An income stream consists of ten $10,000 year-end receipts, discounted at an 8% annual rate. The present value of an annuity factor is approximately 6.710. What is the present value?
Correct Answer
C) $67,100
10000×6.710 = 67100. Common errors: adding the ten receipts without discounting gives $100,000; dividing by the factor instead of multiplying gives $1,490; discounting the total of all receipts as a single year-10 amount gives $46,319.
Why This Is the Correct Answer
Option C is correct because $10,000 × 6.710 = $67,100. The annuity factor already combines the discounting of all ten year-end receipts at 8%, so one multiplication gives the present value.
Why the Other Options Are Wrong
Option A: $1,490
Option A, $1,490, comes from dividing the payment by the factor ($10,000 ÷ 6.710) instead of multiplying. The present value is $10,000 × 6.710 = $67,100.
Option B: $46,319
Option B, $46,319, results from treating all ten receipts as one $100,000 amount received at the end of year 10 and discounting it ($100,000 ÷ 1.08^10). The receipts arrive each year, so the annuity factor applies, giving $67,100.
Option D: $100,000
Option D, $100,000, simply adds the ten $10,000 receipts without discounting. Future receipts are worth less today, so the present value is $67,100.
Payment times factor
For a level series, present value is the payment multiplied by the annuity factor, which is less than the number of payments.
How to use: Use this to check that an annuity answer is below the undiscounted total.
Exam Tip
A present value of a series of positive receipts should be less than their simple total; answers equal to or above the total are not discounted.
Common Mistakes to Avoid
- -Dividing by the annuity factor
- -Adding the payments without discounting
- -Discounting the total as a single lump sum at the end
Concept Deep Dive
Analysis
This question tests use of a present-value-of-an-annuity factor. For a level series of year-end receipts, multiply the periodic payment by the stated factor to get present value.
Background Knowledge
The present value of an annuity is the sum of the present values of a series of equal periodic payments. Annuity factors from tables or calculators combine the discounting of each payment, so present value equals the payment times the factor.
Real-World Application
An appraiser valuing a leased fee interest with level annual rent for ten years multiplies the rent by the present value of an annuity factor at the market discount rate to value the income stream.
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