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To calculate IRR using the formula, one would set NPV equal to zero and solve for the discount rate r, which is here the IRR

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The  internal rate of return (IRR) represents the interest rate in which the  net present value (NPV) of a project's expected total cash flows, both positive and negative, sum to zero. The IRR of a project is used as a  benchmark; if the IRR of a specific project is higher than a company's required rate of return

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Net Present Value(NPV) is a formula used to determine the present value of an investment by the discounted sum of all cash flows received from the project

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The internal rate of return (IRR) is frequently used by corporations to compare and decide between capital projects, but it can also help you evaluate certain financial events in your own life, like lotteries and investments.

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- Investment costs are not the same
- Projected returns are not the same

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IRR is a very popular metric in estimating a project’s profitability, it can be misleading if used alone. Depending on the initial investment costs, a project may have a low IRR but a high NPV, meaning that while the pace at which the company sees returns on that project may be slow, the project may also be adding a great deal of overall value to the company.
A similar issue arises when using IRR to compare projects of different lengths.

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The Internal Rate of Return
The IRR, or internal rate of return, is defined as the discount rate that makes NPV = 0. Like the NPV process, it starts by identifying allCASHinflows and outflows. However, instead of relying on external data (i.e. a discount rate), the IRR is purely a function of the inflows and outflows of that project. The IRR rule states that projects or investments are accepted when the project's IRR exceeds a hurdle rate. Depending on the application, the hurdle rate may be defined as the weighted average cost of capital.

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HBR - A Refresher on Internal Rate of Return

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HBR - A Refresher on Internal Rate of Return

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https://hbr.org/2016/03/a-refresher-on-internal-rate-of-return

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What is 'Net Present Value - NPV'

Net Present Value (NPV) is the difference between the present value of cash inflows and the present value of cash outflows. NPV is used in capital budgeting to analyze the profitability of a projected investment or project.

The following is the formula for calculating NPV:

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Number of years to recover the investment

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Number of years to recover the investment

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Quick information on the number of years that it takes to recover the initial investment in the project.

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The management of Fine Electronics Company is considering to purchase an equipment to be attached with the main manufacturing machine. The equipment will cost $6,000 and will increase annual cash inflow by $2,200. The useful life of the equipment is 6 years. After 6 years it will have no salvage value. The management wants a 20% return on all investments.

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A positive net present value indicates that the projected earnings generated by a project or investment (in present dollars) exceeds the anticipated costs (also in present dollars). Generally, an investment with a positive NPV will be a profitable one and one with a negative NPV will result in a net loss. This concept is the basis for the Net Present Value Rule, which dictates that the only investments that should be made are those with positive NPV values.

When the investment in question is an acquisition or a merger, one might also use the Discounted Cash Flow (DCF) metric.

Apart from the formula itself, net present value can often be calculated using tables, spreadsheets such as Microsoft Excel or Investopedia’s own NPV calculator.

BREAKING DOWN 'Net Present Value - NPV'

Determining the value of a project is challenging because there are different ways to measure the value of future cash flows. Because of the time value of money (TVM), money in the present is worth more than the same amount in the future. This is both because of earnings that could potentially be made using the money during the intervening time and because of inflation. In other words, a dollar earned in the future won’t be worth as much as one earned in the present.

The discount rate element of the NPV formula is a way to account for this. Companies may often have different ways of identifying the discount rate. Common methods for determining the discount rate include using the expected return of other investment choices with a similar level of risk (rates of return investors will expect), or the costs associated with borrowing money needed to finance the project.

For example, if a retail clothing business wants to purchase an existing store, it would first estimate the future cash flows that store would generate, and then discount those cash flows (r) into one lump-sum present value amount of, say $500,000. If the owner of the store were willing to sell his or her business for less than $500,000, the purchasing company would likely accept the offer as it presents a positive NPV investment. If the owner agreed to sell the store for $300,000, then the investment represents a $200,000 net gain ($500,000 - $300,000) during the calculated investment period. This $200,000, or the net gain of an investment, is called the investment’s intrinsic value. Conversely, if the owner would not sell for less than $500,000, the purchaser would not buy the store, as the acquisition would present a negative NPV at that time and would, therefore, reduce the overall value of the larger clothing company.

Let's look at how this example fits into the formula above. The lump-sum present value of $500,000 represents the part of the formula between the equal sign and the minus sign. The amount the retail clothing business pays for the store represents Co. Subtract Co from $500,000 to get the NPV: if Co is less than $500,000, the resulting NPV is positive; if Co is more than $500,000, the NPV is negative and is not a profitable investment.

Drawbacks and Alternatives

One primary issue with gauging an investment’s profitability with NPV is that NPV relies heavily upon multiple assumptions and estimates, so there can be substantial room for error. Estimated factors include investment costs, discount rate and projected returns. A project may often require unforeseen expenditures to get off the ground or may require additional expenditure at the project’s end.

Additionally, discount rates and cash inflow estimates may not inherently account for risk associated with the project and may assume the maximum possible cash inflows over an investment period. This may occur as a means of artificially increasing investor confidence. As such, these factors may need to be adjusted to account for unexpected costs or losses or for overly optimistic cash inflow projections.

Payback period is one popular metric that is frequently used as an alternative to net present value. It is much simpler than NPV, mainly gauging the time required after an investment to recoup the initial costs of that investment. Unlike NPV, the payback period (or “payback method”) fails to account for the time value of money. For this reason, payback periods calculated for longer investments have a greater potential for inaccuracy, as they encompass more time during which inflation may occur and skew projected earnings and, thus, the real payback period as well.

Moreover, the payback period is strictly limited to the amount of time required to earn back initial investment costs. As such, it also fails to account for the profitability of an investment after that investment has reached the end of its payback period. It is possible that the investment’s rate of return could subsequently experience a sharp drop, a sharp increase or anything in between. Comparisons of investments’ payback periods, then, will not necessarily yield an accurate portrayal of the profitability of those investments.

Internal rate of return (IRR) is another metric commonly used as an NPV alternative. Calculations of IRR rely on the same formula as NPV does, except with slight adjustments. IRR calculations assume a neutral NPV (a value of zero) and one instead solves for the discount rate. The discount rate of an investment when NPV is zero is the investment’s IRR, essentially representing the projected rate of growth for that investment. Because IRR is necessarily annual – it refers to projected returns on a yearly basis – it allows for the simplified comparison of a wide variety of types and lengths of investments.

For example, IRR could be used to compare the anticipated profitability of a 3-year investment with that of a 10-year investment because it appears as an annualized figure. If both have an IRR of 18%, then the investments are in certain respects comparable, in spite of the difference in duration. Yet, the same is not true for net present value. Unlike IRR, NPV exists as a single value applying the entirety of a projected investment period. If the investment period is longer than one year, NPV will not account for the rate of earnings in way allowing for easy comparison. Returning to the previous example, the 10-year investment could have a higher NPV than will the 3-year investment, but this is not necessarily helpful information, as the former is over three times as long as the latter, and there is a substantial amount of investment opportunity in the 7 years' difference between the two investments.

Interested in more information on Net Present Value? See: Time Value of Money: Determining Your Future Worth and our Introduction To Corporate Valuation Methods. For more on the relationship between NPV, IRR and associated terms, see the section of our Guide to Corporate Finance called "Net Present Value and Internal Rate of Return."

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