Given:
The after-tax MARR is 25%.
The project life is 5 years.
The combined incremental tax rate is 45%.
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 |
33.33% |
2 |
44.45% |
3 |
14.81% |
Calculate the depreciation charge for 1st year.
Substitute $14000 for B and 33.33% for rt in Equation (I).
dt 1 st=$14000×(33.33100)=$4666.2
Calculate the Net book value for 1st year.
Substitute $14000 for Base book value and $4666.2 for Depreciation charge for year dt in Equation (II).
Net book value1st=$14000−$4666.2=$9333.8
Calculate the depreciation charge for 2nd year.
Substitute $9333.8 for B and 44.45% for rt in Equation (I).
dt 2 nd=$9333.8×(44.45100)=$4148.8
Calculate the Net book value for 2nd year.
Net book value2nd=$9333.8−$4148.8=$5185
Calculate the depreciation charge for 3rd year.
Substitute $5185 for B and 14.81% for rt in Equation (I).
dt 3 rd=$5185×(14.81100)=$767.9
Calculate the Net book value for 3rd year.
Net book value3rd=$5158−$767.9=$4417.1
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 |
$14000 |
$4666.2 |
$9333.8 |
2 |
$9333.8 |
$4148.8 |
$5185 |
3 |
$5185 |
$767.9 |
$4414.1 |
Write the formula to calculate the taxable incomes.
Taxable Incomes=(Before-tax cash flow)−(Depreciation) .... (III).
Calculate the taxable incomes for 1st year.
Substitute $2500 for Before-tax cash flow and $4666.2 for Depreciation in Equation (III).
Taxable Incomes1st=$2500−$4666.2=−$2166.2
Calculate the Income taxes for 1st year.
Income taxes1st=45% of Taxable incomes1st=(45100)×(−$2166.2)=−$974.79
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 $2500 for Before-tax cash flow and −$974.79 for Income taxes in Equation (IV).
After-tax cash flow1st=$2500−(−$974.79)=$3474.79
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 (45% rate) (s)=0.45×(r) |
After-tax cash flow (t)=(p)−(s) |
0 |
−$14000 |
|
|
|
−$14000 |
1 |
$2500 |
$4666.2 |
−$2166.2 |
−$974.79 |
$3474.79 |
2 |
$2500 |
$4148.8 |
−$1648.8 |
−$741.96 |
$3241.96 |
3 |
$2500 |
$767.9 |
$1732.1 |
$779.44 |
$1720.56 |
4 |
$2500 |
|
$2500 |
$1125 |
$1375 |
5 |
$2500 |
|
$2500 |
$1125 |
$1375 |
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 three years is F.
Calculate present worth factor of annuity.
Substitute −$14000 for D, $3474.79 for A, 25% for i, 5 years for n and $5000 for F in Equation (V).
PW=[−14000+($3474.79)( ( 1+ 25 100 )5 −1 ( 25 100 ) ( 1+ 25 100 )5 )+($5000)(1 ( 1+ 25 100 )5 )]=[−$14000+($3474.79)(2.6893)+($5000)(0.3277)]=(−$14000+$9344+$1638.5)=−$3016.75
Thus, the present worth value for Alternate A is −$3016.75.
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 |
33.33% |
2 |
44.45% |
3 |
14.81% |
Calculate the depreciation charge for 1st year.
Substitute $18000 for B and 33.33% for rt in Equation (I).
dt 1 st=$18000×(33.33100)=$5999.4
Calculate the Net book value for 1st year.
Substitute $18000 for Base book value and $5999.4 for Depreciation charge for year dt in Equation (II).
Net book value1st=$18000−$5999.4=$12000.6
Calculate the depreciation charge for 2nd year.
Substitute $12000.6 for B and 44.45% for rt in Equation (I).
dt 2 nd=$12000.6×(44.45100)=$5334.3
Calculate the Net book value for 2nd year.
Net book value2nd=$12000.6−$5334.3=$6666.3
Calculate the depreciation charge for 3rd year.
Substitute $6666.3 for B and 14.81% for rt in Equation (I).
dt 3 rd=$6666.3×(14.81100)=$987.3
Calculate the Net book value for 3rd year.
Net book value3rd=$6666.3−$987.3=$5679
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 |
$18000 |
$5999.4 |
$12000.6 |
2 |
$12000.6 |
$5334.3 |
$6666.3 |
3 |
$6666.3 |
$987.3 |
$5679 |
Calculate the taxable incomes for 1st year.
Substitute $1000 for Before-tax cash flow and $5999.4 for Depreciation in Equation (III).
Taxable Incomes1st=$1000−$5999.4=−$4999.4
Calculate the Income taxes for 1st year.
