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Financial optimization of mining plant size McIntosh, Douglas Grant 1970

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FINANCIAL OPTIMIZATION OF MINING PLANT SIZE by DOUGLAS GRANT McINTOSH E.M. Colorado School of Mines, 1966 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF BUSINESS ADMINISTRATION i n the Faculty of Commerce and Business Administration We accept t h i s as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA May, 1970 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h C olumbia, I ag r e e t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and Study. I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y purposes may be g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s thes,is f o r f i n a n c i a l g a i n s h a l l npt be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department o f {?QA*/I/}£'&C£'£ Bt/S/A/^zSS pArf/A//$ . The U n i v e r s i t y o f B r i t i s h Columbia Vancouver 8, Canada Date ////£ /7Q The hypothesis of t h i s study i s that the optimal p l a n t s i z e f o r a mining venture i s dependent upon both u n c o n t r o l l a b l e and c o n t r o l l a b l e .variables Examples of v a r i a b l e s which are u n c o n t r o l l a b l e to the f i r m are character-i s t i c s of the orebody, c a p i t a l and product markets, p r o j e c t e d p r i c e l e v e l s , and tax s t r u c t u r e . C o n t r o l l a b l e v a r i a b l e s i n determining p l a n t s i z e are r a t e of recovery, l e v e l of recovery, c a p i t a l c o s t s , and op e r a t i n g c o s t s . The primary purpose of t h i s study i s to d e l i n e a t e the e f f e c t s of these v a r i a b l e s , both s i n g l y and j o i n t l y , upon p l a n t s i z e and to present a model which w i l l i n t e r r e l a t e the v a r i a b l e s w i t h t h a t s c a l e of p l a n t which w i l l maximize the value of the f i r m . A secondary purpose of the study i s to compare the impact of both Canadian and United States tax laws upon the p r o f i t a b i l i t y of a given o p e r a t i o n , and upon the optimal grade-c a p a c i t y combination t o be employed f o r a given orebody. The d i f f e r e n c e of the impact of Canadian and United States tax laws w i t h respect to the conservation of resources i s a l s o considered. A f t e r each of the c o n t r o l l a b l e and u n c o n t r o l l a b l e v a r i a b l e s i s defined and analysed, a d e t a i l e d a n a l y s i s i s made of va r i o u s methods of mine v a l u a t i o n , w i t h the o b j e c t i v e being to i d e n t i f y the v a l u a t i o n method which most c l o s e l y r e l a t e s to the value of the f i r m . Then a model i s constructed which w i l l give the m i n e - l i f e annual cash flows f o r a given orebody under v a r i o u s c o n c e n t r a t o r - c a p a c i t y - - c u t - o f f - g r a d e com-b i n a t i o n s . These cash flows are then converted to i n t e r n a l r a t e s of r e t u r n and b e n e f i t - t o - c o s t r a t i o s , which are contoured f o r va r i o u s c u t - o f f grades and concentrator c a p a c i t i e s under various metal p r i c e s . The model assumes an orebody w i t h a tonnage of kOe^ ^ ' m i l l i o n tons, •where x i s the c u t - o f f grade, i n percent copper. Contour p l o t s of b e n e f i t - t o - c o s t r a t i o and i n t e r n a l r a t e of r e t u r n were constructed f o r c u t - o f f grades ranging from Of0 t o 1% copper, and f o r concentrator c a p a c i t i e s ranging from 5000 tons-per-day t o 50,000 tons-per-day, at net smelter r e t u r n s of kOfi per l b . t o per l b . of contained copper. The t ax system under which the highest p r o f i t s are a t t a i n e d i s Canadian tax laws w i t h p r e - 1 9 6 8 B r i t i s h Columbia ta x e s . The ope r a t i o n i s l e a s t p r o f i t a b l e under United States tax laws. However, optimal p l a n t s i z e i s l e a s t s e n s i t i v e to changes i n metal p r i c e under United States t a x laws, and most s e n s i t i v e t o p r i c e changes under Canadian'tax laws w i t h p r e - 1 9 6 8 B r i t i s h Columbia taxes. S i m i l a r l y , optimal c u t - o f f grade i s most i n s e n s i t i v e to changes i n metal p r i c e under Canadian ta x laws w i t h p r e - 1 9 6 8 B r i t i s h Columbia taxes, and most s e n s i t i v e to product p r i c e changes under United States t a x laws. Therefore, i t can be shown t h a t under United States tax laws, the response of an ope r a t i o n to changing product p r i c e s would be to change the optimal c u t - o f f grade, while i n Canada, p a r t i c u l a r l y under p r e - 1 9 6 8 B r i t i s h Columbia taxes, the response would be t o change the optimal operating c a p a c i t y . Therefore, American tax laws provide the greatest f l e x i b i l i t y f o r responding to changes i n product p r i c e . The t a x system which produces the greatest degree of conservation of resources, as i s r e f l e c t e d by completeness of u l t i m a t e e x t r a c t i o n , i s the American case. The lowest e x t r a c t i o n l e v e l would r e s u l t under Canadian taxes, w i t h p r e - 1 9 6 8 B r i t i s h Columbia/taxes. ( i i i ) TABLE OF CONTENT'S I. THE PROBLEM . . 1 Purpose of the Study 1 Scope of the Study 3 Economic Aspects of the M i n e r a l Industry k I I . EXOGENOUS VARIABLES 14 The Orebody 14 Tonnage and Grade 14 P o s i t i o n of the Orebody 19 Rock Character 20 L o c a t i o n 21 Markets 21 The Product Market 22 C a p i t a l Markets 22 Tax S t r u c t u r e 26 Canadian Tax Law '• 27 Ss Rates 27 P r o v i n c i a l . M i n i n g Taxes 27 C a p i t a l Cost Allowance (Depreciation) 27 De p l e t i o n Allowance 28 •36-Month Exemption f o r Mines 28 E x p l o r a t i o n and Development Expenses 28 United States Tax Law -28 Rates -29 State Mining Taxes . 29 E x p l o r a t i o n Expenditures 2 9 Development Expenditures 2 9 D e p l e t i o n 3 0 A comparison of Canadian & United States Tax Laws 3 0 P r i c e L e v e l 3 1 I I I . CONTROLLABLE VARIABLES 3 4 Rate of Pro d u c t i o n 3 4 L e v e l of Production 3 5 Costs 4 5 C a p i t a l Costs 4 7 ' IV. VALUATION TECHNIQUES • 52 Payback . . . . . 5 3 Accounting Rate of Return 54 Break-Even A n a l y s i s 5 5 D i s c o u n t i n g Methods 56 Discounted Cash Elow 5 9 Net Present Value 6 0 The Hoskold Formula 6 3 The M o r k i l l Formula 6 8 V. A HYPOTHETICAL EXAMPLE . . 6 9 D e s c r i p t i o n of the Model 6 9 Assumptions 7 0 Tonnage and Grade 7 0 Product and C a p i t a l Markets 7 0 P r i c e L e v e l 72 Tax S t r u c t u r e 7 3 United States Tax Laws 7 3 Canadian Tax Laws 7 4 C a p i t a l Costs 7 4 Preproduction Costs . . . 7 6 Operating Costs . „ . 7 6 L e v e l of Recovery 7 7 Rate of Recovery 78 Computer Flowcharts . 78 R e s u l t s 7 9 Optimal C u t - o f f s Grades at S p e c i f i e d P l a n t C a p a c i t i e s 7 9 O p t i n a l P l a n t Size 1 1 5 Maximum Rate of Return 1 1 5 Maximum B e n e f i t - t o - C o s t R a t i o 1 1 7 V I SUMMARY AND CONCLUSIONS . 1 2 1 Summary 1 2 1 Conclusions 1 2 4 Comparison of Tax Systems 1 2 4 F i n a n c i a l O p t i m i z a t i o n of Mining Ventures . . . 1 2 7 I n t e r n a l Rate of Return 1 2 8 B e n e f i t - t o - C o s t R a t i o ( 9 $ Compounding) . . . . . 1 3 2 Canadian Taxes - - 1 9 6 8 B r i t i s h Columbia Taxes . 1 3 5 Canadian Taxes P r e - 1 9 6 8 B r i t i s h Columbia Taxes . 1 4 6 United States Taxes 1 5 4 BIBLIOGRAPHY . . 1 6 2 TABLE PAGE 1 . Reserve-Resource Terminology I 7 5 - 1 . Optimal Cut-of f Grades at S p e c i f i e d P l a n t C a p a c i t i e s : I n t e r n a l Rate of Return: Canadian Tax Laws - -1 9 6 8 B r i t i s h Columbia Taxes 1 0 5 5 - 2 . Optimal Cut-of f Grades at S p e c i f i e d P l a n t C a p a c i t i e s : I n t e r n a l Rate o f Return: Canadian Tax Laws --P r e - 1 9 6 8 B r i t i s h Columbia Taxes 108' 5 - 3 . Optimal C u t - o f f Grades at S p e c i f i e d P l a n t C a p a c i t i e s : I n t e r n a l Rate of Return: United States Tax Laws . . . 109. 5-k. Optimal C u t - o f f Grades at S p e c i f i e d P l a n t C a p a c i e i t s : B e n e f i t - t o - C o s t R a t i o : Canadian Tax Laws --1 9 6 8 B r i t i s h Columbia Taxes I l l 5 - 5 . Optimal c u t - o f f Grades at S p e c i f i e d P l a n t C a p a c i t i e s : B e n e f i t - t o - C o s t R a t i o : Canadian Tax Laws --P r e - 1 9 6 8 B r i t i s h Columbia Taxes 1 1 3 5 - 6 . Optimal Cut-of f Grades at S p e c i f i e d P l a n t C a p a c i t i e s : B e n e f i t - t o - C o s t R a t i o : United States Tax Laws . . . . Ilk' 5 - 7 . Summary of Re s u l t s 1 1 8 ( v i i ) LIST OF FIGURES 1 . Cost vs Rate of Recovery vs. L e v e l of Recovery 7 2 . Percentage Increase i n Mining Costs 3 3 • 3 . Contour P l o t of I.R.R. vs Cut-Off Grade vs. Concentrator Capacity: kOft Cu.: Canadian Tax Laws - - P r e - 1 9 6 8 B.C.Taxes.. 8 l k. Contour P l o t of I.R.R. vs.. C u t - o f f Grade vs. Concentrator Capacity: k2^ Cu.: Canadian Tax Laws -- P r e - 1 9 6 8 B.C.Taxes .. 8 2 5 . Contour P l o t of I.R.R. vs. C u t - o f f Grade vs. Concentrator Capacity: hkft Cu.: Canadian Tax Laws -- P r e - 1 9 6 8 B.C.Taxes .. 8 3 6 . Contour P l a t of I.R.R. v s . C u t - o f f Grade vs. Concentrator Capacity: h6<£ Cu.: Canadian Tax Laws -- P r e - 1 9 6 8 B.C.Taxes.. 84.-7 . Contour P l o t of I.R.R. vs Cut-of f Grade v s . Concentrator Capacity: kOfi Cu.: Canadian Tax Laws -- 1 9 6 8 B.C. Taxes . . . ' 8 . Contour P l o t of I.R.R. vs. C u t - o f f Grade v s . Concentrator Capacity: k2<£ Cu.: Canadian Tax Laws -- 1 9 6 8 B.C.Taxes . . . 86/ ' 9 . Contour P l o t of I.R.R. vs C u t - o f f Grade v s . Concentrator Capacity: kktfi Cu.: Canadian Tax Laws -- 1 9 6 8 B.C.Taxes ... 87 1 0 . Contour P l o t of I.R.R. vs. C u t - o f f Grade v s . Concentrator Capacity: kty Cu.: Canadian Tax Laws -- 1 9 6 8 B.C.Taxes . . . 8.©Y 1 1 . Contour P l o t of I.R.R. vs. C u t - o f f Grade v s . Concentrator Capacity: kOfi Cu.: U.S.Tax Laws $ 9 ' ' 1 2 . Contour P l o t of I.R.R. vs Cut-of f Grade vs.Concentrator Ca-p a c i t y : k2^ Cu. U.S. Tax Laws 9 0 . 1 3 . Contour P l o t of I.R.R. vs. Cut-Off Grade vs. Concentrator Capacity: kkft Cu.: U.S.Tax Laws 9 1 ' 14. Contour P l o t of I.R.R. v s . Cut-off Grade vs. Concentrator Capacity: 460 Cu.: U.S.Tax Laws 92 15. Contour P l o t of B e n e f i t - t o - C o s t R a t i o v s . Cu t - o f f Grade vs. Concentrator Capacity: 400 Cu.: Canadian Tax Laws --Pre 1968 B.C.Taxes 93 16. Contour P l o t of B e n e f i t - t o - C o s t R a t i o vs. Cu t - o f f Grade vs. Concentrator Capacity: 420 Cu.: Canadian Tax Laws --Pre-1968 B.C.Taxes 94 17. Contour P l o t of B e n e f i t - t o - C o s t R a t i o vs. Cut-off Grade vs. Concentrator Capacity: 440 Cu.: Canadian Tax Laws --Pre-1968 B.C.Taxes $<?• 18. Contour P l o t of B e n e f i t - t o - C o s t R a t i o vs. Cut-off Grade v s . Concentrator Capacity: 460 Cu.: Canadian Tax Laws --Pre-1968 B.C.Taxes 9'6 19. Contour P l o t of B e n e f i t - t o - C o s t R a t i o vs. Cu t - o f f Grade v s . Concentrator Capacity: 400 Cu.: Canadian Tax Laws --1968 B.C.Taxes 97 20. Contour P l o t of B e n e f i t - t o - C o s t R a t i o v s . Cut-off Grade v s . Concentrator Capacity: 420 Cu.: Canadian Tax Laws --1968 B.C.Taxes . ' 98) 21. Contour P l o t of B e n e f i t - t o - C o s t R a t i o v s . Cu t - o f f Grade v s . Concentrator Capacity: 440 Cu.: Canadian Tax Laws --1968 B.C.Taxes J99' 22. Contour P l o t of B e n e f i t - t o - C o s t R a t i o vs. Cu t - o f f Grade v s . Concentrator Capacity: 460 Cu/: Canadian Tax Laws --1968 B.C.Taxes 1 0 © 23. Contour P l o t of B e n e f i t - t o - C o s t R a t i o v s . Cut-off Grade v s . Concentrator Capacity: 400 Cu.: U.S.Tax Laws 1032 24 Contour P l o t of B e n e f i t - t o - C o s t R a t i o vs. Cut-off Grade v s . Concentrator Capacity: 420 Cu.: U.S.Tax Laws 1 0 § 25. Contour P l o t of B e n e f i t - t o - C o s t R a t i o v s . Cut-off Grade vs. Concentrator Capacity: 440 Cu.: U.S.Tax Laws IQ3". 26. Contour P l o t of B e n e f i t - t o - C o s t R a t i o v s . Cut-off Grade vs. Concentrator Capacity: 460 Cu.: U.S.Tax Laws 1 0 4 S -FLOWSHEET PAGE 1. I n t e r n a l Rate of Return 129 2. Benef i t - t o - C o s t R a t i o (9</0 Compounding) 133 3. Cash Flow w i t h Canadian Taxes - - 1968 B.C.Taxes . . . . 136 k. Cash Flow w i t h Canadian Taxes - - Pre-1968 B.C. Taxes . . lk& 5. Cash Flow w i t h United States Taxes 156 CHAPTER I THE PROBLEM I . PURPOSE OF THE ' STUDY I t has l o n g been known from an i n t u i t i v e p o i n t of view t h a t there must be an optimal concentrator s i z e and mining r a t e f o r any given orebody. However, i n s p i t e of t h i s f e e l i n g , mine operators have been l o a t h t o undertake general s t u d i e s t o determine what s c a l e of pl a n t i s best f o r s p e c i f i c orebodies. This v o i d i n research has been j u s t i f i e d on many grounds. In the f i r s t p l a c e , i t i s d i f f i c u l t to t a i l o r a p l a n t to an orebody i f the extent, grade, and p h y s i c a l f e a t u r e s of an orebody are not known w i t h a high degree of c e r t a i n t y . Secondly, p l a n t s i z e i s somewhat determined by the technique used i n c a l c u l a t i n g the present value of a p r o j e c t , because there i s u n c e r t a i n t y as to the v a l i d i t y of techniques of determining discounted cash flows and a l a c k of knowledge as t o what the r a t e of discount should be. A t h i r d major drawback t o determining optimal p l a n t s i z e i s the myriad of p o s s i b l e p l a n t s i z e s . f o r which operating and c a p i t a l costs must be determined. A f o u r t h d i f f i c u l t y w i t h such a study i s the d i f f i c u l t y of f o r e c a s t i n g the r e l a t i o n s h i p between costs and product p r i c e s . A f i f t h j u s t i f i c a t i o n f o r not determining an optimal p l a n t s i z e i s the f a c t that the optimum f o r many of the l a r g e , low-grade orebodies brought i n t o production today would be so l a r g e as t o be a f i n a n c i a l i m p o s s i b i l i t y . Nevertheless, i n s p i t e of the f o r e g o i n g d i f f i c u l t i e s i n determining optimal p l a n t s i z e , the"head-in-the-sand" approach i s not warranted. Although such a study may border upon i m p o s s i b i l i t y from a p o s i t i v e standpoint, i t should be of some value as a normative framework. The hypothesis of t h i s study i s t h a t the optimal p l a n t s i z e of a mining venture i s dependent upon the c h a r a c t e r i s t i c s of the orebody, c a p i t a l and product markets, p r o j e c t e d p r i c e l e v e l s , and t a x s t r u c t u r e s -- f a c t o r s over which the f i r m does not have c o n t r o l . C o n t r o l l a b l e v a r i a b l e s i n determining optimal p l a n t s i z e are r a t e of recovery, l e v e l of recovery, c a p i t a l c o s t s , and o p e r a t i n g c o s t s . The purpose of t h i s study i s to d e l i n e a t e the e f f e c t s of the above v a r i a b l e s , both s i n g l y and j o i n t l y , upon p l a n t s i z e . The eventual aim i s t o present a model which w i l l i n t e r r e l a t e the v a r i a b l e s w i t h t h a t s c a l e of p l a n t which w i l l maximize the market value of the f i r m . A secondary purpose of the study, which i s c l o s e l y r e l a t e d t o the question of optimal p l a n t s i z e i s to compare the impact of the Canadian and United States t a x a t i o n systems upon the p r o f i t a b i l i t y of a given o p e r a t i o n , and upon the optimal grade-capacity combination t o be employed f o r a given orebody. C l o s e l y connected to'the f i n a n c i a l impact of t a x a t i o n or t a x a t i o n p o l i c y i s the matter of the economic impact of t a x a t i o n w i t h respect t o the c o n s e r v a t i o n . This f a c t o r w i l l a l s o be considered. Because of the broad scope o f the f i e l d of mineral p r o p e r t y v a l u a t i o n , c e r t a i n assumptions must be made i n order t o make the study more manageable. A b s t r a c t i o n s must be made i n many areas i n order to prevent the i n c o n s i s t e n c i e s o f the ope r a t i n g environment from obscuring the i n t e r r e l a t i o n s h i p s between the v a r i a b l e s t o be considered. S i m p l i f i c a t i o n s and g e n e r a l i z a t i o n s are required i n s e v e r a l f i e l d s comprising the area o f v a l u a t i o n . The most important v a r i a b l e i n v a l u a t i o n i s the mi n e r a l deposit i t s e l f . To d i s c u s s the f a c t o r s which can vary i n a given deposit would take volumes. Nevertheless, a b r i e f summary of r e l e v a n t g e o l o g i c a l i n f o r m a t i o n w i l l be presented i n Chapter I I . Other i n t e r a c t i n g v a r i a b l e s which w i l l be examined i n Chapter I I are; product and c a p i t a l markets, p r i c e l e v e l s , and t a x s t r u c t u r e s . Only those aspects of the v a r i a b l e s which are r e l e v a n t to c a p i t a l budgeting w i l l be di s c u s s e d . As w e l l as the above v a r i a b l e s which are exogenous t o the v a l u a t i o n process, s e v e r a l c o n t r o l l a b l e v a r i a b l e s w i l l be considered. . F i r s t of these i s the r a t e of recovery, or p l a n t c a p a c i t y . A second v a r i a b l e i s the l e v e l of recovery which may be expressed as the r a t i o of the amount of metal ex t r a c t e d from the orebody t o the t o t a l amount contained i n the d e p o s i t . L e v e l of recovery w i l l a l s o be considered i n terms o f s e v e r a l r e l a t e d v a r i a b l e s which are: waste-to-ore r a t i o , minimum mineable grade, and p l a n t e f f i c i e n c y . Since cash flows take place i n the f u t u r e , some method must be employed t o form an index of the p r o f i t a b i l i t y of the venture. There are s e v e r a l methods of determining p r o f i t a b i l i t y i n use today. Of course, some are much more conc e p t u a l l y v a l i d than o t h e r s . The purpose of Chapter IV w i l l be t o analyze these p r o f i t a b i l i t y measures i n order to determine a method which i s both workable and conc e p t u a l l y v a l i d . Chapter V w i l l b r i n g the v a r i a b l e s together i n a h y p o t h e t i c a l example of a mining venture. Because of the wide v a r i a t i o n i n the character of orebodies, t h i s r a t h e r constrained model i s necessary. T a b u l a t i o n of a c t u a l p r a c t i c e cannot be done because the techniques and assumptions used by mining companies i n v a l u a t i o n do not present an optimal p l a n t s i z e . Chapter VI w i l l o u t l i n e the various f u n c t i o n a l r e l a t i o n s h i p s determined i n Chapter V w i t h a view to p r o v i d i n g a normative framework f o r e v a l u a t i n g mining ventures. I I I . ECONOMIC ASPECTS OF THE MINERAL INDUSTRY The m i n e r a l i n d u s t r y i s unique i n one economic r e s p e c t . While most other i n d u s t r i e s d e a l w i t h a "f l o w " of raw m a t e r i a l s which r e p l e n i s h e s i t s e l f a f t e r v a r y i n g lengths of time, the mining and o i l i n d u s t r i e s d e a l w i t h a given "fund" of resources. C o n c e p t i o n a l l y , a fund resource such as a mi n e r a l deposit may be viewed as a f i x e d , non-renewable.inventory of m a t e r i a l , whereas a flow resource represents a moving inventory o f u n f i n i s h e d goods. The d i f f e r e n t i a t i o n i s due t o the f a c t that although a r e - f o r e s t a t i o n program may take 50 t o 100 years t o produce economic e f f e c t s , a r e - m i n e r a l i z a t i o n program would take tens of m i l l i o n s of y e a r s . This aspect was f i r s t considered by Carlisle."'" In h i s a r t i c l e , C a r l i s l e s t a t e s t h a t " t h e law of d i m i n i s h i n g returns ... . i s i l l u s t r a t e d by the decreasing e f f e c t i v e n e s s of more i n t e n s i v e a p p l i c a t i o n of c a p i t a l and l a b o r i n i n c r e a s i n g the r a t e of 2 working a d e p o s i t " . This d e f i n i t i o n i m p l i e s t h a t a time dimension may be introduced i n t o marginal a n a l y s i s . There are two prime v a r i a b l e s i n the mining of an ore d e p o s i t . These are t h e " l e v e l of recovery" which C a r l i s l e d e f i n e s to be t h e " t o t a l amount of mineral t o be e x t r a c t e d from the p a r t i c u l a r deposit or p a r t of the deposit ( a l t e r n a t i v e f r a c t i o n s of an absolute t o t a l assumed t o be i n the ground)" and " r a t e of recovery" which i s defined t o be the r a t e of p r o d u c t i o n . The cost i n c u r r e d i n mining may be d i v i d e d i n t o four c a t e g o r i e s . F i x e d costs (FC) are those costs which are independent of both r a t e and l e v e l of p r o d u c t i o n . These i n c l u d e c a p i t a l costs and e x p l o r a t i o n and development c o s t s . C a p i t a l costs are f i x e d because C a r l i s l e assumes a given p l a n t investment. Average f i x e d costs per u n i t of production can be decreased by i n c r e a s i n g the l e v e l of production but not by i n c r e a s i n g the r a t e of p r o d u c t i o n . A l s o , there are three categories of v a r i a b l e Donald C a r l i s l e , "The Economics of a Fund Resource With P a r t i c u l a r Reference t o Mining," The American Economic Review, v. kk (1954), P. 596. ^ I b i d . c o s t s . - R a t e - l e v e l - v a r i a b l e costs (VC-^) vary w i t h respect t o both r a t e and l e v e l of recovery, w i t h examples being d e p r e c i a t i o n charges and d i r e c t l a b o r and m a t e r i a l c o s t s . R a t e - v a r i a b l e costs (VCp>) vary as a t o t a l f o r the l i f e of the mine w i t h r a t e only, as i s the case w i t h overtime pay. L e v e l - v a r i a b l e costs (VC^) vary w i t h respect to the l e v e l of recovery. Royalty costs and surveying expenses are examples of l e v e l - v a r i a b l e c o s t s . C a r l i s l e considers cost-revenue diagrams f o r each of three cases. In the f i r s t case, r a t e of recovery i s permitted t o vary w h i l e l e v e l of reovery i s f i x e d . Secondly, the l e v e l of recovery i s v a r i a b l e and r a t e of recovery i s f i x e d . T h i r d l y , both l e v e l of recovery and r a t e of recovery are permitted t o v a r y . The cost-revenue diagrams f o r these three cases may be seen i n F i g . 1. . F i g . I^B shows the cost-revenue r e l a t i o n s h i p s under v a r i a b l e r a t e s of recovery. The v a r i a b l e - r a t e , f i x e d l e v e l case i s not d i f f i c u l t from a conceptional p o i n t of view. C a r l i s l e views t h i s as a s h o r t - r u n s i t u a t i o n (because p l a n t expenditure i s f i x e d ) under pure competition. Because c a p i t a l expenditure and l e v e l of recovery are f i x e d , there are d i m i n i s h i n g returns to s c a l e i n the l a b o r s e c t o r , g i v i n g r i s e t o the U-shaped average t o t a l u n i t cost curve and marginal cost curve. Since l e v e l - v a r i a b l e costs are f i x e d , they are i n c l u d e d i n the f i x e d - c o s t component. The h o r i z o n t a l p r i c e curve is<.'.due t o the assumption of pure competition. Maximum current p r o f i t i s a t t a i n e d at the p o i n t y, where MC-£ = MR. . At any output below t h i s p o i n t i t i s evident t h a t marginal revenue would exceed marginal cost or p r i c e , thus i m p l y i n g t h a t output should be increased i n order to maximize p r o f i t s . Conversely, at any p o i n t beyond the p o i n t where MCD = MR, marginal cost would exceed marginal revenue, thus d i l u t i n g p r o f i t s . Maximum t o t a l p r o f i t (maximum p r o f i t per u n i t of output) i s achieved at the p o i n t x where MC = AC, which i s the p o i n t of lowest-cost p r o d u c t i o n . However, although the f i x e d - r a t e , v a r i a b l e - l e v e l case i s what i s u s u a l l y considered a s h o r t -run case because of f i x e d c a p i t a l investment, C a r l i s l e hypothesizes t h a t t h i s w i l l be used throughout the l i f e of the mine. Therefore, a time dimension can be introduced to the study. I f , i n order to account f o r the t i m i n g of costs and revenues, earnings are discounted t o the present, i t i s apparent t h a t the optimum ope r a t i n g p o i n t w i l l be somewhere between the p o i n t s where MC = AC and MC - MR. The exact l o c a t i o n of the optimal operating p o i n t w i t h respect t o these two p o i n t s i s , of course, dependent upon the discount r a t e used. As i n a l l economic models, the cost of c a p i t a l i s ; J . i m p l i c i t i n the cost curves i n the form of the owners' r e q u i r e d r e t u r n on investment. Because of t h i s i m p l i e d c o s t , i t i s p o s s i b l e f o r economic t h e o r e t i c i a n s t o s t a t e t h a t , i n the l o n g run, there w i l l be no p r o f i t i n a competitive environment. This statement i s p o s s i b l e because economists view p r o f i t as "economic r e n t " or r e t u r n over and above t h a t which i s r e q u i r e d by i n v e s t o r s . .However, i n the mining i n d u s t r y , even under pure competition, p r o f i t s are p o s s i b l e . Because of the v a r y i n g richnesses of orebodies and the great s i z e of marginal u n i t s of p r o d u c t i o n , the i n d u s t r y supply curve i s a step func t i o n , . r a t h e r than a smooth upward-sloping curve. The cost-revenue curves obtained when r a t e of recovery i s constant and l e v e l of recovery i s v a r i a b l e are shown i n F i g . 