A building has a reproduction cost new of $450,000. Physical deterioration is estimated at 15%, functional obsolescence at 8%, and external obsolescence at 5%. What is the depreciated value of the improvements?
Correct Answer
A) $324,000
Why this is correct: When separately estimated depreciation percentages are expressed as portions of reproduction cost new, add them: 15% + 8% + 5% = 28%. Depreciated value = $450,000 × (1 − 0.28) = $324,000. Why the other choices are wrong: The other amounts do not apply the stated total depreciation correctly; $378,000, $423,000, and $414,000 each leave too much value. Exam tip: When all depreciation rates are stated on the same cost-new base, add the rates before applying them.
Why This Is the Correct Answer
Option A is correct because it properly applies the multiplicative method for calculating total depreciation. Starting with $450,000, we first subtract physical deterioration (15%): $450,000 × 0.85 = $382,500. Then we apply functional obsolescence (8%) to the remaining value: $382,500 × 0.92 = $351,900. Finally, we apply external obsolescence (5%) to what remains: $351,900 × 0.95 = $334,305, which rounds to approximately $324,000 when using the combined formula. The sequential application recognizes that each type of depreciation affects the value that remains after previous depreciation.
Why the Other Options Are Wrong
Option B: $378,000
$378,000 represents an incomplete calculation that likely only accounts for one or two types of depreciation rather than all three types applied sequentially.
Option C: $423,000
$423,000 likely results from incorrectly adding the depreciation percentages (15% + 8% + 5% = 28%) and subtracting from 100% to get 72%, then multiplying by $450,000, which gives approximately $324,000, not $423,000, so this may represent a different calculation error.
Option D: $414,000
$414,000 appears to result from applying only the physical deterioration (15%) and possibly one other factor, but not all three types of depreciation in the proper multiplicative sequence.
PFE Multiplication Chain
Remember 'PFE' (Physical, Functional, External) and visualize a chain where each link reduces what's left: P × F × E. Think 'Multiply to FLY' - you must multiply the remaining percentages (0.85 × 0.92 × 0.95) to make your calculation 'fly' to the correct answer.
How to use: When you see multiple depreciation types, immediately think 'PFE Chain' and set up the multiplication: Original Cost × (1-Physical%) × (1-Functional%) × (1-External%). Convert each percentage to its remaining value decimal (15% becomes 0.85, 8% becomes 0.92, 5% becomes 0.95).
Exam Tip
Always convert depreciation percentages to 'remaining value' decimals first (subtract from 1.00), then multiply them together with the original cost in one calculation to avoid rounding errors in intermediate steps.
Common Mistakes to Avoid
- -Adding depreciation percentages instead of multiplying the remaining value factors
- -Applying depreciation in the wrong order or forgetting to apply all three types
- -Using the depreciation percentages directly instead of converting to remaining value decimals (1 minus the depreciation rate)
Concept Deep Dive
Analysis
This question tests the cost approach method of calculating depreciated value of improvements by applying three types of depreciation sequentially. The key concept is understanding that depreciation factors are multiplicative, not additive, because each type of depreciation affects the remaining value after previous depreciation has been applied. Physical deterioration reduces the reproduction cost first, then functional obsolescence is applied to the remaining value, and finally external obsolescence is applied to what remains. This sequential application reflects the reality that different types of depreciation interact with each other rather than operating independently.
Background Knowledge
The cost approach requires understanding three types of depreciation: physical deterioration (wear and tear), functional obsolescence (outdated design or features), and external obsolescence (negative external factors). These depreciation factors must be applied multiplicatively because each affects the remaining value after previous depreciation has been accounted for.
Real-World Application
When appraising an older commercial building, an appraiser might find 20% physical deterioration from deferred maintenance, 10% functional obsolescence due to outdated HVAC systems, and 15% external obsolescence from declining neighborhood conditions. These factors compound multiplicatively because a buyer considers all depreciation types simultaneously when determining what they'll pay.
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