Think & Done

I will provide an NPV formula to calculate net present value of a perpetuity, annuity, project, future cash flows and loan illustrated with NPV Calculation. Net present value is the sum of discounted future cash flows brought down to reflect their worth as of present day. Any amount of money that is offered to be paid to you in future has lower worth than it's face value. The decline in money's face value may be attributed to interest rate and inflation.

NPV Calculator

Here you will find an online NPV calculator that calculates net present value given that you provide the series of cash flows and the discount rate

NPV of Perpetuity

A perpetuity is defined as a never ending stream of payments or receipts. A classic example of perpetuity is that of a perpetual bond that never matures and makes periodic interest payments for an indefinite time period. You can opt to sell the bond at a future date at a price that is discounted at the market rate of return. To find the present value of the perpetuity, you have to divide the perpetuity amount by the discount rate. Yet if you were considering the net present value of perpetuity that considered the price you paid for the perpetuity then you would have to subtract the present value of perpetuity from the price you paid for it. Let me illustrate this with a example calculation for a perpetuity that you purchased for a $100 and it promised to pay 9% annual interest for indefinite period of time. See the following calculations for present value and net present value of this perpetuity.

PV of Perpetuity = Payment/Interest rate = $10/9% = 10/0.09 = $111
NPV of Perpetuity = -Cost + Payment/Interest rate = -$100 + $10/9% = -100 + 10/0.09 = -100 + 111.11 = $11.11

NPV of an annuity

An annuity is a series of periodic payments or receipts for a fixed or definite period of time. Annuity payments or receipts may occur at either start of period or end of period. Payments in to a pension fund, provident fund, 401K plan, or a savings account require you to make start of period payments; this sort of an annuity is referred to as an annuity due. Mortgage payments and loan repayments require you to make end of period payments and this sort of an annuity is referred to as an ordinary annuity. To find net present value of an annuity, we discount each of payments at an interest rate for the time period t where t ranges from 1 to n for an ordinary annuity and from 0 to n-1 for an annuity due. If the annuity payments or receipts are in constant amounts we can use the shortened formula listed below to find the NPV of an annuity with constant or uniform series of cash flows

Annuity NPV Formula

The first of these formula is for finding NPV of an ordinary annuity where payments or receipts in constant amounts are expected at the end of the period. An example of this type of receipt would be a payment from a pension fund at the end of each month or a payment for a home mortgage at the end of each month or quarter.
NPV = R [1 - {1/(1+i%)^n}]/i%
Here we discount each of the payments or receipts from the annuity at the interest rate, for the time period starting at 1 and ending at n.

Annuity Due NPV Formula

Here each payment or receipt is discounted at interest rate i for time period t-1

Annuity NPV Example

We will illustrate finding net present value of an ordinary annuity by showing you detailed calculation for a hypothetical lottery prize winner who has won 20 million dollars. The state authorities has offered the prize winner two options. The first option is to accept an annual payment of one million dollar for each of the next twenty years starting at the end of this current year. The second option is to take a lump sum payment of 12 million dollars on the spot.

Annuity NPV Calculation

So how would the prize winner decide whether to accept a lump sum payment or to accept annual receipts of one million dollars for the next twenty years. We will show you how the prize winner can make his choice. The winner has to first determine his or her opportunity cost which may be the interest rate offered by a local bank for opening a savings account. Thus the interest rate is taken as the winners opportunity cost that he or she is willing to let go when accepting a lump sum payment. Let us assume an interest rate of 12% compounded annually, see the following table of values that shows the schedule for present value of one million dollars over the next twenty years discounted at 12%. If the net present value of this sum is less than 12 million dollars then the winner is better off taking the lump sum payment of 12 million dollars instead of an annual payment of one million dollars over the next twenty years

Annuity NPV at 12%

Lottery Winner's Choice

As the calculation show that one million dollars paid each year over the next twenty years when discounted at 12% are worth only seven and half million dollars today. Thus the prize winner is better off selecting a lump sum payment of twelve million dollars instead.

Annuity versus Annuity Due

For the sake of complete argument, let us consider the same annuity with start of period payments. The prior example assumed the first payment of lottery prize was due at the end of the year yet in real life a prize winner would get her first payment right after winning the prize and processing of necessary legal documents. Here the net present value will be higher than what we just calculated. The reason behind this is the fact that with start of period payments the first payment is not discounted as it is made at present. I won't overburden you with another lenghty schedule of payments and their discounted values for the calculation of NPV on annuity due but instead summarize the schedule so you be able to see the answer for net present value of the lottery prize winnings when payments start immediately. See the following table for more details.

