PSA — its nature, significance and applications

PSA- its nature, significanceand applications Richard Kotas and StephenWanhill

The authors expound the concept of Profit Sensitivity Analysis (PSA) which examines alterations in the operating accounts of a business solely in terms of their impact on profits. They go on to outline the theoretical foundations of the concept and then indicate a variety of applications.

Profit sensitivity analysis (PSA) is a technique which is, at once, simple in application and (as it is hoped to demonstrate in this paper) useful in a variety of different ways. Although it is a relatively new technique,’ it has already been applied successfully in several types of business operation in the general field of tourism. The example given in the first part of this paper is based on restaurant operations.

Richard Kotas is a Senior Lecturer in accounting and Stephen Wanhill is a Senior Lecturer in economics and tourism at the Department of Hotel, Catering and Tourism Management, University of Surrey, Guildford, Surrey GU2 5XH. UK.

The profit multiplier

The authors would like to thank the Editor and an anonymous comments.

referee for helpful

The central concept in the theory of PSA is that of the profit multiplier, which measures the impact on the net profit of a business of a given change in the relevant key factor (eg price level, sales volume, cost of goods sold, labour costs etc). The basic procedure for the calculation of profit multipliers is as follows. (a) Determine

appropriate key factors. First, it is necessary to determine the key factors. In the example shown in Figure 1 there are altogether six key factors operating to influence the net profit. However, we could have shown two separate key factors (food prices and beverage prices) for the price level. We could also have separated food costs from beverage costs. Overheads are shown as one key factor and they could have been subdivided into separate cost categories such as: sales promotion and advertising; heat, light and power; repairs, renewals and depreciation, etc. Thus in each situation we must decide which of the various elements influencing net profit we wish to designate as our key factors. I. R. Kotas, “Catering profitability - a Profit Sensitivitv Approach”,

Hotel, Catering andSInsiitutional Management Association Journal, June 1975, 42, pages 13-15. See also R. Kotas, “The ABC of PSA : an introduction to Profit Sensitivity Analysis”, Hotel, Catering and

Institutional Management Association Journal, February 74, pages 15-19.

1978.

2. Examples of such consequential changes are: an increase in food and beverage costs resulting from an increase in the number of covers and a decrease in variable (part-time and casual labour) following a decrease in the sales volume.

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(b)

Trace eflect of each key factor on net profit. Having determined the key factors we calculate, in respect of each key factor, a profit multiplier. This is done as follows: we assume a small change (normally 10% for ease of calculation) in one key factor at a time and, holding all other key factors constant (except consequential changes?) we trace the impact of the 10% change on the net profit. The profit multiplier is then obtained by dividing the percentage change in net profit by the percentage change in the key factor. Thus, if we assume a 10% increase in the number of covers and this, holding all other factors constant, raises the net profit by 40%, we have a profit multiplier (PM) value of 4.0. The latter indicates that every 1% change in the number of covers causes a 4% change in net profit. By the same token, a price level PM

0143-2516/81/030176-13$02.00

0 1981 IPC Business

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PSA -

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Figure

1.

Key factors

and

and applications

PROFIT

net

profit

value of 10 would mean that every 1% change in the price level results, again other things being equal, in a 10% change in the net profit - obviously a situation in which the profit of the business is very sensitive to changes in the price level. Numen’cal example Let us assume that we have been given the summary of the Revenue Account of a restaurant as shown in Table 1. From the Revenue A/c we may see that there are six key factors which operate to influence the net profit of the restaurant: ( 1) the number of covers; (2) the price level; (3) food and beverage costs; (4) fixed labour; (5) variable labour; and (6) fixed overheads. The influence of the six key factors on net profit must now be calculated in accordance with the method already explained. The necessary calculations are shown in Table 2 where, it will be observed, we change (increase in this particular case, though a decrease would have produced identical results) each key factor by 10% and, holding other key factors constant, ascertain the effect of each such change on the net profit. When, in the calculations which follow, we hold all other things Table

1.

