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Flexible budgets and the analysis of overhead variances.

Standard costing has become so generally accepted as a useful accounting technique in industry that is unusual to find any of its methods called in question. Yet, while the analysis of variances between actual and standard direct costs--direct materials and direct labor--has given rise to no important differences of theory or practice among writers and practitioners, the same cannot be said of the analysis of overhead variances. Indeed, an examination of the literature of the subject over the last thirty years shows an extraordinary variety of treatments.(1) It would surely be in the best interests of management if some agreement could be reached about the most effective method of analysing overhead variances for general application.

In this paper I shall set myself three objectives. The first is to compare the two principal systems of overhead variance analysis in common use. I shall attempt to do this by the use of a diagrammatic method of representing variances which I have written about previously.(2) From the comparison of the two-variance and three-variance systems, an assessment of what an effective system should achieve is then made. Finally a suggestion is made for an improvement in the flexible budget, leading to a corresponding improvement in the analysis of overhead variances.

The conventional analysis of overhead variances

The conventional analysis of overhead variances is illustrated in diagram I.(3) Diagram I (a) shows the analysis of variance into two parts, a controllable variance, as it is usually called, and a volume variance. In the situation illustrated, the actual man-hours worked fell seriously short of the man-hours budgetted, and the actual efficiency of the work done was also substantially below 100%, so that the output produced (expressed in standard hours) was less than it should have been in the man-hours actually worked. These three quantities, budgetted hours, actual hours and achieved standard hours (i. e. actual output) are represented by the three vertical lines. Budgetted fixed overheads are represented by the horizontal line B.F.O. and on top of this the variable expenses budgetted for varying levels of output are superimposed, to give the line B.T.O. representing budgetted total overheads. This line stands for the flexible budget, and shows at each level of activity the total overhead expenditure allowed by the budget. While this line is shown as a straight line in the diagram, the analysis would in no way be jeopardized, and would almost certainly gain in realism, if the budget line were shown as curved instead of straight, or even as rising in discontinuous jumps.

The standard overhead rate is computed by dividing the total overhead budget allowance appropriate to the budgetted level of activity by the number of man-hours to be worked. This gives the standard hourly rate of overheads, and is represented by a straight line joining the point S (the point where the budget expense line B.T.O. cuts the "budget hours" vertical) to the origin. This line cannot be anything but straight, since the standard rate, once fixed, cannot vary during the period, whatever fluctuations of output or expenditure may occur. Only when there is a revision of standards will the standard overhead rate change.

It is important to make a clear distinction between the budgetary data in the diagram and the information about actual results. To enable this distinction to be kept in mind, budget data is represented by broken lines and "actual" data by unbroken lines. The actual expenditure on overheads during the period under examination is shown by the horizontal line passing through the point A.

With the two-variance system, the flexible budget allowance is regarded as being set by the level of output achieved, and in diagram I (a) it is represented by the point B', allowed expense, the point at which the vertical for actual output cuts the flexible budget line. The excess of actual expense over allowed expense, represented by the vertical distance AB', is the controllable variance, the excess expenditure (in this case) for which the departmental officials and foreman can be held responsible. The difference between the allowed expense and the overheads absorbed into actual production at the standard overhead rate--the absorbed expense--is the non-controllable or volume variance. It represents the fixed overheads which are left unabsorbed because production (expressed in standard hours) fell short of budgetted production. As can be seen from the diagram if output had been as budgetted, the volume variance would have been nil. If output had been zero, then the volume variance would have been equal to--it would in fact have consisted of--the total fixed overheads. The diagram makes it quite clear that there would be no volume variance if there were no fixed costs. It can further be seen that if the achieved standard hours were to exceed the budgetted hours, either because of intensive use of capacity or because of high productivity per hour, then the standard overhead recovery line would have crossed and be above the budget line, and the volume variance would be favorable, i.e. it would represent an over-absorption of fixed costs.

