An income stream of $50,000 per year for 10 years, with a discount rate of 10%, has what present value? (Use PV factor of 6.1446 for 10 years at 10%)
Correct Answer
C) $307,230
Why this is correct: The present value of an annuity (a series of equal annual payments) is calculated as Annual Payment multiplied by the Present Value Factor. Given the factor of 6.1446 for 10 years at 10%, the calculation is: $50,000 * 6.1446 = $307,230. Why the other choices are wrong: "$192,770" might result from using an incorrect factor or misplacing the decimal. "$614,460" is $50,000 * 12.2892, which could be a factor for a different period or rate. "$500,000" is simply the total undiscounted cash flow ($50,000 * 10 years), ignoring the time value of money. Exam tip: Present value factors are provided in exams. Remember: PV of Annuity = Annual Income * PV Factor.
Why This Is the Correct Answer
Option B is correct because it properly applies the present value of annuity formula: Annual Payment × PV Factor = Present Value. The calculation is straightforward: $50,000 × 6.1446 = $307,230. This represents the current worth of receiving $50,000 annually for 10 years when money has a 10% opportunity cost. The PV factor of 6.1446 already incorporates the mathematical complexity of discounting each year's payment back to present value.
Why the Other Options Are Wrong
Option A: $192,770
$192,770 appears to be an incorrect calculation, possibly resulting from using the wrong PV factor or making an arithmetic error in the multiplication.
Option B: $614,460
$614,460 incorrectly multiplies the PV factor by 100 instead of using it as a direct multiplier, suggesting a misunderstanding of how PV factors work.
Option D: $500,000
$500,000 represents the total undiscounted payments ($50,000 × 10 years) without considering the time value of money, which ignores the fundamental principle that future money is worth less than present money.
PAP Formula
Remember 'PAP' - Payment × Annuity factor = Present value. Think of it as giving your future income stream a 'PAP' (pat) to compress it down to today's value.
How to use: When you see an annuity problem, immediately think 'PAP' and look for three components: the annual payment amount, the present value factor (given or to be calculated), and multiply them together for the present value.
Exam Tip
Always check if the PV factor is provided in the question - if so, it's a simple multiplication problem and you don't need to calculate the complex present value formula from scratch.
Common Mistakes to Avoid
- -Adding up all payments without discounting (choosing the simple sum)
- -Confusing present value factors with other financial factors
- -Misplacing decimal points when multiplying large numbers
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 when valuing income-producing properties, as appraisers must convert future rental income streams into current market value. The calculation uses a present value annuity factor, which is a pre-calculated multiplier that accounts for both the discount rate and the time period.
Background Knowledge
Present value of annuity calculations require understanding that money received in the future is worth less than money received today due to opportunity cost and inflation. The present value factor is derived from the formula: PV = PMT × [(1 - (1 + r)^-n) / r], where PMT is the payment, r is the discount rate, and n is the number of periods.
Real-World Application
Appraisers use this calculation when valuing income properties like apartment buildings or commercial properties with long-term leases, where they need to determine what the future rental income stream is worth in today's market for comparison with sale prices of similar properties.
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