The development of various investment analysis methods, which is nothing more than effective planning to determine the most appropriate time for the acquisition of an asset, is a daily work tool for personnel in charge of financial management.
Taking into account the impact on the results of for-profit organizations, the present work lists the Methods used for this purpose, together with a comparative assessment of these.
Introduction
A permanent activity in the business field, it is the analysis of the economic and financial situation of the same, from which to adopt decisions that contribute to improve their performance and thereby maximize their benefits.
To achieve the aforementioned objective, financial forecasts are used: short-term mainly aimed at the preparation of cash budgets and long-term ones that focus on future growth in sales and assets, as well as the financing of said growth..
All of the above shows that a good financial analysis must detect the strength and weaknesses of a business, particularly in the process of evaluating the profitability of investment projects that, regardless of their classification, which may differ between different authors, they are characterized by the occurrence of financial flows over time, they are essential for the entity as they include aspects such as replacement of equipment; replacement of projects; design of new products or services and expansion into other markets, to choose those that contribute to achieving a net increase in capital.
As can be seen, the universe of destination of the projects is very wide to which must be added the impact of the scale of the company's operation and the speed with which a decision must be taken (conjuncture) in an environment of scarce resources.
All this has motivated the development of various investment analysis methods that is nothing more than an effective planning to determine the most appropriate moment for the acquisition of an asset, which is related in the present work, together with a comparative valuation of these.
Theoretical fundament
1. Classification of projects
Companies classify projects in the following categories:
- Replacement: business maintenance, they are intended to replace equipment that is damaged, depreciated in its entirety or morally obsolete. Replacement: cost reduction, which is intended to replace useful but obsolete equipment. The forecast of these expenses is to reduce the cost of labor, materials and other items such as electricity. Expansion of existing products or markets: aims to expand stores or distribution facilities in currently served markets. Expansion into new products or markets:It is used to evaluate the expected expenses and benefits of a new product or service, with which it is intended to expand the company within a geographical area not currently covered. Safety or environmental projects: they relate to the expenses necessary to comply with government regulations, with employment contracts, with the terms of insurance policies. They are called mandatory investments or projects that do not produce income.
At the microeconomic level, the classification of the investment process is as follows:
- Economic: acquisition of assets and rights Financial: placement of savings in the financial market Legal: acquisition of assets and rights that can be the subject of a property right and are likely to form part of the patrimony.
At the macroeconomic level, the classification of the investment process only makes sense for economic investment, since financial and legal investments are mere operations between economic organizations.
2. Financial principles
First Financial Principle: "A monetary unit of today is worth more than a monetary unit of tomorrow." As a corollary of this principle, it can be pointed out that the fundamental work of financial activity is: "to efficiently transfer resources over time, which includes the evaluation and selection of sources and methods of financing."
Second Financial Principle: "A safe monetary unit is worth more than one with risk", which is based on the fact that most investors avoid risk whenever they can, without sacrificing profitability. Therefore, the work content of the financial area established in the previous section must be modified as follows to incorporate this aspect: «transferring resources with the minimum possible risk and efficiently over time, which includes the valuation and selection of sources and methods of financing and protection of resources ”.
III. Methods for evaluating projects
The investment evaluation and selection criteria can be summarized as shown in Table 1.
Table 1. Investment valuation and selection criteria
| Criterion type | Economic Characteristic | Methods |
|
Non-financial (static) |
They do not take into account the chronology of the various cash flows and the value of money over time.
They are simple calculations and are useful for the company. |
Cash Flow.
Accounting Rate of Return. Recovery period (Pay Back). Cost benefit relation. |
|
Financial (dynamic) |
They take into account the chronology of the different cash flows and the value of money over time by updating or discounting.
They are widely used as they homogenize the amounts of money received at different times. |
Net Present Value (NPV).
Internal Rate of Return (IRR). Profitability Index (IR). |
III.1. Non-financial methods
III.1.1. Cash flow method
This method offers information on the dynamics of the company and is an accounting instrument that reflects the flow of funds generated internally, obtained from a relation of money inflows and outflows (income and payable expenses) and provides a measure of self-financing.
Economic Cash Flow = Net Profit + Non-Disbursable Expenses
Note: Non-disbursable expenses are: amortization of intangible fixed assets; depreciation of tangible fixed assets; provision of bad accounts; amortization of deferred expenses; etc.
III.1.2. Accounting Rate of Return
The Accounting Rate of Return (TRC) method consists of comparing the accounting profit with the investment value, choosing the project with the highest TRC.
The TRC is obtained as the average of the profit after taxes divided by the amount of the initial investment as indicated in the following expression.
where:
n: number of periods of the investment.
B t: profit reported by the investment in period t.
B n: average annual net profit.
I: investment.
M i: initial amount of the investment.
The main weaknesses of this method can be summarized as:
- Accounting profits are used and not cash flows, therefore the marginal return on investment is not taken into account It does not take into account the value of money over time According to this criterion, projects with high income returns are preferable short duration, which is not always the case.
This indicator is similar to Return on Assets (ROA) or Return on Capital (ROE).
III.1.3. Recovery period (Pay Back)
It is a simple method, especially for small companies, which is based on determining the investment cost recovery period and selects among mutually exclusive projects the one whose initial recovery period is shorter and the decision to invest or not is made by comparing the Recovery period of the project investment amount with some predetermined standard.
In practice, the Payback Period (P r) is determined by accumulating the successive annual flows until the sum reaches the initial cost of the investment in time (t) that satisfies the condition shown in the following expression:
where:
C j: cash flow in period j
I j: investment in period j
In the case that the flows are constant, the value of P r is determined through the following expression:
Table 2. Advantages and disadvantages of this method
| Advantage | Disadvantages |
| - It is easy to calculate and apply.
- It is cheap, which is why it is currently used to evaluate small capital expenditure decisions when the cost of the other methods is greater than the benefits of choosing better choices among the alternatives. - Provides a measure of the liquidity of the project or the speed with which the invested cash is repaid. - It is useful for companies with little cash availability. |
- Ignore cash flows that extend beyond the payback period, which is a bias for long-term projects.
- It does not consider the value of money in time. - It does not consider all the cash flows of the investment project, and therefore does not include them in the analysis. Likewise, it does not consider the order in which the benefits are obtained, which is of financial interest. - If the company sets a deadline date, only short-term projects will be accepted. |
However, this method can be attractive in investments categorized as very risky, in which funds far away in time are less likely to be carried out.
III.2. Financial methods
III.2.1. Determination of the discount rate (Capital Asset Pricing Model, CAPM)
One of the most important problems in finance is being able to determine the price of risk and thus being able to use an appropriate measure of risk, be it of an investment project, the risk of a company or any financial asset.
Premise: The return on any risky financial asset in equilibrium is a function of its covariance with the risk of the market portfolio return.
The CAPM is a mathematical model that considers the following assumptions about investors and the set of investment opportunities that exist:
- Investors are risk-averse individuals, who always maximize the profit they expect to obtain at the end of a period of time. Investors are price takers (they cannot influence the price formation process) and have homogeneous expectations about the returns of financial assets, which has a normal distribution There is a risk-free asset (government instrument), such that investors can lend or borrow in limited amounts at the risk rate r fFinancial assets exist in limited quantity, are stock exchange (there are always buyers and sellers) and are perfectly divisible. There is no friction in the marketing of financial assets (the interest rate for lending and borrowing is the same), the information does not It has a cost and is available to all investors simultaneously. There are no market imperfections such as taxes, regulations or restrictions on sales.
The Characterization of the market portfolio in terms of the expected return on investment corresponds to that shown in figure 1, where:
E (r p): expected return of a portfolio.
E (r m): performance of the market portfolio.
σ (r p): standard deviation of a portfolio
σ (r m): standard deviation of the market portfolio
II ': risky asset
Under these conditions, the Capital Market Line E (r p) is given by the following expression:
Be a market portfolio structured in:
- risky asset I: a% invested, with a rate of return r i market portfolio M: (1-a%) invested, with a rate of return r m.
Under these conditions, the rate of return of the combined portfolio, composed of the risky asset and the market portfolio, r p, is given by: and the mathematical expectation of r p is given by:
The variance of r p is given by:
Substituting the expressions 4 and 5 in 3 and grouping appropriately, we obtain:
Note the following:
- , represents the variance of the returns of the risk-free asset, hereinafter,., represents the variance of the risk-free market portfolio, hereinafter, represents the covariance of returns between the risk-free asset and the market portfolio , hereinafter,
Therefore, the variance and standard deviation of the profitability of the combined portfolio, r p, is given by the following expressions:
Now the variation in the expectation (mean) and in the standard deviation with respect to the percentage of the portfolio a, invested in assets with risk is obtained from the calculation of the partial derivative of expressions (1) and (6) with respect to the parameter a, as follows:
Sharpe and Treynor's discovery that in equilibrium the market portfolio already has the value I weighted by its value w i, therefore the percentage a of expressions 7 and 8 represents the excess demand for an asset with risk.
