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In finance, the **net present value** (**NPV**) or **net present worth** (**NPW**)^{ [1] } applies to a series of cash flows occurring at different times. The present value of a cash flow depends on the interval of time between now and the cash flow. It also depends on the discount rate. NPV accounts for the time value of money. It provides a method for evaluating and comparing capital projects or financial products with cash flows spread over time, as in loans, investments, payouts from insurance contracts plus many other applications.

**Finance** is a field that is concerned with the allocation (investment) of assets and liabilities over space and time, often under conditions of risk or uncertainty. Finance can also be defined as the art of money management. Participants in the market aim to price assets based on their risk level, fundamental value, and their expected rate of return. Finance can be split into three sub-categories: public finance, corporate finance and personal finance.

The **time value of money** is the greater benefit of receiving money now rather than an identical sum later. It is founded on time preference.

- Formula
- The discount rate
- Use in decision making
- Interpretation as integral transform
- Example
- Common pitfalls
- History
- Alternative capital budgeting methods
- See also
- References

Time value of money dictates that time affects the value of cash flows. For example, a lender may offer 99 cents for the promise of receiving $1.00 a month from now, but the promise to receive that same dollar 20 years in the future would be worth much less today to that same person (lender), even if the payback in both cases was equally certain. This decrease in the current value of future cash flows is based on a chosen rate of return (or discount rate). If for example there exists a time series of identical cash flows, the cash flow in the present is the most valuable, with each future cash flow becoming less valuable than the previous cash flow. A cash flow today is more valuable than an identical cash flow in the future^{ [2] } because a present flow can be invested immediately and begin earning returns, while a future flow cannot.

In finance, **return** is a profit on an investment. It comprises any change in value of the investment, and/or cash flows which the investor receives from the investment, such as interest payments or dividends. It may be measured either in absolute terms or as a percentage of the amount invested. The latter is also called the holding period return.

A **time series** is a series of data points indexed in time order. Most commonly, a time series is a sequence taken at successive equally spaced points in time. Thus it is a sequence of discrete-time data. Examples of time series are heights of ocean tides, counts of sunspots, and the daily closing value of the Dow Jones Industrial Average.

Net present value (NPV) is determined by calculating the costs (negative cash flows) and benefits (positive cash flows) for each period of an investment. The period is typically one year, but could be measured in quarter-years, half-years or months. After the cash flow for each period is calculated, the present value (PV) of each one is achieved by discounting its future value (see Formula) at a periodic rate of return (the rate of return dictated by the market). NPV is the sum of all the discounted future cash flows. Because of its simplicity, NPV is a useful tool to determine whether a project or investment will result in a net profit or a loss. A positive NPV results in profit, while a negative NPV results in a loss. The NPV measures the excess or shortfall of cash flows, in present value terms, above the cost of funds.^{ [3] } In a theoretical situation of unlimited capital budgeting a company should pursue every investment with a positive NPV. However, in practical terms a company's capital constraints limit investments to projects with the highest NPV whose cost cash flows, or initial cash investment, do not exceed the company's capital. NPV is a central tool in discounted cash flow (DCF) analysis and is a standard method for using the time value of money to appraise long-term projects. It is widely used throughout economics, finance, and accounting.

**Capital budgeting**, and **investment appraisal**, is the planning process used to determine whether an organization's long term investments such as new machinery, replacement of machinery, new plants, new products, and research development projects are worth the funding of cash through the firm's capitalization structure. It is the process of allocating resources for major capital, or investment, expenditures. One of the primary goals of capital budgeting investments is to increase the value of the firm to the shareholders.

In finance, **discounted cash flow** (**DCF**) analysis is a method of valuing a project, company, or asset using the concepts of the time value of money. All future cash flows are estimated and discounted by using cost of capital to give their present values (PVs). The sum of all future cash flows, both incoming and outgoing, is the net present value (NPV), which is taken as the value of the cash flows in question.

**Economics** is the social science that studies the production, distribution, and consumption of goods and services.

In the case when all future cash flows are positive, or incoming (such as the principal and coupon payment of a bond) the only outflow of cash is the purchase price, the NPV is simply the PV of future cash flows minus the purchase price (which is its own PV). NPV can be described as the "difference amount" between the sums of discounted cash inflows and cash outflows. It compares the present value of money today to the present value of money in the future, taking inflation and returns into account.

A **coupon** payment on a bond is the annual interest payment that the bondholder receives from the bond's issue date until it matures.

In finance, a **bond** is an instrument of indebtedness of the bond issuer to the holders. The most common types of bonds include municipal bonds and corporate bonds.

