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ESTIMATING RISK IN CREDIT CONDITION ANALYSIS: LATIN HYPERCUBE SIMULATION Pablo Rogers, Universidade Federal de Uberlândia
[email protected] Kárem C. S. Ribeiro, Universidade Federal de Uberlândia
[email protected] Almir Ferreira de Sousa, Universidade de São Paulo
[email protected] Abstract That work has for objective to present a methodology that incorporates the analysis multiperiods in the changes of the credit conditions and that it considers the risk of the estimates in those changes. That last aspect will be gotten by the inclusion of the simulation process by Latin Hypercubes, it will allow to esteem the risk of the change in the credit conditions to produce value for companies. To conclude that the probability analysis as presented supplies useful information to the managers, when incorporating the value of the money in the time and to esteem the risk in credit conditions. Key-Words: Credit Conditions, Latin Hipercube, Monte Carlo Simulation. 1. Introduction The word credit, conceptually speaking, refers to an individual or company’s willingness to give, for a certain period of time, part of its assets or provide a third party with outsourced services, expecting a future payment. According to Lemes Júnior, Rigo and Cherobim (2002, p.442) “it implies in proceeding with receiving the value of the credit in a future date”. Due to the fact that credit concession is a delivery of capital to third parties, it demands for a large amount of cash flow so that it can be financed, because “credit concession is the same as investing in a client, when the investment is associated with a product or service” (ROSS, WESTERFIELD and JAFFE, 1995, p. 574). It becomes common to analyze credit policy in three main aspects: credit conditions or sales terms, credit analysis and selection and collection and monitoring policy (ASSAF NETO, 2003; BREALEY e MYERS, 1992; BRIGHAM e HOUSTON, 1999; GITMAN, 2002 e 2004; GITMAN e MADURA, 2003; LEMES JÚNIOR, RIGO e CCHEROBIM, 2002; ROSS, WESTERFIELD, e JAFFE, 1995; SANVICENTE, 1997; SCHERR, 1989). In the relation to credit conditions, or sales terms, most finance manuals, as mentioned in the previous paragraph, base decisions on the examples of analysis of one only period of time (not considering future cash flows) and do not consider the risks of estimates not being fulfilled as expected (they are based on forecasted numbers). The current paper aims to present a methodology which overcomes these limitations by incorporating to multi-period analysis the HL simulation process. With the help of the Crystal Ball 2000.5 software, the analysis developed will allow us to measure the probability of changes to credit conditions in creating value for the company: the performance measure considered in the analysis was the additional net present value ( NPV) calculated with the difference between the proposed credit condition and the currently used credit conditions. Moreover, we aim to compare, specifically, the value of the ( VPL) obtained through the LH process and the one obtained through the MCS (Monte Carlo Simulation). These two simulation methods are different in theory. The general and specific objective will be
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developed in a later section. The following section will describe the main concepts in regards of credit concession decisions, as well as, the LH simulation process in comparison to the MCS. In section four we’ll list the main conclusions attained in the present article. 2. Bibliographical Review 2.1. Credit Conditions According to Scheer (1989, p. 159-162) the main key-variables affected by credit decisions are: sales collection, investments in stock, sales costs, discount and uncollectible dept expenses, collection costs, capital expenses, effects on revenue taxes (IR) and sales rescue and recovery. Assaf Neto e Silva (2002, p. 109) classified these key-variables in main four: •
Capital investment: sales volume increase, caused by a change in the credit policy may incentive a quicker recovery of investment, increasing its liquidity and reducing the risk.
•
Investment in stock: the smaller the sales volume of a company, the less the need for stock inversion before demand.
•
Collection expenses: Includes all incremental expenses resulting from collection department, letters sent to clients with outstanding debt, administrative staff’s time, legal expenses, need for more employees, etc.
•
Expenses with debtor uncertainty: Probability of losses as a result of total sales in installments.
