FE Mechanical

4. Engineering economics

Our FE Mechanical course is currently in development and is a work-in-progress.

Depreciation, book value & inflation

6 min read

Depreciation

Depreciation is the systematic allocation of an asset’s cost over its useful life. It represents the decrease in value of an asset due to wear, obsolescence, or age. Depreciation is important for:

Tax calculations (depreciation is a tax-deductible expense)
Financial reporting
Replacement analysis
Cost accounting

Definitions

Initial cost ( $C$ ): The purchase price plus installation and setup costs
Salvage value ( $S_{n}$ ): The estimated value at the end of useful life
Useful life ( $n$ ): The expected service life in years
Book value ( $B V$ ): The remaining undepreciated value at any point in time

Straight-line depreciation

The simplest method allocates equal depreciation each year:

$D = \frac{C - S _{n}}{n}$

where $D$ is the annual depreciation.

Book value after j years:

$B V_{j} = C - j D$

Sum-of-years’-digits (SOYD) depreciation

This accelerated method provides higher depreciation in early years:

$D_{j} = \frac{( n - j + 1 ) ( C - S _{n} )}{T}$

where:

$T = \frac{n ( n + 1 )}{2}$

Declining Balance Depreciation This method applies a constant percentage to the remaining book value:

$D_{j} = d \times B V_{j - 1}$

where $d$ is the depreciation rate.

For double declining balance (DDB), the rate is:

$d = \frac{2}{n}$

Book value after j years:

$B V_{j} = C (1 - d)^{j}$

Note: Declining balance methods may not depreciate to the salvage value, requiring a switch to straight-line in later years.

Modified accelerated cost recovery system (MACRS) For U.S. tax purposes, the Modified Accelerated Cost Recovery System (MACRS) is the required depreciation method for most business assets. MACRS specifies:

Recovery periods (3, 5, 7, 10, 15, 20, 27.5, or 39 years)
Depreciation percentages for each year
Half-year or mid-quarter conventions

Engineers should consult current tax regulations or accounting professionals for MACRS calculations.

Book value

Book value represents the asset’s value as recorded in accounting records. It equals the initial cost minus accumulated depreciation:

$B V_{j} = C - k = 1 \sum j D_{k}$

Book value is important for:

Calculating gains or losses on asset disposal
Determining property taxes (in some jurisdictions)
Financial statement preparation
Replacement analysis

Equivalent uniform annual cost (EUAC)

The Equivalent Uniform Annual Cost converts all costs associated with an alternative into an equivalent uniform annual amount. This facilitates comparison of alternatives with different lifespans and cost patterns.

$E U A C = P (A / P, i %, n) - S_{n} (A / F, i %, n) + A_{o p er a t in g}$

where:

$P$ = initial cost
$S_{n}$ = salvage value
$A_{o p er a t in g}$ = annual operating costs

Capitalized cost

Capitalized cost is the present worth of an asset that provides service indefinitely. It represents the amount that, if invested today, would provide enough interest to fund perpetual replacement and maintenance.

For an asset with initial cost $C$ , life $n$ , and no salvage value:

$P_{c a p i t a l i ze d} = C + \frac{C}{( F / P , i % , n ) - 1}$

For perpetual annual costs:

$P_{p er p e t u a l} = \frac{A}{i}$

Inflation

Inflation is the general increase in prices over time, reducing the purchasing power of money. Engineering economic analysis must account for inflation when comparing alternatives over long time horizons.

Real vs. nominal interest rates

Nominal interest rate ( $i$ ): The stated rate, not adjusted for inflation
Real interest rate ( $d$ ): The rate adjusted for inflation
Inflation rate ( $f$ ): The rate of price increase

The relationship between these rates is:

$d = \frac{i - f}{1 + f}$

Or equivalently:

$i = d + f + df$

Constant dollar vs. actual dollar analysis

Actual dollars (current dollars): Include the effects of inflation
Constant dollars (real dollars): Expressed in terms of purchasing power at a reference point

When cash flows are in actual dollars, use the nominal interest rate. When cash flows are in constant dollars, use the real interest rate.

Inflation & project evaluation

Inflation affects project evaluation in several ways:

Future revenues and costs will be higher in nominal terms
Depreciation deductions are based on historical cost, not inflated values
Interest rates typically include an inflation premium
Long-term projects are more affected than short-term projects

Consistency is essential: use actual dollars with nominal rates, or constant dollars with real rates, but do not mix approaches.

Capital budgeting & alternative comparison

Capital budgeting involves evaluating and selecting long-term investments. Several methods are commonly used:

Present worth method

Calculate the present worth of all cash flows for each alternative. Select the alternative with the highest present worth (for revenue projects) or lowest present worth of costs (for cost projects).

