An income stream of $50,000 per year is expected to continue for 10 years. Using a discount rate of 9%, what is the present value? (Present value factor for 10 years at 9% = 6.418)
Correct Answer
C) $320,900
Why this is correct: The present value of a level annuity is calculated as the annual payment multiplied by the present value annuity factor. Given: Annual Income = 50,000, Factor = 6.418. Calculation: 50,000 * 6.418 = 320,900. Why the other choices are wrong: "$21,094" results from dividing the income by the factor. "$45,872" is approximately the income divided by the discount rate (50,000/0.09). "$500,000" is simply the sum of the payments over 10 years without discounting. Exam tip: For annuity PV, multiply payment by factor. For a single future sum, you would multiply by a present value of $1 factor.
Why This Is the Correct Answer
Option A is correct because it properly applies the present value of annuity formula: Annual Income × Present Value Factor. The calculation is straightforward: $50,000 × 6.418 = $320,900. This represents the current worth of receiving $50,000 annually for 10 years when money is discounted at 9% per year. The present value factor of 6.418 already incorporates the 9% discount rate over the 10-year period, making this a direct multiplication problem.
Why the Other Options Are Wrong
Option A: $21,094
Option A ($21,094) is significantly too low and doesn't correspond to any logical calculation related to this problem. It might represent a calculation error or confusion with a different type of present value calculation.
Option B: $45,872
Option B ($45,872) appears to be the present value of only the first year's income ($50,000 ÷ 1.09 ≈ $45,872), but this ignores the remaining 9 years of income streams. This represents a fundamental misunderstanding of the annuity concept.
Option D: $500,000
Option D ($500,000) represents the total undiscounted income over 10 years ($50,000 × 10), but this ignores the time value of money completely. This would only be correct if the discount rate were 0%, which is unrealistic in real-world scenarios.
PV-A Formula: Pay Victory - Annually
Remember 'PV-A = Payment × Victory Factor' where PV-A is Present Value of Annuity, Payment is the annual income, and Victory Factor is the present value factor that 'wins' by doing all the complex math for you.
How to use: When you see an annuity problem, immediately think 'Pay Victory' and look for: (1) the annual payment amount, (2) the present value factor (usually given), then simply multiply them together.
Exam Tip
Present value factors are typically provided in exam questions - don't try to calculate them from scratch. Focus on identifying whether you're dealing with a single sum or an annuity, then apply the correct formula.
Common Mistakes to Avoid
- -Using the total undiscounted income instead of applying the present value factor
- -Calculating present value for only one year instead of the entire annuity stream
- -Confusing present value of annuity with present value of a single sum
Concept Deep Dive
Analysis
This question tests the fundamental concept of present value of an annuity, which is crucial in real estate appraisal for income capitalization approaches. The present value calculation determines what a series of future income payments is worth in today's dollars, accounting for the time value of money through a discount rate. This concept is essential for valuing income-producing properties where rental income streams need to be converted to present value. The calculation requires multiplying the annual income by the appropriate present value factor, which accounts for both the discount rate and the time period.
Background Knowledge
Present value of annuity calculations are fundamental to the income capitalization approach in real estate appraisal. Appraisers must understand that money received in the future is worth less than money received today due to inflation, risk, and opportunity cost, which is why we apply discount rates to future income streams.
Real-World Application
An appraiser valuing a rental property with a 10-year lease at $50,000 annual rent would use this calculation to determine what that income stream is worth today, helping establish the property's value using the income approach.
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