At equilibrium, the excess demand for the risky asset equals zero and prices will adjust until all assets belong to investors. Therefore, if expressions 7 and 8 are evaluated for a equal to zero, it is possible to determine the price relationship in equilibrium as indicated below:
From the previous equations it can be determined that the slope of the curve described by the relationship between the expected return of the asset with risk and the variance of this M r is given by:
Now at the market equilibrium point (M r) it must be equal to the slope of the Capital Market Line (M c), from where, equating expressions (1) and (11), we have:
Simplifying the term in the previous expression, the CAPM equation is obtained:
The above equation indicates that the required rate of return for any asset has two components:
- Risk-free rate, r f. Risk rate: includes the expected return on risk, obtained as the product of the risk premium obtained from the difference by the amount of risk obtained as.
II.2.2. Net Present Value (NPV)
It is an indicator of recovery of values, since it compares the present value of the expected future benefits of a project with the present value of the expected cost.
The Net Present Value (NPV) is the present value of the future returns discounted at the cost of capital of the company, less the cost of the investment and for its determination the following expression is used, where:
C 1, C 2,… C n: Net cash flows in each period.
r i: Appropriate discount rate or cost of capital of the project in each period
C 0: initial cost of the project (initial investment)
n: Number of project duration periods (expected life)
For the purposes of NPV analysis, projects whose NPV is positive are accepted and if it is negative, it must be rejected, while if two or more projects are mutually exclusive, the one with the highest NPV must be chosen, the higher the value. the most attractive NPV is.
A positive NPV indicates that the investment in the project produces surpluses greater, by the amount of the NPV, than those that could be obtained by investing the same amount at the investment rate.
The fundamental advantage of this method is that it considers the value of money over time and its main drawback is the difficulty of specifying the discount or discount rate, r i, which must consider, in addition to the interest rate, the risk it represents. the project.
Another factor that must be considered prior to choosing a portfolio of exclusive projects is whether there are differences between the number of periods of each one, to proceed to homogenize them, assuming that they are repeated in time to infinity.
For this purpose, the following expression can be used that is deduced by considering the NPV of the flow of projects repeated at constant scale in an infinite way.
This alternative, although it homogenizes the projects with different duration, has the disadvantage that it is not real that the discount rate that can be applied for the real duration of the projects, is maintained beyond this period.
III.2.3. Internal Rate of Return (IRR)
This indicator is the maximum benefit that can be expected from the project and is based on obtaining the rate that equals the present value of the benefits with the cost (initial disbursement), that is, it is the discount rate that makes the NPV of the project equal to zero.
Therefore, the Internal Rate of Return (IRR) is the discount rate that equals the present value of the expected future cash flows with the initial cost of the project, therefore it corresponds to the yield to maturity on a bond. It is a discounted cash flow method.
The IRR is the discount rate that equates the present value of the expected future cash flows, or income, with the initial cost of the project, which is mathematically expressed according to the equation where r is a value such that the sum of the discounted income equals the initial cost of the project, which equates the equation to zero.
Mathematically, the value of the IRR is obtained by solving the following equation, where the symbols have the same meaning as in the case of NPV.
The criteria for selecting a project, once the IRR has been obtained by solving the previous equation, corresponds to one of the following three cases:
- IRR> i, and the investment is interesting IRR = i, and the investment is indifferent. IRR <i, and the investment is rejected.
An advantage of this method is that it can be calculated from the projected investment flows, without the need to know the cost of capital of the company, which requires more complex calculations.
The limitations of the use of IRR in project evaluation are fundamentally due to:
- It is based on the hypothesis of reinvestment or financing of the interim net receipts or payments at the rate r, that is, the net payments are reinvested at a yield r and the cost of the net payments is r, which is unreal. existence of various types of profitability in some investments, when loans are required in intermediate periods of the project as illustrated in table 3. In this case, the Rule of Change of Sign of Descartes establishes that there will be as many positive roots for 1 + r, as sign changes in the flow values that define the investment.
