**FORMULA AND DERIVATION**

Net Present value is called so because it considers the time value of money. A cash flow of \(\$1\) today would be worth \(\$1.1\) next year if it accumulates at \(10\%\) per annum. Or in the reverse sense, \(\$1\) cash flow occurring next year would be worth \(\$0.91\) today. Based on this concept all future cash flows are discounted to their present values at a rate called the risk discount rate (which generally represents a company's cost of capital).

For a single cash flow, \(c\), occurring at time \(t\), the present value at risk discount rate \(i\) is:

\[PV=\frac{c}{{\left(1+i\right)}^t}\]

The present value of a series of cash flows \(c_t\) occurring at times \(t=1,2,3,4,\dots ..\) at risk discount rate \(i\) is:

\[PV=\frac{c_1}{{\left(1+i\right)}^1}+\frac{c_2}{{\left(1+i\right)}^2}+\dots \dots \dots +\frac{c_n}{{\left(1+i\right)}^n}\]
\[PV=\sum^n_{t=1}{\frac{c_t}{{\left(1+i\right)}^t}}\]
\[PV=\sum^n_{t=1}{c_tv^t}\]

where, \(v=\frac{1}{\left(1+i\right)}\)

The Initial investment is made at time 0, and thus its present value is the same as the actual amount of investment. This value is incorporated into the formula by deducting it (since it represents a cash outflow) from the present value of the net cash flows arising on all future Days. So, the formula for NPV is:

\[NPV=\sum^n_{t=0}{c_t}v^t\]

Where,

\(c_0\) is the cash flow arising at time \(t=0\) , i.e. it is the initial investment.

Acceptance criteria:

\(NPV>0\)(Accept: Adds value to the company)

\(NPV<0\) (Reject: Reduces value of the company)

\(NPV=0\) (Further scrutiny: Neither adds nor reduces value).

An NPV greater than zero indicates that the project is profitable and is generating returns in excess of its company cost of capital, making the project a better investment option than other alternatives.

An NPV less than zero indicates that the project is eating away the company's value as its returns are not sufficient to meet the companies cost of capital.

If the NPV of a project is zero, further analysis is required to see if it is profitable compared to other alternative investment opportunities available.

If two or more projects show positive NPV's at a suitable discount rate, the yield of the project may be calculated to see which of the two projects would provide the highest return. Though both projects are profitable, the one with the higher yield would be selected.

Determination of an appropriate discount rate

A company generally uses its weighted average cost of capital as the discount rate so that it enters into projects which would enhance the company's value. In practice, companies tend to use a higher discount rate than the WACC to allow for additional risks that might arise from the project.

Accumulated value

Accumulated value is simply the value of the project at a point of time in the future. This is used to find out the actual amount of value that the project would have generated by the time the project comes to an end.

\[Accumulated\ value=NPV\times {\left(1+i\right)}^n\]

**EXAMPLES**

Example 1

Company X makes an investment of \(\$100000\) on its new project. It would require \(\$20000\) at the end of every year for covering the costs relating to the project. The project would in turn generate cash flows of \(\$70000\) , \(\$60000\) , \(\$50000\) , \(\$40000\) and \(\$30000\) at the end of each of the first 5 years. The company decides to use its weighted average cost of capital of \(15\%\) as the discount rate to evaluate the project.

To calculate the NPV, we will have tos first ascertain the net cash flows every year. This is done by deducting the outflow from each year's inflow. Doing so would give us \(\$50000\), \(\$40000\), \(\$30000\), \(\$20000\) and \(\$10000\).

We would now have to find the present value of each cash flow and sum them up.

\[NPV=-100000+\frac{50000}{{\left(1.15\right)}^1}+\frac{40000}{{\left(1.15\right)}^2}+\frac{30000}{{\left(1.15\right)}^3}+\frac{20000}{{\left(1.15\right)}^3}+\frac{10000}{{\left(1.15\right)}^5}\]
\[\ \ \ \ \ \ \ \ \ =-100000+43478.26+30245.75+19725.49+11435.06+4971.77\]
\[\ \ \ \ \ \ \ \ \ =-100000+109856.3\]
\[\ \ \ \ \ \ \ \ \ =9856.33\]

(The initial investment is deducted because it represents outflow of cash).

Suppose the company chose a risk discount rate of \(20\%\) instead of \(15\%\) it would arrive at an NPV of \(\$469.39\) and at \(25\%\) it would arrive at an NPV of \(\$-7571.2\) (indicating a loss of value). Therefore, it is important that a company chooses the right risk discount rate that covers its cost of capital and the additional risk arising from the project.