Chapter 24, Problem 5BTN

To determine

Net Present Value (NPV) Method:

Net present value method is a measure to compare the current worth of cash inflows and current worth of cash outflows to determine the profitability of an investment decision.

Payback Period:

Payback period refers to the time period needed to recover the amount invested in project from the cash inflow received from it.

Formula to calculate payback period,

$\text{Payback period}=\frac{\text{Total investment}}{\text{Amount invested}}$

Cash Flow:

Cash flow is as statement of sum total of income received also referred as cash inflows and expense incurred referred as cash outflow for a particular period of time.

To compute: Payback period and net present value (NPV) of the investment.

Explanation of Solution

Payback Period

Given below is the table for the computation of payback period:

Payback period
Year	Net cash flows ($)	Cumulative net cash flows ($)
0	$\left(15,000\right)$	$\left(15,000\right)$
1	1,000	$\left(14,000\right)$
2	2,000	$\left(12,000\right)$
3	3,000	$\left(9,000\right)$
4	6,000	$\left(3,000\right)$
5	7,000	4,000

Table(1)

As per above table, the cash flow is positive after 4^th year, hence it can be said that cash flow is received after 4^th year.

In order to find the actual payback period, divide the negative cumulative net cash flows of 4^th year by the net cash flow of the 5^th year, then add the given duration to the 4^th year.

Formula to calculate total payback period,

$\text{Total payback period=}\left(4\text{ years}+\frac{{\text{Negative cumulative net cash flows of 4}}^{\text{th}}\text{year}}{{\text{Net cash flow of the 5}}^{\text{th}}\text{year}}\right)$

Substitute $3,000 for negative cumulative cash flows of 4^th year and $7,000 for net cash flow of the 5^th year.

$\begin{array}{c}\text{Total payback period=}\left(4\text{ years}+\frac{\$3,000}{\$7,000}\right)\\ =4.4\text{ years}\end{array}$

Hence, payback period of investment of $15,000 is 4.4 years.

Net Present Value (NPV)

Given below is the table for the computation of Net Present Value (NPV).

Net Present Value
Year	Cash flows ($)	Present value of 1 at 10%	Present value of net cash flows ($)
1	1,000	0.9091	9,09.10
2	2,000	0.8264	1,652.80
3	3,000	0.7513	2,253.90
4	6,000	0.6209	4,098.00
5	7,000		4,346.30
Total cash flows			13,260.10
Invested amount	15,000		$\left(15,000\right)$
Net present value			$\left(1,739.90\right)$

Table(2)

Hence, NPV for investment is $(1,739.90) as the total flows did not recover the invested amount of $15,000.

Payback Period

Given below is the table for the computation of payback period according to the website detail:

Payback period
Year	Net cash flows ($)	Cumulative net cash flows ($)
0	$\left(15,000\right)$	$\left(15,000\right)$
1	7,000	$\left(8,000\right)$
2	6,000	$\left(2,000\right)$
3	3,000	1,000
4	2,000	3,000
5	1,000	4,000

Table(3)

• As per above table, the cash flow is positive after 2^nd year, hence it can be said that cash flow is received after 2^nd year.
• In order to find the actual payback period, divide the negative cumulative net cash flows of 2^nd year by the net cash flow of the 3^rd year, then add the given duration to the 2^nd year.

Formula to calculate total payback period,

$\text{Total payback period=}\left(\begin{array}{l}\left(\text{2 years}\right)\\ +\left(\frac{{\text{Negative cumulative net cash flows of 2}}^{\text{nd}}\text{ year}}{{\text{Net cash flow of the 3}}^{\text{rd}}\text{year}}\right)\end{array}\right)$

Substitute $2,000 for negative cumulative cash flows of 2^nd year and $7,000 for net cash flow of the 3^rd year.

$\begin{array}{c}\text{Total payback period=}\left(\text{2 years}+\frac{\$2,000}{\$3,000}\right)\\ =2.66\text{ years}\end{array}$

Hence, payback period of investment of $15,000 is 2.66 years.

Net Present Value (NPV)

Given below is the table for the computation of Net Present Value (NPV).

Net Present Value
Year	Cash flows ($)	Present value of 1 at 10%	Present value of net cash flows ($)
1	7,000	0.9091	6,363.70
2	6,000	0.8264	4,958.40
3	3,000	0.7513	2,253.90
4	2,000	0.6830	1,366.00
5	1,000	0.6209	620.90
Total cash flows			15,562.90
Invested amount	15,000		$\left(15,000\right)$
Net present value			$562.90$

Table(4)

Hence, NPV for investment is $562.90.

Hence, the project as per book sum has higher payback period and negative NPV as compared to website sum. This will result in rejection of the project.