Several important decision criteria are used to evaluate capital investments. The two most comprehensive and well-understood measures of whether or not a project is profitable are the net present value (NPV) and internal rate of return (IRR) measures. Other measures include the payback period, discounted payback period, average accounting rate of return (AAR), and the profitability index (PI).

**Net Present Value (NPV)**

The Net Present Value (NPV) of a project is the potential change in wealth resulting from the project after accounting for the time value of money. The NPV for a project with one investment outlay made at the start of the project is defined as the present value of the future after-tax cash flows minus the investment outlay.

$$ \text{NPV}=\sum _{ t=1 }^{ n }{ \frac { { CF }_{ t } }{ { \left( 1+r \right) }^{ t } } – } \text{Outlay} $$

Where:

CF_{t} = after-tax cash flow at time t

r = required rate of return for the investment

Outlay = investment cash flow at time zero

Many projects have cash flow patterns in which outflows occur not only at the start of the project (at time = 0) but also at future dates. In these instances, a better formula to use is:

$$ \text{NPV}=\sum _{ t=0 }^{ n }{ \frac { { CF }_{ t } }{ { \left( 1+r \right) }^{ t } } } $$

The decision rule for the NPV is:

- Invest in the project if NPV > 0;
- Do not invest in the project if NPV < 0; and
- Stay indifferent if NPV = 0.

In other words, positive NPV investments are wealth increasing, while negative NPV investments are wealth decreasing.

### Example: Net present value of a project

Suppose Company A is considering an investment of $100 million in a capital expansion project that will return after-tax cash flows of $20 million per year for the first 3 years and another $33 million in year 4, the final year of the project. If the required rate of return for the project is 8%, what would the NPV be and should company undertake this project?

$$ \begin{align*} \text{NPV} & =\frac { 20 }{ { 1.08 }^{ 1 } } +\frac { 20 }{ { 1.08 }^{ 2 } } +\frac { 20 }{ { 1.08 }^{ 3 } } +\frac { 33 }{ { 1.08 }^{ 4 } } -100 \\ \text{NPV} & = 18.519 + 17.147 + 15.877 + 24.256 – 100 \\ & = -$24.201 \text{ million} \\ \end{align*} $$

Since the NPV < 0, the project should not be undertaken.

**Internal Rate of Return**

The internal rate of return (IRR) is the discount rate which makes the net present value (NPV) of all cash flows from a particular project equal to zero. For a project with one initial outlay, the IRR is the discount rate which makes the present value of the future after-tax cash flows equal to the investment outlay.

The IRR solves the equation:

$$ \sum _{ t=1 }^{ n }{ \frac { { CF }_{ t } }{ { \left( 1+IRR \right) }^{ t } } – } \text{Outlay}=0 $$

It looks very much like the NPV equation except that the discount rate is the *IRR* instead of *r*, the required rate of return. Discounted at the IRR, the NPV is equal to zero.

The decision rule for the IRR is to:

- Invest in the project if the IRR exceeds the required rate of return for the project i.e. invest if IRR > r; and
- Do not invest if IRR < r.

In instances where the outlays for a project occur at times other than time 0, a more general form of the IRR equation is:

$$ \sum _{ t=0 }^{ n }{ \frac { { CF }_{ t } }{ { \left( 1+IRR \right) }^{ t } } } =0 $$

### Example: IRR of a project

Following on from the above NPV example, if company A is considering an investment of $100 million in a capital expansion project that will return after-tax cash flows of $20 million per year for the first 3 years and another $33 million in year 4, the final year of the project, what is the IRR for this project and should it be undertaken given that the required rate of return for the project is 8%?

Solve for IRR in the following equation:

$$ -100+\frac { 20 }{ { \left( 1+IRR \right) }^{ 1 } } +\frac { 20 }{ { \left( 1+IRR \right) }^{ 2 } } +\frac { 20 }{ { \left( 1+IRR \right) }^{ 3 } } +\frac { 33 }{ { \left( 1+IRR \right) }^{ 4 } } =0 $$

The solution can be found by trial and error. However, a much simpler approach is to use a financial calculator:

### Step 1: Entering the initial cash outlay

Press the Cash Flow [CF] key to open the cash flow register. The calculator should read CF0=, which tells you to enter the cash flow for time 0. Because you need to send cash out of the company to make the initial $100 investment, this value has to be negative. Type in -100 for CF0, and hit the [ENTER] key.

### Step 2: Entering the cash inflows

Next enter the cash flow values for the subsequent periods. This is done by hitting the down arrow *once*. The calculator should read CF1=. Type in the amount for the first cash flow, 20, and hit [ENTER]. The calculator should now say C01=20.

To enter cash flow from Year 2, hit the down arrow __ twice__. The calculator should read CF2=. If it says F1=, hit the down arrow one more time.

Type in the second year’s cash flow, 20, and hit [Enter]. The calculator should read CF2=20. Hit the down arrow *twice* again and do the same thing for the third cash flow period, CF3.

Do this one more last to enter the last cash flow, 33.

### Step 3: Calculating the IRR

Once the cash flow values have been entered into the calculator you are ready to calculate the IRR.

