# Learning Center

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