Internal rate of return

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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.

To invest is to allocate money in the expectation of some benefit in the future.

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.

The risk-free interest rate is the rate of return of a hypothetical investment with no risk of financial loss, over a given period of time.


It is also called the discounted cash flow rate of return (DCFROR). [1]

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.


The internal rate of return on an investment or project is the "annualized effective compounded return rate" or rate of return that sets the net present value of all cash flows (both positive and negative) from the investment equal to zero. Equivalently, it is the discount rate at which the net present value of the future cash flows is equal to the initial investment, and it is also the discount rate at which the total present value of costs (negative cash flows) equals the total present value of the benefits (positive cash flows).

In finance, the net present value (NPV) or net present worth (NPW) 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.

Interest rate percentage of a sum of money charged for its use

An interest rate is the amount of interest due per period, as a proportion of the amount lent, deposited or borrowed. The total interest on an amount lent or borrowed depends on the principal sum, the interest rate, the compounding frequency, and the length of time over which it is lent, deposited or borrowed.

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.

Speaking intuitively, IRR is designed to account for the time preference of money and investments. A given return on investment received at a given time is worth more than the same return received at a later time, so the latter would yield a lower IRR than the former, if all other factors are equal. A fixed income investment in which money is deposited once, interest on this deposit is paid to the investor at a specified interest rate every time period, and the original deposit neither increases nor decreases, would have an IRR equal to the specified interest rate. An investment which has the same total returns as the preceding investment, but delays returns for one or more time periods, would have a lower IRR.

In economics, time preference is the current relative valuation placed on receiving a good at an earlier date compared with receiving it at a later date.

Uses of IRR

Savings and loans

In the context of savings and loans, the IRR is also called the effective interest rate.

Effective interest rate

The effective interest rate (EIR), effective annual interest rate, annual equivalent rate (AER) or simply effective rate is the interest rate on a loan or financial product restated from the nominal interest rate as an interest rate with annual compound interest payable in arrears.

Profitability of an investment

Corporations use IRR in capital budgeting to compare the profitability of capital projects in terms of the rate of return. For example, a corporation will compare an investment in a new plant versus an extension of an existing plant based on the IRR of each project. To maximize returns, the higher a project's IRR, the more desirable it is to undertake the project. To maximize return, the project with the highest IRR would be considered the best, and undertaken first.

Capital budgeting includes all methods that allow a rational assessment of the calculable aspects of an investment

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 economics, profit in the accounting sense of the excess of revenue over cost is the sum of two components: normal profit and economic profit. All understanding of profit should be broken down into three aspects: the size of profit, the portion of the total income, and the rate of profit. Normal profit is the profit that is necessary to just cover the opportunity costs of the owner-manager or of the firm's investors. In the absence of this profit, these parties would withdraw their time and funds from the firm and use them to better advantage elsewhere. In contrast, economic profit, sometimes called excess profit, is profit in excess of what is required to cover the opportunity costs.

Maximizing net present value

The internal rate of return is an indicator of the profitability, efficiency, quality, or yield of an investment. This is in contrast with the net present value, which is an indicator of the net value or magnitude added by making an investment.

Applying the internal rate of return method to maximize the value of the firm, any investment would be accepted, if its profitability, as measured by the internal rate of return, is greater than a minimum acceptable rate of return. The appropriate minimum rate to maximize the value added to the firm is the cost of capital, i.e. the internal rate of return of a new capital project needs to be higher than the company's cost of capital. This is because only an investment with an internal rate of return which exceeds the cost of capital has a positive net present value.

However, the selection of investments may be subject to budget constraints, or there may be mutually exclusive competing projects, or the capacity or ability to manage more projects may be practically limited. In the example cited above of a corporation comparing an investment in a new plant to an extension of an existing plant, there may be reasons the company would not engage in both projects.

Fixed income

The same method is also used to calculate yield to maturity and yield to call.


Both the internal rate of return and the net present value can be applied to liabilities as well as investments. For a liability, a lower internal rate of return is preferable to a higher one.

Capital management

Corporations use internal rate of return to evaluate share issues and stock buyback programs. A share repurchase proceeds if returning capital to shareholders has a higher internal rate of return than candidate capital investment projects or acquisition projects at current market prices. Funding new projects by raising new debt may also involve measuring the cost of the new debt in terms of the yield to maturity (internal rate of return).

Private equity

IRR is also used for private equity, from the limited partners' perspective, as a measure of the general partner's performance as investment manager. [2] This is because it is the general partner who controls the cash flows, including the limited partners' draw-downs of committed capital.


