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

A deferred annuity is payable annually in advance to a life aged 65 from age 66, with the first payment Rs 5,000 and each later payment 5% larger than the one before. Which is the correct equation for its expected present value, with v the discount factor?

The expected present value is the sum over t from 1 of 5000(1.05)^(t-1) v^t times t_p65. The first payment is at time 1, each payment grows by 5%, it is discounted for t years and is paid only if the life survives to time t.

  1. ASum over t≥1 of 5000(1.05)^(t-1) v^t t_p65Correct
  2. BSum over t≥0 of 5000(1.05)^t v^t t_p65
  3. CSum over t≥1 of 5000(1.05)^t v^t t_p65
  4. DSum over t≥1 of 5000(1.05)^(t-1) v^(t-1) t_p65
  5. Sum over t≥1 of 5000(1.05)^(t-1) v^t q65

Explanation

The first payment of 5000 is at time 1, so the t-th payment occurs at time t and equals 5000(1.05)^(t-1). Discount by v^t and weight by survival probability t_p65. Options that start at t=0 or shift the exponent mismatch the timing, and q65 would be a death probability rather than survival.

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