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Net present value
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Net present value (NPV) or net present worth (NPW)[1] is defined as the total present value (PV) of a time series of cash flows. It is a standard method for using the time value of money to appraise long-term projects. Used for capital budgeting, and widely throughout economics, it measures the excess or shortfall of cash flows, in present value terms, once financing charges are met.
The discounted cash flow is very similar.
Contents
[hide]
• 1 Formula
• 2 The discount rate
• 3 What NPV Means
• 4 Example
• 5 Common pitfalls
• 6 Alternative capital budgeting methods
• 7 External links
• 8 References

[edit] Formula
Each cash inflow/outflow is discounted back to its present value (PV). Then they are summed. Therefore NPV is the sum of all terms , where
t - the time of the cash flow
i - the discount rate (the rate of return that could be earned on an investment in the financial markets with similar risk.)
Rt - the net cash flow (the amount of cash, inflow minus outflow) at time t (for educational purposes, C0 is commonly placed to the left of the sum to emphasize its role as the initial investment.).
[edit] The discount rate
The rate used to discount future cash flows to their present values is a key variable of this process. A firm's weighted average cost of capital (after tax) is often used, but many people believe that it is appropriate to use higher discount rates to adjust for risk for riskier projects or other factors. A variable discount rate with higher rates applied to cash flows occurring further along the time span might be used to reflect the yield curve premium for long-term debt.
Another approach to choosing the discount rate factor is to decide the rate which the capital needed for the project could return if invested in an alternative venture. If, for example, the capital required for Project A can earn five percent elsewhere, use this discount rate in the NPV calculation to allow a direct comparison to be made between Project A and the alternative. Related to this concept is to use the firm's Reinvestment Rate. Reinvestment rate can be defined as the rate of return for the firm's investments on average. When analyzing projects in a capital constrained environment, it may be appropriate to use the reinvestment rate rather than the firm's weighted average cost of capital as the discount factor. It reflects opportunity cost of investment, rather than the possibly lower cost of capital.
NPV value obtained using variable discount rates (if they are known) with the years of the investment duration better reflects the real situation than that calculated from a constant discount rate for the entire investment duration. Refer to the tutorial article written by Samuel Baker[2] for more detailed relationship between the NPV value and the discount rate.
For some professional investors, their investment funds are committed to target a specified rate of return. In such cases, that rate of return should be selected as the discount rate for the NPV calculation. In this way, a direct comparison can be made between the profitability of the project and the desired rate of return.
To some extent, the selection of the discount rate is dependent on the use to which it will be put. If the intent is simply to determine whether a project will add value to the company, using the firm's weighted average cost of capital may be appropriate. If trying to decide between alternative investments in order to maximize the value of the firm, the corporate reinvestment rate would probably be a better choice.
Using variable rates over time, or discounting "guaranteed" cash flows different from "at risk" cash flows may be a superior methodology, but is seldom used in practice. Using the discount rate to adjust for risk is often difficult to do in practice (especially internationally), and is really difficult to do well. An alternative to using discount factor to adjust for risk is to explicitly correct the cash flows for the risk elements, then discount at the firm's rate.
[edit] What NPV Means
NPV is an indicator of how much value an investment or project adds to the value of the firm. With a particular project, if Ct is a positive value, the project is in the status of discounted cash inflow in the time of t. If Ct is a negative value, the project is in the status of discounted cash outflow in the time of t. Appropriately risked projects with a positive NPV could be accepted. This does not necessarily mean that they should be undertaken since NPV at the cost of capital may not account for opportunity cost, i.e. comparison with other available investments. In financial theory, if there is a choice between two mutually exclusive alternatives, the one yielding the higher NPV should be selected. The following sums up the NPVs in various situations.
If... It means... Then...
NPV > 0 the investment would add value to the firm the project may be accepted
NPV < 0 the investment would subtract value from the firm the project should be rejected
NPV = 0 the investment would neither gain nor lose value for the firm We should be indifferent in the decision whether to accept or reject the project. This project adds no monetary value. Decision should be based on other criteria, e.g. strategic positioning or other factors not explicitly included in the calculation.
However, NPV = 0 does not mean that a project is only expected to break even, in the sense of undiscounted profit or loss (earnings). It will show net total positive cash flow and earnings over its life.
[edit] Example
corporation must decide whether to introduce a new product line. The new product will have startup costs, operational costs, and incoming cash flows over six years. This project will have an immediate (t=0) cash outflow of $100,000 (which might include machinery, and employee training costs). Other cash outflows for years 1-6 are expected to be $5,000 per year. Cash inflows are expected to be $30,000 per year for years 1-6. All cash flows are after-tax, and there are no cash flows expected after year 6. The required rate of return is 10%. The present value (PV) can be calculated for each year:
Year Cashflow Present Value
T=0 -$100,000
T=1 $22,727
T=2 $20,661
T=3 $18,783
T=4 $17,075
T=5 $15,523
T=6 $14,112
The sum of all these present values is the net present value, which equals $8,881.52. Since the NPV is greater than zero, it would be better to invest in the project than to do nothing, and the corporation should invest in this project if there is no alternative with a higher NPV.
The same example in an Excel formulae:
• NPV(rate,net_inflow)+initial_investment
• PV(rate,year_number,yearly_net_inflow)

