# Net Present Value

The net present value (NPV) is used in finance and economics to describe or measure today´s value of a stream of future payoffs or payments.

Many decisions that we take will not only have immediate (financial) consequences, but will also lead to future expenses or revenues. For example, if you take out a loan or a mortgage you receive a one-time payment today from your bank and in return you will have to pay back the loan plus interest in monthly installments. An insurance company will receive monthly contributions from their clients but has to pay in case of an accident. If a taxi driver buys a new car, he or she will have pay for the car today but receive a monthly income in the future by using the car to transport clients. Whenever decisions lead to future streams of payments, the NPV can be used as an instrument or measure to help decide whether or not to invest in a certain project or which of several alternatives should be chosen. That is, when the bank decides whether to grant a loan they will need to compare the value of the loan that you receive today, to the sum of monthly instalments it will receive to take this decision. To do so the bank has to determine today´s value of future payments. This is also often referred to as the time value of money. The NPV is a rather simple formula that allows us to compare future expenses or revenues to current expenses and revenues. Consequently, companies that have to choose between several projects they may invest in can use the NPV to determine if an investment is potentially profitable or not and which of several alternatives is the best one.

The net present value of a stream of payments is defined as

$NPV = \sum_{t=0}^{T} \frac{R_{t}}{(1+i)^{t}},$ where $$T$$ is the total number of payments, $$i$$ denotes the interest rate and $$R_{t}$$ is the payment at time $$t$$. Let us look at an example to understand how this formula can be applied. Suppose a firm is considering to invest into a new technology which would require an initial payment $$R_{0}$$ of $$30,000$$. This technology is assumed to increase sales and generate a yearly extra revenue equal to $$11,000$$ for the next 3 years. The interest rate on the capital market that could be earned if the money was kept in a bank account is 3 $NPV = -30,000+\frac{11,000}{1.03}+\frac{11,000}{1.03^{2}}+\frac{11,000}{1.03^{3}}$ $NPV = -30,000 + 31,113.7 = 1,113.7$

The net present value compares the total expenses of the project (30,000) to the sum of the future revenues (11,000 in 3 consecutive years) taking into the interest rate and thereby measuring future payments in current terms. For example, at an interest rate of 3% €11,000 in one year is equivalent to €10,680 today, because if I deposit €10,680 in a bank account today and receive 3% interest per year, I will have €11,000 in one year. The interest rate (or discount factor) helps us to compare money at different points of time, because €100 today is not the same as €100 next year. Think about lending a friend €100 today. If he or she asks you whether you would like to have your money back tomorrow or in one year, what would you answer? Most of us would clearly say that we want the money back by tomorrow. Money that you have today is worth more than money that you will obtain in the future for several reasons: First, you could invest the money and gain interest (e.g., by depositing the money in your bank account). Second, inflation decreases the value of money and the amount of goods and services that you can buy with € 100 in a year's time is not the same as the amount you can buy today. Third, uncertainty plays an important role, because within a year many things can happen. For instance, your friend may lose his or her job and be unable to pay you back. For all of these reasons you would most likely only lend a greater sum if you are compensated for the risk you are taking. That is, you would ask to be paid back with interest so that the value of 100 € in one year is at least the same as the value of € 100 today.

How is the NPV used? If the NPV of an investment is positive, this means that the future streams of payments measured in today´s terms exceed the initial investment. A NPV of zero implies that by making the investment today you will neither make a profit nor a loss. Of course, a negative NPV value means that the initial payment exceeds the future income streams. Please note that while for the examples we assumed that you make an initial payment and receive a future income stream, the NPV can equally be used to measure the opposite. That is, you can also calculate the NPV of a payment that you receive today and compare it to the future expenses (e.g. paying back a loan in monthly instalments).

The NPV is applied by managers or financial analysts to decide whether a firm should invest in a certain project. For the simple example given above, should the firm invest in the new technology? The answer is yes, because the NPV is positive. In general, it is recommended to invest in a project if the NPV is positive and to reject a project if the NPV is negative. Sometimes, firms do not decide between investing into a certain project or not, but they have to choose between several alternative projects. In this case the firms can compare the NPV of the different projects and choose the project(s) with the highest NPV.

The NPV is fairly simplistic and decisions about investments are usually based on several criteria and not only the NPV of the project. For the NPV to be meaningful it is also important that the interest rate and the future payment stream can be predicted with accuracy. The interest rate often fluctuates and if the project is financed using a variable rate this will affect the NPV. Similarly, future revenues (or expenses) that determine the payments $$R_{t}$$ are often characterized by uncertainty and therefore based on estimations themselves, which may make an exact prediction of the NVP of a project difficult in practice.