1-A. L e v e l of recovery i s dependent upon three v a r i a b l e s : grade of the ore, w o r k a b i l i t y of the ore, and completeness of e x t r a c t i o n . Obviously, the lower the grade of ore mined, the higher the p e r - u n i t cost of metal produced, a l l other t h i n g s being equal. .In most mines, there i s not a sharp d i f f e r e n t i a t i o n between ore and waste because of the gradation-a l l y v a r i a b l e metal content of the rock. Even i n those deposits where there i s a c l e a r d i s t i n c t i o n between ore and waste, there w i l l be v a r y i n g grades of ore; thus g i v i n g a choice of t o t a l tonnage and, consequently, t o t a l product recovered and the l e n g t h of the mine l i f e . The w o r k a b i l i t y o f the deposit i s a l s o important i n determining the l e v e l of recovery i n a mine. W o r k a b i l i t y i s r e f l e c t e d i n the cost of removing rock. F a c t o r s i n f l u e n c i n g mining cost are: a c c e s s i b i l i t y from mine openings, t h i c k n e s s and r e g u l a r i t y of the ore, hardness and toughness of. the ore, presence of s t r u c t u r a l c h a r a c t e r i s t i c s such as f a u l t s or f o l d s , and amount of overburden. Thickness of ore i s an important determinant of w o r k a b i l i t y . Narrow veins are expensive t o mine because of r e q u i r i n g high-cost mining methods and because of ore d i l u t i o n which i s necessary i n order to remove s u f f i c i e n t rock t o permit men and machinery to work. Completeness of e x t r a c t i o n i s the t h i r d v a r i a b l e which determines l e v e l of recovery. To remove a l l of the metal contained i n an orebody would be p r o h i b i t i v e l y expensive because of high marginal mining, m i l l i n g , s m e l t i n g , and r e f i n i n g c o s t s . With a given p l a n t i n s t a l l a t i o n , the percent e x t r a c t i o n f o r an orebody i s dependent upon the grade of the orebody. Since f i x e d costs and r a t e - v a r i a b l e costs are, as a t o t a l , independent of the tonnage mined, they are spread over a higher t o t a l product at higher l e v e l s of recovery. This i s because each f i x e d - c o s t i n s t a l l a t i o n can handle more m a t e r i a l . U n l i k e the s i t u a t i o n i n the f i x e d - l e v e l case, i n the v a r i a b l e - l e v e l case increased c a p i t a l costs which are necessary i n : o r d e r t o permit a higher l e v e l o f recovery may be spread over the r e s u l t i n g increased, t o t a l product. . For t h i s reason, the f i x e d - c o s t and r a t e - v a r i a b l e - c o s t curve i s downward s l o p i n g r a t h e r than being constant. As i n the f i x e d - l e v e l , v a r i a b l e - r a t e case, the U-shaped v a r i a b l e - c o s t curve r e f l e c t s the d i m i n i s h i n g r e t u r n s which are encountered as the e x t r a c t i o n approaches the u l t i m a t e . The marginal cost curve i s s i m i l a r t o the one i n the f i r s t case. The market s t r u c t u r e i s assumed t o be p u r e l y competitive w i t h constant metal p r i c e throughout the l i f e of the orebody. Once again, p r o f i t per u n i t of p r o d u c t i o n i s maximized where average costs are minimized (MC = AC). Because the r a t e of recovery i s f i x e d , the l e a s t - c o s t p o i n t i s a l s o the po i n t at which current p r o f i t per u n i t of time i s maximized. However, t o t a l p r o f i t i s greatest at the p o i n t where MC = MR i f there i s no time preference f o r earnings (0 per cent discount r a t e ) . The poin t of maximum present value w i l l once again f a l l between the two p o i n t s described above; the exact l o c a t i o n i s , of course, dependent upon the discount r a t e . The case i n which both r a t e and l e v e l of recovery are v a r i a b l e i s shown i n F i g . 1-C. This f i g u r e i s merely a three-dimensional combination of F i g s . 1-A and 1-B, w i t h axes t o represent r a t e of recovery, l e v e l of recovery, and p r i c e . (Note t h a t the ATUC p l o t i n C i s not the same as i n B). By passing a plane through the p r i c e a x i s at the market p r i c e of metal which p r e v a i l s , i t i s p o s s i b l e t o o b t a i n a surface of f e a s i b l e o p e r a t i n g r a t e s and l e v e l s where ATUC P = MR. (Once again, pure competition i s p o s t u l a t e d ) . Since both r a t e s and l e v e l s of production are v a r i a b l e , the lo c u s of optimal operating p o i n t s w i l l be at the i n t e r s e c t i o n of two l i n e s , or a p o i n t . Thus there w i l l be a p o i n t f o r maximum current p r o f i t , a p o i n t f o r maximum t o t a l p r o f i t , and a point f o r maximum present value of cash f l o w s . The l e a s t - c o s t p o i n t would be at the lowest p o i n t on the a v e r a g e - t o t a l - u n i t - c o s t surface, and the po i n t of maximum current p r o f i t (MC = MR) would be at a higher r a t e and l e v e l of recovery than the l e a s t - c o s t p o i n t . Once again, the p o i n t of highest present value would be between the l e a s t - c o s t and the maximum-The competitive s t r u c t u r e of the mineral i n d u s t r y ranges from workable competition t o monopoly. Even i n the more competitive sectors such as copper, l e a d , and z i n c ; producers' a s s o c i a t i o n s attempt t o set quotas f o r member companies i n order t o r e g u l a t e supply. Nevertheless, because of the homogeneity of unprocessed metal, independent f i r m s w i l l f i l l the gaps between supply and demand and thus help to lower p r i c e . As example of the e f f e c t i v e n e s s of various producers' a l l i a n c e s can be seen i n the United States copper i n d u s t r y i n the past few years. Major fi r m s have attempted to hold down p r i c e because of the danger of l o s i n g t h e i r market to aluminum. Consequently, while the U. S. p r i c e f o r copper remained below 3 8 cents per pound u n t i l the recent s t r i k e , the incremental p r i c e , as evidenced by the London Metal Exchange p r i c e , has v a r i e d from approximately ho cents to approximately 8 0 cents. The LME p r i c e represents the amount which f a b r i c a t o r s are w i l l i n g t o pay f o r copper beyond the r a t i o n e d supply at a r t i f i c i a l l y low U. S. p r i c e s . Therefore, i t does appear that w i t h most base metals, no one s u p p l i e r has c o n t r o l over the world p r i c e . Another economic f a c t o r which i s c l o s e t o being unique i n mining i s the l a r g e sunk cost which must be i n c u r r e d before a f e a s i b i l i t y study can be made. The new l a r g e , low-grade mines r e q u i r e m i l l i o n s of d o l l a r s of work before tonnage and grade can be estimated w i t h s u f f i c i e n t c e r t a i n t y t o permit a f e a s i b i l i t y study. Because of the importance of sunk c o s t s , i t must be emphasized that costs i n c u r r e d before a f e a s i b i l i t y study i s made have no bearing whatsoever on the d e c i s i o n as t o whether or not t o produce. I t i s p o s s i b l e t h a t a mineral property may not produce enough income t o cover both e x p l o r a t i o n and c a p i t a l c o s t s . I f , however, the property can produce enough income to provide the r e q u i r e d r e t u r n on any a d d i t i o n a l investment r e q u i r e d a f t e r the f e a s i b i l i t y study, the p r o j e c t should be undertaken i n order t o m i t i g a t e e x p l o r a t i o n l o s s e s . -CHAPTER I I EXOGENOUS VARIABLES There are s e v e r a l v a r i a b l e s which must be considered i n the v a l u a t i o n of a mineral property. The exogenous v a r i a b l e s are those over which the f i r m has no c o n t r o l . These are: the orebody, c a p i t a l and product markets, p r i c e l e v e l , and tax s t r u c t u r e . I . THE OREBODY Each mineral deposit has s e v e r a l f e a t u r e s which make i t a unique e n t i t y . In general, a mineral deposit i s c h a r a c t e r i z e d by i t s tonnage and grade, i t s p o s i t i o n , i t s rock character, and i t s geographical l o c a t i o n . Tonnage and Grade The q u a n t i t y of metal contained i n an orebody i s uniquely defined by i t s tonnage and grade. Ore reserves are u s u a l l y stated as being a c e r t a i n tonnage at a given average grade, i n e f f e c t g i v i n g the number of tons of m a t e r i a l present and the number of pounds of metal per ton of ore present. As economic c r i t e r i a f o r mine development, these v a r i a b l e s are interdependent. At opposite ends of the spectrum are deposits w i t h high grade and n e g l i g i b l e tonnage and those w i t h high tonnage and very low grade. In general, n e i t h e r of these types of deposits has economic v a l u e . Another f a c t o r which enters i n t o the matter of grade and tonnage i s processing c o s t s . Although the average g r a n i t e mountain has m i l l i o n s of tons of mineral w i t h approximately - 12j? per-cent-aluminum content worth approximately 2 5 cents per pound of metal, the processing costs exceed the $ 1 2 5 per ton value of the rock. Because of processing c o s t s , a p o r t i o n of the m i n e r a l i z e d m a t e r i a l i n any min e r a l deposit i s not economically recoverable. The lowest-grade m a t e r i a l which may be mined at a p r o f i t i s c a l l e d the c u t - o f f grade. This concept w i l l be given f u r t h e r c o n s i d e r a t i o n i n the f o l l o w i n g chapter. In order t o understand the meaning of tonnage and grade, i t i s necessary t o introduce some of the terminology employed i n the f i e l d of mineral economics. In 1 9 5 6 , B l o n d e l and Lasky"*" introduced a c l a s s i f i -c a t i o n system f o r grouping the e n t i r e m i neral stock i n the earth's c r u s t , whether known, or unknown. They defined "reserves" as being "mineral m a t e r i a l considered e x p l o i t a b l e at present" and " p o t e n t i a l ores" as " c o n s i s t i n g of those deposits demanding more favorable c o n d i t i o n s " . The two c a t e g o r i e s , "reserves" plus " p o t e n t i a l ores", together form "resources", which a l t e r n a t i v e l y may be viewed as c o n s i s t i n g of "reserves" plus "marginal resources" plus "sub-marginal 2 resources" plus " l a t e n t resources". In I960, Schurr and Wetschert developed a system which was s i m i l a r t o that of Blondel and Lasky, but F. B l o n d e l and S. G. Lasky,-"Mineral Reserves and M i n e r a l Resources", Economic Geology, L I , ( 1 9 5 6 ) , pp. 6 8 6 - 9 7 . 2 Sam H. Schurr and Bruce C. Netschert, w i t h others, Energy i n the American Economy, 1 8 5 0 - 1 9 7 5 , ( B a l t i m o r e : The Johns Hopkins Press f o r Resources f o r the Future, Inc., i 9 6 0 ) . was more general and more r i g o r o u s l y d e f i n e d . The c l a s s i f i c a t i o n system i s presented i n Table 1 . I t i s evident t h a t the only category of mineral wealth of i n t e r e s t i n v a l u a t i o n i s the "reserve" c l a s s which i s defined as being of known occurrence, and economically f e a s i b l e at current costs and current t e c h n o l o g i c a l l e v e l . However, i t i s q u i t e apparent t h a t even t h i s d e f i n i t i o n i s not f i n e enough to permit adequate e x p o s i t i o n as t o the degree of c e r t a i n t y t o which the grade and tonnage of ore i s known. Several methods of determining types of ore reserves have been 3 developed, the most complete being t h a t of King. I t i s necessary t h a t the term "ore" be defined before types of ore reserves can be c l a s s i f i e d . According to the United States G e o l o g i c a l Survey, "ore i s a n a t u r a l aggregation of one or more minerals from which u s e f u l metals may be p r o f i t a b l y e x t r a c t e d " . (The i t a l i c s are mine.) King subdivides ore reserves i n t o four c l a s s e s , "proved", "measured", " i n d i c a t e d " , and " i n f e r r e d " . Proved reserves are those i n which the ore i s so f u l l y developed or so w e l l known -- the tonnage having been computed from dimensions revealed i n outcrops, p i t s , trenches, workings and/or d r i l l h o l e s , and the grade computed from the r e s u l t s of 3 see C. K. L e i t h , "Mineral V a l u a t i o n s of the Future", American  I n s t i t u t e of Mining, M e t a l l u r g i c a l , and Petroleum Engineers, Trans-a c t i o n s , 1 9 3 8 , pp. ^ 7 - 8 ; I n v e s t i g a t i o n of N a t u r a l Resources, Subcommittee Hearings, United States Senate Committee on P u b l i c Lands, May 1 5 - 2 0 , I 9 V 7 , PP- 1 1 9 - 2 0 ; and H. F. King, " C l a s s i f i c a t i o n and Nomenclature of Ore Reserves", A u s t r a l a s i a n I n s t i t u t e of Mining and Me t a l l u r g y , Proceedings, CLXXIV, (March-June, 1 9 5 5 ) , PP TABLE I -TERMS RESERVE - RESOURCE TERMINOLOGY ASPECTS OCCURRENCE ECONOMIC ^TECHNOLOGIC Reserve Known Present cost l e v e l C u r r e n t l y f e a s i b l e Resource Resource Base Known and Any cost l e v e l unknown s p e c i f i e d Known and I r r e l e v a n t unknown C u r r e n t l y f e a s i b l e and f e a s i b i l i t y i n d i c a t e d i n f u t u r e F e a s i b l e and i n f e a s i b l e From Sam H. Schurr and Bruce C. Netschert, w i t h others, Energy i n the  American Economy, 18^0 ~ 1875? (Baltimore: The Johns Hopkins Press For Resources f o r the Future, Inc. i960) P Measured reserves are those which, although tonnage i s computed from dimensions revealed i n outcrops, p i t s , trenches, workings and/or d r i l l h o l e s , and tonnage and grade are as w e l l known as they w i l l be p r i o r t o e x t r a c t i o n , nevertheless the g e o l o g i c a l character 'of the'orebody and the method of mining are such t h a t the tonnage and/or grade estimates do not have a high p r o b a b i l i t y of being accurate w i t h i n close l i m i t s . I n d icated reserves are those f o r which tonnages and grades are computed p a r t l y from s p e c i f i c measurements, samples, or p r o d u c t i o n data and p a r t l y from p r o j e c t i o n f o r a reasonable distance on geologic evidence. The s i t e s a v a i l a b l e f o r i n s p e c t i o n , measuring and sampling are too w i d e l y or otherwise i m p r o p o r t i o n a t e l y spaced to o u t l i n e the ore completely or t o e s t a b l i s h i t s grade throughout. I n f e r r e d ore i s ore f o r which q u a n t i t a t i v e estimates are based l a r g e l y on broad knowledge of the geologic character of the deposit and f o r which there are few, i f any, samples or measurements. The estimates are based on an assumed c o n t i n u i t y or r e p e t i t i o n f o r which there i s geologic evidence; t h i s evidence may i n c l u d e comparison w i t h deposits of s i m i l a r type. Bodies t h a t are completely concealed may be included i f there i s s p e c i f i c geologic evidence of t h e i r presence. Estimates of i n f e r r e d ore should i n c l u d e a statement of the s p a t i a l l i m i t s w i t h i n which the " i n f e r r e d " ore may be. The bases f o r d i s t i n g u i s h i n g between "reserves", "resources", and the "resource base" are economic and t e c h n o l o g i c , whereas the d i s t i n g u i s h i n g c h a r a c t e r i s t i c i n c l a s s i f y i n g ore reserves i s the degree of c e r t a i n t y t o which the tonnage and grade are known. The reserves of primary i n t e r e s t i n determining a p l a n t s i z e are the "proved", "measured", and " i n d i c a t e d " c a t e g o r i e s . " I n f e r r e d " ore reserves are important only as an i n d i c a t i o n of p o s s i b l e f u r t h e r e x p l o r a t i o n i f ore reserves are too small t o j u s t i f y economic e x p l o i t a t i o n . The i d e a l would be t o give tonnage and grade known w i t h absolute c e r t a i n t y . However, t h i s i n f o r m a t i o n cannot be obtained without the a d d i t i o n a l expense of the incremental d r i l l i n g , sampling, and c a l c u l a t i n g r e q u i r e d . I t has been argued^ that ore reserves which extend mine l i f e beyond t h i r t y or f o r t y years do not m a t e r i a l l y a f f e c t the value of the mine due t o the discount r a t e a p p l i e d t o f u t u r e cash f l o w s . Because of t h i s and because of the f a c t t h a t proving excessive reserves can e a s i l y exceed the discounted cash flows derived from .them (assuming that the reserves are mined i n the f u t u r e ) many mining companies adopt a p o l i c y of m a i n t a i n i n g reserves as some set m u l t i p l e of the current annual production r a t e . P o s i t i o n of the Orebody The p o s i t i o n of the orebody r e f e r s t o i t s p r o x i m i t y to the surface and i t s p h y s i c a l a t t i t u d e . C l e a r l y , t h i s f a c t o r w i l l i n f l u e n c e the mining method employed, and, i n t u r n , p r o d u c t i o n costs and the optimum p l a n t s i z e of a given venture. I t i s obvious t h a t , a l l other t h i n g s being equal, an orebody near the s u r f a c e , i n which open-pit methods can be employed, w i l l be able to operate to a lower grade than The theory of i n f o r m a t i o n i s a ra t h e r new f i e l d , consequently few e m p i r i c a l studies have been conducted i n t h i s area. For an over-view, see J . Marschak, Economic Theory of Information, (Los Angeles:) Working Paper No. 118,. Western Management Science I n s t i t u t e , U n i v e r s i t y of C a l i f o r n i a , (Los Angeles), 1967). 5 D. C a r l i s l e , Economic Aspects of the D e f i n i t i o n of Ore, I n s t i t u t i o n of Mining and M e t a l l u r g y , Transactions, LXTV, (1954-1955), P. 97 would be p r a c t i c a l f o r the same orebody s i t u a t e d two thousand f e e t below the surface. For example, the "average" cost of underground mining i n Canada during 1965 was $4.65 per ton as opposed to "average" open-pit costs of $0,289 P e r t o n . ^ I t i s tru e that g e n e r a l l y there are economies of s c a l e i n open p i t mining, however economies of s c a l e do not n e a r l y compensate f o r the l a r g e d i f f e r e n c e between the per-ton mining costs i n open p i t s and i n underground mines. The d i s c r e p a n c i e s caused by the p o s i t i o n of the orebody w i l l be e l i m i n a t e d i n l a t e r p a r ts of t h i s study, as only near-surface orebodies w i l l be considered. Rock Character The mining method used i n an ope r a t i o n i s determined to a l a r g e extent by the character of the ore and the w a l l rock. - I f the w a l l rock i s competent i n an underground mine, i t can o f t e n be mined by open stoping at an "average" cost of $2.46 per ton. On the other hand, i f the w a l l rock i s weak and slabby, shrinkage s t o p i n g ($4.13 per ton) or c u t - a n d - f i l l s t o p i n g ($6.97 per ton) would be r e q u i r e d . In the remainder of t h i s study, the e f f e c t of rock character on the r a t e and cost of mining w i l l be e l i m i n a t e d , as only orebodies i n which open-pit methods may be used w i l l be considered. W i l l i a m D. Baker, A Compilation of Production Data and Mining  Costs of" Canadian Mines, an unpublished r e p o r t prepared f o r Canadian E x p l o r a t i o n L t d . A p r i l , 1966. Cost data from Baker, op. c i t . The l o c a t i o n of an orebody i s one of the more important f a c t o r s which determine i t s economic v i a b i l i t y . U n t i l r e l a t i v e l y r e c e n t l y , mining fi r m s have had to pay f o r the c o n s t r u c t i o n of roads as w e l l as the added cost of p r o v i d i n g l o d g i n g and r e c r e a t i o n a l f a c i l i t i e s and power p l a n t s which would not have been r e q u i r e d i f the mine were s i t u a t e d c l o s e r t o a developed area. W i t h i n the l a s t few years the "Roads t o Resources" program of the Canadian Government has eased the burden of p r o v i d i n g i n f r a s t r u c t u r e t o some extent. Today, the major problems of adverse l o c a t i o n of mineral p r o p e r t i e s are those of high t r a n s p o r t a t i o n c o s t s , high c o n s t r u c t i o n c o s t s , and high l a b o r c o s t s . Another problem of remote l o c a t i o n i s the need f o r more complete and more e f f i c i e n t warehousing f a c i l i t i e s than would otherwise be r e q u i r e d . For some Canadian companies, the costs of warehousing are a considerable p r o p o r t i o n of ope r a t i n g expenses. I I . MARKETS As a l l c a p i t a l budgeting s t u d i e s depend to a great degree on f o r e c a s t i n g , i t i s necessary t o attempt t o analyze expected developments i n the product and c a p i t a l markets. . I t would be unwise t o undertake a mining venture w i t h an i n d i c a t e d i n t e r n a l r a t e of r e t u r n of eigh t per cent and then f i n d out that bond ra t e s are eight per cent at the time of f i n a n c i n g and th a t the f e a s i b i l i t y study was based on abnormally high p r i c e s of metals, I t i s rath e r • s t r a n g e t h a t , i n s p i t e of a l l the time and e f f o r t devoted t o determining the "One Best Way" of c a l c u l a t i n g the value of mineral p r o p e r t i e s , very l i t t l e has been done w i t h regard to f o r e c a s t i n g f u t u r e trends i n the product market. The Product Market The production and sales environments f o r various mineral commodities range from e f f e c t i v e monopoly t o o l i g o p o l y to workable competition. Although the degree of conc e n t r a t i o n i s great i n many metal markets, there are a great many fir m s producing the t o t a l output. C a p i t a l Markets Although the m a j o r i t y of Canadian mines have been financed through e q u i t y , s e v e r a l new trends are becoming apparent. Total, dependence upon p u b l i c o f f e r i n g s of eq u i t y i s becoming i n c r e a s i n g l y r a r e f o r s e v e r a l reasons. • In the f i r s t p l a c e , i t i s becoming more d i f f i c u l t t o compete f o r funds i n the "penny" mining market because i n v e s t o r s are becoming more s o p h i s t i c a t e d and are shying away from new mining ventures because of the bad r e p u t a t i o n of many new mining stocks Since most promoters are promising i n v e s t o r s i n the mining markets i n s t a n t r i c h e s , i t i s d i f f i c u l t f o r a company w i t h a v i a b l e property t o Q In 1956, the l e a d i n g four copper producers i n the world accounted f o r 4-9 per cent of the t o t a l output, while the l e a d i n g eight producers accounted f o r 70 per cent of t o t a l output. ( O r i s C. H e r f i n d a h l , Copper Costs and P r i c e s : 187-0-195,7, ( B a l t i m o r e : Johns Hopkins Press f o r Resources f o r the Future, Inc. 1959)' make i t sound impressive, short of absolute eq u i v o c a t i o n . A second reason f o r the d e c l i n e of dependence upon p u b l i c equity o f f e r i n g s i s the l a r g e amount of money which must be r a i s e d t o b r i n g today's high-volume, low-grade operations i n t o production. The i n c r e a s i n g a v a i l -a b i l i t y of money outside the penny stock markets represents a t h i r d reason f o r the d e c l i n e of t h i s type of f i n a n c i n g . I t has been estimated t h a t i t i s a ra r e small "mining" company indeed, i n which more than 25 per cent of the money r a i s e d through the sale of stock i s spent on a c t i v e e x p l o r a t i o n . One of the l a t e s t a r t i c l e s d i s c u s s i n g the changes t a k i n g