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IAI Actuarial Core Principles · Actuarial Mathematics for Modelling · Gross random future loss

A whole life policy has sum assured Rs 100,000 payable at the end of the year of death and a level annual premium of Rs 2,000 payable in advance, with i = 5% per year. Initial expenses of Rs 1,000 are incurred and there are no other expenses. Given A_x = 0.25 and 2A_x (at the same rate) = 0.12, what is the variance of the gross future loss at issue?

The variance is Rs 1,159,430,000. The loss is a constant plus (S + P/d) times v^(K+1), so the variance is (142,000)² multiplied by (2A - A²) = 0.0575. The initial expense is a constant and has no effect on the variance.

  1. ARs 1,159,430,000Correct
  2. BRs 575,000,000
  3. CRs 193,430,000
  4. DRs 2,419,680,000
  5. Rs 3,679,930,000

Explanation

Write L = (S + P/d) v^(K+1) - P/d + constant, so the expense does not affect the variance. Here d = 0.05/1.05 and P/d = 42,000, so S + P/d = 142,000. Var = 142,000² × (0.12 - 0.0625) = 20,164,000,000 × 0.0575 = 1,159,430,000. Using S alone gives 575,000,000, which ignores the premium annuity's dependence on K.

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