Break-Even Analysis: Understanding the Relationship between Cost, Volume, and Profit, Study notes of Business

Download Break-Even Analysis: Understanding the Relationship between Cost, Volume, and Profit and more Study notes Business in PDF only on Docsity! A CRITICAL EVALUATION AND TEST OF BREAK-EVEN ANALYSIS AS A TECHNIQUE FOR PROFIT PLANNING by SAY CHONG LIM B.A. (Hons.), University of Malaya, 1963 A Thesis submitted in partial fulfilment of the requirements for the degree of MASTER OF BUSINESS ADMINISTRATION in the Department of COMMERCE AND BUSINESS ADMINISTRATION We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA April, 1965 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that per~ mission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publi- cation of this thesis for financial gain shall not be allowed without my written permission. Departwent of Commerce and Business Administration The University of British Columbia, Vancouver 8, Canada Date April 30th 1965 Chapter Page V. SUMMARY AND CONCLUSION, . 2. 2. 2 2 ee eae 103 Summary . 2. 1 ee ee ee ee tw ee ww 10 Conclusion, . ss. 6 ee ee ee ew ww ww 10! BIBLIOGRAPHY 2. 0 ee we et ee ew ee ew ne 108 APPENDICES 2. 6 1 ee we pe we ee ee ee tw U4 fable II III Iv LIST OF TABLES Number and Percentages of Companies Inter- viewed Indicating Various Types and Frequencies of Break-Even Analysis. . . . « Number and Percentage of Companies Answering Questionnaires Indicating Types and Fre- quency of Preparation of Break-Even Analysis as Reported by 344+ Companies. .. +... 4. Computation of the Standard Deviation of the Universe. «se ee ee ew ewe we ww . Difference between Actual and Forecast Profits - Percentage of Sales Method. . 2.» 2. «+ « « Difference between Actual and Forecast Profits - Break-Even Method . 2. 2 6 se we we we ewe Page 1? 18 87 90 93 Number EXHIBIT 1 LIST OF ILLUSTRATIONS Conventional Break-Even Chart. .. . «6. Profit/Volume Chart. . 1... see 2 ee Scatter Chart. . 2 6 6 2 ee ew ee ew Modified Conventional Break-Even Chart . Curvi-Linear Break-Even Chart. ....-. The Limits of Constant Prices. . .. «+. Constant Average Variable Cost ...4.-. Economists' Cost Curves. . 2. 2. + ee ee Economists' Average Variable Cost Curve. 23 38 43 46 52 59 56 58 -3- future profits, is that 1t is simple, quick and cheap to pre- pare. But, if a prerequisite to the use of break-even analysis is a forecast of all the variables that affect profits and if adjustments have to be made in break-even analysis on the basis of the results of the forecasts of these variables, then the very advantages of break-even analysis will be defeated, If, on the other hand, management does not go to the trouble to make the preliminary forecasts and adjustments but merely makes use of break-even analysis to show profit expectations under a specific set of assumptions regarding external market conditions and internal management strategy, then profits will be shown to vary only with volume. But, in break-even analysis, sales revenue is often used as a measure of volume, on the assumption that volume (output) and sales are synchronised. Under the circumstances, the question arises, "Isn't it simpler, quicker and cheaper to determine the average of profits as a percentage of sales for several past years and apply this percentage to arrive at the expected profits for different volume levels?" This method of forecasting profits is known as the percentage of sales method. With this method, the problem of cost separation and a host of other problems do not arise at all. This study attempts to compare the accuracy of the results of the break-even method with that of the percentage of sales method. On the basis of this comparison, a con- ~4he clusion will be drawn on the merit of using break-even analy- sis for forecasting the operating profits of firms and its managerial implications. Hypothesis and Null Hypothesis The hypothesis is that break-even analysis can be better than the percentage of sales method as a technique for forecasting the future operating profits of firms. Since break-even analysis is more sophisticated, it is expected that, at worst, it will be as good as the percentage of sales method. Therefore, the possibility that the latter may be better than the former will not even be considered here. In order to test the hypothesis, firstly, a sample of firms has to be picked since it is not possible to study the performance of the two methods in all firms.@ Secondly, the average difference between forecast profits and actual profits by each of the two approaches has to be determined and finally a comparison has to be made of the two averages. In establishing the averages, the mean, the median or the mode ean be used. In this study, the mean is preferred to the other two measures since it is least subject to sampling variation.> The alternative to the hypothesis is that there is 2. In chapter IV, an explanation is given of the manner in which random sampling has been used to pick the fifty- seven firms, which form the sample. 3. Frederick E. Croxton and Dudley J. Cowden, Applied General Statistics, ond Ed.; Englewood Cliffs, N.J.: Prentice- Hall, Inc., 1955, p. 197. -5- no difference between break-even analysis and the percentage of sales method as a technique for forecasting the future operating profits of firms. This alternative hypothesis is called the null hypothesis. With the symbol p, used to represent the mean difference between actual and forecast profits obtained by the percentage of sales method, and the symbol Po used to represent that obtained through break-even analysis, the null hypothesis may then be formated as Py = Po- Since the possibility that the less sophisticated percentage of sales method may prove to be more accurate, has been ruled out on the grounds that it is unlikely to happen, the alternative to Py = Po is py > Po: This alternative implies that the average difference between actual and fore- cast profits by break-even analysis is smaller and, therefore, the hypothesis as stated earlier, is correct. From the above discussion, it is obvious that the acceptance of the hypothesis depends on the rejection of the null hypothesis. But, before a decision can be made on the null hypothesis, a test of significance has first to be carried out. The purpose of such a test of significance is to determine whether there is any statistical significance between Py and Po- In order to do this, it is necessary to set a criterion of significance and to determine the proba- bility of 2, where z is the ratio of py - Po to an estimate of the standard error of the difference between the two sample means. The criterion of significance established ~ & = polation, for fifty-seven degrees of freedom, the value of z at the 0.01 level of significance, is 2.667. In this study, a one-tail test is used since the alternative to the null hy- pothesis that pz = Po is one-sided, i.e. Py > Poe With this information, it may be stated that the null hyvothesis should be accepted if 2 & 2.667 and should be rejected if z > 2.667. This is the same as saying that the hypothesis, that break- even analysis can be better than the percentage of sales method as a technique for forecasting the future operating profits of firms, is acceptable if 2 > 2.667 and should be rejected if z <& 2.667. Organisation Chapter IT is the introductory chapter to this thesis. Here, the statement of the problem, the hypothesis and the null hypothesis, the limitations of the study, the organisation and the definitions of terms are given. To pro=- vide a basis for the critical evaluation of break-even analysis, chapter II is devoted to a factual description of the nature of break-even analysis. Briefly, an idea is given of the extent to which break-even analysis is used, the dev~ elopment of break-even analysis, the break-even chart and the break-even point. An idea is also given of how costs of firms are separated for the purposes of break-even analysis. Against this background, a critical evaluation of -9- break-even analysis is made in chapter III, in terms of the assumptions used and the uses to which break-even analysis is put. The evaluation is also in terms of the economic theory of the firm in the short and long run and under per- fect and imperfect competitive conditions. This chapter and chapter II are based primarily on a close examination of the various publications relating to the analysis of cost-volume profit relationships. Chapter IV deals with the test of the hypothesis and the null hypothesis, using data taken from Moody's Indus- trial Manuals. A test of this nature, to be very accurate, requires a very intimate knowledge of the operations of the firms under study. Unfortunately, firms willing to provide such detailed, confidential information are difficult to find. The problem is particularly serious since there are ? as many as fifty-seven firms in the sample, Under