More realistic problems would need to consider other factors, generally including the calculation of taxes, uneven cash flows, and salvage values as well as the availability of alternate investment opportunities.
[edit] Common pitfalls
• If for example the Ct are generally negative late in the project (e.g., an industrial or mining project might have clean-up and restoration costs), then at that stage the company owes money, so a high discount rate is not cautious but too optimistic. Some people see this as a problem with NPV. A way to avoid this problem is to include explicit provision for financing any losses after the initial investment, that is, explicitly calculate the cost of financing such losses.
• Another common pitfall is to adjust for risk by adding a premium to the discount rate. Whilst a bank might charge a higher rate of interest for a risky project, that does not mean that this is a valid approach to adjusting a net present value for risk, although it can be a reasonable approximation in some specific cases. One reason such an approach may not work well can be seen from the foregoing: if some risk is incurred resulting in some losses, then a discount rate in the NPV will reduce the impact of such losses below their true financial cost. A rigorous approach to risk requires identifying and valuing risks explicitly, e.g. by actuarial or Monte Carlo techniques, and explicitly calculating the cost of financing any losses incurred.
• Yet another issue can result from the compounding of the risk premium. R is a composite of the risk free rate and the risk premium. As a result, future cash flows are discounted by both the risk-free rate as well as the risk premium and this effect is compounded by each subsequent cash flow. This compounding results in a much lower NPV than might be otherwise calculated. The certainty equivalent model can be used to account for the risk premium without compounding its effect on present value.[citation needed]
• If NPV<0, the project should not be immediately rejected. Sometimes companies have to execute an NPV-negative project if not executing it creates even more value destruction.
[edit] Alternative capital budgeting methods
• Payback period: which measures the time required for the cash inflows to equal the original outlay. It measures risk, not return.
• Cost-benefit analysis: which includes issues other than cash, such as time savings.
• Real option method: which attempts to value managerial flexibility that is assumed away in NPV.
• Internal rate of return: which calculates the rate of return of a project without making assumptions about the reinvestment of the cash flows (hence internal).
• modified internal rate of return (MIRR): similar to IRR, but it makes explicit assumptions about the reinvestment of the cash flows. Sometimes it is called Growth Rate of Return.
• Accounting rate of return.
• Extended Net Present Value: One of the drawbacks of Net Present Value is it doesnt consider the Discounting of Future Value or the Time Value of Money. This drawback is eliminated by XNPV where the Future Cashflows are first discounted and then the NPV is calculated.

