CHAPTER 5 1 STRAIGHT-LINE (SL) METHOD A constant amount is depreciated each year over the asset's life. N = depreciable life of the asset in years. d k = annual depreciation deduction in year k d k = (B - SV N ) / N for k = 1, 2, ..., N d k * = cumulative depreciation through year k. d k * = k x d k BV k = B - d k *
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CHAPTER 5 1
STRAIGHT-LINE (SL) METHOD
A constant amount is depreciated each year over the asset's life.
N = depreciable life of the asset in years.
dk = annual depreciation deduction in year k
dk = (B - SVN ) / N for k = 1, 2, ..., N
dk* = cumulative depreciation through year k.
dk* = k x dk
BVk = B - dk *
CHAPTER 5 2
DECLINING BALANCE (DB) METHOD
Annual depreciation is a constant percentage of the asset's value at the BOY
R = 2/N 200% declining balance
R = 1.5/N 150% declining balance
d1 = B x R
dk = B(1-R)k-1 (R) = BVk-1 (R)
dk* = B[1 - (1 - R)k ]
BVk = B(1 - R)k
BVN = B(1 - R)N
CHAPTER 5 3
SL and DB Example
The La Salle Bus Company has decided to purchase a new bus for $85,000, with a trade-in of their old bus. The old bus has a trade-in value of $10,000. The new bus will be kept for 10 years before being sold. Its estimated salvage value at that time is expected to be $5,000. Compute the following quantities using (a) the straight-line method, (b) the 200% declining balance method depreciation deduction in the first year and the fourth
Ek = operating expenses in year k plus interest paid on borrowed capital
dk = depreciation allowance for year k
t = effective income tax rate used for computing income taxes
Tk = income tax liability for year k
Income tax represents a significant
cash outflow that we cannot ignore
CHAPTER 5 11
General Tax Procedure
BTCF = Before tax cash flow = R – E
NIBT = Net income before tax = R – E – d
T = tax liability = t ( NIBT ) = t (R – E – d)
NIAT = Net income after tax = NIBT – T
= (R – E – d) – t (R – E – d) = (1 – t)(R – E – d)
ATCF = After tax cash flow = NIAT + d
ATCF = BTCF – t (R – E – d)
CHAPTER 5 12
General Tax Procedure - Example
You invested $113,028 on an asset with the depreciable life of 10 years. You can earn $30,000 per year from this investment for 10 years. Asset has a negligible or zero MV at the end of its useful life. Published income tax rate is 40% on annual taxable income (NIBT). Use after-tax MARR of 15% per year, and straight line depreciation method.
a) NIBT?
b) NIAT?
c) ATCF?
d) Is it profitable investment after taxes?
CHAPTER 5 13
Solution to (a), (b) and (c)
d = 113,028 / 10 = $11,303 (depreciation amount)
(+) Net income 30,000
(-) Deprecation 11,303
(a) NIBT $18,697 (R – E – d)
(-) Income Tax (0.4) 7,479
(b) NIAT $11,218 (1 – t)(R – E – d)
(+) Depreciation 11,303
(c) ATCF $22,521 (1 – t)(R – E – d)+d
CHAPTER 5 14
Solution to part (d)
AW (15%) = ATCF – 113,028 (A/P, 15%, 10)
= 22,521 – 22,521 = $0
CHAPTER 5 15
Typical Before-Tax Cash Flow Diagram:
Typical After-Tax Cash Flow Diagram:
After-Tax Cash Flow Analysis
CHAPTER 5 16
CHAPTER 5 17
After-tax MARR
To perform an after-tax evaluation of a project's after-tax cash flows, we must use an after-tax MARR.
trate, tax income effective - 1
MARRtax -After MARR tax - Before
0.12 0.4) - (1 0.2 MARR t- 1
MARR MARR AT
ATBT
Example: Suppose the before-tax MARR = 20% and t = 40%. What is the approximate after-tax MARR?
CHAPTER 5 18
ATCF Analysis – Example
Investment $10,000
Net Annual Receipts $4,000/yr
Study Period 4 years
Market Value at EOY 4 $5,000
After-tax MARR 15%
Effective income tax rate 40%
Depreciable recovery period 5 years
Is this a worthwhile investment after taxes? Use Straight Line Method for depreciation.
CHAPTER 5 19
Step 1: Find depreciation amounts
for the study period of 4 years:
5 to1 k for $2,000 5
0 - 10,000 d k
EOY, k dk BVk
0 --- 10000
1 2000 8000
2 2000 6000
3 2000 4000
4 2000 2000
CHAPTER 5 20
Step 2: Determine the ATCF with tax
rate of 40%:
Tk
EOY, k BTCFk dk TIk (t = 0.4) ATCFk
0 - 10,000 - 10,000
4b 5000 3000 -1200 3800
1 4000 2000 2000 -800 3200
2 4000 2000 2000 -800 3200
3 4000 2000 2000 -800 3200
4a 4000 2000 2000 -800 3200
MV4 – BV4 = 5000 – 2000
CHAPTER 5 21
Step 3: Use the ATCF to evaluate
this investment @ MARR = 15%:
After-tax cash flow diagram:
PW(15%) = – 10,000 + 3,200 (P/A, 15%, 3)
+ 7,000 (P/F, 15%, 4) = + $1,309
CHAPTER 5 22
Affordable Investment - Example
An asset purchased at the price of (I), and it is depreciated for 10 years, with straight line method. Estimated market value @ EOY 10 is zero, how much can you afford to purchase this asset when it produces annual $30,000 net income for 10 years. Effective income tax is 40% and MARRAT = 15%.