The NPV of a sequence of cash flows takes as input the cash flows and a discount rate or discount curve and outputs a present value, which is the current fair price. The converse process in discounted cash flow (DCF) analysis takes a sequence of cash flows and a price as input and as output the discount rate, or internal rate of return (IRR) which would yield the given price as NPV. This rate, called the yield, is widely used in bond trading.

The **internal rate of return** (**IRR**) is a measure of an investment’s rate of return. The term *internal* refers to the fact that the calculation excludes external factors, such as the risk-free rate, inflation, the cost of capital, or various financial risks.

In finance, the **yield** on a security is the amount of cash that returns to the owners of the security, in the form of interest or dividends received from it. Normally, it does not include the price variations, distinguishing it from the total return. Yield applies to various stated rates of return on stocks, fixed income instruments, and some other investment type insurance products.

Many computer-based spreadsheet programs have built-in formulae for PV and NPV.

A **spreadsheet** is an interactive computer application for organization, analysis and storage of data in tabular form. Spreadsheets developed as computerized analogs of paper accounting worksheets. The program operates on data entered in cells of a table. Each cell may contain either numeric or text data, or the results of formulas that automatically calculate and display a value based on the contents of other cells. A spreadsheet may also refer to one such electronic document.

Each cash inflow/outflow is discounted back to its present value (PV). Then all are summed. Therefore, NPV is the sum of all terms,

where

- – the time of the cash flow
- – the discount rate, i.e. the return that could be earned per unit of time on an investment with similar risk
- – the net cash flow i.e. cash inflow – cash outflow, at time
*t*. For educational purposes, is commonly placed to the left of the sum to emphasize its role as (minus) the investment.

The result of this formula is multiplied with the Annual Net cash in-flows and reduced by Initial Cash outlay the present value but in cases where the cash flows are not equal in amount, then the previous formula will be used to determine the present value of each cash flow separately. Any cash flow within 12 months will not be discounted for NPV purpose, nevertheless the usual initial investments during the first year *R*_{0} are summed up a negative cash flow.^{ [4] }

Given the (period, cash flow) pairs (, ) where is the total number of periods, the net present value is given by:

For constant cash flow , the net present value is a finite geometric series and is given by:

The rate used to discount future cash flows to the present value 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, opportunity cost, 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 5% 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. Re-investment 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.

An NPV calculated using variable discount rates (if they are known for the duration of the investment) may better reflect the situation than one calculated from a constant discount rate for the entire investment duration. Refer to the tutorial article written by Samuel Baker^{ [5] } 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 differently 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 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 using rNPV or a similar method, then discount at the firm's rate.

NPV is an indicator of how much value an investment or project adds to the firm. With a particular project, if is a positive value, the project is in the status of positive cash inflow in the time of *t*. If 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. 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.

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 may 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. |

An alternative way of looking at Net Present Value is that at the given rate of Cost of Capital, whether the project can meet the cost of capital. For example, if the NPV is -$2.5 million (i.e. negative NPV) for a given project, it may mean that at the given Weighted Average Cost of Capital (WACC), the project fails to meet the expectations of the suppliers of capital for the project. On the other hand, the NPV of $2.5 million would add $2.5 million to the wealth of the suppliers of funds over and above their expected returns.

The time-discrete formula of the net present value

can also be written in a continuous variation

where

*r*(*t*) is the rate of flowing cash given in money per time, and*r*(*t*) = 0 when the investment is over.

Net present value can be regarded as Laplace- ^{ [6] }^{ [7] } respectively Z-transformed cash flow with the integral operator including the complex number *s* which resembles to the interest rate *i* from the real number space or more precisely *s* = ln(1 + *i*).

From this follow simplifications known from cybernetics, control theory and system dynamics. Imaginary parts of the complex number *s* describe the oscillating behaviour (compare with the pork cycle, cobweb theorem, and phase shift between commodity price and supply offer) whereas real parts are responsible for representing the effect of compound interest (compare with damping).

A corporation must decide whether to introduce a new product line. The company will have immediate costs of 100,000 at *t = 0*. Recall, a cost is a negative for outgoing cash flow, thus this cash flow is represented as -100,000. The company assumes the product will provide equal benefits of 10,000 for each of 12 years beginning at *t = 1*. For simplicity, assume the company will have no outgoing cash flows after the initial 100,000 cost. This also makes the simplifying assumption that the net cash received or paid is lumped into a single transaction occurring *on the last day* of each year. At the end of the 12 years the product no longer provides any cash flow and is discontinued without any additional costs. Assume that the effective annual discount rate is 10%.