Table 3. Projected cash flows for four projects with multiple IRRs
| Draft | Periods | ||||||
| 0 | one | two | 3 | 4 | 5 | 6 | |
| TO | -100 | Four. Five | 25 | fifteen | 40 | 30 | 30 |
| B | -100 | 40 | 33 | 30 | 30 | 28 | -fifteen |
| C | -100 | 56 | Four. Five | -twenty | 37 | 30 | 25 |
| D | -100 | 54 | 48 | 37 | -25 | 40 | -twenty |
: Indicates sign changes in updated flows
Table 4. Descartes criterion for the example in table 3.
| Draft | Periods | Sign changes | Number of real roots | ||||||
| 0 | one | two | 3 | 4 | 5 | 6 | |||
| TO | - | + | + | + | + | + | + | one | a |
| B | - | + | + | + | + | + | - | two | two or none |
| C | - | + | + | - | + | + | + | 3 | three or one |
| D | - | + | + | + | - | + | - | 4 | four; two or none |
Based on this behavior, investments can be classified as:
- Simple: when there is only one value of r and therefore there are no sign changes in the updated flows. Not simple: when there are two or more positive roots. In these cases there are several sign changes in the cash flows and in practice they can be considered as the sum of several independent investments. Mixed: are those investments in which, if they have multiple roots, in one of the intermediate periods the updated flow becomes negative, which occurs in projects that receive most of their return at a given time as illustrated in the example collected in table 5.
Table 5. Example of the behavior of cash flows in mixed investments
| Flow: | Periods | Σ | ||||||
| 0 | one | two | 3 | 4 | 5 | 6 | ||
| Of box | -1500 | 600 | 700 | 1200 | -900 | 150 | 143.75 | - |
| Discounted at 15% | -1500 | 521.74 | 529.30 | 907.37 | -680.53 | 113.42 | 108.70 | 0 |
The example shows that the discounted flows at the end of the second year are positive and that the amount of capital employed from the second year is negative, since it is the project that finances the company. In this case there will be multiple positive roots or a single value of the IRR that is not economically significant.
In this case, the extended IRR method can be used, which consists in that cash flows are discounted at the cost of capital of the company and not at the type of profitability of the project until they are offset by positive flows.
In practice, obtaining the IRR value in any of the previous cases is equivalent to determining the IRR in the following equation, obtained by multiplying the expression (1) by the magnitude (1 + r) n, where for simplicity r is used to represent the IRR value.
(1 + r) n C 0 + (1 + r) n-1 C 1 + (1 + r) n-2 C 2 +… + (1 + r) C n-1 + C n = 0
If the substitution x = 1+ r is carried out, the following working expression is finally obtained, which mathematically corresponds to a polynomial of degree n, whose coefficients constitute the net flows of each period.
P (x) = C 0 x n + C 1 x n-1 + C 2 x n-2 +… + C n-1 x + C n = 0
Therefore, the determination of the IRR corresponds to the search for the real and positive roots (the complex ones and the negative values do not make economic sense) of a polynomial of degree n. Mathematically it is shown that a polynomial of degree n with real coefficients has n roots in the field of complex numbers, which leads to the following three questions:
- How many values of IRR are mathematically possible? The answer to this question is provided by Descartes ' Rule «the number of positive roots of the equation P (x) = 0 is not greater than the number of sign variations of the polynomial P (x) and can be differentiated from this number by a even quantity '. Therefore, there may be projects with multiple IRR values in the mathematical sense. When there are multiple mathematical values of the IRR, what is their interpretation? This situation may be an indication that the nature of the project consists of more than one stage and its division is recommended for analysis or that it requires a greater initial investment to make its behavior unique. How to determine the IRR values? To determine the magnitude of the IRR, various methods can be used according to the characteristics, which can be grouped into the five cases that are analyzed separately below.
Case I: If there is a unique IRR value for the project, regardless of the number of periods that the project consists of, it can be calculated using the IRR () function of the EXCEL Electronic Spreadsheet that has as an argument the projected values of flow and an initial IRR value, which is used for the internal calculation algorithm and can be omitted, as illustrated in Figure 2.
Case II: If the project consists of a period. In this case, the problem statement corresponds to the following equation:
The value of r can be obtained by solving its value in the previous equation, it is obtained:
Case III: When the investment is for two periods. For these conditions, the IRR is given by:
Multiplying the previous expression by the term (1 + r) 2 and substituting (1 + r) for x, the following is obtained as a working expression:
C 0 x 2 + C 1 x + C 2 = 0
The solution of this equivalent equation corresponds to the general solution of the quadratic equation which, applied to the IRR, takes the following form:
Note that if in the expression for the determination of x, the magnitude of the expression contained in the radical is negative, there are no real values of the IRR for the analyzed project, which can happen if the following two conditions are met:
- In the second period a loan is required (C 2: negative). The absolute value of 4C 0 C 2 is higher.
Another particular case of interest is one in which the sum of the cash flows of the three periods is zero, that is,. Under these conditions it is true that.
