Given:
After-tax minimum attractive rate of return is 12%.
Combined incremental income tax rate is 42%.
Calculation:
Alternate A:
Write the formula to calculate the depreciation charge for the property at any year.
dt=B×rt .... (I).
Here, depreciation charge in any year t is dt, cost of the property is B and appropriate MACRS percentage rate is rt.
Write the formula to calculate the Net book value.
Net book value=(Base book value)−(Depreciation charge for year dt) .... (II).
Determine the values of rt .throughout the recovery period.
A table would be most suitable to calculate the values of rt.
Recovery year, t (years) |
MACRS percentage rate for the recovery year t, rt (in %) |
1 |
20% |
2 |
32% |
3 |
19.2% |
4 |
11.52% |
5 |
11.52% |
Calculate the depreciation charge for 1st year.
Substitute $11000 for B and 20% for rt in Equation (I).
dt 1 st=$11000×(20100)=$2200
Calculate the Net book value for 1st year.
Substitute $11000 for Base book value and $2200 for Depreciation charge for year dt in Equation (II).
Net book value1st=$11000−$2200=$8800
Calculate the depreciation charge for 2nd year.
Substitute $8800 for B and 32% for rt in Equation (I).
dt 2 nd=$8800×(32100)=$2816
Calculate the Net book value for 2nd year.
Net book value2nd=$8800−$2816=$5984
Calculate the depreciation charge for 3rd year.
Substitute $5984 for B and 19.2% for rt in Equation (I).
dt 3 rd=$5984×(19.2100)=$1148.93
Calculate the Net book value for 3rd year.
Net book value3rd=$5984−$1128.93=$4855.07
Calculate the depreciation charge for 4th year.
Substitute $4855.07 for B and 11.52% for rt in Equation (I).
dt 4 th=$4855.07×(11.52100)=$559.30
Calculate the Net book value for 4th year.
Net book value4th=$4855.07−$559.30=$4295.77
Calculate the depreciation charge for 5th year.
Substitute $4295.77 for B and 11.52% for rt in Equation (I).
dt 5 th=$4295.77×(11.52100)=$494.87
Calculate the Net book value for 5th year.
Net book value5th=$4295.77−$494.87=$3800.9
Tabulate the values of annual depreciation charge and Net book value.
Year, t (years) |
Base book value(a) |
Depreciation charge for year dt (b) |
Net book value(a-b) |
1 |
$11000 |
$2200 |
$8800 |
2 |
$8800 |
$2816 |
$5984 |
3 |
$5984 |
$1148.93 |
$4855.07 |
4 |
$4855.07 |
$559.30 |
$4295.77 |
5 |
$4295.77 |
$494.87 |
$3800.9 |
Write the formula to calculate the taxable incomes.
Taxable Incomes=(Before-tax cash flow)−(Depreciation) .... (III).
Calculate the taxable incomes for 1st year.
Substitute $3000 for Before-tax cash flow and $2816 for Depreciation in Equation (III).
Taxable Incomes1st=$3000−$2816=$184
Calculate the Income taxes for 1st year.
Income taxes1st=42% of Taxable incomes1st=(42100)×($184)=$77.28
Write the formula to calculate the after-tax cash flow.
After-tax cash flow=(Before-tax cash flow)−(Income taxes) .... (IV).
Calculate the after-tax cash flow.
Substitute $3000 for Before-tax cash flow and $77.28 for Income taxes in Equation (IV).
After-tax cash flow1st=$3000−($77.28)=$2922.72
Calculate the After-tax cash flow for the remaining years in tabular form.
Period |
Before-tax cash flow(p) |
MACRS Depreciation(q) |
Taxable Incomes (r)=(p−q) |
Income taxes (42% rate) (s)=0.42×(r) |
After-tax cash flow (t)=(p)−(s) |
0 |
−$11000 |
|
|
|
−$11000 |
1 |
$3000 |
$2200 |
$800 |
$336 |
$2664 |
2 |
$3000 |
$2816 |
$184 |
$77.28 |
$2922.72 |
3 |
$3000 |
$1148.93 |
$1851.07 |
$777.45 |
$2222.55 |
4 |
$3000 |
$559.30 |
$2440.7 |
$1025.09 |
$1974.91 |
5 |
$3000 |
$494.87 |
$2505.13 |
$1052.15 |
$1947.85 |
Write the equation for present worth factor of annuity (PW).
PW=D+A(PA,i,n)+F(PF,i,n)=D+A( ( 1+i) n−1i ( 1+i) n)+F(1 ( 1+i) n) .... (V).
Here, initial payment is D, present value of the sum of the money is P, interest rate is i, number of years is n, After-tax cash flow per year is A and net salvage amount after five years is F.
Calculate present worth factor of annuity.