Finance Manuals simplify variables, which affect credit decisions, where there are changes in the credit policy as: sales volumes, investments in receivables and expenses with debtor uncertainty. Credit conditions include the period for which the credit is granted, discount for cash purchase, and the credit tool type (ASSAF NETO, 2003; BREALEY e MYERS, 1992; BRIGHAM e HOUSTON, 1999; GITMAN, 2002 e 2004; GITMAN e MADURA, 2003; LEMES JÚNIOR, RIGO e CCHEROBIM, 2002; ROSS, WESTERFIELD, e JAFFE, 1995; SANVICENTE, 1997). In general, perhaps for didactic purposes, manuals analyze changes in credit conditions (especially discount and term) and its influence on sales volume investments on receivables and expenses with debtor uncertainty in one only period, as exemplified in chart 1. As we can see in chart 1, this methodology measures the net profit considering the opportunity cost for the company in the period before changes in the credit conditions and the period of time after these changes: Changes in credit conditions are treated as any other investment decisions. However, for this decision making process the only cash flow considered relevant in measured in one period of time only. Actually, since changes in credit affect the value of receivables, we must treat these changes as investments or redeem of investments made by the company in its clients, therefore, studied with investment analysis techniques commonly accepted, specially through NPV and IRR (interest rate of return), once these express the economic reality of an investment. Moreover, as all investment analysis processes include estimates, there’s a possibility of prediction error, meaning, we must incorporate the risk in investment decisions through techniques such as: scenario analysis, sensitivity analysis, (certainty equivalents), discount rates adjusted to risk and simulation methods.
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Consider that the Simulated Company is planning to introduce a 2% discount for payments to be made within 10 days after the purchase. The average collection time is thirty days, non-cash sales make up for a total of 6.000 units at a R$ 100,00 price per unit. Variable costs make up for R$ 60,00 per unit. The company estimates that by introducing the discount, sales will increase in 5%, and 50% will be of noncash sales. We estimate that the average collection time will be reduced to fifteen days, and losses with uncollectable will decrease from 2% to 1% of sales. The return required by the company on investments with the same risk is 10%. Receivable flow Original Plan =
360 = 12 30
Proposeded Plan =
Net Profit Increase Simplified Calculation
360 = 24 15
Increased Contribution Margin [ 300 units × (100 – 60)] Increase Cost in Receivables
(60 × 6.300) 24 (60 × 6.000) B) Investiment with Original Plan = 12 Cost of Increased Investiment [(B-A) × 0,10] A) Investiment with Proposed Plan =
Increased Cost with debtor uncertainty C) Cost with Proposed Plan (0,01 × 6.300 × 100) D) Cost with Original Plan (0,02 × 6.000 × 100) Increased Cost with Uncollectables (C-D) Cost of Financial Discount (0,02 × 0,50 × 100 × 6.300) = Profit Variation result of the proposal
12.000 (15.750)
30.000 1.425 (6.300) 12.000
5.700 (6.300) 12.825
Chart 1 – Traditional Methodology for Credit Condition Decisions Source: Rogers, Dami e Ribeiro (2004, p. 6). Simulation methods application becomes more robust when basing investment decisions on results obtained within a confidence interval, thus allowing managers to visualize as infinity of possible scenarios. 2.2. Latin Hypercubes and the Monte Carlo Simulation In order for the simulation process to be present in the analysis, all we have to do is verify if any variables of the problem assume the randomness condition. Specifically in the case of credit condition decisions, which can be analyzed as investments in clients’ cash flow, the simulation tool becomes a formal and efficient technique. The simulation is an attempt to replicate a real system, through the construction of a mathematical model as similar to reality as possible. Opposed to the analytical deterministical methods which aim at finding great solutions for problems, simulation aims at modeling a system and observing how entry parameter variations affect its outputs variables / inputs variables. A practical visualization of the stages in the computer simulation process are described in Chart 2. With the advancements in computers the simulation process has become quite accessible for the analysis of various types of problems.
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We can highlight among many simulation methods, specifically applied to investment analysis, the following: Latin Hypercubes and Monte Carlo Simulation. The Latin Hypercube simulation method aims at generating samples which describe, in a more accurate manner, a probability distribution. Such method “consists on a complete stratification of the presented distribution, in equiprobable stracts, and also in the random selection of a value for each stract” (FARIA, MELO and SALIBY, 1999. p. 4). Similar to the Monte Carlo Simulation, in the Latin Hypercube method, values for a probability distribution are generated through a inverse accumulated distribution procedure, obtained with: X = F-1 ( R)
[1]
Where: X = is a random number to be generated through the accumulated distribution defined by:
F(X) = Pr(X ≤ x)
[2]
So that R is a random value, informally distributed in the unitary interval, defined by a pseudo-random number.
Define the problem Establish independent and dependent variables and dependence relations
Present important variables associated with the problem
Build the model Have a model in na excell sheet or entry values in to specific softwares
Choose the simulation method to be used MCS or LH
Use supplementary packages for the electronic excel sheet such as Crystal Ball 200.5 or @Risk softwares
Execute the variable with the help of computers
Generate reports and charts (graphs) needed for the analysis
Evaluate simulation results
To make decisions
Chart 2 – Simulation Process Stages Source: Rogers, Rogers and Ribeiro (2004, p.11) adapted.
Model’s defined entry variables
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However, according to Faria, Melo and Saliby (1999, p.4), in order to ensure an independence among the values generated for the different random entry variables, in the LH method a random exchange of each group of values generated is made. According to the authors, five steps are needed for the sampling value generation in the LH method: 1)
definition of sample size (n);
2)
stratification of accumulated distribution in n equiprobable stracts;
3)
Selection of one of the stracts defined in step 2, by using random sampling without replacement;
4)
random generation of a sampling value for the selected stract in accordance with the inverse transformation method defined in equations 1 and 2; and
5)
repetition of steps 3 and 4 until the number of values to be generated is completed, meaning, n observations.
By using the LH method, one ensures that each part of the probability distribution will be present in the sample, which leads to a quick convergence of the sample in relation to the distribution represented. According to Faria, Melo and Saliby (1999, p.4), as a consequence, the results obtained by the LH method will be more accurate. The accuracy, as understood by the authors leads us to believe that in comparison to the MCS method, the LH produces less variability, or, ultimately estimates of equal variance, however, never bigger.
3. Latin Hypercubes Application to Changes on Credit Conditions Practical application of the LH method on credit conditions changes will be exemplified according to exercise developed by Rogers, Dami and Ribeiro (2004). These authors have applied the MCS method to the credit policy changes, developing a multi-period analysis aiming at incorporating the investment value in time and the risk inherent to the estimate exercise. In the present paper we have developed the same example of the authors, once one of our specific objectives is to compare the values obtained through the MCS method and the LH one. In that sense, we must consider the following practical development: Example, Inc sold in the last period 5.000 units of product X at R$ 10,00 each, and 50% of this total will be paid in short / long terms (not cash sales). Variable unitary costs and expenses represent 50% of product price. Fixed costs and expenses reach a total of R$ 10.000,00 for the period and the loss with uncollectible represented 1% of the non-cash sales. The directors have considered the events which have taken place over the course of the last period the basis for comparison for changes in credit conditions proposed as follows. Example, Inc is currently studying the possibility of changing payment term from 30 to 60 days. The directors estimate that if changed, such policy will increase sales in 20%, with 40% of cash sales. As a consequence of the flexibility in the credit policy, they estimate that losses with uncollectible will increase by 1,5%. Prices have remained the same. The average weighted capital cost, used as a discount rate is approximately 10%. The way to obtain needed investment, using company’s marginal cash flow analysis for three subsequent periods is presented in Table 1 below. According to Rogers, Dami and Ribeiro (2004, p. 9):
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the value of receivables must be seen as company’s investment, and as such, na investment decision is made when the value present in the cash flow is superior to the one obtained in its absence, meaning that, variation between the proposed situation obtained with the new credit policy and the current situation with the current policy must be higher than zero.
