A price index rises from 100 to 121 over two years. What compound annual rate does this represent?
Correct Answer
C) 10.0 percent a year
Why this is correct: The index rose from 100 to 121, a 21% total increase over two years. This represents a 10% compound annual rate because 1.10 * 1.10 = 1.21. Simply dividing 21% by 2 years (10.5%) ignores compounding. Why the other choices are wrong: 21.0 percent a year would yield a much higher result. 10.5 percent a year is the simple average, not the compound rate. 12.1 percent a year does not mathematically result in 121 from 100 in two years. Exam tip: For compound growth, find the annual multiplier: (Ending Value / Starting Value)^(1/Number of Years) - 1.
Why This Is the Correct Answer
Option C is correct because 10.0 percent compounded over two years turns an index of 100 into 121: the first year reaches 110, and 10 percent of 110 adds 11 more to reach 121. The result can be verified in either direction, by compounding forward or by taking the square root of 1.21, and both confirm 1.10. This is the recognized method for annualizing a multi-period change. The clean numbers in this item are a deliberate signal that the compound relationship is what is being tested.
Why the Other Options Are Wrong
Option A: 21.0 percent a year
Twenty-one percent per year is the cumulative two-year change misread as an annual rate. Applied for two years it would produce an index of about 146, far above the stated 121. This error treats the total change as if it occurred in each period rather than across both.
Option B: 10.5 percent a year
Ten and a half percent is 21 divided by 2, the simple average, and it ignores that the second year's growth builds on the first year's result. Because compounding does part of the work, the true annual rate must be less than the simple average, which is a useful directional check. The gap looks small over two years but widens sharply over longer horizons.
Option D: 12.1 percent a year
Twelve and one tenth percent appears to be constructed by moving a decimal in the ending index, and it has no mathematical relationship to the growth question asked. Compounded twice it would produce roughly 125.7, overshooting the stated index. Distractors built by rearranging digits from the stem are common, and testing the candidate figure by compounding it forward exposes them immediately.
Two Tens Make Twenty-One
Ten percent twice does not give twenty percent; it gives twenty-one, because the second ten percent is charged on the bigger number. Memorize the pair 1.10 squared equals 1.21, along with 1.05 squared equals about 1.1025. Seeing 121 over two years should trigger 10 percent instantly.
How to use: When an index or price change spans multiple periods, never divide by the number of periods. Take the ratio, take the appropriate root, and subtract one. To check your answer under exam pressure, compound the candidate rate forward and see whether it lands on the stated ending value.
Exam Tip
Compound the answer choices forward rather than computing roots by hand; multiplying 1.10 by 1.10 is faster and less error-prone than extracting a square root on scratch paper.
Common Mistakes to Avoid
- -Dividing a cumulative percentage change by the number of years to get an annual rate
- -Applying an annual market conditions rate to a comparable that sold only a few months earlier
- -Adjusting from the sale closing date when the price was actually set at contract, months earlier
Concept Deep Dive
Analysis
This question tests compound versus simple growth, which is the arithmetic behind every time adjustment applied over more than one period. Moving from 100 to 121 is a 21 percent cumulative rise, but that increase did not occur on a fixed base: the second year's growth applied to the already-grown level from the first year. The correct annual rate is the value that, applied twice, reproduces the ending index, which is the square root of 1.21. Since 1.10 times 1.10 equals 1.21 exactly, the compound annual rate is 10.0 percent. The general formula is the ending value divided by the beginning value, raised to the power of one over the number of periods, minus one. Appraisers need this whenever a market conditions adjustment spans multiple months or years, because applying a simple average rate compounds a small error into a large one over a long interval, and because rates quoted in the market are almost always annual while adjustments are often applied monthly.
Background Knowledge
You need to know the compound growth formula, ending divided by beginning raised to the one over n power minus one, and to be able to invert it by compounding a candidate rate forward as a check. You should also know that a compound annual rate is always lower than the simple average of the same cumulative change, and how to convert an annual rate to a monthly one when applying market conditions adjustments over partial years.
Real-World Application
Deriving a market conditions adjustment, an appraiser finds the local median price index moved from 100 to 121 over 24 months, concludes a 10 percent annual compound rate, converts it to roughly 0.80 percent per month, and applies that monthly factor to each comparable according to its exact elapsed time since contract.
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