the cir~ cumstances, resort has to be made to published data. For the purposes of this study, Moody's Industrial Manuals pro- vide the most detailed, published information and have therefore been chosen as the source of data . Chapter V provides the summary and conclusion of the study. Limitations of the Study In this study, in forecasting profits by the break- 7. A detailed explanation of the estimation of the sample size is given in chapter IV. - 10 ~ even method, no adjustments were made to take into account the effects of possible changes in the determinants of profits. This is due to the fact that the necessary data for such adjustments are not available. It is admitted here that failure to make the adjustments will. most certainly reduce the accuracy of the forecasts. Again, in the break-even method of forecasting operating profits, total costs have to be separated into fixed costs and variable costs. fo achieve this, as des= eribed in chapter IV, the statistical approach using the method of least squares is used. But, it has been found in preliminary studies that this approach is possible only if the data for the firms show that the firms have experienced losses over some of the periods in the analysis. This draw- back can be avoided if the accounting or engineering approach” is used, but since the statistical approach is likely to give the most reliable results in terms of the data available, it is preferred to the other two techniques.” Hence, the uni- verse of firms, from which the sample is chosen, is restricted to only those firms in Moody's Industrial Manuals, which have had lesses at some time in the periods under study. The data of some firms do not appear in Moody's Industrial Man- uals for the period prior to 1951, These firms have also 8. This is explained in chapter II. 9. The data in Moody's Industrial Manuals are not in sufficient detail to enable the use of either the accounting or engineering methods, with reasonably accurate results. -13- payments for raw and semi-finished materials and payments for overhead costs of, all kinds. Implicit costs; The implicit costs of a firm are the costs of self-owned, self-employed resources. Examples of implicit costs are the returns to labour provided by the owner himself (implicit wages) and the returns to the land and capital, which are provided by the owner himself rather than hired from outside owners (implicit rent and interest). Margin of safety: Margin of safety refers to the excess of actual (or budgeted) sales over the break-even sales volume. Operating costs: . The operating costs of a firm are the costs incurred by the firm in conducting its regular major activities. They include the costs of goods sold, commercial costs, interest and depreciation and amortization. But, they exclude all ether costs which are not subject to the controls exercised through everyday operating procedures. Hence, income taxes, losses from sales of plants and other property disposals, losses on foreign exchange and flood, fire and other extraordinary losses are excluded. Operating revenue: The operating revenue of a firm refers to the gross revenue or gross sales from the firm's regular major activities less returns, allowances and cash discounts. Operating profits: The operating profits of a firm are the -m- excess of the operating revenue over the operating costs. If operating costs exceed operating revenue, the excess is called operating losses. Summary In this chapter, an attempt has been made to bring out the purpose of this study, its limitations and its organ~ isation. The hypothesis and null hypothesis have also been discussed in detail and the definitions of some of the more important terms in this thesis have been given, The nature of break-even analysis is presented in the next chapter. CHAPTER IT THE NATURE OF BREAK-EVEN ANALYSIS Introduction The term, ‘breakeeven' has become a part of the standard vocabulary of economists, accountants and managers in general. Each of these three classes of people have contributed in no small way to the increasing popularity of this subject in recent years. In fact, it may be said that, today, the best discussions on break-even analysis are to be found in economics, accounting and management books and journals. But, although a great deal of attention has been given to this subject, there is still some vagueness as to what the area involves. To many people, when break-even analysis is mentioned, the first thing that comes to mind is a simple cross-over chart, indicating total sales revenue and total costs, with the cross-over point representing the break-even point. In reality, break-even analysis is more than the mere determination of the volume level at which revenue equals cost. Rather, it exposes the effect on profits resulting from the interplay of such factors as prices, costs, volume and product mix. Use of Break-Even Analysis To date, very little has been done to determine -18- TABLE II Number and Percentage of Companies Answering Questionnaires Indicating Types and Frequency of Preparation of Break-Even Analysis as Reported by 344 Companies “Less than one per cent. No. of Per Cent of Use of Break-Even Analysis Companies 344+ Companies Companies using break-even analysis 175 51 Companies not indicating the use of break-even analysis 169 ms) aes Per Cent of 175 No. of Companies using Frequency of Preparation Companies B~H Analysis Prepare monthly 45 “13 Prepare quarterly 7 2 Prepare semi-annually 1 * Prepare annually 106 31 Prepare as needed 51 15 No indication as to frequency of preparation 13h 39 au 00 Per Cent of 175 ae No. of Companies using Types of Break-Kven Analysis Companies B-% Analysis -Prepare for company as a whole 182 3 Prepare for various divisions 147 3 Prepare for plants 106 31 Prepare for product lines 134 39 Prepare for sales branches 2k 7 “the number of companies dees not total 344 and the percentages do not total 100 because some companies use more than one type of break-even analysis. Source: Burnard H. Sord and Glenn A, Welsch, Business Budgeting, New York: Controllership Foundation Thos i9e8 ; > P. 282, Table 83. -19- conceived as a result of attempts by university teachers and businessmen to develop tools to provide a scientific approach 2 to business. very little is known about the early stages of 3 the development of this tool. Some writers” have tried to throw some light on this area but views tend to conflict. The first attempt to separate costs into their fixed and variable components was made in 1832 and the man who, supposedly, holds the distinction of having made that attempt is a certain Charles Babbage. * fo the extent that the separation of costs is the basis of break-even analysis, it may be said that Charles Babbage made the first contribu- tion to the development of break-even analysis. But anything close to the formulas or charts that are used in break-even analysis today, was not known till 1897 when Henry Hess wrote on "Time Saving and Its Relation to Profits" in Volume XX (Dec. 16, 1897) of the Engineering Magazine. In December, 1903, Henry Hess contributed another article, "Manufacturing: Capital, Costs, Profits, and Dividends", to the Engineering Magazine (Volume XXVI) and, in this article, he included charts which are quite similar to those used in break-even 2. Sord and Welsch, op. cit., p. 281. 3. Ned Chapin, "The Development of the Break-Even Chart: A Bibliographical Note", Journal of Business, Vol, XXVIII, No. 2, April 1955, pp. 146-149, Raymond Villers, "Communications - The Origin of the Break-Even Chart", Journal of Business, Vol. XXVIII, No. 4, Oct. 1955, pp. 296-297, 4, Villers, ibid., p. 296. - 20- analysis today, although he did not designate his presenta- 5 tion as break-even charts. During the period just prior to World War I, two other industrial engineers, Walter Rautenstrauch and C, BE. Knoeppel, were prominent in the development of the break- even chart. Rautenstrauch was the first to use the name, 'Break-even charts’ .° This terminology is, today, univer- sally accepted. It might also be mentioned that in Rauten~ strauch's charts, the functional relationship between costs and volume was brought out, for the first time. Such a relationship was never depicted in the charts Henry Hess used.” The Break-Even Chart The break-even chart is a portrayal in graphic form of the relationship of production, costs and sales to profit. It shows the amount of fixed and variable costs and the sales revenue at different levels of operation. Various names have been given to this chart. It is sometimes known as a crossover chart, a profit realisation chart or a profit- graph. The break-even chart may be pictted in several different forms, depending on the kind of information desired. 5. Chapin, op, cit., p. 149. 