X restaurant

-

revenue

A/c

No of covers Average price Sales volume

20 000

f5 00 f 100 000

Less

Food and beverage Fixed labour Variable labour Fixed overheads

costs

f 90 000 flOOOO

Net profit

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f 30 000 f25 000 flOOOO f25 000

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PSA Table

its nature, significance 2.

Determination

and applications

of profit

multipliers Effect

on net profit of 10% change

in:

Key factors Base

No of

figures

covers

No of covers Price per cover

20 000 f 5.00 flOOOOO

Sales

Price level

22 000 f 5.00 fllOOOO

Fixed labour

F&B costs

20 000 f5.50 fllOOOO

20 000 f 5.00 f100000

Variable labour

20 000

20 000

20 000

f 5.00 f100000

f 5.00 flOOOOO

f 5.00 f100000

f 30 000

F 8 B costs Fixed labour Variable labour Fixed overheads

f30 000 25 000 10000 25 000

f33 25 11 25

000 000 000 000

f 30 000 25 000 10000 25 000

f33 000 25 000 10000 25 000

f 30 000 27 500 10000 25 000

Total

f 90 000

f94

000

f 90 000

f93

f92

f10

fl6

000 60% 6.0

f20

cost

Net profit % change in net profit Profit multiplier

000

000 100% 10.0

000

Fixed overheads

f7 000 30% 3.0

500

f7 500 25% 2.5

25 000 11 000 25 000 f91

000

f9 000 10% 1.0

f30

000

25 000 10000 27 500 f92

500

f7 500 25% 2.5

constant, we do not imply that in real-life business situations there is a change in one key factor at any time. Our sole purpose here is, simply, to ascertain the effect of a change in each separate key factor on net profit. Changes in several key factors simultaneously could be considered without loss of generality in the concept of PSA, but the dynamics would entail unnecessary complications. By ascertaining the PM values we have now quantified the impact of each key factor on the net profit of the restaurant. We are now able, in consequence, to redraw Figure 1 by adding to each key factor the relevant profit multiplier, as shown in Figure 2.

NO Cf

i-i

Figure 2. multipliers

178

Key factors, and

net

covers

profit

profit

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PSA - its nature, significance and applications Table

Ranking

3.

Key factor

PM

Price

10.0

level

No of covers Food

and

6.0

beverage

costs

3.0

Fixed

labour

2.5

Fixed

overheads

2.5

Variable

1.o

labour

The PM pro_file It is difficult to draw any conclusions from the PM values unless we arrange the key factors in some appropriate order. We therefore show a ranking of the key factors based on the magnitude of the relevant PM values (see Table 3). From the ranking shown it may be seen that the positioning of revenue-based and cost-based key factors is both significant and interesting. The two revenue-based key factors, represented by the highest profit multipliers, are at the top of our list. This indicates that the revenue side of the business has a much greater influence on profitability than the cost side. The costbased key factors are, in this example, represented by relatively low PM values and, in any case, two of these four key factors are, in practice, uncontrollable from one trading period to another. The PM profile has certain implications for the kind of accounting and control strategy that should be employed. Traditionally, accounting and control methods are cost-oriented, and this is quite clear from the terminology used : we employ a cost accountant to practise cost control and he then chooses the right costing method which probably necessitates the use of cost sheets, etc. The profile obtained above points in the opposite direction and suggests that we should adopt a ‘revenue accounting’ approach: the surest way to success in our restaurant (and the application of PSA to hotels, motels, conference centres, theatres etc, will lead to similar conclusions) is through the control of the revenue inflow, and this immediately points to the important determinants of total revenue, such as the number of covers, average spending power, sales mix, gross profit margins etc. Our restaurant is clearly market-oriented, by which we mean heavily dependent on its sales revenue. Quite obviously then it would make little sense to employ cost-oriented control methods in this situation. Thus the right road to success in our restaurant is through revenuebased control techniques such as volume forecasting, sales mix management, waiters’ sales analysis, frequent reviews of gross profit margins and menu prices. The control of variable costs, though undoubtedly important, will not - because of the relatively low PM values - yield nearly as great benefits in terms of net profit.

