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. t Volume^31 -^ Nurnger^4
I Summer,^1986
The E n g i n e e r i n g Economist 4
Equivalent Real Annual Costs. Evaluating
Investment Alternatives with Unequal Lives Under
S c o t t L. Lummer Texas A&M U n i v e r s i t y and M a n u f a c t u r e r s Hanover T r u s t Company A3smCI
options vich^ ??lis^ paper provides d i f f e r e n t economic^ a c r i t e r i o n Llves^ chati n an^ can be applied i n f l a c i o ~ r y^ environment.Co^ evaluate^ a u c u ~ l l yThis c r i t e r i o ne x c l w i v e -^ investment denoted as :he
"replacenenc^ equivalenc^ real annual chain'' approach.^ cosc^ procedure Uoreover.^ -^ i sthe^ aore equivalenc^ computaciolully real annual^ e f f i c i e n t c o s t mechod^ chacn i sche remarkably^ alcernacive s i n i l a r i n i c s calculations and i n iCs intuitive appeal Co che v e l l known aquivalenc annual cosc mechod. I. IKTRODUCTION T h i s p a p e r d e s c r i b e s a p r o c e d u r e t h a t c a n b e employed t o s e l e c t between m u t u a l l y e x c l u s i v e i n v e s t m e n t a l t e r n a t i v e s w i t h d i f f e r e n t economic l i v e s d u r i n g a
a -1 , p e r i o d o f r i s i n g p r i c e s. Firms o c c a s i o n a l l y h a v e t o d i s c r i m i n a t e between v a r i o u s
;a, a s s e t s t h a t c a n b e used t o produce a p a r t i c u l a r v e n d i b l e good. Moreover, companies
f r e q u e n t l y must d e c i d e w h e t h e r t h e y s h o u l d r e p l a c e e x i s t i n g a s s e t s , and must choose between v a r i o u s a l t e r n a t i v e r e p l a c e m e n t c a n d i d a t e s. T h e s e t y p e s o f d e c i s i o n s a r e j g e n e r a l l y r e f e r r e d t o a s r e p l a c e m e n t problems. A c o m p l i c a t i o n a r i s e s when t h e a s s e t s u n d e r c o n s i d e r a t i o n h a v e d i f f e r e n t economic l i v e s. I f r e p l a c e m e n t c o s t s a r e c o n s t a n t o v e r time, t h e a n a l y t i c a l p r o c e d u r e is n o t e x t r e m e l y cumbersome. Many t e x t b o o k s and s e v e r a l a r t i c l e s ( f o r example, s e e [ I ] , [ Z ] , 131, [ 4 ] , a n d [ 6 ] ) h a v e d e s c r i b e d two methods t h a t c a n b e u t i l i z e d t o s e l e c t t h e o p t i m a l a l t e r n a t i v e among a s s e t s w i t h non-uniform l i v e s : ( i ) t h e r e p l a c e m e n t c h a i n a p p r o a c h , and ( i i ) t h e e q u i v a l e n t a n n u a l c o s t (EAC) p r o c e d u r e. Given c e r t a i n a s s u m p t i o n s -- p a r t i c u l a r l y , t h a t no i n f l a t i o n e x i s t s -- t h e two methods a r e c o n c e p t u a l l y e q u i v a l e n t. However, t h e EAC a p p r o a c h is by f a r more c o m p u t a t i o n a l l y e f f i c i e n t. Many r e c e n t s t u d i e s , i n c l u d i n g Rappaport and T a g g a r t [ 9 ] , LeRoy and Fowler [ 7 ] , E z z e l l and K e l l y [ 5 ] , Mehta, Curley and Fung [ a ] , a n d Shashua and Goldschmidt [ l o ] h a v e e x p l o r e d t h e i m p l i c a t i o n s t h a t r i s i n g p r i c e s h a v e f o r c a p i t a l b u d g e t i n g methods. However, t h e e f f e c t o f i n f l a t i o n on r e p l a c e m e n t d e c i s i o n s p e r s e h a s l a r g e l y b e e n i g n o r e d. I f^ p r i c e s^ o f^ r e p l a c e m e n t^ a s s e t s^ a r e^ r i s i n g^ o v e r^ t i m e ,^ t h e E A c method is n o t c o n c e p t u a l l y v a l i d. T h i s^ p o i n t^ w i l l^ b e^ a d d r e s s e d^ more^ t h o r o u g h l y i n S e c t i o n Three a n d i n t h e Appendix. Textbooks t h a t d i s c u s s t h i s problem p o i n t
Ex.-PSC-Sirohi- Docket 5-CE- Witness: Udaivir Singh Sirohi February 10, 2012 Page 1 of 10
PSC REF#: Public Service
(^) Commission
(^) of Wisconsin
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EXHIBIT 3 5-CE- 3/5/2012 (aff)
