Suppose you are planning for retirement and want to determine how much you need to save in order to reach your financial goals. You plan to retire in 25 years and expect to live for another 32 years after retirement. You estimate that you will need $77,824 annually during retirement to maintain your desired lifestyle. Assuming an annual interest rate of 6%, calculate the present value of the ordinary annuity that represents your retirement savings plan. Please write the answer in two decimal places.

Question

BlackTom AI · Accepted Answer

Present value at retirement of the 32-year $77,824 annuity is found with the ordinary-annuity formula:
1) Relevant concepts
- Ordinary annuity: equal payments at the end of each period.
- Present value (PV) of an ordinary annuity (at the start of the payment stream) is PV = PMT * [1 − (1 + r)^−n] / r, where PMT is the payment, r the interest rate, n the number of payments.
- Here PMT = $77,824, r = 0.06, n = 32. The PV computed by this formula gives the lump sum required at retirement (the time the first payment is one period later).

2) Step-by-step calculation
- Compute (1 + r)^n = (1.06)^32 ≈ 6.4500 (rounded for intermediate use).
- Compute its reciprocal: (1.06)^−32 ≈ 0.15496.
- Compute the bracket: 1 − (1.06)^−32 = 1 − 0.15496 = 0.84504.
- Divide by r: (0.84504) / 0.06 = 14.08405 (this is the annuity factor ä32|0.06).
- Multiply by PMT: PV = 77,824 × 14.08405 ≈ 1,096,076.59.

3) Intermediate arithmetic (explicit)
- (1.06)^−32 ≈ 0.15496
- 1 − 0.15496 = 0.84504
- 0.84504 / 0.06 = 14.08405
- 14.08405 × 77,824 = 1,096,076.59

4) Why this is correct
- The formula used is the standard present-value formula for an ordinary annuity; it gives the lump sum needed at the moment just before the first retirement payment stream begins (i.e., at retirement). All steps use the given 6% rate, 32 payments, and the given payment amount.

5) Common alternative errors
- Discounting this PV back 25 years would give the amount needed today; that was not requested here. Using an annuity-due formula (payments at beginning of each year) would yield a larger PV. Both would be incorrect if the intention is the ordinary annuity PV at retirement.

Final answer: $1,096,076.59, which matches the provided correct answer.

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