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Sensitivity of the rate of return
034EGA. The Int. II o f Ntgmt Sci,, Vol. 4, No. 3, 1976. Pcrgataoa Press. Printed in Great Britain
Sensitivity of the Rate of Return t h a n v does. In m o s t industrial operations 0 < a < 0 - 5 , so that the 'pecking order' (in terms o f the expected effect o n r) is first p, t h e n c a n d finally v, but for large profit m a r g i n s when a > 0 . 5 the order i s p , v, c, a n d w h e n a is negative the order becomes c, p, v. Note that for relatively small values o f p*, c* a n d v* e q u a t i o n (4a) m a y be replaced by the a p p r o x i m a t i o n
IN AN EARLIER note [1] G o l d ' s m o d e l for managerial control ratios [2, p23] was expressed as (1)
r = ( p - - c) e k = 7rcek
where r = rate o f return o n total i n v e s t m e n t = profit/total i n v e s t m e n t ; p ---- unit price for the o u t p u t ; c = unit cost for the o u t p u t ; e = o u t p u t / c a p a c i t y = capacity utilisation; k = capacity/total i n v e s t m e n t ; = = unit profit relative to cost = (p -
r* = v *
c)/c.
In order to e x a m i n e the relative effects o f c h a n g e s in unit price, unit cost a n d v o l u m e o n the rate o f return, it m a y be c o n v e n i e n t as an alternative to (1) to express r as r =
(? -
e)v
(2)
where v = e k = o u t p u t / t o t a l i n v e s t m e n t . If the unit price c h a n g e s f r o m p to p -k- 8p, the unit cost f r o m c to c + 8c a n d v to v + 8v the resultant c h a n g e in r c a n be s h o w n to be 8r = ( p - c) 8v + (v + By) (Sp - 8c). (3)
If the profit m a r g i n is d e n o t e d by a
=
( p -- c)/p the relative c h a n g e in r d e n o t e d as r * ( = 8r/r) m a y be expressed as r* = v* + ! +.V* [ p . _ c . (l _ a)]
(4a)
ct
which for s o m e purposes conveniently f o r m u l a t e d as 1 +
r*
1 +v*
-- 1 +
p*
m a y be m o r e 1 -- -
c*.
(4b)
a
Alternatively, these expressions m a y be written in terms o f ~r (by s u b s t i t u t i n g a = ,~/(. + 1)). E q u a t i o n (4) highlights the relative effects o f c h a n g e s i n p , c a n d v. Since a < 1 it follows that a n y given c h a n g e in p has a greater effect on r t h a n when a c h a n g e o f the s a m e m a g n i tude occurs in c or in v, w h e n a > 0 ; also, if a < 0"5 then a given change in c affects r m o r e
+ p*
1-ac.
(5)
a n d each term in this expression s h o w s the relative effect on r by one p a r a m e t e r w h e n the others are zero (for example, r* = v * for p * = c * = 0, etc.), thereby providing a useful guide as to the expected results for given possible c h a n g e s in the system a n d for a given value a. E q u a t i o n (4) c a n also be used to c o m p u t e the effect o n r o f several s i m u l t a n e o u s c h a n g e s in the parameters p, c a n d v. A s s u m e that each m a y either change by & A ~ (upwards or downwards) or remain u n c h a n g e d ; the n u m b e r o f c o m b i n a t i o n s o f sets of changes in the three parameters is 27, s h o w n in Table 1, where the cases are ranked in terms of the effect on r* (the first 12 cases s h o w i n g all the possible c o m b i n a t i o n s for increasing r a n d the last 12 cases involving a decline). T h e Table gives the resultant value of r* for two possible values of A (I ~o a n d 5 ~ ) a n d for a = 0"1 a n d 0-2 respectively. F o r example, for a = 0-2 w h e n p * = I ~ , c* = 0 a n d v* = - - i ~ (case 8) the rate of return improves by 4 ~ (all the results are r o u n d e d off to the nearest integer). Table I reveals interesting, a n d in s o m e cases intriguing, results. First, we find that while certain cases are symmetrical (not only when a single p a r a m e t e r is subject to c h a n g e a n d where s y m m e t r y is to be expected, but s o m e t i m e s w h e n two parameters c h a n g e simultaneously, as in cases 2 a n d 26), in the majority o f cases there is no s y m m e t r y a n d increases in p a r a m e t e r values tend to have a greater effect t h a n decreases. Secondly, in s o m e cases the value of a has a significant
350
Omega, Vol. 4, No. 3 T ~
1. R~LAmVF CS.~'qGF (in ~.) IN THE gAT~ OF ~ T t r ~ ' ; (r*) FOR GIVEN C ~ ; G ~ p * , AND t~*
a = 0-i Case
p*
c*
v*
~x=
a = 0"2
1~
5~
1%
5 52 45 38 31 26 25 20 19 14 10 5 5 0 0 0 -5 --5 --10 --16 --20 --21 --24 --25 --29 --42 --45 --48
1
+
--
+
2 3 4 5 6 7
q+ + 0 ÷ 0
--0 -0 --
0 -+ + 0 0
20 19 18 11 10 10 9
105 95 85 58 52 50 45
10 9 8 6 5 5 4
8
+
0
--
9
42
4
9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27
0 + 0 q0 q0 -. 0 0 -0 ------
-+ 0 + 0 + 0 --
-+ q0 0 q0
8 2 1 1 0 0 0 -1 -- 1 2 --8 --9 --9 --10 --10 --11 -- 18 --19 --20
38 10 5 5 0 0 0 -5 --5 --10 --42 --45 --48 --48 --50 --52 --95 --95 --95
3 2 1 1 0 0 0 -1 -- 1 --2 --3 --4 --4 --5 --5 --6 --8 --9 --10
.
. -k qO+ 0 0 q"b -b
. + 0 + -0 --~ 0 --
c*
The change in p*, c* or v* is s h o w n in the second, third and fourth columns as follows: + denotes an increase by A %; - - denotes a decrease by A ~ ; 0 denotes no change. effect on the results, in others none (as in cases 11, 12, 17 and others). But p e r h a p s the most noteworthy result is that the effect of combined changes in parameter values can be extremely large c o m p a r e d with w h a t a p p e a r to be small changes in the parameters themselves. F o r example, in case 1 for a = 0"1 when p and o increase by 5 ~ and c decreases by the same a m o u n t , the combined effect on the rate of r e t u r n is to increase it by 1 0 5 ~ . I n 18 cases of the 27 listed in the Table the changes in r are relatively high, thereby indicating h o w sensitive r is, particularly w h e n a is small. This result is of some significance b o t h for prediction and for 351
planning purposes, in that it suggests that when forecasts for changes in p, c and v are given in the form of ranges rather than by single values, the resultant change in r is likely to be very variable. This was corroborated in a case study using risk simulation for the analysis of plant performance, where it was found that while the distribution of unit cost was fairly compact for a given set of forecast changes, the corresponding distrib u t i o n for the return on investment was very wide [2, pp. 141-2]. It should be noted that when it is desirable to decompose the parameter v into several ratios, equation (4) can easily be expanded
Memoranda 2. EILON S, GOLD B and SO~A,,," J (1976)
by substituting the appropriate value of v*. For example, i f v = ek as in equation (1), it can be shown (following the method in [1]) that
Applied Productivity Analysis for Industry. Pergamon Press, Oxford. Samuel Eiion
(February 1976)
v* = e * + k* + e ' k * and similarly when v is expressed as the product of three ratios (as irt Gold's model in [2]) v* can be easily found prior to the use of equation (4).
Department of Management Science Imperial College of Science and Technology Exhibition Road London S W 7 2BX UK
REFERENCES 1. EILON S (1975) Changes in profitability components. Omega 3 (3), 353-4.