To do this press the [IRR] key. The screen will read IRR=0.000. To display the IRR value for the data set, press the [CPT] key at the top left corner of the calculator. If you have followed this process correctly, the calculator will display the correct IRR. The IRR is computed to be -2.626%. Since -2.626% < 8%, the project should not be undertaken.

**Payback Period**

The payback period refers to the number of years required to recover the original investment in a project. It is very simple to compute and explain. It, however, ignores the time value of money and the risk of a project by not discounting cash flows at the project’s required rate of return. It also ignores cash flows that occur after the payback period is reached. It may be used as an indicator of a project’s liquidity but not of its profitability.

### Example: Payback Period

The exhibit below provides data on the cash flows of a project. How long would the project take to recover the initial investment (payback period)?

$$ \textbf{Exhibit 1: Payback Period Example} $$ $$ \begin{array}{c|c|c|c|c} \hline \text{Year} & {0} & {1} & {2} & {3} & {4} \\ \hline \text{Cash flow} & {-5,000}& {1,500} & {3,500} & {4,000} & {4,000} \\ \hline \text{Cumulative cash flow} & {-5,000} & {-3,500} & {0} & {4,000} & { 4,000} \\ \hline \end{array} $$

After the first year, 1,500 of the initial investment of 5,000 is recovered, with 3,500 still unrecovered. In year 2, the project earns 3,500, which means that the initial investment is now fully recovered. The payback period is therefore 2 years. Payback period ignores the cash flows which occur in years 3 and 4.

**Discounted Payback Period**

The discounted payback period refers to the number of years for the cumulative discounted cash flows from a project to equal to the original investment. By factoring in a discount rate, the discounted payback period is a slight improvement over the payback period. It, however, ignores cash flows which occur after the discounted payback period is reached.

### Example: Discounted Payback Period

Following on from the example in Exhibit 1 above, what would be the discounted payback period assuming a discount rate of 10%?

$$ \begin{array}{c|c|c|c|c} \hline \text{Year} & {0} & {1} & {2} & {3} & {4} \\ \hline {\text{Cash flow }(\text{CF})} & {-5,000} & {1,500.00} & {3,500.00} & {4,000.00} & {4,000.00} \\ \hline \text{Cumulative CF} & {-5,000} & {-3,500.00} & {0} & {4,000.00} & {4,000.00} \\ \hline \text{Discounted CF} & {-5,000} & {1,363.64} & {2,892.56} & {3,005.26} & {2,732.05} \\ \hline \text{Cumulative discounted CF} & {-5,000} & {-3,636.36} & { -743.80} & {2,261.46} & {4,993.51} \\ \hline \end{array} $$

The discounted payback period is between 2 and 3 years. More precisely, it is two years plus (the cumulative discounted CF after 2 years divided by the discounted cash flow in year 3) i.e. 2 + 743.80/3,005.26 = 2.25 years.

**Profitability Index (PI)**

The profitability index (PI) refers to the present value of a project’s future cash flows divided by the initial investment.

In the form of an equation, it is –

$$ PI=\frac { \text{PV of future cashflows}}{\text{Initial investment}} =1+\frac { \text{NPV} }{ \text{Initial investment} } $$

Whenever the NPV > 0, the PI will be greater than 1.0; conversely, whenever the NPV is negative, the PI will be less than 1.0.

The decision rule for the PI is; Invest in the project if PI>1.0; do not invest in the project if PI<1.0.

### Example: Profitability index

If company A has a project with an initial outlay of $100 million and an NPV of -$24.201 million, the profitability index is computed as:

$$ PI=1+\frac { \text{NPV} }{ \text{Initial investment} } =1+\frac { -24.201 }{ 100000 } =0.758 $$

Since the PI < 1.0, undertaking the project would not be a profitable investment.

QuestionGiven the following cash flows for a capital project, compute the NPV and IRR of the project. The required rate of return is 8 percent.

$$ \begin{array}{c|c|c|c|c} \hline \text{Year} & {0} & {1} & {2} & {3} & {4} & {5} \\ \hline {\text{Cash flow } ($)} & {-50,000} & {10,000} & {10,000} & {15,000} & {15,000} & {15,000} \\ \hline \end{array} $$

A. NPV: $1,023; IRR: 10.64%

B. NPV: $974; IRR: 8.68%

C. NPV: $2,400; IRR: 7.12%

SolutionThe correct answer is B.

$$ \begin{align*} \text{NPV} &=-50,000+\frac { 10,000 }{ { 1.08 }^{ 1 } } +\frac { 10,000 }{ { 1.08 }^{ 2 } } +\frac { 15,000 }{ { 1.08 }^{ 3 } } +\frac { 15,000 }{ { 1.08 }^{ 4 } } +\frac { 15,000 }{ { 1.08 }^{ 5 } } \\ \text {NPV} & =-50,000+9259.26+8573.39+11,907.48++11,025.45+10,208.75 \\ & =$974.33 \text{ million} \end{align*} $$

*Reading 32 LOS 32d: *

*Calculate and interpret net present value (NPV), internal rate of return (IRR), payback period, discounted payback period, and profitability index (PI) of a single capital project*