# Measures of Profitability

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).

## 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:

CFt = 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. ## Internal Rate of Return 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: $$\sum _{ t=0 }^{ n }{ \frac { { CF }_{ t } }{ { \left( 1+IRR \right) }^{ t } } } =0$$ ### Example: IRR of a project Here is a follow-up on the above NPV example. If company A is considering an investment of100 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. ### 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 once more and for the last time to enter the last cash flow, 33. ### Step 3: Calculating the IRR 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. ## Payback Period 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. ### 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)? $$\textbf{Exhibit 1: Payback Period Example}$$ $$\begin{array}{c|c|c|c|c} \text{Year} & {0} & {1} & {2} & {3} & {4} \\ \hline \text{Cash flow} & {-5,000}& {1,500} & {3,500} & {4,000} & {4,000} \\ \hline \text{Cumulative cash flow} & {-5,000} & {-3,500} & {0} & {4,000} & { 4,000} \\ \end{array}$$ 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. ## Discounted Payback Period 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. ### Example: Discounted Payback Period 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. ## 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. ## Question Given 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%

Solution

\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*}

Shop CFA® Exam Prep

Offered by AnalystPrep

Featured Shop FRM® Exam Prep Learn with Us

Subscribe to our newsletter and keep up with the latest and greatest tips for success

Sergio Torrico
2021-07-23
Excelente para el FRM 2 Escribo esta revisión en español para los hispanohablantes, soy de Bolivia, y utilicé AnalystPrep para dudas y consultas sobre mi preparación para el FRM nivel 2 (lo tomé una sola vez y aprobé muy bien), siempre tuve un soporte claro, directo y rápido, el material sale rápido cuando hay cambios en el temario de GARP, y los ejercicios y exámenes son muy útiles para practicar.
diana
2021-07-17
So helpful. I have been using the videos to prepare for the CFA Level II exam. The videos signpost the reading contents, explain the concepts and provide additional context for specific concepts. The fun light-hearted analogies are also a welcome break to some very dry content. I usually watch the videos before going into more in-depth reading and they are a good way to avoid being overwhelmed by the sheer volume of content when you look at the readings.
Kriti Dhawan
2021-07-16
A great curriculum provider. James sir explains the concept so well that rather than memorising it, you tend to intuitively understand and absorb them. Thank you ! Grateful I saw this at the right time for my CFA prep.
nikhil kumar
2021-06-28
Very well explained and gives a great insight about topics in a very short time. Glad to have found Professor Forjan's lectures.
Marwan
2021-06-22
Great support throughout the course by the team, did not feel neglected
Benjamin anonymous
2021-05-10
I loved using AnalystPrep for FRM. QBank is huge, videos are great. Would recommend to a friend
Daniel Glyn
2021-03-24
I have finished my FRM1 thanks to AnalystPrep. And now using AnalystPrep for my FRM2 preparation. Professor Forjan is brilliant. He gives such good explanations and analogies. And more than anything makes learning fun. A big thank you to Analystprep and Professor Forjan. 5 stars all the way!
michael walshe
2021-03-18
Professor James' videos are excellent for understanding the underlying theories behind financial engineering / financial analysis. The AnalystPrep videos were better than any of the others that I searched through on YouTube for providing a clear explanation of some concepts, such as Portfolio theory, CAPM, and Arbitrage Pricing theory. Watching these cleared up many of the unclarities I had in my head. Highly recommended.