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

A 15-year increasing term assurance on a life aged 35 pays Rs 1,00,000 x k at the end of the year of death in policy year k (k = 1 to 15), and nothing on survival. Which expression gives the expected present value of the benefit, with v the discount factor and the usual notation?

The correct expression is 1,00,000 times the sum from k=1 to 15 of k x v^k x (k-1)p35 x q(35+k-1). Death in year k requires survival for k-1 years, then death during year k, with payment of k times Rs 1,00,000 discounted by v^k.

  1. A1,00,000 x sum over k = 1 to 15 of k x v^k x (k-1)p35 x q(35+k-1)Correct
  2. B1,00,000 x sum over k = 1 to 15 of k x v^(k-1) x kp35 x q(35+k)
  3. C1,00,000 x sum over k = 1 to 15 of v^k x kp35
  4. D1,00,000 x 15 x v^15 x 15p35
  5. 1,00,000 x sum over k = 1 to 15 of k x v^k x kp35 x q(35+k)

Explanation

Death in year k needs survival k-1 years then death in that year, with probability (k-1)p35 x q(35+k-1). The benefit is paid at the end of year k, so discounting is v^k, and the benefit is 1,00,000 x k. Option 1 uses the wrong survival, mortality ages and discount. Option 4 shifts the survival period by one year.

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