Chapter 12, Problem 39P

To determine

The present worth of the first $4$ year of after tax cash flow from the barge.

Answer to Problem 39P

The present worth of the first $4$ year of after tax cash flow from the barge is $\$5815022$.

Explanation of Solution

Given:

Cost of the barge is $\$13.2\text{million}$.

Expected income from the barge is $\$7.5\text{million}$ for the first year.

Growth in income is $\$1\text{million}$ per year.

Expenses for the first year is $\$2.6\text{million}$.

Growth in expenses is $\$400,000$ per year.

The tax rate is $38\%$.

MARR is $12\%$.

Concept used:

Write the equation for the base book value.

${\left(\text{Base}\text{book}\text{value}\right)}_{n^{th}year}=\left[\begin{array}{l}{\left(\text{Book}\text{value}\right)}_{{\left(n-1\right)}^{th}year}\\ -\left(\left(\text{MACRS}\text{rate}\right)\times {\left(\text{Book}\text{value}\right)}_{{\left(n-1\right)}^{th}year}\right)\end{array}\right]$

Write the formula to calculate the taxable income.

$\text{TI}=\text{BTCF}-\text{DI}$

Here, the taxable income is $\text{TI}$, the depreciation is $\text{DI}$ and the before tax cash flow is $\text{BTCF}$.

Write the formula to calculate the taxable amount.

$\text{TAX}=0.38\times \text{TI}$

Here, the taxable amount is $\text{TAX}$.

Write the formula to calculate the after tax cash flow.

$\text{ATCF}=\text{TAX}+\text{BTCF}$

Here, the after tax cash flow is $\text{ATCF}$.

Calculation:

The projected revenue for the first year is $\$7,500,000$.

The revenue of the project is growing $\$1,000,000$ for each year.

Calculate the project revenue for second year.

$\begin{array}{c}{\text{Revenue}}_{2^{nd}}={\text{Revenue}}_{1^{st}}+\$1000000\\ =\$7500000+\$1000000\\ =\$8500000\end{array}$

Calculate the project revenue for third year.

$\begin{array}{c}{\text{Revenue}}_{3^{rd}}={\text{Revenue}}_{2^{nd}}+\$1000000\\ =\$8500000+\$1000000\\ =\$9500000\end{array}$

Calculate the project revenue for fourth year.

$\begin{array}{c}{\text{Revenue}}_{4^{th}}={\text{Revenue}}_{3^{rd}}+\$1000000\\ =\$9500000+\$1000000\\ =\$10500000\end{array}$

The operating and maintenance cost for first year is $\$2,600,000$.

The expenses of operating and maintenance cost is increasing by $\$400,000$ for each year.

Calculate the operating and maintenance cost for second year.

$\begin{array}{c}{\left(\text{Operating and Maintenance cost}\right)}_{2^{nd}}=\left[\begin{array}{l}{\left(\text{Operating and Maintenance cost}\right)}_{1^{st}}\\ +\$400000\end{array}\right]\\ =\$2600000+\$400000\\ =\$3000000\end{array}$

Calculate the operating and maintenance cost for third year.

$\begin{array}{c}{\left(\text{Operating and Maintenance cost}\right)}_{3^{rd}}=\left[\begin{array}{l}{\left(\text{Operating and Maintenance cost}\right)}_{2^{nd}}\\ +\$400000\end{array}\right]\\ =\$3000000+\$400000\\ =\$3400000\end{array}$

Calculate the operating and maintenance cost for fourth year.

$\begin{array}{c}{\left(\text{Operating and Maintenance cost}\right)}_{4^{th}}=\left[\begin{array}{l}{\left(\text{Operating and Maintenance cost}\right)}_{3^{rd}}\\ +\$400000\end{array}\right]\\ =\$3400000+\$400000\\ =\$3800000\end{array}$

Calculate the net revenue as shown in table below.

Year	Projected revenue (a)	Operating and maintenance cost (b)	Net revenue $\left(\text{a}-\text{b}\right)$
$1$	$\$7500000$	$\$2600000$	$\$4900000$
$2$	$\$8500000$	$\$3000000$	$\$5500000$
$3$	$\$9500000$	$\$3400000$	$\$6100000$
$4$	$\$10500000$	$\$3800000$	$\$6700000$

Calculate the book value by using MACRS percentage rate as shown in table below.

Year	MACR rate (a)	Base book value (b)	Depreciation charge $\left(\text{a}\times \text{b}\right)$	Net book value $b-\left(\text{a}\times \text{b}\right)$
$1$	$10\%$	$\$13200000$	$\$1320000$	$\$11880000$
$2$	$18\%$	$\$11880000$	$\$2138400$	$\$9741600$
$3$	$14.40\%$	$\$9741600$	$\$1402790.4$	$\$8338809.6$
$4$	$11.52$	$\$8338809.6$	$\$960630.9$	$\$7378178.7$

Calculate the gain on the salvage value as shown below.