Substituting the previous expression in (1) we obtain:
In correspondence with the fact that the model problem is an inversion, C 0 is negative, therefore r 2 is given by:
As a summary of this case, it can be pointed out that one of the two values of the IRR is always zero, while the sign (positive or negative) of the other value depends on the relationship between the amount of cash flow in the first period (C 1) and the initial investment (C 0), is greater or less than two, that is, for there to be a positive IRR in the project it is necessary that the flow in the first period is, at least, the double the initial investment.
Case IV: When the investment is for three periods. For these conditions, the IRR is obtained by solving the following equation:
Multiplying the previous expression by the term (1 + r) 3 and substituting (1 + r) for x, the following is obtained as a working expression:
C 0 x 3 + C 1 x 2 + C 2 x + C 3 = 0
The previous equation is equivalent to the determination of the roots of a third degree polynomial, for which the Cardano procedure (Annex E2) can be used in combination, which leads to one of the following three variants:
- Variant # 1: One real solution and two complex conjugate, you are the latter with no financial value. In this case the only problem is when the real root is negative, since it does not make economic sense and the formulation of the problem and the calculations must be analyzed. Mathematically, this case corresponds to the condition and the value of the IRR is given by the expression, where: Variant # 2: Three real solutions, two of them equal. In practice, this case provides two IRR values, among which the analyst must choose by applying additional criteria that provide rationality to the selected magnitude. Mathematically, this case corresponds to the condition and the IRR values are given by the expressions:; and.Variant # 3: Three real solutions unequal to each other. This case requires, like the previous one, a complementary financial analysis, to choose which of the three solutions makes economic sense. The analytical determination of the three values requires the extraction of the cube root of a complex number, which is illustrated in Annex D. In this case the possible values of a and b are given by expressions in table 6, in which q 0 for simplicity is assumed to be zero.
Table 6. Expressions for the determination of a and b.
| k | Possible a values: | Possible b values: |
| 0 | ||
| one | ||
| two |
Case V: When the investment consists of four periods, there are four, two or no roots with positive NPV. For these conditions, the IRR is obtained by solving the following equation:
Multiplying the previous expression by the term (1 + r) 4 and substituting (1 + r) for x, the following is obtained as a working expression:
C 0 x 4 + C 1 x 3 + C 2 x 2 + C 3 x + C 4 = 0
The previous equation is equivalent to determining the roots of a third degree polynomial, for which Ferrari expressions can be used, which in practice is cumbersome and it is preferable from this number of roots to use an iterative algorithm. For quantities of periods greater than four, there are no algebraic procedures that allow obtaining the roots of a polynomial in terms of its coefficients.
III.2.4. Profitability index
The Profitability Index is used to decide between alternatives with similar NPV and IRR when there is a shortage of resources, since this indicator measures how much each invested monetary unit reports. For its determination the following expression is used:
III.2.5. Comparison between NPV and IRR Methods
The NPV method clearly and accurately indicates whether the completion of a project is justified, since its benefits exceed its costs (initial investment) evaluated at a discount rate that reflects the cost of capital. It is very useful to select from a group of projects, the one that provides the greatest benefit, since it provides comprehensive information on the project and does not lead to an evaluation of the characteristics of cash flow throughout the project, which is of particular interest in the case of long-term projects.
In the case of IRR, an advantage is that it can be obtained using the data corresponding to the project's cash flows without having to know the company's cost of capital.
From the foregoing, it can be seen that the NPV and IRR criteria can lead to different choices because both criteria measure different things: the IRR provides the relative profitability of the project and the NPV the absolute profitability.
- If two projects are independent, the NPV and IRR criteria coincide. If the projects are mutually exclusive, a conflict occurs when the cost of capital is less than the IRR and the NPV greater than zero. There are two fundamental conditions that can cause conflicts between NPV and IRR criteria: when there are differences in the size (scales) of the projects, that is, when the cost of one project is higher than the other and when there are differences in opportunity, that is, the timing of the flows of cash from projects differs in such a way that most of the cash flows of a project are presented in the first years and in the other at the end. These factors advise that when mutually exclusive projects are evaluated,especially those with difference in scale and opportunity in time, the NPV should be used.
III.2.6. Selection criteria based on the capital available in the company
The amount and types of projects that can be chosen vary depending on the capital available in the company for investment in new projects, identifying the following four possible alternative types:
- Constant capital company and independent projects: The projects with the highest NPV and IRR are chosen from among the proposed projects, until the amount of available capital is reached. Constant capital company and mutually exclusive projects: The project with the highest NPV or IRR is chosen, the amount of which does not exceed the constant capital available. Unrestricted capital company and independent projects: All projects that meet the condition of NPV greater than zero and IRR greater than the cost of capital of the company are chosen. Unrestricted capital company and mutually exclusive projects: The one with the highest NPV and IRR is chosen.