6. Villers, loc. cit. 7. Ibid., p. 296-297. Profit » $3 » EXHIBIT IT A PROFIT/VOLUME CHART Thousands of Dollars 25 PROFIT AREA gf aq 20 Loss AREA Sales Volume (Theusands of Dellars) -~ 2h - relationship between profits and volume is the profit/volume chart. It is sometimes referred to as a profitgraph, a marginal income chart, or a profit-volume analysis graph. It is said to have been develeped by C. E. Knoeppel.© Although profit/volume charts are generally simpler than break-even charts, they are not always preferred to break- even charts because they do not show the relationships between costs, revenue and volume. Nickerson, however, argues that since profit is the residue of cost and revenue, profit/volume charts, therefore, really reflect cost- volume-revenue relationships.” The profit/volume chart indicates the path of profit and consists of two areas, the profit area and the loss area, both of which are created by the horizontal axis which represents the total sales volume. The less area is composed of the fixed costs which is marked off on the vertical axis. The profit area indicates the amount of profit earned as the profit line passes over the horizon- tal axis. The points of the profit line are computed by 8. Knoeppel explained the reason for its development as follows: "So far the financial statement has been a finan- cial tool rather than a management tool. It is a historical document and not in the least prophetic. It is static rather than dynamic. It performs only a part of the function of which it is capable. Few accountants have crossed the line between accounting and engineering, while many engineers have jumped the fence between the two" - C. HE. Knoeppel and Edgard C, Seybold, Managing for Profit, New York: McGraw- Hill Book Company, Inc., 1957, pp. 53-94. 9. Clarence B, Nickerson, Cost Accounting, Toronto: MeGraw-Hill Book Company, Inc., 1954, p. 272. - 5 - subtracting from sales income the total costs indicated for each sales volume.1° The break-even point is the point at which the profit line intersects the horizontal axis. The Break-Even Point Various definitions have been advanced for the break-even point. Generally, it may be said that the break- even point is the volume level at which total revenue ex= actly equals total cost and neither profits nor losses are made. Other definitions are merely variations. Mathematic- ally, the break-even point can be determined by using the following. equation: Fixed Costs Break-even point = L- Variable Costs Sales To illustrate, the following budget data of a fictitious company, the ABC Company, may be used. 10. The slope of the profit line is also equal to the profit-volume (P/V) ratio, which is the rate at which profit increases with increases in volume and is given by the formula: = 1 - Variable Costs Sales © Sales - Variable Costs ales Fixed Costs + Profits Sales = o aad abo du ato it or The profitevolume ratio is sometimes called the marginal income ratio or variable profit ratio. ~ 2B Although the break-even point may have some use- fulness, its determination should not be over-emphasised,. It should always be remembered that it does not remain fixed at all times, for any particular enterprise, but varies from time to time as the factors affecting it undergo change. Howard Stettler, a professor of Business Administration in the University of Kansas states that: Although statistical confirmation is not available, there appears to be con- siderable justification for concluding that most people who are familiar with break-even charts and the analysis of break-even informa- tion assume that the reason for seeking such information is to determine the break-even point for all or some segment of a profit~- making enterprise. True, break-even analy- sis is capable of living up to its name and showing the volume level at which expenses and revenues are equal, but if, as con~ tended, this is the only use generally made of such information, the use is not only deficient, but my, involve an actual dis- service as well. It must also be stressed that the break-even point is, at best, only an approximation because of the many re- strictive assumptions that have to be made in its computa- tion. 2 Under the circumstances, even the appropriateness of the term, ‘break-even point’ is questionable. The word, ‘point', carries the connotation of great exactness. A better term might be ‘break-even area' to indicate that the 12. Howard F. Stettler, "Break-Even Analysis: Its Uses and Misuses", Accounting Review, Vol. 37, July 1962, p. 460. 13. This is discussed in Chapter 3. = 29.- precise location of the break-even volume is not known and can be estimated only roughly. Cost Breakdown From the discussion so far, it may already be evident that cost behaviour constitutes the central problem in break-even analysis. If costs cannot be classified as fixed or variable, no break-even chart can be constructed. This can easily be explained. If all costs are variable, there cannot be any break-even point so long as the total revenue and the total costs are represented by straight lines, starting from the intersection of the X and ¥ axes. The two lines will never intersect to provide a situation whereby losses turn into profits or whereby profits turn into losses. At the same time, all costs cannot be fixed because this would be an unrealistic situation. Therefore, total costs have to be separated into their fixed and var- iable components. Fixed costs may be defined as those costs whose amount is not at all influenced by the level of activity in the short-run and within the expected range of activity. On the other hand, variable costs may be defined as those costs, whose amount is a function of activity, increasing or decreasing in the same direction as activity. The change in the total variable costs may or may not be proportional to the change in the level of activity. However, it is - 30+ usually assumed in conventional break-even analysis that straight line relationships exist and, consequently, it is not uncommon to find variable costs being defined as those costs which increase or decrease proportionately with in- ereases or decreases in volume. The assumption of linearity is justified if the change in output is not too great ~ assuming that there is no change in technology and advertising and sales promotion are absent. If output changes are too large, the variable costs may not be linear (constant per unit). This could be caused by changing prices or increasing or diminishing returns. This follows from the fact that, with a large change in the demand for factors of production, the prices of the factors of production may also change and this may result in a change in the variable costs per unit. Purther, the level of effic- iency at which variable resources work, may differ when different amounts of them are used with given quantities of the fixed resources. The increasing or diminishing returns that result may also change the average variable costs. This will be explained in greater detail in chapter III. In our definition of fixed costs, reference was made to the 'short run' which may be defined as a period of time within which a firm cannot alter or add on to items such as its capital equipment and buildings.- It now becomes clear that the classification of costs into fixed and variable 14, Paul Yacobian, “A Practical Evaluation of Break~Even Analysis) NAA. Bulletin, Vol. 40, Sec. 1, Jan. 1959, p. 24. -33- men), while regulated costs would refer to costs which, though fixed, are nevertheless subject to the discretion of manage- ment (for example, the bonus given to salesmen). This kind of approach seems to agree with the advice given by Wally George in 1941. As of today, a great deal has already been written in textbooks and journals about the inappropriateness of the term, ‘fixed costs'. In the years to come, there is no doubt that even more will be written about it. Admittediy, it is not the ideal term to describe the kind of costs to which it refers. But, its critics should realise that it, nonetheless, is perhaps a better term than any that they can suggest, to the extent that it has the advantage of widely established usage. In the past, many people were dissatisfied with the classification of costs into fixed and variable. Today, many people are still dissatisfied with this class- ification, In the years to come, more people may join the ranks of this dissatisfied group. It is true that the classification of cost behaviour in this way is far from being perfect; but, it is also true that for years, costs have been classified in a similar way for budgeting pur- poses and although the techniques used in the separation were simple ones, the ultimate results have been quite satisfactory.2t There is no doubt that the alarm raised 21. William J, Vatter, “Accounting Measurements of In- eremental Cost", Journal of Business, Vol, XVIII, No. 1, Jan. 1945, pp. 147-148. ~ 34 - is not a false one. But, the intensity of the excitement is perhaps greater than is warranted by the