In a hypothetical subdivision cash-flow model, a supported remediation payment of $350,000 occurs exactly six months after the valuation date. Using a 14% effective annual discount rate with compound fractional-year discounting, what is that payment’s present value, rounded to the nearest $100? This asks only for the payment’s present value, not the property’s total environmental value effect.
Correct Answer
C) $327,800
Present cost = $350,000/(1 + 0.14)^(6/12) = $327,805.03; nearest $100 = $327,800. The indicated value is $327,800. This is the result under the stated inputs, not a universal market ratio or an approval of a proposed development. Common errors: discounting for a full year instead of six months gives $307,000; using simple half-year discounting (÷ 1.07) gives $327,100; compounding forward instead of discounting gives $373,700.
Why This Is the Correct Answer
Option C is correct because PV = $350,000 / (1.14)^0.5 = $327,805.03, which rounds to $327,800 at the nearest $100. The payment occurs six months out, so the exponent is 0.5. Compound discounting of an effective annual rate requires the fractional exponent rather than a simple half-rate.
Why the Other Options Are Wrong
Option A: $307,000
Option A, $307,000, discounts for a full year: $350,000 / 1.14 = $307,017.54, rounded to $307,000. The payment is only six months away, so the correct figure is $350,000 / 1.14^0.5 = $327,800.
Option B: $327,100
Option B, $327,100, uses simple half-year discounting: $350,000 / 1.07 = $327,102.80, rounded to $327,100. The stem calls for compound fractional-year discounting, $350,000 / 1.14^0.5 = $327,800.
Option D: $373,700
Option D, $373,700, compounds forward instead of discounting: $350,000 × 1.14^0.5 = $373,697.74, rounded to $373,700. A present value of a future payment must be less than the payment, $327,800 here.
Future to present: divide
Money due later is worth less today, so divide by the growth factor. For half a year at an annual effective rate, use the square root of (1 + i).
How to use: Use this to check that a present value answer is smaller than the future amount.
Exam Tip
Read the discounting convention in the stem: compound fractional-year means a fractional exponent, while simple means scaling the rate.
Common Mistakes to Avoid
- -Discounting a six-month payment for a full year
- -Using half the annual rate when compound discounting is specified
- -Multiplying by the growth factor instead of dividing
Concept Deep Dive
Analysis
This question tests discounting a single future amount with a fractional-year period. The present value formula is PV = FV / (1 + i)^n, with n = 6/12 = 0.5 year and i = 0.14. Because the rate is an effective annual rate with compound fractional-year discounting, you take the square root of 1.14, not half the rate.
Background Knowledge
Present value converts a future amount to today's dollars by dividing by (1 + i) raised to the number of periods. With an effective annual rate, a partial year uses a fractional exponent. In subdivision and discounted cash flow models, the timing of each cost or revenue affects its present value.
Real-World Application
An appraiser builds a cash-flow model for a proposed subdivision where environmental cleanup must be paid shortly after acquisition. Each cost is discounted from its expected payment date so the model reflects the time value of money.
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