###### How Taxes Affect the Cost of Capital

Taxes can have a significant impact on the weighted average cost of capital... **Read More**

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). Other measures include the payback period, discounted payback period, average accounting rate of return (AAR), and the profitability index (PI).

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.

** **

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 the 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.

The internal rate of return (IRR) is the discount rate that 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 that 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:

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

** **

Here is a follow-up on 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 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 arrived at through trial and error. However, a simpler approach is to use a financial calculator:

** **

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. Since 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.

** **

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 once more and for the last time to enter the last cash flow, 33.

** **

Once the cash flow values have been fed 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.

** **

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

** **

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

After the first year, 1,500 of the initial investment of 5,000 is recovered. 3,500 remains 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. The payback period ignores the cash flows which occur in years 3 and 4.

The discounted payback period refers to the number of years it takes the cumulative discounted cash flows from a project to equal 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 that occur after the discounted payback period is attained.

** **

Here is a follow-up on the example in Exhibit 1 above. What would be the discounted payback period assuming a discount rate of 10%?

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

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.

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.

** **

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.

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

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