The subject property is a 14-unit apartment building with NOI of $140,000, valued using a 7% overall rate. A client asks how value would change if NOI falls 5% and the overall rate for similar properties rises to 8%. What percentage decrease in value would result?
Correct Answer
A) 16.9%
Today's value is $140,000 ÷ 0.07 = $2,000,000. Forecast NOI is $133,000, so the new value is $133,000 ÷ 0.08 = $1,662,500, a decline of $337,500, or 16.9%. Adding the 5% to a 12.5% rate effect gives 17.5%, and adding it to the rate's own 14.3% rise gives 19.3%. The 20.3% figure inverts the calculation by comparing today's value with the lower forecast value.
Why This Is the Correct Answer
Option A is correct because today's value is $2,000,000 and the forecast value is $133,000 ÷ 0.08 = $1,662,500. The decline of $337,500 divided by $2,000,000 is 16.875%, or 16.9%. Equivalently, 0.95 × (7 ÷ 8) = 0.83125, which is a 16.9% drop.
Why the Other Options Are Wrong
Option B: 17.5%
Option B, 17.5%, adds the 12.5% effect of the rate change to the 5% income decline. These effects compound rather than add, since 0.95 × 0.875 = 0.83125, so the decline is 16.9%.
Option C: 19.3%
Option C, 19.3%, adds the 5% income decline to 14.3%, the percentage rise in the rate from 7% to 8%. Value moves inversely with the rate, so the rate effect alone is 7 ÷ 8, a 12.5% drop, and the combined decline is 16.9%.
Option D: 20.3%
Option D, 20.3%, divides the $337,500 decline by the forecast value of $1,662,500. A decline from today's value must use today's $2,000,000 as the base, which gives 16.9%.
Multiply the changes, do not add
Multiply the income factor by the old rate over the new rate: 0.95 × 7/8. One minus the result is the percentage decline.
How to use: Use this to check any problem in which income and the overall rate change together.
Exam Tip
If two variables change at once, calculate the before and after values rather than adding percentages. Then measure the change from the starting value.
Common Mistakes to Avoid
- -Adding the percentage effects of income and rate
- -Treating a one-point rate rise as a one-percent value drop
- -Measuring the decline from the lower forecast value
Concept Deep Dive
Analysis
This tests whether you know how two changes combine in V = I/R. The income change and the rate change compound, so their percentage effects cannot simply be added. The dependable path is to compute both values and measure the decline from today's value.
Background Knowledge
Value is income divided by rate, so a rise in the rate lowers value and a fall in income lowers it further. Because the rate is a small number in the denominator, small variations in it can produce large movements in value. Forecasts of both variables are therefore usually tested together.
Real-World Application
A lender asks an appraiser how a loan's collateral would hold up if rents soften while investors demand higher returns. Comparing the current value with the stressed value shows the percentage cushion needed under that scenario.
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Previous Question
The subject property is a fourplex with potential gross income of $120,000. Stabilized vacancy/credit loss in its neighborhood is 10% of PGI. A similar fourplex in an adjoining neighborhood with lower vacancy sold for $855,000 cash equivalent, with PGI of $100,000 and stabilized vacancy/credit loss of 5% of PGI. What value does an effective gross income multiplier indicate for the subject?
Next Question
The subject property has annual NOI of $112,000. Comparable sales indicate a price-to-income multiplier of 12.5, based on annual NOI. What is the indicated value?