12-29. In January Gerald Adair bought a small house and lot for $99,700. He estimated that $9,700 of this amount represented the value of the land. He rented the house for $6,500 a year during the four years he owned the house. Expenses for property taxes, maintenance, and so forth were $500 per year. For tax purposes the house was depreciated by MACRS depreciation (27.5 year straight line depreciation with mid-month convention is used for rental property). At the end of four years the property was sold for $105,000. Gerald is married and works as an engineer. He estimates that his incremental state and federal combined tax rate is 24%. What after tax rate of return did Gerald obtain on his investment in the property?
Solution:
First let us find out the depreciation schedule amount for the four years
Cost base for depreciation = 99,700 - 9,700 = 90,000 (building only)
Depreciation for a full year = 90,000/27.5 = 3272.72
As purchased in the month of January he will get 11.5 months prorated depreciation for the
first year according to mid month convention of MACRS = 11.5/12*3272.72 = 3136.36
Assuming the property was sold in December in the 4th year, the depreciation
that year is also for 11.5 months = 3136.36
Before tax cash flow:
Purchase Price = 99,700
Annual benefit (before Depreciation) = 6500-500 = $ 6000 per year
Salvage value = $105,000
As the property is sold at a price higher than the cost base, in the 4th year in
addition to the tax on his income from the property, he will have to pay tax for Capital
gain and depreciation recapture ( or ) ordinary gain due to the sale of the
property.
Capital gain = 105,000 - 99,700 = 5300, and
Ordianry gain = total depreciation = 3136.36+3272.72+3272.72+3136.36 = 12818.16
These amounts will be additional taxable income
in the fourth year.
Year |
Before tax cash flow |
Depreciation |
Taxable income |
Tax @ 24% |
After tax cash flow |
(1) |
(2) |
(3)=(1)-(2) |
(4)=.24*(3) |
(5)=(1)+(4) |
|
0 |
$ (99,700.00) |
$ (99,700.00) |
|||
1 |
$ 6,000.00 |
3136.36 |
2863.64 |
-687.27 |
$ 5,312.73 |
2 |
$ 6,000.00 |
3272.72 |
2727.28 |
-654.55 |
$ 5,345.45 |
3 |
$ 6,000.00 |
3272.72 |
2727.28 |
-654.55 |
$ 5,345.45 |
4 |
$6000+105,000.00 |
3136.36 |
2863.64+12818.16 |
-3763.63 |
$106,176.37 |
Net present benefit at (NPB) = 5312.73*(P/F, i, 1) +5345.45*(P/F,i,2)+
5345.45*(P/F,i,3)+ 106,176.37*(P/F,i,4)
Try i =5%
NPB =5312.73*(P/F, 5%, 1) +5345.45*(P/F,5%,2)+ 5345.45*(P/F,5%,3)+ 106,176.37*(P/F,5%,4)
=5312.73*(.9524) +5345.45*(.9070)+
5345.45*(.8636)+ 106176.37*(.8227)
=101,875.80
Try i =6%
NPB =5312.73*(P/F, 6%, 1) +5345.45*(P/F,6%,2)+ 5345.45*(P/F,6%,3)+ 106176.37*(P/F,6%,4)
=5312.73*(.9434) +5345.45*(.8900)+
5345.45*(.8396)+ 106176.37*(.7921)
= 98,359.82
As NPC= $99,700 this investment generates a after tax rate of return btween 5 and 6%.
The rate of return for the After tax cash flow is 5.61% from Excel
12-18. A firm is considering the following investment project:
Year |
Before Tax Cash Flow |
0 |
-$1000 |
1 |
+500 |
2 |
+340 |
3 |
+244 |
4 |
+100 |
5 |
+100 +125 Salvage value |
The project has a five-year useful life with a $125 salvage value as shown. Double declining balance depreciation will be used assuming a $125 salvage value. The income tax rate is 34%. If the firm requires a 10% rate of return, should the project be undertaken?
Solution:
To solve this problem, first we have to find out the DDB depreciation schedule
DDB depreciation in any year = (2/N)*(the last year's book value). The calculations are shown in the following table:
Year |
Book value |
DDB |
0 |
1000.00 |
|
1 |
1000-400=600.00 |
2/5*1000=400.00 |
2 |
600-240=360.00 |
2/5*600=240.00 |
3 |
360-144=216.00 |
2/5*360=144.00 |
4 |
216-86.4=129.60 |
2/5*216=86.40 |
5 |
125.00 |
129.60-125=4.60*** |
***Note the adjustment in the last year, where the depreciation is adjusted to match the Book Value to the salvage value. In case of DDB this adjustment is required when the Book Value and the salvage value is not the same after last years depreciation.
