A property produces annual NOI of $160,000. The equity interest has a known value of $500,000 and a 12% current dividend rate, and the annual mortgage constant is 8%. Using the mortgage residual technique, what is the total property value?
Correct Answer
C) $1,750,000
Equity income=500000×0.12=60000; mortgage=(160000−60000)÷0.08=1250000; total=1750000. Common errors: leaving out the known equity value gives $1,250,000; not allocating income to equity before capitalizing gives $2,500,000; capitalizing the remaining income at the equity rate gives $1,333,333.
Why This Is the Correct Answer
Option C is correct because equity income is $500,000 × 12% = $60,000, the residual income of $100,000 capitalized at the 8% mortgage constant gives $1,250,000, and adding the $500,000 equity gives $1,750,000. Residual techniques can be applied to financial components such as debt and equity. Each component's income is capitalized at its own rate.
Why the Other Options Are Wrong
Option A: $1,250,000
Option A, $1,250,000, leaves out the known equity value: ($160,000 − $60,000) ÷ 0.08 = $1,250,000. That is only the mortgage component; adding the $500,000 equity gives $1,750,000.
Option B: $1,333,333
Option B, $1,333,333, capitalizes the remaining income at the equity rate instead of the mortgage constant: ($160,000 − $60,000) ÷ 0.12 + $500,000 = $1,333,333. The residual income belongs to the mortgage and is capitalized at 8%, giving $1,250,000 + $500,000 = $1,750,000.
Option D: $2,500,000
Option D, $2,500,000, skips the equity income allocation and capitalizes all NOI at the mortgage constant before adding equity: $160,000 ÷ 0.08 + $500,000 = $2,500,000. The equity's $60,000 must be removed first, giving $100,000 ÷ 0.08 + $500,000 = $1,750,000.
Known first, residual second, add
Take income for the known part at its rate, capitalize what is left at the other part's rate, then add both values.
How to use: Use this sequence for any land, building, mortgage or equity residual problem.
Exam Tip
In residual problems, track which rate goes with which component and remember to add the known value back at the end.
Common Mistakes to Avoid
- -Forgetting to add the known component back
- -Capitalizing residual income at the wrong component's rate
- -Capitalizing total NOI without deducting the known component's income
Concept Deep Dive
Analysis
This question tests a financial residual technique where equity value is known and the mortgage component is the residual. First allocate income to equity at its dividend rate, then capitalize the remaining income at the mortgage constant, then add the two components. Equity income is $500,000 × 0.12 = $60,000, mortgage value is ($160,000 − $60,000) ÷ 0.08 = $1,250,000, and the total is $1,750,000.
Background Knowledge
Residual techniques split income between two components, value the known component, and capitalize the leftover income to find the unknown component. In a mortgage residual, the equity's required income is deducted and the remainder is capitalized by the mortgage constant. The mortgage constant is the ratio of annual loan payment to loan principal.
Real-World Application
An investor knows how much equity she will contribute and the cash return she requires on it. An appraiser estimates how much mortgage the remaining NOI can support at the lender's constant. Adding the supportable loan to the equity indicates the total value the income can carry.
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