Substituting figures for the diagram,(1) suppose that the budget for the month calls for 2000 standard hours of output in 2000 man-hours. At this level of activity, budgetted expenses are $ 1000 fixed expense and $ 1000 variable expense for the month. At the end of the month, it is found that only 1800 man-hours were worked and only 1600 standard hours were produced, while actual overhead expenditure was $ 2150.

|Mathematical Expression Omitted~

Turning to diagram I (b), we have an illustration of the conventional three-variance system. The data is the same as in diagram I (a), and the notation is almost the same. As can be seen from the diagram, with this system the budget allowance for overheads is determined by reference to the actual man-hours worked, at point B.

The three variances can now be identified. The vertical distance A B, the excess of actual overheads over the budget allowance, represents the budget variance. This variance will usually be in part a price variance, due to differences between the standard and actual prices of overhead services or indirect materials, and in part a spending variance, due to divergences between actual and budget expenditures unrelated to price variations. There is no technical difficulty whatever in making this division between the price element and the "spending" element in the budget variance, for all that needs to be done is to have all overhead services, indirect materials and indirect labor priced at both actual and standard prices, in the same way as is done for direct materials and direct labor. But in practice it is not often thought worthwhile to go to these lengths.

The vertical distance B C is the capacity variance. As can be seen, it is the difference between the budget allowance for the hours actually worked, and the cost which would have been absorbed, at the standard overhead rate, if every hour worked had been 100% effective in producing output. This latter amount, the actual hours worked evaluated at the standard overhead rate per hour, which we might appropriately call the time-absorbed expense, is represented by point C. Like the volume variance of diagram I (a), the capacity variance can be seen to be equal to the full fixed costs when output is zero, falling to a nil variance when actual hours are equal to budget hours, and then reversing its sign and becoming a favorable variance when actual hours exceed budget hours.

The third member of the trinity is the efficiency variance. This is the vertical distance C D. The point D is merely a projection on the "actual hours" vertical of the point F, which represents the standard cost recovered or absorbed by actual output. It will be seen that the efficiency variance, so measured, is the difference between the standard overhead cost absorbed by actual output and what we have called the time-absorbed expense--the standard overhead cost which would have been absorbed if the actual hours worked had been fully effective in producing output.

|Mathematical Expression Omitted~

As may be supposed, the two methods of variance analysis just explained are not unrelated to each other. The controllable variance of the two-variance method is equal to the budget variance plus part of the efficiency variance of the other method--a part equal to the difference between the budget allowance for actual hours and the budget allowance for achieved standard hours, represented on diagram I (a) by the vertical distance |Mathematical Expression Omitted~. The same quantity is shown on diagram I (b) as the vertical distance E D, a part of the efficiency variance. The position of E is determined by drawing a straight line from F, the absorbed expense, parallel to the flexible budget line until it cuts the "actual hours" vertical. The slope of E F represents the variable overhead rate per hour, and E D therefore represents the extra variable cost incurred because it took more actual hours to get the output achieved than it would have done had efficiency been 100%. Since diagrams I (a) and I (b) are basically identical, it is easy to prove that |Mathematical Expression Omitted~, since the triangles FED and |Mathematical Expression Omitted~ are congruent.

This relationship is significant in any assessment of the relative merits of the two systems. The chief defect of the two-variance system is that it does not distinguish between the "spending" element and the "efficiency" element of the controllable variance; it does not, that is to say, distinguish between excessive expenditure due to loose control of spending or price increases and excessive expenditure arising from man-hours wasted through inefficiency.(1) The three-variance system, on the other hand, does yield a separate efficiency variance, but it does so in a very questionable manner; for the efficiency variance, as conventionally determined, does not represent the true extra cost which results from inefficiency or the cost saving which results from high efficiency, but rather the difference between two hypothetical figures of "absorbed" expense.

Another criticism, which can be levelled at both methods, is that the budget allowance for overheads, as set by the flexible budget, is regarded as a function of a single variable. All expenses are regarded as varying, if they vary at all, either with the number of man-hours worked (as in the three-variance system) or with the level of output produced (as in the two-variance system), whereas the facts of cost behavior clearly make any such simple assumption unrealistic.