In mathematical terms:
Table 1 – Cash Flow Calculation for Exemplo, Inc. Original Situation Begin Cash Flow (+) Receiveables Cash Sales Non-cash Sales (-) Losses with Uncollectibles (-) Payments Costs with Variable Expenses Costs with Fixed Expenses End Cash Flow Proposed Situation Begin Cash Flow (+) Receiveables Cash Sales Non-cash Sales (-) Losses with Uncollectibles (-) Payments Costs with Variables Expenses Costs with Fixed Expenses End Cash Flow
Period 1
Period 2
Period 3
0
(10.000)
4.750
25.000 0 0
25.000 25.000 (250)
25.000 25.000 (250)
(25.000) (10.000) (10.000)
(25.000) (10.000) 4.750
(25.000) (10.000) 19.500
0
(16.000)
(32.000)
24.000 0 0
24.000 0 0
24.000 36.000 (540)
(30.000) (10.000) (16.000)
(30.000) (10.000) (32.000)
(30.000) (10.000) (12.540)
Source: Rogers, Dami e Ribeiro (2004, p. 10). ∆NPV = NPVP − NPVA > 0
[3]
Where: NPVP = net present value of the proposed situation and NPVA = is the net present value of the present situation. Considering the stocking term of one month and that the costs and expenses are incurred and paid cash, the cash flow for the present situation can be expressed according to Chart 3 below. For a discount rate of 10% and considering the cash flow for the second month to be the always the same, the NPV of the original situation would be: NPV = −35.000 −
10.000 (1 + 0,1)
1
+
14.750 (1 + 0,1)
1
×
1 0,1
= 90.000
Representation on Chart 4 expresses the cash flow for the proposed situation.
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14.750
14.750
14.750
1
0
2 35.000
3
n
(...)
10.000
Chart 3 –Representation of Proposed Situation Cash Flow Source: Rogers, Dami and Ribeiro (2004, p.9). 19.460
19.460
1
0
2 3
40.000
16.000
n
(...)
16.000
Chart 4 – Representation of Proposed Situation Cash Flow Source: Rogers, Dami and Ribeiro (2004, p.10). Proposed situation’s NPV is calculates as follows: NPV = −40.000 −
16.000 (1 + 0,1)
1
−
16.000 (1 + 0,1)
2
+
19.460 (1 + 0,1)
2
×
1 0,1
≅ 93.058
In such case, the positive different in the NPV (93.058 – 90.000 = 3.058) of the proposed situation in relation to the original credit policy leads us to conclude that such change in credit condition is economically attractive. However, in order to estimate risk in changing the credit policy as proposed, we incorporate into the example developed above an analysis (in probability terms) by using the LH simulation process. Considering the study of Example, Inc’s historical data, the credit conditions of similar lines of products and an analysis of specialized publications, we were able to establish the current situation, and objectively estimated the proposed one, probability distribution for the random variables presented in Chart 5 below. The definition for probability distribution can be found through the following tests: Kolmogorov-Smirnov (COSTA NETO, 2002, p.135), Anderson-Darling (MINITAB, 2000; SPSS, 2003), Ryan-Joiner (MINITAB, 2000) and Adherence Test by Qui-Square (COSTA NETO, 2002; DOWNING and CLARK, 1999; SPSS, 2001 and 2003; TRIOLA, 1999). Softwares such as BestFit 4.5 and Crystal Ball 2000.5 can be used to find out which probability distribution best fits the data, so that afterwards such distribution can be used as the entry value for the value to be simulated. The output value or, the random variable which will be used as performance measure to compare both proposals will be the marginal NPV. The difference between the NPVs found
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for the original and proposed situation in each simulation will be disposed in a frequency distribution where descriptive statistics will be found for the marginal NPV (ROGERS, DAMI, and RIBEIRO, 2004, p.12). Sales Volume Original Situation through Sales history we were able to find a period average of 5.000 units with an average variation of 250 units. Sales volume has assumed a normal distribution with a maximum of 6.000 and a minimum of 4.000 units. Proposed Situation by changing sales term from 30 to 60 days they expected an increase to na average of 6.000 units sold , and the possibility of reaching a maximum of 7.000 and a minimum of 5.500 units. The variable will continue assuming a normal distribution and an average variation of 250 units. Losses and Uncollectib les Original Situation a detailed analysis of company’s uncollectible history has shown that such variable has shown a triangular probability distribution with a most probable value 1% of sales. The maximum value found for such variable was 1,2%, and the minimum of 0,8%. Proposed Situation by focusing on the analysis of the company’s market as set in the proposed situation, we estimate the most probable value for losses with uncollectible to be 1,5%, not exceeding the maximum of 2% and the minimum of 1%. Percentage of Non-Cash Sales Original Situation by assessing company’s sales history we have found an average of 50% of non-cash sales (term sales) from the total, with