A property sells for $400,000 and resells for $463,050 exactly three years later. Compute the compound annual growth rate of the observed prices, without attributing the change solely to market conditions.
Correct Answer
A) 5.0 percent a year
CAGR = (463,050 / 400,000)^(1/3) − 1 = 0.05, or 5.0% per year. Simple gain divided by three is about 5.25% and does not compound; 15.7625% is the total three-year increase. This is an observed price-growth calculation. Attribution solely to market conditions would also require checking financing, concessions, rights, transaction circumstances and other relevant differences.
Why This Is the Correct Answer
Option A is correct because CAGR = ($463,050 ÷ $400,000)^(1/3) − 1 = (1.157625)^(1/3) − 1 = 0.05, or 5.0 percent a year. Checking the result, $400,000 × 1.05³ = $463,050. Attributing all of this growth to market conditions would require checking financing, concessions, property changes and other differences between the sales.
Why the Other Options Are Wrong
Option B: 5.3 percent a year
Option B, 5.3 percent a year, comes from simple averaging without compounding: ($463,050 − $400,000) ÷ $400,000 = 15.7625%, divided by 3 ≈ 5.25%, rounded to 5.3%. The compound rate is (1.157625)^(1/3) − 1 = 5.0%.
Option C: 15.8 percent a year
Option C, 15.8 percent a year, is the total three-year increase of 15.7625% rounded and mislabeled as an annual rate. Spread over three compounding years, the annual rate is (1.157625)^(1/3) − 1 = 5.0%.
Option D: 4.8 percent a year
Option D, 4.8 percent a year, inverts the ratio by dividing the starting price by the ending price: ($400,000 ÷ $463,050)^(1/3) = 0.9524, and 1 − 0.9524 ≈ 4.8%. That is a discount rate back from the resale price, not growth from the first sale; the correct rate is ($463,050 ÷ $400,000)^(1/3) − 1 = 5.0%.
End over start, root, minus one
For compound growth, divide the ending price by the starting price, take the root for the number of years, then subtract one.
How to use: Use this three-step sequence whenever a question asks for an annual growth rate over several years.
Exam Tip
Simple division of the total change by the number of years overstates the compound rate, and putting the starting price on top understates it. Check your answer by compounding the starting price forward.
Common Mistakes to Avoid
- -Dividing the total percentage change by the number of years
- -Reporting the total change as an annual rate
- -Dividing the starting price by the ending price
Concept Deep Dive
Analysis
This question tests the compound annual growth rate (CAGR) between two sale prices. Divide the ending price by the starting price, take the root for the number of years, and subtract 1. The question also cautions that the observed growth may reflect factors besides market conditions.
Background Knowledge
Compound growth assumes each year's change builds on the prior year's value, so the annual rate is found with a root rather than simple division. Paired sales and resales of the same property are commonly used to analyze market trends. Before attributing the change to market conditions, the appraiser verifies that the property, financing and terms of sale were comparable.
Real-World Application
An appraiser finds a comparable that sold for $400,000 three years ago and resold this month for $463,050. Before using the 5.0% annual rate as a market conditions trend, she confirms neither sale involved concessions and the house was not renovated between sales.
More Statistics Questions
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