Realism demands that we recognise at least three types of expense:

(a) those that vary with the volume of output, e.g. manufacturing supplies, certain costs of materials handling and maintenance of equipment, royalties, inspection costs.(1)

(b) those that vary with the number of man-hours worked, e.g. welfare expenditure, supervision, heating and lighting.

(c) those that vary only with the length of the period, and not with the amount of work done or the output obtained, e.g. rent, fire insurance.

This points to the need to make the flexible budget flexible in more than one direction, taking each expense separately and making appropriate adjustments to the original budget figures in the light of what is known, at the end of the period, about the number of man-hours worked and the level of output achieved.(2)

Recognition of the fact that at least two independent variables control the level of overheads suggest that many more variables could be introduced into the function, bringing greater realism but also, unfortunately, greater complexity. For the more or less routine purpose of expenditure control with which we are now concerned, it is doubtful whether the greater complexity would be worthwhile.

An improved method of overhead variance analysis--four variances

The shortcomings of the conventional methods of variance analysis which were pointed out above can be met, while retaining their virtues, by recognising at least four separate variances, viz:

1. A budget variance. This is, as normally, the difference between actual expenditure on overheads and the allowance set by the flexible budget; but by making the flexible budget more flexible, as has already been suggested, the budget variance can be made much more meaningful than it commonly is. The flexible budget must be made to register the allowance for overheads appropriate to the actual number of hours worked and the actual level of output achieved.

Nothing that is said here precludes more detailed analysis of the budget variance such as, for instance, the separation of the effects of price and quantity variations, although nothing more will be said about such further analysis. Again, it may be presumed that in practice the budget variance will be examined item by item rather than in terms of total overhead or of broad categories of overhead, as is done here.

2. A true efficiency variance--true, because it really measures the gain or loss in overhead expenditure attributable to variations away from the level of efficiency stipulated in the budget. One of the figures necessary in this calculation is the flexible budget allowance just mentioned. The other figure, which can easily be derived from the flexible budget, is the budget allowance appropriate to the production of the achieved output in the standard number of man-hours, as distinct from the actual number of man-hours. Clearly, those expenses which vary with output and not with working time (and also, of course, those which are fixed in relation to both) will be unaffected by excess man-hours or savings in man-hours as compared with the standard time required to produce the actual output. Such expenses are determined by the actual output, not by the time it takes to produce it. The expenses linked to man-hours, on the other hand, will be directly affected by excesses or savings in man-hours. It is these expenses, and these alone, which should enter into the overhead efficiency variance, which now becomes the difference between the flexible budget allowance for the actual output obtained in the actual man-hours worked and the budget allowance which would have been set if the actual output had been obtained in the standard number of man-hours.(1,2)

3. A volume (capacity) variance. This is one part of the unabsorbed (or over-absorbed) fixed costs--the part attributable to the loss or gain in output resulting from the difference between the budgetted number of man-hours and the actual man-hours worked.

4. A volume (efficiency) variance, being the remaining part of the unabsorbed or over-absorbed fixed costs. This part is attributable to the difference between actual output and the output which would have been obtainable from the actual number of man-hours worked if these had all been worked at standard efficiency.

It is not difficult to see that this system combines the advantages of the two-variance and the three-variance systems, without their shortcomings. All we need to do is to build into the system a capacity to distinguish between the two types of variable expense just discussed--the time-variable expenses, as they might be called, and the output-variable expenses. This is not a difficult thing to do. The method is perhaps best explained by means of an illustration, using the same basic data as in the earlier illustrations, but making the further assumptions that (1) the original overhead budget allowance of $1000 for variable expenses is divisible into $600 for output-variable items and $400 for the time-variables, while (2) the actual expenditure of $2150 is made up of $1050 in fixed expenses, $675 in output-variables and $425 in time-variable items.