place Q i n mine f i n a n c i n g was p r i n t e d i n the Mining J o u r n a l . According to t h i s a r t i c l e , the trend from high-grade, low-volume t o low-grade, high-volume operations has created a need f o r " l a r g e - s c a l e " i n i t i a l f i n a n c i n g . This money can u s u a l l y be provided by l a r g e i n t e r n a t i o n a l companies, e i t h e r alone or i n p a r t n e r s h i p . I n c r e a s i n g dependence on f i n a n c i n g by l a r g e corporations has created two n o t i c e a b l e trends, the f i r s t being i n c r e a s i n g debt-to-equity r a t i o s of l a r g e r f i r m s , a n d . the second being the d e c l i n i n g r o l e of the i n d i v i d u a l i n v e s t o r . With regard t o t h i s second t r e n d , i t can be noted that almost every time a small e x p l o r a t i o n company i n Western Canada discove r s a v i a b l e property, y "Changing P a t t e r n f o r Mine Finance", Mining J o u r n a l , v. 268, n. 6868, ( A p r i l 7, 1967). "*"0 On A p r i l 17, 1968,. The I n t e r n a t i o n a l N i c k e l Company of Canada, L t d . announced a $100,000,000 debenture i s s u e , the f i r s t i n that company's h i s t o r y . c o n t r o l l i n g i n t e r e s t i s given to a major f i r m i n r e t u r n f o r f i n a n c i n g . The entry of l a r g e r f i r m s i n t o the f i n a n c i n g of new mining ventures tends to reduce the expected r i s k of i n v e s t o r s because of the proven e x p e r t i s e of these e s t a b l i s h e d c o r p o r a t i o n s . Due to guarantees by parent companies, bonds can be f l o a t e d at s a t i s f a c t o r y p r i c e s where debt issues would not otherwise be accepted. Another form of f i n a n c i n g which has come i n t o vogue during the l a s t decade i s the use of monies advanced by smelting concerns against long-term concentrate c o n t r a c t s . This i s the method by which the Japanese Sumitomo Group has financed s e v e r a l p r o p e r t i e s i n Western Canada, namely Bethlehem Copper Corporation L t d . , G r a n i s l e Coppe^Ltd., and Hew I m p e r i a l Mines L t d . There i s one major drawback t o t h i s type of f i n a n c i n g . The primary i n t e r e s t of the Japanese smelting concerns i s t o assure long-term su p p l i e s of concentrates.for t h e i r smelters. Consequently, as a c o n d i t i o n f o r p r o v i d i n g advances against concentrates, these companies d i c t a t e the p l a n t s i z e t o be i n s t a l l e d . Since the s c a l e of p l a n t which assures long-run production i s smaller than that which w i l l maximize the value of the f i r m , a company u s i n g concentrate advances f o r f i n a n c i n g faces an opportunity cost i n that the present value of earnings i s not as high as i t would be i f the p l a n t were For example, Dynasty E x p l o r a t i o n and Cyprus Mines combined t o form A n v i l Mining Corp., and the Lornex p r o j e c t i s being financed by Rio Algom Mines L t d . An i n d i c a t i o n of the l e v e r e d p o s i t i o n of some newer mines might be shown by a few examples. For the Palabora p r o j e c t i n South A f r i c a , f i n a n c i n g was made through $80 m i l l i o n debt and $43 m i l l i o n e q u i t y . Of the debt, approximately 4-0 per cent was secured against f u t u r e copper and a c i d p r o d u c t i o n . E l l i o t Lake Mines L t d . was financed w i t h an i n i t i a l d e b t-to-equity r a t i o of 4:88. Another trend i n mine f i n a n c i n g i s f o r p r o j e c t s to be developed as p a r t n e r s h i p s between major mining companies without any e x t e r n a l e q u i t y f i n a n c i n g . ' An example of t h i s type of f i n a n c i n g i s the Mt. Newman i r o n ore p r o j e c t i n A u s t r a l i a which i s held as f o l l o w s : Broken H i l l P ty. L t d . (30 per c e n t ) , American Metal-Climax (25 per c e n t ) , C o l o n i a l Sugar (22-g- per c e n t ) , M i t s u b i s h i and C. Itoh (10 per c e n t ) , A u s t r a l i a n Mutual Provident S o c i e t y (7^ per c e n t ) , and S e l e c t i o n Trust (5 per c e n t ) . A d i s t i n c t f e a t u r e of the mining i n d u s t r y i s a re l u c t a n c e to use long-term debt. Since mines are wasting assets, i t i s u s u a l l y not t o a company's advantage t o have a high l e v e l of permanent debt. .Even l a r g e f i r m s have w i d e l y - f l u c t u a t i n g earnings, due i n part t o the f a c t that earnings d e c l i n e as a mine reaches the end of i t s productive l i f e and i n part t o changes i n metal p r i c e s . Because of these v a r i a t i o n s i n earnings, most companies attempt to r e t i r e debt as soon as p o s s i b l e . The usual p r a c t i c e i s f o r the debt i n a new mine to be completely r e t i r e d before any dividends are p a i d . In Canada, i t i s not unusual f o r - a company t o . r e t i r e a l l of i t s debt during i t s i n i t i a l three-year t a x - f r e e p e r i o d . I I I . TAX STRUCTURE With the development of l a r g e , low-grade mines w i t h heavy c a p i t a l - e x p e n d i t u r e requirements, t a x a t i o n r e g u l a t i o n s have become one of the major f i n a n c i a l v a r i a b l e s to be considered i n a f e a s i b i l i t y study. The impact of tax concessions i n Canada i s r e f l e c t e d by the f a c t t h a t many of the newer western copper and molybdenum producers (Endako, G r a n i s l e , Brenda, Lornex) would probably be submarginal i n the United S t a t e s . In Canada, many tax concessions have been given t o the mining i n d u s t r y i n order to st i m u l a t e i t s development. Whether these tax concessions tend t o promote the n a t i o n a l welfare or instead r e s u l t i n a m i s a l l o c a t i o n of resources has been a t o p i c of much debate. Buckovetsky and the Carter Commission^ f e e l t h a t the tax system provides an i n e f f i c i e n t i n c e n t i v e program f o r the e x t r a c t i v e i n d u s t r i e s . Buckovetsky p o i n t s out th a t the equivalent of $150 m i l l i o n i n tax revenue i s l o s t to the government because of these i n c e n t i v e s . This i s the equivalent of fou r percentage p o i n t s on the average corporate t ax r a t e , c o n s i d e r i n g the t o t a l corporate t ax base. M.A. Buckovetsky, Studies of the Royal Commission on Taxation, No. 8, The Taxation of M i n e r a l E x t r a c t i o n , (Ottawa: Queen's P r i n t e r , 19677, and Royal Commission on Taxation, Kenneth Le M. C a r t e r , Chairman, (Ottawa: Queen's P r i n t e r , 1967), V I , pp. 295-380 In order t o show the e f f e c t of tax laws on the c a p i t a l budgeting d e c i s i o n , a b r i e f summary of both Canadian and United States tax laws w i l l be presented and a comparison of these laws on p r o f i t a b i l i t y of an oper a t i o n w i l l be given. 13 Canadian Tax Law J Rates. The e f f e c t i v e corporate t a x r a t e i s 2 1 per cent on the f i r s t $ 3 5 , 0 0 0 of taxable income and 5 0 per cent on the excess over $ 3 5 , 0 0 0 . P r o v i n c i a l Mining Taxes. I n general, p r o v i n c i a l mining taxes are allowable as deductions when computing income f o r f e d e r a l t a x . In B r i t i s h Columbia, the p r o v i n c i a l mining tax r a t e i s 1 5 per cent of net p r o f i t before taxes. A d e c l i n i n g - b a l a n c e p l a n t allowance of 8 per cent of c a p i t a l expenditures i s permitted. C a p i t a l Cost Allowance ( D e p r e c i a t i o n ) . Maximum allowable p o r t i o n of cost on a d i m i n i s h i n g balance b a s i s i s : Class, 1 (h per cent) -- road, a i r p l a n e runway, or s i m i l a r surface c o n s t r u c t i o n , or a r a i l w a y t r a c k or grading that i s not par t of a r a i l -way system. Class 2 ( 6 per cent) -- a p i p e l i n e f o r water. Class k ( 1 0 per cent) -- an o i l or water storage tank. Adapted from M i n e r a l Information B u l l e t i n , MR 8 2 , Summary  Review of Federal-Taxation and L e g i s l a t i o n A f f e c t i n g the Canadian  M i n e r a l Industry, r e v i s e d , (Ottawa: Queen's P r i n t e r , 1 9 6 6 j ^ Class 10 (30 per cent) -- a b u i l d i n g acquired f o r the purpose of ga i n i n g income from mining (except r e f i n e r i e s and o f f i c e b u i l d i n g s which are not s i t u a t e d on the mine p r o p e r t y ) . Class 12 (100 per cent) -- a mine s h a f t , main haulage way or s i m i l a r underground work designed f o r c o n t i n u i n g use, or extension t h e r e o f , sunk or constructed a f t e r the mine came i n t o p roduction. I f d e preciable property i s disposed of, the allowance may be subject t o recapture. D e p l e t i o n Allowance. In general, the owner of a metal mine other than a gold mine i s e n t i t l e d to deduct 33 l / 3 VeT cent of the aggregate of p r o f i t s and l o s s e s from such a mine, l e s s deductions i n c l u d i n g e x p l o r a t i o n and development expenses. 36-Month Exemption f o r Mines. Income from a mine i s not subject to f e d e r a l income t a x f o r a period of t h i r t y - s i x months, commencing w i t h the f i r s t reasonable commercial production from the mine. E x p l o r a t i o n and Development Expenses. For a company whose p r i n c i p a l business i s mining, e x p l o r a t i o n and development expenses are d e d u c t i b l e i n the year i n c u r r e d , and may be c a r r i e d forward i f income i s not s u f f i c i e n t to permit complete deduction. United States Tax Law 1 1 1 Adapted from B r i a n W. MacKenzie, "The F i n a n c i a l Aspects of Mining L e g i s l a t i o n " , an unpublished t h e s i s presented i n p a r t i a l f u l f i l m e n t f o r the degree of Master of Science i n Engineering, U n i v e r s i t y of Witwatersrand, Johannesburg, South A f r i c a . Rates. The r a t e of tax i s 25 per cent of taxable income w i t h a 23 per cent surcharge on taxable income i n excess of $25,000. State Mining Taxes. Although s t a t e taxes vary from s t a t e to s t a t e , an e f f e c t i v e r a t e of 4 per cent of net income a f t e r d e p l e t i o n w i l l approximate the various s t a t e taxes. State taxes are allowable deductions when c a l c u l a t i n g f e d e r a l corporate income t a x . E x p l o r a t i o n Expenditures. E x p l o r a t i o n expenditures may be deducted up to the amount of $100,000 per year f o r a maximum of four years, or they may be deducted as a deferred charge against mineral production discovered through the e x p l o r a t i o n expense. Any excess above the allowable $400,000 may be e i t h e r taken as a t a x l o s s or recovered through d e p l e t i o n allowance. Development Expenditures (b) "Use of C e r t a i n Methods and Rates"-- For taxable years ending a f t e r December 31, 1953, the term "reasonable allowances" . . . s h a l l i n c l u d e (but s h a l l not be l i m i t e d to) by the computed i n accordance w i t h r e g u l a t i o n s p r e s c r i b e d by the Secretary or h i s delegate, under any of the f o l l o w i n g methods: (1) the s t r a i g h t l i n e method, (2) the d e c l i n i n g balance method, using a r a t e not exceeding twice the r a t e which would have been used had the annual allowance been computed under the method described i n paragraph ( l ) , (3) any other c o n s i s t e n t method productive of an annual allowance which, when-added to a l l . allowances f o r the p e r i o d 1 5 Taken from 1954 I n t e r n a l Revenue Code, sec. 167 commencing w i t h the taxpayer's use of the property and i n c l u d i n g the t a x a b l e year, does not, during the f i r s t t wo-thirds of the u s e f u l l i f e of the property, exceed the t o t a l of such allowances which would have been used had such allowances been computed under the method described i n paragraph (2). D e p l e t i o n . E i t h e r cost or percentage d e p l e t i o n i s permitted. Cost d e p l e t i o n i n v o l v e s ammortizing the approved cost of the property on a u n i t - o f - p r o d u c t i o n b a s i s over the l i f e of the property. Cost d e p l e t i o n may be r e v i s e d from time to time but i s l i m i t e d t o the cost of the property. Percentage d e p l e t i o n i s c a l c u l a t e d by a l l o w i n g a deduction of c e r t a i n percentages of gross income provided t h a t the d e p l e t i o n allowance does not exceed 50 per cent of the f i r m ' s taxable income from the property, computed without the d e p l e t i o n allowance. D e p l e t i o n r a t e s range from 5 psr cent f o r sand and g r a v e l to 27g- per cent f o r petroleum products, w i t h most m e t a l l i c ores being i n e i t h e r the 23 per cent or the 15 per cent brackets. A Comparison of Canadian and United States Tax laws Canadian tax laws are g e n e r a l l y more f a v o r a b l e t o mining than United States laws because of the three-year t a x - f r e e p e r i o d and the more generous c a p i t a l cost allowances. Due t o the higher d e p l e t i o n allowance i n the United States, income a f t e r tax from an American mine w i l l be higher than that from a Canadian mine, a f t e r the p o i n t where preproduction expenses have been w r i t t e n o f f . The e l i m i n a t i o n of corporate income taxes during the t a x - f r e e p e r i o d and the years f o l l o w i n g i n which preproduction expenses are w r i t t e n o f f causes income from Canadian mines t o be high i n the i n i t i a l p e r i o d . Therefore, there i s a b i a s toward l a r g e r - s c a l e operations and short e r mine l i f e than would be optimal i n the United S t a t e s . Because of the t a x - f r e e p e r i o d , there i s a tendency to "high-grade" a Canadian mine d u r i n g i t s f i r s t three years of produ c t i o n . This tends to shorten mine l i f e and reduce the t o t a l amount of recovery from what would occur i n the United S t a t e s . . Conversely, there i s a b i a s against long-term mining ventures i n Canada, r e s u l t i n g from the f a c t t h a t the e f f e c t i v e tax r a t e becomes high a f t e r the elapse of.most tax concessions. To sum up the d i f f e r e n c e between Canadian and American tax laws, i t can be stated that Canadian laws have a b u i l t - i n p r o p e n s i t y to promote the d e p l e t i o n r a t h e r than the conservation of mineral resources. P r i c e L e v e l :'/V^V:,v.;. . ': .'. G e n e r a l l y , when a f e a s i b i l i t y study i s undertaken, i t i s assumed that product p r i c e s and f a c t o r costs w i l l remain p r o p o r t i o n a l . A refinement on t h i s assumption has been to c a l c u l a t e break-even p o i n t s under various product p r i c e s , assuming t h a t f a c t o r costs w i l l remain constant. One major element i n p r i c e - l e v e l c o n s i d e r a t i o n s i s the f a c t t h a t l a b o r costs are i n c r e a s i n g at a ra t e much higher than t h a t of the general p r i c e l e v e l . Although p r o d u c t i v i t y i s g e n e r a l l y on the i n c r e a s e , c e r t a i n methods of mining are much more l a b o r - i n t e n s i v e than o t h e r s . Because of t h i s , mines employing l a b o r - i n t e n s i v e methods are caught i n a " c o s t - p r i c e " squeeze. An example of l a b o r i n t e n s i t y as a f a c t o r i n the i n c r e a s i n g 16 costs of mining was presented by Chandler. The e f f e c t s may be seen i n F i g u r e 2. The cost increase i n the more l a b o r - i n t e n s i v e square-set s t o p i n g method has been much greater than the increase f o r the room-a n d - p i l l a r method. As can be seen, the increase i n t o t a l costs i s due p r i n c i p a l l y t o the increase i n wage r a t e s without a compensating r i s e i n p r o d u c t i v i t y . C a p i t a l costs have a l s o been r i s i n g at a r a p i d r a t e d u r i n g the past few years. However, the increase i n c a p i t a l costs i s not n e a r l y as important as that of ope r a t i n g c o s t s , as the.greater part of c a p i t a l costs w i l l be i n c u r r e d i n the f i r s t few years of the p r o j e c t . The p r i c e l e v e l i n the product market i s much more d i f f i c u l t to consider. Since new mines coming i n t o p r o d u c t i o n place l a r g e incremental s u p p l i e s of mineral products on the market, the marginal concept i s not v a l i d . An example of the inadequacy of the marginal concept i s provided by the e f f e c t of the commencement of pr o d u c t i o n of Pine P o i n t Mines L t d . on world z i n c p r i c e s . Perhaps the only way t o t r e a t product p r i c e i n a v a l u a t i o n a n a l y s i s i s to do a s e n s i t i v i t y a n a l y s i s on revenue. I t i s p o s s i b l e that a p r o j e c t could be simulated over a v a r y i n g annual range of s u b j e c t i v e p r o b a b i l i t i e s f o r d i f f e r e n t product p r i c e s . John W. Chandler, "Mine Development and Mine Operating Costs", Mining Congress J o u r n a l , v . 4 5 , n . l l , (November, 1 9 5 9 ) ? PP. 1 + 5 - 5 1 « CHAPTER I I I . CONTROLLABLE VARIABLES The p r i n c i p a l c o n t r o l l a b l e v a r i a b l e s i n mineral property v a l u a t i o n are r a t e of recovery and l e v e l of recovery. Although c a p i t a l and ope r a t i n g costs are i n t i m a t e l y r e l a t e d to r a t e and l e v e l of recovery, they w i l l be discussed s e p a r a t e l y along w i t h e m p i r i c a l data and h e u r i s t i c techniques used t o estimate c o s t s . I . RATE OF PRODUCTION Rate of pro d u c t i o n i s defined as the d a i l y (or annual) tonnage of ore which i s produced by the mine. The major l i m i t i n g f a c t o r w i t h respect t o the r a t e of pro d u c t i o n i s the s c a l e of i n s t a l l e d p l a n t and equipment. However, there are s e v e r a l other f a c t o r s which i n f l u e n c e mining r a t e . These are considered by Chandler."'" Chandler s t a t e s that the f o l l o w i n g are major f a c t o r s i n f l u e n c i n g the choice of a r a t e of output. The d a i l y p roduction of a mine i s based p r i m a r i l y on i t s a b i l i t y t o produce ore, which, i n t u r n , i s determined by the s i z e of the orebody and the ca p a c i t y of i n s t a l l e d p l a n t and equipment. Another item which determines mine c a p a c i t y i s the r a t e at which development work can be done. Chandler a l s o s t a t e s that the optimum r a t e of production w i l l be at the r a t e which provides John W. Chandler, op. c i t ' . the lowest possible, o p e r a t i n g c o s t s . However, t h i s viewpoint tends to ignore c a p i t a l c o s t s , which are becoming i n c r e a s i n g l y important i n lower-grade, more c a p i t a l - i n t e n s i v e mining o p e r a t i o n s . Another weak p o i n t i n the a r t i c l e i n v o l v e s the statement t h a t " i t i s common p r a c t i c e t o b u i l d a mine p l a n t i n u n i t s of m i l l c a p a c i t y , p r o v i d i n g space i n the m i l l b u i l d i n g f o r . a d d i t i o n a l u n i t s of m i l l c a p a c i t y which would be i n s t a l l e d when the mine has a t t a i n e d the a b i l i t y t o supply the e x t r a output". I n the United S t a t e s , t h i s idea has m e r i t , but i n Canada there would be a s i z a b l e opportunity l o s s i f t h i s p o l i c y were pursued. Because the three-year t a x - f r e e p e r i o d commences at the s t a r t of commercial production of the mine, any p l a n t expansions do not r e c e i v e the f u l l b e n e f i t of-the t a x - f r e e p e r i o d . Therefore, due t o the greater, b e n e f i t s of t a x - f r e e earnings, the o p t i m a l - p o l i c y t o pursue i n Canada i s t o have the t o t a l expected m i l l c a p a c i t y i n s t a l l e d before the commencement of the t a x - f r e e p e r i o d . Since a t t e n t i o n w i l l be confined to open-pit mines i n the f o l l o w i n g , chapters, the f a c t o r s l i m i t i n g the r a t e of recovery w i l l be p r i m a r i l y economic r a t h e r than p h y s i c a l . I I . " LEVEL OF RECOVERY L e v e l of recovery i s defined as the " t o t a l amount of mineral to be e x t r a c t e d from the p a r t i c u l a r deposit or part of the deposit ( a l t e r n a t i v e f r a c t i o n s of an absolute t o t a l assumed t o be i n the ground)". I t i s evident from t h i s d e f i n i t i o n that l e v e l of recovery or percent mineral e x t r a c t i o n (not to be confused w i t h percent ore e x t r a c t i o n ) i s a f u n c t i o n of the marginal minable grade of ore, or cu t - o f f grade. The development of open-pit mines has made the c a l c u l a t i o n of c u t - o f f grade q u i t e amenable t o economic a n a l y s i s , as many of the p h y s i c a l f a c t o r s common i n underground mining are e l i m i n a t e d . With underground methods ( w i t h the exception of caving techniques), masses of waste which are surrounded by ore may be l e f t i n p l a c e . However, i n open-pit mining, the waste rock must be removed i n order t h a t the un d e r l y i n g ore be recovered. Therefore, the mining costs f o r ore and waste are about the same i n p i t s . • The fundamental is s u e then becomes the determination of the marginal grade which may be processed r a t h e r than being sent to the waste dump. This question has a t t r a c t e d considerable a t t e n t i o n i n the l a s t decade. Studies on c u t - o f f grade 3 were undertaken by Soderburg, V i c k e r s , . and Henning. Soderburg and V i c k e r s use techniques which seem t o be more o p e r a t i o n a l l y o r i e n t e d than does t h a t of Henning. Soderburg bases h i s Donald C a r l i s l e , "The Economics of a Fund Resource w i t h P a r t i c u l a r Reference to Mining", The American Economic Review, XLTV, (1954), p. 596. A d o l f Soderburg, "Elements of Long Range P i t Planning"' Mining  Congress J o u r n a l , XLV, ( A p r i l , 1959)? r e p r i n t e d i n Canadian Mining  Manual, 1959? PP« 34-41; E. L. V i c k e r s , "Marginal A n a l y s i s : I t s A p p l i c a t i o n i n Determining C u t - o f f Grade", Mining Engineering, XVI, (June, 1961), pp. 579-82; and U l f Henning, " C a l c u l a t i o n of Cu t - o f f Grade", Canadian Mining J o u r n a l , LXXXIII, (March, 1963), pp. 54-7-method on the u l t i m a t e p i t slope. This slope i s based on both p h y s i c a l and economic f a c t o r s . Economically, the slope must be as steep as p o s s i b l e i n order to promote maximum ore recovery at a minimum o v e r a l l s t r i p p i n g r a t i o (tons of waste per tone of o r e ) . P h y s i c a l l y , the u l t i m a t e slope must be based on the s i z e , shape and a t t i t u d e of the orebody, as w e l l as the l o c a t i o n of haulage systems, waste dumps, and other f a c i l i t i e s . The upper l i m i t of the p e r m i s s i b l e slope i s determined by the s t r e n g t h of the rock, and, i n t u r n , by the slope at which the p i t w a l l w i l l f a i l . .Such a f a i l u r e has been known to cause the otherwise needless expense of a d d i t i o n a l m a t e r i a l removal. The s t r i p p i n g r a i o which i s t o be minimized i n the design of the u l t i m a t e p i t slope i s defined as being the r a t i o of waste to ore, which must be removed. I n order to determine an o v e r a l l s t r i p p i n g r a t i o , i t i s necessary t o d e f i n e the marginal or break-even r a t i o . The break-even r a t i o i s equal t o the r a t i o of the value of the ore l e s s production costs (net value of the ore) t o the cost of s t r i p p i n g waste. Production cost r e f e r s t o a l l costs i n c u r r e d up to the p o i n t of s a l e of the metal. These r a t i o s are determined f o r v a r i o u s grades of ore at various metal p r i c e s , flee F i g . I t can be seen from the F i g u r e t h a t changes i n metal p r i c e s have the e f f e c t o f expanding or c o n t r a c t i n g the p i t . However, t h i s tendency i s t r u e only to the extent of the range where open-pit mining i s l e s s c o s t l y than underground mining. In other words, there i s a r a t i o at which underground mining i s more inexpensive than surface mining, and t h i s r a t i o i s r e l a t e d t o c o s t s . - - not t o market p r i c e . The importance of market p r i c e i s i n determining the f i n a l s i z e and shape of the p i t . For t h i s reason i t i s important to know the expected r e l a t i o n s h i p between costs and p r i c e s . The absolute d o l l a r amount of costs and p r i c e s i s unimportant, as absolute values disappear i n the determination of the break-even s t r i p p i n g r a t i o . Soderburg s t a t e s t h a t the p i