One $800,000 receipt occurs at the end of year 10, with no other cash flows. At an annually compounded 9% discount rate, what is its present value, rounded to the nearest dollar?
Correct Answer
B) $337,929
800000÷(1+0.09)^10 = 337928.65. Common errors: discounting with simple instead of compound interest gives $421,053; subtracting 9% per year from the receipt gives $80,000; discounting for nine years instead of ten gives $368,342.
Why This Is the Correct Answer
Option B is correct because PV = $800,000 ÷ (1.09)^10 = $800,000 ÷ 2.3674 = about $337,929. Each year's discount compounds on the prior year's, which is what the exponent captures.
Why the Other Options Are Wrong
Option A: $80,000
Option A, $80,000, comes from subtracting 9% of the receipt per year: $800,000 × (1 − 0.09 × 10) = $80,000. Discounting divides by a compound factor; the correct value is $800,000 ÷ (1.09)^10 = $337,929.
Option C: $368,342
Option C, $368,342, results from discounting for nine years instead of ten: $800,000 ÷ (1.09)^9 = $368,342. The receipt occurs at the end of year 10, so the exponent is 10, giving $337,929.
Option D: $421,053
Option D, $421,053, comes from using simple instead of compound interest: $800,000 ÷ (1 + 0.09 × 10) = $421,053. With annual compounding, PV = $800,000 ÷ (1.09)^10 = $337,929.
Divide by growth, raised to time
Present value is the future amount divided by (1 + rate) raised to the number of periods. The exponent equals the year of receipt.
How to use: Use this formula for any single future sum, such as a reversion.
Exam Tip
Match the exponent to the year the cash arrives; a receipt at the end of year 10 uses n = 10.
Common Mistakes to Avoid
- -Using simple interest instead of compound discounting
- -Using the wrong number of periods
- -Subtracting a percentage per year instead of dividing by a compound factor
Concept Deep Dive
Analysis
This question tests the present value of a single future amount with annual compounding. Divide the $800,000 receipt by (1.09) raised to the 10th power. The key is using compound discounting for the full ten years.
Background Knowledge
The present value of one dollar is 1 ÷ (1 + i)^n, where i is the periodic rate and n the number of periods. It reflects the time value of money, since a dollar received later is worth less today. This factor underlies discounting of reversions in DCF analysis.
Real-World Application
An appraiser valuing a leased fee interest estimates the property's resale value at the end of a ten-year lease. The appraiser discounts that reversion back to today at a market-derived yield rate using compound discounting.
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The subject is a 10-year-old dwelling with improvement cost new of $640,000. A highly comparable 7-year-old dwelling recently sold for $580,000; at the sale date its site was worth $150,000 and its improvement cost new was $500,000. Market evidence shows depreciation accruing evenly with actual age. What is the depreciated value of the subject improvements?