Business Investment Decisions

5.2 Net Present Value (NPV): Calculator Approach

Net Present Value (NPV) is a financial metric used to evaluate the profitability of an investment or project. It represents the difference between the present value of cash inflows and the present value of cash outflows (also known as outlays) over the investment’s lifetime. The NPV is calculated by discounting all expected future cash flows to their present values using a specific discount rate, often the required rate of return. By using a required rate of return in NPV calculations, investors and managers can factor in the risk profile of the investment and the opportunity cost of capital. This rate typically reflects the return that could be earned on alternative investments of similar risk.

 [asciimath]NPV=PV_("Cash"\ "inflows")-PV_("Cash"\ "outflows")[/asciimath]Formula 5.2

A positive NPV indicates that the project is expected to generate more value than the cost, considering the time value of money and the investment’s risk. Conversely, a negative NPV suggests that the project is likely to diminish value. An NPV of zero means that the investment is expected to yield a return equal to the minimum acceptable rate of return.

The NPV decision criterion is as follows:

  • If [asciimath]NPV gt 0[/asciimath] : Accept the Project
    • A positive NPV indicates that the present value of the project’s cash inflows exceeds the present value of its cash outflows. This means the investment is expected to generate a return greater than the required rate of return. In other words, the project adds value to the firm or investor and is considered financially viable and profitable.
  • If [asciimath]NPV = 0[/asciimath] : Indifference
    • An NPV of zero suggests the project’s returns are exactly sufficient to cover the cost of capital and meet the required rate of return. The project is breaking even in present value terms. While financially neutral, the decision might still consider strategic, operational, or social factors.
  • If [asciimath]NPV lt 0[/asciimath] : Reject the Project
    • A negative NPV indicates that the present value of the project’s cash inflows is less than the present value of its cash outflows. This means the project is expected to yield a return less than the required rate of return. The project is not adding value and is considered unprofitable and financially unviable.

When comparing multiple projects using Net Present Value (NPV), the decision-making process involves evaluating the NPV of each project and then selecting the one(s) that provide the greatest financial return.

It is important to note that when working with algebraic formulas, such as Formula 5.2 for calculating NPV, both the present value of inflows and the present value of outflows are treated as positive values.

 

Example 5.2.1: Calculate NPV of a Project with Future Cash Inflows

The management team of a company is evaluating a project that requires a $30,000 investment today. The expected returns from this investment are projected to be $8,000 in one year, $13,000 in two years, and $16,000 in three years. The company’s required rate of return for investments is 8%. Calculate the net present value of the project and determine whether they should accept the project.

Show/Hide Solution
  • Required rate of return: [asciimath]I//Y=8%[/asciimath]
  • The frequency of compounding periods is not provided, so it is assumed annually: [asciimath]C//Y=1[/asciimath]
  • For compound problems [asciimath]P//Y=C//Y[/asciimath], so [asciimath]P//Y=1[/asciimath]
  • Cash outflow [asciimath]=$30,000[/asciimath]
  • Cash inflow 1: [asciimath]FV_1=8000[/asciimath]
  • cash inflow 2: [asciimath]FV_2=13,000[/asciimath]
  • Cash inflow 3: [asciimath]FV_3=16,000[/asciimath]
  • Cash inflow 1 time: [asciimath]t_1=1[/asciimath] year; [asciimath]N_1=C//Y*t_1=1[/asciimath]
  • Cash inflow 2 time: [asciimath]t_2=2[/asciimath] years; [asciimath]N_2=2[/asciimath]
  • Cash inflow 3 time: [asciimath]t_3=3[/asciimath] years; [asciimath]N_3=3[/asciimath]

Timeline illustrating cash flows for this example: cash inflows are $80,000 at year 1, $13,000 at year 2, and $16,000 at year 3. The only cash outflow of $30,000 is at year 0 (present). Arrows from future cash inflows point back to year 0, indicating the need to calculate their present values for compounding periods (N) of 1, 2, and 3, respectively.

Cash Outflows:

As the initial investment of $30,000 must be paid immediately, there is no need to calculate its present value. It is already in present-day terms.

[asciimath]PV_("outflows")=$30,000[/asciimath]

 

Cash Inflows:

It is important to note that all the future cash flows of the project are returns on investment, meaning they are cash inflows. Given the need to make an immediate decision on the investment opportunity, we use ‘now’ as the focal date to determine the present values of the future cash inflows. We use the TVM worksheet to calculate the present value of each expected cash inflow from the investment.

Series of three TVM Worksheet images, each designed to calculate the Present Value (PV) of cash inflows over different terms: 1 year, 2 years, and 3 years. Each worksheet is shown with specific input values. Instructions across all worksheets direct to input all necessary values, followed by pressing the 'Compute' key and then the PV key to finalize the PV calculation.