nature of the problem. In break-even analysis, it is assumed that it is reasonable to classify costs as fixed or variable. This brings us to the problem of semi-variable costs. Such costs vary with volume though not in direct proportion to it. They possess the characteristics of both fixed and variable eosts and are sometimes called fixed-variable costs or semi- fixed costs. Examples include items such as supervision, power and maintenance costs. In break-even analysis, one is faced with the problem of separating these costs into their fixed and variable components, One way out of this problem is to measure the variability of these costs. Generally, there are three approaches to the measurement of cost variability. They are: (a) The Accounting Approach - Inspection of accounts. (b) The Statistical Approach- Statistical analysis of historical costs. (ce) The Engineering Approach- Industria} engineering studies. The accounting approach is, by far, the simplest of the three. It also requires the least time. By this 22, Joel Dean,"Methods and Potentialities of Break- Even Analysis'in David Solomons (ed), Studies in Costing, London: Sweet & Maxwell, Ltd., 1952, pp. 232-233. ~ 35 = method, a careful study has first to be made of the chart of accounts. On the basis of this study, costs which are either fixed or wholly variable, are then picked out, leav- ing behind the so-called semi-variable costs. The statis~ tical technique and/or the engineering technique may then be applied to the semi-variable costs to separate their fixed and variable components. It is obvious that the accounting approach requires a thorough knowledge of the behaviour of the costs in each of the accounts. Unless a fairly complete knowledge of the operations and activi- ties of the enterprise is obtained beforehand, the results arrived at, by using this method, may be misleading. In any case, the very fact, that this approach requires the exercise of judgement, means that it is far from being an infallible one. The statistical approach, involving the statis- tical analysis of historical costs is probably more thor- ough and scientific than the accounting method. But, unless computers are used, it can also be more time consum- ing and expensive. It can be carried out by using statis~ tical correlation techniques which relate each cost component to some measure of activity. The best example of this approach is the scatter chart technique under which the historical cost and volume (production or sales dollars as a measure of activity) during each of several past months or years are plotted on a chart with volume as the horizon- - 38 - The main difficulty with the statistical approach is that historical cost data often show poor correlation with volume. This is so mainly because costs often vary not solely because of volume but also because of many other factors. These factors include changes in plant, equipment, materials used, methods of manufacturing, personnel, work- ing hours, factor prices and managerial policy. In study- ing cost behaviour for purposes of break-even analysis, it is necessary to assume that these non-volume factors, which affect costs, will remain constant. The engineering approach is the only feasible method, when historical data are unavailable or too un- reliable but it can also be used for supplementing statis- tical or accounting analysis when it is desired to project cost behaviour beyond the range of past output experience or when it is necessary to estimate the effect of major changes in technology or plant size upon cost behaviour over a familiar or unfamiliar output range. In essence, this method attempts to determine the physical inputs nec- essary to achieve certain levels of output and then con- vert these to dollars at current or anticipated prices. ‘The superiority of this method lies in the fact that it attempts to work with relationships between various physical inputs and the volume of activity rather than with observed histor- ical patterns, which may be distorted by certain non-volume factors. However, like the statistical approach, it also - 39 = can involve very high expenses. In addition, it suffers from the drawback that the practical feasibility of its estimates cannot be pretested. These three approaches to the measurement of cost ' variability are not necessarily mutually exclusive. In fact, it is often a good practice to use them to supplement ah each other. Summary Break-even analysis does not merely involve the determination of the break-even point. It also shows the effect on profits resulting from the interplay of such fae- tors as prices, costs and volume. Although there is very little agreement among writers regarding the development of break-even analysis, it may generally be said that Charles Babbage, Henry Hess, Walter Rautenstrauch and C. E, Knoeppel were the pioneers. Henry Hess was responsible for the basic idea of the break-even chart but its universally accepted terminology has been credited to Professor Rautenstrauch. The break-even chart is basically a portrayal in graphic form of the relationship of production, cost and sales to profit, though it may be plotted in several differ- ent ways. In this chapter, only the conventional break- even chart and the profit/volume chart have been presented. 24. Dean, op, cit., p. 231. -~ ko - In the discussions on break-even charts, in later chapters, reference will be made mainly to these two charts. In any break-even chart, there must be a break~ even point, which may be defined as the volume level or point of time in the budgetary period when losses turn into profits. The break-even point is useful beeause it is a prerequisite to the determination of the margin of safety, which is a useful reference device for action. It is also useful because it indicates the point of time in the bud- getary period when contributions to profits begin. However, it must be realised that the break-even point is not as exact as its name implies and that it does not remain fixed at all times. Therefore, its usefulness should not be over- emphasised. One of the most important steps in break-even analysis is the classification of costs as fixed or variable. Three approaches may generally be used. They are the account- ing, statistical and engineering approaches. Of these three, the statistical method is likely to give the most reliable results in terms of the data available and will be used in the test, which will be described in chapter IV. From the facts that are presented on break-even analysis in this chapter, a critical evaluation of break-even analysis is made in the next chapter. loop gor ~ Y L Mog ve 06 UO eet U4 £0 v4 if o 9 Pr > 3 uo wu Ss E Ne zat ~43 + EXHIBIT IV MODIFIED CONVENTIONAL BREAK-EVEN CHART Jolol Revenue Total costs »| Relevant Range tt 20 40 GO 80 loo Volume (Thousands of Dollars) A Short-Run Concept The static situation that break-even analysis assumes cannot exist for long periods of time because the longer the period covered in the projection, the less reliable is the forecast of revenue and costs. In the short-run, it may be true that there exists a unique functional relationship between the profits of a firm and its volume and that, given the volume, the corresponding level of profit could be deter- mined. But, this is progressively less true as the time period inereases because, realistically, profit is dependent on a great many other factors, apart from volume and, in the long-run, dynamic forces continually work to shift and modify these other factors as well as volume. It, therefore, becomes clear that break-even analysis is essentially a short-run con~ cept and is more useful in short-run, as opposed to long-run, financial planning. In fact, if a long-run concept is attached to break-even analysis, its usefulness immediately becomes dubious. Professor Neuner states that: Break-even analysis and charts must be kept current and not attempt to reflect probable operating circumstances over a period longer than a year because not only the mixture of variable cost and income elements may change but also fixed costs gradually shift over extended periods of time. Linear and Curvi-Linear Charts There is an interesting and perhaps deceptive resem- 2. John J. W. Neuner, Cost Accounting, Homewood, Illinois: Richard D. Irwin Company, 1957, p. 790. -45 = blance between the linear and curvi-linear break-even charts. The basis for the construction of the latter stems from the cost-volume and revenue-volume functions of the economic theory of the firm, as illustrated in Exhibit V. Presented in this form, the curvi-linear chart closely resembles the linear chart, as described in chapter II, except for the nature of its total cost and total revenue functions. This will be discussed in detail later. Meanwhile, it must be pointed ont that where- as the linedr chart has only one break-even point, the curvi- linear chart reveals