[edit] External links
• Net present value (NPV) explained in simple terms
[edit] References
1. ^ Lin, Grier C. I.; Nagalingam, Sev V. (2000). CIM justification and optimisation. London: Taylor & Francis, 36. ISBN 0-7484-0858-4.
2. ^ Baker, Samuel L. (2000). "Perils of the Internal Rate of Return". Retrieved on Jan 12, 2007.
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Payback period
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Payback period in business and economics refers to the period of time required for the return on an investment to "repay" the sum of the original investment. For example, a $1000 investment which returned $500 per year would have a two year payback period. It intuitively measures how long something takes to "pay for itself"; shorter payback periods are obviously preferable to longer payback periods (all else being equal). Payback period is widely used due to its ease of use despite recognized limitations, described below.
The expression is also widely used in other types of investment areas, often with respect to energy efficiency technologies, maintenance, upgrades, or other changes. For example, a compact fluorescent light bulb may be described of having a payback period of a certain number of years or operating hours (assuming certain costs); here, the return to the investment consists of reduced operating costs. Although primarily a financial term, the concept of a payback period is occasionally extended to other uses, such as energy payback period (the period of time over which the energy savings of a project equal the amount of energy expended since project inception); these other terms may not be standardized or widely used.
Payback period as a tool of analysis is often used because it is easy to apply and easy to understand for most individuals, regardless of academic training or field of endeavour. When used carefully or to compare similar investments, it can be quite useful. As a stand-alone tool to compare an investment with "doing nothing", payback period has no explicit criteria for decision-making (except, perhaps, that the payback period should be less than infinity).
The payback period is considered a method of analysis with serious limitations and qualifications for its use, because it does not properly account for the time value of money, risk, financing or other important considerations such as the opportunity cost. Whilst the time value of money can be rectified by applying a weight average cost of capital discount, it is generally agreed that this tool for investment decisions should not be used in isolation. Alternative measures of "return" preferred by economists are net present value and internal rate of return. An implicit assumption in the use of payback period is that returns to the investment continue after the payback period. Payback period does not specify any required comparison to other investments or even to not making an investment.
[edit] Basic formula
Payback = Days/Weeks/Months x Initial Investment / Total cash received
Internal rate of return
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This article does not cite any references or sources.
Please help improve this article by adding citations to reliable sources. Unverifiable material may be challenged and removed. (June 2007)
The internal rate of return (IRR) is a capital budgeting metric used by firms to decide whether they should make investments. It is an indicator of the efficiency or quality of an investment, as opposed to net present value (NPV), which indicates value or magnitude.
The IRR is the annualized effective compounded return rate which can be earned on the invested capital, i.e., the yield on the investment.
A project is a good investment proposition if its IRR is greater than the rate of return that could be earned by alternate investments of equal risk (investing in other projects, buying bonds, even putting the money in a bank account). Thus, the IRR should be compared to any alternate costs of capital including an appropriate risk premium.
In general, if the IRR is greater than the project's cost of capital, or hurdle rate, the project will add value for the company.
In the context of savings and loans the IRR is also called effective interest rate.
Contents
[hide]
• 1 Method
o 1.1 Example
• 2 Graph of NPV as a function of r for the example
• 3 Problems with using internal rate of return (IRR)
• 4 Mathematics
• 5 See also
• 6 References
• 7 External links
• 8 Further reading

[edit] Method
Given a collection of pairs (time, cash flow) involved in a project, the internal rate of return follows from the net present value as a function of the rate of return. A rate of return for which this function is zero is an internal rate of return.
Thus, in the case of cash flows at whole numbers of years, to find the internal rate of return, find the value(s) of r that satisfies the following equation:
Note that instead of converting to the present we can also convert to any other fixed time; the value obtained is zero if and only if the NPV is zero.
In the case that the cash flows are random variables, such as in the case of a life annuity, the expected values are put into the formula.
[edit] Example
Calculate the internal rate of return for an investment of 100 value in the first year followed by returns over the following 4 years, as shown below:
Year Cash Flow
0 -100
1 40
2 59
3 55
4 20

Solution:
We use an iterative solver to determine the value of r that solves the following equation:
The result from the numerical iteration is .
[edit] Graph of NPV as a function of r for the example

This graph shows the changing of NPV in relation to r (labelled 'i' in the graph)

[edit] Problems with using internal rate of return (IRR)
As an investment decision tool, the calculated IRR should not be used to rate mutually exclusive projects, but only to decide whether a single project is worth investing in.

NPV vs discount rate comparison for two mutually exclusive projects. Project 'A' has a higher NPV (for certain discount rates), even though its IRR (=x-axis intercept) is lower than for project 'B' (click to enlarge)
In cases where one project has a higher initial investment than a second mutually exclusive project, the first project may have a lower IRR (expected return), but a higher NPV (increase in shareholders' wealth) and should thus be accepted over the second project (assuming no capital constraints).
IRR makes no assumptions about the reinvestment of the positive cash flow from a project (if it is reinvested at a lower rate of return, the overall IRR is lower). IRR is best used for projects with singular positive cash flows at the end of the project period. This makes IRR a suitable (and popular) choice for analyzing venture capital and other private equity investments, as these strategies usually require several cash investments, but only see one cash outflow (e.g. via IPO or M&A).
Since IRR does not consider cost of capital, it should not be used to compare projects of different duration. Modified Internal Rate of Return (MIRR) does consider cost of capital and provides a better indication of a project's efficiency in contributing to the firm's discounted cash flow.
In the case of positive cash flows followed by negative ones (+ + - - -) the IRR is a rate for lending/owing money, so the lowest IRR is best. This applies for example when a customer makes a deposit before a specific machine is built.
In a series of cash flows like (-10, 21, -11), one initially invests money, so a high rate of return is best, but then receives more than one possesses, so then one owes money, so now a low rate of return is best. In this case it is not even clear whether a high or a low IRR is better. There may even be multiple IRRs for a single project, like in the example 0% as well as 10%. Examples of this type of project are strip mines and nuclear power plants, where there is usually a large cash outflow at the end of the project.
In general, the IRR can be calculated by solving a polynomial equation. Sturm's theorem can be used to determine if that equation has a unique real solution. In general the IRR equation cannot be solved analytically but only iteratively.
A potential shortcoming of the IRR method is that it does not take into account that the intermediate positive cash flows possibly come at inconvenient moments. Their reinvestment may have a lower yield. In that case it may be more realistic to compute the IRR of the project including the reinvestments until e.g. the end date of the project.[1] Accordingly, MIRR is used, which has an assumed reinvestment rate, usually equal to the project's cost of capital.
Despite a strong academic preference for NPV, surveys indicate that executives prefer IRR over NPV. Apparently, managers find it easier to compare investments of different sizes in terms of percentage rates of return than by dollars of NPV. However, NPV remains the "more accurate" reflection of value to the business. IRR, as a measure of investment efficiency may give better insights in capital constrained situations. However, when comparing mutually exclusive projects, NPV is the appropriate measure.
[edit] Mathematics
Mathematically the value of the investment is assumed to undergo exponential growth or decay according to some rate of return (any value greater than -100%), with discontinuities for cash flows, and the IRR of a series of cash flows is defined as any rate of return that results in a net present value of zero (or equivalently, a rate of return that results in the correct value of zero after the last cash flow).
Thus internal rate(s) of return follow from the net present value as a function of the rate of return. This function is continuous. Towards a rate of return of -100% the net present value approaches infinity with the sign of the last cash flow, and towards a rate of return of positive infinity the net present value approaches the first cash flow (the one at the present). Therefore, if the first and last cash flow have a different sign there exists an internal rate of return. Examples of time series without an IRR:
• Only negative cash flows - the NPV is negative for every rate of return.
• (-1, 1, -1), rather small positive cash flow between two negative cash flows; the NPV is a quadratic function of 1/(1+r), where r is the rate of return, or put differently, a quadratic function of the discount rate r/(1+r); the highest NPV is -0.75, for r = 100%.
In the case of a series of exclusively negative cash flows followed by a series of exclusively positive ones, consider the total value of the cash flows converted to a time between the negative and the positive ones. The resulting function of the rate of return is continuous and monotonically decreasing from positive infinity to negative infinity, so there is a unique rate of return for which it is zero. Hence the IRR is also unique (and equal). Although the NPV-function itself is not necessarily monotonically decreasing on its whole domain, it is at the IRR.
Similarly, in the case of a series of exclusively positive cash flows followed by a series of exclusively negative ones the IRR is also unique.
• Extended Internal Rate of Return: The Internal rate of return calculates the rate at which the investment made will generate cash flows. This method is convenient if the project has a short duration, but for projects which has an outlay of many years this method is not practical as IRR ignores the Time Value of Money. To take into consideration the Time Value of Money Extended Internal Rate of Return was introduced where all the future cash flows are first discounted at a discount rate and then the IRR is calculated. This method of calculation of IRR is called Extended Internal Rate of Return or XIRR.
[edit] See also
• Accounting
• Capital budgeting
• Cost of capital
• Finance
• Net present value
• Discounted cash flow
[edit] References
1. ^ Internal Rate of Return: A Cautionary Tale
[edit] External links
• Internal rate of return (IRR) in simple details
• Using Excel to find IRR
• Using the web to find IRR

*from wikipedia