Chapter 17, Problem 51P

(a):

To determine

Calculate the present worth.

Explanation of Solution

The recovery rate for year 1 (RR1) is 0.3333, year 2 (RR2) is 0.4445, year 3 (RR3) is 0.1481, and year 4 (rr4) is 0.0741. Time period is indicated by ‘n’ and MARR is indicated by ‘i’.

Depreciation (D) can be calculated using the following formula:

$\text{D}=\text{First cost}\times \text{RR}$        (1)

Substitute the respective value in Equation (1) to calculate the depreciation in year 1.

$\begin{array}{c}\text{D}=\text{8,000}\times \text{0}\text{.3333}\\ =2,666\end{array}$

The first year’s depreciation is $2,666.

The taxable income (TI) can be calculated using the following formula:

$\text{TI}=\left(\text{GI}-\text{OE}\right)-\text{D}$        (2)

Substitute the respective value in Equation (2) to calculate the taxable income in year 1.

$\begin{array}{c}\text{TI}=3,500-2,666\\ =834\end{array}$

The first year’s taxable income is $834.

Tax (T) can be calculated using the following formula:

$\text{T}=\text{TI}\times {\text{T}}_{\text{e}}$        (3)

Substitute the respective value in Equation (3) to calculate the tax in year 1.

$\begin{array}{c}\text{T}=834\times 0.4\\ =333\end{array}$

The first year’s tax is $333.

The cash flow after tax (CFATI) can be calculated using the following formula:

$\text{CFAT}=\left(\text{GI}-\text{OE}\right)-\text{T}$        (4)

Substitute the respective value in Equation (4) to calculate the cash flow after tax in year 1.

$\begin{array}{c}\text{CFAT}=3,500-333\\ =3,167\end{array}$

The first year’s cash flow after tax is $3,167.

Table 1 shows the depreciation, taxable income, tax, and CFAT obtained using Equations (1), (2), (3) and (4).

Table 1

Year	P	GI-OE	D	TI	T	CFAT
0	-8,000					-8,000
1		3,500	2,666	834	333	3,167
2		3,500,	3,556	-56	-22	3,522
3		3,500	1,185	2,315	926	2,574
4	0	0	593	-593	-237	237

The present worth (PW) can be calculated as follows:

$\begin{array}{c}\text{PW}={\text{CFAT}}_{\text{0}}+\frac{{\text{CFAT}}_{\text{2}}}{{\left(1+\text{i}\right)}^2}+\frac{{\text{CFAT}}_{\text{2}}}{{\left(1+\text{i}\right)}^2}+\frac{{\text{CFAT}}_{\text{3}}}{{\left(1+\text{i}\right)}^3}++\frac{{\text{CFAT}}_{\text{4}}}{{\left(1+\text{i}\right)}^4}\\ =-8,000+\frac{3,167}{{\left(1+0.08\right)}^1}+\frac{3,522}{{\left(1+0.08\right)}^2}+\frac{2,574}{{\left(1+0.08\right)}^3}+\frac{237}{{\left(1+0.08\right)}^4}\\ =-8,000+\frac{3,167}{1.08}+\frac{3,522}{1.1664}+\frac{2,574}{1.259712}+\frac{237}{1.360489}\\ =169\end{array}$

The present worth of alternate X is $169.

Table 2 shows the depreciation, taxable income, tax, and CFAT of alternate Y obtained using Equations (1), (2), (3), and (4).

Table 2

Year	P	GI-OE	D	TI	T	CFAT
0	-13,000					-13,000
1		5,000	4,333	667	267	4,733
2		5,000	5,779	-779	-311	5,311
3		5,000	1,925	3,075	1,230	3,770
4	0	2,000	963	1,037	415	1,585

The present worth (PW) can be calculated as follows:

$\begin{array}{c}\text{PW}={\text{CFAT}}_{\text{0}}+\frac{{\text{CFAT}}_{\text{2}}}{{\left(1+\text{i}\right)}^2}+\frac{{\text{CFAT}}_{\text{2}}}{{\left(1+\text{i}\right)}^2}+\frac{{\text{CFAT}}_{\text{3}}}{{\left(1+\text{i}\right)}^3}++\frac{{\text{CFAT}}_{\text{4}}}{{\left(1+\text{i}\right)}^4}\\ =-13,000+\frac{4,733}{{\left(1+0.08\right)}^1}+\frac{5,311}{{\left(1+0.08\right)}^2}+\frac{3,770}{{\left(1+0.08\right)}^3}+\frac{1,585}{{\left(1+0.08\right)}^4}\\ =-8,000+\frac{4,733}{1.08}+\frac{5,311}{1.1664}+\frac{3,770}{1.259712}+\frac{1,585}{1.360489}\\ =93\end{array}$

The present worth of project Y is $93. Since the present worth of project X is greater than project Y, select project X.

(a):

To determine

Calculate the time period.

Explanation of Solution

The present worth of alternate X and Y can be calculated using spreadsheet as follows: