An appraiser is valuing land for a proposed townhouse subdivision using the subdivision development method. The site requires a $350,000 environmental remediation cost that will be incurred 6 months after project commencement. The developer’s required annual yield is 14%. What is the present value of the remediation cost to be deducted from gross lot sales?
Correct Answer
A) $327,800
Why this is correct: A cost incurred in the future is deducted at its present value, discounted at the developer's required yield for the actual time until it is incurred. Present value = future amount / (1 + r)^t, with r = 0.14 and t = 0.5 years. The half-year factor is the square root of 1.14, which is 1.0677078, so $350,000 / 1.0677078 = $327,805, or $327,800 rounded to the nearest hundred. Why the other choices are wrong: '$332,000' discounts at roughly 5.4% for the period, well short of a half year at 14%, and understates the discount. '$337,500' is close to a 3.6% deduction and understates it further; it is the figure a candidate gets by treating the half-year discount as far gentler than 14% per year implies. '$326,500' overshoots the compound figure slightly and matches no supportable convention: compound discounting gives $327,805, a simple-interest deduction of half of 14% gives $350,000 x 0.93 = $325,500, and discounting a full year by mistake gives $307,018. Exam tip: For a partial-year discount, raise (1 + r) to the fractional power rather than prorating the rate. Six months at 14% annually is division by 1.0677, not by 1.07.
Why This Is the Correct Answer
With a future amount of $350,000, an annual yield of 14% and a term of half a year, the factor is 1.14 raised to the 0.5 power, which is 1.0677078. Dividing gives $350,000 / 1.0677078 = $327,805, reported as $327,800. The result must be slightly below the undiscounted $350,000 and well above a full year's discount of $307,018, which is a quick sanity check on any answer you pick.
Why the Other Options Are Wrong
Option B: $332,000
$332,000 implies a discount of only about 5.4% over the six months, which corresponds to an annual rate near 11%, below the 14% the developer requires. Any figure this high leaves part of the developer's required return unearned on the money tied up in remediation.
Option C: $337,500
$337,500 implies roughly a 3.6% deduction for the half year, less than a third of the compound discount the stated yield calls for. This is the answer a candidate reaches by discounting far too gently, as though the six-month wait were nearly costless.
Option D: $326,500
$326,500 is close to the right figure but corresponds to no supportable convention. Compound discounting gives $327,805, a simple-interest deduction of half of 14% gives $325,500, and a full-year discount gives $307,018. A near-miss that cannot be reproduced by any method is not a defensible answer.
Fraction the power, not the rate
For part of a year, put the fraction in the exponent, never in the rate. Six months at 14% is 1.14 to the one-half power, about 1.0677, so the half-year discount is a shade under 6.4%, not the 7% a prorated rate suggests.
How to use: Whenever a stem gives a cost or receipt at a non-annual interval, write the exponent first: three months is 0.25, six months is 0.5, eighteen months is 1.5. Then compute the factor and divide.
Exam Tip
Bracket your answer before choosing. A six-month discount must fall between the undiscounted amount and the full-year discounted amount, which usually eliminates two options immediately.
Common Mistakes to Avoid
- -Prorating the annual rate instead of using a fractional exponent
- -Discounting a six-month cost as though it were a full year away
- -Accepting a near-miss option and calling the difference rounding
Concept Deep Dive
Analysis
The subdivision development method values raw land as the present value of projected lot sales less the present value of every cost of getting those lots to market, including development costs, holding costs, marketing, and the developer's profit. Because the revenues and the costs occur at different times, each has to be discounted at the developer's required yield for the actual interval before it occurs. A remediation cost due in six months is discounted for half a year, and a half-year discount at an annual rate means raising one plus the rate to the 0.5 power, not halving the rate. The distinction matters because compounding is not linear, and exam items are written so that the prorated shortcut lands on a different answer.
Background Knowledge
You need the mechanics of present value at a stated yield, including fractional periods, and the structure of the subdivision development method: gross lot sales, absorption period, development and holding costs, marketing, entrepreneurial profit, and discounting everything to the date of value. You also need to know that the discount rate in this analysis is the developer's required yield, not a capitalization rate.
Real-World Application
A land appraiser valuing a townhouse tract nets out a remediation obligation the developer will incur once grading starts. The workfile shows the timing assumption, the yield rate drawn from developer interviews, and the fractional-period factor, so a reviewer can reproduce the deduction exactly.
More Land/Site Questions
Under which condition is the land residual technique most applicable?
Why can the same physical parcel carry different values in two assignments?
In a built-up area where no vacant land has sold for years, which approach to site value is the usual fallback?
How is entrepreneurial profit treated in the subdivision development method?
A land comparable sold 18 months ago in a market rising about 4 percent a year. What adjustment direction applies?
Why does a developer's required profit rise for a longer subdivision project?
How does holding cost enter the valuation of land bought for future development?
Excess land is best described as land that has which characteristic?
Plottage value arises in which of the following situations?
Which of the following is an off-site improvement rather than a site improvement?
People Also Study
Real Estate Market
13.6% of exam
Property Description
11.8% of exam
Sales Comparison Approach
16.4% of exam
Cost Approach
13.6% of exam
Income Approach
8.2% of exam