318 The^ E n g i n e e r i n g^ Economist
o u t t h a t t h e EAC p r o c e d u r e c a n n o t b e employed, and s u g g e s t t h a t a n a n a l y s t r e v e r t t o t h e r e p l a c e m e n t c h a i n a p p r o a c h. However, t h e r e i s a n a l t e r n a t i v e t o t h e compli- c a t e d c a l c u l a t i o n s which r e p l a c e m e n t c h a i n s e n t a i l - t h i s a l t e r n a t i v e is t h e e q u i v a l e n t r e a l a n n u a l c o s t method. The n e x t s e c t i o n b r i e f l y r e v i e w s EAC and d i s c u s s e s why i t is a well-founded d e c i s i o n c r i t e r i o n. Given t h a t d i s c u s s i o n , s e c t i o n t h r e e p o s e s t h e problem of d i f f e r e n t - l i v e d a s s e t s e l e c t i o n u n d e r i n f l a t i o n , and d e v e l o p s t h e e q u i v a l e n t r e a l a n n u a l c o s t p r o c e d u r e. A f i n a l s e c t i o n summarizes t h e i m p l i c a t i o n s of t h e p a p e r.
- CONCEPT OF EQUIVALENT ANNUAL COST The u s e o f t h e r e p l a c e m e n t c h a i n approach i n v o l v e s comparing t h e c o n t i n u a l r e p l a c e m e n t c y c l e s o f two a s s e t s. The comparison is made o v e r t h e number o f y e a r s t h a t is e q u a l t o t h e l e a s t common m u l t i p l e o f t h e two a s s e t s ' economic l i v e s. The much less cumbersome a l t e r n a t i v e t o t h e r e p l a c e m e n t c h a i n p r o c e d u r e is t h e e q u i v a l e n t a n n u a l c o s t t e c h n i q u e. The EAC o f a n a s s e t is s i m p l y t h e r a t i o of i t s n e t p r e s e n t v a l u e t o a r e l e v a n t a n n u i t y f a c t o r. An a s s e t ' s n e t p r e s e n t v a l u e is t h e p r e s e n t v a l u e o f b e n e f i t s r e l a t e d t o t h e a s s e t , i n c l u d i n g i n v e s t m e n t t a x c r e d i t s , d e p r e c i a - t i o n s h i e l d s and s a l v a g e v a l u e , minus t h e p r e s e n t v a l u e o f a l l a s s e t s p e c i f i c c o s t s. S i n c e t h e r e v e n u e s produced by t h e a s s e t a r e t y p i c a l l y i g n o r e d , t h e NPV is u s u a l l y n e g a t i v e. M a t h e m a t i c a l l y , EAC i s d e f i n e d a s
EAC = -^ NPV
A f l J k
where k i s a n a p p r o p r i a t e d i s c o u n t r a t e , N is t h e economic l i f e o f t h e a s s e t , and AiiJ i is^ t h e^ p r e s e n t^ v a l u e^ a n n u i t y^ f a c t o r. When a n a n a l y s t is comparing s e v e r a l a s s e t a l t e r n a t i v e s , a n i m p l i c i t assumption is t y p i c a l l y made. Given a p a r t i c u l a r a s s e t is c h o s e n , i t is assumed t h a t t h e same t y p e o f a s s e t w i l l b e s e l e c t e d a s a s u p p l a n t m e n t a t t h e end o f i t s economic l i f e. I f t h e a s s u m p t i o n is e x t e n d e d a d - i n f i n i t u m ( o r more s p e c i f i c a l l y , i f t h e assumption i s e x t e n d e d t o t h e least common m u l t i p l e o f t h e a s s e t ' s economic l i v e s ) , t h e a n a l y s t is a c t u a l l y comparing v a r i o u s p e r p e t u a l r e p l a c e m e n t c h a i n s. (^) I f p r i c e s a r e e x p e c t e d t o remain c o n s t a n t , t h e e x p e c t e d n e t p r e s e n t v a l u e o f e a c h f u t u r e r e p l a c e m e n t c y c l e is e q u a l t o t h e NPV o f t h e f i r s t c y c l e. NPV* is d e f i n e d a s t h e n e t p r e s e n t v a l u e o f a l l a s s e t r e l a t e d c o s t s and b e n e f i t s f o r t h e f u t u r e i n f i n i t e c h a i n o f
r e p l a c e m e n t cyc-les. The o p t i m a l c h o i c e among s e v e r a l a s s e t s-.-- -. - - - .-is - - t h e- a l t e r n a t i v e
w i t h t h e h i g h e s t ( o r l e a s t n e g a t i v e ) NPV*. S i n c e , by d e f i n i t i o n , t h e v a l u e a t t h e
begin- v a l u e EACs - a S S m s e l e c d i f f e t h e E r e s p e
meani o n e 1 (NPV.