Theory of profit multipliers In this section we look at the micro-foundations of PM analysis, dealing with price adjustments, cost adjustments, cost changes leading to a price adjustment and volume changes respectively. But first we make two assumptions which are standard to profit volume analysis: l l

3. The assumption of linearity here is based on the generally accepted practice in industry: in all short- and medium-term profit planning and control procedures it is normally expected that direct costs will account for a constant proportion of output.

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Supply adjusts to demand at constant marginal cost per unit.3 Over the relevant range, there are no indivisibilities in the elements of variable cost and fixed costs remain fixed.

Price level PA4 We start from the basic business model, where net profit (rr) is by definition equal to the difference between revenue (R) and costs (C’), ie

R=R-C but R = PQ, where: P = price; Q = sales volume; and Management

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(1)

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PSA -

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I

I

Figure pliers

3.

Price

level

profit

multi-

-4

0

I

4

I

8 Pnce

level

12

I

16

I 20

PM

c = c, + c, where: C, = variable costs; C, = fixed costs; and

G=MQ where: M = marginal cost per unit. Performing the necessary substitutions in equation (1) gives 7~ =

(2)

PQ ---MQ -C’

Differentiating

equation (2) with respect to P yields

(3)

is the economist’s definition of elasticity of demand which may be drawn from empirical estimation of the demand curve or simply a subjective estimate based on ‘market feel’ as to how the sales volume is likely to respond to a given percentage change in price. Taking differentials of equation (3) gives dr=

(lie--Fe]

Qdp

and dividing through by 7~and then by dP/P produces the result

where the expression on the left-hand side of equation (4) is our definition of the price level profit multiplier (Ph$,). In Figure 3 we graph profit multipliers for differing values of e on the assumption that M is 40% of P.

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Using equation (4), some polar results follow immediately. If e = 0 then PM, = R/n, and since revenue is the largest element in the profit and loss account, it pays to raise price rather than cut costs. For Pl”rP to be equal to zero, requires that

jl+e--Fe)!=0 from which it follows that e must be 1

e=--

M

.

The denominator of equation (5) is known as the unit contribution margin : the lower the contribution margin, ie the higher the variable cost element, the higher must be the price sensitivity of market demand before PM, = 0. This is analogous to the Margin of Safety (MS) which is defined as

MS=;.l

(6) 1-F

From (6), the lower the contribution margin, the greater is the margin of safety between actual sales revenue and break-even sales revenue. Manipulating equation (5) produces

I-!!

ie

=

P

-r

e

-!

p-M=

e

(7) The left-hand side of equation (7) is the marginal revenue from selling an additional unit, and the right-hand side is, by definition, the marginal cost. This is the traditional profit maximizing rule of economic theory : thus PMl, = 0 at the point where profits are maximized. Finally if we put e = - 1 in equation (4) we have PM

P

z!!!.!! P

a

=- MQ

(8)

lr

When the absolute value of the demand elasticity is unity, revenue is constant, irrespective of the level at which price is set. However, from equation (8) we see that PM, is still positive for a price rise. This is due to the reduction in variable costs caused by the fall in sales volume on an increase in price. Cost level PA4 We commence from the absorbed by the business alterations. Thus, taking changes in unit variable International

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position that all variable cost changes are and are not passed on to customers as price differentials of equation (2) with respect to costs gives 181

PSA -

its nature, significance and applications V,

net

pr3flt

margin

25

2or

1

15-

10 -

5-

Figure

4.

Cost

level

profit

I

I

multi-

-8

I

-6

I

-4

pliers

Cost

-2 level

I 0

2

PM

dr = -- QdM

(9)

Dividing equation (9) through by 7~and then by dM/M produces the cost level profit multiplier (PM,) PM

c

=dn.!!_=--!!Q A

dM

(10)

71

In Figure 4 we graph this result for an initial value of M at 40% of P and dM positive. The only result of interest to note here is that if e = - 1 then PM, = -PM, as can be seen from equation (8). It also follows that although we have only shown the multiplier effects of changes in marginal costs, a similar result to equation ( 10) may be obtained for changes in fixed costs. Cost-price level PM Starting once more with the relationship defined by equation (2), we now assume a change in the marginal cost (dM) which is then passed on to the customer, to a larger or smaller extent, in the form of a price change (dP). Taking differential results in a somewhat more complicated expression d77 =

