 ̅ =

́ :̅̅̅|

Recursive Formulas for Increasing and Decreasing Insurance 14.10 � ́ :̅̅̅| = ́ :̅̅̅| + � +́ :̅̅̅̅̅̅̅ − | � 14.11 � ́ :̅̅̅| = ́ : ̅| + ́ :̅̅̅̅̅̅̅ + − | 14.12 ́ :̅̅̅| = ́ : ̅| + +́ :̅̅̅̅̅̅̅ − | 14.13 = + ̅̅̅ ̅̅̅ ̅̅̅̅̅̅̅ ́: | ́: | ́: − |

Monica E. Revadulla EXAM MLC – Models for Life Contingencies

15.6

̅

|

̅

�

́ :̅̅̅|

=

:̅̅̅|

=

̅ = =

̅ =

�

�

�

+

�

|

, ̅

,

́ :̅̅̅| ́ :̅̅̅|

+

́| :̅̅̅

⁡

Claims Acceleration Approach ̅ = + + ́ :̅̅̅| = ̅ = + | ̅ :̅̅̅| = + . ̅

= ̅ =

+ +

.

.

.

́ :̅̅̅| −

|

́ :̅̅̅|

+

́| :̅̅̅

⁡

Formula Summary of ASM 2014

9 Lesson 17 – Annuities: Discrete, Expectation

= − + − ̅̅̅| − + − + :̅̅̅| = ∑ = − ∑ = :̅̅̅| = ∞ ∑ = = | 17.15 Constant Force of Mortality + =

17.12 17.13 17.14

Annuities-Due Whole Life Annuities − 17.1 = 17.2 = −

Temporary Life Annuities −

:̅̅̅| = − :̅̅̅| =

17.3 17.4

:̅̅̅|

̅̅̅̅̅̅̅̅̅ + |

−

Whole life annuities − = +

− � +

⁡⁡⁡

Temporary life annuities = :̅̅̅| + :̅̅̅| + 17.16 = − ̅̅̅ ̅̅̅ ́: | : | :̅̅̅|

n-year certain-and-life annuity-due 17.7 17.8

̅̅̅|

=

̅̅̅̅̅̅̅̅̅ + |

−

=

⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡

− � +

−

⁡⁡⁡⁡

Table 17.2: Actuarial notation for standard types of annuity-due Name Whole Life Temporary Life

Annual pmt at k ⁡⁡⁡⁡⁡ ⁡⁡⁡ min ⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡ > min

17.9 17.10 17.11

PV , − , −

⁡⁡⁡⁡⁡⁡⁡⁡

⁡⁡⁡ < n⁡or⁡k > K x ⁡⁡⁡ K x ⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡ ⁡⁡⁡ min⁡ , ⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡ ⁡⁡⁡⁡ n⁡or⁡t > T ⁡⁡⁡⁡ < ⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡ ⁡⁡⁡⁡ n⁡or⁡t > T ⁡⁡⁡⁡ < + ⁡ ⁡ ⁡⁡⁡⁡ > +

18.2 ̅ = 18.3 ̅ =

⁡⁡⁡⁡ max⁡ , ⁡⁡⁡⁡ > max⁡ T, − ̅

18.5 ̅ = ∫

18.6 ̅ = 18.9

18.10

:̅̅̅|

:̅̅̅|

|̅

|̅

18.11 ̅

� �+�

−

=

:̅̅̅|

+

� �+�

=

� � − ̅ :̅̅̅| �

=

>

|̅

>

< > <

+

+

| ̅ :̅̅̅|

̅̅̅̅̅̅̅̅ :̅̅̅||

Whole Life and Temporary Life 19.1 19.2 19.3

− ̅

− ̅ �

−

:̅̅̅| − ̅ :̅̅̅| �

̅ =

= ̅ ( −

CFM: ̅ = ̅ + Relationships: ̅ = ̅ :̅̅̅| + ̅ ̅ ̅ ̅̅̅̅̅̅̅̅ :̅̅̅|| = ̅̅̅| + |

�+�

19.6 19.7

−

�+�

�+�⁡

)=

̅

̅ =

⁡⁡⁡

− − �+� �+�

=

=

=

−

� ̅ −

+

̅

� ̅ ̅̅̅ − ̅ ̅̅̅ : | : | �

̅

(̅ − ̅ ) �

−

µ

+

− ̅

̅

:̅̅̅|

̅ :̅̅̅| − ̅ :̅̅̅| �

(

̅̅̅̅̅̅̅̅ + |)

̅ =

=

−

=

=

(

Other Annuities 19.13 [ ] = ∑∞= 19.14

⁡⁡⁡

−

µ

− (̅

:̅̅̅| )