men t cons show used men t
an o c a l c R ~ P F The is 1
The c om! d i f
NPV
f lo.
Docket 5-CE- Witness: Udaivir Singh Sirohi February 10, 2012 Page 2 of 10
The E n g i n e e r i n g Economist (^) qolum
vhert
So l v
A n : i n f : t h e
f
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Docket 5-CE- Witness: Udaivir Singh Sirohi February 10, 2012 Page 4 of 10
(st - Volume 3 1 -Number 4
I D e f i n e a c a s h flow s t r e a m , X I , ..., X N , t h a t s a t i s f i e s t h e f o l l o w i n g c r i t e r i a : I i (^) ( 1 ) Each s u c c e s s i v e c a s h f l o w is l t i h i g h e r t h a n t h e p r e c e d i n g c a s h f l o w ;
j i. e. ,^ Xt^ =^ ( l k ) X t^ - ( 2 ) The v a l u e of t h e c a s h flow s t r e a m a t t i m e z e r o is e q u a l t o t h e n e t ! (^) p r e s e n t v a l u e of t h e asset's f i r s t c y c l e ; i. e. ,
A t
NPVI = It (^) ( 1 t k l t = l ,^ ..., N
. where k is t h e a p p r o p r i a t e nominal^ d i s c o u n t^ r a t e.^ Xu^ is^ d e f i n e d a s^ x l / ( l + i ).
: E q u a t i o n ( 2 ) c a n t h e n be r e w r i t t e n a s
1 S o l v i n g f o r X (^) 0 y i e l d s
? An^ i n t e r e s t i n g^ r e s u l t^ i s^ now^ o b t a i n e d.^ I f^ t h e^ d e f i n i t i o n o f^ X t^ is^ e x t e n d e d^ ad- : i n f i n i t u r n (XNtl = ( l t i l X N , ..., X (^) s = I l k l X s (^) - f o r a l l s ) , t h e f o l l o w i n g is t r u e f o r t h e f u t u r e c y c l e s ' N P V s :
Docket 5-CE- Witness: Udaivir Singh Sirohi February 10, 2012 Page 5 of 10
!volume 31 --Number - 4 "" 1 ;,?4C, and t h e n u s i n g E q u a t i o n (5) t o c a l c u l a t e t h e i n f i n i t e - l i f e J P V *. T h i s p r o c e d u r e i s s i g n i f i c a n t l y l e s s c o m p l i c a t e d t h a n t h e a l t e r n a t i v e r e p l a c e m e n t c h a i n 1 approach Some b a s i c i n t u i t i v e a p p e a l can b e g i v e n t o t h e ERAC by f u r t h e r e x a m i n a t i o n o f
Equation ( 3 ). The^ d e f i n i t i o n o f^ a^ r e a l^ i n t e r e s t^ r a t e^ r^ is^ g i v e n by^ r = ( l + k ) / ( l + i )^ - 1.