two break-even points, i.e., two levels of output at which the firm's revenue just covers its costs so that net profit is zero. These are the points By and Bs (Exhibit V). Point BL is analogous to the break-even point in the linear chart and point B, is the logical result of the 2 curvi-linear nature of the total cost and total revenue func- tions. Another basic difference between the two analyses is in the point of maximum profits. Profits may be defined as the excess of total revenue over total costs. It is clear, therefore, that the largest profits, which a firm could make, will be earned when the vertical distance between the total cost and the total revenue curves is at its greatest. This is indicated by MP at volume K, in Exhibit V. fhe linear break-even chart, on the other hand, shows profit maximised at full capacity. This tends to give the impression that the curvi~linear analysis has a slight advantage over the - 48 - someone else in.a similar capacity. To the economist, a salary for the proprietor equal to the value of his services in his next best alternative employment may be considered as a part of the firm's costs. It is an implicit cost. In linear charts, however, implicit cost is ig- nored and a firm's total costs are considered to include only the explicit obligations to resource owners. Under the eircumstances, a firm's net income becomes the remainder of gross revenue after operating and financial expenses have been deducted. No consideration is given to implicit costs such as interest and dividend payments equal to what investors could earn had they invested elsewhere in the economy. Separation of Costs In chapter II, it was stated that in break-even analysis, it is necessary to separate total costs into fixed costs and variable costs. Unless such a classification is made, it is impossible to construct a break-even chart. But, if as defined earlier, fixed costs equal those costs which remain fixed, irrespective of volume and variable costs equal those costs which vary in direct proportion to volume, and if costs can only be classified as fixed or variable, then there are bound to be some costs which are beyond classifi- cation. Sidney Robbins states that, “many costs and the components of these costs do not fall into neat black or white, fixed or variable categories, but are rather grey- -49 - hued, partaking of the characteristics of both types..." Some of these costs, popularly known as semi-variable costs, inelude costs for such items as supervision labour, power, maintenance, and accounting services. In break-even analysis, as indicated in chapter II, these costs are usually broken down into their fixed and variable components by either the accounting, statistical or engineering methods. None of these methods can produce com- pletely accurate results but there is also no reason to sus- pect their ability to produce satisfactory results," Under the circumstances, the assumption made in break-even analysis that all costs can be reasonably separated into their fixed and variable components should not provide any cause for alarm. What is important is recognition of the fact that irrespective of the method used in the division of the costs, the result will not be completely accurate and the more in- accurate the division of the costs, the more inaccurate will the results of the break-even analysis be. Constant Selling Prices The presentation of total cost and total revenue 3. Sidney M. Robbins, "Emphasizing the Marginal Factor in Break-Even Analysis", NAA. Bulletin, Vol. 43, Oct. 1961, De 59. 4, William J. Vatter, "Accounting Measurements of Incre- mental Cost", Journal of Business, Vol. XVIII, No. 1, Jan. 1945, pp. 147-148. -~ 50 ~ functions as straight lines has often been questioned. The linearity of the total revenue curve implies a constant selling price over the entire range of output. This is not unusual if conditions of pure competition are assumed. In a pure market, all competitors sell an insignificant pro- portion of the total output of a homogenous product and no single seller can, by his own efforts, influence price. Every seller must accept the same market price, determined as it is by the overall interaction of supply and demand in the market. In addition, every seller can sell all his out- put at the market price. Unfortunately, such conditions of pure competition are rare or impossible to achieve in the real world. This, therefore, tends to suggest that the presentation of the total revenue function as a straight line is not valid. Under any other market condition, other than pure competition, a firm can increase its sales volume only by lowering its selling price, when all other determinants of demand - consumer incomes, consumer tastes and preferences, number of consumers and range of goods available to con- sumers - remain unchanged and if advertising and sales pro- motion are assumed to be absent. In other words, the demand curve slopes downward to the right when the seller has any degree of monopolistic control over price, implying that for each possible selling price, there is a corresponding sales volume. Under such circumstances, the total revenue function - 53-- long as the demand does not change, the firm can charge a constant price, OP and sell any output up to the level 0Q. Qbviously, for quantities less than 0Q, the firm could have charged a higher price and still sell the whole of its out- put. For example, for quantity OM, the firm could have charged OP,. But, it is not unusual to find a firm fixing its price at OP even though it is willing to sell only OM quantities, with the given demand curve DD. This is so because firms tend to feel that their customers prefer stable prices and hence once price is set and shown to be profit- able, it is likely to be retained until some major change in conditions causes an inroad into the desired profit goal.? Since PP and DD, in Exhibit VI, have to intersect somewhere, it therefore follows that the total revenue curve cannot con- tinue indefinitely as a straight line but, sooner or later, must fall quite steeply. Beyond the sales volume 0Q, the price line PP is above the demand line DD and any desired increase in sales must therefore necessarily be preceded by a reduction in prices. From the above it may logically be concluded that, in the vast majority of non-agricultural, industrial enterprise situations, which are characterised by conditions of imperfect or monopolistic competition, the linear break-even chart is incorrect on the revenue side, except for small range of sales volume over which it is 5. Robert F. Lanzillotti, "Pricing Objectives in Large Companies", American Economic Review, VOL. XLVIII, No. 5, Dec. 1958, p. 937. - 5h - possible to have an unchanged price. Total Cost and Constant Unit Variable Costs The linear break-even chart also carries the assumption that cost-volume relationships are usually char- acterised by straight lines and since the fixed cost com- ponent is always taken as given, it therefore follows that it is the shape of the variable cost function that determines the shape of the total cost function. If this is the case, then to draw a linear total cost function from zero to 100 percent of productive capacity is to suggest that variable cost per unit is constant for all volumes of activity up to full capacity and that marginal cost is also constant and equal to variable cost per unit, as illustrated in Exhibit VII. A linear total cost function also suggests, as the same Exhibit shows, that total cost per unit declines con- tinuously over the entire volume range up to full capacity and is always higher than variable costs per unit or marginal costs. Diseconomies of scale are supposedly non-existent. This disturbs economists because it conflicts with the economic theory of the firm. Economists have generally drawn the total cost function as a curve which rises first at a declining rate and then at an accelerating rate, as illus- trated in Exhibit VIII. They believe that, as the volume of output of a firm increases from zero level to 'optimum' - 55 - 80 EXHIBIT VII CONSTANT AVERAGE VARIABLE COST Ter ed 5or 7] rs) "i 0 gob VU Bor 2or Average Nariable cost Or Marginal Cost lor AWera ge Nariab le Cast ° \o 20 30 ao 50 GO Volume (Output) Costs TO 60 40] 20 \o ~ 58 - EXHIBIT IX ECONOMISTS! AVERAGE VARIABLE COST CURVE A verage Variable cost Average Fixed Cost Volume (Output) - 59 = Meticulous statistical investigation by Joel Dean, R. A. Lester, R. H. Whitman and others, however, do not seem to support the arguments of the economists.” In an article in an N.A.C.A. bulletin, John Kempster mentioned that: In the economic research which has been done on cost, one of the important points which has been at stake is the question of whether unit variable costs fall and then rise with expanding output or are constant in their variability. Putting it another way, this is the same question as whether total variable costs would be expressed as a curve or a straight line in diagrammatic presentations. Some- what contrary to theory, the research investigations of economists have concluded, in general, that unit variables are con- stant throughout the relevant ranges of volume, that is, total varjable costs in- erease at a constant rate. Summarising from the above discussion, it may be said that since fixed costs remain fixed at all volumes, it is the variable cost function that determines the shape of the total cost function. In the linear break-even analysis, the total cost function is drawn as a straight line, giving the impression that unit variable costs remain constant at all volumes. This is contradictory to the economic theory of the firm. In economic analysis, unit variable costs are described as having a 'U' shape. The research investigations of some economists, however, support the impression of constant 8. J. Johnston, Statistical Cost Analysis, New York: McGraw-Hill Book Company, Inc., 1960, pp. 136-168. 