Given a collection of pairs (time, cash flow) representing a project, the net present value is a function of the rate of return. The internal rate of return is a rate for which this function is zero, i.e. the internal rate of return is a solution to the equation NPV = 0.

Given the (period, cash flow) pairs (, ) where is a non-negative integer, the total number of periods , and the , (net present value); the internal rate of return is given by in:

Note that in this formula, (≤0) is the initial investment at the start of the project. The period is usually given in years, but the calculation may be made simpler if is calculated using the period in which the majority of the problem is defined (e.g., using months if most of the cash flows occur at monthly intervals) and converted to a yearly period thereafter.

Any fixed time can be used in place of the present (e.g., the end of one interval of an annuity); 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 above formula.

Often, the value of that satisfies the above equation cannot be found analytically. In this case, numerical methods or graphical methods must be used.


If an investment may be given by the sequence of cash flows

Year ()Cash flow ()

then the IRR is given by

In this case, the answer is 5.96% (in the calculation, that is, r = .0596).

Numerical solution

Since the above is a manifestation of the general problem of finding the roots of the equation , there are many numerical methods that can be used to estimate . For example, using the secant method, is given by

where is considered the th approximation of the IRR.

This can be found to an arbitrary degree of accuracy. Different accounting packages may provide functions for different accuracy levels.

The convergence behaviour of by the following:

  • If the function has a single real root , then the sequence converges reproducibly towards .
  • If the function has real roots , then the sequence converges to one of the roots, and changing the values of the initial pairs may change the root to which it converges.
  • If function has no real roots, then the sequence tends towards +∞.

Having when or when may speed up convergence of to .

Numerical solution for single outflow and multiple inflows

Of particular interest is the case where the stream of payments consists of a single outflow, followed by multiple inflows occurring at equal periods. In the above notation, this corresponds to:

In this case the NPV of the payment stream is a convex, strictly decreasing function of interest rate. There is always a single unique solution for IRR.

Given two estimates and for IRR, the secant method equation (see above) with always produces an improved estimate . This is sometimes referred to as the Hit and Trial (or Trial and Error) method. More accurate interpolation formulas can also be obtained: for instance the secant formula with correction


(which is most accurate when ) has been shown to be almost 10 times more accurate than the secant formula for a wide range of interest rates and initial guesses. For example, using the stream of payments {4000, 1200, 1410, 1875, 1050} and initial guesses and the secant formula with correction gives an IRR estimate of 14.2% (0.7% error) as compared to IRR = 13.2% (7% error) from the secant method.

If applied iteratively, either the secant method or the improved formula always converges to the correct solution.

Both the secant method and the improved formula rely on initial guesses for IRR. The following initial guesses may be used:


Here, refers to the NPV of the inflows only (that is, set and compute NPV).

Exact dates of cash flows

A cash flow may occur at any time years after the beginning of the project. may not be a whole number. The cash flow should still be discounted by a factor .

Problems with using internal rate of return

As a tool applied to making an investment decision, to decide whether a project adds value or not, comparing the IRR of a single project with the required rate of return, in isolation from any other projects, is equivalent to the NPV method. If the appropriate IRR (if such can be found correctly) is greater than the required rate of return, then using the required rate of return to discount cash flows to their present value, the NPV of that project will be positive, and vice versa. However, using IRR to sort projects in order of preference does not result in the same order as using NPV.

Maximizing net present value

One possible investment objective is to maximize the total net present value of projects.

When the objective is to maximize total value, the calculated IRR should not be used to choose between mutually exclusive projects.

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) Exclusive investments.png
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).

When the objective is to maximize total value, IRR should not be used to compare projects of different duration. For example, the net present value added by a project with longer duration but lower IRR could be greater than that of a project of similar size, in terms of total net cash flows, but with shorter duration and higher IRR.

Practitioner preference for IRR over NPV

Despite a strong academic preference for NPV, surveys indicate that executives prefer IRR over NPV. [3] 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 to maximize value.

Maximizing long-term return

Maximizing total value is not the only conceivable possible investment objective. An alternative objective would for example be to maximize long-term return. Such an objective would rationally lead to accepting first those new projects within the capital budget which have the highest IRR, because adding such projects would tend to maximize overall long-term return.


To see this, consider two investors, Max Value and Max Return. Max Value wishes her net worth to grow as large as possible, and will invest every last cent available to achieve this, whereas Max Return wants to maximize his rate of return over the long term, and would prefer to choose projects with smaller capital outlay but higher returns. Max Value and Max Return can each raise up to 100,000 US dollars from their bank at an annual interest rate of 10 percent paid at the end of the year.

