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

A life aged 40 buys a whole life assurance paying a sum assured of Rs 1,000 at the end of the year of death in year 1, Rs 2,000 in year 2, Rs 3,000 in year 3 and so on, increasing by Rs 1,000 each year. Which expression gives the expected present value, where K is the curtate future lifetime and v the discount factor?

The expected present value is E[1000(K+1)v^(K+1)]. Death in year K+1 triggers a benefit of 1000(K+1), paid at the end of that year, so it is discounted by v^(K+1). Taking the expectation over K gives the increasing assurance.

  1. AE[1000 (K+1) v^(K+1)]Correct
  2. BE[1000 K v^K]
  3. CE[1000 (K+1) v^K]
  4. DE[1000 K v^(K+1)]
  5. E[1000 (K+1)] v^(K+1)

Explanation

If death occurs in policy year K+1 the benefit is 1000(K+1), paid at time K+1, so the present value is 1000(K+1)v^(K+1). Taking expectation gives 1000 (IA)_40. Options using K give the wrong benefit, and v^K gives the wrong discounting.

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