- Case discussion
Example 1. Let projects A, BC, D, E, F and G be alternative investments of the company Mesa & PP SA with the cash flows shown in table 7. Select the most attractive project using the following methods:
- a) Cash Flow b) Accounting Rate of Return c) Payback Period (Pay Back) d) Present Value (Present Value) e) Present Value with Different Durationsf) Profitability Indexg) Internal Rate of Return (IRR) h) Discuss in a comparative way the results achieved in the previous sections.
Table 7. Cash flows of the six projects under study.
| B | C | D | AND | F | G | H | I | |
| period | Example 1. Project cash flows | |||||||
| Project A | Project B | Project C | Project D | Project E | Project F | Project G | ||
| 4 | 0 | -1200.00 | -1200.00 | -1200.00 | -6000.00 | -10000.00 | -6000.00 | -1200.00 |
| 5 | one | 200.00 | 300.00 | 1050.00 | 4300.00 | 3500.00 | 4900.00 | 400.00 |
| 6 | two | 1000.00 | 850.00 | 100.00 | 1400.00 | 1500.00 | 1050.00 | 850.00 |
| 7 | 3 | 550.00 | 450.00 | 570.00 | 500.00 | 1000.00 | 950.00 | 400.00 |
| 8 | 4 | 370.00 | 550.00 | 100.00 | 500.00 | 400.00 | 700.00 | 485.00 |
| 9 | 5 | -100.00 | 580.00 | -1500.00 | ||||
| 10 | 6 | 300.00 | 600.00 | 3000.00 | ||||
| eleven | 7 | 6000.00 |
Reply:
- a) Table 8 shows the results obtained when applying the Cash Flow Methods to the six projects evaluated, using the EXCEL Electronic Spreadsheet, indicating the calculation equation used.
Table 8. Results by the Cash Flow Method.
| index | Example 1. Cash Flow Method | ||||||
| Project A | Project B | Project C | Project D | Project E | Project F | Project G | |
| Calculation B: | SUM (C4: C8) | SUM (D4: D8) | SUM (E4: E10) | SUM (F4: F10) | SUM (G4: G11) | to) | to) |
| B: | 920.00 | 950.00 | 820.00 | 1880.00 | 3900.00 | 1600.00 | 935.00 |
| Selection: | * |
- a) The calculations for these projects are carried out in a similar way to the previous ones.
Selected project: E.
- b) Table 9 shows the results for the Accounting Rate of Return method in the case of the six projects evaluated.
Table 9. Results by the Accounting Rate of Return Method.
| B | C | D | AND | F | G | H | I | |
| index | Example 1. Accounting Rate of Return method | |||||||
| Project A | Project B | Project C | Project D | Project E | Project F | Project G | ||
| 2. 3 | Calculation I: | ABS (C4) | ABS (D4) | ABS (E4) | ABS (F4) | ABS (G4) | ABS (H4) | ABS (I4) |
| 24 | Calculation Bm: | AVERAGE (C5: C11) | to) | to) | to) | to) | to) | to) |
| 25 | TRC calculation: | C27 / C26 | D27 / D26 | E27 / E26 | F27 / F26 | G27 / G26 | H27 / H26 | I27 / I26 |
| 26 | I: | 1200.00 | 1200.00 | 1200.00 | 6000.00 | 10000.00 | 6000.00 | 1200.00 |
| 27 | Bm: | 530.00 | 537.50 | 336.67 | 1313.33 | 1985.71 | 1900.00 | 533.75 |
| 28 | TRC: | 0.44 | 0.45 | 0.28 | 0.22 | 0.20 | 0.32 | 0.44 |
| 29 | Selection: | * |
- a) The calculations for these projects are carried out in a similar way to the previous ones.
Selected project: B.
- c) Table 10 shows the results for the Recovery Period Method
Table 10. Results by the Recovery Period Method.
| period | Example 1. Payback Period Method | ||||||
| Project A | Project B | Project C | Project D | Project E | Project F | Project G | |
| one | -1000.00 | -900.00 | -150.00 | -1700.00 | -6500.00 | -1100.00 | -800.00 |
| two | 0.00 | -50.00 | -50.00 | -300.00 | -5000.00 | -50.00 | 50.00 |
| 3 | 550.00 | 400.00 | 520.00 | 200.00 | -4000.00 | 900.00 | 450.00 |
| 4 | 920.00 | 950.00 | 620.00 | 700.00 | -3600.00 | 1600.00 | 935.00 |
| 5 | 520.00 | 1280.00 | -5100.00 | 1600.00 | |||
| 6 | 820.00 | -2100.00 | 1600.00 | ||||
| 7 | $ 3 900.00 | $ 1,600.00 | |||||
| Selection: | * |
Selected project: A.