Investors Max Value and Max Return are presented with two possible projects to invest in, called Big-Is-Best and Small-Is-Beautiful. Big-Is-Best requires a capital investment of 100,000 US dollars today, and the lucky investor will be repaid 132,000 US dollars in a year's time. Small-Is-Beautiful only requires 10,000 US dollars capital to be invested today, and will repay the investor 13,750 US dollars in a year's time.


The cost of capital for both investors is 10 percent.

Both Big-Is-Best and Small-Is-Beautiful have positive NPVs:

and the IRR of each is (of course) greater than the cost of capital:

so the IRR of Big-Is-Best is 32 percent, and

so the IRR of Small-Is-Beautiful is 37.5 percent.

Both investments would be acceptable to both investors, but the twist in the tale is that these are mutually exclusive projects for both investors, because their capital budget is limited to 100,000 US dollars. How will the investors choose rationally between the two?

The happy outcome is that Max Value chooses Big-Is-Best, which has the higher NPV of 20,000 US dollars, over Small-Is-Beautiful, which only has a modest NPV of 2,500, whereas Max Return chooses Small-Is-Beautiful, for its superior 37.5 percent return, over the attractive (but not as attractive) return of 32 percent offered on Big-Is-Best. So there is no squabbling over who gets which project, they are each happy to choose different projects.

How can this be rational for both investors? The answer lies in the fact that the investors do not have to invest the full 100,000 US dollars. Max Return is content to invest only 10,000 US dollars for now. After all, Max Return may rationalize the outcome by thinking that maybe tomorrow there will be new opportunities available to invest the remaining 90,000 US dollars the bank is willing to lend Max Return, at even higher IRRs. Even if only seven more projects come along which are identical to Small-Is-Beautiful, Max Return would be able to match the NPV of Big-Is-Best, on a total investment of only 80,000 US dollars, with 20,000 US dollars left in the budget to spare for truly unmissable opportunities. Max Value is also happy, because she has filled her capital budget straight away, and decides she can take the rest of the year off investing.

Multiple IRRs

When the sign of the cash flows changes more than once, for example when positive cash flows are followed by negative ones and then by positive ones (+ + − − − +), the IRR may have multiple real values. 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 real 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.

The IRR satisfies 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 by iteration.

With multiple internal rates of return, the IRR approach can still be interpreted in a way that is consistent with the present value approach if the underlying investment stream is correctly identified as net investment or net borrowing. [4]

See [5] for a way of identifying the relevant IRR from a set of multiple IRR solutions.

Modified internal rate of return (MIRR)

Modified Internal Rate of Return (MIRR) considers cost of capital, and is intended to provide a better indication of a project's probable return. It applies a discount rate for borrowing cash, and the IRR is calculated for the investment cash flows. This applies in real life for example when a customer makes a deposit before a specific machine is built.

When a project has multiple IRRs it may be more convenient to compute the IRR of the project with the benefits reinvested. [6] Accordingly, MIRR is used, which has an assumed reinvestment rate, usually equal to the project's cost of capital.

Average internal rate of return (AIRR)

Magni (2010) introduced a new approach, named AIRR approach, based on the intuitive notion of mean, that solves the problems of the IRR. [7] However, the above-mentioned difficulties are only some of the many flaws incurred by the IRR. Magni (2013) provided a detailed list of 18 flaws of the IRR and showed how the AIRR approach does not incur the IRR problems. [8]

Doesn't account for all variables

Traditional IRR calculations only consider the financial aspects of a decision but does not not fully "capture the short- or long-term importance, value, or risks associated with natural and social capital" [9] because it does not account for the environmental, social and governance performance of an organization. Without a metric for measuring the short and long term environmental, social and governance performance of a firm, decision makers are planning for the future without considering the extent of the impacts associated with their decisions.


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:

In the case of a series of exclusively negative cash flows followed by a series of exclusively positive ones, the resulting function of the rate of return is continuous and monotonically decreasing from positive infinity (when the rate of return approaches -100%) to the value of the first cash flow (when the rate of return approaches 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.

Finally, by Descartes' rule of signs, the number of internal rates of return can never be more than the number of changes in sign of cash flow.

The reinvestment debate

It is often stated that IRR assumes reinvestment of all cash flows until the very end of the project. This assertion has been a matter of debate in the literature.