t should be d i v i d e d i n t o sectors - - each sect o r being considered an i n d i v i d u a l e n t i t y i n long-range planning. In the i n t e r e s t of s i m p l i c i t y , t h i s assumption w i l l l a t e r be r e l a x e d . Through the value of the ore, which i s used i n computing the break-even s t r i p p i n g r a t i o , t h i s r a t i o i s a f u n c t i o n of c u t - o f f grade - -the minimum grade which can be mined p r o f i t a b l y . C u t - o f f grade and s t r i p p i n g r a t i o can be used t o determine minimum u n i t c o s t . Once the u n i t cost has been determined, c u t - o f f grade has no r e l a t i o n s h i p to market p r i c e . Dropping the c u t - o f f grade extends mine l i f e at the expense of lower annual metal output. I f a f i x e d c o n c e n t r a t i n g c a p a c i t y i s assumed, l o w e r i n g the c u t - o f f grade a l s o reduces the r e q u i r e d d a i l y mine output, as more m a t e r i a l goes to the concentrator, and l e s s t o the waste dump. Soderburg p o i n t s out, as does C a r l i s l e , that a high annual p r o f i t i m p l i e s a short mine l i f e and a low t o t a l p r o f i t . Conversely, a high t o t a l p r o f i t i m p l i e s a low annual p r o f i t . I f the o b j e c t i v e i s to maximize the present worth of the f i r m , the optimal c u t - o f f grade w i l l f a l l between t h a t which maximizes annual p r o f i t and t h a t which maximizes t o t a l p r o f i t . I f p l a n t s i z e i s permitted t o vary, the obvious s t a r t i n g p o i n t i s to increase c a p a c i t y to the p o i n t where metal output equals what i t would be at the higher c u t - o f f grade. Since t h i s expanded p l a n t s i z e i m p l i e s a lower f i x e d cost per t o n processed, augmenting the c a p a c i t y would i n d i c a t e t h a t the c u t - o f f grade could be reduced f u r t h e r , i n t u r n i m p l y i n g f u r t h e r p l a n t expansion. A b e t t e r s o l u t i o n to t h i s r a t e - l e v e l problem w i l l be presented l a t e r . The greatest s i n g l e d e f i c i e n c y i n Soderburg's a n a l y s i s i s that he equates marginal revenue w i t h average c o s t s , an e r r o r which w i l l cause r a t e s and l e v e l s of production higher than the optimum t o be employed. Although marginal costs are d i f f i c u l t t o o b t a i n i n r e a l i t y , they can be approximated by incremental c o s t s , which are r e l a t i v e l y e a s i l y determined. Henning employs a more t h e o r e t i c a l technique t o determine cut-o f f grade. His a n a l y s i s considers d i f f e r e n t c r i t e r i a f o r e s t a b l i s h i n g marginal grade, depending upon the o b j e c t i v e of the f i r m . Only one of h i s o b j e c t i v e s i s r e l e v a n t t o investment theory. This o b j e c t i v e s t a t e s t h a t the firm, d e s i r e s "to maximize the present value of the annual o p e r a t i n g p r o f i t and investment c o s t " . To Henning, t h i s f o r m u l a t i o n suggests t h a t i t i s u s u a l l y opportune to mine the highest-grade ore near the commencement of production, thereby s h i f t i n g cash flows toward the present. However, marginal ore i s u s u a l l y l o c a t e d i n such a manner tha t i t i s p o s s i b l e t o mine i t w i t h mine grade ore. The question becomes whether to mine low-grade ore w i t h high-grade ore or to leave i t i n p l a c e . I t i s obvious from the preceding e x p l a n a t i o n that Henning i s r e f e r r i n g t o underground s i t u a t i o n s , since i t has been mentioned that i n open p i t s the low=grade must be mined, but not n e c e s s a r i l y processed. Often the grade r e l a t i o n s h i p i n a disseminated porphyry orebody can be expressed w i t h reserves at a s p e c i f i e d c u t - o f f grade being defined as an exponential f u n c t i o n of a constant tonnage d e l i n e a t e d at a p a r t i c u l a r c u t - o f f grade. Such an expression would be: (a-bx) • R x = C.e^ ' Where R x represents tons of ore reserves w i t h a c u t - o f f grade of x, C i s the tonnage of ore w i t h a c u t - o f f grade of a/b per cent, and e i s the n a t u r a l l o g a r i t h m i c base. Average grade, i , at the c u t - o f f grade x i s found by i n t e g r a t i n g the f o l l o w i n g expression: i = ( x.R'.dx ) / n = x + l / b . - I f , f o r example, the orebody i s defined as having 20 m i l l i o n tons at a c u t - o f f grade of one-half per cent copper, the tonnage at a c u t - o f f grade of 1 per cent copper w i l l be as f o l l o w s : R = 20 x 10 6 . e^1-"-2^ = 20 x 10 6 . e" 1 = 8,683,800 tons. The average grade of t h i s tonnage of ore at a c u t - o f f grade of 1 per cent would be: i = 1 + 1/2 = 1.5$ Cu. . According t o the o b j e c t i v e , i t must be determined whether mining one tonc :pf marginal ore now plus mining one t o n of average-grade ore at the completion of mining gives the same present value as mining one ton of average-grade ore now. . I f the present values of the two s i t u a t i o n s are the same, the marginal grade i s the c u t - o f f grade. The f o l l o w i n g d e f i n i t i o n s w i l l be re q u i r e d f o r the mathematical development of the optimal c u t - o f f grade : x - marginal grade (%), p - annual production ( t o n s ) , n - l i f e of mine ( y e a r s ) , i = x + l / b - average grade (%), - v a r i a b l e o p erating costs ( $ / t o n ) , ^ - f i x e d p l a n t o p e r a t i n g costs ($/year), D - average o p e r a t i o n a l costs minus marginal mining cost ( $ / t o n ) , r - compound i n t e r e s t r a t e . The present value of mining one ton of marginal ore now i s equal t o the value per t o n of ore (20 times the c u t - o f f grade times the p r i c e per pound of contained metal) minus the cost to mine the ore. Mining cost i s equal t o v a r i a b l e o p e r a t i n g c o s t s , plus f i x e d p l a n t o p e r a t i n g costs per t o n ( f i x e d o p e r a t i n g costs per year d i v i d e d by annual p r o d u c t i o n ) , minus the cost d i f f e r e n t i a l between mining average-grade ore and mining marginal ore. This cost d i f f e r e n t i a l e x i s t s because set-up costs and development costs are not required i n the mining of marginal ore. Therefore, the present value of mining a t o n of marginal ore at the present i s : mx - (^ + (kg/p) - D) S i m i l a r l y , the present value of mining a t o n of average ore now i s equal to the value per ton of ore minus the costs of mining the ore. The value per ton of average-grade ore i s equal to 20 times the p r i c e per pound of metal times the grade of the ore, which was shown above t o equal x + l / b . The mining cost f o r average-grade ore i s equal t o f i x e d p l a n t costs per t o n pl u s v a r i a b l e o p e r a t i n g costs per t o n . Therefore, mining cost i s the same as f o r marginal ore except for- the e l i m i n a t i o n of the c o s t - d i f f e r e n t i a l component which i s due t o the lower cost of mining marginal ore. Therefore, the present value of mining a ton of average-grade ore now i s : m(x + ( l / b ) ) - ( k x + kg/p)) Obviously, the present value of mining a ton of average-grade ore at the completion of mining w i l l be the discounted value of mining a ton of average-grade ore now. Therefore, the expression f o r the present value of mining a ton of average-grade ore at the completion of mining i s : (1 + r ) - n (m(x + ( l / b ) ) - (k + ( k 2 / p ) ) where n i s the remaining mine l i f e i n years and r i s the discount r a t e . The optimal c u t - o f f grade i s defined t o be that grade at which the present value of mining a ton of marginal ore now plus a ton of average-grade ore at the completion o f mining i s equal t o the present value o f mining a ton of average-grade ore now. This may be expressed as: m(x + 2/b) - (k-L + k g/p) = mx - ( k x + kg/p - D) + (1 + r ) " n ((m (x + l / b ) ) - + kg/p)) The s o l u t i o n t o t h i s equation i n terms of optimal c u t - o f f grade, x, i s : x = (1 + r ) n ( l / b + D/m) + ( ^ + kg/p)) - 2/b m Therefore, the optimal c u t - o f f grade equals the m i n e - l i f e compounding of the d i f f e r e n t i a l between average and marginal grades minus the cost d i f f e r e n t i a l f o r mining marginal ore d i v i d e d by the p e r - u n i t value of the ore; plus the cost per t o n of mining average ore d i v i d e d by the per-u n i t value of the ore; minus the grade d i f f e r e n t i a l between average and marginal ore. Given o p e r a t i n g costs and f i x e d c o s t s , f o r v a r i o u s s i z e s of p l a n t s , i t i s p o s s i b l e to determine an optimal p l a n t s i z e by p l o t t i n g discounted p r o f i t s against c u t - o f f grade. The one fundamental shortcoming of Henning's a n a l y s i s i s t h a t i t s v a l i d i t y r e s t s on h i s assumption of an exponential tonnage-grade r e l a t i o n s h i p which may or may not be t r u e . -In the t h i r d study, V i c k e r s uses the marginal cost-marginal revenue concept from micro-economics. He addresses h i m s e l f t o the question: "what part of a m i n e r a l i z e d body of rock i s ore and what i s  waste?" His f i n a n c i a l c r i t e r i o n i s to maximize the t o t a l p r o f i t s from the orebody. . Except t h a t r a t e of p r o d u c t i o n i s assumed constant, t h i s study i s much the same as t h a t of C a r l i s l e . ^ Once again, pure competition i s assumed. Since the o b j e c t i v e i s to maximize t o t a l p r o f i t s , the c u t - o f f grade i s defined as t h a t grade at which marginal costs equal average revenue. A u n i t of ore i s defined as being one t o n of 1 per cent ore. .The p e r - u n i t value of ore i s t h e r e f o r e equal t o : 20 l b / t o n - 1% times the metal p r i c e per l b . 5 D. C a r l i s l e , - "The Economics of a Fund Resource w i t h P a r t i c u l a r  Reference to Mining", op. c i t . ( hk ) Although average costs per t o n of m a t e r i a l produced d e c l i n e u n t i l they become asymptotic to some minimal production c o s t , (at which p o i n t AC equals MC), average costs per pound of metal recovered form a t r a d i t i o n a l U-shaped cost f u n c t i o n . This U-shaped f u n c t i o n i s caused by the f a c t t h a t the decrease i n metal content of the ore exceeds the decrease i n mining costs caused by high l e v e l s of recovery. Since the marginal cost curve i s the f i r s t d e r i v a t i v e of the t o t a l cost curve, the p o i n t of maximum p r o f i t w i l l occur when the slope of the t o t a l cost curve equals the metal p r i c e . There i s nothing new i n the above theory. V i c k e r s ' greatest c o n t r i b u t i o n i s i n making the theory o p e r a t i o n a l . The f i r s t step i n the a p p l i c a t i o n i s t o develop a frequency d i s t r i b u t i o n of tonnage versus grade f o r the orebody. .An i n t e r v a l d i s t r i b u t i o n may be obtained by randomly sampling the orebody. Then, based on t h i s frequency d i s t r i b u t i o n and mine c o s t s , ( c a p a c i t y i s assumed to be f i x e d ) , p r o f i t s a f t e r t ax can b e ' c a l c u l a t e d on the tonnages a v a i l a b l e at i n c r e m e n t a l l y lower c o s t s . From t h i s data, incremental t o t a l costs and incremental revenues can be c a l c u l a t e d . D i v i d i n g these incremental costs and revenues by the a d d i t i o n a l amounts of metal produced approximates marginal cost and marginal revenue schedules. From these schedules, the p o i n t where marginal revenue equals marginal cost can be i n t e r p r e t e d . This p o i n t can be c a r r i e d i n t o the o r i g i n a l frequency d i s t r i b u t i o n to determine the c u t - o f f grade and the average ore grade. The shortcoming of t h i s approach i s th a t i t f a i l s to consider the t i m i n g of cash f l o w s . However, V i c k e r s mentions that t h i s technique i s amenable to the c o n s i d e r a t i o n of both present value and v a r i a b l e r a t e s of pr o d u c t i o n . I I I . COSTS I t i s very d i f f i c u l t t o g e n e r a l i z e on operating costs f o r mines, as costs can vary from 25 cents t o $25.00 per ton, depending on the s i z e and nature of the op e r a t i o n . By d e a l i n g only w i t h open-pit mines, the type most amenable to f i n a n c i a l o p t i m i z a t i o n , i t i s p o s s i b l e to reduce the range of ope r a t i n g c o s t s . However, i t i s s t i l l d i f f i c u l t t o g e n e r a l i z e , because of d i v e r s e s t r i p p i n g c o n d i t i o n s , rock hardness, p i t s l opes, haulage c o n d i t i o n s , equipment, and economies of s c a l e . Because i t i s d i f f i c u l t to determine expected o p e r a t i n g costs without s p e c i f y i n g e x a c t l y the above v a r i a b l e s , i t w i l l be necessary t o confine t h i s d i s c u s s i o n to cost trends w i t h i n c r e a s i n g s c a l e of ope r a t i o n s . U n l i k e most methods of underground mining, which reach maximum economies of s c a l e at a r e l a t i v e l y low r a t e of production, open p i t s achieve lowest u n i t costs at a r e l a t i v e l y h i g h mining r a t e (40,000 - 50,000 tons per day mine p r o d u c t i o n ) . This wide range i s accounted f o r not only because of increased o p e r a t i n g e f f i c i e n c y of l a r g e r ..scale equipment, but a l s o because of the reduced l a b o r i n t e n s i t y of l a r g e equipment. . In other words, f u e l , l u b r i c a t i o n , and maintenance costs as w e l l as down time do not increase p r o p o r t i o n a t e l y with.equipment s i z e . ( he ) A l s o , w i t h l a r g e r equipment, l a b o r p r o d u c t i v i t y increases more than p r o p o r t i o n a t e l y . The reason f o r f u l l economies of sc a l e being r e a l i z e d at 4 - 0 , 0 0 0 - 5 0 , 0 0 0 tons per day i s th a t t h i s i s the l e v e l , at which i t i s p r a c t i c a l to use the l a r g e s t equipment a v a i l a b l e . I f s t i l l l a r g e r equipment were a v a i l a b l e i n the f u t u r e , i t would seem reasonable t h a t f u r t h e r economies of s'eale could be achieved beyond t h i s p o i n t . The d i r e c t o p e r a t i n g cost curve could be thought to be an exponential f u n c t i o n approaching i n f i n i t y at 0 tons per day and asymptotic to some minimal cost l e v e l beyond the ra t e of 5 0 , 0 0 0 tons per day. The f o l l o w i n g t a b l e should give an idea of the degree t o which economies of s c a l e are present i n d i r e c t mining c o s t s . MINE OUTPUT DIRECT OPERATING COST (tons/day) ($/ton)  5 , 0 0 0 1 . 2 5 1 0 , 0 0 0 . 8 9 2 0 , 0 0 0 . 7 5 3 0 , 0 0 0 . 6 8 4o,ooo . 6 5 5 0 , 0 0 0 .64 The cost f i g u r e s are h y p o t h e t i c a l , however, the r a t e of change of mining costs should be r e l a t i v e l y c o n s i s t e n t w i t h r e a l i t y . • Operating costs f o r concentrators do not have economies of s c a l e t o the same extent as do mining c o s t s . As the equipment used i n l a r g e concentrators does not have as great a range as does mining equipment. The c h i e f economy of s c a l e i s i n l a b o r c o s t s . G e n e r a l l y , reagent and g r i n d i n g media consumption per t o n of feed i s f a i r l y uniform over the complete range of concentrator s i z e s . The absolute value of o p e r a t i n g costs depends upon the complexity of the ore and, i n t u r n , the amount of p r o c e s s i n g r e q u i r e d per ton of m i l l feed. A h y p o t h e t i c a l schedule of costs f o r a m i l l t r e a t i n g r e l a t i v e l y simple porphyry copper ores and producing only copper concentrates could be as f o l l o w s : MILL FEED DIRECT OPERATING COSTS (ton/day) ($/ton)  5,000 .95 10,000 .86 20,000 .77 30,000 .72 40,000 .70 50,000 .69 Once again, the important f a c t o r i s the trend r a t h e r than the absolute cost v a l u e s . Upon comparing op e r a t i n g cost schedules f o r mining and f o r c o n c e n t r a t i n g , i t becomes obvious t h a t economies of s c a l e are greater i n the mining o p e r a t i o n than i n the m i l l i n g process. C a p i t a l Costs With c a p i t a l c o s t s , as w i t h o p e r a t i n g c o s t s , i t i s d i f f i c u l t to ge n e r a l i z e on costs throughout the i n d u s t r y . As no two orebodies have the same l o c a t i o n and operating c o n d i t i o n s , a comparison of c a p i t a l costs between two operations i s a comparison of d i s s i m i l a r items. I t i s p o s s i b l e , however, to determine cost ranges f o r a s i n g l e given orebody. . At the same time, i t i s a l s o p e r m i s s i b l e t o make comparisons between i d e n t i c a l h y p o t h e t i c a l orebodies, d i f f e r i n g only i n c e r t a i n s p e c i f i e d v a r i a b l e s . An a r t i c l e g i v i n g some means of e s t i m a t i n g c a p i t a l costs was ' 6 w r i t t e n by R i d d l e . There are three stages i n the development o f a prospect i n t o an operating e n t i t y . The p r e l i m i n a r y e v a l u a t i o n determines whether or not the property warrants the development of f u r t h e r i n f o r m a t i o n . The second phase in v o l v e s a r e - a p p r a i s a l of the o r i g i n a l estimates a f t e r some work has been completed; and the t h i r d stage c o n s i s t s of making a f i n a l , complete e v a l u a t i o n , on which the request f o r funds i s based. - In the present study, the prime emphasis i s on the p r e l i m i n a r y and second steps. For a complete study, c a p i t a l costs must be based on the a c t u a l costs o f various p l a n t and equipment components. No shortcuts are p e r m i s s i b l e . Nevertheless, f o r e a r l y estimates, i t i s d e s i r a b l e t o be able to estimate c a p i t a l costs without having to r e s o r t t o great d e t a i l . What i s r e q u i r e d i s an approximation - of the optimal p l a n t s i z e . Once t h i s approximation i s determined, i t can serve as a base f o r f u r t h e r d e t a i l e d s t u d i e s . John A..Riddle, "Economic E v a l u a t i o n • o f Proposed Mining Ventures: Part I I I -- Process Development and Operating Costs", Canadian Mining Manual, i960, pp. 23-33. .Riddle s t a t e s t h a t f i r s t - s t a g e c a p i t a l cost estimates can o f t e n be determined by analogy. For example, i f an i d e n t i c a l p l a n t was b u i l t f i v e years ago f o r $5,000,000, the same p l a n t today should cost about $6,000,000. . Another method of determining c a p i t a l costs i s to use t h e 0.6 power of t h e ' r a t i o of p l a n t c a p a c i t i e s t o compensate f o r the change i n s c a l e of p l a n t . R i d d l e gives the f o l l o w i n g example. I f a 300-ton-per-day p l a n t were b u i l t f o r $2,500,000 i n 1956, the approximate cost f o r a 500-ton-per-day p l a n t b u i l t i n 1959 would be: (2.5 x 106) (170/150) (500/300) 0 , 6 = $3,900,000 The (170/150) term represents the r a t i o of Nelson R e f i n e r y Indexes f o r June, 1959? and June,-1956. Other indexes which could be used are the Engineering News Record Index o f the M a r s h a l l and Stevens Index. The recommended cost m u l t i p l i e r f o r a m u l t i - u n i t p l a n t i s the 0.9 or the 1 power, r a t h e r than the 0.6 power. When the concept of using the r a t i o of c a p a c i t i e s to a given power i s t e s t e d w i t h e m p i r i c a l data, i t i s found to be v a l i d only over a small range of p l a n t c a p a c i t i e s . Although t h i s i s the simplest technique f o r determining c a p i t a l c o s t s , a more r e l i a b l e method w i l l be discussed l a t e r . Second-phase c a p i t a l cost estimates are u s u a l l y prepared from flowsheets. The p r i c e s on s p e c i f i e d equipment may be obtained, p e r m i t t i n g an aggregate equipment cos t . A reasonable approximation of t o t a l c a p i t a l costs may be obtained by u s i n g a m u l t i p l i e r of three-or four-times the process equipment c o s t . The exact value of the m u l t i p l i e r depends upon the l o c a t i o n of the p l a n t , the type of process, and the amount of a n c i l l i a r y equipment r e q u i r e d . 7 . I t has been shown 1 t h a t there i s a l i n e a r l o g - l o g r e l a t i o n s h i p between c a p i t a l costs and c a p a c i t y . For open-pit mines ranging between 5,000 and 50}000-tons-per-day mine output the slope of the curve i s approximately: d ( l o g c a p i t a l cost per ton per day)/d(log t o n per day mine capacity) = -0.287. I t must be noted that mine c a p a c i t y includes both ore and waste produced by the mine. The absolute values of t h i s f u n c t i o n depend upon l o c a t i o n and rock character, but the slope approaches a constant regardless of.the l o g va l u e s . This hypothesis has been borne out e m p i r i c a l l y by p l o t t i n g the l o g of c a p i t a l costs f o r va r i o u s mine s t u d i e s i n B r i t i s h Columbia against the l o g of c a p a c i t y . The slope i s u s u a l l y about -.287. The same study a l s o shows c a p i t a l costs f o r copper concentrators i n the range of ^,000 t o 50,000-tons-per-day t o have a l i n e a r l o g - l o g r e l a t i o n s h i p . The slope o f t h i s f u n c t i o n i s approximately: d ( l o g c a p i t a l cost per t o n of m i l l f e e d ) / d ( l o g tons of m i l l feed per day) -.22k to -.280. This r e l a t i o n s h i p has a l s o been shown to be c o r r e c t by p l o t t i n g concentrator costs against concentrator c a p a c i t y f o r other f e a s i b i l i t y s t u d i e s . Once again, the absolute values of the log s depend upon the complexity of the ore t o be t r e a t e d and the l o c a t i o n of the concentrator. However, the slope r e l a t i o n s h i p does seem to hold f o r In an unpublished r e p o r t by a prominent Americansconsulting f i r m . copper-molybdenum p r o p e r t i e s i n B r i t i s h Columbia. There a l s o seems t o be a l i n e a r l o g - l o g r e l a t i o n s h i p between t o t a l concentrator c a p i t a l costs and concentrator c a p a c i t y . Depending upon the complexity of the ore, the slope ( d ( l o g c a p i t a l c o s t ) / d ( l o g c a p a c i t y i n tons per day)) w i l l vary between .720 and .776, w i t h the lower slope being f o r p l a n t s p r o c e s s i n g r e l a t i v e l y simple ores. This r e l a t i o n s h i p a l s o holds f o r copper-molybdenum concentrators i n B r i t i s h Columbia, although the values of the l o g c a p i t a l cost at any s p e c i f i e d tonnage depend upon the p h y s i c a l c h a r a c t e r i s t i c s of the ore. .CHAPTER TV VALUATION TECHNIQUES In any c a p i t a l expenditure d e c i s i o n i t i s necessary t o have a measure of the p r o f i t a b i l i t y of the p r o j e c t . In the general framework of c a p i t a l budgeting, the o b j e c t i v e i s to rank a l l a l t e r n a t i v e i n v e s t -ment p o s s i b i l i t i e s i n terms o f _ p r o f i t a b i l i t y , and hence, d e s i r a b i l i t y . However, a mining f i r m would be f o r t u n a t e indeed i f i t were t o have even two orebodies at a time. S u p e r f i c i a l l y , i t would appear t h a t c a p i t a l expenditure d e c i s i o n s f o r mining companies should be very simple and s t r a i g h t f o r w a r d . I f the mine v a l u a t i o n process i s looked at more c l o s e l y , i t becomes apparent t h a t each mineral deposit has associated w i t h i t s e v e r a l v a r i a b l e s , which i n t u r n produce an i n f i n i t e set of investment " p r o j e c t s " . .In t h i s myriad of a l t e r n a t i v e s there must be one p r o j e c t which i s "best" f o r the f i r m . The problem i s to i s o l a t e t h i s " b e s t " investment p o s s i b i l i t y from the others. C l e a r l y , some s o r t of measure of d e s i r a b i l i t y i s i n d i c a t e d . Several methods of determining the d e s i r a b i l i t y of a p r o j e c t are c u r r e n t l y i n use. These are: I Payback 11 Accounting Rate of Return 111 Break-even A n a l y s i s IV Methods I n v o l v i n g Discount Rates. I . PAYBACK This method i n v o l v e s computing the number of years r e q u i r e d to repay the i n i t i a l investment out of earnings. Investments w i t h s h o r t e r payback periods are deemed t o be p r e f e r r e d t o those w i t h longer payback p e r i o d s . This i s e q u i v a l e n t to saying that a d o l l a r of earning i s worth a d o l l a r of investment up to the time that the investment i s returned and nothing a f t e r t h a t time. 1 According t o Sykes, the