9. John H. Kempster, "Break-Even Analysis - Common Ground for the Economist and the Cost Accountant", N,A.C.A. Bulletin, Feb. 15, 1949, p. 712. i - 60 - unit variable costs given in linear break-even analysis, for relevant ranges of volume. Production Equals Sales So far, various assumptions in break-even analysis, relating to the total cost and revenue functions, have been made. To this list, may be added the further assumption that sales and production are synchronised and there is no sig- nificant amount of production for inventory or no substantial amount of sales from inventory. All fixed costs incurred by the firm are, therefore, considered as period costs and charged against the revenue realised in the same period. This assumption is obviously not entirely true. At times, firms produce more than what they can sell, as a result of which inventories are built up and, at other times, they may produce less than what they can sell and consequently, inventories are depleted. In fact, in practice, firms sel- dom find that their sales exactly equal their production. In many periods, however, firms may find that the difference between sales and production is not very significant and this is the position that is generally taken in discussions on break-even analysis. One writer stated that: Inventories, though, are usually very small in comparison to total pro- duction and, for practical purposes, are ignored in comparing sales at various levels of production....The least probable error, then, is obtained by disregarding the inventory problem in determining sales - 63:-- for the period, then it means that, at the break-even level of activity, revenue for the period equals the expenses incurred in realising this revenue plus the expenses in- curred in realising the revenue of later periods, when the inventories from current production are sold. This tends to distort the picture of the profitability of the business for the current period as well as for those periods in the future, whose sales include inventories from prior produc- tion. Although conventional break-even analysis elimin- ates this problem by assuming that production equals sales, it is wise to be aware of the existence of this problem. The greater the difference between production and sales, the more serious is the problem. In fact, in firms in which there exists a significant difference between production and sales, it may be advisable not to consider the use of break-even analysis. Sales Mix The synchronization of production and sales is, however, not the last of the basic assumptions of break-even analysis. An executive, who intends to make use of break- even analysis, is also faced with the problem of product mix or sales mix. This problem arises so long as the firm is a multi-product firm and if, in addition, the various products have different margins of return over variable - 64 - costs. ‘This becomes clear when we consider the fact that, ina firm, if the total sales revenue is made up of the revenue of products with high margins over variable costs, the break-even point will. be lower than if total sales revenue is composed of the revenue of low margin items, This being the case, each time the sales mix changes, the break~ even point and the profit pattern will also change. Hence, other things being equal, management is generally considered to be making a good move, profitwise, if it tries to in- erease the sales of a high-profit margin product at the expense of a less profitable item. The sales mix is, therefore, an important factor in break-even analysis. With a given total cost function and a given total revenue function, an increase in total sales, from a sales volume above the break-even volume, may not produce the expected increase in profits, if there is a change in the sales mix. The increase in profits may be greater or less than what is expected, depending on whether the change in sales mix is from the higher margin products to the lower margin products or the reverse. To overcome this problem, the users of break-even analysis assume a given mix or that the sales mix will remain constant as sales volume changes. This assumption, however, presents a serious limitation when the composition of demand for the products of the firm changes. To avoid this assumption and to make break-even - 65 - analysis more tseful, various writers have advanced many possible solutions to this problen.?* Perhaps, the approach which has received the greatest attention, is the one which requires a separate calculation or graph for each product. Fixed costs, therefore, have to be appropriately allocated to the various products and this is where the difficulty lies with this method. It has already been mentioned earlier that the separation of costs as fixed costs or variable costs is fraught with difficulties. The job of allocating fixed costs among the various products is even more trying. Some costs may be common costs, the allocation of which is just not practicable. This means that the sum of the individual break-even points will not equal the break-even point for the firm as a whole. The assumption of a constant sales mix, made in conventional break-even analysis, is thus necessary only in a multi-product firm; but then the single-product firm is, today, a distinct rarity in the real world of business, None 14. Paul May recommends the use of a profit polygraph - P, A. May "Profit Polygraph for Product Mix Evaluation", NACA. Bulletin, Vol. 37, Sec. 1, Nov. 1955, pp. 307-318. Richard Conway suggests the method of sequential con- sideration on a single chart or the use of multi-dimensional analysis - R. W. Conway, "Breaking out of the Limitations of Break-Even Analysis", NACA. Bulletin, Vol. 38, Sec. 1, June 1957, pp. 1265-1272. Joel Dean suggests the use of a family of product-mix lines - Joel Dean, Managerial] Economics, Englewood Cliffs, N,J,2 Prentice-Hail, Inc., 1951, p. 335. These methods may produce more accurate results but, usually this is achieved at the expense of the advantages of break-even analysis, such as, ease of understanding, inexpen- siveness and quickness in preparation, - 68 - the reduction of fixed costs or of variable costs or whether the efforts should be exerted to increase volume. If the fixed costs of a firm constitute a very high proportion of total costs, then it must operate at a substantial percentage of capacity to cover such costs but, once the break-even volume is reached, profits increase at a very rapid rate, with increases in volume. On the other hand, if the total costs of a firm are made up mainly of variable costs, then a relatively low volume is sufficient to cover fixed costs but, even after the break-even volume has been reached, profits will not increase at a fast rate. On the financial side, if a firm has a high percentage of fixed costs, an increase in volume may not cause a serious demand for cash but, if the firm has a high percentage of variable costs, an increase in volume is likely to cause an increase in variable costs and eventually a drain on cash. In control, break-even analysis is useful for detecting any insidious upward creep of costs, which might otherwise go unnoticed. It can also be used to compare actual and planned performances and to show the logical points of attack to effect improvement. A common error made by manage- ment is to overemphasize the importance of volume as a deter- minant of profits. Some management people may assume that an increase in volume will automatically increase profits. Actually, this happens above the break-even point only if prices remain unchanged and only if variable costs are kept - 69 - under control. Unfortunately, an increase in volume very often is accompanied by an increase in costs, which is fre- quently large anough to more than offset the beneficial volume effect. Break-even analysis comes