NPV of Annuity Calculation (Alternative Formula)

I illustrated NPV Calculation using the complete payment schedule yet if the number of payments are large this way of finding net present value can get tedious and waste of time. Instead you can use a short form NPV formula for annuity when payments are in constant amounts. Staying with the lottery prize winner's example, the following shows NPV calculation using a shortend mathematical formula:

Effect of continuous compounding

The last example assumed that interest is compounded per period in this case 12% is compounded per year. Yet in good number of situations in finance and banking, the interest rate is compounded continuously. When interest is compounded continuously the formula to calculate net present value of an annuity is different from the one we have used thus far. The difference is attributed to the interest factor. Staying with the lottery example I discussed earlier, if interest of 12% were to be compounded continuously the net present value of the 20 year annuity will be worth less than what it was with discrete compounding of interest. The following table shows the annual payments of the annuity discounted with continuous compounding of interest.

NPV Annuity with continuous compounding of interest

Continuous compounding formula for NPV

Instead of showing a complete schedule of discounted payments for the 20 years for NPV calculation with continuous compounding of interest, we could have instead used the shorthand mathematical formula as follows:

NPV of a Project

NPV or net present value of a project is oen of the DCF methods that is defined as the difference between discounted benefits and discounted costs associated with a project. A positive NPV value is acceptable where as an NPV of zero yields the internal rate of return. A negative value for NPV suggests that investment is not worthy of the money we are about to invest.

NPV Formula

NPV Example

Let us examine finding Net Present Value or NPV with an example investment proposal. Let us say we were offered a series of cash inflows at the end of each of the next four years as $5000, $4000, $3000, and $1000. And the initial cash outlay for this proposal is $10,000 and weighted average cost of capital or WACC is 12%.

NPV Calculation at 12%

NPV Calculation Online

This online tool will perform NPV calculation and will display step by step workout similar to one you have seen in this discussion.

NPV Calculation at 15%

NPV Acceptance criteria

We usually accept a project if it has a positive NPV and one that is highest amongst the projects we are evaluating. This example project results in a NPV of $425 at the weighted average cost of capital of 12%, thus this is the amount in present value terms that we would gain if we were to invest in it. Yet if this company had a cost of capital of 15%, at this rate the project would yield no gains and we would discard the idea of putting our money in it.

NPV with different discount rates

So far the examples we have looked at assumed that interest rates were constant throughout the lifetime of the annuity or the capital budgeting project. Yet in real life, rates vary from period to period. An example where rates change from one year to the next is that of a ARM or adjusted rate mortgage loans. When annuity rates differ from one year to the next, we have to discount each of the net cash flows or the annuity payments at a different interest rate. To find net present value with different discount rates, we have to create the complete annuity payment schedule with discounted cash flows for each time period. There is no short hand mathematical formula that we can use to find NPV with different discount rates. Here we calculate discounted amounts individually and then find the net present value as a sum of the discounted cash flows. I will illustrate this sort of NPV calculation for the lottery prize winner example yet the same method applies for ordinary annuity where we start the discounting process with the initial cash flow. The following schedule shows NPV calculation for annuity due where the interest rates change from one year to the next.

Frequently Asked Questions

Here I will attempt to answer some of the frequently asked questions pertaining to net present value. This will provide you with an ample material to discern the topic in detail.

Q: What is NPV in finance?

A: NPV refers to the net worth of an investment that results when you lessen sum of discounted costs from sum of discounted benefits. The resulting difference of costs from benefits represents either a loss or profit from an investment. Yet this result does not represent the money amount in present term. Since face value of money erodes with passage of time, one can only get a fair picture of the amounts due in future is when those amounts are discounted to brought back to it's present value.

Q: What is NPV analysis?

