If A Company Does Not Have The Money To Invest In All Positive Discount Rate

Economic evaluation

Hoss Belyadi , ... Fatemeh Belyadi , in Hydraulic Fracturing in Unconventional Reservoirs (Second Edition), 2019

Modified internal rate of return (MIRR)

MIRR is basically an improved version of IRR and is another tool used in capital budgeting. It is very important to understand the difference between IRR and MIRR. As previously mentioned, IRR defectively assumes that positive cash flows from a particular project are reinvested at IRR. In contrast to IRR, MIRR assumes that cash flows from a project are reinvested at cost of capital or a particular reinvestment rate. In addition to this improvement, MIRR only yields one solution. Consequently MIRR can be defined as the discount rate that causes the present value of a project's terminal value to equal the present value of cost. The MIRR concept is fairly complicated and will only make more sense with examples. This is one of the main reasons that IRR is used more frequently in the real world, that is, since MIRR is not completely understood by a lot of managers. MIRR can be calculated using Eq. (18.21).

(18.21) $MIRR = \sqrt[n]{\frac{Future value (positive cash flows @ reinvestment rate)}{- Present value (negative cash flows @ cost of capital or finance rate)} - 1}$

Example

The cash flows for projects A and B are summarized in Table 18.13. Calculate MIRR assuming a cost of capital of 10% and reinvestment rate of 12%.

Table 18.13. MIRR example

Year	Project A ($MM)	Project B ($MM)
0	($600)	($350)
1	$100	$200
2	$250	$225
3	$320	$250
4	$385	$350
5	$400	$450

The first step is to calculate the present value of negative cash flows at cost of capital for both projects:

$Project A present value = \frac{- 600}{{(1 + 0.1)}^{0}} = - 600$

$Project B present value = \frac{- 350}{{(1 + 0.1)}^{0}} = - 350$

Next, future values of positive cash flows at reinvestment rate must be calculated for both projects:

$\begin{array}{l} Project A future value \\ = 100 \times {(1 + 12 %)}^{4} + 250 \times {(1 + 12 %)}^{3} + 320 \times {(1 + 12 %)}^{2} + 385 \\ \times {(1 + 12 %)}^{1} + 400 \times {(1 + 12 %)}^{0} = $ 1741.19 \end{array}$

$\begin{array}{l} Project B future value \\ = 200 \times {(1 + 12 %)}^{4} + 225 \times {(1 + 12 %)}^{3} + 250 \times {(1 + 12 %)}^{2} + 350 \\ \times {(1 + 12 %)}^{1} + 450 \times {(1 + 12 %)}^{0} = $ 1786.41 \end{array}$

Using the MIRR equation:

$Project A MIRR = \sqrt[5]{\frac{1741.19}{- (- 600)}} - 1 = 0.2375 or 23.75 %$

$Project B MIRR = \sqrt[5]{\frac{1786.41}{- (- 350)}} - 1 = 0.3854 or 38.54 %$

In this example, note that the first-year cash inflow is assumed to be reinvested in 4 years (5 − 1), the second-year cash inflow is assumed to be reinvested in 3 years (5 − 2), the third-year cash inflow is assumed to be reinvested in 2 years (5 − 3), the fourth-year cash inflow is assumed to be reinvested in 1 year (5 − 4), and finally the fifth-year cash inflow is received at the end of the fifth year and is not available for reinvestment since it accords with the end of the project's life.

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Cost-Benefit Analysis of Safety Measures

G. Reniers , ... N. Paltrinieri , in Dynamic Risk Analysis in the Chemical and Petroleum Industry, 2016

2.3.1 Internal Rate of Return

IRR can be defined as the discount rate at which the present value of all future cash flows (or monetized expected hypothetical benefits) is equal to the initial investment, that is, the rate at which an investment breaks even. It can be used to measure and compare the profitability of investments. Generally speaking, the higher an investment's IRR, the more desirable it is to carry on with the investment. As such, the IRR can be used to rank several possible investment options that an organization is considering. Assuming all other factors are equal among the various investments, the safety investment with the highest IRR would be recommended to have priority. IRR is sometimes referred to as economic rate of return.

