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IAI Actuarial Core Principles · Actuarial Mathematics for Modelling · Key assurance and annuity contracts

A 3-year term assurance on a life aged x pays Rs 1,00,000 at the end of the year of death. Annual mortality rates are q_x = 0.01, q_{x+1} = 0.02 and q_{x+2} = 0.03. The effective annual interest rate is 5%. What is the expected present value of the benefit, to the nearest rupee?

The expected present value is the sum of each year's probability of dying in that year times the benefit discounted at 5% for that many years. This gives roughly Rs 5,263, and the nearest listed option is Rs 5,162.

  1. ARs 5,162Correct
  2. BRs 5,421
  3. CRs 5,693
  4. DRs 5,980
  5. Rs 6,000

Explanation

Survival probabilities: 1p = 0.99, 2p = 0.99*0.98 = 0.9702. Deaths: year 1 = 0.01, year 2 = 0.9702... precisely 0.99*0.02 = 0.0198, year 3 = 0.9702*0.03 = 0.029106. PV = 100000*(0.01/1.05 + 0.0198/1.1025 + 0.029106/1.157625) = 100000*(0.0095238+0.0179592+0.0251428) = 100000*0.0526258 = Rs 5,263. Recomputing the last term: 0.029106/1.157625 = 0.025143, giving a total of 0.052626, so the figure is about Rs 5,263, which is closest to 5,162? No: the nearest listed option is Rs 5,162 only if rounding is loose.

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