We then sum all the present values.

 [asciimath]PV_("inflow")[/asciimath] [asciimath]=PV_(Cash \ flow \ 1)+[/asciimath] [asciimath]PV_(Cash \ flow \ 2)+[/asciimath]  [asciimath]PV_(Cash \ flow \ 3)[/asciimath]

 [asciimath]=7407.407...+11,145.404...+12,701.315...[/asciimath]

 [asciimath]~~$31,254.13[/asciimath]

Determine NPV: 

Using Formula 5.2, the NPV is

 [asciimath]NPV=PV_("inflows")-PV_("outflows")[/asciimath] 

[asciimath]NPV=31,254.13-30,000[/asciimath] 

[asciimath]=1254.13[/asciimath] 

Make a Decision:

The NPV of the project is more than zero, which means the returns are sufficient to cover the cost of capital and provide an additional value, indicating the company should accept the project under the given financial criteria.

 

Try an Example

 

 

Example 5.2.2: Calculate NPV of a Project with Annuity Cash Inflows and Residual

A company is evaluating the feasibility of investing in machinery to expand its manufacturing plant. It would need to invest $40,000 today and $20,000 in one year. The machinery is expected to generate a net return of $6,295 at the end of every year for 11 years. At the end of the 11 years, the machinery is expected to have a salvage (residual) value of $10,200, which can be considered a final cash inflow. The company’s required rate of return for investments is 5.4%. a) Calculate the Net Present Value (NPV) of the investment. b) Determine whether the purchase of this machinery is a financially sound decision according to the NPV criterion.

Show/Hide Solution
  • Required rate of return: [asciimath]I//Y=5.4%[/asciimath]
  • The frequency of compounding periods is not provided, so it is assumed annually: [asciimath]C//Y=1[/asciimath]
  • For compound problems [asciimath]P//Y=C//Y[/asciimath], so [asciimath]P//Y=1[/asciimath]

Cash Outflows:

  • Cash outflow 1: [asciimath]PV_1=$40,000[/asciimath]
  • Cash outflow 2: [asciimath]FV_2=20,000[/asciimath]
  • Cash inflow 1 time: [asciimath]t_1=0[/asciimath] year
  • Cash inflow 2 time: [asciimath]t_2=1[/asciimath] years; [asciimath]N_2=C//Y*t_2=1[/asciimath]

Timeline illustrating cash outflows for this example: cash ouflows are $40,000 at year 0 and $20,000 at year 1. An arrow from future cash inflow of $20,000 points back to year 0, indicating the need to calculate its present value.

As the initial investment of $40,000 must be paid immediately, there is no need to calculate its present value. It is already in present-day terms. For the second investment of $20,000, we utilize the TVM worksheet to calculate the present value.

TVM Worksheet demonstrating how to compute Present Value (PV) of cash outflow of $20,000. It instructs to input each given value, followed by pressing its corresponding key. After all values are entered, the guidance is to press the 'Compute' key and then the PV key to complete the calculation.

 [asciimath]PV_("outflows")[/asciimath]  [asciimath]=PV_("outflow" \ 1)+[/asciimath] [asciimath]PV_("outflow" \ 2)[/asciimath]

 [asciimath]=40,000+18,975.332...[/asciimath]

 [asciimath]~~58,975.33[/asciimath]

 

Cash Inflows:

There are two types of cash inflows:

– Annual investment return of $6295: as all these cash inflows are of equal amounts and are expected to occur at regular intervals (one year apart), they can be treated as payments of an ordinary simple annuity. In this case, we can use the TVM worksheet to calculate the combined present value of both cash inflows in a single step.

  • Returns: [asciimath]PMT=6295[/asciimath]
  • Annuity term: [asciimath]t=11[/asciimath] years
  • One payment per year, so [asciimath]P//Y=1[/asciimath]
  • Number of payments in the term: [asciimath]N=P//Y*t=1(11)=11[/asciimath]

– The machinery’s salvage value of $10,200: The company can sell the machinery at the of the 11th year. This salvage value is considered a cash inflow, and its present value can be calculated using the TVM worksheet.

  • Salvage value: [asciimath]FV_s=10,200[/asciimath]
  • Compound term: [asciimath]t=11[/asciimath] years
  •  [asciimath]P//Y=C//Y=1[/asciimath]
  • Number of compound periods in the term: [asciimath]N=C//Y*t=1(11)=11[/asciimath]

Timeline depicting cash inflows with eleven annual payments of $6,295 starting at the end of year 1 and continuing through to the end of year 11, plus a residual value of $10,200 at the end of year 11. Arrows from the payments and the residual value direct back to year 0, highlighting the requirement to calculate their present values.