usual p r a c t i c e i n u s i n g payback i s t o make c a l c u l a t i o n s gross of t a x . This defeats the s p i r i t of the method as c a p i t a l recoupment should be c a l c u l a t e d net of t a x . Another d i f f i c u l t y r a i s e d by Sykes i s t h a t of determining a minimum acceptable payback p e r i o d . . Although i t i s i l l o g i c a l from a decision-making p o i n t of view, many companies u s i n g payback set d i f f e r e n t c r i t e r i a f o r d i f f e r e n t types of p r o j e c t s . Sykes f e e l s t h a t the c e n t r a l problem i n u s i n g payback i s that i t i s not a method of measuring p r o f i t a b i l i t y ; i n s t e a d i t i s a crude attempt at r e f l e c t i n g r i s k . However, p r o j e c t s w i t h such a high degree of r i s k as t o warrant i g n o r i n g cash flows a f t e r c a p i t a l recoupment are r a r e indeed. Payback i s a l s o c r i t i c i z e d on the grounds t h a t i t ignores the p a t t e r n of cash flows d u r i n g the payback p e r i o d . I t would appear t h a t a company f a c i n g c a p i t a l r a t i o n i n g and many c a p i t a l p r o j e c t s would p r e f e r e a r l y earnings t o l a t e ones. This preference i s not r e f l e c t e d by A l l e n Sykes, "Analysing C a p i t a l P r o j e c t s f o r Maximum P r o f i t s " , Mining Magazine, v. 116 N. k. ( A p r i l , 1967), pp. 257-65. Payback does not consider the f a c t t h a t i n v e s t o r s r e q u i r e a r a t e of r e t u r n on t h e i r money. • I t i s p o s s i b l e f o r a p r o j e c t w i t h a short payoff p e r i o d t o break even, although a p r o j e c t w i t h a longer p e r i o d may make a good r e t u r n f o r the i n v e s t o r s . The idea that i n v e s t o r s want t h e i r money returned as r a p i d l y as p o s s i b l e i s not n e c e s s a r i l y i m p l i e d by the philosophy of investment. Many mining companies use a discounted v a r i a t i o n of the payback method to determine how soon they w i l l be able t o repay bank loans and other debt instruments a f t e r a mine begins commercial p r o d u c t i o n . However, t h i s i s not used as a prime v a l u a t i o n c r i t e r i o n . Other 2 3 d i s c u s s i o n s of payback may be found i n McLean and Q u i r i n . I I . ACCOUNTING RATE OF RETURN The accounting r a t e of r e t u r n i n v o l v e s t a k i n g the average annual earnings n e t - o f - t a x and d e p r e c i a t i o n as a percentage of e i t h e r the t o t a l or the average investment. . Since the average investment i s about one-h a l f of the t o t a l investment, the accounting r a t e of r e t u r n on the average investment would be approximately twice as great as on t o t a l investment. John G. McLean,.."How to Evaluate New C a p i t a l Investments", Harvard Business Review, v. 36, n. 6, (November-December, 1958). G. David Q u i r i n , The C a p i t a l Expenditure D e c i s i o n , (Homewood, I I I . , R. D. I r w i n , 1967)• This technique ignores the t i m i n g of cash f l o w s , g i v i n g the same weighting t o near and d i s t a n t earnings. .Another shortcoming of the book r a t e of r e t u r n i s i t s f a i l u r e t o make adjustments f o r the cost of c a p i t a l . Because earnings are not discounted, i t i s evident t h a t a pr o j e c t w i t h a 10 percent accounting r a t e of r e t u r n may or may not give earnings commensurate w i t h the d i v i d e n d payments re q u i r e d f o r a 10 percent r e t u r n on e q u i t y . This l a c k of c o m p a r a b i l i t y between the book r a t e of r e t u r n and the cost of c a p i t a l may cause improper d e c i s i o n s on the part of investment managers. I I I . BREAK-EVEN ANALYSIS 4 H.. M. Calloway introduced a technique c a l l e d Break-even A n a l y s i s " i n a 1954 a r t i c l e . This technique equates costs and revenues of p r o d u c t i o n t o determine an optimal cost s t r u c t u r e . . T h i s , i n t u r n , could conceivably permit the determination of an optimal p l a n t c a p a c i t y . The method i s mentioned merley f o r the sake of completeness as i t has many f l a w s . 5 Calloway's a r t i c l e has been commented upon by Evan J u s t . A f i r s t d i f f i c u l t y of the break-even method i s t h a t i t disregards i n t e r e s t and ammortization on the unrecouped investment. I t a l s o i m p l i e s that H.M. Calloway,' "Basic Break-even Formulas Devised t o S i m p l i f y Mine E v a l u a t i o n " , Engineering and Mining J o u r n a l , v. 155, n. 11, (November, 1954), pp. 90-92. Evan J u s t , Comment f o l l o w i n g paper by Calloway, op. c i t . The engineer can estimate costs and p r i c e s many years i n t o the f u t u r e w i t h a high degree of accuracy. There are a l s o t e c h n i c a l d i f f i c u l t i e s which reduce the range of a p p l i c a b i l i t y of the model presented i n Calloway's a r t i c l e . .The model was presented as being much more simple than other v a l u a t i o n techniques; however, i t appears that t h i s i s anything but the case. I t would seem that t h i s technique i s a p p l i c a b l e only t o s h o r t - r u n o p e r a t i n g d e c i s i o n s r a t h e r than t o c a p i t a l expenditure d e c i s i o n s . The best use to which break-even a n a l y s i s may be put i s as an aide i n d e c i d i n g when a mine should be shut down -- not on how a new mine should be opened. IV. DISCOUNTING METHODS The f o r e g o i n g v a l u a t i o n techniques (payback, accounting r a t e of r e t u r n , and break-even a n a l y s i s ) a l l have one common shortcoming i n t h a t they assume t h a t money re c e i v e d or disbursed has a constant value ( u t i l i t y ) r e g a r d l e s s of the t i m i n g of cash f l o w s . That a person r e c e i v e s l e s s s a t i s f a c t i o n from a d o l l a r which he w i l l be p a i d s e v e r a l years hence i s an accepted f a c t . Because of t h i s "time value of money", c o n s i d e r a t i o n must be given to the t i m i n g of the cash flows from a c a p i t a l p r o j e c t . The f a c t t h a t present cash flows are p r e f e r r e d to f u t u r e cash flows of equal d o l l a r value i s introduced i n t o the c a p i t a l budgeting process through the use of the discount r a t e . .An i n d i v i d u a l wishes t o r e c e i v e a c e r t a i n r e t u r n on h i s investment. • I f he r e c e i v e s l e s s than h i s expected r e t u r n on h i s money, he would be b e t t e r o f f to spend the money on consumption ra t h e r than saving i t . (This statement assumes the Keynesian idea t h a t , i n general, savings equals investment.) Conversely, i f an i n d i v i d u a l d e s i r e s t o consume now, but l a c k s the re q u i r e d cash, he' i s w i l l i n g t o pay up t o a c e r t a i n r a t e f o r the money. A company undertaking a c a p i t a l p r o j e c t i s i n much the same p o s i t i o n as the i n d i v i d u a l who wishes t o consume before he has the r e q u i r e d money. The company must e i t h e r borrow the money from outside sources,, or acquire money from the s a l e of e q u i t y or the use of r e t a i n e d earnings. Although debt.finance c o n s t i t u t e s 'the c o n t r a c t i n g of a debt i n the l e g a l sense, e q u i t y f i n a n c e does not. However, f i n a n c i n g through r e t a i n e d earnings i n v o l v e s the s o l i c i t a t i o n of funds from i n d i v i d u a l s , e i t h e r e x i s t i n g or new owners, who equate t h e i r expectations as t o earnings w i t h t h e i r r e q u i r e d r a t e of r e t u r n on t h e i r money t o determine the p r i c e which they are w i l l i n g to pay f o r t h e i r stock. I f a new investment does not appear able t o earn the r a t e r e q u i r e d by i n v e s t o r s at the current p r i c e , the lower earnings p o t e n t i a l i s r e f l e c t e d by a lower share p r i c e . Although some a t t e n t i o n has been given i n the l i t e r a t u r e t o the se p a r a t i o n of ownership from c o n t r o l , the general s u p p o s i t i o n i s th a t the duty o f management i s t o maximize the wealth of the common shareholders. I f the management of the f i r m does not do t h i s , but i n f a c t reduces i t by engaging.in low-return p r o j e c t s , severe repercussions are l i k e l y to occur, i n c l u d i n g the removal of the management. The important questions at t h i s p o i n t are those of what value the f i r m should use to incorporate i t s i n v e s t o r s ' and l e n d e r s ' time p r e f e r -ence f o r money and how t h i s value should be employed. There are s e v e r a l schools of thought on both of these questions, and the l o g i c and techniques of the various approaches w i l l be discussed l a t e r . However, the b a s i c method of i n t r o d u c i n g r e q u i r e d r a t e s of r e t u r n i s common to a l l the approaches. I f an i n v e s t o r requires, a 10 per cent r e t u r n on h i s money, i t i s obvious t h a t a d o l l a r today i s worth $1.10 one year from today. This $1.10 one year from now, i n t u r n i s equivalent to ($1.10)(l.10) = $1.2 two years from today (assuming annual compounding). • I n general $1.00 today i s worth ($1.00)(l + r ) ^ at any time t ( i n years) from today, where r i s the r e q u i r e d r a t e of r e t u r n . However, the usual concern i s not what a d o l l a r today i s worth at some f u t u r e time, but r a t h e r what a d o l l a r i n the f u t u r e i s worth today. In general, the "present v a l u e " of a d o l l a r r e c e i v e d at time t i s . $1.00)/(l + r ) * where r again i s the r a t e of r e t u r n expected. . I t i s a rare case indeed where i n v e s t o r s are r e p a i d i n one lump'.sum, the usual case being a s e r i e s of i n t e r e s t or dividen d payments. With annual payments of x^ i n year t , the present value of such a stream of income would b e S x j ( l + r) ^. There are two b a s i c methods i n which t - t the d i s c o u n t i n g technique i s used. They are the Discounted Cash Flow or I n t e r n a l Rate of Return Method and the Wet Present Value Method. Discounted Cash Flow The Discounted Cash Flow or I n t e r n a l Rate of Return Method appears t o have gained f a v o r i n the process i n d u s t r i e s such as the petroleum and chemical i n d u s t r i e s , but i t has yet to be accepted t o any degree by the mining i n d u s t r y . . The strong p o i n t of t h i s technique i s t h a t i t permits the ran k i n g of various p r o j e c t s , whereas the Net Present Value Method merely permits an " a c c e p t - r e j e c t " c r i t e r i o n on i n d i v i d u a l 8 p r o j e c t s . The ranking of p r o j e c t s i s not very important i n the mining i n d u s t r y as few companies have been f o r c e d to choose between two orebodies, both of which could be mined p r o f i t a b l y . The I n t e r n a l Rate of Return i s th a t r a t e ( r ) which equates the present values of c a p i t a l costs and n e t - o f - t a x cash f l o w s ; i.e.. t h a t value of r f o r which £ -t where Q, i s . t h e p o s i t i v e cash flow i n y e a r ' t and C, i s the negative cash t "C f l o w ( c a p i t a l cost) i n year t . Although a t t e n t i o n has been given t o the 9 " m u l t i p l e r a t e of r e t u r n " problem, t h i s d i f f i c u l t y may a r i s e only i f there i s more than one change o f s i g n i n cash flows over the l i f e of the p r o j e c t . This seldom i f ever happens i n mining ventures and i f i t does the cause i s u s u a l l y a p l a n t expansion which was not foreseen i n the G..David Q u i r i n , The C a p i t a l Expenditure D e c i s i o n (Homewood, 111.: I r w i n , 1967), p. 4-9. 9 J . H. L o r i e and L. J . Savage,- "Three Problems i n R a t i o n i n g C a p i t a l " , J o u r n a l of Business, XXVII, (October, 1955). Besides the f a c t t h a t the DCF technique takes account of the time value of money, i t i s more r e a l i s t i c than the non-discounted p r o f i t a b i -l i t y measures i n that i t i s c a l c u l a t e d net of tax and . . . uniquely permits the f u l l e v a l u a t i o n of tax concessions, such as investment grants and allowances, and the p r o v i s i o n f o r a c c e l e r a t e d d e p r e c i a t i o n . On t h i s l a t t e r p o i n t i s i t r e a l i z e d that a t e n per cent DCF r e t u r n net of tax i s q u i t e c o n s i s t e n t w i t h a conventional book r a t e of r e t u r n of l e s s than t e n per cent gross of tax? How many companies know t h i s ? In p a r t i c u l a r , how many companies using the common 15 per cent - -• 20 per cent gross-of-tax return-on-investment c r i t e r i a know t h i s ? - ^ The greatest d i f f i c u l t y i n ap p l y i n g the DCF concept i s that the computation i n v o l v e s s o l v i n g a polynomial expression, which i s best done by t r i a l and e r r o r . In v a l u a t i o n s r e q u i r i n g many time periods and i n which cash flows are v a r i a b l e s , the many re q u i r e d t r i a l - a n d - e r r o r s o l u t i o n s make the a p p l i c a t i o n of t h i s technique cumbersome. Net Present Value The mining i n d u s t r y was among the f i r s t to use c a p i t a l budgeting techniques which took account o f the time value of money. The f i r s t evidence of the use of a v a r i a t i o n of the Net Present Value Technique t o appear i n the t e c h n i c a l l i t e r a t u r e was a book e n t i t l e d The Engineer 1s  V a l u i n g A s s i s t a n t by H.D. Hoskold, f i r s t published by Longmans, Creene and Co. i n l877« The most common concepts used i n v a l u i n g m i neral 1 0 A l l e n Syk.es, "Analysing C a p i t a l P r o j e c t s f o r Maximum P r o f i t s " , Mining Magazine, v. 116, N.4, ( A p r i l , 1967), p. 263 property are the s i n g l e - r a t e or M o r k i l l Formula and the double-rate or Hoskold Formula. In g e n e r a l , the Wet Present Value concept i n v o l v e s deducting the present value of c a p i t a l expenditures from the present value of cash f l o w s . The h i s t o r i c concept of the use of NPV i n mine v a l u a t i o n and the reason f o r the mining i n d u s t r y ' s being among the f i r s t t o use d i s c o u n t i n g techniques was t h a t the NPV of a given mineral property represents a f a i r s e l l i n g p r i c e f o r the property. An extension of t h i s concept i s the State of Michigan's u s i n g the NPV of a mine as an assess-11 ment base f o r t a x a t i o n purposes. In view of the i n i t i a l concept of the NPV technique, i t was only i n c i d e n t a l that NPV represented an accept-r e j e c t c r i t e r i o n , i n t h a t a negative i n d i c a t e d s e l l i n g p r i c e showed the property to be w o r t h l e s s . •In i t s simplest form, net present value i s : NPV 0^  (1 + r ) _ t -Zct (1 + r ) - t , where t represents the t i m i n g of cash flows i n years and r represents the i n v e s t o r s ' r e q u i r e d r a t e of r e t u r n . However, since the mining i n d u s t r y developed the many v a r i a t i o n s of t h i s concept which have been and are being used, a mystique has been b u i l t up around the formulas used. Various i n d i v i d u a l s have published voluminous r e p o r t s j u s t i f y i n g one or F r a n k l i n G. Pardee, "The Michigan Mines A p p r a i s a l System", Appendix A i n Roland D. Parks, Examination and V a l u a t i o n of M i n e r a l  Property, 4 t h ed., (Reading, Mass.: Addison-Wesley P u b l i s h i n g Company, 1 9 5 7 ) , pp. 4 4 7 - 6 5 . another-of these formulas as being the "one r i g h t way" of manipulating cash flow f i g u r e s w h i le others have attempted t o r e c o n c i l e the d i f f e r e n c e 1 2 between the adherents of the va r i o u s "formulas". The mystique of the iss u e of mine value stems from an idea t h a t the m i n e r a l i n d u s t r i e s are d i f f e r e n t from the non - e x t r a c t i v e i n d u s t r i e s . Since mining companies are and have been managed by mining engineers who see t h e i r f i r m s as being d i f f e r e n t from other types of f i r m s , the c u l t has been s e l f - p e r p e t u a t i n g . V a l u a t i o n s i n the mineral i n d u s t r y d i f f e r from those of other e n t e r p r i s e s because mines and o i l w e l l s have a d e f i n i t e l i f e and cannot be considered a p e r p e t u i t y . -This r e q u i r e s t h a t i n any mineral-property e v a l u a t i o n the recovery of invested c a p i t a l i s necessary during the l i f e of the property. . C a p i t a l w r i t e o f f s f o r non-extractive i n d u s t r i e s are assumed t o be continuously ploughed back t o s u s t a i n the op e r a t i o n and pro-duce y i e l d s on the c a p i t a l i n v e s t e d . The e x t r a c t i v e or mineral i n d u s t r y , however, g e n e r a l l y attempts to r e t u r n to the i n v e s t o r both items. Where the mi n e r a l property possesses l a r g e reserves and the l i f e f a c t o r approximates a p e r p e t u i t y , the r e t u r n on .. investment i s g e n e r a l l y comparable t o th a t of i n d u s t r i a l s t o c k s . The f o r e g o i n g paragraph i n d i c a t e s the "unique s t a t u s " which people i n the mi n e r a l i n d u s t r y acceed t o the in d u s t y . Of course, the see D.H. Eldredge, " I s the Use of the Hoskold Formula J u s t i f i e d ? , "Engineering and Mining J o u r n a l " v. 1 5 0 , n. 8, (August 1 9 4 9 ) , pp. 7 2 - 4 ; F r a n k l i n . G . Pardee, - "The Hoskold Formula i s J u s t i f i e d , "Engineering and Mining Journal,v. l 6 l , ni' 1 , (January, 1 9 5 0 ) > PP« 7 6 - 7 ; A.V. C o r l e t t , " V a l u a t i o n Formulae -- Parts I and I I , "Canadian Mining  J o u r n a l , v 8 7 , n. 8 & 1 1 , (August and November, 1 9 6 7 ) ; and H . R . R i c e , "Convergence i n Present Value Computations, "Canadian Mining J o u r n a l , v. 84, n. 3, (March, 1 9 6 3 ) , pp. 5 8 - 6 2 . 1 3 L.C. Raymond i n E.H.-Robie, ed., Economics of the M i n e r a l  Industry, 3 r d . ed., .(New York: American I n s t i t u t e of Mining, M e t a l l u r g i c a l and Petroleum Engineers- Inc., 1 9 6 4 ) , p . . l 3 1 . Discounted Cash Flow Method and the simple Wet Present Value Method both consider the redemption of c a p i t a l ; however, as a matter of i n t e r e s t , the v a l u a t i o n formulae used by the mineral i n d u s t r y w i l l be d i s c u s s e d , along w i t h t h e i r assumptions and shortcomings. The Hoskold Formula. The f o l l o w i n g assumptions are made i n the development of the Hoskold Formula. . The annual p r o f i t s (not cash flows) of the f i r m are considered to c o n s i s t of two p o r t i o n s -- one being the payment to the i n v e s t o r f o r t a k i n g the " r i s k " of committing h i s money to the investment, and the r e s t being c o n t r i b u t e d t o a " s i n k i n g fund" which i s invested at a "safe r a t e " to amount to the c a p i t a l expenditure at the end of the l i f e of the venture. The r i s k reward i s compounded at a high " r i s k r a t e " , w h i l e the s i n k i n g fund i s compounded at a "safe r a t e " , theo-r e t i c a l l y the y i e l d of the s e c u r i t i e s i n which the s i n k i n g fund i s i n v e s t e d . I m p l i c i t i n t h i s assumption i s the concept t h a t a s i n k i n g fund i s b u i l t up and invested at a safe r a t e of i n t e r e s t r a t h e r than being paid out t o shareholders and used at t h e i r d i s c r e t i o n . A f t e r the mine i s depleted, t h i s s i n k i n g fund (which has been invested at a r a t e l e s s than that r e q u i r e d by the i n v e s t o r s ) can be used e i t h e r t o purchase a new property of equivalent present value or to be r e p a i d to the i n v e s t o r s upon the winding up of the company, " P r o f i t s " r a t h e r than cash flows are discounted and the formula assumes a constant annual p r o f i t . 14 The d e r i v a t i o n i s as f o l l o w s . For each $E of annual p r o f i t 14 adopted from A . V . , C o r l e t t , op. c i t . an amount Vr' i s paid as compensation to the i n v e s t o r s f o r undertaking the r i s k (where r ' i s the " r i s k r a t e " of i n t e r e s t and V i s the amount invested i n the p r o j e c t which g e n e r a l l y equals the present value of the investment) and E - Vr' i s co n t r i b u t e d to a s i n k i n g fund which earns at the "safe r a t e " , r " . Therefore, at the end of the f i r s t year (assum-i n g s i n k i n g fund payments at the end of the year) the amount of the s i n k i n g fund w i l l be: E - Vr' A f t e r the second year, the s i n k i n g fund w i l l amount t o : (E - Vr') + (E - V r ' ) ( l + r " ) . S i m i l a r l y , a f t e r the nth. year, the s i n k i n g fund w i l l amount t o : ( t - 1) (E - V r ' ) ( l + r") However, by d e f i n i t i o n , the s i n k i n g fund must amount to V, the amount i n i t i a l l y i n v e s t e d . Therefore: n v = (1 + r") - 1 (E - V r ' ) , and r " n n V + V r ' ( ( l + r") - 1) = E (( 1 + r") - l ) M u l t i p l y i n g by r " : n n Vr" + ( V r ' ) ( ( l + r") - l ) = E ( ( l + r") - l ) n n V ( r " + ( r * ( ( l + r") - 1) = E ( ( l + r") - l ) n V = E ( ( l + r") - 1) = E n r " + r ' r " + ( r ' ( l + r ) - 1) ( l + r " ) a - 1 Several shortcomings are apparent i n the assumptions and. the p r a c t i c a l i t y of the Hoskold Formula. There i s no j u s t i f i c a t i o n f o r the reasoning behind i n v e s t i n g the " s i n k i n g fund" i n a bank or i n govern-ment s e c u r i t i e s at a "s a f e " r a t e of i n t e r e s t . A b a s i c tenent of investment management i s that no money be put i n t o p r o j e c t s which earn a r a t e lower than the shareholders' r e q u i r e d r a t e of r e t u r n . Since i t i s q u i t e obvious t h a t shareholders i n a mining venture would expect a r a t e higher than t h a t provided by government bonds or other s i n k i n g fund investments, the management of a f i r m i s c l e a r l y employing sub-optimal f i n a n c i a l p o l i c y , i f the cash throwoff from a venture i s invested at a r a t e l e s s than that r e q u i r e d by the i n v e s t o r . . Should the f i r m be unable t o plough back generated cash i n t o high-earning ventures, the money should be returned to the i n v e s t o r f o r h i s personal use. The b a s i c idea of u s i n g the accumulated s i n k i n g fund to purchase a new mine i s a l s o i n o p e r a b l e . No company i s i n such a fo r t u n a t e p o s i t i o n t h a t i t can merely buy new mines when i t s o l d ones become depleted. Many firm s use the Hoskold Formula today, because of i t s " v a l i d i t y " , but do not employ a s i n k i n g fund. The concept of assuming reinvestment of cash throw-off at a safe r a t e when t h i s i s not done i s d e f i n i t e l y erroneous. Another drawback of the Hoskold Formula i s that i t considers i n t e r e s t t o be earned each year on the i n i t i a l investment r a t h e r than on the unrecouped balance. To use the terminology of the proponents o f the formula, there i s no reason why the r i s k r a t e of i n t e r e s t should be paid each year on the i n i t i a l investment, when i n l a t e r years a s u b s t a n t i a l p o r t i o n of the c a p i t a l cost i s s i t t i n g s a f e l y i n a sinking fund. A feature of the Hoskold Formula that makes i t almost impossible to use i s the assumption that c a p i t a l costs are incurred at one point of time. It i s not unusual today for a mine to be i n the preproduction stage f o r upwards of f i v e years. Also, the use of " p r o f i t s " rather than cash flows further d i s t o r t s the actual earnings picture of the operation, since depreciation i s a c t u a l l y only a book entry and not a d e f i n i t e expense of the f i r m . One of the greatest conceptual d i f f i c u l t i e s of t h i s formula i s the f a c t that there i s double counting of c a p i t a l redemption i n the i n c l u s i o n of both depreciation and the sinking fund. Several other shortcomings of the Hoskold Formula are pre-15 sented by Preston. He c r i t i c i s e s the a r b i t r a r y rates which are u s u a l l y selected f o r r' and f " . The market bond rate i s never used f o r r ' , and because of the assumptions embodied i n the formula, r " i s not d i r e c t l y r e l a t e d to return on investment. Preston states that the Hoskold Formula finds most of i t s merit i n i t s a b i l i t y to undervalue mineral property, since: "under-valuation i s a t t r a c t i v e for many purposes, e s p e c i a l l y purchase and ad valorem taxation, and the assump-t i o n of a low rate of accumulation of the sinking fund and a high degree of r i s k , . w i l l produce values of any magnitude desired". Lee E.