Once the depreciation is known, we can analyze the before and after tax cash flow in the following table.
Year |
BTCF |
DDB |
Taxable Income |
Tax @34% |
ATCF |
0 |
-1000.00 |
-1000.00 |
|||
1 |
500.00 |
400.00 |
100.00 |
-34.00 |
466.00 |
2 |
340.00 |
240.00 |
100.00 |
-34.00 |
306.00 |
3 |
244.00 |
144.00 |
100.00 |
-34.00 |
210.00 |
4 |
100.00 |
86.40 |
13.60 |
-4.62 |
95.38 |
5 |
100.00+ 125.00** |
4.60 |
95.40 |
-32.44 |
67.56+ 125.00** |
**Note that salvage value is not taxable, unless you sell it at a higher price, that is unless you make any profit by selling it. As DDB with salvage value of 125K is used to depreciate the asset, thus, the book value and the market value is same and hence there will be no tax consequence at the last year due to the sale of the property.
To examine if the After Tax Cash Flow is generating more than 10%, let us compute the NET PRESENT BENEFIT (NPB) at 10% rate.
NPB= 466(P/F,10%,1) + 306(P/F,10%,2) + 210(P/F,10%,3)
+ 95.83(P/F,10%,4) + 192.56(P/F,10%,5) = $1,019.02
As NPB is more than the net present cost (NPC= $1000), the project is economically justified.
(Also, using EXEL, the after tax IRR = 10.94%, which is more than the required 10%, and hence justified.)
12-47. A house and a lot are for sale for $155,000. It is estimated that $45,000 is the value of the land and $110,000 is the value of the house. If purchased, the house can be rented to provide a net income of $12,000 per year after taking all expenses, except depreciation, into account. The would be depreciated by straight-line depreciation using a 27.5 year depreciable life and zero salvage value.
Mary Silva, the prospective purchaser, wants 10% after-tax rate of return on her investment after considering both annual income taxes and a capital gain when she sells the house and lot. At what price would she have to sell the house at the end of ten years to achieve her objective? You may assume that Mary has an incremental income tax rate of 28% in each of the ten years.
Solution:
With straight line depreciation (SLD), 27.5 years depreciable life and zero salvage
value,
the depreciation charge for each year = (P-S)/n = 110,000/27.5 = $4000
P = The cost base = value of the building = $110,000
Total depreciation in 10 years = 10*(4000) = $40,000
Then the book value of the property = $155,000 - $40,000 = $115,000
Let us assume that the price at which the property will be sold after 10 years = X dollars.
Assuming X is more than 155,000 his capital gain = X-155000
Thus he needs to pay capital gain tax @20% at the tenth year=
0.2*(X-155000)
As the property is sold at a price higher than it original price, depreciation recapture in 10th year =
$ 40,000
Thus additional tax due to ordinary gain/depreciation recapture at the 10th year
= 40,000*.28 = 11,200
The before and after tax cash flow table:
Year |
BTCF |
Depreciation |
Taxable Income |
Tax
@27% |
ATCF |
0 |
-155000 |
-155000 |
|||
1 |
12000 |
4000 |
8000 |
-2160 |
12000-2160 = 9840 |
2 |
12000 |
4000 |
8000 |
-2160 |
9840 |
3 |
12000 |
4000 |
8000 |
-2160 |
9840 |
4 |
12000 |
4000 |
8000 |
-2160 |
9840 |
5 |
12000 |
4000 |
8000 |
-2160 |
9840 |
6 |
12000 |
4000 |
8000 |
-2160 |
9840 |
7 |
12000 |
4000 |
8000 |
-2160 |
9840 |
8 |
12000 |
4000 |
8000 |
-2160 |
9840 |
9 |
12000 |
4000 |
8000 |
-2160 |
9840 |
10 |
12000 + X |
4000 |
8000 Dep recap = 40,000 |
-2160 Tax for Dep recap (@27%) = |
9840 |
Now, if 10% after tax rate of return is desired, then the future cost of the first cost and the after tax incomes for the 10 years =
-155000*(F/P,10%,10)+9840*(F/A,10%,10)
= -155000*(2.594)+9840*(15.937) = -402070+156820 = - $ 245,250
So the sales price after paying all taxes due to depreciation recapture and capital gain must be equal to 246,524.88 to make 10% rate of return. Thus,
X - 10800 - 0.2*(X-155000) = $245,250
or, 0.8X -10800+31000 = $245,250
or, X = ($245,250+10800-31000)/0.8 =
$ 281,313