352
Sensitivity of the Rate of Return t h a n v does. In m o s t industrial operations 0 < a < 0 - 5 , so that the 'pecking order' (in terms o f the expected effect o n r) is first p, t h e n c a n d finally v, but for large profit m a r g i n s when a > 0 . 5 the order i s p , v, c, a n d w h e n a is negative the order becomes c, p, v. Note that for relatively small values o f p*, c* a n d v* e q u a t i o n (4a) m a y be replaced by the a p p r o x i m a t i o n
IN AN EARLIER note [1] G o l d ' s m o d e l for managerial control ratios [2, p23] was expressed as (1)
r = ( p - - c) e k = 7rcek
where r = rate o f return o n total i n v e s t m e n t = profit/total i n v e s t m e n t ; p ---- unit price for the o u t p u t ; c = unit cost for the o u t p u t ; e = o u t p u t / c a p a c i t y = capacity utilisation; k = capacity/total i n v e s t m e n t ; = = unit profit relative to cost = (p -
r* = v *
c)/c.
In order to e x a m i n e the relative effects o f c h a n g e s in unit price, unit cost a n d v o l u m e o n the rate o f return, it m a y be c o n v e n i e n t as an alternative to (1) to express r as r =
(? -
e)v
(2)
where v = e k = o u t p u t / t o t a l i n v e s t m e n t . If the unit price c h a n g e s f r o m p to p -k- 8p, the unit cost f r o m c to c + 8c a n d v to v + 8v the resultant c h a n g e in r c a n be s h o w n to be 8r = ( p - c) 8v + (v + By) (Sp - 8c). (3)
If the profit m a r g i n is d e n o t e d by a
=
( p -- c)/p the relative c h a n g e in r d e n o t e d as r * ( = 8r/r) m a y be expressed as r* = v* + ! +.V* [ p . _ c . (l _ a)]
(4a)
ct
which for s o m e purposes conveniently f o r m u l a t e d as 1 +
r*
1 +v*
-- 1 +
p*
m a y be m o r e 1 -- -
c*.
(4b)
a
Alternatively, these expressions m a y be written in terms o f ~r (by s u b s t i t u t i n g a = ,~/(. + 1)). E q u a t i o n (4) highlights the relative effects o f c h a n g e s i n p , c a n d v. Since a < 1 it follows that a n y given c h a n g e in p has a greater effect on r t h a n when a c h a n g e o f the s a m e m a g n i tude occurs in c or in v, w h e n a > 0 ; also, if a < 0"5 then a given change in c affects r m o r e
+ p*
1-ac.
(5)
a n d each term in this expression s h o w s the relative effect on r by one p a r a m e t e r w h e n the others are zero (for example, r* = v * for p * = c * = 0, etc.), thereby providing a useful guide as to the expected results for given possible c h a n g e s in the system a n d for a given value a. E q u a t i o n (4) c a n also be used to c o m p u t e the effect o n r o f several s i m u l t a n e o u s c h a n g e s in the parameters p, c a n d v. A s s u m e that each m a y either change by & A ~ (upwards or downwards) or remain u n c h a n g e d ; the n u m b e r o f c o m b i n a t i o n s o f sets of changes in the three parameters is 27, s h o w n in Table 1, where the cases are ranked in terms of the effect on r* (the first 12 cases s h o w i n g all the possible c o m b i n a t i o n s for increasing r a n d the last 12 cases involving a decline). T h e Table gives the resultant value of r* for two possible values of A (I ~o a n d 5 ~ ) a n d for a = 0"1 a n d 0-2 respectively. F o r example, for a = 0-2 w h e n p * = I ~ , c* = 0 a n d v* = - - i ~ (case 8) the rate of return improves by 4 ~ (all the results are r o u n d e d off to the nearest integer). Table I reveals interesting, a n d in s o m e cases intriguing, results. First, we find that while certain cases are symmetrical (not only when a single p a r a m e t e r is subject to c h a n g e a n d where s y m m e t r y is to be expected, but s o m e t i m e s w h e n two parameters c h a n g e simultaneously, as in cases 2 a n d 26), in the majority o f cases there is no s y m m e t r y a n d increases in p a r a m e t e r values tend to have a greater effect t h a n decreases. Secondly, in s o m e cases the value of a has a significant