CHAPTER 5 23
Solve for I?
d = (I – 0) / 10
ATCF = 30000 – 0.4 (30000 – d)
AW(15%) = – I(A/P,15%,10)
+ 30000(1 – 0.4) + 0.4 (I/10)
AW(15%) = 0
I = 30000(1 – 0.4) / ((A/P,15%,10) – 0.04)
I = $113,028 0.1993
CHAPTER 5 24
Non-depreciable asset - Example Construction cost of the facility $600,000
Purchasing cost of land $550,000
Annual gross income $230,000
Operating expenses per year $30,000
Facility will be depreciated for 5 years, using 200% DB
MARRAT 12% Tax rate 40%
Is the investment worthwhile after taxes for the study period of 5 years?
Note: Land will be kept after the five-year operation! Facility has a market value of zero @EOY5
CHAPTER 5 25
200% DB R = 2/5 = 0.4
EOY, k dk BVk
cost basis for only
depreciable asset
0 --- 600,000
1 240,000 360,000
2 144,000 216,000
3 86,400 129,600
4 51,840 77,760
5 31,104 46,656
600,000 x 0.4
600,000 – 240,000
BV5 MV5 = 0
CHAPTER 5 26
ATCF analysis with t = - 0.4
ATCF
-1,150,000
216,000
177,600
154,560
140,736
132,442
Tax
16,000
-22,400
-45,440
-59,264
-67,558
TI
-40,000
56,000
113,600
148,160
168,896
d
240,000
144,000
86,400
51,840
31,104
EOY
0
1
2
3
4
5a
5b
BTCF
-1,150,000
200,000
200,000
200,000
200,000
200,000
550,000 - 46,656
MV5 – BV5 = 0 – 46,656
18,662
550,000 + 18,662
568,662
CHAPTER 5 27
ATCF analysis with MARRAT = 12%
ATCF
-1,150,000
216,000
177,600
154,560
140,736
132,442
568,662
EOY
0
1
2
3
4
5a
5b
+ (132,442 + 568,662)(P/F, 12%, 5)
= - $218,283 < 0 reject !
PW(12%) = - 1,150,000
+ 216,000(P/F, 12%, 1)
+ 177,600(P/F,12%, 2)
+ 154,560(P/F, 12%, 3)
+ 140,736(P/F, 12%, 4)
CHAPTER 5 28
Lease versus Purchase - Example Determine the more economic means of acquiring a
copier in your business if you may either:
a) purchase the copier for $5,000 with a probable resale value of $1,000 at the end of 5 years or
b) rent the copier for an annual fee of $900 per year for 5 years with an initial deposit of $500 refundable upon returning the copier in good condition.
If you own the copier, you will depreciate it by using the 150% DB method (class life of 5 years). All rental fees are deductible for income tax purposes. As the owner or lessee, you will pay all expenses associated with the operation of the copier. A deposit does not affect taxes when paid out or received back.
Compare these alternatives by using the equivalent uniform annual cost method. The after-tax MARR is 10% per year, and the effective income tax rate is 40%.
CHAPTER 5 29
Option A – Purchase Copier
EOY BTCF
R=0.3
d TI
t= -0.4
Tax ATCF
0 -5000 -5000
1 1500 -1500 600 600
2 1050 -1050 420 420
3 735 -735 294 294
4 515 -515 206 206
5a 360 -360 144 144
5b 1000 840 160 -64 936
CHAPTER 5 30
Option A – Purchase Copier EOY ATCF
0 -5000
1 600
2 420
3 294
4 206
5a 144
5b 936
AW(10%) = [-5000
+ 600(P/F,10%,1)
+420(P/F,10%,2)
+294(P/F,10%,3)
+206(P/F,10%,4)
+(144+936)(P/F,10%,5)](A/P,10%,5)
= - 3075(A/P,10%,5) = - 811.185
CHAPTER 5 31
Option B – Rent (lease) Copier
EOY BTCF
d TI
t = -0.4
Tax ATCF
0 -500 -500
1 -900 -900 360 -540
2 -900 -900 360 -540
3 -900 -900 360 -540
4 -900 -900 360 -540
5a -900 -900 360 -540
5b 500 500
CHAPTER 5 32
Option B – Rent (lease) Copier EOY ATCF
0 -500
1 -540
2 -540
3 -540
4 -540
5a -540
5b 500
AW (10%) = -500(A/P,10%,5)
+ 500(A/F,10%,5) – 540
= - 590
Option B (lease) is the least cost
alternative for having the coppier
CHAPTER 5 33
Before tax leasing cost? Over what range of before-tax leasing costs
would you choose the purchase option based on an after-tax analysis?
AW_lease =AW_purchase
AW_lease = - 811.185
-500(A/P,10%,5) + 500 (A/F,10%,5) + (1 – t) L
say L = before tax leasing cost
0.6 L = 811.185 + 500(A/P,10%,5) – 500 (A/F,10%,5)
L = 1269 (if before tax leasing cost is greater than 1269,