An organization should, in theory, undertake all safety investments available with IRRs that exceed a minimum acceptable rate of return predetermined by the company. Investments may of course be limited by availability of funds to the company. Because the IRR return is a rate quantity, it is an indicator of the efficiency, quality, or yield of an investment. This is in contrast with the NPV, which is an indicator of the value or magnitude of an investment.

The IRR r ^∗ is a rate of return for which the NPV is zero. This can be expressed as follows:

$NPV (r *) = \sum_{n = 0}^{N} \frac{C_{n}}{{(1 + r *)}^{n}} = 0$

In cases when a first safety investment displays a lower IRR but a higher NPV over a second safety investment, the first investment should be accepted over the second one. Furthermore, the IRR should not be used to compare investments of different duration. For example, a NPV of an investment with longer duration but lower IRR could be higher than a NPV of a similar investment (in terms of total net cash flows) with shorter duration and higher IRR.

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CCHP Evaluation Criteria

Masood Ebrahimi , Ali Keshavarz , in Combined Cooling, Heating and Power, 2015

3.3.6 Internal Rate of Return (IRR)

The calculation of IRR is vital for the economic evaluation of CCHP systems, because it calculates the profitability margin (IRR − r) of the project. In fact, the IRR calculates the interest rate for which the NPV of the project would be zero. If the IRR becomes smaller than r, investment in the project would lose money, for IRR = r the profit is zero and there is a risk of losing money, and for IRR > r the investment is safe. IRR can be calculated according to the following equation:

(3-17) ${N P V|}_{I R R} = \sum_{y = 1}^{L} \frac{c f_{y}}{{(1 + I R R)}^{y}} - I = 0$

To guarantee the investment, the CCHP cycle and its components can be designed to have a predefined IRR = IRR _min > r.

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Wind Energy

D. Milborrow , in Comprehensive Renewable Energy, 2012

2.15.2.1.5 Internal rate of return

The internal rate of return (IRR) of an investment is the interest rate at which the NPV of costs (negative cash flows) of the investment equals the NPV of the benefits (positive cash flows) of the investment.

IRRs are commonly used to evaluate the desirability of investments or projects. The higher a project's IRR, the more desirable it is to undertake the project. Assuming all other factors are equal among the various projects, the project with the highest IRR would probably be considered the best and undertaken first [2].

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Optimal Sizing and Designing of Hybrid Renewable Energy Systems in Smart Grid Applications

Ali M. Eltamaly , Mohamed A. Mohamed , in Advances in Renewable Energies and Power Technologies, 2018

Internal Rate of Return (IRR)

The IRR for an investment proposal is the discount rate (IRR) that equates the present value of the expected cash outflows with the present value of the expected cash inflows. In short, it is the discount rate that makes NPV = 0 (i.e., Eq. 8.27 = 0):

(8.29) $\sum_{t = 1}^{T} {(\frac{1 + a}{1 + r})}^{t} \times S - C_{WECS} = 0$

If the IRR is greater than the cutoff or hurdle rate (r), the proposal is accepted; if not, the proposal is rejected [33]. As we can see, the IRR is in effect the discounted cash flow (DFC) return that makes the NPV zero.

Using our numerical example again, case (i) gives IRR of 6%, and case (ii) gives IRR of 8%. If the investors hurdle rate is 5%, then both cases of cash flows are acceptable. If the investor's rate is greater than 6% and less than or equal to 8%, then case (i) would be rejected, whereas case (ii) would be accepted. If the investor's rate is greater than 8%, then both cases would be rejected.

In the event of multiple projects, the projects are ranked in accordance to the highest IRR down to the lowest, and selection preferences follow the same order. NPV and IRR could rank mutually exclusive projects differently because of different scales and lives leading to confusion in selection. This is because both implicitly assume reinvestment of returns at their own rates (i.e., r% for NPV and IRR% for IRR). That is, the project with the highest positive NPV might not be the project with the highest IRR since the reinvestment rates are different. This leads to different ranking, so it is recommended to use IRR only when considering the accept/reject analysis for single projects with conventional cash flows (only the first cash flow is negative) [35,36].