350
Omega, Vol. 4, No. 3 T ~
1. R~LAmVF CS.~'qGF (in ~.) IN THE gAT~ OF ~ T t r ~ ' ; (r*) FOR GIVEN C ~ ; G ~ p * , AND t~*
a = 0-i Case
p*
c*
v*
~x=
a = 0"2
1~
5~
1%
5 52 45 38 31 26 25 20 19 14 10 5 5 0 0 0 -5 --5 --10 --16 --20 --21 --24 --25 --29 --42 --45 --48
1
+
--
+
2 3 4 5 6 7
q+ + 0 ÷ 0
--0 -0 --
0 -+ + 0 0
20 19 18 11 10 10 9
105 95 85 58 52 50 45
10 9 8 6 5 5 4
8
+
0
--
9
42
4
9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27
0 + 0 q0 q0 -. 0 0 -0 ------
-+ 0 + 0 + 0 --
-+ q0 0 q0
8 2 1 1 0 0 0 -1 -- 1 2 --8 --9 --9 --10 --10 --11 -- 18 --19 --20
38 10 5 5 0 0 0 -5 --5 --10 --42 --45 --48 --48 --50 --52 --95 --95 --95
3 2 1 1 0 0 0 -1 -- 1 --2 --3 --4 --4 --5 --5 --6 --8 --9 --10
.
. -k qO+ 0 0 q"b -b
. + 0 + -0 --~ 0 --
c*
The change in p*, c* or v* is s h o w n in the second, third and fourth columns as follows: + denotes an increase by A %; - - denotes a decrease by A ~ ; 0 denotes no change. effect on the results, in others none (as in cases 11, 12, 17 and others). But p e r h a p s the most noteworthy result is that the effect of combined changes in parameter values can be extremely large c o m p a r e d with w h a t a p p e a r to be small changes in the parameters themselves. F o r example, in case 1 for a = 0"1 when p and o increase by 5 ~ and c decreases by the same a m o u n t , the combined effect on the rate of r e t u r n is to increase it by 1 0 5 ~ . I n 18 cases of the 27 listed in the Table the changes in r are relatively high, thereby indicating h o w sensitive r is, particularly w h e n a is small. This result is of some significance b o t h for prediction and for 351
planning purposes, in that it suggests that when forecasts for changes in p, c and v are given in the form of ranges rather than by single values, the resultant change in r is likely to be very variable. This was corroborated in a case study using risk simulation for the analysis of plant performance, where it was found that while the distribution of unit cost was fairly compact for a given set of forecast changes, the corresponding distrib u t i o n for the return on investment was very wide [2, pp. 141-2]. It should be noted that when it is desirable to decompose the parameter v into several ratios, equation (4) can easily be expanded
Memoranda 2. EILON S, GOLD B and SO~A,,," J (1976)
by substituting the appropriate value of v*. For example, i f v = ek as in equation (1), it can be shown (following the method in [1]) that
Applied Productivity Analysis for Industry. Pergamon Press, Oxford. Samuel Eiion
(February 1976)
v* = e * + k* + e ' k * and similarly when v is expressed as the product of three ratios (as irt Gold's model in [2]) v* can be easily found prior to the use of equation (4).
Department of Management Science Imperial College of Science and Technology Exhibition Road London S W 7 2BX UK
REFERENCES 1. EILON S (1975) Changes in profitability components. Omega 3 (3), 353-4.
352