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A fair value of a life insurance premium corresponds to the discounted value of the value of the insurance that will be paid in case of death of the insured person multiplied by the chance of paying the life insurance at a given age. To illustrate how the Actuarial Toolbox handles with this subject, we will show the direct calculus of the premium via the function “sActuarialLifeInsurance” for insurance of $100,000 for a 40-year-old person. Just for the completeness of the explanation, we will also present the calculus using the chance of someone dying at the exact age times the value of the insurance.

For the direct calculus, simply select the Products/Assurance (liability) option at the Actuarial Toolbox tab and change the inputs accordingly. Let’s suppose that the table UP 1994 - Female is the relevant table to represent the mortality force that acts upon the insured person’s life. The results can be seen below:

A

B

C

1

Life insurance example (direct calculus)

 

 

2

 

 

 

3

Forward (real) interest rate curve

0.05

 

4

Normal life table

6

UP1994F

5

Current age

40

 

6

Number of payments

131

 

7

Assurance value

 $ 100,000.00

 

8

Annuity (present value)

 $    13,699.36

=sActuarialAssuranceLiability(sActuarialFlatRateVector($B$3),$B$4,$B$5,$B$6)*$B$7

9

 

 

 

The appropriate function for the indirect calculus is the “sActuarialqxnpxVector” that returns the desired probabilities. It can be inserted selecting the option Common operations/qxnpx (vector) at the Actuarial Toolbox tab. Also, the discounting factors can also be inserted through the option Financial/Discount factor (vector) available at the actuarial tab. Now, multiplying and summing the columns as presented in cell B7 (see below) gives the desired result.

A

B

C

D

E

F

G

1

Life insurance example (indirect calculus)

 

 

 

 

2

 

 

 

 

 

 

 

3

Actuarial table

6

UP1994F

 

 

 

 

4

Age

40

 

 

 

 

 

5

Interest rate

0.05

 

 

 

 

 

6

Assurance value

$ 100,000.00

 

 

 

 

 

7

Annuity (present value)

$ 13,699.36

=SUM(F50:F140)

 

 

 

 

8

 

 

 

 

 

 

 

9

Age

Probability of dying on the exact age

 

 

 

 

 

10

0

0

 

 

 

 

 

11

1

0

 

 

 

 

 

...

...

...

...

...

...

...

...

49

39

0

 

Discount factors

 

 

 

50

40

0.000763

={sActuarialqxnpxVector($B$3,$B$4)}

0.952380952

={sActuarialForwardInterestRatesToDiscountFactorForecastVector(

sActuarialFlatRateVector($B$5))}

72.66667

=B50*D50*$B$6

51

41

0.00082537

Note: after entering the above formula click on the "Multiple Values Formula" button.

0.907029478

Note: after entering the above formula click on the "Multiple Values Formula" button.

74.86347

 

52

42

0.00088659

 

0.863837599

 

76.58694

 

53

43

0.000940666

 

0.822702475

 

77.38883

 

54

44

0.000988612

 

0.783526166

 

77.46031

 

55

45

0.001041393

 

0.746215397

 

77.71036

 

56

46

0.00110495

 

0.71068133

 

78.52673

 

57

47

0.001188166

 

0.676839362

 

80.41972

 

58

48

0.001286963

 

0.644608916

 

82.95877

 

59

49

0.001395292

 

0.613913254

 

85.65881

 

60

50

0.001519993

 

0.584679289

 

88.87086

 

...

...

...

...

...

...

...

...

127

117

8.57297E-07

 

0.022245116

 

0.001907

 

128

118

4.28649E-07

 

0.021185825

 

0.000908

 

129

119

2.14324E-07

 

0.020176976

 

0.000432

 

130

120

2.14324E-07

 

0.019216167

 

0.000412

 

131

121

0

 

0.018301112

 

0

 

132

122

0

 

0.01742963

 

0

 

133

123

0

 

0.016599648

 

0

 

134

124

0

 

0.015809189

 

0

 

135

125

0

 

0.01505637

 

0

 

136

126

0

 

0.0143394

 

0

 

137

127

0

 

0.013656571

 

0

 

138

128

0

 

0.013006259

 

0

 

139

129

0

 

0.012386913

 

0

 

140

130

0

 

0.01179706

 

0

 

141

 

 

 

 

 

 

 

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