a normal distribution and an average variation of 5% and maximum and minimum values not higher than 60% and 40%, respectively, of total sales. Proposed Situation a careful analysis of the market has allowed the company to assume the following distributions for the new credit policy: average of 60%, standard variation of 3%, maximum of 65% and minimum of 50% of total sales. Fixed Costs and Expenses Original Situation The company is prepared for a sales volume of 6.500, once the company history shows that never has the company sold more than 6.000 units, fixed costs and expenses remain the same, approximately R$ 10.000,00. Proposed Situation with a new credit policy sales volume might increase to 7.000 units, automatically requiring a R$ 2.000,00 investment in fixed costs and expenses, such as costs with more employees and more administrative expenses – meaning that such variable assumes the logical function in which we understand that fixed costs and expenses will reach R$ 12.000,00 if the demand is higher than 6.500. Average Weighted Capital Cost Original Situation the average weighted capital cost, used as discount rate on cash flows, has shown great variability due to instability in internal interest rates. Such value has assumed a uniform distribution of between 9% and 11% over the last periods. Proposed Situation we do not expect the proposed situation to change the average weighted capital cost, once this variable is mostly affected by market’s interest rate variations, which we’d expect to follow the same pattern
Chart 5 – Presumed Values for Model’s Random Variable Source: Rogers, Dami and Ribeiro (2004, p. 12).
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We have decided to consider as random variables those proposed in the original situation, once the current situation will not happen deterministically in the future, as it has in the last period. It means that the marginal contribution for wealth maximization must be analyzed, considering that current policy will also produce random values in the future, the same as the proposed policy. The Crystal Ball 2000.5 software was used for the simulation; it works as a supplementary package to the Excell sheet. Probability distribution found in the marginal NPV (output) is presented in Chart 6. The main statistics found for the group of simulated values for the marginal VPL are shown in Table 2. Output: Marginal NPV 300
Frequency
250 200 150 100 50 0 -30.336
-16.674
-3.011
10.651
24.313
R$
Chart 6 – Probability Distribution of the Marginal NPV Source: authors.
Table 2 – Statistic Summary of the Probability Distribution of the Marginal NPV Statistic Simulation Type Simulation Number Minimum Maximum Mean Mean Std. Error Variance Median Mode
Value Latin Hypercubes 10.000 (R$ 42.815,86) R$ 56.171,60 R$ 3.478,21 R$ 13.136,76 R$ 172.574.474,95 R$ 3.294,92 ---
Percentil% Value 10% (R$ 42.815,86) 20% (R$ 13.181,82) 30% (R$ 7.607,99) 40% (R$ 3.580,95) 50% (R$ 15,80) 60% R$ 3.294,92 70% R$ 6.596,57 80% R$ 10.419,52 90% R$ 14.504,64
Source: authors. Considering the probabilistic model incorporated into the LH method, and highlighting the fact that the analysis focuses only on financial terms, acceptance of the new credit policy would depend on the risk the company would be willing to take. We notice that in Chart 6 the group of values for the random variables marginal NPV present a normal probability distribution, which allows us to use the standard formula of normal curve as approximation, to infer on a marginal NPV value greater than zero, which also means that we will estimate investment risk by assessing values to be received due to changes in credit conditions, thus creation of value to the shareholder ( ∆NPV > 0 ). Therefore:
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Z=
X −µ
σ
=
0 − 3.478, 21 13.136, 76
≅ −0, 2648
Considering the value here for a score Z and using a normal probability distribution chart, we can conclude that: P ( ∆NPV > 0) ≅ 60, 44%
It means that, according to the applied methodology there’s a 60,44% probability that the proposed change in credit policy will be financially feasible, and consequently, a 39,56% possibility that the proposal will destroy value for the shareholder. Company’s decision will be based on its aversion to risk, as well as on its defined strategies – for example, the company may believe that it’s plausible to take on such risk in order to gain market share. By comparing the values found in the present study and the ones found by Rogers, Dami, and Ribeiro (2004) we notice that differences have been minimal. According to the authors, by developing the same example, except for the fact that the MCS was used instead of the LH, the probability of the marginal NPV being greater that zero is 59,8%. These results reinforce the idea that the difference between the MCS and the LH methods reside only on theory, whereas in practical both methods present