$P W = t = 0 \sum n C F_{t} (P / F, i %, t)$

Advantages:

Considers all cash flows
Accounts for time value of money
Easy to understand

Disadvantages:

Requires specifying an interest rate
Difficult to compare projects with different lives

Annual worth method

Convert all cash flows to equivalent uniform annual amounts. Select the alternative with the highest annual worth (or lowest annual cost).

$A W = P W (A / P, i %, n)$

Advantages:

Easy to compare projects with different lives
Results are intuitive (annual cost or benefit)

Disadvantages:

Requires specifying an interest rate

Rate of return method

Find the interest rate that makes the present worth of cash flows equal to zero. This rate is the internal rate of return (IRR):

$t = 0 \sum n C F_{t} (P / F, I RR, t) = 0$

Accept projects with IRR greater than the minimum attractive rate of return (MARR).

Advantages:

Does not require specifying an interest rate
Results are intuitive (percentage return)

Disadvantages:

May have multiple solutions for non-conventional cash flows
Can give misleading results for mutually exclusive alternatives
Requires incremental analysis for comparing alternatives

Benefit-cost ratio method

Calculate the ratio of benefits to costs:

$B / C = \frac{P W _{b e n e f i t s}}{P W _{cos t s}}$

Accept projects with B/C ratio greater than 1.0.

Advantages:

Commonly used for public projects
Easy to understand

Disadvantages:

Sensitive to how benefits and costs are defined
Requires incremental analysis for comparing alternatives

Payback period method

Determine the time required to recover the initial investment:

$P a y ba c k = \frac{I ni t ia l I n v es t m e n t}{A nn u a l N e t C a s h Fl o w}$

Note: Simple payback ignores the time value of money. Discounted payback accounts for interest but is more complex to calculate.

Advantages:

Simple to calculate and understand
Useful for screening projects

Disadvantages:

Ignores time value of money (simple payback)
Ignores cash flows after payback period
Does not measure profitability

Next | 4.2 Break-even analysis, risk analysis & accounting principles

Depreciation, book value & inflation

Depreciation

Depreciation is the systematic allocation of an asset’s cost over its useful life. It represents the decrease in value of an asset due to wear, obsolescence, or age. Depreciation is important for:

Tax calculations (depreciation is a tax-deductible expense)
Financial reporting
Replacement analysis
Cost accounting

Definitions

Initial cost ( $C$ ): The purchase price plus installation and setup costs
Salvage value ( $S_{n}$ ): The estimated value at the end of useful life
Useful life ( $n$ ): The expected service life in years
Book value ( $B V$ ): The remaining undepreciated value at any point in time

Straight-line depreciation

The simplest method allocates equal depreciation each year:

$D = \frac{C - S _{n}}{n}$

where $D$ is the annual depreciation.

Book value after j years:

$B V_{j} = C - j D$

Sum-of-years’-digits (SOYD) depreciation

This accelerated method provides higher depreciation in early years:

$D_{j} = \frac{( n - j + 1 ) ( C - S _{n} )}{T}$

where:

$T = \frac{n ( n + 1 )}{2}$

Declining Balance Depreciation This method applies a constant percentage to the remaining book value:

$D_{j} = d \times B V_{j - 1}$

where $d$ is the depreciation rate.

For double declining balance (DDB), the rate is:

$d = \frac{2}{n}$

Book value after j years:

$B V_{j} = C (1 - d)^{j}$

Note: Declining balance methods may not depreciate to the salvage value, requiring a switch to straight-line in later years.

Recovery periods (3, 5, 7, 10, 15, 20, 27.5, or 39 years)
Depreciation percentages for each year
Half-year or mid-quarter conventions

Engineers should consult current tax regulations or accounting professionals for MACRS calculations.

Book value

Book value represents the asset’s value as recorded in accounting records. It equals the initial cost minus accumulated depreciation:

$B V_{j} = C - k = 1 \sum j D_{k}$

Book value is important for:

Calculating gains or losses on asset disposal
Determining property taxes (in some jurisdictions)
Financial statement preparation
Replacement analysis

Equivalent uniform annual cost (EUAC)

$E U A C = P (A / P, i %, n) - S_{n} (A / F, i %, n) + A_{o p er a t in g}$

where:

$P$ = initial cost
$S_{n}$ = salvage value
$A_{o p er a t in g}$ = annual operating costs

Capitalized cost

For an asset with initial cost $C$ , life $n$ , and no salvage value:

$P_{c a p i t a l i ze d} = C + \frac{C}{( F / P , i % , n ) - 1}$

For perpetual annual costs:

$P_{p er p e t u a l} = \frac{A}{i}$

Inflation

Real vs. nominal interest rates

Nominal interest rate ( $i$ ): The stated rate, not adjusted for inflation
Real interest rate ( $d$ ): The rate adjusted for inflation
Inflation rate ( $f$ ): The rate of price increase

The relationship between these rates is:

$d = \frac{i - f}{1 + f}$

Or equivalently:

$i = d + f + df$

Constant dollar vs. actual dollar analysis

Actual dollars (current dollars): Include the effects of inflation
Constant dollars (real dollars): Expressed in terms of purchasing power at a reference point

When cash flows are in actual dollars, use the nominal interest rate. When cash flows are in constant dollars, use the real interest rate.

Inflation & project evaluation

Inflation affects project evaluation in several ways:

Future revenues and costs will be higher in nominal terms
Depreciation deductions are based on historical cost, not inflated values
Interest rates typically include an inflation premium
Long-term projects are more affected than short-term projects

Consistency is essential: use actual dollars with nominal rates, or constant dollars with real rates, but do not mix approaches.

Capital budgeting & alternative comparison

Capital budgeting involves evaluating and selecting long-term investments. Several methods are commonly used:

Present worth method

Calculate the present worth of all cash flows for each alternative. Select the alternative with the highest present worth (for revenue projects) or lowest present worth of costs (for cost projects).

$P W = t = 0 \sum n C F_{t} (P / F, i %, t)$

Advantages:

Considers all cash flows
Accounts for time value of money
Easy to understand

Disadvantages:

Requires specifying an interest rate
Difficult to compare projects with different lives

Annual worth method

Convert all cash flows to equivalent uniform annual amounts. Select the alternative with the highest annual worth (or lowest annual cost).

$A W = P W (A / P, i %, n)$

Advantages:

Easy to compare projects with different lives
Results are intuitive (annual cost or benefit)

Disadvantages:

Requires specifying an interest rate

Rate of return method

Find the interest rate that makes the present worth of cash flows equal to zero. This rate is the internal rate of return (IRR):

$t = 0 \sum n C F_{t} (P / F, I RR, t) = 0$

Accept projects with IRR greater than the minimum attractive rate of return (MARR).

Advantages:

Does not require specifying an interest rate
Results are intuitive (percentage return)

Disadvantages:

May have multiple solutions for non-conventional cash flows
Can give misleading results for mutually exclusive alternatives
Requires incremental analysis for comparing alternatives

Benefit-cost ratio method

Calculate the ratio of benefits to costs:

$B / C = \frac{P W _{b e n e f i t s}}{P W _{cos t s}}$

Accept projects with B/C ratio greater than 1.0.

Advantages:

Commonly used for public projects
Easy to understand

Disadvantages:

Sensitive to how benefits and costs are defined
Requires incremental analysis for comparing alternatives

Payback period method

Determine the time required to recover the initial investment:

$P a y ba c k = \frac{I ni t ia l I n v es t m e n t}{A nn u a l N e t C a s h Fl o w}$

Note: Simple payback ignores the time value of money. Discounted payback accounts for interest but is more complex to calculate.

Advantages:

Simple to calculate and understand
Useful for screening projects

Disadvantages:

Ignores time value of money (simple payback)
Ignores cash flows after payback period
Does not measure profitability

Depreciation

Depreciation is the systematic allocation of an asset’s cost over its useful life. It represents the decrease in value of an asset due to wear, obsolescence, or age. Depreciation is important for:

Tax calculations (depreciation is a tax-deductible expense)
Financial reporting
Replacement analysis
Cost accounting

Definitions

Initial cost ( $C$ ): The purchase price plus installation and setup costs
Salvage value ( $S_{n}$ ): The estimated value at the end of useful life
Useful life ( $n$ ): The expected service life in years
Book value ( $B V$ ): The remaining undepreciated value at any point in time

Straight-line depreciation

The simplest method allocates equal depreciation each year:

$D = \frac{C - S _{n}}{n}$

where $D$ is the annual depreciation.

Book value after j years:

$B V_{j} = C - j D$

Sum-of-years’-digits (SOYD) depreciation

This accelerated method provides higher depreciation in early years:

$D_{j} = \frac{( n - j + 1 ) ( C - S _{n} )}{T}$

where:

$T = \frac{n ( n + 1 )}{2}$

Declining Balance Depreciation This method applies a constant percentage to the remaining book value:

$D_{j} = d \times B V_{j - 1}$

where $d$ is the depreciation rate.

