Ma. Brida Lea D.

Diola
UP-Institute of Civil Engineering
CE 22: Engineering Economy
Objectives:
 to illustrate the concepts and mechanics of
depreciation and depletion
 to describe their role in Engineering Economy,
particularly on after-tax analysis
 to apply different depreciation methods
1. Straight Line (SL) Method
2. Accelerated Methods
a) Declining Balance (DB) Method
b) Double Declining Balance (DDB) Method
c) Sum-of-the-Years’ Digit (SYD) Method
d) Sinking Fund Method 2
 Depreciation
❑ Definition: Loss of  Changes in Market Value
value for a fixed asset
End of Market Loss of
❑ Example: You Year Value Value
purchased a vehicle 0 $25,000
worth $25,000 at the

Depreciation
1 19,000 $6,000
beginning of year 2 16,000 3,000
3 2,000
2010. 14,000
4 12,000 2,000
5 10,000 2,000

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5
Depreciation of Smart Phones (2017)

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Depreciation of Smart Phones (2018)

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Depreciation of heavy equipment

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Classification of Types of Depreciation

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Types of of Accounting Depreciation
◼ Book Depreciation
❑ In reporting net income to investors/stockholders
❑ In pricing decision
❑ used by businesses for internal accounting purposes
◼ Tax Depreciation
❑ In calculating income taxes
❑ In engineering economics, we use depreciation in
the context of tax depreciation
Why Do We Consider Depreciation?
Business Gross Income -Expenses:
Expense: (Cost of goods sold)
Depreciation is (Depreciation)
viewed as a (operating expenses)
part of business
expenses that Taxable Income
reduce taxable - Income taxes
income.
Net income (profit)

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Classification of Assets
 Tangible
 Personal property
 Real property

 Intangible
 Personal property

 Tangible assets can be depreciated.

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What Can Be Depreciated?
✓ Assets used in business or held for production of
income
✓ Assets having a definite useful life and a life longer
than one year
✓ Assets that must wear out, become obsolete or
lose value
A qualifying asset for depreciation must satisfy all of
the three conditions above.

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Factors to Consider in Asset Depreciation

❑ Cost basis (depreciation basis)

❑ Depreciable life (how long?)

❑ Salvage value (disposal value)

❑ Method of depreciation (how?)

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Important Terms:
First Cost or Basis
 First cost, basis,
unadjusted basis FC– Cost of a new hole-punching
initial cost of acquiring machine (Invoice price) $62,500
an asset, plus sales tax, + Freight 725
transportation, and
normal costs of making + Installation labor 2,150
asset serviceable + Site preparation 3,500
 Initial purchase price +
Cost basis to use in depreciation $68,875
all costs incurred in
calculation
placing the asset in
service

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Important Terms:
 Salvage Value (S) → Estimated value of property at
the end of useful life.
-- net salvage value used when expenses incurred in
disposing of property; cash outflows must be deducted
from cash inflows for final net salvage value

 Market Value (MV) → Amount paid by willing buyer

to willing seller for property where no advantage and
no compulsion to transact

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Depreciation Concepts
 Book Value (BVk) → Remaining undepreciated
capital investment on the books after total amount of
depreciation charges to date have been subtracted
from the basis, can be drastically different to MV

(Book Value)k= adjusted cost basis - Skj=1 (depreciation

deduction)j
 Useful Life, n → Expected (estimated) period of time
property will be used in trade or business or to
produce income;
 depreciable life
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Estimating Useful Life & Salvage Value
 Salvage value – Asset’s estimated value at the end of
its useful life. Every effort should be made to
estimate a reasonable residual value of the asset, but
if not possible, a 10% rule (10% of the initial value)
could be adopted for depreciation purpose.
 Depreciable life – Adopt the Asset Depreciation
Ranges (ADR) published by the IRS.

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19
Depreciation of Properties Used in
Petroleum Operations
 useful life shall be ten (10) years of such shorter life as
may be permitted by the Commissioner.
 Properties not used directly in the production of
petroleum shall be depreciated under the straight-line
method on the basis of an estimated useful life of five
(5) years.

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Properties Used in Mining Operations
 At the normal rate of depreciation if the expected life is ten
(10) years or less; or
 (b) Depreciated over any number of years between five (5)
years and the expected life if the latter is more than ten (10)
years, and the depreciation thereon allowed as deduction
from taxable income: Provided, That the contractor notifies
the Commissioner at the beginning of the depreciation
period which depreciation rate allowed by this Section will
be used.