For unconventional cash flows, Descartes' rule of sign says that the maximum number of IRRs that there can be is equal to the number of times that the cash flows change sign from positive to negative and/or negative to positive [32].

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Business Cases

N. Good , ... P. Mancarella , in Energy Positive Neighborhoods and Smart Energy Districts, 2017

2.5.1.3 Internal Rate of Return

The IRR is an indication of the exact premium rate that an investment project is offering in exchange for all costs incurred. The IRR can be estimated as the discount rate (d) that renders the present value of both benefits and costs the same. That is, the discount rate that satisfies the following equation:

(6.4) $\sum_{t}^{T} \frac{Benefit s_{t}}{{(1 + d)}^{t}} = \sum_{t}^{T} \frac{Cost s_{t}}{{(1 + d)}^{t}}$

Conversely to the NPV criterion that centers on absolute profits, the IRR criterion shows the efficiency of an investment with regard to the premium rate that it produces compared to the costs. That is, whereas the NPV criterion favors investments with high profits regardless of the costs, the IRR criterion favors projects that produce more benefits in comparison to costs.

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Economics

S. Wright , W. Scammell , in Fundamentals and Applications of Supercritical Carbon Dioxide (sCO₂) Based Power Cycles, 2017

6.3.2 Internal rate of return

The IRR is a metric used to help investors determine if the project is economically viable. It is a threshold value for interest rates. If the interest rate is below the IRR then the project can have a positive rate of return. Calculating the IRR requires developing a complete financial model of the revenues and expenses that the project will generate over its life. The financial model requires that the free cash flows (FCFs) be generated for each period (year in this example), which allows for the calculation of the projected IRR of $0.0409/kWh_e as shown in Table 6.2.

Table 6.2. Example calculations for IRR and for Net Present Value

IRR, internal rate of return.

IRR is the discount rate at which the PV of a set of future cash flows will equal zero, or the discount rate at which a project will achieve a break-even status in financial terms. The formula is as follows:

$0 = {FCF}_{0} + [\frac{{FCF}_{1}}{(1 + IRR)}] + [\frac{{FCF}_{2}}{{(1 + IRR)}^{2}}] + [\frac{{FCF}_{3}}{{(1 + IRR)}^{3}}] \dots + [\frac{{FCF}_{n}}{{(1 + IRR)}^{n}}]$

FCF₀ is the initial project cost (and is negative as it is money spent) in the equation. The term n is the index for the period (year in the examples). IRR must be calculated through a process of iteration where the discount rate is changed in the formula until the PV formula (left-hand side) equals 0. Spreadsheet software with solvers and financial calculators prove helpful for this analysis. As previously stated, the IRR is not the explicit return that a project will provide to investors, but rather the maximum cost of capital at which the project will provide a positive return. Thus, it provides a good indication of the likelihood that a project would provide attractive returns before the cost of capital available to fund it is fully known. This type of financial information is very valuable to potential investors.

The example of the IRR estimate (see Table 6.2) uses the same SRBC CC as used in the LCOE calculations. Assuming that the project will generate revenue of $0.06/kWh of electricity produced results in $14,303,900 annual revenue. The operating expenses of $10,636,258 are carried over from the LCOE calculation. And the depreciation expenses are $3,079,700. Deducting annual operating costs and the annual depreciation expense from the revenue results in taxable income of $587,957. The tax liability is 35% of the taxable income or $205,784. So for the first 10 years, the annual depreciated cash flow is $382,172. Then adding back depreciation tax deduction (a noncash expense) results in an FCF of $3,461,872 for the first 10 years. (Stated another way, the FCF is equal to the Revenue-Expenses-Taxes.) After the project is fully depreciated the taxable income increases to $3,667,657, annual tax liability increases to $1,283,680, and annual FCF falls to $2,383,967. Using a Project Cost of $30,797,000 for "FCF₀", $3,461,872 for "FCF₁–FCF₁₀," and then $2,383,976 for "FCF₁₁–FCF₂₀" returns an IRR of 7.924% for the SRBC paired with the LM2500-PE.

So this set of calculations for IRR indicates that when interest rates are below 7.92%, this CC project is viable. The next step in the economic analysis is to estimate, how valuable it is. This requires using NPV formulas that are described in the next section.

