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IAI Actuarial Core Principles · Actuarial Mathematics for Modelling · Gross random future loss

A whole life policy issued to a life aged x pays a sum assured S at the end of the year of death. Level annual premiums P are payable in advance for as long as the life survives. Let K be the curtate future lifetime and v the annual discount factor. Which expression gives the gross future loss random variable at issue, ignoring expenses?

The future loss is S v^(K+1) − P ä_(K+1 |). The death benefit is paid at the end of the year of death, at time K+1. Premiums fall at times 0 to K, which is K+1 payments in advance. Loss is outgo minus income, so the premiums are subtracted.

  1. AS v^(K+1) − P ä_(K+1 |)Correct
  2. BS v^K − P ä_(K |)
  3. CS v^(K+1) − P a_(K+1 |)
  4. DS v^(K+1) − P ä_(K |)
  5. S v^(K+1) + P ä_(K+1 |)

Explanation

The loss is the present value of outgoings less the present value of income. The benefit is paid at time K+1, so its present value is S v^(K+1). Premiums are paid in advance at times 0, 1, ..., K, which is K+1 payments, so the income is P ä_(K+1 |). Using ä_(K |) or an arrears annuity miscounts the premiums, and adding the premiums is a sign error.

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