̅ is 1 moment at twice FOI

19.12

�

:̅̅̅|

=

st

19.11

̅ ̅̅̅ − ̅ : |

∞

( ̅ ̅̅̅| ) =

19.10

⁡⁡⁡

=

[ ̅ ̅̅̅̅| ] = ∫

19.5

19.9

=

∞

[ ̅ ̅̅̅̅| ] = ∫ ̅ ̅̅̅|

19.4

Note: �

∞

= =

Symbol ̅ ̅ :̅̅̅|

19.8

�

− ̅ ∞ 18.4 ̅ = ∫ ̅ ̅|

18.7 ̅ 18.8 ̅

PV ̅ ̅̅̅| ̅ ̅̅̅| ⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡ ⁡ ̅̅̅̅| ⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡ ⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡ ̅ ̅̅̅| − ̅̅̅̅| ⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡ ⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡ ̅ ̅̅̅| − ̅̅̅̅| ⁡⁡⁡⁡⁡⁡⁡⁡⁡ ̅̅̅̅̅̅̅̅̅ + | − ̅̅̅̅| ⁡⁡⁡⁡ ̅̅̅̅| ⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡ ̅̅̅̅̅̅̅̅̅̅ + | ⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡

[

:̅̅̅| ]

=∑

:̅̅̅| −

−

=

−

:̅̅̅|

=

)

+

̅̅̅| − |

̅̅̅| − |

−

− +

−

̅̅̅|

=∑

− =

̅̅̅|

− |

+

−

̅̅̅|

+

Monica E. Revadulla EXAM MLC – Models for Life Contingencies

Formula Summary of ASM 2014

11 Lesson 20 – Annuities: Probabilities and Percentiles For the continuous whole life annuity PVRV Y, the relationship of as follows: = Pr⁡ − ⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡= Pr⁡ ⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡= Pr⁡ ⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡= Pr⁡ ln ⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡= Pr⁡ − ⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡= Pr

⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡=

(−

to

21.2

= ̅ =

− ln − ln − ln −

ln

−

̅

To calculate a probability for an annuity, calculate the for which ̅ has the desired property. Then calculate the probability is in that range. , then

Some adjustments may be needed for discrete annuities or non-whole-life annuities, as discussed in the lesson. If forces of mortality and interest are constant, then the probability that the present value of payments on a continuous whole life annuity will be greater than its expected present value is Pr( ̅ ̅̅̅|

�

� � ) > ̅ )=( �+

Monica E. Revadulla EXAM MLC – Models for Life Contingencies

=

Temporary

)

To calculate a percentile of an annuity, calculate the percentile of calculate ̅ ̅̅̅|

Lesson 21 – Annuities: Varying, Recursive Formulas Whole Life 21.1 = + +

:̅̅̅|

:̅̅̅|

:̅̅̅|

= = =

Deferred life = | = | |̅ =

̅

+

+

̅

+

+

+ +̅

: ̅|

+ :̅̅̅̅̅̅̅ − |+ :̅̅̅̅̅̅̅ − |+ ̅

+ :̅̅̅̅̅̅̅ − | + +

− |

− |

: ̅|

+

+

− |̅ +

n-year certain and life + ̅̅̅̅̅̅ ̅̅̅̅̅̅̅ − |+ :̅̅̅| = + ̅̅̅̅̅̅ ̅̅̅̅̅̅̅ − |+ :̅̅̅| = + ̅ ̅̅̅̅̅̅̅ = ̅ ̅| ̅̅̅̅̅̅̅ − |+ :̅̅̅|

Increasing/decreasing annuities � � + = + ̅� ̅ = ⁡⁡⁡⁡ �+μ ̅� ̅ :̅̅̅| + ⁡ ̅ ̅ :̅̅̅| = ̅ :̅̅̅| �

:̅̅̅|

+⁡

:̅̅̅|

=

+

̅̅̅̅̅̅̅̅̅̅̅̅̅̅ + :̅̅̅̅̅̅̅ − |

̅̅̅̅̅̅̅̅̅̅̅̅̅̅ + :̅̅̅̅̅̅̅ − |

̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅ + :̅̅̅̅̅̅̅ − |

:̅̅̅|

Formula Summary of ASM 2014

12

Lesson 24 – Premiums: Net Premiums for Discrete Insurances – Part I Future Loss = PV(Benefits) – PV(Gross Premiums) Equivalence Principle: EPV(Premiums) = EPV(Payments)  E[FutureLoss]=0 Net Premium  E[PVFB] = E[PVFP]

Lesson 22 – Annuities: 1/m-thly Payments − 22.1 ≈ − ≈

−

−

Uniform Distribution of Deaths (UDD) ⁡ = 22.2 22.3 22.4

|

= = :̅̅̅|

Woolhouse’ Fo

22.8

|

22.9 ̅ ≈ 22.10

≈

≈

−

−

ln

22.6 µ ≈ − :̅̅̅|

|

ula

≈

22.7

−

:̅̅̅|

=

22.5

−

=

≈

:̅̅̅|

|

−

− −

−

+ −

−

−

−

+ ln

−