, S u b s t i t u t i n g t h i s d e f i n i t i o n i n E q u a t i o n ( 3 ) , t h e ERAC is c a l c u l a t e d by
.. g: ;where A is t h e p r e s e n t v a l u e a n n u i t y f a c t o r f o r r e a l r a t e r and economic l i f e N.
present^47 N J^ r
: Equation (5) t h e n s i m p l i f i e s t o
PPV* = - ERAC
r
(^4) : i! The^ i n i t u i t i v e a p p e a l^ o f^ t h e ERAC a p p r o a c h^ and^ i t s^ a n a l o g o u s r e l a t i o n s h i p^ t o
. ..^ ; t h e.^ EAC method^ a r e now^ c l e a r.^ The^ E,WC^ is^ o b t a i n e d^ by^ d i v i d i n g^ t h e^ f i r s t^ c y c l e ' s
-i I .PT/ by t h e a p p r o p r i a t e r e a l a n n u i t y f a c t o r. Comparisons between i n v e s t m e n t o p t i o n s 1 can b e made o n a n ERAC b a s i s i f t h e i n v e s t m e n t s ' a p p l i c a b l e r e a l d i s c o u n t r a t e s a g r e e. 5 ) h n u m e r i c a l example d e s c r i b e d i n t h e a p p e n d i x d e m o n s t r a t e s t h e s i m p l i c i t y w i t h which ZiiAC c a n b e a p p l i e d. I f t h e a p p r o p r i a t e r e a l d i s c o u n t r a t e s f o r t h e i n v e s t m e n t s a r e a n not t h e same, t h e r e s p e c t i v e ERAC's c a n be a d j u s t e d by d i v i d i n g by t h e a p p r o p r i a t e , r e a l d i s c o u n t r a t e s , a s s u g g e s t e d by E q u a t i o n ( 7 ). Of c o u r s e , i f t h e i n f l a t i o n r a t e f:' the is z e r o , t h e r e a l d i s c o u n t r a t e w i l l be e q u a l t o t h e nominal r a t e and t h e ER4C w i l l e i v e. be e q u a l t o t h e EAC.
:: and ; IV. CONCLUSION
dc-s! T h i s p a p e r h a s d e v e l o p e d a n " e q u i v a l e n t r e a l a n n u a l c o s t " ( E F? C ) p r o c e d u r e. g the! :This p r o c e d u r e , d e s c r i b e d by E q u a t i o n s ( 6 ) a n d ( 7 ) , c a n be used t o compare a s s e t : a l t e r n a t i v e s w i t h d i f f e r e n t economic l i v e s w h e t h e r o r n o t p r i c e s a r e e x p e c t e d t o i
Docket 5-CE- Witness: Udaivir Singh Sirohi February 10, 2012 Page 7 of 10
324 The^ E n g i n e e r i n g^ Economist
i n c r e a s e. The a p p e a l o f t h i s approach l i e s i n its c o m p u t a t i o n a l and i n t u i t i v e s i m p l i c i t y. The end^ r e s u l t^ o f^ t h i s^ s t u d y ' s^ a n a l y s i s^ shows^ t h a t^ t h e^ ERAC^ d i f f e r s from t h e t r a d i t i o n a l e q u i v a l e n t a n n u a l c o s t method o n l y i n t h e d i s c o u n t r a t e which i s used i n t h e c a l c u l a t i o n s. I t is hoped t h a t t h i s p r o c e d u r e w i l l s i m p l i f y t h e a n a l y s i s r e q u i r e d .by d e c i s i o n makers t o choose between m u t u a l l y e x c l u s i v e investment a l t e r n a t i v e s. APPENDIX The f o l l o w i n g n u m e r i c a l example w i l l b e used t o d e m o n s t r a t e t h e a p p l i c a t i o n of r e p l a c e m e n t c h a i n a n a l y s i s , t h e e q u i v a l e n t a n n u a l c o s t method, and t h e e q u i v a l e n t r e a l a n n u a l c o s t p r o c e d u r e. A f i r m is c o n s i d e r i n g t h e u s e o f o n e o f two m u t u a l l y e x c l u s i v e a s s e t s. Net p r e s e n t v a l u e s of a s s e t s A and B a r e -$1,00OSand -$1,800, r e s p e c t i v e l y. The two a s s e t s have economic l i v e s o f f o u r and n i n e y e a r s , r e s p e c t i v e l y. Net p r e s e n t v a l u e s are e x p e c t e d t o i n f l a t e a t a r a t e o f t e n p e r c e n t p e r annum, w h i l e t h e a p p r o p r i a t e a n n u a l nominal d i s c o u n t rate f o r b o t h a s s e t s is f i f t e e n p e r c e n t. E q u i v a l e n t a n n u a l c o s t s o f t h e a s s e t s a r e c a l c u l a t e d i n t h e f o l l o w i n g manner:
Thus, t h e EAC method judges a s s e t A t o b e t h e more e f f i c i e n t o f t h e two (by a f a c t o r of a p p r o x i m a t e l y 7%). The r e p l a c e m e n t c h a i n approach e n t a i l s c a l c u l a t i n g a l l o f t h e c y c l e n e t present v a l u e s f o r b o t h assets o v e r a t h i r t y - s i x y e a r t i m e h o r i z o n. The p r e s e n t v a l u e s of t h e two r e p l a c e m e n t c h a i n s a r e then o b t a i n e d. The asset w i t h t h e h i g h e r NPV of the c h a i n is t h e more e f f i c i e n t o f t h e two. T a b l e 1 d i s p l a y s t h e c o m p u t a t i o n s involved i n c a l c u l a t i n g t h e c h a i n WVs. The t o t a l NWs o f a o f a s s e t s A and B o v e r a t h i r t y - s i x y e a r p e r i o d a r e -$4900 and -$4357, r e s p e c t i v e l y , T h e r e f o r e , asset B is a c t u a l l y f a r more e f f i c i e n t t h a n a s s e t A (by a f a c t o r o f 1 2 2 ).