Substitute −$11000 for D, $2922.72 for A, 12% for i, 5 years for n and $2000 for F in Equation (V).
PW=[−11000+($2922.72)( ( 1+ 12 100 )5 −1 ( 12 100 ) ( 1+ 12 100 )5 )+($2000)(1 ( 1+ 12 100 )5 )]=[−$11000+($2922.72)(3.6043)+($2000)(0.5674)]=(−$11000+$10534.36+$1134.8)=$669.16
Thus, the present worth value for Alternate A is $669.16.
Alternate B:
Determine the values of rt .throughout the recovery period.
A table would be most suitable to calculate the values of rt.
Recovery year, t (years) |
MACRS percentage rate for the recovery year t, rt (in %) |
1 |
20% |
2 |
32% |
3 |
19.2% |
4 |
11.52% |
5 |
11.52% |
Calculate the depreciation charge for 1st year.
Substitute $33000 for B and 20% for rt in Equation (I).
dt 1 st=$33000×(20100)=$6600
Calculate the Net book value for 1st year.
Substitute $33000 for Base book value and $6600 for Depreciation charge for year dt in Equation (II).
Net book value1st=$33000−$6600=$27000
Calculate the depreciation charge for 2nd year.
Substitute $27000 for B and 32% for rt in Equation (I).
dt 2 nd=$27000×(32100)=$8640
Calculate the Net book value for 2nd year.
Net book value2nd=$27000−$8640=$18360
Calculate the depreciation charge for 3rd year.
Substitute $18360 for B and 19.2% for rt in Equation (I).
dt 3 rd=$18360×(19.2100)=$3525.12
Calculate the Net book value for 3rd year.
Net book value3rd=$18360−$3525.12=$14834.88
Calculate the depreciation charge for 4th year.
Substitute $14834.88 for B and 11.52% for rt in Equation (I).
dt 4 th=$14834.88×(11.52100)=$1708.98
Calculate the Net book value for 4th year.
Net book value4th=$14834.88−$1708.98=$13125.9
Calculate the depreciation charge for 5th year.
Substitute $13125.9 for B and 11.52% for rt in Equation (I).
dt 5 th=$13125.9×(11.52100)=$1512.10
Calculate the Net book value for 5th year.
Net book value5th=$13125.9−$1512.1=$11613.8
Write the values of annual depreciation charge and Net book value in tabular form.
Year, t (years) |
Base book value(a) |
Depreciation charge for year dt (b) |
Net book value(a-b) |
1 |
$33000 |
$6600 |
$27000 |
2 |
$27000 |
$8640 |
$18360 |
3 |
$18360 |
$3525.12 |
$14834.88 |
4 |
$14834.88 |
$1708.98 |
$13125.9 |
5 |
$13125.9 |
$1512.1 |
$11613.8 |
Calculate the taxable incomes for 1st year.
Substitute $9000 for Before-tax cash flow and $8640 for Depreciation in Equation (III).
Taxable Incomes1st=$9000−$8640=$360
Calculate the Income taxes for 1st year.
Income taxes1st=42% of Taxable incomes1st=(42100)×($360)=$151.2
Calculate the after-tax cash flow.
Substitute $9000 for Before-tax cash flow and $151.2 for Income taxes in Equation (IV).
After-tax cash flow1st=$9000−($151.2)=$8848.8
Calculate the After-tax cash flow for the remaining years in tabular form.
Period |
Before-tax cash flow(p) |
MACRS Depreciation(q) |
Taxable Incomes (r)=(p−q) |
Income taxes (42% rate) (s)=0.42×(r) |
After-tax cash flow (t)=(p)−(s) |
0 |
−$33000 |
|
|
|
−$33000 |
1 |
$9000 |
$6600 |
$2400 |
$1008 |
$7992 |
2 |
$9000 |
$8640 |
$360 |
$151.2 |
$8848.8 |
3 |
$9000 |
$3525.12 |
$5474.88 |
$2299.45 |
$6700.55 |
4 |
$9000 |
$1708.98 |
$7291.02 |
$3062.22 |
$5937.78 |
5 |
$9000 |
$1512.1 |
$7487.9 |
$3144.92 |
$5855.08 |
Calculate present worth factor of annuity.
Substitute −$33000 for D, $8848.8 for A, 12% for i, 5 years for n and $3000 for F in Equation (V).
PW=[−33000+($8848.8)( ( 1+ 12 100 )5 −1 ( 12 100 ) ( 1+ 12 100 )5 )+($3000)(1 ( 1+ 12 100 )5 )]=[−$33000+($8848.8)(3.6043)+($3000)(0.5674)]=(−$33000+$31893.73+$1702.2)=$595.93
Thus, the present worth value for Alternate A is $595.93.
Conclusion:
Alternative A will have much greater positive value of $669.16.
Thus, choose Alternative A.