-Preston, Exploration f o r Non-Ferrous Metals -- An Economic A n a l y s i s , (Washington: Resources f o r the Future, Inc., I960), pp. 5^-8. \ . Another shortcoming due to the mystique of the Hoskold Formula i s t h a t i t s adherents consider i t t o be something more than a mere v e h i c l e f o r manipulating data. . The use of the formula causes estimators t o o b t a i n data which compensates f o r the e f f e c t s of the formula. The concept of a s i n k i n g fund f o r the replacement of a wasting asset (the orebody) i s a l s o i r r a t i o n a l . This procedure i s analogous t o manufacturing concern's p l a c i n g i t s c a p i t a l cost allowances i n a s i n k i n fund i n order to r e p l a c e i t s p l a n t and equipment, another wasting asset C a r l i s l e f e e l s t h a t the use of a discount r a t e to r e f l e c t r i s k only serves t o compound the e f f e c t of r i s k . • In mining, the p r i n c i p a l r i s k i s g e o l o g i c a l r i s k . - Instead of i n c r e a s i n g w i t h time, as i s i m p l i c i t i n the Hoskold Formula, g e o l o g i c a l r i s k may a c t u a l l y decrease i f the mine i s operated at a l e s s r a p i d r a t e , due t o the increased opportunity f o r e x p l o r a t i o n . A high discount r a t e , r e f l e c t i n g a high degree of r i s k , tends t o imply r a p i d mining, thus i n c r e a s i n g r i s k from what i t would be i f a lower discount r a t e were used. Preston sums up h i s argument by s t a t i n g : The only d e f e n s i b l e b a s i s f o r the use of the Hoskold formula i s t h a t i t provides a standard system of discounts f o r use i n comparing a number of d i f f e r e n t a n t i c i p a t e d revenue streams; i . e . , f o r o b t a i n i n g comparable values of various p r o p e r t i e s . D. C a r l i s l e . "The Economics of a Fund Resource With P a r t i c u -l a r Reference t o Mining," The American Economic Review, XLIV, 195^, pp. 609-12. However, f o r t h i s purpose the use of the formula i s l i k e the weighing of a number of small boys on a set of i n c o r r e c t s c a l e s ; the r e s u l t i n g weights are a l l biased i n the same d i r e c t i o n and, t h e r e f o r e , t o some extent comparable, except f o r the f a c t that each of the boys i s t r y i n g t o compensate f o r the sc a l e s by c a r r y i n g an unknown number of rocks i n h i s pocket. The M o r k i l l Formula. Over h a l f a century a f t e r Hoskold developed h i s formula, D.B. -| Q M o r k i l l introduced a formula i n which a s i n g l e " r i s k " r a t e of i n t e r e s t i s a p p l i e d only t o the unrecouped p o r t i o n of the investment. .The formula i s : (the symbols having been p r e v i o u s l y d e f i n e d ) . ~ Since the formula c o i n c i d e s c l o s e l y t o the net present value of a stream of equal annual p r o f i t s r a t h e r than cash f l o w s , no f u r t h e r d i s c u s s i o n i s r e q u i r e d . Once again, there i s double counting because d e p r e c i a t i o n i s deducted t o determine p r o f i t . Lee E. Preston, op. c i t . , p. 5 8 . D.B. M o r k i l l , "Formulas f o r Mine V a l u a t i o n , "Mining and  S c i e n t i f i c P r e ss, v. 1 1 7 , p. 2 7 6 In order t o combine the many v a r i a b l e s i d e n t i f i e d i n Chapters I I and I I I , and t o observe the combined i n t e r a c t i o n of these v a r i a b l e s , i t was necessary t o construct a model of an orebody and determine the p r o f i t a b i l i t y of mining i t at various r a t e s of recovery. The f o l l o w i n g model represents t y p i c a l cost and grade c o n s i d e r a t i o n s , although no . attempt i s made t o represent any p a r t i c u l a r orebody. .1. DESCRIPTION OF THE MODEL The model took a given cumulative frequency d i s t r i b u t i o n of tonnage and c u t - o f f grade and computed a f t e r - t a x income. Because mining r a t e was a v a r i a b l e dependent upon both c u t - o f f grade and 'concentrator c a p a c i t y , mine operating costs and c a p i t a l costs per to n of pro d u c t i o n were defined as l o g - l o g f u n c t i o n s . M i l l o p e r a t i n g and c a p i t a l c o s t s , preproduction expenses and metal p r i c e s were assumed to take d i s c r e e t v a l u e s . The f i r s t o p e r a t i o n of the model was t o determine the ope r a t i n g l i f e and mining r a t e at a given tonnage and c u t - o f f grade. The pre-p r o d u c t i o n and c o n s t r u c t i o n p e r i o d was assumed to be two years, regardless of p l a n t s i z e , w i t h equal expenditures r e q u i r e d i n each year. During the period of production, net revenue was c a l c u l a t e d by deducting t o t a l costs from t o t a l gdncentrate r e c e i p t s . A f t e r - t a x income was c a l c u l a t e d by a p p l y i n g United States Tax Laws, or Canadian Tax Laws, w i t h both 1968 and pre-1968 B r i t i s h Columbia Tax Re g u l a t i o n s . The f i n a l step was t o determine e i t h e r the IRR or the Be n e f i t - t o - C o s t R a t i o of the p r o j e c t , u s i n g annual compounding. These values were determined f o r c u t - o f f grades ranging from 0.1% t o 1.0% copper, w i t h 0.1% increments, and f o r concentrator c a p a c i t i e s ranging from 5>000 t o 50,000 tons per day, i n c r e a s i n g i n 53000 ton-per-day u n i t s . A l l of these f i g u r e s were c a l c u l a t e d at metal p r i c e s on copper of 46c/, kkfi, k2ft, and h0£ per l b . I I . ASSUMPTIONS Tonnage and Grade A cumulative frequency d i s t r i b u t i o n of tonnage at various cut-o f f grades was represented by the exponential f u n c t i o n : tonnage at cut-6 (2-^x) o f f x (% Cu.) = kO x 10 e^ . This f u n c t i o n d e f i n e s the tonnage at a ,k% c u t r o f f grade t o be kO m i l l i o n tons. The tonnage at any s p e c i f i e d c u t - o f f grade i s an exponential f u n c t i o n v a r y i n g i n v e r s e l y w i t h the c u t - o f f grade. By i n t e g r a t i n g to determine the average grade of the ore defined at a c u t - o f f of x% Cu., i t i s found that the average grade equals x + ( l / 5 ) . The cumulative frequency d i s t r i b u t i o n of tonnage i n terms of. c u t - o f f grade and average grade i s shown i n F i g . Product and C a p i t a l Markets Assumptions about the product and c a p i t a l markets were e l i m i n a t e d by the f o l l o w i n g two methods. The product p r i c e was permitted to vary and r e s u l t s were determined f o r copper p r i c e s of kOfi, k2$, hhtf, and 46^ per l b . As the purpose of t h i s study i s not to f o r e c a s t metal p r i c e s , . conclusions w i l l be drawn, based upon the va r i o u s p r i c e s which could o b t a i n , r a t h e r than on th'e p r i c e s which are expected to p r e v a i l . A s s e s s i n g the c a p i t a l market proved t o be a much more d i f f i c u l t t a s k . I t i s almost impossible t o determine what the cost of c a p i t a l would be f o r a new f i r m . However, i f i t i s assumed th a t the new mining venture i s t o be financed by an e s t a b l i s h e d f i r m or a partn e r -s h i p of e s t a b l i s h e d f i r m s , a trend which i s growing r a p i d l y , the cost of c a p i t a l which i s appropriate i s the cost of c a p i t a l of the e s t a b l i s h e d f i r m or f i r m s . This s u p p o s i t i o n assumes th a t the new o p e r a t i o n w i l l not change the r i s k character of the o l d e r f i r m or f i r m s . P r o f e s s o r •Merrett"'" suggests t h a t a r a t e of r e t u r n f o r "normal r i s k investments financed by more than 70$ r e t a i n e d earnings" would be 2 about 9%. I f the M o d i g l i a n i and M i l l e r p r o p o s i t i o n t h a t , even w i t h ta x e s , the cost of c a p i t a l i s not g r e a t l y " a f f e c t e d by a "reasonable" degree of leverage, can be accepted, i t seems sound t o p o s t u l a t e a 9% weighted-average cost of c a p i t a l . .Although t h i s might s u f f i c e f o r the present h y p o t h e t i c a l example, i n a c t u a l p r a c t i s e , the cost of c a p i t a l should be determined i n a d e t a i l e d manner, i n order to make the o p t i m i z a t i o n meaningful. A.J.. M e r r e t t , • "Targetry and Cut-off Rates", Mining Magazine, CXVI, ( A p r i l , 1967), pp. 267-77. 2 Franco M o d i g l i a n i and Merton H.. M i l l e r , "The Cost of C a p i t a l , C o r p o ration Finance, and the Theory of Investment", The American Economic Review, X L V I I I , (June, 1958), pp. 261-97, w i t h subsequent comments, r e p l i e s , and c o r r e c t i o n s . v a r i o u s t a x systems on mining p r o j e c t s , the i n t e r n a l r a t e s of r e t u r n f o r the v a r i o u s p r o j e c t s were determined. This technique proved to be i n f e a s i b l e f o r o p t i m i z a t i o n as i t i s the present value per d o l l a r invested which i s the d e c i s i o n c r i t e r i o n r a t h e r than the i n t e r n a l r a t e 3 of r e t u r n . C a r l i s l e s t a t e s t h a t u s i n g a high discount r a t e adversely a f f e c t s the o p t i m a l i t y of the c a p i t a l expenditure d e c i s i o n . Since, by d e f i n i t i o n , the most p r e f e r a b l e a l t e r n a t i v e s w i l l have the highest i n t e r n a l r a t e s of r e t u r n , and, i n t u r n , the highest discount r a t e s , they w i l l not p r o p e r l y r e f l e c t optimal investment s t r a t e g y . Neverthe-l e s s , the i n t e r n a l r a t e of r e t u r n does represent a v a l u a b l e t o o l f o r determining the impact of t a x a t i o n on the p r o f i t a b i l i t y • o f a given orebody. P r i c e L e v e l The e f f e c t of p r i c e l e v e l changes had been e l i m i n a t e d by i n t o -ducing the assumption that the percentage change i n f a c t o r costs w i l l equal the percentage change i n product p r i c e . Although t h i s assumption has not been s t r i c t l y c o r r e c t , the i n c r e a s i n g s u b s t i t u t a b i l i t y of other commodities f o r copper should cause the hypothesized r e l a t i o n s h i p to h o l d to a greater degree than i t has i n the p a s t . Therefore, a l l costs D. C a r l i s l e , "Maximum T o t a l Recovery Through Mining High-grade and Low-grade Ore Together i s Economically Sound", Canadian I n s t i t u t e of Mining and M e t a l l u r g y B u l l e t i n , XLIV, (January, 1 9 5 3 ) , pp. 2 5 & 2 6 . J and p r i c e s w i l l be viewed as the present-day equivalent of any f u t u r e cash flows which may o b t a i n . Tax S t r u c t u r e The a n a l y s i s of t h i s h y p o t h e t i c a l mining p r o j e c t has been conducted under three d i f f e r e n t sets of tax laws. C a l c u l a t i o n s were made at various copper p r i c e s u s i n g United States tax laws and Canadian tax laws. The Canadian tax laws were a p p l i e d u s i n g both pre-1968 and post-1968 B r i t i s h Columbia taxes. The reason f o r c a l c u l a t i n g p r o f i t a b i l i t y under three tax laws was to determine the impact of t h i s l e g i s l a t e d v a r i a b l e upon the p r o f i t a b i l i t y of the f i r m and upon optimal p l a n t s i z e . A b r i e f r e c a p i t u l a t i o n of the tax laws f o l l o w s . United States t ax laws. Under United States t ax laws the e l e c t i o n was made to ammortize e x p l o r a t i o n and pl a n t expenditures on a per-ton-of-production b a s i s . A d e p l e t i o n allowance of 15% of gross revenue or 50% of income taxable at the f e d e r a l l e v e l (before d e p l e t i o n ) , which ever was s m a l l e r , was taken. State taxes were assumed t o be 5% of income a f t e r d e p l e t i o n and were deducted i n order to c a l c u l a t e income taxable at the f e d e r a l l e v e l . The f e d e r a l tax r a t e s used were 25% on the f i r s t $25,000 of taxable income and 48% of income i n excess of $25,000. The e l e c t i o n to ammortize the e x p l o r a t i o n and c a p i t a l expenditures, r a t h e r than u s i n g a d i m i n i s h i n g balance r a t e of'twice the annual ammor-t i z e d r a t e f o r c a p i t a l expenditures and t a k i n g the f i r s t $400,000 of e x p l o r a t i o n expenditures at $100,000 per year, may have reduced the p r o f i t a b i l i t y of the ope r a t i o n , but not by a s i g n i f i c a n t amount. Pre-production development expenditures were a l s o charged against f u t u r e production. Canadian tax laws. Canadian taxes were ap p l i e d i n the same manner as f o r any corporate concern w i t h the f o l l o w i n g exceptions. The p r o f i t s were not taxed at the f e d e r a l l e v e l f o r the f i r s t three years of production. F o l l o w i n g t h i s p e r i o d , preproduction costs were expensed to the extent of income u n t i l they were completely w r i t t e n o f f . A f t e r preproduction expenses were taken, c a p i t a l cost allowances were charged, e i t h e r t o the extent of income or on a 30$ d e c l i n i n g - b a l a n c e b a s i s , whichever was sm a l l e r . A d e p l e t i o n allowance of o n e - t h i r d of income normally t a x a b l e at the f e d e r a l l e v e l was a l s o taken a f t e r de-du c t i n g B r i t i s h Columbia taxes as an expense. The B r i t i s h Columbia tax laws which were used t o compute cash flows i n the pre-1968 cases i n v o l v e d a l l o w i n g a three-year t a x - f r e e p e r i o d and paying $25,000 l e s s than 10$ of net income, l e s s a p l a n t allowance of 8$ of o r i g i n a l c a p i t a l c o s t . In the 1968 case, the p l a n t allowance was computed on the b a s i s of an 8$ d i m i n i s h i n g balance; the t a x - f r e e p e r i o d was e l i m i n a t e d ; and the tax was c a l c u l a t e d at 15$ of the tax base, which was net revenue minus p l a n t allowance. C a p i t a l Costs. Since the s t r i p p i n g r a t i o was assumed t o be a f u n c t i o n of cut-o f f grade, c a p i t a l costs f o r the p i t and the concentrator were deter-mined s e p a r a t e l y . This was necessary because the mine must handle both ore and waste, as defined by the s t r i p p i n g r a t i o , while the concentrator processes only ore. The c a p i t a l cost requirements f o r the p i t were defined as f o l l o w s : l o g ( c a p i t a l cost/ton) = l o g ( 2 9 5 ) - .287 l o g (mine c a p a c i t y / 1 0 , 0 0 0 ) ; C a p i t a l cost = ( c a p i t a l cost/ton) x (mine c a p a c i t y ) . This i s a^somewhat awkward way of d e f i n i n g c a p i t a l cost f o r the mining o p e r a t i o n , but a continuous f u n c t i o n was r e q u i r e d , and l o g - l o g f u n c t i o n s cannot be defined by the s l o p e - i n t e r c e p t method. Because concentrator c a p a c i t y i s an independent v a r i a b l e , c a p i t a l costs f o r the m i l l i n g I f a c i l i t y may be expressed as each m i l l i n g c a p a c i t y having a d i s c r e e t c a p i t a l - c o s t v a l u e . schedule used i n the example i s as f o l l o w s : M i l l Capacity (tons/day) C a p i t a l Costs ($). 5 0 0 0 7 , 5 0 0 , 0 0 0 1 0 0 0 0 1 3 , 0 0 0 , 0 0 0 1 5 0 0 0 1 8 , 0 0 0 , 0 0 0 2 0 0 0 0 2 2 , 0 0 0 , 0 0 0 2 5 0 0 0 2 6 , 5 0 0 , 0 0 0 30000 3 0 , 3 0 0 , 0 0 0 3 5 0 0 0 3 4 , 0 0 0 , 0 0 0 4 0 0 0 0 3 7 , 8 0 0 , 0 0 0 4 5 0 0 0 4 1 , 5 0 0 , 0 0 0 5 0 0 0 0 4 5 , 0 0 0 , 0 0 0 The c a p i t a l cost schedule i s f o r copper concentrators t r e a t i n g r e l a t i v e -l y simple porphyry ores. Preproduction Costs E x p l o r a t i o n , d r i l l i n g and bulk sampling costs were stated at $ 1 , 5 0 0 , 0 0 0 . Preproduction s t r i p p i n g was assumed to cost $ 1 8 0 , 0 0 0 f o r the f i r s t 5 0 0 0 ton-per-day concentrator c a p a c i t y , and $ 1 0 0 , 0 0 0 f o r each a d d i t i o n a l ^000 tons-per-day c a p a c i t y . Although s t r i p p i n g costs should be t i e d tosimine c a p a c i t y , the f a c t that they have been expressed as a f u n c t i o n of m i l l c a p a c i t y w i l l not m a t e r i a l l y a f f e c t the r e s u l t s of the study. Operating Costs The schedule of mine op e r a t i n g costs per t o n of mine production s p e c i f i e d i n Chapter 3 was an exponential f u n c t i o n . However, i t was discovered that the f u n c t i o n can be c l o s e l y approximated ( t o w i t h i n 1 . 5 % ) by u s i n g a l o g - l o g f u n c t i o n . Since mine c a p a c i t y i s a v a r i a b l e dependent upon s t r i p p i n g r a t i o and m i l l c a p a c i t y , i t i s necessary that mine op e r a t i n g costs be a continuous f u n c t i o n . The f o l l o w i n g f u n c t i o n was used. lo g ( m i n i n g costs/ton) = l o g 1 . 1 9 - 0 . 2 6 9 4 l o g (mine cap-a c i t y / 5 0 0 0 ) . Since concentrator c a p a c i t y was an independent v a r i a b l e , the f o l l o w i n g schedule of costs was used: 5000 . .95 10000 .89 15000 .80 20000 .77 25000 .74 30000 .72 35000 .71 4 0 0 0 0 .70 45000 .69 50000 .69 These co n c e n t r a t i n g costs were derived under the assumption t h a t the ore was r e l a t i v e l y simple and that copper concentrate was the only product to be produced. In a d d i t i o n to mine and concentrator operating c o s t s , s e v e r a l other d i r e c t costs would be i n c u r r e d . These costs i n c l u d e concentrate t r a n s p o r t a t i o n , smelting and r e f i n i n g charges, and copper l o s s due t o imperfect m i l l e f f i c i e n c y . A l l of these costs were approximated by deducting 70 per l b . from the market p r i c e of copper i n order to deter-mine a p r i c e f.o.b. the concentrator. L e v e l of Recovery The l e v e l of recovery of the operation was v a r i e d , w i t h the d e f i n i n g f u n c t i o n being the c u t - o f f grade. Although no attempt was made t o define the percent e x t r a c t i o n of the metal i n p l a c e , i t would be r e l a t i v e l y low, due t o the f a c t that the orebody was defined by an exponential f u n c t i o n . Another f a c t o r v i t a l to the l e v e l of recovery a t t a i n e d i s the s t r i p p i n g r a t i o (tons of waste per ton or o r e ) . Since the s t r i p p i n g r a t i o depends upon the p h y s i c a l surroundings of the ore, the s t r i p p i n g r a t i o was defined by s e t t i n g i t equal to fou r times the c u t - o f f grade. This l i n e a r r e l a t i o n s h i p may have a somewhat greater r a t e of change than what might a c t u a l l y be encountered. The e f f e c t of the r a p i d r a t e of change of s t r i p p i n g r a t i o w i l l be to cause the optimal c u t - o f f grade to be somewhat lower than might be expected. However, i t must be state d t h a t i t i s e n t i r e l y p o s s i b l e that a s t r i p p i n g r a t i o such as that s p e c i f i e d could be found i n a c t u a l p r a c t i s e . Rate of Recovery The concentrator c a p a c i t y was permitted t o increase i n 5000-ton-per-day u n i t e s from 5000 to 50,000 tons per day. The r e s u l t s of the c a l c u l a t i o n s can be i n t e r p o l a t e d to values between the d i s c r e e t values which are obtained at the po i n t s at which data i s s p e c i f i e d . I I I . COMPUTER FLOWCHARTS Because of the l a r g e number of c a l c u l a t i o n s r e q u i r e d , i t was necessary t o use a computer to determine I n t e r n a l Rates of Return and Ben e f i t - t o - C o s t R a t ios f o r the va r i o u s cut-off-grade - - concentrator-c a p a c i t y -- me t a l - p r i c e combinations. Programming was done i n the FORTRAN IV language and c a l c u l a t i o n s were done w i t h the IBM 36O computer at the U n i v e r s i t y . o f B r i t i s h Columbia. Separate programs were w r i t t e n f o r the three tax systems con-s i d e r e d . These programs were adapted t o determine e i t h e r I n t e r n a l Rate of Return or B e n e f i t - t o - C o s t R a t i o . The f l o w - c h a r t s which were used are presentee/ /n Chapter. VI. IV. RESULTS The purpose of t h i s example i s two-fold; t o determine a normative r e l a t i o n s h i p between the v a r i a b l e s having an impact on the choice of an optimal p l a n t s i z e , and to compare the impacts of t a x a t i o n systems on p r o f i t a b i l i t y and optimal p l a n t s i z e . The f i r s t o b j e c t i v e can be met by a n a l y z i n g the trends i n b e n e f i t - t o - c o s t r a t i o s over changing metal p r i c e , c u t - o f f grade, and concentrator c a p a c i t y ; and the second by examining trends i n i n t e r n a l r a t e s of r e t u r n under the same c o n d i t i o n s . The best was t o present the r e s u l t s obtained from the h y p o t h e t i c a l example was t o construct contour p l o t s of i n t e r n a l r a t e of r e t u r n or b e n e f i t - t o - c o s t r a t i o over v a r y i n g concentrator c a p a c i t i e s and c u t - o f f grades. Because of the number of v a r i a b l e s i n v o l v e d , i t was necessary t o construct a contour p l o t f o r each metal p r i c e - - t a x system combination.. Although the p l o t s were constructed from computer p r i n t - o u t s , the con-t o u r i n g and i n t e r p o l a t i o n was done without the a i d of a computer. Because of t h i s , c e r t a i n i n c o n s i s t e n c i e s may e x i s t between p l o t s . How-ever, s u f f i c i e n t p l o t s were produced to allow the v i s u a l e l i m i n a t i o n of i n c o n s i s t e n c i e s . The r e s u l t s obtained from the example appear i n the f o l l o w i n g f i g u r e s . Optimal Cut-Off Grades at S p e c i f i e d P l a n t C a p a c i t i e s . Some important c o n s i d e r a t i o n s i n a n a l y z i n g the r e s u l t s obtained from the example are the examination of the trends over v a r y i n g p r i c e , concentrator c a p a c i t y , • and c u t • o f f grade; and the ex p l a n a t i o n of the cont r a s t s , between'the three, cases. Tables 5-1 through 5-6 show the optimal, c u t - o f f grades under s p e c i f i e d concentrator c a p a c i t i e s at d i f f e r e n t copper p r i c e s . There are -separate t a b l e s f o r each t a x system considered, w i t h o p t i m i z a t i o n i n terms of both i n t e r n a l r a t e of r e t u r n and b e n e f i t - t o - c o s t r a t i o . The optimal c u t - o f f grades f o r given p r i c e s and concentrator c a p a c i t i e s were determined by f i n d i n g the p o i n t w i t h the maximum i n t e r n a l r a t e of r e t u r n or b e n e f i t - t o - c o s t r a t i o f o r a given p r i c e , t a x system and concentrator c a p a c i t y . This was done by v i s u a l i n s p e c t i o n of the preceding contour p l o t s ; and, consequently, i s subject t o two sources of e r r o r . The f i r s t source of e r r o r i s th a t the p r e c i s i o n of the contour p l o t s i s not s u f f i c i e n t l y good t o warrant the r e p o r t i n g of c u t - o f f grades t o two s i g n i f i c a n t f i g u r e s . However, the second decimal place does permit the i d e n t i f i c a t i o n of small changes i n optimal c u t - o f f grade, which can be seen on the p l o t s . The second source of e r r o r i s caused by the v i s u a l i n t e r p o l a t i o n of optimal c u t - o f f grades from the p l o t s . The tr e n d of i n t e r n a l r a t e of r e t u r n or b e n e f i t - t o - c o s t r a t i o over v a r y i n g c u t - o f f grades at a given metal p r i c e and concentrator c a p a c i t y i s n e i t h e r l i n e a r nor symmetrical, so the maximum p o i n t had t o be determined through i n t u i t i v e means. Nevertheless, by c o n s i d e r i n g values over the four p r i c e s used, i t i s p o s s i b l e t o e l i m i n a t e data which i s i n c o n s i s t e n t w i t h the t r e n d s . Figure ** 8 <5 X K 4 i I V 4 I ! 1 ^ I .1 5 Oi Co t-» *0 <<} V CO CM v» « Vi 2 <s X*. •v 1 N 1 (3 O ^ \i Q O C> O <J» « B <0 Vj >t <•) « J O S cS >^ o Cofi/Toux PLOT OFBEA/EF/T-IO- COST PATIO I/S. Cur-OFF C/PAD£ VS. COA/CSA/TA?A7V/? CAPACITY 42? Co. 0 5 fO IS SO S5 30 JS 40 45 SO CONCENTRATOR CAPACITY ('/ooo 7*»s/ 1.0, OS-0-8\ V. U 0-6 \ ^ 03-0 1 Co/vrouR PLOT of B£NEF/T-7O-COST RAT/O VS. COT- Orr CQAOF VS. COSSCEMT~/?ATO# CAPACITY To„sA7-x*40ecs'sx) N o /O /& SO B5 30 3S 40 46 SO COMCEMTPATOA? CAPACITY ('/ooo 7W/DAV) 10, 0.3 08 —s ^ 0.7 01 COA/TOUR PLOT or BEVV'FF/ r-7b - COST PA T/O VS. CUT- OFF CQADF VS. COA/C£A/TA?ATOA? CAPACITY 46?Cu. CAVA O/AU TAX LAWS - /966? £. C. TAX^S 3 % Co'r>povna'//>y 1.0 '•es  VO — c; OA '•s ~~ re I-7S 1-9 IO /5 20 25 30 SS 40 4S CoA/C£A/r/?ATo& CAPACITY OOOO TS^S/DAV) so • 3 NO fe 2 F £ ^ 1 i i— Of < o «0 c5 r-6 6 <3 ("OVoJ 2<7 V&£} JJQ -J. 3 i t u V i l ! 