in handy here since it is capable of bringing to the attention of management the profit determinant that has been responsible for offsetting the volume effect. With this brief introduction to the uses of break- even analysis, we can now go on to examine more specific areas of management planning and control, in which break-~ even analysis is capable of playing a significant role. Pricing Policies Pricing a product is one of the most delicate prob- lems of management. A poor pricing policy may lead a business into bankruptcy. Many factors influence the pricing decisions of management but the most important factor is probably cost. Some firms adopt the policy of selling some of their minor products telow cost, in order to attract customers, There is, however, hardly any profit-making firm which can afford to sell consistently below cost. In order to be successful, firms have to recover not only their costs but also a profit that is adequate to maintain the incentive for their continued operation, Break-even analysis can provide some help to man- agement in the establishment of prices. Break-even charts - 70- ean be drawn to show the effect on profits of different price levels, These charts may then be compared with one drawn under existing conditions to show the volume of sales that would be necessary to achieve the same level of profits. A higher price, ceteris paribus, has the effect of raising the profit/volume ratio and accelerating the recovery of fixed costs. Hence, a lower volume of sales would be“ sufficient to attain the profit objective. Conversely, a lower price would lower the profit/volume ratio and reduce the rate of recovery of fixed costs. Attainment of the profit objéctive, in this case, would require a higher volume of sales. The usefulness of break-even analysis, in pricing decisions, arises mainly from the fact that it can ably show the cost-volume-revenue structure of a business. But, one should never overestimate the usefulness of break-even analysis in pricing decisions because the effect on profits of a change in price depends not only on the cost-volume- revenue structure of the business but also on the effect on volume of the change in price, that is, on the price elas- ticity of demand. In actual fact, in any pricing decision, the latter would appear to be, as important as, if not more important than the former. Unfortunately, break-even analy- sis does net, in any way, tell us what the price elasticity of demand for a product is like. Capital Expenditures Capital expenditures usually involve large sums -73- have to worry about these policy factors, then the answer to the make or buy problems would probably revolve around the question of costs. This means that proper cost information would be required so that the cost of making can be compared with the cost of buying. If a firm has unused productive capacities in the short-run, the cost of making may be based on the additional costs that it will have to incur if the orders were kept in the company. In the long-run, however, the firm's cost of making should include direct materials, direct labour, the variable costs involved, a share of fixed costs and a profit figure. Break-even analysis is useful in the comparison of the cost of making and the cost of buying since it can show the effects on profits, at different volume levels, of the two alternatives and thereby help management to make its decision. Cost Control Cost control is one of the most important aspects of the management of a business. Operating profits, as de- fined in chapter I, are equal to operating revenue minus operating costs. But, management does not have too much control over operating revenue since there is a limit to the amount that a business can sell and selling prices are, to a large extent, established by competition. Hence, the Tyo profit-making capacity of a business depends largely on the efficiency with which costs are controlled. One of the ways in which cost control can be achieved is through the use of flexible budgets, which ‘peflect the amount of cost that is reasonably necessary to achieve each of several specified volumes of activity, "18 For purposes of cost control, the predetermined costs are based on standards set for materials, Labour and expenses. These predetermined costs may then be compared with actual eosts and the differences, called variances, may be analysed. From the analysis of the variances, management may introduce measures to check the unfavourable trends and departures from the predetermined costs. In this way, flexible budgets aid in the control of costs. There is a greal deal of similarity between flex- ible budgets and break-even charts. In fact, it may be said that whereas flexible budgets are tabular variable income statements, break-even charts are graphic variable income statements .29 The construction of breakeeven charts is very often based on the data of flexible budgets; and just as flex- ible budgets are useful for cost control, so are break-even charts. For purposes of cost control, the predetermined costs and the actual cost may be plotted on a break-even chart to 18. Shillinglaw, op. cit., p. 217. 19. Adolf Matz, Othel J. Curry and George W. Frank, Cost Accounting, Cincinnati: South-Western Publishing Company, 1952, p. 678. -~75- bring out the variances and on the basis of the analysis of these variances, corrective actions may be taken by management. Summary In this chapter, it has been shown that break-even analysis can be used in decision-making involving make or buy problems and in problems related to capital expenditures, cost control and pricing decisions. These are, by no means, the only uses to which break-even analysis can be put. In fact, break-even analysis has been used in the solving of many other problems concerning alternatives, which involve cost, volume and profit relationships. It was also pointed out, in this chapter, that in using break-even analysis, many restrictive assumptions have to be made. The assumptions include the following: (a) All costs can be reasonably separated into their fixed and variable components and whereas fixed costs remain fixed at all volumes, variable costs vary in direct proportion to volume. (b) Selling prices remain constant at all volumes. (ce) Production equals or closely follows sales and all fixed costs incurred in a period are, therefore, deducted from that period's revenue. (d) There is only one product or if several products are being produced and sold, the sales mix will remain eonstant, . These assumptions are more valid for some firms than for others. In those firms in which these assumptions are very unrealistic, break-even analysis is virtually useless, unless - 78 =! A more sophisticated way of forecasting profits involves the use of the breakseven technique. Break-even analysis - its nature and its pros and cons ~ needs no further comment here. The purpose in this chapter is to test the hypothesis and the null hypothesis, as detailed in chapter I. Source of Data The data for the test are taken from Moody's Industrial Manuals. As indicated in chapter I, the statis- tical approach (least squares method) is used to separate the total costs of the firm into fixed costs and variable costs, since in terms of the data available, it is likely to give more reliable results than the accounting or engin- eering methods. Included in the universe of firms are only those firms in Moody's Industrial Manuals, which have had losses at some time or other over the period covered in the study. This is so because preliminary studies to this test showed that, in the case of those firms, which had never suffered any losses, it was not possible te separate their total costs into their fixed and variable components, by the statistical approach. In this test, the year chosen for the forecast is 1956, Any other year could have been chosen so long as it is a past year; otherwise, it would not be possible to 1. Detailed in chapter I. - 79 - compare the forecast profits with the actual profits, to deter-~ mine the accuracy of the forecasts. For the test, in order to measure cost variability, the behaviour of costs of each firm in the sample is studied for a maximum period of ten years, from 1946 to 1955, As indicated in chapter I, Moody's Industrial Manuals from 1946 to 1958 are used to obtain the data.” A close examination of the Manuals for this period showed that 589 firms could be included in the universe. Poreeasting Operating Profits Ag Break-Even Method in forecasting the operating profits of the sample firms by the break-even method, the following assumptions 3 are made: (a) All costs can reasonably be classified as fixed or variable. (b) Selling prices remain constant at all volumes, (e) Production and sales are synchronised and (d) The sales mix remains constant. The operating profits of firms at any given volume Level is equal to the operating