Using this data, we can see from the table below how the variances would be arrived at.(3)

Column (1) of the table sets out the original budget figures, classified under types of expense, and based on a plan to produce 2000 standard hours of output in 2000 manhours. The actual expenditure for the period, as ascertained at the end of the period, and following the same classification of expenses, in shown in column (2). Column (3) shows how the original budget is adjusted to take account of the actual time worked and the actual output produced in that time. Fixed expenses call for no adjustment. The expenses which vary with output are reduced to 1600/2000 of the original budget figure, and the expenses which vary with man-hours worked are reduced to 1800/2000 of the original budget. The total adjusted budget for the actual output in the actual time is therefore $1840. Column (4) shows what the budget allowance would have been if the actual output had been produced in the standard time. Fixed expenses and those that vary with output would have been as shown in column (3) but the allowance for expenses varying with time has to be reduced to only 1600/2000 of the original budget. The total adjusted budget for the actual output in the standard time is therefore $1800. Finally, column (5) shows the absorbed overheads as $1600, i.e. 1600 standard hours of output at the standard rate of $1 an hour.

The determination of the budget and efficiency variances from the table is sufficiently obvious to need no comment. The budget variance of $310 is the excess of col. (2) over col. (3) and the efficiency variance of $40 is the excess of col. (3) over col. (4). The treatment of the volume variance of $200 is not so obvious, however, and calls for explanation.

There is really only one difference between col. (4) and col. (5), and that is in the fixed expenses. In col. (4) only the fixed expenses are unchanged from the original budget in col. (1). The other types of expense have shrunk to 1600/2000 of the original figures. In col. (5), all the figures, in effect, have shrunk to 1600/2000 of the original budget. The difference between col. (4) and col. (5), therefore, consists of the 400/2000 of $1000, or $200, by which the fixed expenses have failed to shrink with the shortfall of output below the quantity originally budgetted, and this $200 is the volume variance. But we can go further than this. Using the fixed overhead rate of $1000/2000 or $0.50 per standard hour, we can say that $0.50 x (2000-1800) or $100 of fixed expenses remain unabsorbed because less capacity was used than was expected, and that $0.50 x (1800-1600) or a further $100 are left unabsorbed because the capacity that was used was used with less than standard efficiency. Thus the volume variance of $200 can be split into a volume (capacity) variance of $100 and a volume (efficiency) variance of a further $100.

Diagrammatic representation of the revised system

As overheads now have to be regarded as a function of two independent variables--man-hours and output--instead of one--man-hours or output--this system cannot be represented on a simple two-dimensional diagram. But it may be illuminating to compare it diagrammatically in a partial way with the conventional systems, and this can be done if we concentrate our attention on those costs which are fixed and those which vary with the number of man-hours worked, ignoring those costs which vary with output. By so doing, we can represent the system on a two-dimensional diagram.

Diagram II (a) merely repeats diagram I (b), to facilitate comparison between the three-variance system there shown and the four-variance system represented in diagram II (b). The underlying data are again the same as before, so that all the lines have the same slope as in the previous diagram, and the actual overhead expenditure for the period is the same as before also.

Since in diagram II (b) we are dealing only with expenses which are fixed or which vary with the man-hours worked, the flexible budget allowance for the period is indicated by the point B, where the budget line is cut by the "actual hours" vertical. Point B is projected across to the left to give |B.sub.1~, and the vertical distance A B is the budget variance. It is, of course, equal to A B, the budget variance in diagram II (a).

The second of the four variances in diagram II (b) is the efficiency variance. This is the vertical distance |Mathematical Expression Omitted~; for |B.sub.1~ (equal to B) is the budget expense allowance for the man-hours actually worked, and |Mathematical Expression Omitted~ is the allowance for the hours which the output achieved ought to have taken. |Mathematical Expression Omitted~ is therefore the cost increment resulting from wasted hours(1), and is truly an efficiency variance. As already noted when we were discussing diagram I above,|Mathematical Expression Omitted~ in diagram II (b) is equal to ED, a part of the efficiency variance, in diagram II (a).