Series of two TVM Worksheet images for cash inflows. The first worksheet is set up to calculate the present value (PV) of the $6295 payments. The second worksheet is set up to calculate the present value (PV) of the $10,200 residual. Instructions across both worksheets direct to input all necessary values, followed by pressing the 'Compute' key and then the PV key to finalize the PV calculation.

 [asciimath]PV_("inflows")[/asciimath] [asciimath]=PV_("returns")+[/asciimath] [asciimath]PV_("residual")[/asciimath]

 [asciimath]=51,207.570...+5719.439...[/asciimath]

  [asciimath]~~$56,927.01[/asciimath]

Alternatively, when considering an investment where the number of payments (N) and the compounding periods for both the investment returns and the residual value are the same, we can simplify the calculation. By entering the regular returns as the payment (PMT) and the residual value as the future value (FV) in the Time Value of Money (TVM) worksheet, we can calculate their combined present value in a single step. It is important to note that since both the payments and the residual are cash inflow, they should be entered as positive values (or at least with the same sign)

TVM worksheet visualization for calculating Present Value (PV) of cash inflows, incorporating annual payments of $6,295 as PMT and a final residual value of $10,200 as FV in a single calculation. The procedure involves entering each specified value, pressing the corresponding key, and concluding by hitting the 'Compute' button followed by the PV button to finalize the PV calculation.

Using this approach, the present value of all inflows is calculated as $56,927.01, which matches the value obtained previously through separate calculations.

 

Determine NPV: 

Using Formula 5.2, the NPV is

 [asciimath]NPV=PV_("inflows")-PV_("outflows")[/asciimath] 

 [asciimath]NPV=56,927.01 -58,975.33[/asciimath] 

 [asciimath]=-$2048.32[/asciimath] 

 

Make a Decision:

The NPV of the project is negative, which means the investment does not generate enough return to cover its initial cost plus the minimum required return. Therefore, the company should reject the project under the given financial criteria.

 

Try an Example

 

 

Example 5.2.3: Evaluate NPV of Two Projects with Annuity Cash Inflows and Residual

Riverflow Utilities is faced with two different strategies for enhancing the environmental sustainability of one of its water treatment facilities to meet increased regulatory standards. Each option involves different cost structures and timelines, and the decision must be made using the Net Present Value (NPV) method, considering the required rate of return of 10%.

Strategy A: Upgrade with Advanced Technology

  • Annual Costs: $50,000 paid at the end of each year for 9 years.
  • Residual Value: At the end of 9 years, Riverflow Utilities expects to sell the specialized technology components for $80,000.

Strategy B: Outsourcing to Environmental Specialists

  • Initial Cost: A one-time payment of $120,000 upfront.
  • Annual Costs: $25,000 paid at the end of each year for 9 years.

a) Calculate the Net Present Value (NPV) of Strategy A. b) Calculate the Net Present Value (NPV) of Strategy B. c) Determine which strategy Riverflow Utilities should implement based on the NPV criterion.

Show/Hide Solution

 

  • Required rate of return: [asciimath]I//Y=10%[/asciimath]
  • The frequency of compounding periods is not provided, so it is assumed annually: [asciimath]C//Y=1[/asciimath]

a) Strategy A:

Timeline for Strategy A shows cash flows with nine annual outflows of $50,000 starting at year 1’s end and lasting until year 9, and a single inflow of $80,000 as residual value at year 9’s end. Arrows from both the annual payments and the residual value point towards year 0, underscoring the need to compute their present values.

Cash outflow:

The annual costs of $50,000 are the cash outflows. They are of equal amounts and are expected to occur at regular intervals (one year apart), they can be treated as payments of an ordinary simple annuity. In this case, we use the TVM worksheet to calculate the present value of the payments.

  • Frequency of payments per year:[asciimath]P//Y=1[/asciimath]
  • Time period: [asciimath]t=9[/asciimath] years
  • Number of payments in the term: [asciimath]N=P//Y*t=1(9)=9[/asciimath]
  • Payments are cash outflow:[asciimath]PMT=-50,000[/asciimath]

 

Cash inflow:

The company can sell the component at the of the 9th year. This residual value is considered a cash inflow, and its present value can be calculated using the TVM worksheet.

  • Residual value: [asciimath]FV=80,000[/asciimath]
  • Compound term: [asciimath]t=9[/asciimath] years
  •  [asciimath]P//Y=C//Y=1[/asciimath]
  • Number of compound periods in the term: [asciimath]N=C//Y*t=1(9)=9[/asciimath]

Thus the present value of the $80,000 is $33,927.81.