PI i
volu
cyc l
exi i?? 1 ye: A l l a c r it CO! I n 0 f
Docket 5-CE- Witness: Udaivir Singh Sirohi February 10, 2012 Page 8 of 10
The E n g i n e e r i n g Economist
Volt
The E f l C method l e a d s t o t h e same d e c i s i o n a s t h e r e p l a c e m e n t c h a i n a p p r o a c h. Morevoer, t h e p e r c e n t a g e o f t h e d i f f e r e n c e s between t h e a s s e t ' s e f f i c i e n c i e s a r e i d e n t i c a l f o r t h e two p r o c e d u r e s. However,^ t h e^ c o m p u t a t i o n s^ i n v o l v e d^ i n^ t h e^ ERAC^ 1 a p p r o a c h a r e much s i m p l e r. I ACKNOWLEDGEMENT I would l i k e t o thank K e i t h Brown, K e r r y Cooper, K l a u s F i s c h e r , J a y e S m i t h , and t h e two r e f e r e e s f o r t h i s j o u r n a l f o r t h e i r many h e l p f u l s u g g e s t i o n s.
REFERENCES (^) I
1. Bierman, H. and S. Smidt, The Capitul Budgeting Decision, New York, Macmillan,
2. B r e a l e y , R. and S. Myers, Principles of Corporate Finance, New York, McGraw-Hill,
3. Brigham, E., Financial Management: Theory and Practice, New York, Dryden,
- Emery, G. , "Some G u i d e l i n e s f o r E v a l u a t i n g C a p i t a l I n v e s t m e n t A l t e r n a t i v e s
w i t h Unequal L i v e s , " Financial Management 11 ( S p r i n g 1 9 8 2 ) , pp. 14-19.
- E z z e l l , J. and W. K e l l y , "An A P V A n a l y s i s o f C a p i t a l B u d g e t i n g Under I n f l a t i o n , "
Financial :.lcmagenent 1 3 (Autumn 1 9 8 4 ) , pp. 49-54.
- Lee, C. , Financial AnaZysis and Planning, Reading, ?lA, Addison-IJes l e y , 1985.
- LeRoy, R. and D. Fowler, " P r e s e n t Value A n a l y s i s , I n f l a t i o n , and C a p i t a l Recoucr:
F a c t o r s , " Engineering Economist 28 ( F a l l 1 9 8 2 ) , pp. 31-52.
- Mehta, D. , M. C u r l e y and H. Fung, " I n f l a t i o n , Cost o f C a p i t a l , and C a p i t a l
B u d g e t i n g P r o c e d u r e s , " Financial Management 1 3 ( W i n t e r 1 9 8 4 ) , pp. 48-54.
- R a p p a p o r t , A. and R. T a g g a r t , " E v a l u a t i o n o f C a p i t a l E x p e n d i t u r e P r o p o s a l s Under
I n f l a t i o n , " financial Idanagsnent 11 ( S p r i n g 1 9 8 2 ) , pp. 5-13.
- Shashua, L. and Y. Goldschmidt, "Costing D e p r e c i a b l e Asset's S e r v i c e s During
I n f l a t i o n , " Engineering Economist 30 (Winter 1 9 8 5 ) , pp. 157-171.
m
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