Sources stating that there is such a hidden assumption include those cited below. [6] [10] Sources arguing that there is no IRR reinvestment assumption include [11] [12] [13] [14] [15] [16]

When comparing investments, making an implicit assumption that cash flows are reinvested at the same IRR would lead to false conclusions. If cash flows received are not reinvested at the same rate as the IRR, a project with a relatively short duration and a high IRR does not necessarily add more value over a longer time span than another project with a longer duration and a lower IRR. This is why IRR should be not used on a stand-alone basis, but in combination with NPV.

The Modified Internal Rate of Return (MIRR) addresses this issue by allowing for the inclusion a second investment at a potentially different rate of return, to calculate a portfolio return without external cash flows over the life of the project. However, for capital budgeting, when the objective is to maximize value, finance theory holds that NPV using the firm's cost of capital is the optimal metric.

The internal rate of return in personal finance

The IRR can be used to measure the money-weighted performance of financial investments such as an individual investor's brokerage account. For this scenario, an equivalent, [17] more intuitive definition of the IRR is, "The IRR is the annual interest rate of the fixed rate account (like a somewhat idealized savings account) which, when subjected to the same deposits and withdrawals as the actual investment, has the same ending balance as the actual investment." This fixed rate account is also called the replicating fixed rate account for the investment. There are examples where the replicating fixed rate account encounters negative balances despite the fact that the actual investment did not. [17] In those cases, the IRR calculation assumes that the same interest rate that is paid on positive balances is charged on negative balances. It has been shown that this way of charging interest is the root cause of the IRR's multiple solutions problem. [18] [19] If the model is modified so that, as is the case in real life, an externally supplied cost of borrowing (possibly varying over time) is charged on negative balances, the multiple solutions issue disappears. [18] [19] The resulting rate is called the fixed rate equivalent (FREQ). [17]

Unannualized internal rate of return

In the context of investment performance measurement, there is sometimes ambiguity in terminology between the periodic rate of return, such as the internal rate of return as defined above, and a holding period return. The term internal rate of return or IRR or Since Inception Internal Rate of Return (SI-IRR) is in some contexts used to refer to the unannualized return over the period, particularly for periods of less than a year. [20]

Integrated rate of return (IntRR)

To address the lack of integration of the short and long term importance, value and risks associated with natural and social capital into the traditional IRR calculation, companies are valuing their environmental, social and governance (ESG) performance through an Integrated Management approach to reporting that expands IRR to Integrated Rate of Return. [9] This allows companies to value their investments not just for their financial return but also the long term environmental and social return of their investments. By highlighting environmental, social and governance performance in reporting, decision makers have the opportunity to identify new areas for value creation that are not revealed through traditional financial reporting. [21] The social cost of carbon is one value that can be incorporated into Integrated Rate of Return calculations to encompass the damage to society from greenhouse gas emissions that result from an investment. This is an integrated approach to reporting that supports Integrated Bottom Line (IBL) decision making, which takes triple bottom line(TBL) a step further and combines financial, environmental and social performance reporting into one balance sheet. This approach provides decision makers with the insight to identify opportunities for value creation that promote growth and change within an organization. [22]

See also

Related Research Articles


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.

Time value of money value of current money with interest after time

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.

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.

Economic value added

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.

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.

In finance, the equivalent annual cost (EAC) is the cost per year of owning and operating an asset over its entire lifespan. It is calculated by dividing the NPV of a project by the "present value of annuity factor":

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.

Cash-flow return on investment (CFROI) is a valuation model that assumes the stock market sets prices based on cash flow, not on corporate performance and earnings.

The modified Dietz method is a measure of the ex post performance of an investment portfolio in the presence of external flows.

In finance, the T-model is a formula that states the returns earned by holders of a company's stock in terms of accounting variables obtainable from its financial statements. The T-model connects fundamentals with investment return, allowing an analyst to make projections of financial performance and turn those projections into a required return that can be used in investment selection. Mathematically the model is as follows:

The Penalized Present Value (PPV) is a method of Capital Budgeting under risk developed by Fernando Gómez-Bezares in the 1980s.

The time-weighted return (TWR) is a method of calculating investment return. To apply the time-weighted return method, combine the returns over sub-periods, by compounding them together, resulting in the overall period return. The rate of return over each different sub-period is weighted according to the duration of the sub-period.

Corporate finance area of finance dealing with the sources of funding and the capital structure of corporations

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.

Public Market Equivalent designs a set of analyses used in the Private Equity Industry to evaluate the performance of a Private Equity Fund against a public benchmark or index. The analysis is also referred by the acronym ICM for Index Comparison Method. More than a specific calculation method, PME encompasses a number of different analyses sharing the same objective. The first PME analysis was proposed by Long and Nickels in 1996.


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