An investment will provide you with $100 at the end of each year for the next 10 years. What is the present value of that annuity if the discount rate is 8% annually?

Question

An investment will provide you with $100 at the end of each year for the next 10 years. What is the present value of that annuity if the discount rate is 8% annually?

BlackTom AI · Accepted Answer

Present value of an ordinary annuity is given by PV = PMT * [1 - (1 + r)^(-n)] / r. Here PMT = $100, r = 0.08, n = 10.

1) Relevant concepts
- Time value of money: future payments are worth less today; discounting converts future cash flows to present value.
- Ordinary annuity: payments occur at the end of each period, so the standard annuity formula above applies.

2) Step-by-step reasoning
- Substitute the numbers into the annuity formula.
- Compute (1 + r)^n, then its reciprocal, then 1 minus that reciprocal, divide by r, and multiply by PMT.

3) Intermediate calculation steps
- (1 + r) = 1.08
- (1.08)^2 = 1.1664
- (1.08)^3 = 1.259712
- (1.08)^4 = 1.36048896
- (1.08)^5 = 1.4693280768
- (1.08)^6 = 1.586874323744
- (1.08)^7 = 1.71382426964352
- (1.08)^8 = 1.8509302112150016
- (1.08)^9 = 1.9990046281122017
- (1.08)^10 = 2.1589249983611779
- (1.08)^(-10) = 1 / 2.1589249983611779 = 0.4631930216388225
- 1 - (1.08)^(-10) = 1 - 0.4631930216388225 = 0.5368069783611775
- Divide by r: 0.5368069783611775 / 0.08 = 6.710087229514719
- Multiply by PMT: PV = 100 * 6.710087229514719 = 671.0087229514719

4) Why this answer is correct
- The calculation follows the exact ordinary annuity present-value formula, applying continuous arithmetic with the correct discount rate and number of periods. Discounting each of the ten $100 payments yields the sum 671.0087, which rounds to $671.01.

5) Common incorrect approaches (why they are wrong)
- Summing payments without discounting (100 * 10 = $1,000) ignores time value of money and overstates PV.
- Using the future-value formula would compute the value at the end of 10 years, not today.
- Using an annuity-due formula (payments at beginning of each period) would give a larger PV (multiply by 1.08) and thus be incorrect for end-of-year payments.

6) Conclusion
The present value of the $100 annuity for 10 years at 8% is $671.01, which matches the provided answer.

An investment will provide you with $100 at the end of each year for the next 10 years. What is the present value of that annuity if the discount rate is 8% annually?数值题

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