For double declining balance (DDB), the rate is:

$d = \frac{2}{n}$

Book value after j years:

$B V_{j} = C (1 - d)^{j}$

Note: Declining balance methods may not depreciate to the salvage value, requiring a switch to straight-line in later years.

Recovery periods (3, 5, 7, 10, 15, 20, 27.5, or 39 years)
Depreciation percentages for each year
Half-year or mid-quarter conventions

Engineers should consult current tax regulations or accounting professionals for MACRS calculations.

Book value

Book value represents the asset’s value as recorded in accounting records. It equals the initial cost minus accumulated depreciation:

$B V_{j} = C - k = 1 \sum j D_{k}$

Book value is important for:

Calculating gains or losses on asset disposal
Determining property taxes (in some jurisdictions)
Financial statement preparation
Replacement analysis

Equivalent uniform annual cost (EUAC)

$E U A C = P (A / P, i %, n) - S_{n} (A / F, i %, n) + A_{o p er a t in g}$

where:

$P$ = initial cost
$S_{n}$ = salvage value
$A_{o p er a t in g}$ = annual operating costs

Capitalized cost

For an asset with initial cost $C$ , life $n$ , and no salvage value:

$P_{c a p i t a l i ze d} = C + \frac{C}{( F / P , i % , n ) - 1}$

For perpetual annual costs:

$P_{p er p e t u a l} = \frac{A}{i}$

Inflation

Real vs. nominal interest rates

Nominal interest rate ( $i$ ): The stated rate, not adjusted for inflation
Real interest rate ( $d$ ): The rate adjusted for inflation
Inflation rate ( $f$ ): The rate of price increase

The relationship between these rates is:

$d = \frac{i - f}{1 + f}$

Or equivalently:

$i = d + f + df$

Constant dollar vs. actual dollar analysis

Actual dollars (current dollars): Include the effects of inflation
Constant dollars (real dollars): Expressed in terms of purchasing power at a reference point

When cash flows are in actual dollars, use the nominal interest rate. When cash flows are in constant dollars, use the real interest rate.

Inflation & project evaluation

Inflation affects project evaluation in several ways:

Future revenues and costs will be higher in nominal terms
Depreciation deductions are based on historical cost, not inflated values
Interest rates typically include an inflation premium
Long-term projects are more affected than short-term projects

Consistency is essential: use actual dollars with nominal rates, or constant dollars with real rates, but do not mix approaches.

Capital budgeting & alternative comparison

Capital budgeting involves evaluating and selecting long-term investments. Several methods are commonly used:

Present worth method

Calculate the present worth of all cash flows for each alternative. Select the alternative with the highest present worth (for revenue projects) or lowest present worth of costs (for cost projects).

$P W = t = 0 \sum n C F_{t} (P / F, i %, t)$

Advantages:

Considers all cash flows
Accounts for time value of money
Easy to understand

Disadvantages:

Requires specifying an interest rate
Difficult to compare projects with different lives

Annual worth method

Convert all cash flows to equivalent uniform annual amounts. Select the alternative with the highest annual worth (or lowest annual cost).

$A W = P W (A / P, i %, n)$

Advantages:

Easy to compare projects with different lives
Results are intuitive (annual cost or benefit)

Disadvantages:

Requires specifying an interest rate

Rate of return method

Find the interest rate that makes the present worth of cash flows equal to zero. This rate is the internal rate of return (IRR):

$t = 0 \sum n C F_{t} (P / F, I RR, t) = 0$

Accept projects with IRR greater than the minimum attractive rate of return (MARR).

Advantages:

Does not require specifying an interest rate
Results are intuitive (percentage return)

Disadvantages:

May have multiple solutions for non-conventional cash flows
Can give misleading results for mutually exclusive alternatives
Requires incremental analysis for comparing alternatives

Benefit-cost ratio method

Calculate the ratio of benefits to costs:

$B / C = \frac{P W _{b e n e f i t s}}{P W _{cos t s}}$

Accept projects with B/C ratio greater than 1.0.

Advantages:

Commonly used for public projects
Easy to understand

Disadvantages:

Sensitive to how benefits and costs are defined
Requires incremental analysis for comparing alternatives

Payback period method

Determine the time required to recover the initial investment:

$P a y ba c k = \frac{I ni t ia l I n v es t m e n t}{A nn u a l N e t C a s h Fl o w}$

Note: Simple payback ignores the time value of money. Discounted payback accounts for interest but is more complex to calculate.

Advantages:

Simple to calculate and understand
Useful for screening projects

Disadvantages:

Ignores time value of money (simple payback)
Ignores cash flows after payback period
Does not measure profitability