The present value (value at *t = 0*) can be calculated for each year:

Year | Cash flow | Present value |
---|---|---|

T = 0 | −100,000 | |

T = 1 | 9,090.91 | |

T = 2 | 8,264.46 | |

T = 3 | 7,513.15 | |

T = 4 | 6,830.13 | |

T = 5 | 6,209.21 | |

T = 6 | 5,644.74 | |

T = 7 | 5,131.58 | |

T = 8 | 4,665.07 | |

T = 9 | 4,240.98 | |

T = 10 | 3,855.43 | |

T = 11 | 3,504.94 | |

T = 12 | 3,186.31 |

The total present value of the incoming cash flows is 68,136.91. The total present value of the outgoing cash flows is simply the 100,000 at time *t = 0*. Thus:

In this example:

Observe that as *t* increases the present value of each cash flow at *t* decreases. For example, the final incoming cash flow has a future value of 10,000 at *t = 12* but has a present value (at *t = 0*) of 3,186.31. The opposite of discounting is compounding. Taking the example in reverse, it is the equivalent of investing 3,186.31 at *t = 0* (the present value) at an interest rate of 10% compounded for 12 years, which results in a cash flow of 10,000 at *t = 12* (the future value).

The importance of NPV becomes clear in this instance. Although the incoming cash flows (10,000 × 12 = 120,000) appear to exceed the outgoing cash flow (100,000), the future cash flows are not adjusted using the discount rate. Thus, the project appears misleadingly profitable. When the cash flows are discounted however, it indicates the project would result in a net loss of 31,863.09. Thus, the NPV calculation indicates that this project should be disregarded because investing in this project is the equivalent of a loss of 31,863.09 at *t = 0*. The concept of time value of money indicates that cash flows in different periods of time cannot be accurately compared unless they have been adjusted to reflect their value at the same period of time (in this instance, *t = 0*).^{ [2] } It is the present value of each future cash flow that must be determined in order to provide any meaningful comparison between cash flows at different periods of time. There are a few inherent assumptions in this type of analysis:

- The
*investment horizon*of all possible investment projects considered are equally acceptable to the investor (e.g. a 3-year project is not necessarily preferable vs. a 20-year project.) - The 10% discount rate is the appropriate (and stable) rate to discount the expected cash flows from each project being considered. Each project is assumed equally speculative.
- The shareholders cannot get above a 10% return on their money if they were to directly assume an equivalent level of risk. (If the investor could do better elsewhere, no projects should be undertaken by the firm, and the excess capital should be turned over to the shareholder through dividends and stock repurchases.)

More realistic problems would also need to consider other factors, generally including: smaller time buckets, the calculation of taxes (including the cash flow timing), inflation, currency exchange fluctuations, hedged or unhedged commodity costs, risks of technical obsolescence, potential future competitive factors, uneven or unpredictable cash flows, and a more realistic salvage value assumption, as well as many others.

A more simple example of the net present value of incoming cash flow over a set period of time, would be winning a Powerball lottery of $500 million. If one does not select the "CASH" option they will be paid $25,000,000 per year for 20 years, a total of $500,000,000, however, if one does select the "CASH" option, they will receive a one-time lump sum payment of approximately $285 million, the NPV of $500,000,000 paid over time. See "other factors" above that could affect the payment amount. Both scenarios are before taxes.

- If, for example, the
*R*_{t}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 following: if some risk is incurred resulting in some losses, then a discount rate in the NPV will reduce the effect 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 ]} - Another issue with relying on NPV is that it does not provide an overall picture of the gain or loss of executing a certain project. To see a percentage gain relative to the investments for the project, usually, Internal rate of return or other efficiency measures are used as a complement to NPV.
- Non-specialist users frequently make the error of computing NPV based on cash flows after interest. This is wrong because it double counts the time value of money. Free cash flow should be used as the basis for NPV computations.

Net present value as a valuation methodology dates at least to the 19th century. Karl Marx refers to NPV as fictitious capital, and the calculation as "capitalising," writing:^{ [8] }

The forming of a fictitious capital is called capitalising. Every periodically repeated income is capitalised by calculating it on the average rate of interest, as an income which would be realised by a capital at this rate of interest.

