Depreciation Methods for the FE Exam
Depreciation is a method of allocating the cost of a tangible asset over its useful life. It appears in engineering economics problems on the FE exam, particularly in after-tax analysis and capital budgeting. Understanding the different methods and when to apply each one is essential.
Why Depreciation Matters
Depreciation is not a cash flow; it is an accounting concept that:
- Reduces taxable income, which reduces taxes (a real cash flow impact)
- Reflects the declining value of an asset over time
- Determines the book value of an asset for accounting purposes
- Affects the gain or loss when an asset is sold or disposed of
Key Terminology
- Cost basis (P or B): The initial cost of the asset, including purchase price, shipping, and installation.
- Salvage value (S): The estimated value of the asset at the end of its useful life.
- Useful life (n): The estimated number of years the asset will be in service.
- Book value (BV): The remaining undepreciated value. BV = Cost basis minus accumulated depreciation.
- Depreciation (D_t): The amount of depreciation charged in year t.
Straight-Line Depreciation
The simplest method: equal depreciation each year.
D = (P - S) / n
Book value at end of year t: BV_t = P - t * D
Depreciation rate: d = 1 / n
Example: An asset costs $100,000, has a salvage value of $10,000, and a useful life of 10 years.
D = ($100,000 - $10,000) / 10 = $9,000 per year
Book value after year 3: BV_3 = $100,000 - 3 * $9,000 = $73,000
Declining Balance Depreciation
An accelerated method where more depreciation is taken in early years. The depreciation rate is a fixed percentage of the current book value.
D_t = d * BV_(t-1)
Where d is the depreciation rate. Common rates:
- 150% declining balance: d = 1.5 / n
- 200% declining balance (double declining balance, DDB): d = 2 / n
Important: Declining balance depreciation does not use salvage value in the calculation formula. However, the book value cannot be depreciated below the salvage value.
Book value after year t: BV_t = P * (1 - d)^t
Sum-of-Years-Digits (SOYD)
Another accelerated method. The depreciation in year t is:
D_t = (remaining useful life at beginning of year t / sum of years digits) * (P - S)
Sum of years digits: SYD = n * (n + 1) / 2
For year t: D_t = (n - t + 1) / SYD * (P - S)
Example: For an asset with a 5-year life, SYD = 5 * 6 / 2 = 15.
- Year 1: D_1 = 5/15 * (P - S)
- Year 2: D_2 = 4/15 * (P - S)
- Year 3: D_3 = 3/15 * (P - S)
- Year 4: D_4 = 2/15 * (P - S)
- Year 5: D_5 = 1/15 * (P - S)
MACRS (Modified Accelerated Cost Recovery System)
MACRS is the depreciation method required by the IRS for tax purposes in the United States. Key features:
- Salvage value is assumed to be zero
- Uses specific recovery periods (3, 5, 7, 10, 15, 20 years for personal property)
- Uses a half-year convention (assumes the asset is placed in service at the midpoint of the first year)
- Based on 200% declining balance switching to straight-line (for most property classes)
The MACRS percentages are given in the reference handbook. For common property classes:
- 3-year property: 33.33%, 44.45%, 14.81%, 7.41%
- 5-year property: 20.00%, 32.00%, 19.20%, 11.52%, 11.52%, 5.76%
- 7-year property: 14.29%, 24.49%, 17.49%, 12.49%, 8.93%, 8.92%, 8.93%, 4.46%
Note: MACRS percentages always sum to 100% and have one more year than the recovery period (due to the half-year convention).
D_t = MACRS percentage for year t * Cost basis
Switching from Declining Balance to Straight-Line
Sometimes it is optimal to switch from declining balance to straight-line depreciation partway through the asset's life. The switch occurs when the straight-line depreciation on the remaining book value exceeds the declining balance depreciation. MACRS tables already incorporate this switch.
After-Tax Analysis
Depreciation reduces taxable income, which reduces taxes:
Tax savings from depreciation = D_t * tax rate
After-tax cash flow = Before-tax cash flow - Taxes
Taxes = (Revenue - Expenses - Depreciation) * tax rate
Disposal and Gain/Loss
When an asset is sold or disposed of:
- Selling price greater than book value: Gain on disposal = Selling price - Book value. The gain is taxed as ordinary income (up to the amount of depreciation taken; any excess is capital gain).
- Selling price less than book value: Loss on disposal = Book value - Selling price. The loss provides a tax benefit.
- Selling price equals book value: No gain or loss.
Common FE Exam Problem Types
- Calculating depreciation for a given year using any of the four methods
- Finding the book value at a specific point in time
- Determining the tax savings from depreciation
- Comparing alternatives on an after-tax basis
- Calculating gain or loss on disposal of an asset
Practice Tips
- Know how to apply all four methods; the exam could ask about any of them
- For MACRS, use the percentages directly from the reference handbook
- Practice after-tax cash flow calculations
- Remember that MACRS assumes zero salvage value
Depreciation problems on the FE exam are straightforward once you know the method. Practice each method until you can set up and solve problems quickly, and know where to find the MACRS tables in the handbook.