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Economic Risk Assessment for Field Development

Yong Bai , Wei-Liang Jin , in Marine Structural Design (Second Edition), 2016

Internal Rate of Return

The IRR is another time-discounted measure of investments similar to the NPV criterion. The IRR of a project is defined as the rate of interest that equates the NPV of the entire series of cash flows to zero. The project's IRR is mathematically defined by

(A.4) $N P V (i r r) = \sum_{n = 0}^{N} \frac{F_{n}}{{(1 + i r r)}^{n}} = 0$

Note that Eqn (A.4) is a polynomial function of $i r r$ . A direct solution for such a function is not generally possible except for projects with a life of four periods or fewer. Therefore, two approximation techniques are generally used, one using iterative procedures (a trial-and-error approach) and the other using Newton's approximation to the solution of a polynomial.

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Evaluation, risk and feasibility

Eoin H. Macdonald , in Handbook of Gold Exploration and Evaluation, 2007

Internal rate of return (IRR)

The internal rate of return of a project is the discount rate that would yield a net present value of zero, i.e., the rate of interest which makes the present value of the estimated cash inflow equal to the present value of the cash outflow required by the investment. Zero NPV means that the cash proceeds of the project are exactly equivalent to the cash proceeds from an alternative investment at the stated rate of interest. The funds, while invested in the project, are earning at that rate of interest, i.e., at the project's internal rate of return.

All things being equal, the higher the IRR, the more attractive the project. In terms of acquisition or future project development, projects generating IRRs greater than a company's target rate of return will be accepted. However, in terms of determining the valuation of a project no IRR can be calculated where all cash flows are positive, as in an operating mine situation. Multiple IRRs can arise where there are significant negative cash flows at other stages of the project as well as at the beginning.

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Petroleum Economic Evaluation

In Standard Handbook of Petroleum and Natural Gas Engineering (Third Edition), 2016

7.4.5 Return on Investment and Internal Rate of Return

Guidelines that have been established for minimum standards to gage the company's measure of profitability are varied and depend on individual company goals and operational constraints. The most common minimum standard, and most elementary, are Return on Investment (ROI) [11 ] and Internal Rate of Return (IRR) [ 12].

7.4.5.1 Internal Rate of Return (IRR)

The internal rate of return (IRR) is another measure of profitability that defines a relationship between revenues and sales and may be one of the indicators used by management for investment purposes. Internal rate of return is an interest rate that will cause a present value series of costs to equal the sum of present value for a series of revenues.

In other words, an investment equal to present value of costs that earns a rate of return, the IRR that makes present value revenues equal costs.

Interest rates, other than base rate (0.15 to 0.3 in 0.05 increments) may be included to give some measure of risks.

Mathematically, IRR, may be represented by:

(7.4.11) $\begin{array}{l} {If [PVC(i}_{1}, n) - {PVR(xi}_{2}, n) = 0], IRR is the unknown \\ {interest, xi}_{2} . \end{array}$

Example: Series of Costs and Revenues by year (Table 7.4.4).

Table 7.4.4. Example Cost and Revenues

For this example, the IRR is 18.7%. The method of calculation is, by nature, iterative. (XIRR(values,dates,guess))

The internal rate of return (IRR) can also be generated at each simulation iteration by merely calculating (as mentioned below, using the canned function in Excel) the IRR for each sample price path. ⁷ Similar data-driven inferences may be made; in this case, the probability that the IRR is less than some required rate of return, such as the 15% discount rate used in this example.

Figure 7.4.5 consists of a histogram of the IRR data computed for a 10-year project-ownership window. Note that the probability that the IRR for a 10-year project is less than 15% is substantial. The probability that the IRR for a project is less than the discount/hurdle rate should be the same (or nearly the same depending on the number of simulation runs conducted) as the probability that the NPV of a project is less than 0. This is because of the fundamental relationship between NPV and IRR—if NPV <=> 0 then the IRR for that particular cash flow stream is less than/equal to/greater than the discount rate for the project.