- d) Table 11 shows the results for the Present Value Method.
Table 11. Results by the Present Value Method.
| B | C | D | AND | F | G | H | I | |
| index | Example 1. Present Value Method (NPV) | |||||||
| Project A | Project B | Project C | Project D | Project E | Project F | Project G | ||
| 46 | NPV calculation: | NAV ($ C $ 48 C5: C11) + C4 | to) | to) | to) | to) | to) | to) |
| 47 | GO: | $ 474.20 | $ 488.96 | $ 440.99 | $ 482.10 | - $ 713.00 | $ 514.17 | $ 497.90 |
| 48 | r (%): | 10.00% | ||||||
| 49 | Selection: | * |
- a) The calculations for these projects are carried out in a similar way to the previous ones.
Selected project: F.
- e) Table 12 shows the results for the Present Value Method with Different Durations.
Table 12. Results by the Present Value Method with Different Durations.
| B | C | D | AND | F | G | H | I | |
| index | Example 1. Present Value Method for Projects with Different Durations | |||||||
| Project A | Project B | Project C | Project D | Project E | Project F | Project G | ||
| 54 | Infinite VAN calculation: | to) | to) | to) | to) | to) | to) | to) |
| 55 | VAN infinite: | 281.76 | 290.52 | 281.88 | 308.15 | -471.20 | 305.51 | 295.84 |
| 56 | N: | 4 | 4 | 6 | 6 | 7 | 4 | 4 |
| 57 | Selection: | * |
- a) C47 * POWER (1 + $ C $ 48 C56) / (1 + POWER (1 + $ C $ 48 C56))
Selected project: D.
- f) Table 13 shows the results for the Profitability Index Method.
Table 13. Results by the Profitability Index Method.
| B | C | D | AND | F | G | H | I | |
| period | Example 1. Profitability Index Method | |||||||
| Project A | Project B | Project C | Project D | Project E | Project F | Project G | ||
| 62 | 0 | -1200.00 | -1200.00 | -1200.00 | -6000.00 | -10000.00 | -6000.00 | -1200.00 |
| 63 | one | 181.82 | 272.73 | 954.55 | 3909.09 | 3181.82 | 4454.55 | 363.64 |
| 64 | two | 826.45 | 702.48 | 82.64 | 1157.02 | 1239.67 | 867.77 | 702.48 |
| 65 | 3 | 413.22 | 338.09 | 428.25 | 375.66 | 751.31 | 713.75 | 300.53 |
| 66 | 4 | 252.71 | 375.66 | 68.30 | 341.51 | 273.21 | 478.11 | 331.26 |
| 67 | 5 | -62.09 | 360.13 | -931.38 | ||||
| 68 | 6 | 169.34 | 338.68 | 1693.42 | ||||
| 69 | 7 | 3078.95 | ||||||
| 70 | IR calculation: | to) | to) | b) | to) | c) | to) | to) |
| 71 | GO: | 1.40 | 1,407 | c) | 1.08 | 0.93 | 1.09 | 1,415 |
| 72 | r: | 10.00% | ||||||
| 73 | Selection: | * |
- a) SUMA (C63: C66) / ABS (C62) b) (SUMA (E63: E66) + E68) / (ABS (E62 + E67)) c) (SUMA (G63: G66) + G68 + G69) / (ABS (G62 + G67))
Selected project: F.
- g) Table 14 shows the results for the IRR Method.
Table 14. Results by the IRR Method.
| B | C | D | AND | F | G | H | I | |
| index | Example 1. Internal Rate of Return (IRR) method | |||||||
| Project A | Project B | Project C | Project D | Project E | Project F | Project G | ||
| 78 | TIR calculation: | IRR (C4: C11 $ C $ 80) | IRR (D4: D11 $ C $ 80) | to) | to) | to) | to) | to) |
| 79 | IRR: | 26% | 27% | 30% | fifteen% | 8% | 16% | 28% |
| 80 | r_initial (%): | 8.00% | ||||||
| 81 | Selection: | * |
Selected project: C.
- h) Comparative discussion.
The first result that is evident from the analysis of the selection obtained when evaluating the seven projects of the example for each of the methods collected in the literature is that: «the same set of projects analyzed by different methods leads to different selections that, in the limit, as illustrated in the case study, the selection does not coincide by any of the methods. Therefore, method selection is a high-impact process and must be done carefully.