$\text{Gain}=\left(\text{Expected}\text{salvage}\text{value}-\text{Net}\text{book}\text{value}\text{after}\text{3}\text{year}\right)$

Substitute $\$0$ for expected salvage value and $\$7378178.7$ for net book value.

$\begin{array}{c}\text{Gain}=\left(\$0-\$7378178.7\right)\\ =-\$7378178.7\end{array}$

Calculate the tax effect as shown below.

$\text{Tax}\text{effect}=\text{Gain}\times \text{Tax}\text{rate}$

Substitute $-\$7378178.7$ for Gain and $38\%$ for Tax rate.

$\begin{array}{c}\text{Tax}\text{effect}=-\$7378178.7\times 0.38\\ =-\$2803707.9\end{array}$

Calculate the net salvage value after fourth year as shown below.

$\text{Net}\text{salvage}\text{value}=\text{Expected}\text{salvage}\text{value}-\text{Tax}\text{effect}$

Substitute $\$0$ for expected salvage value and $-\$2803707.9$ for tax effect.

$\begin{array}{c}\text{Net}\text{salvage}\text{value}=\text{\$0}-\left(-\$2803707.9\right)\\ =\$2803707.9\end{array}$

Take before tax cash flow for the first year is $\$4900000$.

Calculate the taxable income for the first year.

$\text{Taxable}\text{income}=\text{Before}\text{tax}\text{cash}\text{flow}-\text{depreciation}$

Substitute $\$4900000$ for before tax cash flow and $\$1320000$ for depreciation.

$\begin{array}{c}\text{Taxable}\text{income}=\$4900000-\$1320000\\ =\$3580000\end{array}$

Calculate income tax for first year as shown below.

$\text{Income}\text{tax}=\text{38\%}\text{of}\text{taxable}\text{income}$

Substitute $\$3580000$ for taxable income.

$\begin{array}{c}\text{Income}\text{tax}=\text{0}\text{.38}\times \$3580000\\ =\$1360400\end{array}$

Calculate the after tax cash flow for the first year as shown below.

$\text{After}\text{tax}\text{cash}\text{flow}=\text{Before}\text{tax}\text{cash}\text{flow}+\text{Income}\text{tax}$

Substitute $\$4900000$ for before tax cash flow and $\$1360400$ for income tax.

$\begin{array}{c}\text{After}\text{tax}\text{cash}\text{flow}=\$4900000+\$1360400\\ =\$6260400\end{array}$

Calculate the after tax cash flow as shown in table below for all the subsequent years.

Year	BTCF	Depreciation	Taxable income	Income tax	ATCF
$0$	$-\$13200000$				$-\$13200000$
$1$	$\$4900000$	$\$1320000$	$\$3580000$	$\$1360400$	$\$6260400$
$2$	$\$5500000$	$\$2138400$	$\$3361600$	$\$1277408$	$\$6777408$
$3$	$\$6100000$	$\$1402790.4$	$\$4697210$	$\$1784939.65$	$\$7884939.65$
$4$	$\$6700000$	$\$960630.9$	$\$5739369$	$\$2180960.26$	$\$8880960.26$

Write the equation for present worth factor of annuity (PW).

$\begin{array}{c}PW=D+A\left(\frac{P}{A},i,n\right)+F\left(\frac{P}{F},i,n\right)\\ =D+A\left(\frac{{\left(1+i\right)}^n-1}{i{\left(1+i\right)}^n}\right)+F\left(\frac{1}{{\left(1+i\right)}^n}\right)\end{array}$

Here, the present worth is PW, initial payment is D, present value of money is P, interest rate is i, the after tax cash flow is A, the net salvage amount is F, and the number of year is n.

Substitute $-\$13200000$ for D, $\$6260400$ for A, $12\%$ for i, $4\text{years}$ for n and $\$0$ for F.

$\begin{array}{c}PW=\left(-\$13200000+\$6260400\left(\frac{{\left(1+0.12\right)}^4-1}{0.12{\left(1+0.12\right)}^4}\right)+\$0\left(\frac{1}{{\left(1+0.12\right)}^4}\right)\right)\\ =\left(-\$13200000+\$6260400\left(3.037\right)\right)\\ =\left(-\$13200000+\$19015022\right)\\ =\$5815022\end{array}$.

Conclusion:

Therefore, the present worth of the first $4$ year of after tax cash flow is, $\$5815022$.