(11)

=

Carrying out the usual manipulation cost-price level multiplier (PMC,)

on equation (11) produces a

(12)

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PSA -

I Figure 5. multipliers

Cost-price

level

profit

I

-8

its nature, significance and applications

I

-6

-4 Cost

-2 -prce

level

I

I

0

2

J 4

PM

In Figure 5 we graph equation ( 12) for a cost increase on the assumption that price is marked-up one and a half times the cost change and for differing values of e, still retaining the initial position of M being 40% of P. Thus dP = 1.5 dM : if dP = dM then PMC, would be zero for e = 0 and negative for any value of Ie I >O. Alternatively, if e = 0, then PM

(13)

CP

according to whether i:

5 1. Ife=--

_L-L-M

which

is the

profit maximizing position for this business for a given P and M,then equation (12) collapses to PM

CP

=

-‘!iQ

(14)

lr

which is the cost level multiplier and is true for all values of dP/dM. Putting PMCp equal to zero in equation ( 12) and solving for dP/dM yields, 1 dP = --dM 1 +e-:e = ----

1

(15)

l+e

l-$ ( 1 Given that the unit contribution margin

will always be less

than one, then for I e I < 1 there will always exist a positive dP/ditl for which PMCp is zero. For Ie I > 1, the existence of a positive dP/dM for which PMCp > 0 depends on whether

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PSA -

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l+e

1-E >0 i p, 1 It may be easily seen that for realistic values of M/P, a high elasticity of demand implies that the correct policy is not to raise the price on a cost increase, but rather reduce it. In other words, the business is operating on the wrong part of its demand curve in the first place and there is potential for expansion by cutting prices. Sales volume PM Estimation of the sales volume profit multiplier (PM), follows classical lines by taking differentials of equation (2) with respect to Q. That is dn=

k(l+f)-M]

dQ

Dividing through by x and then by dQ/Q gives

(16) As with the price level profit multiplier, equation (16) equals zero at the point of profit maximization. When e = 0, it follows that (17) which is the unit contribution times the inverse of the net profit margin. Taking the ratio of equation (4) to equation (16) yields

EL PM,,

l+e---iCIe P-

1+‘-M e

P

=e

that is, PM, = e PM,,

(18)

Thus the price level profit multiplier is related to the sales volume profit multiplier through the elasticity of demand. This result shows, given PM, and PM,,, what the value of e must be for a price fall to be just worthwhile in terms of an increase in sales volume without damaging profitability. In this section we have considered the effects on profitability of alterations in price, unit costs, and sales volume together with combinations of these factors. We have shown that even if the elasticity of demand for the product is greater than one in absolute terms the profit multiplier from raising price may still be positive, depending on the unit contribution margin. The converse may also be true for a price fall. Equation (4) gives the exact result. For cost changes that are absorbed by the business, then Equation ( 10) shows that the cost level profit multiplier is equal to the ratio of the cost element to net profit. Where a price increase is contemplated following a cost increase, it has been shown that for products for which the demand is inelastic the price increase normally has to be greater than the cost increase in order to maintain profitability. If the 184

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PSA Table

Budgeted

4.

profit

its nature,

and

loss

and applications

account

Sales

Department

significance

Variable

Contribution

cost

f

f Rooms Food

and

Minor

aThe s&called ‘departmental controllable expenses’ (eg departmental wages and salaries, domestic supplies, cleaning, laundry, repairs and replacement of equipment etc) are, in fact, a mixed cost consisting of fully fixed and fullv variable elements. It is assume& for the present purpose, that of the total of such exoenses 80% is the fixed and 20% is the variable element. bThese include rent, insurance of buildings and contents, interest payable and depreciation.

beverage

operated

600

000

450

000

70 000

depts

1 120000

40 000 180

000

40 000 260

000

130

000

f 560 000 270

000

30 000 860

000

760

000

100

000

Less Undistributed

operating

Administration

and

expenses:

general

Marketing

40 000 45 000

Energy Property

operation

Departmental Other Net

profit

fixed

and

fixed

maintenance

costs’

chargesb

before

income

tax

65 000 330

000

150

000

demand for the product is elastic than the range of possibility by which a price increase may counter a cost increase to maintain profits is severely limited. Finally, we demonstrated that the link between the price level profit multiplier and the sales volume profit multiplier is through the elasticity of demand.

