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

A whole life annuity-immediate pays Rs 10,000 at the end of each year while the life aged x survives, with interest at 5% per annum effective. The annuity-due of 1 per annum for the same life has present value variance 25.00. What is the standard deviation of the present value of the Rs 10,000 annuity-immediate, in rupees?

The standard deviation is Rs 50,000. The annuity-immediate equals the annuity-due minus 1, so its variance is unchanged at 25 and its standard deviation per unit is 5. Scaling by the payment of Rs 10,000 gives 5 × 10,000 = Rs 50,000.

  1. A47,619.05
  2. B50,000.00Correct
  3. C52,500.00
  4. D250,000.00
  5. 500,000.00

Explanation

The annuity-immediate satisfies a_K = ä_{K+1} − 1, so subtracting the constant 1 leaves the variance unchanged at 25.00 and the standard deviation per unit is 5. Scaling by 10,000 gives 50,000. Multiplying by v gives 47,619.05, which wrongly treats the annuity-immediate as the annuity-due discounted. Using 10,000 × 25 confuses variance with standard deviation.

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