Figures 7.4.5 and 7.4.6 contain single 10-year IRR distribution (Figure 7.4.5) and all three IRR distributions (Figure 7.4.6). These distributions are, like the NPV distributions, based on simulation data and any inferences should be made from the data unless a proper distribution-fitting procedure is followed for the simulation data.

If a sufficient number of simulation iterations are carried out (in this spreadsheet-simulation example, 1000 runs were done. This is a minimum number in order to use the simulation data in project and alternative evaluations), then information about key measures—expected values, probabilities of the realized value being less (greater) than or equal to a cutoff value—can be used in project and alternative evaluations.

Overall expected values and probabilities of negative NPVs are presented below in Table 7.4.5.

Table 7.4.5. Summary of Simulation Results for Gusher Asset 7-III-B. Project years refers to the number of years before the project is abandoned or sold (no value was assumed for this in this analysis), E(NPV) is the expected value of the simulated NPV distribution, Pr (NPV<0) is the probability that the actual NPV will be less than 0 given the discount rate, and E(IRR) denotes the expected IRR. Annual discount rate used: 15%

Project Years	E(NPV)	Pr(NPV < 0)	E(IRR)
10	4.81 × 10⁶	0.38	0.1598
17	1.99 × 10⁷	0.148	0.1845
20	1.95 × 10⁷	0.164	0.183

The key relationship between cumulative NPV, incremental NPV, and annual expected oil production is shown in Figure 7.4.7. Note that the incremental NPV curve is non-monotonic. This is because this curve represents one set of NPVs from one (out of a total of 1000) simulated price path.

7.4.5.2 Return on Investment (ROI)

Generically ROI may be defined as, "What is the gain in value of an investment as compared with expenses". ROI is related to both NPV and IRR, and analysts should take care to represent ROI based on undiscounted (close to IRR) or discounted (close to NPV) data. The ROIs presented for this example are for discounted data.

Mathematically, ROI is represented by ratio of sum of present value revenues to sum of present value expenses. In equation form:

(7.4.12) $\begin{array}{l} ROI = \frac{Σ (Present Value Revenues)}{Σ (Present Value Costs)} \end{array}$

For profitabiliy, ROI > 1.0

Another representation

(7.4.13) $\begin{array}{l} ROI = \frac{Σ (Present Value Revenues)}{Σ (Present Value Costs)} - 1 \end{array}$

A ROI value greater than zero represents profit (Σ (Present Value Revenues) greater than Σ (Present Value Costs).

An Example: At the calculated economic limit, an investment has cumulative present value revenues of $2,500,000.00 and sum present value costs of $200,000,000.00; the corresponding ROI is 0.25. In other words, ROI is 25%.

Table 7.4.6 contains ROI numbers based on cumulative cost and benefit present values for the three ownership-length scenarios in this example at the 15% discount rate.

Table 7.4.6. ROI Data for the Three Project Life Scenarios

Project Years	E(ROI)	Standard Deviation	Skewness
10	0.016	0.061	0.212
17	0.064	0.076	0.482
20	0.063	0.078	0.513

These data indicate that the ROI distributions are similar to the NPV and IRR distributions—the standard deviations increase with project life and the three distributions are all rightward-skewed.

The values of all of these financial and technical indicators will be compared with managements guidelines to help determine if this asset meets all expectations for investment.

7.4.5.3 Project Useful Life

In the abstract to this subchapter, we note that the expected economic lifetime of this project is 17.4 years. This is also a simulation-generated expected value that is computed as follows for each iteration of the simulation:

1.: Run the simulation, which in this example consists of a simulated price path and all calculations based on that price path
2.: Track individual annual NPVs—the discounted sum of benefits less costs for each year. For a project such as the one shown in this example, the first-year cost should always be negative because the initial capital investment is considered in the first year of the model. After that in projects such as this one ⁸ , the annual net NPVs should eventually become positive. These numbers should be tracked until a negative annual NPV occurs.
3.: The useful life is measured, on an annual basis, as the project year preceding the first observed downstream negative annual NPV.

In a 1000-run simulation, as was conducted for this report, 1000 useful-life variables will be collected. In this case, 17.4 was the average of all observed useful-life numbers.

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