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IMPORTANT TERMS
The following terms are used in the classical (historical)
depreciation method equations:
n = depreciable life of the asset in years
FC = cost basis, including allowable adjustments
dk = annual depreciation deduction in year k (1< k <N)
Dk = cumulative depreciation through year k
BVk = book value at the end of year k
BVN = book value at the end of the depreciable (useful) life
SV = salvage value at the end of year N
R or a = the ratio of depreciation in any one year to the BV at the
beginning of the year

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Types of Depreciation Methods:
❑ Straight-Line Method
❑ Accelerated Methods
❑ Units-of-Production Method
Straight-Line (SL) Method
 Simplest depreciation method
❑ Principle: A fixed asset is providing its service in a
uniform fashion over its life
❑A fixed amount is removed each year by depreciation
dk = ( FC - SVN ) / N
Dk = kdk for 1 < k < N
BVk = FC – Dk = BVk-1 - dk
 This method requires an estimate of the final SV ( also
the final book value at the end of year N )

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 n dk BVk
0 $10,000
1 $1,600 8,400

Straight – Line 2
3
1,600
1,600
6,800
5,200

(SL) Method 4 1,600 3,600

5 1,600 2,000

❑Example:
A company owns an
industrial machine which
they bought for $10,000. If
the machine is to be sold for
$2,000 after 5 years,
calculate the book value of
the machine all throughout
its useful life using SL
Method. 25
Example – SL Method
 A new electric saw for cutting small pieces of lumber
in a furniture manufacturing plant has a cost basis of
$4,000 and a 10-year depreciable life. The estimated SV
is zero at the end of 10 years.
a) Determine the annual depreciation amounts
Ans: $400
b) Book value of the saw at the end of year 5
Ans: $2,000

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Accelerated Method:
Sum-of-the-Years-Digits (SYD)
❑ Principle:
• The depreciation charge is a function of the sum of the years
(SYD), 1,2,3…..N where N is the estimated years of useful life.

❑ Formula:
SYD=1+2+3+4……+N = N(N+1)/2

N-k+1
dK = SYD (FC-SV)

FC – First Cost
SV – Salvage Value
N – Useful Life
Accelerated Method:
Sum-of-the-Years-Digits (SYD)
1. Number for each permissible year of life are listed in reverse order
2. Sum the digits of this reverse order
3. The depreciation factor for any year is the corresponding number
from the reverse order listing divided by the sum of those digits, or
the following
2 (N - k + 1 ) N–k+1
df = ----------------- = -------------
N(N+1) SYD

SYD = ½ N(N+1)
4. Depreciation for any year is the product of that year’s SYD
depreciation factor and the difference between FC and the final
estimated SV
d k = ( FC – SV ) • df
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Sum-of-the-Years-Digits
 The cumulative depreciation through the kth year

k(2n - k +1)
Dk = (FC - SV )
2 * SYD
 Book value at the end of year k

BVk = FC - Dk

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EXAMPLE - SYD Method
❑ A company owns an industrial machine which
they bought for $10,000. If the machine is to be
sold for $2,000 after 5 years, calculate the book
value of the machine all throughout its useful life
using SYD Method.

EOY Df Dep BV
0 10000.00
1 5/15 2666.67 7333.33
2 4/15 2133.33 5200.00
3 3/15 1600.00 3600.00
4 2/15 1066.67 2533.33
5 1/15 533.33 2000.00
SYD 15
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Example – SYD Method
 A new electric saw for cutting small pieces of lumber
in a furniture manufacturing plant has a cost basis of
$4,000 and a 10-year depreciable life. The estimated SV
is zero at the end of 10 years.
a) Determine the depreciation at the end of year 5
Ans: $436.36
b) Book value of the saw at the end of year 5
Ans: $1,090.91

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Accelerated Method:
Declining Balance (DB) Method
❑Constant percentage method / Matheson formula
❑Principle:
annual cost of depreciation is a fixed percentage of the BV
at the beginning of the year
• a is constant = 2/N when 200% DB (Double DB)
= 1.5/N when 150% DB
❑ Formula: dk = a BVk-1 = a FC(1- a )k-1
Dk = FC[1- (1- a )k ]
BVk = FC (1 - a )k
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Declining Balance (DB) Method
k BVk-1 dk BVk

❑ Example:
❑FC = $10,000
❑S = $2,000
❑N = 5 years
Compute the annual
depreciation and BVs
using double declining
balance method

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Example – Double DB Method
 A new electric saw for cutting small pieces of lumber
in a furniture manufacturing plant has a cost basis of
$4,000 and a 10-year depreciable life. The estimated
SV is zero at the end of 10 years.
a) Determine the depreciation at the end of year 5
Ans: $327.68
b) Find the total depreciation at the end of year 5
Ans: $2,689.28
c) Book value of the saw at the end of year 5
Ans: $1,310.72
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 Adjustments to the DB Method
 No further depreciation
after n”  Given:
 FC = $10,000
 SV = $2,000
End of Year Depreciation Book Value