similar results. An important aspect to be highlighted is that each simulation processed with computers will produce different values, where the comparison between two simulations can not be considered as conclusive about the exact values of the outputs statistics. However, these values, specially the average and the average variation, tend to be very close in the n simulation processes developed, which has been proved in the example used and various different tests produced by authors. We can also notice in the present paper that, according to theory, the LH really produces a smaller variance than the one obtained in the MCS. As an example, only, the average variation in applying the LH is R$ 13.136,76, whereas Rogers, Dami and Ribeiro (2004) by applying the MCS found an average variation of R$ 13.166,01. Additionally, after the simulation process, we can conduct an analysis of outputs sensitiveness (Marginal NPV) in relation to the inputs variables (a new proposal demand, previous proposal demand, discount rate, etc). Three sensitiveness measures stand out in this analysis, where the same tend to produce similar results: correlation of beta coefficient and contribution to variance. The Crystal Ball 2000.5 software has resources which present a correlation coefficient and contribution to variance whose values of the example in question, for the 4 main variables which contribute to the marginal NPV variability, shown in chart 7. Sensitiveness analysis is important because it allows management to identify where we have to focus change efforts in case of a credit policy. According to the example, 60,7% of estimated demand for the previous situation contributes to the VPL variance; whereas 29% of the marginal estimated demand for the proposed situation contributes to the marginal VPL variance. The other variables contribute minimally to the marginal VPL variance as shown in Table 2. These results show that if on one hand the demand variability, as it has happened over time, is the main influencing variable in the case of the marginal VPL, on the other hand we can see that the Example Inc. estimates that the change in the credit policy will create a more stable demand (with smaller variability). In this sense, as an example of the efforts deployed by company’s management, we can highlight complementary strategies to the credit policy change which will try to turn the demand into a more stable one, effectively, so that all effects may take place as expected.
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Output: Marginal NPV -100,0 -80,0 % %
-60,0 %
-40,0 %
-20,0 %
0,0% 20,0% 40,0% 60,0% 80,0%
100,0 %
Last demand
Proposed demand
Non-cash sales
IRR
Chart 7: Contribution to Marginal NPV Variance Source: author.
5. Conclusions Changes to credit conditions must be treated as many other investment by the company, because it influences sales volume, receivables, and expenses with uncollectible debtor, alter the cash flow need a consequently the cash flow itself. Cash flows must be measured in terms of increase, meaning, in relation to cash flow obtained in the present situation. Cash flows estimated for the proposed changes in credit conditions, must be analyzed through techniques commonly used in investment analysis: VPL and IR. Most finance manuals, which have been used in the development of the present paper, highlight the need for increase analysis. However, they consider the effects of changes in credit policy for one only period. This paper aims at presenting an analysis which considers the effects of changes in credit conditions in more than one period (multi-period analysis), besides making use of the current net value in order to study an investment economical feasibility. We have enhanced the analysis by adding the Latin Hypercube Simulation aiming at estimating the risk for credit condition changes, so as to create value for the company. The increase in cash flow estimation process is subject to estimates which can become real or not, or even become partially real, with the possibility of significantly affecting economical results inherent to the investment value, therefore, justifying the incorporation of a simulation method. By developing a practical example we are able not only to exemplify the methodology developed, but also to compare values obtained through the Latin Hypercube Simulation and the ones obtained through a commonly used method, the Monte Carlo Simulation. (MCS). We have concluded that even though different in theory, these two methods do not present practical differences, meaning that, in practical terms they present the same results. In general,
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the analysis developed allows for management to base its decisions in regards of the economical feasibility of credit condition changes, in probabilistic terms, by measuring the risk of those in generating value for the company.
