One $150,000 receipt occurs at the end of year 5, with no other cash flows. At an annually compounded 8% discount rate, what is its present value, rounded to the nearest dollar?
Correct Answer
B) $102,087
150000÷(1+0.08)^5 = 102087.48. Common errors: discounting with simple instead of compound interest gives $107,143; subtracting 8% per year from the receipt gives $90,000; compounding forward instead of discounting gives $220,399.
Why This Is the Correct Answer
Option B is correct because PV = $150,000 ÷ (1.08)^5 = $150,000 ÷ 1.469328 = $102,087.48, or $102,087 rounded. Compound discounting reduces the amount for each of the 5 years at 8%.
Why the Other Options Are Wrong
Option A: $90,000
Option A, $90,000, subtracts 8% of the receipt for each year: $150,000 × (1 − 0.08 × 5) = $90,000. Present value uses compound discounting: $150,000 ÷ (1.08)^5 = $102,087.
Option C: $107,143
Option C, $107,143, uses simple interest: $150,000 ÷ (1 + 0.08 × 5) = $150,000 ÷ 1.40 = $107,143. With annual compounding the divisor is (1.08)^5 = 1.469328, giving $102,087.
Option D: $220,399
Option D, $220,399, compounds forward instead of discounting: $150,000 × (1.08)^5 = $220,399. A future receipt is worth less today, so divide: $150,000 ÷ (1.08)^5 = $102,087.
Future to present: divide
Money moving back in time shrinks, so divide by (1 + i) raised to the number of years. Money moving forward grows, so multiply.
How to use: Use this to choose between dividing and multiplying by the compound factor.
Exam Tip
A present value must be less than the future amount when the rate is positive; discard any answer larger than the receipt.
Common Mistakes to Avoid
- -Using simple interest instead of compound discounting
- -Multiplying by the compound factor instead of dividing
- -Subtracting a flat percentage per year
Concept Deep Dive
Analysis
This question tests discounting a single future amount to present value with annual compounding. The present value factor is 1 ÷ (1 + i)^n. The distractors use simple interest, straight-line reductions or compounding forward instead of discounting.
Background Knowledge
The present value of $1 is 1 ÷ (1 + i)^n, where i is the discount rate per period and n is the number of periods. Discounting converts future benefits, such as a reversion, into present value. It is the reverse of compounding to future value.
Real-World Application
In a discounted cash flow analysis, an appraiser estimates the sale price of a building at the end of a five-year holding period and discounts that reversion back to today at a market-supported discount rate.
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