A property sold for $400,000 and resold three years later for $463,050 with no physical change. What compound annual rate does this indicate?
Correct Answer
A) 5.0 percent a year
Why this is correct: The original explanation shows the calculation: $400,000 * (1 + r)^3 = $463,050. Solve for r: (1+r)^3 = $463,050/$400,000 = 1.157625. The cube root of 1.157625 is 1.05, so r = 0.05 or 5%. The governing concept is compound annual growth rate (CAGR). Why the other choices are wrong: 5.3 percent a year might come from simple interest calculation: ($63,050 gain / 3 years) / $400,000 = 5.25%. 15.8 percent a year is the total appreciation over 3 years ($63,050/$400,000), not annualized. 4.2 percent a year is not supported by the math. Exam tip: For compound growth, use: (End Value / Start Value)^(1/n) - 1, where n is years.
Why This Is the Correct Answer
Option A is correct because 5.0 percent compounded three times reproduces the resale price exactly: $400,000 grows to $420,000, then $441,000, then $463,050. Verification by forward compounding is easier and more reliable than extracting a cube root under exam conditions, and it confirms the answer to the dollar. The exact match also confirms the item intends compound rather than simple growth. No other rate offered reproduces the ending figure.
Why the Other Options Are Wrong
Option B: 5.3 percent a year
Five and three-tenths percent comes from treating the growth as simple, dividing the 15.76 percent total by three years to get about 5.25 percent and rounding up. That approach ignores that the second and third years grew on an already larger base. Because compounding contributes part of the total, the true annual rate must always be below the simple average, which makes this choice detectable without any calculation.
Option C: 15.8 percent a year
Fifteen point eight percent is the cumulative three-year change reported as though it were annual. Compounded over three years it would produce roughly $620,000, far above the actual resale price. Mislabeling a total as a rate per year is the most consequential version of this error, since it would triple a market conditions adjustment.
Option D: 4.2 percent a year
Four and two-tenths percent understates the growth and does not reconcile to the numbers: compounded three times it reaches only about $452,600, roughly $10,000 short. It has no derivation from the stem's figures. Testing a candidate rate by compounding it forward exposes unsupported distractors like this in a few seconds.
Grow It Forward to Check
Do not fight the cube root; grow the candidate forward instead. Five percent on $400,000 gives $420,000, then $441,000, then $463,050. If the third step lands on the stated price, that rate is the answer. Multiplying three times is faster and safer than extracting a root by hand.
How to use: Identify the number of periods first, then test the roundest answer choice by compounding it forward. Remember that the compound rate is always less than the simple average of the same total, so eliminate any choice at or above the simple average. Also confirm the question wants an annual rate and not the cumulative change.
Exam Tip
When compound growth choices include one round number and several awkward ones, test the round number first; these items are built so the correct rate comes out clean.
Common Mistakes to Avoid
- -Dividing total appreciation by the number of years and calling the result a compound rate
- -Using a resale pair where the property was renovated between the two sales
- -Reporting the cumulative percentage change as though it were an annual rate
Concept Deep Dive
Analysis
This question tests compound annual growth applied to a resale pair, which is one of the cleanest ways to support a market conditions adjustment. Because the property is described as physically unchanged between the two transactions, the entire price movement is attributable to the passage of time, which is exactly the isolation a matched-pair resale analysis seeks. The ratio of the two prices is $463,050 divided by $400,000, or 1.157625, representing a 15.76 percent cumulative rise over three years. Converting that to an annual figure requires the cube root rather than division, and the cube root of 1.157625 is exactly 1.05, giving 5.0 percent per year. The clean result is the tell, since exam writers construct these numbers so that the correct compound rate resolves to a round figure while the simple-division alternative does not. In practice a single resale pair is thin evidence, so an appraiser would seek several such pairs before concluding a rate.
Background Knowledge
You need the compound growth formula, ending divided by beginning raised to the power of one over the number of periods, minus one, and the ability to verify by compounding forward. You should also know how resale and repeat-sales analysis supports market conditions adjustments, and that the analysis requires the property to be unchanged physically and both transactions to be arm's length.
Real-World Application
Supporting a time adjustment in a slow market, an appraiser assembles seven resale pairs with no permits filed between transactions, computes compound annual rates ranging from 4.6 to 5.4 percent, concludes 5.0 percent, and converts it to roughly 0.41 percent per month for application to each comparable's elapsed time.
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