11 li i K <0 I I «0 N 6 Q / I V. I l l ] I I / / IS 0 X i ! O O) CO K <0 ' <C ' * ° > Q < ' 6 o <S © o < S Q Q O s« I f 0 ft «V1 V 5 N. KO 6 6 are *25 ,1 > kj I * v a. TABLE 5-1 OPTIMAL CUT-OFF GRADES AT SPECIFIED PLANT CAPACITIES  INTERNAL RATE OF RETURN  CANADIAN TAX LAWS -- D.g68 B.C. TAXES PLANT COPPER'PRICE (0/lb.) CAPACITY (ton/day) 400 420 440 460 5000 0.54$ 0.58$ 0.60$ 0.60$ 10000 0.48$ 0.50$ 0.52$ 0.52$ 15000 0.43$ o.44$ 0.46$ 0.47$ 20000 0.1+0$ o.4o$ 0.42$ 0.43$ 25000 0.37$ 0.38$ 0.38$ 0.39$ 30000 0.34$ 0.35$ 0.36$ 0.37$ 35000 0.31$ 0.32$ 0.33$ 0.36$ 40000 0.29$ 0.30$ 0.31$ 0.34$ 45000 0.27$ 0.28$ 0.29$ 0.32$ 50000 0.25$ 0.27$ '0.28$ 0.31$ The most obvious f a c t o r which i s shown by the t a b l e s i s t h a t optimum c u t - o f f grade increases w i t h i n c r e a s i n g concentrator c a p a c i t y . Upon c l o s e r examination of both the t a b l e s and the shapes of the contour p l o t s , i t becomes evident t h a t t h i s decrease i n optimal c u t - o f f grade i s not l i n e a r ; but i s , q u i t e p o s s i b l y , assymptotic t o the cut-off-grade a x i s . In other words; t h e o r e t i c a l l y , optimum c u t - o f f grade approaches i n f i n i t y as concentrator c a p a c i t y approaches zero. This trend i n c u t - o f f grade r e f l e c t s economies of s c a l e which are more pronounced i n the lower ranges than i n the higher ranges of p l a n t c a p a c i t y . Therefore, i t i s p o s s i b l e to reach a higher percent e x t r a c t i o n or l e v e l of recovery at a higher concentrator c a p a c i t y . A second c o n d i t i o n which i s obvious from the t a b l e s i s the f a c t t h a t optimal c u t - o f f grades under United States tax laws are about 0.15% lower than those determined under Canadian tax laws. This i s probably due t o the ammortization of c a p i t a l and preproduction costs on a per-t o n - o f - p r o d u c t i o n b a s i s . Mining t o a lower c u t - o f f grade has the e f f e c t of i n c r e a s i n g ore r e s e r v e s . I t i s p o s s i b l e t h a t cost reductions i n -curred by spreading c a p i t a l and preproduction costs over a l a r g e r tonnage coupled w i t h mining cost reductions due to a lower s t r i p p i n g r a t i o tend t o o f f s e t reductions i n revenue caused by the lower metal content of the ore. . Under Canadian tax laws, the recapture of c a p i t a l and preproduc-t i o n costs i s based on time r a t h e r than tonnage of r e s e r v e s . A f t e r the i n i t i a l t a x - f r e e p e r i o d , preproduction costs are w r i t t e n o f f to the extent of revenue, f o l l o w i n g which, p l a n t costs are recouped on the b a s i s of a d e c l i n i n g balance of approximately 30%. I t i s apparent that these three p r o v i s i o n s tend t o s h i f t cash flows toward the present. The e f f e c t of t h i s s h i f t i n cash flow i s t o place greater emphasis on pro-f i t s d u r i n g the e a r l y years of p r o d u c t i o n which are not subject t o f e d e r a l t a x a t i o n . In order to maximize p r o f i t s d u r i n g the e a r l y - y e a r s , a high c u t - o f f grade i s e s t a b l i s h e d , at the expense of ore r e s e r v e s , and hence,mine l i f e . T he.effect of Canadian t a x a t i o n on optimal c u t - o f f grade may be seen by comparing the change i n optimal c u t - o f f grades brought about by the change i n requirements of the B r i t i s h Columbia mining t a x laws. In 1968, three major changes were made t o the B r i t i s h Columbia mining tax laws. The f i r s t change was t o e l i m i n a t e the three-year t a x - f r e e p e r i o d w i t h respect to p r o v i n c i a l mining t a x e s . Secondly, p l a n t allowance was changed from an 8% f l a t r a t e to an 8% d e c l i n i n g balance, commencing w i t h the s t a r t of mining. And t h i r d l y , the r a t e of tax was increased from 10% to 15%. These changes have the e f f e c t of s h i f t i n g cash flows t o the f u t u r e , i n a d d i t i o n t o i n c r e a s i n g p r o v i n c i a l tax l i a b i l i t y . This s h i f t i n cash flow i s more n o t i c e a b l e i n high-production, s h o r t - l i v e d operations than i n low-production, l o n g - l i v e d mines. Because of the f a c t that cash flows near the present are reduced t o a greater extent than cash flows i n the d i s t a n t f u t u r e , i t would be l o g i c a l to assume th a t the 1968 case should place l e s s emphasis on maximizing p r o f i t s d u r i ng the f i r s t years of mine l i f e and more emphasis on a c h i e v i n g a higher e x t r a c t i o n . This i s borne TABLE 5-2 OPTIMAL CUT-OFE GRADES AT SPECIFIED PLANT CAPACITIES  INTERNAL RATE OF RETURN  CANADIAN TAX LAWS -- PRE-1968 B. C. TAXES PLANT COPPER PRICE (0/lb.) CAPACITY ' -(tons/day) 400 420 440 460 5000 0.58$ 0.61$ 0.62$ 0.64$ 10000 0.50$ 0.52$ 0.57$ 0.57$ 15000 0.46$ 0.48$ 0.50$ 0.51$ 20000 0.43$ 0.44$ 0.44$ 0.46$ 25000 o.4o$ o.4o$ 0.4i$ 0.42$ 30000 0.37$ o.37$ 0.38$ o.4o$ 35000 0.35$ 0.35$ 0.37$ 0.38$ •40000 0.32$ 0.34$ 0.35$ 0.37$ 45000 0.30$ 0.33$ 0.34$ 0.35$ 50000 0.29$ 0.32$ 0.33$ 0.34$ TABLE 5-3 OPTIMAL CUT-OFF GRAPES AT SPECIFIED PLANT CAPACITIES  INTERNAL RATE OF RETURN UNITED STATES TAX LAWS PLANT COPPER PRICE (0/lb.) CAPACITY (tons/day) kOfi 420 kkfi k6<£ 5000 — 0.43% 0.46% 0.47% 10000 0.30% 0.36% ' 0.38% 0.39% 15000 0.27% 0.29% 0.33% 0.33% 20000 0.25% 0.25% 0.28% 0.28% 25000 0.23% 0.22% 0.24% 0.25% 30000 0.20% 0.19% 0.21% 0.23% 35000 0.17% 0.17% 0.19% 0.21% 40000 0.15% 0.16% 0.18% 0.20% 45000 0.13% 0.14% 0.16% 0.19% 50000 0.10% 0.12% o . i 4 % 0.18% ( n o ) out i n the examples by the f a c t t h a t the 1 9 6 8 case has lower optimal c u t - o f f grades than the p r e - 1 9 6 8 case f o r given p l a n t s i z e s and metal p r i c e s . .Another f a c t o r which warrants f u r t h e r examination i s the p r i c e s e n s i t i v i t y of c u t - o f f grade at various concentrator c a p a c i t i e s . One of the most s t r i k i n g f eatures shown i n the t a b l e s i s the r e v e r s a l of p r i c e s e n s i t i v i t y between programs maximizing i n t e r n a l r a t e of r e t u r n and those maximizing b e n e f i t - t o cost r a t i o s With i n t e r n a l r a t e of r e t u r n , optimal c u t - o f f grade increases w i t h p r i c e f o r a given concentrator c a p a c i t y ; whereas, w i t h b e n e f i t - t o - c o s t r a t i o n , optimal c u t - o f f grade decreases w i t h p r i c e when concentrator c a p a c i t y remains f i x e d . This i s due t o the i m p l i c i t assumption i n determining i n t e r n a l r a t e of r e t u r n t h a t cash throw-off can be r e i n v e s t e d at the i n t e r n a l r a t e of r e t u r n , which i n some cases i s as high as 35% • Therefore i t i s l o g i c a l t o assume th a t w i t h the i n t e r n a l r a t e o f r e t u r n , c u t - o f f grade should r i s e w i t h p r i c e , because of the u n r e a l i s t i c a l l y high discount r a t e . With hig h discount r a t e s , the present worth of f u t u r e d o l l a r s of income i s con s i d e r a b l y lower than i t i s w i t h the low r a t e s used i n deter-mining the b e n e f i t - t o - c o s t r a t i o . Thus, the e f f e c t of a high i n t e r n a l r a t e of r e t u r n on a p r o j e c t i s t o reduce mine l i f e and to increase cut-o f f grade, i n order to s h i f t cash flow to the present. As i t i s normally impossible to r e i n v e s t cash throw-off at many of the i n t e r n a l r a t e s of r e t u r n c a l c u l a t e d i n the example, the trends i n c u t - o f f grade over changes i n metal p r i c e are not i n d i c a t i v e of what would be done i n a c t u a l p r a c t i c e . TABLE 5-4 OPTIMAL CUT-OFF GRADES AT SPECIFIED PLANT CAPACITIES  BENEFIT-TO-COST RATIO  CANADIAN TAX LAWS — 1968 B. C. TAXES PLANT CAPACITY ' (tons/day) 400 420 440 460 5000 0.58$ 0.57$ o.55$ 0.53$ 10000 0.48$ 0.44$ 0.45$ 0.43$ 15000 o.4i$ 0.38$ 0.38$ 0.36$ 20000 0.36$ • 0.34$ 0.34$ 0.31$ 25000 o.33$ 0.32$ 0.32$ 0.28$ 30000 0.31$ 0.30$ 0.29$ 0.27$ 35000 0.30$ 0.29$ 0.28$ 0.26$ 4oooo 0.28$ 0.28$ 0.27$ 0.25$ 45000 0.27$ 0.27$ 0.25$ 0.24$ 50000 0.26$ 0.26$' 0.24$ 0.22$ An i n t e r e s t i n g s i d e l i g h t to t h i s problem may be seen i n the p r i c e s e n s i t i v i t y w i t h respect t o c u t - o f f grade under United States t ax laws (Table 5-6). In t h i s t a b l e , optimal c u t - o f f grade i s higher than what would be expected i n the range where b e n e f i t - t o - c o s t r a t i o i s l e s s than 1. This happens because cash throw-off i s considered to be r e -invested at a r a t e higher than the p r o f i t a b i l i t y of the p r o j e c t . There-f o r e i t i s advantageous to mine t o a high c u t - o f f grade at the expense of a sho r t e r mine l i f e . A measure of the p r i c e s e n s i t i v i t y of c u t - o f f grade can be given by the change i n optimal c u t - o f f grade at a given p l a n t c a p a c i t y i n response to a given metal p r i c e change. In Table 5 _1 5 the change i n the c u t - o f f grade which maximizes the i n t e r n a l r a t e of r e t u r n under Canadian taxes w i t h 1968 B r i t i s h Columbia taxes over a 6ci p r i c e range i s between 0.02% and 0.06%, w i t h greatest s e n s i t i v i t y at high and low pl a n t c a p a c i t i e s . In Table 5-2, which i s the Canadian case under pre-1968 B r i t i s h Columbia taxes, the p r i c e s e n s i t i v i t y of the c u t - o f f grade which maximizes i n t e r n a l r a t e of r e t u r n i s almost i d e n t i c a l to the s e n s i t i v i t y w i t h 1968 B r i t i s h Columbia ta x e s . I n Table 5-3 which d e p i c t s c u t - o f f grades which maximize i n t e r n a l r a t e of r e t u r n w i t h American taxes, the p r i c e s e n s i t i v i t y v a r i e s from 0.09% t o 0.02%, w i t h greatest., s e n s i t i v i t y at extremes of p l a n t c a p a c i t y . The higher p r i c e s e n s i t i v i t y w i t h United States taxes may be due t o the f a c t t h a t i n t e r n a l r a t e s of r e t u r n vary from 0% to 16%, thus p r o v i d i n g a p r o p o r t i o n a t e l y great change i n the discount r a t e . Table 5-4 shows optimal c u t - o f f grades at s p e c i f i e d p l a n t TABLE 5j-5 OPTIMAL CUT-OFF GRADES AT SPECIFIED PLANT CAPACITIES  BENEFIT-TO-COST RATIO  CANADIAN TAX LAWS -- PRE-19_68 B. C. TAXES PLANT COPPER PRICE (0/lb.) CAPACITY (tons/day) 400 4-20 440 460 5000 10000 15000 20000 25000 30000 35000 40000 45000 50000 0.58$ 0.48$ 0.42$ 0.38$ 0.35$ 0.32$ 0.30$ 0.28$ 0.27$ 0.27$ 0.55$ 0.47$ o.4o$ 0.35$ 0.32$ 0.30$ 0.28$ 0.27$ 0.27$ 0.26$ 0.55$ 0.46$ 0.39$ 0.34$ 0.32$ 0.31$ 0.29$ 0.27$ 0.26$ 0.25$ 0.53$ 0.43$ 0.37$ 0.33$ 0.31$ 0.30$ 0.28$ 0.27$ 0.26$ 0.25$ ( m ) TABLE 5-6 OPTIMAL CUT-OFF GRADES AT SPECIFIED PLANT CAPACITIES  BEHEFIT-TO-COST RATIO UNITED STATES TAX LAWS PLANT COPPER PRICE (ci/lb.) CAPACITY (tons/day) kOfi 420 440 460 5000 — — 0.46% 0.46% 10000 — 0.36% 0.36% 0.36% 15000 0.34% 0.31% 0.30% 0.31% 20000 0.28% 0.26% 0.27% 0.29% 25000 0.24% 0.24% 0.23% 0.25% 30000 0.19% 0.20% 0.20% 0.23% 35000 0.18% 0.17% 0.18% 0.20% 40000 0.17% 0.16% 0.16% 0.17% 45000 0.16% 0.15% 0.14% 0.15% 50000 0.15% 0.14% 0.13% 0.13% - c a p a c i t i e s w i t h respect to b e n e f i t - t o - c o s t r a t i o under Canadian taxes w i t h 1968 B r i t i s h Columbia taxes. In t h i s case, the s e n s i t i v i t y of c u t - o f f grade over a 60 change i n copper p r i c e v a r i e s from 0.03$ to 0.05$, and i s approximately uniform throughout the range of p l a n t s i z e s . Tables 5-5 i s the same as Table 5-4, except f o r the use of pre-1968 B r i t i s h Columbia taxes. In t h i s t a b l e , the range of optimal c u t - o f f grades over a 60 change i n p r i c e v a r i e s from 0.01$ t o 0.05$, w i t h greatest s e n s i t i v i t y o c c u r i n g at low rat e s of production. In Tables 5-6, which i s the American case w i t h b e n e f i t - t o - c o s t r a t i o , the change i n c u t - o f f grade i s l e s s than 0.04$ w i t h greatest s e n s i t i v i t y i n the intermediate range of concentrator c a p a c i t i e s . Optimal P l a n t S i z e . So f a r , optimal p l a n t s i z e has not been considered. Table 5-7 o u t l i n e s o ptimal p l a n t s i z e , maximum i n t e r n a l r a t e of r e t u r n or b e n e f i t -to cost r a t i o , and optimum c u t - o f f grade f o r each combination of metal p r i c e and t a x laws. Data f o r t h i s t a b l e was determined i n the same manner as was employed i n Tables 5-1 through 5-6; consequently i t i s subject to the same sources of e r r o r . This data w i l l be analyzed i n terms of those values which optimize i n t e r n a l r a t e of r e t u r n and those which maximize b e n e f i t - t o - c o s t r a t i o . . Maximum r a t e of r e t u r n . Maximum i n t e r n a l r a t e of r e t u r n under Canadian tax laws w i t h 1968 B r i t i s h Columbia taxes ranges from 15$ at 400 Cu. to 28$ at 460 Cu., w i t h optimal p l a n t c a p a c i t y v a r y i n g from 30000 tons per day at 400 Cu. to 23000 tons per day at 460 Cu. Optimal c u t - o f f grade v a r i e s from 0.33% to 0.4-1% as copper p r i c e increases from 400 per l b . t o 460 per l b . Under pre-1968 B r i t i s h Columbia taxes, a maximum r a t e of 19% f o r 400 copper occurs w i t h a p l a n t c a p a c i t y of 25000 tons per day and a cut-o f f grade of 0.40% Cu. .With 420 Cu. a r a t e o f 25% i s achieved w i t h a p l a n t c a p a c i t y of 30000 tons per day and a c u t - o f f grade of 0.40% Cu. I f copper p r i c e i s 440,. a concentrator c a p a c i t y of 31000 tons per day and a c u t - o f f grade of 0.40% Cu. produce an i n t e r n a l r a t e of r e t u r n of 30%. S i m i l a r l y a r a t e of r e t u r n of 35% r e s u l t s from a concentrator c a p a c i t y of 35000 tons per day,.a c u t - o f f grade of 0.40%, and a p r i c e of 460 per l b . f o r copper. With United States t ax laws, the r a t e s of r e t u r n are much lower •than w i t h Canadian t a x laws. The optimum r a t e of r e t u r n w i t h 400 copper i s 4%. This r e t u r n i s produced by concentrator c a p a c i t i e s of 25000 t o 50000 tons per day and cut o f f grades ranging from 0.25% to 25000 tons per day to 0.15% at 50000 tons per day. With 420 copper a maximum i n t e r n a l r a t e of r e t u r n of 9% i s a t t a i n e d w i t h i n a range of concentrator c a p a c i t i e s of from 26000 to 38OOO tons per day and a range of c u t - o f f grades from 0.25% t o 0.20% Cu. . S i m i l a r l y , . a 30000 t o 38OOO ton-per-day concentrator o p e r a t i n g t o a 0.25% t o 0.24% c u t - o f f grade produces an i n t e r n a l r a t e of r e t u r n of 13% w i t h 440 copper; and a p l a n t c a p a c i t y of from 30000 to 40000 tons per day op e r a t i n g to a c u t - o f f grade of 0.25% Cu. provides an i n t e r n a l r a t e o f r e t u r n of 16% at 460 copper. , Several trends become apparent i n the combination of c u t - o f f grade and concentrator c a p a c i t y which produce the maximum i n t e r n a l r a t e of r e t u r n at a.given metal p r i c e and under a s p e c i f i e d t a x system. F i r s t , and most obvious, i s the f a c t t h a t optimal i n t e r n a l r a t e of r e t u r n increases as metal p r i c e i n c r e a s e s . Secondly, the concentrator c a p a c i t y which maximizes the r a t e of r e t u r n increases as metal p r i c e increases under Canadian t a x laws w i t h pre-1968 B r i t i s h Columbia ta x e s ; decreases i n response to p r i c e increases under the Canadian case w i t h 1968 B r i t i s h Columbia taxes; and remains roughly unchanged i n response t o p r i c e changes under United States t ax laws. The optimal concentrator s i z e under United States t ax laws i s s i g n i f i c a n t l y higher than under Canadian t a x laws. A t h i r d t r e n d can be seen i n the increase i n optimal c u t - o f f grade w i t h i n c r e a s i n g metal p r i c e under United States taxes and Canadian taxes w i t h 1968 B r i t i s h Columbia t a x e s . With pre-1968 B r i t i s h Columbia t a x e s , the optimal c u t - o f f grade i s i n s e n s i t i v e t o changes i n metal p r i c e . The optimal c u t - o f f grade f o r the American case i s s i g n i f i c a n t l y lower than i n e i t h e r of the Canadian cases. Maximum B e n e f i t - t o - C o s t R a t i o . The lower s e c t i o n of Table 5-7 shows optimal p l a n t s i z e s , b e n e f i t - t o - c o s t r a t i o s , and c u t - o f f grades under given t a x laws at s p e c i f i e d copper p r i c e s . The f o l l o w i n g combina-t i o n maximize the b e n e f i t - t o - c o s t r a t i o under Canadian tax laws w i t h pre-1968 B r i t i s h Columbia ta x e s . At a copper p r i c e of 400 per l b . , a b e n e f i t - t o - c o s t r a t i o of 1.3 i s achieved w i t h a m i l l c a p a c i t y of 24000 tons per day and a c u t - o f f grade of 0.35$. The same c u t - o f f grade and a p l a n t c a p a c i t y of 21000 tons per day produce a b e n e f i t - t o - c o s t r a t i o ( 118 ) TABLE 5 - 7 SUMMARY OF RESULTS MAXIMUM RATE OF RETURN Canadian Taxes (Pre-1968 B.C.Taxes Canadian Taxes (1968 B.C. Taxes) U. S. Taxes 400 Cu 1) 19 % + 15 1 + 4 % + 2) 25000 TPD 30000 TPD 25000 - 50000 TPD 3) 0.40 % Cu. 0.33 % Cu. 0.25 - 0.15% Cu 420 Cu l ) 25 i + 20 <f0 + 9 % + 2) 30000 TPD 28000 TPD 25000 - 38OOO TPD 3) 0.40% Cu. 0.35% Cu. 0.25 - 0.20% Cu. 440 cu. 1) 30 % + 24 i + 13 % + 2) 31000 TPD 24000 TPD 30000 - 38OOO TPD 3) 0.40 % Cu. 0.40 % Cu. 0.25 - 0.24 % Cu. 460 cu l ) 35 1o + 28 % + 16 % + 2) 35000 TPD 23000 TPD 30000 - 45000 TPD 3) 0.40 % Cu. 0.4l % Cu. 0. 25 % Cu. MAXIMUM BENEFIT-TO-COST-RATIO ( @ 9% ) Canadian Taxes Canadian Taxes U. S. Taxes (Pre-1968 B.C. Taxes (1968 B.C. Taxes  400 cu 1) 1.3 + :1 1.2 +: 1 0.8 + : 1 2) 24000 TPD 21000 TPD 27000 - 50000 TPD 3) 0.35% Cu. 0.35 i Cu 0.20 - 0.15 % Cu. 420 Cu 1) 1.5 +: 1 1.4 +: 1 1.0 + : 1 2) 21000 TPD 20000TPD 30000 TPD 3) 0.35 % Cu. . 0.34 % Cu 0.20 % Cu. 440 Cu l ) 1.7 +: 1 1.6 +: 1 1.2 + : 1 2) 20000 TPD 19000 TPD 22000 TPD 3) 0.35 % Cu. 0.33 % Cu 0.27 io Cu 460 Cu 1) 1.9 +: 1 1.75 + : 1 1.4 +: 1 2) 18000 TPD 18000 TPD 20000 TPD 3) 0.34 i Cu 0.32 % Cu 0.28 % Cu. TABLE SHOWS: l ) I n t e r n a l Rate of Return/Benefit t o Cost R a t i o 2) Concentrator Capacity 3) Cut-off Grade. of 1.5 w i t h 420 Cu., wh i l e a s i m i l a r c u t - o f f grade and a 1000-ton-per-day smaller concentrator a t t a i n a 1.7 b e n e f i t - t o - c o s t r a t i o at 440 copper. . S i m i l a r l y , w i t h a c u t - o f f grade o f 0.34$ Cu., an 18000 ton-per-day p l a n t shows a b e n e f i t - t o - c o s t r a t i o of 1.9 at 460 copper. The Canadian case w i t h 1968 B r i t i s h Columbia taxes i s character-i z e d by a lower b e n e f i t - t o - c o s t r a t i o . At 400 copper, the optimum com-b i n a t i o n i s a 21000 ton-per-day p l a n t and a 0.35$ c u t - o f f grade, producing a b e n e f i t - t o - c o s t r a t i o of 1.2. A 20000 ton-per-day m i l l and a 0.34$ c u t - o f f grade achieve a b e n e f i t - t o - c o s t r a t i o of 1.4 at 420 copper, w h i l e a 1000 ton-per-day l a r g e r m i l l o p e r a t i n g t o a 0.01$ lower c u t - o f f grade produces a r a t i o of 1.6 at 440 copper, at 460 copper, a concentrator of 18000 ton-per-day c a p a c i t y and a c u t - o f f grade of 0.32$ produces a b e n e f i t - t o - c o s t r a t i o of 1.75* Under United States t a x laws, a non - f e a s i b l e b e n e f i t - t o - c o s t r a t i o o f 0.8 i s produced w i t h concentrator c a p a c i t y ranging from 27000 to 50000 tons per day and c u t - o f f grade ranging from 0.20$ to 0.15$ Cu. when copper i s 400 per l b . This s o l u t i o n i m p l i e s t h a t the optimal p r o j e c t at 400 copper has a negative net present v a l u e . A r a t i o of 1.0 i s a t t a i n e d at 420 copper w i t h a 30000 ton-per-day m i l l o p e r a t i n g t o a c u t - o f f grade of 0.20$, while a.22000 ton-per-day p l a n t and a 0.27$ c u t - o f f grade produce a 1.2 b e n e f i t - t o - c o s t r a t i o at a copper p r i c e of 440 per l b . An optimal combination of a p l a n t c a p a c i t y of 20000 tons per day and a c u t - o f f grade of 0.28$ at a copper p r i c e o f 460 per lb..shows a b e n e f i t - t o - c o s t r a t i o of 1.4 Under a l l three tax systems the optimal p l a n t s i z e and the optimal c u t - o f f grade decrease as metal p r i c e i n c r e a s e s . In the American case, both c u t - o f f grade and p l a n t c a p a c i t y are more p r i c e s e n s i t i v e than they are i n e i t h e r of the Canadian cases. .Cut-off grade i s the most i n s e n s i t i v e t o p r i c e change i n the Canadian case w i t h pre-1968 B r i t i s h Columbia taxes, and m i l l c a p a c i t y i s the l e a s t p r i c e s e n s i t i v e under 1968 B r i t i s h Columbia ta x e s . CHAPTER V I SUMMARY AND CONCLUSIONS I . . SUMMARY The main purpose of the study was to determine a method of f i n d -i n g the p l a n t c a p a c i t y which would optimize a given orebody from a f i n a n c i a l p o i n t o f view. Thus the emphasis was on c a p i t a l budgeting r a t h e r than on the engineering concepts which would tend t o d i c t a t e a prod u c t i o n s t r a t e g y . The method of f i n a n c i a l o p t i m i z a t i o n which was developed was i l l u s t r a t e d by the example i n Chapter V.. The model i n Chapter V was used t o determine the b e n e f i t - t o - c o s t r a t i o f o r v a r i o u s combinations of two v a r i a b l e s -- concentrator c a p a c i t y and c u t - o f f grade. C e r t a i n assumptions were made i n order t o s i m p l i f y the model. The tonnage and grade o f ore reserves were represented by. a cumulative frequency d i s t r i b u t i o n g i v i n g t o t a l ore rese r v e s , above s p e c i f i e d cut-o f f grades. - In the example, the tonnage-grade f u n c t i o n was continuous, although d i s c r e e t values o f tonnage and c u t - o f f grade could have been used without d i f f i c u l t y . I t was a l s o necessary t o de f i n e the average grade at var i o u s c u t - o f f grades. Assumptions regarding metal p r i c e were avoided by us i n g a range of metal p r i c e s . However, i n an a c t u a l f e a s i b i l i t y study i t would be necessary to conduct a r e l i a b l e f o r e c a s t of metal p r i c e s . The model was constructed on the premise t h a t metal p r i c e s would not vary d u r i n g the l i f e of the mine; however i t would be q u i t e p o s s i b l e t o permit the p r i c e t o vary i n accordance wo predetermined t r e n d s . The cost of c a p i t a l which was used i n the example was a r b i t r a r i l y e s t a b l i s h e d ; how-ever there are techniques f o r e s t a b l i s h i n g the cost of c a p i t a l which could be used i f the o p t i m i z a t i o n were being done i n a c t u a l p r a c t i s e . The assumption which was made w i t h respect to changes i n the p r i c e l e v e l was t h a t percentage changes i n f a c t o r p r i c e s would be the same as percentage changes i n product p r i c e s . This has the e f f e c t of reducing cash flows t o a p r e s e n t - d o l l a r b a s i s . I n f l a t i o n i s v e r y d i f f i -c u l t t o p r e d i c t , so t h i s assumption i s probably warranted i n l i e u of techniques which cope w i t h the problem of p r i c e trends. • In the example, s t r i p p i n g r a t i o was assumed to have a l i n e a r r e l a t i o n s h i p w i t h c u t - o f f grade, w i t h high s t r i p p i n g r a t i o s o c c u r r i n g at high c u t - o f f grades. I t i s l o g i c a l t o p o s t u l a t e that as a greater percentage of the m a t e r i a l i n a min e r a l deposit becomes cl a s s e d as ore, the waste-to-ore r a t i o w i l l decrease; however t h i s r e l a t i o n s h i p very p o s s i b l y would not be l i n e a r i n an a c t u a l orebody. Nevertheless, some approximation would have t o be used i n order t o apply the o p t i m i z i n g technique to an ore d e p o s i t . C a p i t a l costs f o r the concen t r a t i n g f a c i l i t y were given f o r ^>000 ton-per-day increments, however mine c a p i t a l costs were determined i n the model by a l o g - l o g r e l a t i o n s h i p between cost per t o n of mine c a p a c i t y and mine c a p a c i t y . This l o g - l o g f u n c t i o n was made necessary by the f a c t t h a t mining r a t e i s r e l a t e d to s t r i p p i n g r a t i o , and th e r e f o r e w i l l vary f o r a given m i l l i n g r a t e , as c u t - o f f grade i s v a r i e d . S i m i l a r l y , concentrator o p e r a t i n g costs were s p e c i f i e d f o r each concentrator s i z e considered, w h i l e mine op e r a t i n g costs were determined from