revenue minus the operating 2, There is a time lag in Moody':s Industrial Manuals. The data of some companies appear in the manuals one or two years after the end of their fiscal year. 3. The need for these assumptions have been discussed in chapter III. Their validity varies among the sample firms. For most of the sample firms, the first three assumptions are quite valid. ‘he fourth assumption, however, is not valid for almost all the firms but has to be made in order to carry out the test. - 80 - costs at that volume level. In this study, the forecast volume level is given and is equal to that at which the actual operating profits are realised. The operating revenue is also given and is equal to the volume level, on the assump- tion that production and sales are synchronised. Therefore, in order to forecast the operating profits, it is only nec~ essary to determine the operating costs. The determination of the operating costs, at a given volume level, can be attempted in many ways. The accounting, engineering and statistical approaches have already been explained in chapter II. The statistical approach, with the scatter chart technique and the method of least squares, is used here. The reason for this has been discussed in chapter I. By this method, the operating cost’ figures of all the sample firms are collected for as many as possible of the years between 1946 and 1955 (inclusive). These figures are then plotted on scatter charts with sales volume as the horizontal axis and operating costs as the vertical axis. The idea here is to achieve an estimate of the correlation between costs and sales volume. Shilling- law advises that the statistical approach "must be regarded as first approximations. If there are strong common-sense reasons for doubting that the resulting cost-volume pattern is reasonable, then the conclusion of the statistical analy- sis should be supplemented by the application of judgement. "* 4, Gordon’ Shillinglaw, Gost Accounting: Analysis and Control, Homewood, Illinois: Richard D. Irwin, Inc.,°1961, Dp. 235. - 83 - as in the case of break-even analysis, is to compare the forecast operating profits with the actual operating profits, therefore, it is assumed that the sales volume in the forecast year, 1956, is given and is equal to the sales volume at which the actual operating profits are realised in 1956. Once the sales volume is known, a forecast of the operating profits can be made by applying to the given sales volume, the average percentage of profits as a percentage of sales for the years 1946 to 1955, The Sample Before making a decision on the firms to be in- cluded in the sample, a decision has to be made on the number of firms to be included in the sample. For this purpose, it is necessary to state the desired degree of accuracy. In this study, it is asserted with a probability of 0.95 that the estimated mean will be within $0.10 of the true mean, The confidence interval is arbitrarily fixed, depending on what is felt to be reasonable, under the cir- cumstances. In this case, consideration was given to the fact that the sample mean of the exploratory study,” Xa> is only $0.26m. (Table IIL). The t-table (Appendix II) shows that for a 95 percent degree of confidence, with 9 degrees of freedom (n-1), the standard error of the mean 9. This is explained in the next page. - 8 - 19 is equal to 2.262. From this, the following formula may be used to determine the size of the sample: 2.262 Ge) = §$0.10m where 6 = standard deviation of the universe (population) and n size of the sample. In order to determine the standard deviation of the universe, a start has to be made with an exploratory study of some firms, picked at random from the universe. In this case, 10 firms are used for the exploratory study. The basic principle behind random sampling is that every firm in the universe must have an equal chance of being chosen. To achieve this, use can be made of prepared tables of random numbers. Firstly, all the 589 firms in the universe are listed in alphabetical order and numbered from 1 to 589. A decision is then made to use Kendall and Smith's "Tables of Random Sampling Numbers, Tracts for Computers No. xxryett ‘(Appendix I). To avoid any possibility that the choice of a starting point might be nonrandom, it is arbitrarily decided, before examining the Random Number Tables, to start picking 10 firms from Row 12, and columns 6, 7 and 8 of the random numbers shown on page 15 of the tables. This would give the numbers: 10. John E. Freund and Frank J. Williams, Modern Business Statistics, Englewood Cliffs, N.J.: Prentice-Hall, Inc., 1958, p. 193. 11.M. G. Kendall and B. B. Smith, Tables of Random Sampling Numbers, Tracts for Computers No. XXIV, Cambridge: Cambridge University Press, 1951. - 85 - 377 339 218 O43 157 Ty 451 soe 070 525 Since the universe is made up of 589 firms, any number exceeding 589 is ignored. In the same way, any number which appeared more than once is also ignored after it had appeared for the first time. This happened in the case of the number, 043. Once the firms for the exploratory study have been picked, the standard deviation of the universe X (6) can be determined by using the formula: where: © = standard deviation of the universe = sigma, sum of, summation = difference between actual and forecast profits mean of the difference in profits p ol gf oh = size of the sample in the exploratory study 3.2 Table III shows that 4(Xqg ~ Xq)° = $0.9127m. Therefore, the standard deviation of the universe,©® is equal to: [i net = /$0.1014n. Earlier, the formila for the size of the sample u $0.3184m. In determining the differences between actual and forecast profits ( p, - signs are ignored because we are only interested in the magnitude of tHe ence and not in the direction of the differences. dite ), er-~ - 9g - - 89 - sample is known, the firms can be drawn at random from the universe of 589 alphabetically listed firms. Here, again, use can be made of Kendall and Smith's "Tables of Random Sampling Numbers, Tracts for Computers No. xxtyet3 (Appen- dix I), following the same procedure as that used to obtain the firms for the exploratory study. Method of Analysis In order to determine whether break-even analysis or the percentage of sales method can provide a better fore- east of operating profits, a comparison must first be made of the foreeasts of the two methods with the actual operating profits. A comparison of the forecast of the percentage of sales method with the actual operating. profits is given in fable IV and a comparison of the forecast of break-even analysis with the actual operating profits is given in Table v. The method which gives a smaller difference be- tween forecast operating profits and actual operating profits should be the more accurate method. Table V shows that, for the 57 firms shown in the sample, the difference between the forecast operating profits and actual operating profits, by the percentage of sales method, totals $24.27m. This gives a mean difference of $0.4257m, that is $24.27m divided by s7ett Table V shows that the difference between the forecast 13. Kendall, ioc. cit. 14, The mean is used instead of the mode or the median because it is least subjected to sampling variation. This was discussed in chapter I. TABLE IV DIFFERENCE BETWEEN ACTUAL AND FORECAST PROFITS - PERCENTAGE OF SALES METHOD ($ Amounts in Millions) Av.Percentage Difference of Profits Between as a Per- Sales Rore- Actual and centage of Volume” cast Actual Forecast Name of Company Sales in 1956 bese Profits Profits (a) (>) ax 100 Pa € PawPe ) 1 Baush Machine Tool Co. 9.8% $4.03 $0.39 $0.44 $0.05 2 Bell Company - 055 6.64 (0.04) (0.28) 0.24 3 Bishop and Babcock Manufacturing Co. 3.69 5.72 0.21 (0.15) 0.36 Brown-McLaren Manufacturing Co, 3. 1.62 0.05 (0.10) 0.1 5 Carpenter (L.E.) & Co. = 2.52 3.45 (0.09) (0.43) 0.3 6 Chief Consolidated Mining Co, 0.74 0.56 0.00 (0.07) 0.07 7 Cleveland-Sandusky Brewing Corp. 5.76 1.38 0.08 0.02 0.06 t 8 Consolidated Retail Stores, Inc. 2.59 21,04 0-38 (1.77) 2.32 0 9 Cooper Tire and Rubber Co, 3.13 23.74 0.7 1.07 0.33 3° 10 Curtis Lighting, Inc. 2.01 3.73 0.08 0.15 0.07 1 11 Dixon (Joseph) Crucible Co, 3.20 12.65 O41 0.98 0.57 12 E. & B, Brewing Co. Inc, 1.35 0.92 0.01 0.05 0.04 13 Flagg-Utica Corp. - 1.17 17.18 (0.20) 0.64) 0.84 14 Flotill Products, Inc. 123 21.41 0.37 1.92 1.55 15 Flour Mills of America, Inc. Q. 18.95 0.21 0.97 0.76 16 Gerotor May Corp. - 2322 1.3 (0.07) (0.25) 0.18 17 Gum Product, Inc. ~ 413 2.10 (0.09) 0.13 0,22 18 Hathaway Bakeries, Inc. 3.81 18,89 0.53 (1,00) 1.53 19 Hiller Helicopters 4.39 9.83 0.43 0.32 0.11 20 Jacob Ruppert 0.03 49.57 0.01 (0.19) 0.20 21 Jeannette Glass Co. 5.69 5.18 0.30 Oy 0.14 22 Jessop Steel Co. 3.06 24.85 0.76 3.47 2.71 23 Lanston Industries, Inc. 10,21 2.91 0.30 (0.02) 0.32 24 Longchamps, Inc. 2.16 7.73 0.17 (0,02) 0.19 25 Macmillan Petroleum Corp. 2.63 14,16 0.37 0.53 0.16