# Net Present Value

## What is Net Present Value?

The basic financial concept of time value money states that the money you have known is more valuable than the money you collect later on. This is because you can use it now to earn more money by running a business or buying something now and selling it later for more, or simply putting it in the bank and earning more interest. The money received in the future is also less valuable because inflation erodes its purchasing power. But how do you compare the value of money now with the value of money in the future? This is where net present value plays an important role. Let us discuss what net present value is.

Net Present Value or NPV is the sum of the present value of cash inflows and outflows. In other words, it is the difference between the present values of cash inflows and the present value of cash outflows over some time.

### Net Present Value Formula

NPV is a strong approach to determine if the project is profitable or not. It considers the interest rate, which is generally equivalent to the inflation rate, Hence, the real value of money now at each year of operation is considered.

Following are the formulas used to calculate NPV.

In case of even cash flows, the following NPV formula can be used:

$NPV = R \times \frac{1-(1+i)^{-n}}{i} - Initial \, Investment$

• Here, n is the total life of the project in months, years, etc.

• i is the required rate of return per period.

• R is the estimated periodic net cash flows.

In the case of even cash flows, the following NPV formula can be used:

$NPV = \int_{i=1}^{n} \frac{R}{(1+i)^{n}} - Initial \, Investment$

• Here, R is the assumed cash flows of the investment for the ith period

• i is the required rate of return per period.

• n is the total life of the project in months, years, etc.

NPV can also be calculated as:

NPV = Present Value of expected cash flows - Present value of cash invested.

### NPV Decision Rule

The following NPV signs explain whether the investment is good or bad.

• NPV > 0 - The present value of cash inflows is more than the present value of cash outflows. The money earned on the investment is more than the money invested. Hence, it is a good investment.

• NPV = 0 - The present value of cash flows is more than the present value of cash outflows. The money earned on the investment is equal to the money invested. Therefore, there is no difference between cash inflows and cash outflows.

• NPV < 0 - The present value of cash inflows is less than the present value of cash outflows. The money earned on the investment is less than the money invested. Hence, it is not a fruitful investment.

Following are the NPV decisions which can be made by looking at the above NPV signs:

In the case of the standalone project, accept the project if NPV is positive or greater than 0, reject a project if NPV is negative or less than 0, and stay indifferent between accepting or rejecting the project if NPV is 0. In the case of competing projects (mutually exclusive projects), accept the project with greater NPV.

### Role of NPV

Net present value (NPV) is the difference between the present value of an investment and the cost resulting from an investment. The points given below define the role of NPV accurately.

• A positive NPV indicates that the investor’s financial position will be improved by undertaking a project.

• A negative NPV indicates the financial loss of an investor.

• Null or zero NPV indicates that the present value of all the benefits over useful time is equivalent to the present value of cost.

### How to Calculate Net Present Value Example?

As we know, money is worth more than it is later. For example, ＄1000 dollar today is worth more than ＄1000 in three years. This is because you can take ＄1000 today, and invest it at a rate of 4% each year. In three years, ＄1000 will be worth ＄1124.86. It means the present value of investment ＄1000 will be ＄1124.86 after 3 years without considering the inflation rate.

The most important factor that should be considered is the dynamic inflation rate. If you will not invest your money, your ＄1000 will be ＄915.14 in three years. These numbers can be calculated by using the following present value formula.

Present Value = (Future Value)/(1 + r)n

Here,

• r is the interest rate.

• n is the number of years.

With this, we can easily calculate NPV by adding and subtracting all the present values:

• Subtract all the present value that you pay.

Let us now understand net present value calculation examples to understand the concept appropriately.

### Net Present Value Calculation Example

One of your friends needs ＄500 now and promised to pay you back ＄500 in a year. Is that a fruitful investment when you can invest at 10% elsewhere?

Solution:

Money Invested Now = ＄500

So PV = -＄500

Money Received After a Year = ＄570

$So, PV = \frac{FV}{(1+r)^{n}}$

$PV = \frac{570}{(1+0.10)^{1}}$

$PV = \frac{570}{1.10^{1}} =  518.1$

Net Present Value = ＄518.18 - ＄500 = ＄18.18

Therefore, at 10%, the investment is worth ＄18.18.

In other words, it states that ＄18.18 is better than a 10% investment in today’s value of money.

### Net Present Values Problems With Solutions

Let us understand a few net value problems to understand the concept precisely.

1. Suppose a project requires an initial investment of ＄2000 and it is expected to generate a cash flow of ＄100 for 3 years plus ＄12500 in the third year. The target rate of return of the project is 10% per annum. Calculate the net present value of the project.

Solution:

Money Invested Now = ＄2000

So, PV now = - ＄2000

$Year \, 1: PV = \frac{FV}{(1+r)^{n}} = \frac{100}{(1+0.10)^{1}} = \frac{100}{1.10^{1}} = 90.91$

$Year \, 2: PV = \frac{FV}{(1+r)^{n}} = \frac{100}{(1+0.10)^{2}} = \frac{100}{1.10^{2}} = 82.64$

$Year \, 3: PV = \frac{FV}{(1+r)^{n}} = \frac{100}{(1+0.10)^{3}} = \frac{100}{1.10^{3}} = 75.13$

$Year \, 3 (Final \, Payment) = \frac{FV}{(1+r)^{n}} = \frac{250}{(1+0.10)^{3}} = \frac{250}{1.10^{3}} =  1878.29$

= ＄1878.29

Adding Total Cash Inflows = ＄90.91 + ＄82.64 + ＄75.13 + ＄1878.29 = ＄2126.97

NPV = ＄2126.97 - ＄2000 = ＄126.97

Therefore, NPV of the project at 10% is ＄126.97

It seems like a good investment.

