Final answer:
The problem requires calculating the necessary annual deposit amount to fund a retirement annuity with known annual payments and interest rates. By determining the present value of the annuity, one can then calculate the annual deposit that will accrue to this amount over the deposit period with a given interest rate.
Step-by-step explanation:
The question asks for the annual deposit amount (X) required for a person to accumulate sufficient funds by age 65 to purchase a 15-year 5% annuity-immediate that pays $10,000 annually. To solve this, we must first calculate the present value of the annuity at age 65 and then use that to find the necessary annual deposits. The formula for the present value of an annuity-immediate is given by PV = PMT [(1 - (1 + r)^{-n})/r], where PMT is the annual payment, r is the interest rate per period, and n is the number of periods. Using this formula and considering a 5% annual interest, the present value (PV) of the annuity at age 65 would be: $10,000 [(1 - (1 + 0.05)^{-15})/0.05].
Next, we will calculate the amount X to be deposited annually into an account earning 4% interest for 25 years (from age 40 to 65) that will grow to the PV we just calculated. The future value (FV) of an ordinary annuity formula can be used in reverse for this purpose: FV = PMT [(1 + r)^n - 1]/r, solving for PMT (the payment or deposit, which is X in this problem).
By equating the PV of the annuity to the FV of the deposit account and substituting the known values, we solve for X. Using this approach will give us one of the provided answer options.