An appraiser estimates a building's value as its $4,000,000 cost new less $800,000 of depreciation. Total annual NOI for the property is $500,000. Market data indicate capitalization rates of 8% for both the land and the building. Using the land residual technique, what is the land value?
Correct Answer
B) $3,050,000
Building=4000000−800000=3200000; land=(500000−3200000×0.08)÷0.08=3050000. Common errors: allocating building income on cost new instead of depreciated value gives $2,250,000; capitalizing all of the NOI as land income gives $6,250,000; allocating building income on the depreciation amount gives $5,450,000.
Why This Is the Correct Answer
Option B is correct because building value is $4,000,000 − $800,000 = $3,200,000, building income is $3,200,000 × 0.08 = $256,000, and the residual $244,000 capitalized at 8% gives $3,050,000. In the land residual technique, income attributable to the building (building value times building rate) is deducted from total net income. The remainder is capitalized at the land rate.
Why the Other Options Are Wrong
Option A: $2,250,000
Option A, $2,250,000, allocates building income on cost new instead of depreciated value: ($500,000 − $4,000,000 × 0.08) ÷ 0.08 = $2,250,000. The building value is $3,200,000 after depreciation, which gives $3,050,000.
Option C: $5,450,000
Option C, $5,450,000, allocates building income on the depreciation amount: ($500,000 − $800,000 × 0.08) ÷ 0.08 = $5,450,000. Building income should be based on the $3,200,000 depreciated value, giving $3,050,000.
Option D: $6,250,000
Option D, $6,250,000, capitalizes all of the NOI as land income: $500,000 ÷ 0.08 = $6,250,000. The building's $256,000 share must be removed first, leaving $244,000 ÷ 0.08 = $3,050,000.
Known first, residual second
Take out income for the part you know (the depreciated building), then capitalize what is left for the part you seek (the land).
How to use: Use this sequence for land, building, mortgage or equity residual problems.
Exam Tip
In a land residual, base building income on the building value, which here is cost new less depreciation.
Common Mistakes to Avoid
- -Using cost new rather than depreciated building value
- -Capitalizing total NOI as land income
- -Applying the building rate to the depreciation amount
Concept Deep Dive
Analysis
This question tests the land residual technique. The building's value is depreciated cost, $4,000,000 − $800,000 = $3,200,000, and its income is $3,200,000 × 8% = $256,000. The residual land income is $500,000 − $256,000 = $244,000, and land value is $244,000 ÷ 0.08 = $3,050,000.
Background Knowledge
Physical residual techniques split NOI between land and improvements. In the land residual technique, the building value is known, its income is deducted, and the leftover income is capitalized into land value. The building residual technique reverses the roles, deducting land income to find building value.
Real-World Application
An appraiser needs a land value for a site in an area with few vacant land sales. She estimates the depreciated cost of the existing income-producing building, deducts the income that building requires, and capitalizes the remaining income at a land rate. She treats the result as one land value indication alongside any available land sales.
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