Applications The aim of this section is to describe several applications of PSA.4 Whereas some examples given point to possible applications, others are based on actual applications in the general field of tourism during 1980 and 1981. Use of PM” in profit planning Let us assume that a budgeted profit and loss account (shown in Table 4) has been submitted by the accountant of a hotel to the board of directors. The directors, having examined the budgeted profit and loss account, have decided that: l l

the hotel should produce a net profit of El 30 000; the increase of f30 000 in the net profit should be achieved through an appropriate adjustment in the price level.

We may now apply PSA in order to decide on the various alternative ways of achieving the additional net profit of 230 000. The first step is to calculate the relevant price level profit multipliers. For this particular purpose we assume that: l

l

4. More examples of applications PSA will be found in R. Kotas,

of

Accounting in the Hotel and Catering Industry (International Textbook Co, 1981) chapter 24.

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the elasticity of demand is nil (where elasticity is other than zero the procedure would be as indicated earlier in this paper); the undistributed operating expenses are a fully fixed cost.

The price level profit multipliers would be calculated as shown in Table 5. Note that for each key factor we assume a change in the price level of 10% and then trace the effect of the change to the total net profit of the hotel.

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PSA Table

its nature, signijicance 5.

Price

level

profit

and applications

multiplier

calculation

Sales

Total

Net

% A

Key factor

volume

cost

profit

profit

Room sales

1180000

1 020

000

160

000

60.0

6.0

F Et B sales

1 165000

1 020

000

145

000

45.0

4.5

MOD Total

1 127000 1 232 000

1 020 1 020

000 000

107

000

212

000

f

f

sales sales

f

in net

Profit multiplier

%

0.7

7.0

11.2

112.0

Some alternative methods of securing the additional net profit may now be enumerated as follows: l l l l

Raise all charges by 2.7%, as -2.7% X 11.2 (PM) increase in net profit. Raise room rates only by 5%, as -5.0% X 6.0 (PM) increase in net profit. Raise all F & B prices by 6.7%, as -6.7% x 4.5 (PM) increase in net profit. Raise all MOD charges by 42.9%, as -42.9% X 0.7 30% increase in net profit.

Similarly l

combinations

of above price adjustments

Raise F & B prices by 5.0% X 4.5 Raise MOD prices by 10.7% X 0.7 Leave room rates unchanged Total

l

230 000 in

increase

increase

= 30% (PM) =

eg

= 22.5% = 7.5% = -

(PM)

in net profit

in net profit

= 30%

are possible,

(PM)

30.0%

Raise room rates by 2.5% X 6.0 (Pfif) Raise F & B prices by 3.3% X 4.5 (JW) Leave MOD prices unchanged Total

= 130%

= 15.0% = 15.0% = 30.0%

Now that we have established a multiplicity of price level adjustments which will be instrumental in raising the net profit to El 30 000, it is for top management to decide which prices should, in the circumstances, be the most appropriate to increase. Readers will have noticed that relatively small changes in the price level of the hotel in our example have a powerful impact on total net profit. This illustrates the high degree of market orientations of hotels and indeed several other types of business enterprise in the general field of tourism.

5. For a more detailed consideration of the concept of market orientation see R. Kotas, Market Orientation in the Hotel and Caten’ng Industrv (Guildford, UK, Surrey University Press, 1975).

186

Use of PiUs in pricing tactics and revenue management Different types of business on the supply side of the tourist industry operate at different cost structures (different respective proportions of fixed and variable costs) and are, therefore, subject to the operation of different patterns of profit multipliers. This has, in turn, important implications for their approach to pricing tactics (special discounts, off-season rates, cover charges, minimum charges, reduced rates for children, etc). Similarly, different patterns of PM values point to different approaches to profit control, suggest appropriate pricing tactics, and give a great deal of insight into the way a business makes its profit. Let us consider an example involving two bars. The City Bar is, for this purpose, an ordinary bar. The Airport Bar, on the other hand,

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PSA Table

6.

