An income stream of $50,000 per year for 10 years, with a discount rate of 8%, has what present value? (Use PV factor of 6.71)
Correct Answer
D) $335,500
Why this is correct: The present value of a series of equal annual payments (an annuity) is calculated as: Present Value = Annual Payment × Present Value Factor. Given the factor of 6.71 for the specified term and rate, the calculation is: $50,000 × 6.71 = $335,500. Why the other choices are wrong: The choice "$50,000" is just the single annual payment, ignoring the 10-year stream. The choice "$400,000" might be a rough estimate using a simple 8% cap rate ($50,000/0.08 = $625,000) or another incorrect calculation. The choice "$500,000" is the undiscounted sum of the payments over 10 years ($50,000 × 10), which ignores the time value of money. Exam tip: When a PV factor is provided, use it. The present value of an annuity will always be less than the sum of the payments (if the discount rate is positive).
Why This Is the Correct Answer
Option A is correct because it properly applies the present value formula for an annuity: Present Value = Annual Income × PV Factor. The calculation is straightforward: $50,000 × 6.71 = $335,500. The PV factor of 6.71 already incorporates the 8% discount rate over the 10-year period, representing the sum of individual discount factors for each year. This methodology correctly converts the future income stream into its equivalent present value.
Why the Other Options Are Wrong
Option A: $50,000
Option A ($50,000) only represents one year's income and completely ignores both the 10-year duration and the present value calculation. This answer fails to consider the annuity nature of the income stream.
Option B: $400,000
Option B ($400,000) appears to be an arbitrary figure that doesn't correspond to any logical calculation method. It's neither the undiscounted total nor the properly discounted present value, making it an incorrect distractor.
Option C: $500,000
Option C ($500,000) represents the total undiscounted income over 10 years ($50,000 × 10), which ignores the time value of money completely. This fails to account for the fact that future dollars are worth less than present dollars due to the 8% discount rate.
PV-AI Formula
Remember 'PV = AI × PVF' where PV is Present Value, AI is Annual Income, and PVF is Present Value Factor. Think 'Pay Very Attention to Income' to remember the multiplication relationship.
How to use: When you see an annuity present value question, immediately identify the three components: annual income amount, the given PV factor, and multiply them together. Don't get distracted by calculating individual year discounts when a factor is provided.
Exam Tip
Always use the provided PV factor rather than trying to calculate it manually during the exam - this saves time and reduces calculation errors. Double-check that you're multiplying, not dividing, the annual income by the factor.
Common Mistakes to Avoid
- -Using total undiscounted income instead of applying the PV factor
- -Dividing instead of multiplying the annual income by the PV factor
- -Trying to manually calculate individual year present values instead of using the provided annuity factor
Concept Deep Dive
Analysis
This question tests the fundamental concept of present value calculation for an annuity in real estate appraisal. Present value represents the current worth of a future stream of income payments, discounted back to today's dollars using a specific discount rate. The calculation requires multiplying the annual income by a present value factor, which represents the sum of individual present value factors for each year of the income stream. This concept is essential in income approach valuations where appraisers must convert future rental income or cash flows into current market value.
Background Knowledge
Present value calculations are fundamental to the income approach in real estate appraisal, where future income streams must be converted to current market value. The present value factor represents the mathematical relationship between future cash flows and their current worth, incorporating both the discount rate and time period.
Real-World Application
Appraisers use this calculation when valuing income-producing properties like rental buildings, where they need to convert projected rental income over a holding period into current market value for comparison with sales data.
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