An appraiser is updating appraisals of two duplexes in different market areas, each with unchanged NOI of $33,000. Since the prior appraisals, the overall rate indicated for Duplex A has fallen from 6.0% to 5.5%, and the overall rate indicated for Duplex B has fallen from 10.0% to 9.5%. What is the percentage increase in the indicated value of each duplex?
Correct Answer
A) Duplex A 9.1%; Duplex B 5.3%
With NOI unchanged, value moves with the ratio of the old rate to the new rate. Duplex A rises 6.0 ÷ 5.5 − 1 = 9.1% ($550,000 to $600,000), and Duplex B rises 10.0 ÷ 9.5 − 1 = 5.3% ($330,000 to $347,368). Dividing the 0.5 point by the old rate gives 8.3% and 5.0%, and dividing it by the old rate plus 0.5 gives 7.7% and 4.8%. Swapping the results gives the reverse pair.
Why This Is the Correct Answer
Option A is correct because with NOI unchanged, the value increase equals the old rate divided by the new rate, less 1. Duplex A rises 6.0 ÷ 5.5 − 1 = 9.1%, from $550,000 to $600,000. Duplex B rises 10.0 ÷ 9.5 − 1 = 5.3%, from $330,000 to $347,368.
Why the Other Options Are Wrong
Option B: Duplex A 8.3%; Duplex B 5.0%
Option B, 8.3% and 5.0%, divides the 0.5 point change by the old rates: 0.5 ÷ 6.0 and 0.5 ÷ 10.0. That is the percentage change in the rate, while value moves by the old rate over the new rate: 6.0 ÷ 5.5 − 1 = 9.1% and 10.0 ÷ 9.5 − 1 = 5.3%.
Option C: Duplex A 5.3%; Duplex B 9.1%
Option C, 5.3% for Duplex A and 9.1% for Duplex B, reverses the correct results. The same 50 basis points is a larger share of the lower rate, so Duplex A at 6.0% gains 9.1% and Duplex B at 10.0% gains 5.3%.
Option D: Duplex A 7.7%; Duplex B 4.8%
Option D, 7.7% and 4.8%, comes from 0.5 ÷ 6.5 and 0.5 ÷ 10.5, which measures the value loss from a rate rise of 50 basis points. These rates fell, so the correct results are 6.0 ÷ 5.5 − 1 = 9.1% and 10.0 ÷ 9.5 − 1 = 5.3%.
Old Rate Over New Rate
When only the rate changes, the value multiple is the old rate over the new rate. Small denominators make the same 50 basis points count for more.
How to use: Use the old-over-new ratio whenever NOI is unchanged and only the overall rate has moved.
Exam Tip
When income does not change, you can skip the dollar values and work from the ratio of the rates. Check that a falling rate gives an increase measured against the old value.
Common Mistakes to Avoid
- -Assuming equal basis point moves produce equal percentage changes in value
- -Dividing the basis point change by the old rate instead of comparing old and new rates
- -Expecting the higher-rate property to be the more sensitive one
Concept Deep Dive
Analysis
Because value equals income divided by rate, value responds to the proportional change in the rate, not to the number of basis points. A 50 basis point drop is a larger share of a 6.0% rate than of a 10.0% rate, so the same move lifts value more at the lower level. NOI cancels out of the percentage, so the ratio of old rate to new rate gives the answer directly.
Background Knowledge
Direct capitalization divides income by a rate, so value and rate move inversely and the relationship is not linear. A fixed change in basis points has a larger proportional effect when the rate is low. This is why value is more sensitive to rate changes in low-rate markets.
Real-World Application
An appraiser updates files on two small rental properties, one in a low-rate metropolitan market and one in a higher-rate small-town market. Both markets show rates down 50 basis points, but the owner in the low-rate market sees a noticeably larger percentage gain with the same income.
More Income Approach Questions
Which model describes yield capitalization of property benefits?
An appraiser is valuing a duplex. A market survey of recent duplex sales reports prices at 10.8 times potential gross income, with stabilized vacancy and credit loss of 10% of PGI and no other income. The subject's rent schedule is unavailable, but the property manager's records show collected income of $54,000 a year, in line with market rent and the 10% market loss. What value is indicated?
The subject property is a fourplex with annual EGI of $72,000 and operating expenses of $28,800. Of those expenses, $18,000 is fixed and $10,800 varies in direct proportion to EGI. The owner's annual mortgage payments are $21,000. An appraiser projects that EGI will fall 10% because of higher vacancy. What is the projected NOI?
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The subject property has annual potential rent of $88,000, with combined vacancy and collection loss of 5% of rent only. The owner also collects expense reimbursements of $6,400 per year, with no loss taken against them. Total property expenses of $15,000 include the reimbursed costs and exclude debt service. The overall rate is 10%. What value is indicated by direct capitalization?