For equal-length annual periods with rates greater than −100%, why average the positive growth factors 1+r geometrically and then subtract 1?
Correct Answer
A) To obtain the constant annual rate reproducing the compounded endpoint change
For equal-length periods with positive growth factors, multiply the factors 1+r, take their geometric mean and subtract 1. That constant rate reproduces the same compounded change. Taking a geometric mean of raw signed percentage rates is generally invalid. The procedure does not eliminate outliers, use fewer observations or necessarily exceed the arithmetic average rate.
Why This Is the Correct Answer
Option A is correct because the geometric mean of the growth factors, minus 1, gives the constant annual rate that reproduces the compounded endpoint change. For example, factors of 1.10 and 0.95 multiply to 1.045, and the square root, 1.02225, gives a 2.225% annual rate that compounds back to 1.045. The arithmetic average of 2.5% would compound to about 1.0506, which does not match.
Why the Other Options Are Wrong
Option B: To produce a larger average rate than the arithmetic mean of the annual rates
Option B says the purpose is to produce a larger average rate than the arithmetic mean of the rates. The goal is to match the compounded change, not to produce a larger number, and the geometric rate does not exceed the arithmetic average; in the example it is lower, 2.225% versus 2.5%.
Option C: To remove the extreme growth observations from the series
Option C says the method removes the extreme growth observations from the series. The geometric mean uses all the growth factors; it does not eliminate outliers.
Option D: To avoid needing the annual observations
Option D says the method avoids needing the annual observations. Each annual growth factor is multiplied into the calculation, so the observations are still used.
Compounding calls for geometric
Changes that multiply need an average that multiplies. Convert rates to factors 1+r, take the geometric mean, then subtract 1.
How to use: Use it whenever a question asks for an average annual rate over several compounding periods.
Exam Tip
Test the claimed average by compounding it over the periods and checking that it reproduces the total change. The method that passes this test is the geometric one.
Common Mistakes to Avoid
- -Averaging annual rates arithmetically for a compounded series
- -Taking a geometric mean of signed percentages instead of factors
- -Forgetting to subtract 1 after taking the geometric mean
Concept Deep Dive
Analysis
This question tests why compound growth rates are averaged geometrically. Because annual changes compound, the average that reproduces the total change over the period is found by taking the geometric mean of the growth factors 1+r and subtracting 1. The arithmetic mean of the rates does not reproduce the compounded endpoint when the rates vary.
Background Knowledge
When values change by compounding, the overall change is the product of the period growth factors. The geometric mean of those factors, minus 1, is the single constant rate that produces the same overall change over equal-length periods. Taking a geometric mean of raw signed percentage rates is generally invalid, which is why the method uses factors 1+r.
Real-World Application
An appraiser analyzing market conditions finds that paired resales show prices up 10% one year and down 5% the next. To express a constant annual trend, the appraiser multiplies the factors and takes the geometric mean rather than averaging 10% and minus 5%. The resulting rate correctly reproduces the two-year change when applied to a sale date adjustment.
More Statistics Questions
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An appraiser estimates a market-wide price trend using only sales data from one brokerage that specializes in luxury homes, without checking how other segments are represented. What is the principal concern?
Compared with analyzing one matched sale pair for one feature, what can a suitably specified multiple regression do?
The base-period median sale price is $400,000 and the current-period median is $460,000. Compute the price relative defined as current ÷ base, as a decimal to three places.
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Assessment-to-sale ratios are 0.94, 0.96, 0.98, 1, 1.01. Compute the arithmetic mean. Report as a decimal ratio to three places.
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Sale prices are $200,000, $220,000, $240,000, $260,000, $280,000. Define percentile rank as (number strictly below the selected value ÷ total observations) × 100. What is the percentile rank of $240,000?