Income taxes1st=45% of Taxable incomes1st=(45100)×(−$4999.4)=−$2249.7
Calculate the After-tax cash flow.
Substitute $1000 for Before-tax cash flow and −$2249.7 for Income taxes in Equation (IV).
After-tax cash flow1st=$1000−(−$2249.7)=$3429.7
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 (45% rate) (s)=0.45×(r) |
After-tax cash flow (t)=(p)−(s) |
0 |
−$18000 |
|
|
|
−$18000 |
1 |
$1000 |
$5999.4 |
−$4999.4 |
−$2249.7 |
$3249.7 |
2 |
$1000 |
$5334.3 |
−$4334.3 |
−$1950.4 |
$2950.4 |
3 |
$1000 |
$987.3 |
$12.7 |
$5.72 |
$994.3 |
4 |
$1000 |
|
$1000 |
$450 |
$550 |
5 |
$1000 |
|
$1000 |
$450 |
$550 |
Calculate present worth factor of annuity.
Substitute −$18000 for D, $3249.7 for A, 25% for i, 5 years for n and $10000 for F in Equation (V).
PW=[−18000+($3249.7)( ( 1+ 25 100 )5 −1 ( 25 100 ) ( 1+ 25 100 )5 )+($10000)(1 ( 1+ 25 100 )5 )]=[−$18000+($3249.7)(2.6893)+($10000)(0.3277)]=(−$18000+$8739.4+$3277)=−$5983.6
Thus, the present worth value for Alternate B is −$5983.6.
Alternate C:
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 |
33.33% |
2 |
44.45% |
3 |
14.81% |
Calculate the depreciation charge for 1st year.
Substitute $10000 for B and 33.33% for rt in Equation (I).
dt 1 st=$10000×(33.33100)=$3333
Calculate the Net book value for 1st year.
Substitute $10000 for Base book value and $3333 for Depreciation charge for year dt in Equation (II).
Net book value1st=$10000−$3333=$6667
Calculate the depreciation charge for 2nd year.
Substitute $6667 for B and 44.45% for rt in Equation (I).
dt 2 nd=$6667×(44.45100)=$2963.5
Calculate the Net book value for 2nd year.
Net book value2nd=$6667−$2963.5=$3703.5
Calculate the depreciation charge for 3rd year.
Substitute $3703.5 for B and 14.81% for rt in Equation (I).
dt 3 rd=$3703.5×(14.81100)=$548.5
Calculate the Net book value for 3rd year.
Net book value3rd=$3703.5−$548.5=$3155
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 |
$10000 |
$3333 |
$6667 |
2 |
$6667 |
$2963.5 |
$3703.5 |
3 |
$3703.5 |
$548.5 |
$3155 |
Calculate the taxable incomes for 1st year.
Substitute $5000 for Before-tax cash flow and $3333 for Depreciation in Equation (III).
Taxable Incomes1st=$5000−$3333=−$1667
Calculate the Income taxes for 1st year.
Income taxes1st=45% of Taxable incomes1st=(45100)×($1667)=$750.15
Calculate the After-tax cash flow.
Substitute $5000 for Before-tax cash flow and $750.15 for Income taxes in Equation (IV).
After-tax cash flow1st=$5000−($750.15)=$4249.85
Calculate the After-tax cash flow for the remaining years in tabular form.
Period |
Before-tax cash flow(p) |
MACRS Depreciation(q) |
Taxable Incomes(p-q) |
Income taxes (45% rate) |
After-tax cash flow |
0 |
−$10000 |
|
|
|
−$10000 |
1 |
$5000 |
$3333 |
−$1667 |
$750.15 |
$4249.85 |
2 |
$5000 |
$2963.5 |
$2036.5 |
$916.42 |
$4083.58 |
3 |
$5000 |
$548.5 |
$4451.5 |
$2003.2 |
$2996.8 |
4 |
$5000 |
|
$5000 |
$2250 |
$2750 |
5 |
$5000 |
|
$5000 |
$2250 |
$2750 |
Calculate present worth factor of annuity.
Substitute −$10000 for D, $4249.85 for A, 25% for i, 5 years for n and $0 for F in Equation (V).
PW=[−10000+($4249.85)( ( 1+ 25 100 )5 −1 ( 25 100 ) ( 1+ 25 100 )5 )+($0)(1 ( 1+ 25 100 )5 )]=[−$10000+($4249.85)(2.6893)+($0)(0.3277)]=(−$10000+$11429.12+$0)=$1429.12
Thus, the present worth value for Alternate C is $1429.12.
Conclusion:
Alternative C will have much greater positive value of $1429.12.
Thus, choose Alternative C.