1 0.4($10,000) = $4,000 $10,000 - $4,000 = $6,000

2 0.4(6,000) = 2,400 6,000 – 2,400 = 3,600
3 0.4(3,600) = 1,440 3,600 –1,440 = 2,160
4 0.4(2,160) = 864 > 160 2,60 – 160 = 2,000
5 0 2,000 – 0 = 2,000

Note: Tax law does not permit us to

depreciate assets below their salvage
value.
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Declining Balance with Switchover to Straight Line
❑Asset: ❑ Without Switching
❑ Invoice Price $9,000
Book
❑ Freight 500
n Depreciation Value
❑ Installation 500
1 10,000(0.4) = 4,000 $6,000
❑Depreciation Base $10,000 2 6,000(0.4) = 2,400 3,600
❑Salvage Value 0 3 3,600(0.4) = 1,440 2,160
❑Depreciation 200% DB 4 2,160(0.4) = 864 1,296
5 1,296(0.4) = 518 778
❑Depreciable life 5 years

Note: Without switching, we have not

depreciated the entire cost of the asset

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Declining balance with switchover to
straight line method
 Used in situations where assets initially render higher
benefits but later becomes stable.
 Another technical reason to use this method is that
declining balance method has a basic flaw i.e. declining
balance method can never diminish the book value to
expected residual value.
Declining Balance with Switchover to Straight Line
Switching Rule
 If depreciation by DB Method in any year is less than
or equal to depreciation by SL Method, switch to SL
Method
 Straight line depreciation in any year k

dk = BVk-1 – SV______
remaining useful life at BOY k
N-k+1

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 Declining Balance with Switchover
to Straight Line  Without switching
Book
n Depreciation Value
 Switch from DB to SL after n’
1 10,000(0.4) = 4,000 $6,000
2 6,000(0.4) = 2,400 3,600
3 3,600(0.4) = 1,440 2,160
4 2,160(0.4) = 864 1,296
5 1,296(0.4) = 518 778

Book
n Depreciation Value
1 $4,000 $6,000
2 6,000/4 = 1,500 < 2,400 3,600
3 3,600/3 = 1,200 < 1,440 2,160
4 2,160/2 = 1,080 > 864 1,080
5 1,080/1 = 1,080 > 518 0
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Example – DDB Method
 A new electric saw for cutting small pieces of lumber in a
furniture manufacturing plant has a cost basis of $4,000
and a 10-year depreciable life. The estimated SV is zero at
the end of 10 years. Compute the annual depreciation and
BVs using double declining balance method (w/ switch-
over to SL method).

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Double Declining Balance with switch-over to Straight-Line
Method
Given
FC = 4000 a= 0.2
SV = 0
n= 10

EOY, k dk (DDB) BV dk (SL) Dep BV

0 4000.00 4000.00
1 800.00 3200.00 400.00 800.00 3200.00
2 640.00 2560.00 355.56 640.00 2560.00
3 512.00 2048.00 320.00 512.00 2048.00
4 409.60 1638.40 292.57 409.60 1638.40
5 327.68 1310.72 273.07 327.68 1310.72
6 262.14 1048.58 262.14 262.14 1048.58
7 209.72 838.86 262.14 262.14 786.43
8 167.77 671.09 262.14 262.14 524.29
9 134.22 536.87 262.14 262.14 262.14
10 107.37 429.50 262.14 262.14 0.00
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Accelerated Method: Sinking Fund
Method
❑ Formula:
( FC − SV )i ( k −1)
AnnualDepreciation, d k = (1 + i )
(1 + i )N − 1
Total Depreciation Dk = ( FC − SV )( A / F , i %, n)( F / A, i %, k )
after k years

BVk = FC - Dk
A = sinking fund deposit
FC = First cost
n = useful life of the asset
i = interest rate
Dk = Total depreciation after k years DEPRECIATION METHODS
BVk = Book value at the end of k years 42
Example:
A Factory is constructed at an initial cost of $160,000
and with an estimated salvage value of $10,000 at the
end of 10 years. Find the depreciation and its appraisal
value at the end of 5 years using Sinking fund method
and the interest rate is 15%.

DEPRECIATION METHODS
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Example:
A Factory is constructed at an initial cost of $160,000 and
with an estimated salvage value of $10,000 at the end of 10
years. Find its appraisal value at the end of 5 years using
Sinking fund method if the interest rate is 15%.

DEPRECIATION METHODS
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Summary: Reasons for Using
Accelerated Depreciation
 An asset is more useful earlier in its life than later,
and the useful life may be difficult to estimate.
 Depreciation expense is deductible in computing
taxable income and income taxes.
The second reason is the most common
reason for using accelerated depreciation.

 The total amount of taxes to pay remains unchanged

regardless of depreciation methods adopted. It only
changes the timing of the payment.
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Summary
❑ Many firms select straight-line depreciation for
book depreciation because of its relative ease of
calculation.
❑ Given the frequently changing nature of
depreciation and tax law, we must use whatever
percentages, depreciable lives, and salvage
values mandated at the time an asset is acquired.