In mainstream neo-classical economics, NPV was formalized and popularized by Irving Fisher, in his 1907 *The Rate of Interest* and became included in textbooks from the 1950s onwards, starting in finance texts.^{ [9] }^{ [10] }

- Adjusted present value (APV): adjusted present value, is the net present value of a project if financed solely by ownership equity plus the present value of all the benefits of financing.
- Accounting rate of return (ARR): a ratio similar to IRR and MIRR
- Cost-benefit analysis: which includes issues other than cash, such as time savings.
- Internal rate of return (IRR): which calculates the rate of return of a project while disregarding the absolute amount of money to be gained.
- 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.
- Payback period: which measures the time required for the cash inflows to equal the original outlay. It measures risk, not return.
- Real option: which attempts to value managerial flexibility that is assumed away in NPV.
- Equivalent annual cost (EAC): a capital budgeting technique that is useful in comparing two or more projects with different lifespans.

**Discounting** is a financial mechanism in which a debtor obtains the right to delay payments to a creditor, for a defined period of time, in exchange for a charge or fee. Essentially, the party that owes money in the present purchases the right to delay the payment until some future date. The **discount**, or **charge**, is the difference between the original amount owed in the present and the amount that has to be paid in the future to settle the debt.

In economics and finance, **present value** (**PV**), also known as **present discounted value**, is the value of an expected income stream determined as of the date of valuation. The present value is always less than or equal to the future value because money has interest-earning potential, a characteristic referred to as the time value of money, except during times of negative interest rates, when the present value will be more than the future value. Time value can be described with the simplified phrase, "A dollar today is worth more than a dollar tomorrow". Here, 'worth more' means that its value is greater. A dollar today is worth more than a dollar tomorrow because the dollar can be invested and earn a day's worth of interest, making the total accumulate to a value more than a dollar by tomorrow. Interest can be compared to rent. Just as rent is paid to a landlord by a tenant without the ownership of the asset being transferred, interest is paid to a lender by a borrower who gains access to the money for a time before paying it back. By letting the borrower have access to the money, the lender has sacrificed the exchange value of this money, and is compensated for it in the form of interest. The initial amount of the borrowed funds is less than the total amount of money paid to the lender.

The **weighted average cost of capital** (**WACC**) is the rate that a company is expected to pay on average to all its security holders to finance its assets. The WACC is commonly referred to as the firm's cost of capital. Importantly, it is dictated by the external market and not by management. The WACC represents the minimum return that a company must earn on an existing asset base to satisfy its creditors, owners, and other providers of capital, or they will invest elsewhere.

In corporate finance, as part of fundamental analysis, **economic value added** (**EVA**) is an estimate of a firm's economic profit, or the value created in excess of the required return of the company's shareholders. EVA is the net profit less the capital charge ($) for raising the firm's capital. The idea is that value is created when the return on the firm's economic capital employed exceeds the cost of that capital. This amount can be determined by making adjustments to GAAP accounting. There are potentially over 160 adjustments but in practice only several key ones are made, depending on the company and its industry. EVA is a service mark of Stern Value Management.

The APV was introduced in 1974 bij Stewart Myers. According to Myers, the value of the levered firm is equal to the value of the firm with no debt plus the present value of the tax savings due to the tax deductability of interest payments, the so called value of the tax shield (VTS). Myers proposes calculating the VTS by discounting the tax savings at the cost of debt (Kd). The argument is that the risk of the tax saving arising from the use of debt is the same as the risk of the debt. The method is to calculate the NPV of the project as if it is all-equity financed. Then the base-case NPV is adjusted for the benefits of financing. Usually, the main benefit is a tax shield resulted from tax deductibility of interest payments. Another benefit can be a subsidized borrowing at sub-market rates. The APV method is especially effective when a leveraged buyout case is considered since the company is loaded with an extreme amount of debt, so the tax shield is substantial.

A **swap** is a derivative in which two counterparties exchange cash flows of one party's financial instrument for those of the other party's financial instrument. The benefits in question depend on the type of financial instruments involved. For example, in the case of a swap involving two bonds, the benefits in question can be the periodic interest (coupon) payments associated with such bonds. Specifically, two counterparties agree to exchange one stream of cash flows against another stream. These streams are called the *legs* of the swap. The swap agreement defines the dates when the cash flows are to be paid and the way they are accrued and calculated. Usually at the time when the contract is initiated, at least one of these series of cash flows is determined by an uncertain variable such as a floating interest rate, foreign exchange rate, equity price, or commodity price.

**Rational pricing** is the assumption in financial economics that asset prices will reflect the arbitrage-free price of the asset as any deviation from this price will be "arbitraged away". This assumption is useful in pricing fixed income securities, particularly bonds, and is fundamental to the pricing of derivative instruments.

**Bond valuation** is the determination of the fair price of a bond. As with any security or capital investment, the theoretical fair value of a bond is the present value of the stream of cash flows it is expected to generate. Hence, the value of a bond is obtained by discounting the bond's expected cash flows to the present using an appropriate discount rate.