A: NPV analysis refers to the set of procedures we follow to find net present value of an investment. The analysis starts with the steps of finding incremental cash flows. We first use methods to approximate the expected benefits or savings from an investment. Both fixed and variable costs are removed from the expected benefits. Depreciation allowance is taken off this last amount thus resulting in EBIT or earnings before interest and taxes. The tax expense is removed from EBIT and depreciation is added back to find free cash flows. If there were to be any increase in net working capital that gets added to the initial costs and are finally recovered in the terminal cash flow. NPV analysis requires that an organization knows how to compute it's cost of capital also referred to as the opportunity cost. The cost of capital or the WACC is used as the discount rate when finding net present value from the free cash flows. The resulting amount may be either positive, negative or zero. The acceptance criteria for NPV suggests approval of a capital budgeting project with the highest positive net present value.

Q: What is NPV test?

A: The NPV test is used to analyze capital budgeting projects that are undertaken by an organization. The NPV test is based on discounted cash flow analysis where each of the cash flow in the series of future returns from investment is discounted at the cost of capital and it's resulting sum is known as the net present value. The NPV test would then be used in deciding whether the resulting NPV is positive and the highest amongst the projects under consideration. If a project passes the NPV test, the management will give it go ahead otherwise the project is discarded.

Q: What is NPV profile?

A: In capital budgeting, net present value is calculated by summing up the series of discounted cash flows. The discount rate that is used in this process is referred to as the cost of capital. IRR is another method used in capital budgeting and it is the interest rate at which net present value results in zero. The NPV profile refers to a visual presentation of NPV and the discount rate. The NPV is represented vertically on y-axis and the discount rate is represented horizontally on x-axis. The resulting line graph is called the NPV profile. NPV profile shows the IRR as a point of intersection on x-axis where NPV is zero.

Q: What is NPV in financial management?

A: The term NPV is used interchangeably with net present worth. NPV and NPW are almost identical in meaning in terms of financial management yet their meaning may refer to two different concepts. In one context, we may only be referring to the present value of expected future payments or receipts. This case can be highlighted with an example where a lottery prize winner would have to either decide on taking a lump sum of prize money or to take a series of annual payments over time. As there is no outgoing cash flow, the NPV would simply refer to discounting of the series of annual payments to find it's present value. In the second context where an organization takes on capital budgeting projects that incur initial and interim costs, the net present value will refer to the difference between the sum of discounted costs from sum of discounted benefits. Do note that there is no difference in the method used to calculate NPV in both these instances. However one thing to point out is the timing of the expected cash flows. If the cash flow occur at the end of period, for example, first payment is made at the end of year and the subsequent payments are made at the end of the each year in the future, such form of payments will be considered an ordinary annuity or simply an annuity. When payments are made at the start of the period for example first payment due today and the subsequent payments are made at the start of each of the years in to the future, such an annuity is referred to as annuity due.

Q: What are NPV and IRR?

A: NPV and IRR are two popular methods used by organizations in evaluating investments that require capital budgets. These two techniques are used in conjunction, however that said, each of these methods has its pros and cons. IRR is an interest rate that provides an organization with a measure to compare it against it's cost of capital. If the IRR is found to be higher than cost of capital, an organization is likely to accept the project. The NPV on the other hand is the money amount that a company stands to make or lose from an investment. Both methods are widely used and in some instances NPV is seen as a better and superior measure for the reason that in some cases there may not exist an IRR or if does it may not be unique.

Q: What are NPV, IRR and Payback Period?

A In short NPV, IRR and Payback period are three methods or capital budgeting techniques used in evaluating investment projects. I have already discussed the NPV and IRR methods a bit earlier, here I will talk about the payback period. The payback period refers to the time period required to recover the initial cost incurred. The time period may refer to days, weeks, months, quarters, or years depending on the frequency of the cash flows. The point to note here is that payback period does not take into consideration the time value of money thus it is not a true payback period. A true payback period is referred to as discounted payback period where the expected future cash flows are discounted to reflect their present value. The discount rate used to discount the future cash flows is called the cost of capital or the WACC.

Q: What are NPV, IRR and MIRR?

A: NPV, IRR and MIRR are techniques used in evaluating capital budgeting projects. I have already discussed NPV and IRR in previous discussion, here I will attempt to summarily describe the MIRR. MIRR is short for modified internal rate of return; the need for modification arises for the reason that IRR may not exist in some cases or in others it may not be unique. To overcome this difficulty and to still be able to get a rate of return, an adjusted or modified internal rate of return makes possible getting a rate of return. Unlike IRR, MIRR is calculated using a closed form formula which almost always produces a solution when at least one cash flow is negative and at least one cash flow is positive. IRR assumes that the cash flows can only be reinvested at the IRR. This leads to problems as it may not be always possible to reinvest at the IRR that is quite higher than the cost of capital. This issue does not arise with MIRR that assumes that the cash flows can be reinvested at the company's cost of capital.