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❖ Depreciation
❖ Book and Tax depreciation
❖ Book depreciation Methods
❖ Straight Line Method
❖ Accelerated Method
❖ SYD Method
❖ Declining Balance Method
❖ Sinking Fund factor Method
❖ Units-of-Production Method

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 Units-of-Production Method
❑Principle:
Service units will be consumed in a non-time-phased
fashion

❑ Formula:

I = Initial investment
S = Salvage value

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Example - Units-of-Production Method
 A truck hauling coal has an estimated net cash of $55,000,
and is expected to give service for 250,000 miles, resulting
in a $5,000 salvage value.
 Compute the allowed depreciation amount for truck usage
of 30,000 miles.
 Solution:
30,000
Dn = ($55,000 − $5,000)
250,000
 3 
=   ($50,000)
 25 
= $6,000

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Example – Units-of-Production
Method
 A piece of equipment used in a business has a basis of
$50,000 and is expected to have a $10,000 salvage value
when replaced after 30,000 hours of use.

 Find its depreciation rate per hour of use

 Find its book value after 10,000 hours of operation

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Depletion
 Unlike depreciation, which mainly describe the
deduction of expenses due to the aging of equipment
and property, depletion is the physical reduction of
natural resources.
 Payments to owners (gains):
 earned profit
 portion of owner’s capital returned, depletion allowance
 Two types of depletion:
 Cost depletion
 Percentage depletion

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Cost Depletion
❑Concept: Units-of-production method
❑Cost depletion formula:
Depletion allowance

Depletion allowance  Adjusted Basis 

=  
 Total number of recoverable units 
 Adjusted Basis 
=    (Number of units sold)
 Total number of recoverable units 
 (Number of units sold)

Depletion Unit

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Example - Cost Depletion
You bought a timber tract with land for $200,000, and the land
was worth $80,000. The tract has an estimated 1.5 million board
feet (1.5 MBF) of standing timber. If you cut 0.5 MBF of timber,
determine your depletion allowance.

$120,000
Depletion allowance = 0.5 MBF 
1.5 MBF
= $40,000

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 Percentage Depletion
❑Concept: Based on a prescribed percentage of the gross income
from the property during the tax year

❑Note:
❑The total depletion on a property may exceed the total cost of
the property
❑Annual allowance under this method cannot be more than 50%
of the taxable income (before deduction of depletion allowance)
from the property.

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Percentage depletion allowances for
mineral properties
Deposits Percentage
Oil and gas wells (only for certain domestic and gas production) 15
Sulfur and uranium (deposits from US only) 22
Asbestos, lead, zinc, nickel, mica and certain other ores and 22
minerals
Gold, silver, copper, iron ore, and oil shale (deposits from US 15
only)
Coal, lignite and sodium chloride 10
Clay and shale to be used in making sewer pipe or bricks 7.5
Clay (used for roofing tile), gravel, sand and stone 5
Most other minerals; CO2 produced from a well and metallic ores 14

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Example - Percentage Depletion
A gold mine with an  Solution:
estimated deposit of 300,000
ounces of gold has a basis of Calculation
30 million dollars (cost minus
Gross income from sale of $16,425,000
land value). The mine has a 18,000
45,000 ounces
ounces × 15%
Depletion percentage
gross income of $16,425,000
Computed percentage $2,463,750
for the year from selling depletion
45,000 ounces of gold (unit
price 0f $365 per ounce). Gross income from sale of $16,425,000
18,000
45,000 ounces
ounces 12,250,000
Less mining expenses 4,175,000
Taxable income from mine × 50%
Mining expenses before Deduction limitation

depletion equal $12,250,000. Maximum depletion

deduction
$2,087,500

Compute the percentage

depletion allowance.
Is cost depletion applicable?
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Calculating the Allowable Depletion
Deduction for Federal Tax

$2,087,500

$2,463,750

$2,087,500
 $30, 000, 000 
Cost depletion =   (45, 000)
 300, 000 
= $4,500, 000
$4,500,000

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Seatwork
1. A bowling alley costs $500,000 and has a useful life of ten years.
Its estimated salvage value at the end of year ten is $20,000.
Determine the depreciation and book value for years 1 to 10 using:
a) Straight-line method
b) Sum-of-the years digits method
c) Double declining balance method (with switchover to SL
method, if needed)
d) Sinking fund method with i = 10% (For practice na lang)

2. A gold mine that is expected to produce 30,000 ounces of gold is

purchased for $2.4M. The gold can be sold for $450 per ounce;
however, it costs $265 per ounce for mining and processing costs. If
3,500 ounces are produced this year, what will be the depletion
allowance for a) unit depletion, and b) percentage depletion?
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