**Return on capital** (**ROC**), or **return on invested capital** (**ROIC**), is a ratio used in finance, valuation and accounting, as a measure of the profitability and value-creating potential of companies after taking into account the amount of initial capital invested. The ratio is calculated by dividing the after-tax operating income (NOPAT) by the book value of both debt and equity capital less cash/equivalents.

The **modified internal rate of return** (**MIRR**) is a financial measure of an investment's attractiveness. It is used in capital budgeting to rank alternative investments of equal size. As the name implies, MIRR is a modification of the internal rate of return (IRR) and as such aims to resolve some problems with the IRR.

The **Z-spread**, **ZSPRD**, **zero-volatility spread** or **yield curve spread** of a mortgage-backed security (MBS) is the parallel shift or spread over the zero-coupon Treasury yield curve required for discounting a pre-determined cash flow schedule to arrive at its present market price. The Z-spread is also widely used in the credit default swap (CDS) market as a measure of credit spread that is relatively insensitive to the particulars of specific corporate or government bonds.

The **dividend discount model** (**DDM**) is a method of valuing a company's stock price based on the theory that its stock is worth the sum of all of its future dividend payments, discounted back to their present value. In other words, it is used to value stocks based on the net present value of the future dividends. The equation most widely used is called the **Gordon growth model** (**GGM**). It is named after Myron J. Gordon of the University of Toronto, who originally published it along with Eli Shapiro in 1956 and made reference to it in 1959. Their work borrowed heavily from the theoretical and mathematical ideas found in John Burr Williams 1938 book "The Theory of Investment Value."

**Credit valuation adjustment** (**CVA**) is the difference between the risk-free portfolio value and the true portfolio value that takes into account the possibility of a counterparty’s default. In other words, CVA is the market value of counterparty credit risk. This price depends on counterparty credit spreads as well as on the market risk factors that drive derivatives’ values and, therefore, exposure. CVA is one of a family of related valuation adjustments, collectively xVA; for further context here see Financial economics #Derivative pricing.

The **Datar–Mathews method** is a method for real options valuation. The method provides an easy way to determine the real option value of a project simply by using the average of positive outcomes for the project. The method can be understood as an extension of the net present value (NPV) multi-scenario Monte Carlo model with an adjustment for risk aversion and economic decision-making. The method uses information that arises naturally in a standard discounted cash flow (DCF), or NPV, project financial valuation. It was created in 2000 by Professor Vinay Datar, Seattle University, and Scott H. Mathews, Technical Fellow, The Boeing Company.

**Corporate finance** is an area of finance that deals with sources of funding, the capital structure of corporations, the actions that managers take to increase the value of the firm to the shareholders, and the tools and analysis used to allocate financial resources. The primary goal of corporate finance is to maximize or increase shareholder value. Although it is in principle different from managerial finance which studies the financial management of all firms, rather than corporations alone, the main concepts in the study of corporate finance are applicable to the financial problems of all kinds of firms.

- ↑ Lin, Grier C. I.; Nagalingam, Sev V. (2000).
*CIM justification and optimisation*. London: Taylor & Francis. p. 36. ISBN 0-7484-0858-4. - 1 2 Berk, DeMarzo, and Stangeland, p. 94.
- ↑ erk, DeMarzo, and Stangeland, p. 64.
- ↑ Khan, M.Y. (1993).
*Theory & Problems in Financial Management*. Boston: McGraw Hill Higher Education. ISBN 978-0-07-463683-1. - ↑ Baker, Samuel L. (2000). "Perils of the Internal Rate of Return" . Retrieved January 12, 2007.
- ↑ Grubbström, Robert W. (1967). "On the Application of the Laplace Transform to Certain Economic Problems".
*Management Science*.**13**: 558–567. doi:10.1287/mnsc.13.7.558. - ↑ Steven Buser: LaPlace Transforms as Present Value Rules: A Note,
*The Journal of Finance*, Vol. 41, No. 1, March, 1986, pp. 243-247. - ↑ Karl Marx,
*Capital,*Volume 3, 1909 edition, p. 548 - ↑ Bichler, Shimshon; Nitzan, Jonathan (July 2010),
*Systemic Fear, Modern Finance and the Future of Capitalism*(PDF), Jerusalem and Montreal, pp. 8–11 (for discussion of history of use of NPV as "capitalisation") - ↑ Nitzan, Jonathan; Bichler, Shimshon (2009),
*Capital as Power. A Study of Order and Creorder.*, RIPE Series in Global Political Economy, New York and London: Routledge

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