Q: What does NPV mean?

A: NPV can either mean the present value of a lump sum, the present value of expected payments or the net present value of a capital investment. In case of a lump sum NPV would simply be the present value of the lump sum discounted at the interest rate i for n periods. In case of a series of periodic payments, NPV would mean the sum of discounted periodic payments at interest rate i for time period t where t ranges from 1 to n. In case of capital budgeting, NPV is the difference of sum of discounted costs and sum of discounted benefits. So depending on the context it's used in, NPV can mean either of the 3 cases I listed. It may also be that NPV refers to the net present value of an infinite stream of payments or receipt usually referred to as a perpetuity.

Q: What does NPV stand for?

A: NPV stands for net present value. It refers to future sums of money discounted at interest rate i for n periods to reflect such amounts in it's present value.

Q: What does NPV tell me?

A: NPV can tell you the true worth of any money you are promised in the future. Using NPV, you can find the true reflection of money that you either would have to pay or someone promises to pay you. NPV is the vehicle that eliminates effect of inflation and interest rates from any sum of money promised at a later date.

Q: What does NPV of zero mean?

A: A NPV of zero means that you or an organization that plans to invest in a capital project stands to make nothing from the investment. You only break even at an NPV of zero thus no gains are made from the investment. An investment is only worthwhile if it's NPV is a positive amount and the higher the NPV more sense it makes to invest.

Q: What does NPV show?

A: NPV shows the actual value of an investment. It provides you with a yard stick that lets you measure future investments in it's present value. A dollar promised to be paid to you in the future would not buy the same amount of goods as a dollar can buy you today. Thus time value of money dictates that NPV represents the actual worth of money that is due in the future.

Q: Why NPV is better?

A: NPV is by far a better method to evaluate capital investments as compared to the other methods. The main reason NPV is preferred is the fact that one can always calculate NPV given the discount rate. Compare this to the IRR which may not always exist or it may not be unique. NPV provides an investor with a money amount that an organization tends to gain or lose by investing in a project. Other measures such as payback period provide time periods as compared to money amounts.

Q: Why NPV over IRR?

A: This question was partially answered in the last discussion. The main advantage NPV has over IRR is that one is always able to calculate it with a discount rate whereas IRR may not yield a value in certain instances or there may be multiple IRR values. Another issue that renders IRR useless is when there is a ranking issue with the cash flows and comparative projects have opposing values for NPV and IRR. One project may have higher IRR and lower NPV whereas the other project may have lower IRR and higher NPV. In case of such a conflict one can always make a decision about the investment solely based on NPV. The same holds true with projects with different lives, in such cases IRR does not provide any help yet we can alter the behavior of NPV with either replacement chain NPV or the annual annuity method to make the decision.

Q: Why NPV is important?

A: NPV is important to an organization that is about to undertake a capital budgeting project since the organization will be able to judge how much of its capital investment will make a return on investment. And a money figure as a representative of ROI is more important than a interest rate.

Q: Why NPV and IRR conflict?

A: NPV and IRR conflict may arise when the timings of cash flows are not at par with each other whilst comparing more than one project. As discussed earlier this leads to conflicting results for IRR and NPV where one project seems to have a lower IRR and higher NPV whereas another project has a higher IRR and lower NPV. In such a conflict making a decision based on NPV is more reliable as compared to making one based on the IRR.

Q: Why NPV is best?

A: NPV is best and beats the rest. That is so true when comparing mutually exclusive projects or when budget rationing is the option due to scarce resources an organization has at it's disposal. IRR method may not result in a solution in certain cases or it may have multiple solutions. MIRR like IRR is a rate of interest yet NPV provides a money amount that is either made or lost if an organization were to invest in a capital project. As most organizations would keep an eye on the bottom line thus they will be more inclined to approve of a profitable investment when NPV is positive.

Related DCF analysis methods

1. Internal Rate of Return
2. Modified Internal Rate of Return
3. Profitability index
4. Payback period
5. Discounted payback period