On the other hand, each of the described methods exhibits a rationality in their proposal, by taking an accounting or financial approach, which leads to another conclusion derived from the example: «using more than one method provides levels of comparison of approaches and can be very useful, as long as it does not constitute a financial burden and an appreciable delay in obtaining the information necessary to make the decision.
Finally, another lesson can be drawn from this case: "no method replaces the analysis of the Finance group, they only represent a tool that facilitates decision-making."
Example # 2. Let projects I, II, III, IV and V be investment alternatives of the company Rich & Poor SA with the cash flows shown in table 15. Determine:
- a) The Internal Rates of Return using the general solution of the Equation of a Second as well as the IRR and the NPV through the EXCEL spreadsheet. b) Analyze the results obtained in the previous section.
Table 15. Cash flows of the projects to evaluate.
| B | C | D | AND | F | G | |
| period | Example # 2. Cash flows | |||||
| Project I | Project II | Project III | Project IV | Project V | ||
| 4 | 0 | -1900 | -900 | -2500 | -6000 | -6000 |
| 5 | one | 2800 | 2800 | 8000 | 8000 | 5500 |
| 6 | two | -900 | -1900 | -6000 | -2500 | 2500 |
Answer:
- a) The IRR values obtained by the requested methods are shown in table 16.
Table 16. IRR values for the projects analyzed.
| B | C | D | AND | F | G | |
| index | Example # 2. Internal Rate of Return (IRR) method | |||||
| Project I | Project II | Project III | Project IV | Project V | ||
| 13 | Calculation sum flows: | SUM (C4: C6) | SUM (C4: C6) | SUM (C4: C6) | SUM (C4: C6) | SUM (C4: C6) |
| 14 | TIR1 calculation: | (-C5 + ROOT (C5 * C5-4 * C6 * C4)) / (2 * C4) -1 | to) | to) | to) | to) |
| fifteen | TIR2 calculation: | (-C5-ROOT (C5 * C5-4 * C6 * C4)) / (2 * C4) -1 | b) | b) | b) | b) |
| 16 | TIR EXCEL: | IRR (C4: C6,0.1) | IRR (D4: D6,0.1) | c) | c) | c) |
| 17 | Calculation VAN # 1: | NAV ($ C $ 25 C5: C6) + C4 | NAV ($ C $ 25 D5: D6) + D4 | d) | d) | d) |
| 18 | Calculation VAN # 2: | NAV ($ C $ 27 C5: C6) + C4 | NAV ($ C $ 27 D5: D6) + D4 | and) | and) | and) |
| 19 | Calculation VAN # 3: | NAV ($ C $ 29 C5: C6) + C4 | NAV ($ C $ 29 D5: D6) + D4 | F) | F) | F) |
| twenty | Sum flows: | 0.00 | 0.00 | -500.00 | -500.00 | 2000.00 |
| twenty-one | IRR1: | -52.63% | 0.00% | 20.00% | -50.00% | -133.33% |
| 22 | IRR2: | 0.00% | 111.11% | 100.00% | -16.67% | 25.00% |
| 2. 3 | IRR according to EXCEL: | 0.00% | 0.00% | 20.00% | -16.67% | 25.00% |
| 24 | VAN1: | - $ 145.75 | $ 98.11 | - $ 80.34 | - $ 933.84 | $ 672.97 |
| 25 | Rate1: | 15.00% | ||||
| 26 | VAN2: | - $ 191.67 | $ 113.89 | $ 0.00 | - $ 1,069.44 | $ 319.44 |
| 27 | Rate2: | 20.00% | ||||
| 28 | VAN3: | - $ 236.00 | $ 124.00 | $ 60.00 | - $ 1,200.00 | $ 0.00 |
| 29 | Rate3: | 25.00% |
- b) Analysis of the results
The results shown in table 16 reveal the following aspects:
- It is verified that in those projects whose final cash flow is zero (I; II), one of the IRR values is zero. Likewise, it is appreciated that the other value of the IRR can be negative to positive. It is verified that when a project is evaluated at a value of the IRR, its NPV is null (III; V) Projects whose two values of the IRR are positive (III) or null (I), they can have a negative NPV. The algorithm used by EXCEL always chooses the IRR closest to zero, which corresponds to the most rational value.
Conclusions
As a result of this work, it can be noted that the different Methods for the evaluation of Investment Projects were presented, as well as their advantages and limitations, which are illustrated in the examples developed for this purpose.
Similarly, it is clear from the issues raised the importance of the irreplaceable "intelligent" evaluation of the results and the selection of the evaluation method used is confirmed.
Bibliography
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