The distance |Mathematical Expression Omitted~ is the volume (capacity) variance. It is, by construction, equal to BC, since the point |Mathematical Expression Omitted~ is determined by the intersection of the "achieved standard hours" vertical with a line through C drawn parallel to the budget line, so that |Mathematical Expression Omitted~ is a parallelogram, the opposite sides of which are equal. |Mathematical Expression Omitted~, because it is equal to BC on diagram II (b), is also equal to BC on diagram II (a). It represents the fixed overheads which are unabsorbed by reason of the fact that the actual man-hours worked fell short of the budgetted hours by reference to which the standard overhead rate was fixed.

The fourth variance is the volume (efficiency) variance. It is the balance of the unabsorbed fixed costs, the portion which is unabsorbed because the actual output (which can alone really absorb costs) fell short of the output which the actual man-hours worked would have achieved if the standard level of efficiency had been maintained. It is represented by the vertical distance |Mathematical Expression Omitted~.

1 Reference may be made to The Cost Accountants' Handbook, ed. Theodore Lang (Ronald Press, 1945) pp. 78-95, for a review of a number of methods developed prior to 1945. The second (1960) edition of this publication (now called The Accountants' Cost Handbook edited by Robert Dickey) does not find much to add.

2 "A Diagrammatic Representation of Standard Cost Variances", by David Solomons: Accounting Research, Vol. 2, No.1 (Jan. 1951) pp. 46-51. In "The Mathematics of Variance Analysis - II", Accounting Research, Vol. 4, No. 4 (October 1953) pp. 329-350, Mr. Gilbert Amerman uses a similar method to analyse overhead variances. While the results of his analysis are similar to mine, he does not form any judgment about the relative merits of existing systems, nor does he make any proposals for their improvement.

3 See page 92.

1 In the diagrams, the variances have been exaggerated to make the demonstrations more effective. The diagrams are therefore not drawn on a scale corresponding to the numerical illustrations. It need hardly be pointed out that the method of analysis used is equally applicable, whether any or all of the variances are favorable or unfavorable.

1 What is said here in terms of unfavorable variances of course applies, mutatis mutandis, to cost savings which give rise to favorable variances.

1 The assertion of a functional relationship between cost and volume or cost and any other independent variable does not imply that the relationship is necessarily a simple one, and certainly not that it is one of direct proportionality. This point will be emphasised again below.

2 Prof. W. J. Vatter gave a pointer in the same direction some years ago: "Rate of activity is often (and I think wrongly) taken as output or capacity or some related concept. Costs for planning and control purposes are related to decisions. Decisions have to do with inputs, not output. It would be better to talk and think about break-even charts, budgets, and other planning devices in terms of the input factors which must be controlled, rather than the output bases on which we can write up our post-mortems."

(N.A.C.A. Conference Proceedings, August 1954, p. 1700). I would rather say that we have to have regard to both inputs and outputs.

1 The standard number of man-hours is not to be confused with budgetted man-hours. The standard man-hours are arrived at by multiplying actual output by the standard time per unit of output. Budget man-hours are the result of multiplying budgetted output by the standard time per unit of output.

2 This definition of the efficiency variance corresponds in effect to that used by Lang, McFarland & Schiff (Cost Accounting, 1953, p. 376), except that they do not recognise the distinction drawn above between output-variable and time-variable expenses. The same may be said of Matz, Curry and Frank (Cost Accounting, 2nd Edition, 1957, p. 581).

3 In this illustration, the variable expenses (of both types) are regarded as being proportionately variable with man-hours or output. As was noted above in connection with Diagram I, it is not really necessary to make any such simple assumption. The variations might be continuous but disproportional, or they might be discontinuous. In either case, the flexible budget would be embodied in a table, only a little more complex than the kind commonly used at present, and the figures could be read off quite simply.

1 See footnote on page 87
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Title Annotation:Control
Author:Solomons, David
Publication:Management International Review
Date:Jan 1, 1992
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