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IAI Actuarial Core Principles · Actuarial Mathematics for Modelling · Means and variances of assurance and annuity payments

A 10-year pure endowment pays Rs 100,000 on survival to the end of the term. The survival probability is 10p_x = 0.85 and the 10-year discount factor is v^10 = 0.60. What is the variance of the present value of the benefit?

The variance is Rs 459,000,000. The benefit is a two-point random variable, so its variance is the benefit squared times (v^10)² times p times (1 − p). That is 10^10 × 0.36 × 0.85 × 0.15, which equals 459,000,000.

  1. ARs 765,000,000
  2. BRs 459,000,000Correct
  3. CRs 2,499,000,000
  4. DRs 3,060,000,000
  5. Rs 21,424

Explanation

E[Z] = 100,000 × 0.60 × 0.85 = 51,000. E[Z²] = 10^10 × 0.36 × 0.85 = 3,060,000,000. Var = 3,060,000,000 − 51,000² = 3,060,000,000 − 2,601,000,000 = 459,000,000. This agrees with 10^10 × 0.36 × 0.85 × 0.15. Using v^10 instead of v^20 in the variance gives 765,000,000, which is wrong.

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