2. Assume that ABC Inc is considering two projects namely Project X and Project Y and wants to calculate the NPV for each project. Both project X and project Y is four-year project and cash flows of both the projects for four years are given below:

 Year Project A Cash Flows Project B Cash Flows 1. ＄5000 ＄1000 2. ＄4000 ＄3000 3. ＄3000 ＄4000 4. ＄1000 ＄6750

The firm's cost of capital is 10% for each project and the initial investment amount is ＄10,000. Calculate the NPV of each project and determine in which project the firm should invest.

Solution:

Following is the calculation of NPV for project X and project Y.

Project X NPV Calculation

Money Invested Now = ＄2000

So, PV now = - ＄10,000

$Year \, 1: PV = \frac{FV}{(1+r)^{n}} = \frac{5000}{(1+0.10)^{1}} = \frac{5000}{1.10^{1}} =  4545.45$

$Year \, 2: PV = \frac{FV}{(1+r)^{n}} = \frac{4000}{(1+0.10)^{2}} = \frac{4000}{1.10^{2}} =  3305.78$

$Year \, 3: PV = \frac{FV}{(1+r)^{n}} = \frac{3000}{(1+0.10)^{3}} = \frac{3000}{1.10^{3}} =  2253.94$

$Year \, 4: PV = \frac{FV}{(1+r)^{n}} = \frac{1000}{(1+0.10)^{4}} = \frac{1000}{1.10^{4}} =  683.01$

Total Cash Inflows = ＄4545.45 + ＄3305.78 + ＄2253.94 + ＄683.01 = ＄2126.97

NPV = ＄10,788 - ＄10000 = ＄788.38

Therefore, NPV of the project X at 10% is ＄788.38

Project Y NPV Calculation

Money Invested Now = ＄10000

So, PV = - ＄10,000

$Year \, 1: PV = \frac{FV}{(1+r)^{n}} = \frac{1000}{(1+0.10)^{1}} = \frac{1000}{1.10^{1}} =  909.09$

$Year \, 2: PV = \frac{FV}{(1+r)^{n}} = \frac{3000}{(1+0.10)^{2}} = \frac{3000}{1.10^{2}} =  2479.33$

$Year \, 3: PV = \frac{FV}{(1+r)^{n}} = \frac{4000}{(1+0.10)^{3}} = \frac{4000}{1.10^{3}} =  3053.43$

$Year \, 4: PV = \frac{FV}{(1+r)^{n}} = \frac{6750}{(1+0.10)^{4}} = \frac{6750}{1.10^{4}} =  4610.34$

Total Cash Inflows = ＄909.09 + ＄2479.33 + ＄3005.25 + ＄4610.34 = ＄11004.01

NPV = ＄11,004.01- ＄10000 = ＄1004.01

Therefore, NPV of the project Y at 10% is ＄1004.01

We can see, the NPV of project Y is greater than the NPV of project X. Hence, the firm should invest in project Y.

As we go through the basic financial concepts of time, value and money, we realize that the money we have now is more valuable than the money we will collect in the future. This is because the money we have right now is the money we will use to invest in various junctures like opening a business, or buying a property now and selling it later for a higher price, or simply just putting that money in the bank and earning interest. The money received in the future is also less valuable as inflation will erode its purchasing power in the future. How we calculate the value of that money and why Net Present Value is a term that will be useful for us in calculating that value is what we will study on this page.

## FAQs on Net Present Value

1. How do the Results of NPV Maximise the Shareholder’s Wealth?

The NPV measures the present value of future cash flows that a project will produce. A positive NPV indicates that the investment will increase the value of the firm and lead to maximizing the shareholder's wealth. A positive NPV provides a return that is more than enough to compensate for the required return on investment. Hence, using NPV as a guideline for capital budgeting decisions is dependable to maximize shareholders' wealth.

2. How will we calculate our Net Present Value in real life?

So, as we all know that the money we have right now is worth more than the money we will have later. For example, if we take $1000 today and the same amount of money in the future, let us say three years, its value would be worth more now. This happens because if we take$1000 today and invest that money at a 4% rate each year, it will be worth $1124.86 in the future. This means the present$1000 will turn into $1124.86, which will be more than$1000 in the future.

The most vital factor we observe by looking at this example is that Dynamic Inflation Rate is something we should always take into consideration. This is because if we do not invest our $1000 right now, due to inflation that$1000 will turn into \$915.14 in the future.

We calculate these amounts using the following formula.

Present Value =  (Future Value)/(1 + r)n

In this case,

We take r as the interest rate.

We take n as the number of years

With the help of this formula, we can also calculate the NPV very easily by adding or subtracting all the present values. This will be done by:

• Subtracting all the present values that we pay.

3. How can the shareholders' wealth increase due to an NPV?

• The Net Present Value (NPV) is a tool that will measure the present value of the future cash flows that a project will produce. In this case, a positive NPV will tell a shareholder that their investment will increase the firm's value and maximize their wealth.

• This positive NPV will provide a return that would be more than the required amount to compensate for the investment return.

• Therefore, using NPV as a guideline for capital budgeting reasons is beneficial for a shareholder’s wealth.

4. Are there any assumptions that the NPV is based on?

Net Present Value mainly assumes that the cash flows received by a person or a business are reinvested at a cost of capital that is more realistic and practical than any other methods of calculating possible future benefits.

The Net Present Value (NPV) is based on two following assumptions:

• The cash that is received by a project is immediately reinvested to get a return that will be equal to the discount rate using product value analysis.

• The cash inflows and cash outflows should be apart from the starting investment that occurs at the end of each period.

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