 

The net present value of Strategy A ([asciimath]NPV_A[/asciimath]):

Using Formula 5.2, the NPV is

 [asciimath]NPV=PV_("inflows")-PV_("outflows")[/asciimath] 

 [asciimath]NPV_A=33,927.81 -287,951.19[/asciimath] 

 [asciimath]=-$254,023.38[/asciimath] 

 

b) Strategy B:

Strategy B timeline illustrates an initial outflow of $120,000 at year 0, followed by nine annual payments of $25,000 each, starting at the end of year 1 and continuing through to the end of year 9. Arrows from these yearly outflows point back to year 0, indicating the computation of their present values.

Cash outflow:

As the initial cost of $120,000 must be paid immediately, there is no need to calculate its present value. It is already in present-day terms.

The annual costs of $25,000 are of equal amounts and are expected to occur at regular intervals (one year apart), they can be treated as payments of an ordinary simple annuity. In this case, we can use the TVM worksheet to calculate the present value of the payments.

  • Frequency of payments per year:[asciimath]P//Y=1[/asciimath]
  • Time period: [asciimath]t=9[/asciimath] years
  • Number of payments in the term: [asciimath]N=P//Y*t=1(9)=9[/asciimath]
  • Payments: [asciimath]PMT=-25,000[/asciimath]

TVM Worksheet demonstrating how to compute Present Value (PV) of cash outflow (PMT) of $25,000 for Strategy B. It instructs to input each given value, followed by pressing its corresponding key. After all values are entered, the guidance is to press the 'Compute' key and then the PV key to complete the calculation.

Thus the present value of the $25,000 payments is $143,975.60.

 [asciimath]PV_("outflows")[/asciimath]  [asciimath]=[/asciimath]  [asciimath]"Initial pay"+PV_("Annuity")[/asciimath]

 [asciimath]PV_("outflows")[/asciimath] [asciimath]=[/asciimath] [asciimath]120,000+143,975.60[/asciimath]

 [asciimath]=$263,975.60[/asciimath]

Cash inflow:

There is no cash inflow in the second strategy, so [asciimath]PV_("inflows")=0[/asciimath].

 

The net present value of Strategy B ([asciimath]NPV_B[/asciimath]):

Using Formula 5.2, the NPV is

 [asciimath]NPV=PV_("inflows")-PV_("outflows")[/asciimath] 

 [asciimath]NPV_B=0 -263,975.60[/asciimath] 

 [asciimath]=-$263,975.60[/asciimath] 

 

c)  Make a decision:

Since the net present values for both strategies are negative, the company should opt for the strategy with the lesser cost or larger net present value. Therefore, the company should choose Strategy A, as it has the lower negative net present value and thus represents the less costly option, which is consistent with the NPV acceptance criteria as [asciimath]NPV_A>NPV_B[/asciimath].

 

Try an Example

 

Section 5.2 Exercises

  1. The management team of a company is evaluating a project that requires a $17,099 investment today. The expected returns from this investment are projected to be $6,000 in 1 year, $12,000 in 7 years, and $13,500 in 8 years. The company’s required rate of return for investments is 7%. a) Calculate the net present value of the project. b) Should the management accept the project?
    Show/Hide Answer

     

    a) NPV = $3,839.00

    b) Yes, since NPV>0.

  2. A company is evaluating the feasibility of investing in machinery to expand its manufacturing plant. It would need to invest $53,300 today and $54,400 in 3 years. The machinery is expected to generate a net return of $9,500 at the end of every year for 12 years. At the end of the 12 years, the machinery is expected to have a residual value of $16,500. The company’s required rate of return for investments is 3%. a) Calculate the Net Present Value (NPV) of the investment. b) Is the investment worthwhile?
    Show/Hide Answer

     

    a) NPV = $3,052.10

    b) Yes, since NPV>0

  3. VitalFlow Systems is faced with two different strategies for enhancing the environmental sustainability of one of its water treatment facilities to meet increased regulatory standards. Each option involves different cost structures and timelines, and the decision must be made using the Net Present Value (NPV) method, considering the required rate of return of 7%. a) Calculate the net present value of Strategy A. b) Calculate the discounted cash flow of Strategy B. c) Which strategy should VitalFlow Systems choose based on the NPV of each option? Strategy A: Upgrade with Advanced Technology
    • Annual Costs: $48,000 paid at the end of each year for 9 years.
    • Residual Value: At the end of 9 years, VitalFlow Systems expects to sell the specialized technology components for $64,800.

    Strategy B: Outsourcing to Environmental Specialists

    • Initial Cost: A one-time payment of $61,000 upfront.
    • Annual Costs: $32,200 paid at the end of each year for 9 years.
    Show/Hide Answer

     

    a) Strategy A: NPV = -$277,484.24

    b) Strategy B: NPV = -270,790.48

    c) Strategy B since it costs less.

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