Typical

weekly

its nature, si,oniJcance and applications

figures

of two

bars

City

Bar

Airport

f Sales

Bar

f

10 000

10 000

4 000

4 000

Less: Cost

variable

Fixed

costs

Total Net

of sales

Other

costs

cost profit

Table

7.

Derived

Price

level volume

Fixed

3 000

4 000

2 000

9 000

9 000

1 000

Sales Variable

1 000

PM

cost cost

1 000

10 000

10000

City

Airport

values Bar

10.0

10.0

5.0

3.0

5.0

7.0

4.0

2.0

Bar

is run by a firm of caterers, who pay the airport authority a concession rent based on the sales volume of the bar. The airport authority provides, in return, the necessary space, plant and equipment. In terms of the cost structure of the Airport Bar the effect of the arrangement is a high percentage of variable costs and a low percentage of fixed costs. The typical weekly figures of the two bars are shown in Table 6. From these figures we may calculate the PM values shown in Table 7. The two PM profiles throw some interesting light on the business orientation of the two bars. The City Bar seems to be market- rather than cost-oriented as its highest PM values relate to the revenue side of the business. The position of the Airport Bar is quite different. As some readers will know, catering contractors operating at airports negotiate periodical price increases with the relevant airport authority. From one week to another the price level must, therefore, be regarded as uncontrollable (though there is normally no reason why the caterer should not decrease his prices!). Thus the manager of the Airport Bar has little, if any, scope for price manipulation. His volume of sales is, largely, dependent on passenger flow. By definition his fixed costs are fixed and, in the short term, uncontrollable. He is only able to influence the level of his variable costs; and the high PM value here of 7.0 makes his operation cost-orientedpar excellence. Success in this business will only be achieved through strict (variable) cost control and the implementation of a wide repertoire of control techniques: standard recipes, yield tests, strict requisitioning of materials, use of optic *measures etc. The respective differences between the price level and sales volume profit price level and sales volume profit multipliers indicate what scope exists for the pricing tactics of the business. In the case of the City Bar the PM values for the price level and the sales volume are 10.0 and 5.0 respectively. Here a price level decrease of 10% requires an increase of at least 20% in the sales volume to make this exercise worthwhile. In other words, from result (18), unless the elasticity of demand is at least 2.0 a price decrease will not have the

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PSA -

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effect of increasing the net profit of the business. In the case of the Airport Bar, a price decrease of 10% will not be instrumental in raising total net profit unless the elasticity of demand is at least 3.3. Thus it seems that a price decrease designed to boost the sales volume and hence increase profits is more likely to be successful in the case of the City Bar. There are numerous examples of ‘product improvement’ programmes designed to increase the volume of sales. The product may be improved by (a) substitution of existing raw materials by more costly and, therefore, more acceptable materials (eg more expensive cuts of meat in restaurant meals, linen rather than nylon sheets in guest bedrooms); (b) providing larger quantities of existing materials (eg increasing the size of the steak from 802 to 1002); or(c) improving the packaging and/or presentation of the product. Of course, various combinations of (a), (b) and (c) are also possible. Almost all such product improvement measures have the effect of increasing the variable cost of the product concerned. In the case of the City Bar if we add 10% to the variable cost by way of product improvement we have to be sure that this will, at least, increase the sales volume by 10%. Any resulting increase in the sales volume over and above 10% will add to total net profit. The total variable cost of the Airport Bar is 27 000 and this includes the concession rent. If the concession rent is 10% of the sales volume then the variable cost profit multiplier relevant to any product improvement programme is 6.0. In this situation a 10% increase in variable costs would require a resulting sales volume increase of 16.6%. Our general conclusion is this: where there is a low variable cost PM value in relation to the sales volume PM value product improvement measures stand a better chance of being successful than in cases where the reverse is true.

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