6. Bibliographical References ASSAF NETO, Alexandre. Finanças Corporativas e Valor. São Paulo: Atlas, 2003. ASSAF NETO, A. e SILVA, C. A T. Administração do capital de giro. 3º Ed, São Paulo: Atlas, 2002. BESTFIT Probability Distribuition Fitting for Microsoft Windows Version 4.5.3 – Industrial Edition. Sistema de Ajuda do Software. USA: Palisade Corporation, 2004. BREALEY, R. A. e MYERS, S. C. Princípios de Finanças Empresariais. 3º Edição, Lisboa: Editora McGraw-Hill de Portugal Lda, 1992. BRIGHAM, E. F. e HOUSTON, J. F. Fundamentos da moderna administração financeira. 4º Ed, Rio de Janeiro: Campus, 1999. COSTA NETO, P. L. O. Estatística. 2º Ed, São Paulo: Edgard Blücler, 2002. CRYSTAL Ball 2000 Professional Edition 5.2.2. Sistema de Ajuda do Software. Denver, USA: Decisioneering Inc., 2002. DOWNING, D. e CLARK, J. Estatística Aplicada. 1º Ed, São Paulo: Saraiva, 1999. FARIA, H. D; MELO, S. S.;SALIBY, E. Análise de risco: uma comparação de diferentes métodos de amostragem. XXIII Encontro da Associação Nacional de Pós-Graduação em Administração (ENANAP), 1999, Florianópolis. Anais ... Rio de Janeiro: ANPAD, 1999 (CDROM). GITMAN, L. J.. e MADURA, J. Administração financeira: uma abordagem gerencial. São Paulo: Addison Wesley, 2003. GITMAN, L. J. Princípios de Administração Financeira. 7º Ed, São Paulo: Editora Harbra, 2002. ________. Princípios de Administração Financeira. 10º Ed, São Paulo: Pearson Education, 2004. LEMES JÚNIOR, A. B. & RIGO, C. M. e CHEROBIM, A. P. M. S. Administração Financeira: princípios fundamentos e práticas brasileiras. 5º ed. Rio de Janeiro: Elsevier, 2002. MINITAB Release 13.0. Sistema de Ajuda do Software. BCIS Lab St. Cloud State University: Minitab Inc.,2000. RISK Analysis Add-in for Microsoft Excel Version 4.5.3 – Industrial Edition. Sistema de Ajuda do Software. USA: Palisade Corporation, 2004. ROGERS, P.; ROGERS, D.; RIBEIRO, K. C. S. Avaliando o risco na gestão financeira de estoques. In: VII SIMPOI, 2004, São Paulo, Anais..., São Paulo: FGV-SP, 2004 (CD-ROM) ROGERS, P.; DAMI, A. B. T. e RIBEIRO, K. C. S. Avaliando o risco das decisões de crédito. In: 4º Congresso de Controladoria e Contabilidade, 2004, São Paulo, Anais..., São Paulo: FEA/USP, 2004 (CD-ROM).
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ROSS, S. A.; WESTERFIELD, R. W. e JAFFE, J. F. Administração Financeira: Corporate Finance . São Paulo: Atlas, 1995. SANVICENTE, A. Z. Administração Financeira. 3º Ed, São Paulo: Editora Atlas, 1997. SCHERR, F. C. Modern Working Capital Management. New Jersey: Prentice-Hall, 1989. SPSS 11.5 for Windows. Sistema de Ajuda do Software. United States of América: SPSS Inc., 2003. SPSS Inc. Statistical Analysis Using SPSS. Version 11 – Chicago, Illinois: SPSS Traning, 2001. TRIOLA, M. F. Introdução à Estatística. 7º Ed, Rio de Janeiro: Editora LTC, 1999.