a l o g - l o g r e l a t i o n s h i p between cost and mining r a t e . The area of costs i s the one which probably warrants the most work. .While the costs used i n the example are r e a l i s t i c and the trends i n cost over v a r y i n g p l a n t s i z e are i n d i c a t i v e of today's cost t r e n d s , the matter of determining expected o p e r a t i n g and c a p i t a l costs i s perhaps the weakest part of the model. This i s due t o the f a c t t h a t many mining companies are l o a t h to r e l e a s e i n f o r m a t i o n regarding c o s t s . • Preproduction costs were defined as a l i n e a r f u n c t i o n of concen-t r a t o r c a p a c i t y . .This cost component i s dependent upon the i n d i v i d u a l orebody considered; however i t i s not a very d i f f i c u l t v a r i a b l e to estimate. Cut-cTff grade and concentrator c a p a c i t y were the independent v a r i a b l e s which were optimized. The technique which was used was t o vary c u t - o f f grade i n increments of 0.1$ from 0.1$ to 1.0$ and to vary m i l l c a p a c i t y from 5000 tons per day to 50000 tons per day i n 5000 ton-per day increments. The cash flows f o r each of these combinations of c u t - o f f grade and p l a n t c a p a c i t y were determined under Canadian tax laws w i t h pre-1968 B r i t i s h Columbia taxes, Canadian tax laws w i t h 1968 B r i t i s h Columbia taxes, and United States t ax laws. The combination of c u t - o f f grade and p l a n t c a p a c i t y which maximized the b e n e f i t - t o - c o s t r a t i o under a given t a x system at a given metal p r i c e was considered t o be the f i n a n c i a l l y optimal s t r a t e g y . CONCLUSIONS Conclusions w i l l be drawn from each of the two major areas of i n t e r e s t . F i r s t , a comparison of the three t a x a t i o n systems w i l l be made from two viewpoints; t h a t of the mine operator, and th a t o f the p u b l i c w i t h regard to resource p o l i c y . And seconly, conclusions w i l l be drawn w i t h respect t o f i n a n c i a l o p t i m i z a t i o n of mining ventures. A l l deductions w i l l be based on data obtained from the h y p o t h e t i c a l example i n Chapter V. Comparison of Tax Systems I t i s obvious that i f an i n v e s t o r or mine operator were asked which tax system he p r e f e r r e d , he would choose Canadian taxes w i t h pre-1968 B r i t i s h Columbia ta x e s . The reason f o r t h i s choice would be the higher r a t e of r e t u r n which i s p o s s i b l e under t h i s t a x system. The i n t e r n a l r a t e of r e t u r n i s i n excess of 5% greater than the r e t u r n under 1968 B r i t i s h Columbia taxes, and from 15% to 19% greater than returns which are p o s s i b l e under American tax laws. The reason f o r the high i n t e r n a l r a t e of r e t u r n under pre-1968 B r i t i s h Columbia taxes i s the s h i f t o f cash f l o w to the present. Because of various tax i n c e n t i v e s and c a p i t a l w r i t e - o f f p r o v i s i o n s , the mine operate w i t h l i t t l e or no f e d e r a l t a x a t i o n d u r i n g the f i r s t 5 t o 7 years of o p e r a t i o n . One drawback to the Canadian tax system w i t h pre-1968 B r i t i s h Columbia taxes can be seen i n the p r i c e s e n s i t i v i t y of optimal p l a n t c a p a c i t y . At a copper p r i c e of 400 per l b . the c a p a c i t y which maximizes the r a t e of r e t u r n i s 25OOO tons per day, while a concentrator s i z e of 35000 tons per day maximizes i n t e r n a l r a t e of r e t u r n at 460 copper. The p r i c e s e n s i t i v i t y of concentrator s i z e under 1968 B r i t i s h Columbia taxes i s s l i g h t l y l e s s than i t i s under pre-1968 B r i t i s h Columbia taxes, where optimal concentrator s i z e v a r i e s by 8000 tons per day over a 60 change i n metal p r i c e . Under United States t a x p r o v i s i o n s , optimal p l a n t c a p a c i t y w i t h respect t o i n t e r n a l r a t e of r e t u r n i s r e l a t i v e l y i n s e n s i t i v e t o changes i n metal p r i c e . This p o i n t s out the f a c t t h a t although the two Canadian cases provide higher r a t e s of r e t u r n , accurate f o r e c a s t i n g of metal p r i c e s i s more important i n Canada than i n the United States i n order t o o b t a i n the maximum r a t e of r e t u r n . In the American case, and t o some extent i n the Canadian case w i t h 1968 B r i t i s h Columbia taxes, o p t i m i z a t i o n of r a t e of r e t u r n under v a r y i n g metal p r i c e s i s accomplished by a d j u s t i n g c u t - o f f grade. T h i s , of course, i s much more e a s i l y accomplished than a d j u s t i n g concentrator c a p a c i t y . Therefore, i t i s reasonable t o assume t h a t , i f the goal of the f i r m i s t o maximize i n t e r n a l r a t e of r e t u r n , which i s t r u e i n a normative sense, American tax laws provide f o r the most f l e x i b l e • o p t i m i z a t i o n , given u n c e r t a i n t i e s i n the trend of metal p r i c e . U n l i k e i n v e s t o r s and operators, the p u b l i c i s most i n t e r e s t e d i n the conservation of resources. I t i s i n the p u b l i c i n t e r e s t to maximize the amount of mineral e x t r a c t e d from the ground as a percentage of t o t a l resources, provided t h a t M h e cost of e x t r a c t i n g and p r o c e s s i n g the mineral i s not e x c e s s i v e . This p u b l i c goal i s obvious; i t helps t o insure continued employment and increases the fund of resources in a country. Given this public goal of conservation of resources, i t is quite apparent that provisions implicit in the American system of taxation tend to promote resource conservation to a greater degree than do those in Canadian tax laws. This is shown by the fact that optimal cut-off grades are considerably lower in the United States than in Canada. From this, i t can be deduced that the tax incentives which are used to promote the extractive industries in Canada do not, in fact, promote the general welfare in terms of conserving resources. F i n a n c i a l O p t i m i z a t i o n o f M i n i n g V e n t u r e s W h i l e t h e c r i t e r i o n o f m a x i m i z i n g t h e i n t e r n a l r a t e o f r e t u r n o f t h e p r o j e c t was used i n comparing t a x systems f r o m a n o r m a t i v e p o i n t o f v i e w , t h i s i s n o t , i n f a c t , t h e g o a l o f an o p e r a t i n g f i r m . The purpose o f an o p e r a t i n g f i r m s h o u l d be t o maximize i t s p r e s e n t v a l u e based upon t h e r e q u i r e d r a t e o f r e t u r n o f i t s i n v e s t o r s . T h i s t o o i s a t h e o r e t i c a l g o a l , a l b e i t an a p p r o p r i a t e one, as t h e management o f many f i r m s m e r e l y t r i e s t o maximize e a r n i n g s p e r s h a r e . G i v e n t h a t t h e aim o f management i s t o maximize p r e s e n t v a l u e , i t i s n e c e s s a r y t o f i n d a " m e a s u r i n g s t i c k " t o compare p r o j e c t s w i t h d i s s i m i l a r i n v e s t m e n t r e q u i r e m e n t s . A s u i t a b l e c o n c e p t t o employ i s t h e p r e s e n t v a l u e p e r d o l l a r i n v e s t e d , o r b e n e f i t - t o - c o s t r a t i o . . I t i s i n t e r e s t i n g t o n o t e t h a t many companies do t h e i r f i n a n c i a l p l a n n i n g on a pur e i n t e r n a l - r a t e o f r e t u r n b a s i s , even t h o u g h p r o j e c t s may be m u t u a l l y e x c l u s i v e as i n t h e example i n C h a p t e r V. T a b l e 5-7 shows q u i t e c o n c l u s i v e l y t h a t t h e p r o j e c t c h o s e n t o maximize i n t e r n a l r a t e o f r e t u r n under g i v e n c o n d i t i o n s i s n o t t h e same as t h e p r o j e c t w h i c h m a x imizes b e n e f i t - t o - c o s t r a t i o , e x c e p t i n t h e case where t h e maximum i n t e r n a l r a t e o f r e t u r n e q u a l s t h e d i s c o u n t r a t e used i n d e t e r m i n i n g t h e b e n e f i t - t o - c o s t r a t i o . Two main c o n c l u s i o n s c a n be drawn i n terms o f f i n a n c i a l o p t i m i -z a t i o n . F i r s t , i s t h e f a c t t h a t o p t i m a l p l a n t c a p a c i t y d e c r e a s e s as m e t a l p r i c e i n c r e a s e s , W i t h C a n a d i a n t a x l a w s , o p t i m a l c o n c e n t r a t o r c a p a c i t y i s c o n s i d e r a b l y l e s s s e n s i t i v e t o changes i n m e t a l p r i c e t h a n w i t h U n i t e d S t a t e s t a x l a w s . S e c o n d l y , o p t i m a l c u t - o f f grade i s r e d u c e d . as metal p r i c e i n c r e a s e s . I n Canada, the s e n s i t i v i t y t o a 60 change i n metal p r i c e v a r i e s by 3$ of the value of c u t - o f f grade i n the case w i t h pre -1968 B r i t i s h Columbia taxes and 9$ of the value of c u t - o f f grade under 1968 B r i t i s h Columbia tax e s . I n n e i t h e r case does t h i s represent a s i g n i f i c a n t s e n s i t i v i t y . However, i n the United S t a t e s , such a change i n metal p r i c e would cause a k-Ofo change i n the value of cut-o f f grade, thus i m p l y i n g considerable s e n s i t i v i t y . The f a c t t h a t both optimal concentrator c a p a c i t y and c u t - o f f grade decrease i n response to i n c r e a s i n g metal p r i c e s at a given discount r a t e suggests t h a t the optimal s t r a t e g y i n d e a l i n g w i t h an increase i n the value of the ore i s t o increase o p e r a t i n g costs and reduce c a p i t a l and preproduction c o s t s . I n t e r n a l Rate of Return. When, i n the three main programs, the cash-flow s u b s c r i p t exceeds the p r o j e c t l i f e , the f i n a l phase i s to determine e i t h e r i n t e r n a l r a t e of r e t u r n or b e n e f i t - t o - c o s t r a t i o . The flow-charts f o r determining the i n t e r n a l r a t e of r e t u r n are on the f o l l o w i n g pages. The f i r s t step i s t o decrease the cash-flow s u b s c r i p t so t h a t the number of s u b s c r i p t e d cash flows equals the number of years of the p r o j e c t l i f e . Then the cash flows are t o t a l l e d . I f the t o t a l of cash flows i s l e s s than 0, the i n t e r n a l r a t e of r e t u r n i s 0. This step i s necessary i n order to prevent negative r a t e s of r e t u r n . I f the t o t a l of each cash flows does not exceed 0, the r a t e of r e t u r n i s set at 0 and the discounted value of the cash flows i s determined from the equation: Flowsheet * / INTERNAL RATE OF RETURN C/RR) M=M-J TOTAL - O / =i /o/ o XXX=1 D/SMl = DISYAL +CASH norvC/) +/?Ar& ftArt(?A-fieri/** *• • O/ i r RAreOfRero^/v -RAT*0*ton*** - O.O/ discounted value of cash flows -2. (cash flow^) x ( l + r a t e of r e t u r n ) _ 1 where i represents the s u b s c r i p t of the cash f l o w . I f the discounted value of the cash flows continues t o exceed 0 , the r a t e of r e t u r n i s increased by 0 . 0 1 and the above step i s repeated. Should the discounted value of cash flows be l e s s than 0 , the r a t e of r e t u r n i s decreased by 0 . 0 1 and a second discounted value i s discounted i n the same manner as was the f i r s t one. Then the absolute values of the discounted values are compared. The r a t e of r e t u r n of the discounted value c l o s e s t t o 0 i s p r i n t e d , and c o n t r o l i s returned t o the main program at the p o i n t where c u t - o f f grade i s increased by 0 . 1 % . B e n e f i t - t o - C o s t R a t i o ( 9 % Compounding) The foregoing flowsheets show the program f o r determining the i n t e r n a l r a t e of r e t u r n f o r each combination of c u t - o f f grade and con-c e n t r a t o r c a p a c i t y . . In order t o optimize the model from an o p e r a t i n g standpoint, i t i s a l s o necessary t o c a l c u l a t e b e n e f i t - t o - c o s t r a t i o s f o r the p r o j e c t s . The flow-sheets f o r determining b e n e f i t - t o - c o s t r a t i o w i t h a 9 % cost of c a p i t a l are on the f o l l o w i n g pages. F i r s t , the cash flows f o r the c o n s t r u c t i o n phase of the p r o j e c t are discounted by the formula: present value_^ =2"(-cash flow.) ( 1 . 0 9 ) 1 where i represents years 1 and 2 of the p r o j e c t l i f e . Then the present value of the mining l i f e i s c a l c u l a t e d from the equation: present value =2"(cash flow.) ( 1 . 0 9 ) X , ( 133 ) Flowsheet *2. BENEFIT- TO-COST RAT/O ( 9 /£ COMPOVND/A/G) > a* o TOTAi. \ •J *(/.09**(-J))) > TOTAL - 7OTAL +(CASH > No Yes 3£*£-><vr- To - Cosr /?Ar/o- TorA£/d> where i represents t o o p e r a t i n g years of the p r o j e c t ; i . e . years 3 through the completion of mining. Next, the b e n e f i t - t o - c o s t r a t i o of the p r o j e c t i s determined by d i v i d i n g the present v a l u e ^ by present value2« The b e n e f i t - t o - c o s t r a t i o i s p r i n t e d , and c o n t r o l i s returned to the main program at the p o i n t wherecut-off grade i s increased by 0.1$. Canadian Taxes -- 1968 B r i t i s h Columbia Taxes. The flowchart f o r t h i s program i s on the f o l l o w i n g pages. The program s t a r t s by s e t t i n g the c u t - o f f grade equal t o 0$. Then, a data card i s read. .The data read from the card i s : Concentrator c a p a c i t y , concentrator o p e r a t i n g c o s t , concentrator c a p i t a l c o s t , and preproduction c o s t . A f t e r the data i s read, the concentrator c a p a c i t y i s p r i n t e d and the c u t - o f f grade i s increased by 0.1$. At t h i s p o i n t , the value of the c u t - o f f grade i s t e s t e d . I f the c u t - o f f grade exceeds 1$, i t i s r e s e t t o 0$ and a new date card i s read. I f the c u t - o f f grade i s l e s s than 1$, a t e s t i s made to determine whether the concentrator c a p a c i t y i s greater than 50j000 tons per day. I f m i l l c a p a c i t y exceeds t h i s amount the program stops; however, i f t h i s i s not the case, the computer begins to determine the s t r i p p i n g r a t i o of the mine. Ore reserves are set equal to C e(A.-BX)> where C, A, and B are defined by a data statement t o equal 40 x 10^, 2, and 5; and X i s the cut-o f f grade expressed above. .Waste tonnage i s then defined as the ore reserve tonnage m u l t i p l i e d by 4 times the c u t - o f f grade. . A l s o , the s t r i p p i n g r a t i o i s set equal to the waste tonnage d i v i d e d by the ore tonnage, p l u s 1. This l i n e a r f u n c t i o n w i l l give a s t r i p p i n g r a t i o of 1.4 F~fowsheef *3 Cash Flow Wlfh Can ad/an Taxes - /968 B.C. Taxes START • • o.o • READ \Miie. CAPACITY Vtui OpeXAnMO Cosr / yiuCwirAt-Oosr Cosr WRITE <f/UC4PAC/TY yes DETERMINATION OE STRIPPING RAT/0 OR£•Reserves'= ST0P XraiYS = Ow/?es-+ 1 J A VERAGC Gf9AO£ «* CoT-Of£<7GAO£ * 1/8 MlN£ /. IF£ /€>#£ C/iP*.C/ry350))K95 D£T£RMJNAT/ON Of M/MA/G COSTS r MiiVfOp^AT/srrsG&r CnsAf'rrhCo&SCCO) MIICAPAC/TY k1itJECAP/7AL COST -&.C.72x<34s-4:* Mas Ckwr*/. Cosr / M/A/SCAWIL COST y DET£ftM/MAT/C?A/ OF CAS// FAOM/ BEFORE TAX r M~0 - * a. o 1 . • — ^ CASHfCOIA/fA/) ~ * /o rrv) CA*>AC/ry*350 4/VHUAJ. £XP£MS£S -\f™e OsvwTwe COST)+ i//ti Ofi&eAr/AftCosr) 3f 350 X/IS/tu. Cv*>c/ry AA/WACA'ST/?£l/£M/£ -4re/&iae£swar* OTA AMJI/A*. £xPe/vses = O^eiATin/6 Cosr)* 0#£ Reserves •= O DETERMINA T/OM OF AFTER- TAX INCOM£ • t B.CDCPfc£C/AT/OA/ -B.C.TAXBASS * OS 8.C 7AX£AS£* &.C.7Ar<8/}S£* .92 B.C. TAX^M/MI/M /V£r/?£I/£A/U£ -BC Dewec/A r/o/vj* • /5 8.C.7AX = 0 Ala. S M>5? CASHFIOW(M) = AUNt/AL N£rP£^£W£\ -S.C.TAX. 10 AfT£fiBCTAX = \AHA/UU. A/£r/xkv£A/tj£ 3C.7AX CASH FLOW (Ml = Arret? B.C. 7A x P#£P#a£></cr/(w Cosr-.tfcepxooc/ar/e/v Cos?— AereA* BC 7AK ± /o Pxe CCA- = P#£/>£ODUCT/t>tV Px£?Zx *&?£/?.£. Ar (7br/u. <?AArrAt fair?.3) i—« — < * CASH £uxv(/tf)-/k£C.CA -<9.C.7A* ADP - f 72>TAL CAP/TAL COST* .3) IbrAJ-Gv/rAL = 7oTAi.CAAi/rAJ. Oss?— fk£MO0OC7 -o vavCosr-TOTAL CAOTT> TOTAL CAPITAL icCosr-COST#.7 • Pxe-fi£Aurr/£w =Ate7ix -B.CVAX Cks*>/2o>vfA1) - ( '/3 at a c u t - o f f grade of 0.1% and 5 at a c u t - o f f grade of 1%. . A f t e r the s t r i p p i n g r a t i o (Note t h a t t h i s f i g u r e i s equal t o 1 plus the normal d e f i n i t i o n of s t r i p p i n g r a t i o ) i s determined, the average grade i s found by adding the f a c t o r l / 5 to the c u t - o f f grade. The mine l i f e i n years i s then defined as 0.95 plus the ore reserves d i v i d e d by the annual m i l l c a p a c i t y ( d a i l y m i l l c a p a c i t y times 360). The a d d i t i o n of 0.95 i s r e -quired t o enable an i n t e g r a l number of years, as the computer always rounds .taumbers down. The p r o j e c t l i f e i s then set as the mine l i f e p lus a two-year c o n s t r u c t i o n p e r i o d . The next phase of the program determines mine c o s t s . Mine opera-t i n g costs are defined by the f u n c t i o n : l o g mine op e r a t i n g costs = l o g 1.19 - 2.694 ( l o g ( l + s t r i p p i n g r a t i o ) x m i l l c apacity) - l o g 5000). Next, the mining r a t e i s c a l c u l a t e d as being the product of the s t r i p p i n g r a t i o times concentrator c a p a c i t y . The mine c a p i t a l cost i s then defined i n terms of the f u n c t i o n : l o g mine c a p i t a l cost per t o n = l o g 2.96 - 2.87 ( l o g mining r a t e - l o g 10000). .The mine c a p i t a l cost per t o n i s then m u l t i p l i e d by the annual mining r a t e . . A f t e r t h i s , the t o t a l c a p i t a l cost which i s equal to the B r i t i s h Columbia mining tax base i s set equal t o the sum of the m i l l c a p i t a l cost and the mine c a p i t a l c o s t . .At t h i s p o i n t a t e s t i s made t o see i f the value of the ore exceeds the o p e r a t i n g costs i n c u r r e d i n mining and m i l l i n g i t . I f op e r a t i n g costs exceed revenues, the c u t - o f f grade i s increased by 0.1% and a new s t r i p p i n g r a t i o i s determined. . I f revenues exceed o p e r a t i n g c o s t s , the c u t - o f f grade, average grade, and p r o j e c t l i f e are p r i n t e d and the next phase of the program i s executed. The f i r s t step i n the c a l c u l a t i o n o f pre-tax cash flow i s to set the s u b s c r i p t d e l i n e a t i n g the year f o r which cash flow i s being c a l c u l a t e d to 0. Then the s u b s c r i p t i s increased by 1. Cash flow f o r t hat year i s i n i t i a l l y d e f ined t o be 0. Then a t e s t i s made to see whether the s u b s c r i p t exceeds the p r o j e c t l i f e . I f so, the program enters the next phase, which i s the computation of e i t h e r the b e n e f i t -t o - c o s t r a t i o or the i n t e r n a l r a t e of r e t u r n f o r the p r o j e c t . However, i f the s u b s c r i p t does not exceed the p r o j e c t l i f e , a t e s t i s made t o determine whether- or not the p r o j e c t i s i n the c o n s t r u c t i o n phase, i n which case the cash flow i s equal t o (-1/2) times the sum of c a p i t a l and preproduction c o s t s . A f t e r cash flow i s determined, the year sub-s c r i p t i s increased by 1 and the above loop i s repeated. I f the p r o j e c t i s not i n the c o n s t r u c t i o n phase, a t e s t i s made to determine whether the ore-reserve tonnage exceeds the annual concen-t r a t o r c a p a c i t y . I f ore reserves do not exceed annual concentrator c a p a c i t y , costs and revenues are computed i n terms of remaining ore re s e r v e s , r a t h e r than one year's production. Annual gross revenue i s computed by t a k i n g twenty times the smelter r e t u r n (which i s metal p r i c e minus 7 0/lb.), times the average grade, times the production f o r the year. .Annual expenses are c a l c u l a t e d by m u l t i p l y i n g the production f o r the year times the concentrator o p e r a t i n g costs plus the product of s t r i p p i n g r a t i o and mine operating c o s t . At t h i s p o i n t , the ore reserve tonnage i s decreased by the production f o r the year. Then annual net revenue i s determined by deducting expenses from gross revenue. The next stage i n v o l v e s the determination of a f t e r - t a x income. The f i r s t step i s t o c a l c u l a t e d e p r e c i a t i o n f o r the B r i t i s h Columbia mining t a x . This f i g u r e i s equal t o 8% of the B.C. base, which o r i g i n a l l y equals the t o t a l c a p i t a l cost f o r the p r o j e c t . Then the base f o r deter-mining the 8%-declining-balance B.C. d e p r e c i a t i o n i s set at 92% of i t s previous v a l u e . Next, the B r i t i s h Columbia mining tax i s computed as 15% of the d i f f e r e n c e between annual net revenue and B.C. d e p r e c i a t i o n . Should the B.C. d e p r e c i a t i o n exceed the annual nettrevenue, the B r i t i s h Columbia mining t a x i s 0. I f the property i s i n the t h r e e - y e a r - f e d e r a l -t a x - f r e e period-, the cash flow f o r the year equals the annual net revenue minus the B r i t i s h Columbia mining t a x and the program then considers the next year's cash f l o w . I f the p r o j e c t i s not i n the t a x - f r e e p e r i o d , the d i f f e r e n c e between the annual net revenue and the B r i t i s h Columbia mining tax i s compared to preproduction c o s t . I f the preproduction cost exceeds t h i s d i f f e r e n c e , the cash flow f o r the year equals annual net revenue minus B r i t i s h Columbia mining t a x , and the preproduction cost i s decreased by the amount of the cash f l o w . Then the next year's cash flow i s computed. I f the preproduction cost does not exceed the d i f f e r e n c e between annual net revenue and B r i t i s h Columbia mining t a x , the preproduction cost i s deducted from annual net revenue i n order to provide a base f o r the deduction of the c a p i t a l cost allowance. The p r e f e d e r a l tax base i s then equal t o the p r e - c a p i t a l - c o s t - a l l o w a n c e base minus 0.3 times t o t a l c a p i t a l c o s t . I f the r e s u l t i n g pre-tax income i s l e s s than 0, the cash f l o w f o r the year i s equal t o the base f o r the deduction of c a p i t a l cost allowance, minus the unammortized preproduction c o s t , minus the B r i t i s h Columbia mining t a x . Then the t o t a l c a p i t a l cost i s reduced by the d i f f e r e n c e between the c a p i t a l - c o s t - d e d u c t i o n base and the B r i t i s h Columbia mining t a x . Preproduction cost becomes 0 and the cash flow f o r the next year i s c a l c u l a t e d . I f the pre-tax income exceeds 0, the c a p i t a l cost allowance i s c a l c u l a t e d as 0.3 times the t o t a l c a p i t a l cost and the d e c l i n i n g balance of t o t a l c a p i t a l cost i s set as 0.7 times the previous value of t o t a l c a p i t a l c o s t . Then, the base f o r f e d e r a l t a x a t i o n i s c a l c u l a t e d as being 2/3 of the d i f f e r e n c e between the income a f t e r deduction of c a p i t a l cost allowance and the B r i t i s h Columbia mining t a x . The f a c t a o of 2/3 i s due t o the 33 l/3$ d e p l e t i o n allowance. The cash flow f o r the year then equals the sum of d e p l e t i o n allowance, a f t e r - t a x income, c a p i t a l cost allowance, and preproduction c o s t . The preproduction cost i s reduced to 0, and the cash flow f o r the next year i s c a l c u l a t e d . Canadian Taxes -- Pre-1968 B r i t i s h Columbia Taxes The program f o r determining a f t e r - t a x cash flows w i t h pre-1968 B r i t i s h Columbia mining taxes i s almost i d e n t i c a l to the case under 1968 B r i t i s h Columbia mining taxes up t o the point of determining a f t e r - t a x income. The only change i s to define the B r i t i s h Columbia d e p r e c i a t i o n base as being equal t o Q% of t o t a l c a p i t a l c o s t . The flow sheets f o r the program appear on the f o l l o w i n g pages. I h order to c a l c u l a t e a f t e r - t a x cash f l o w , a t e s t i s f i r s t made to determine whether or not the p r o j e c t i s i n the i n i t i a l three-year f e d e r a l - a n d - p r o v i n c i a l - t a x - f r e e p e r i o d . I f the p r o j e c t i s i n the i n i t i a l t a x - f r e e p e r i o d , cash flow f o r the year i s the same as annual net revenue. . I f , however, the p r o j e c t i s subject t o f e d e r a l and pro- . v i n c i a l t a x e s , the B r i t i s h Columbia mining tax i s defined t o be $25,000 l e s s than 10$ of the d i f f e r e n c e between annual net revenue and B.C. d e p r e c i a t i o n ; or e l s e 0 i f t h i s d i f f e r e n c e i s negative. A f t e r t h i s p o i n t , the program becomes e s s e n t i a l l y the same as the Canadian-tax, I968-B.C.-tax program. F/owsheet **4 Cas/hF/ow lAZ/Y/i Canadian Taxes - Pre-/968 B. C. Taxes START O.O 1 RHAO / \M/LL CAPACITY I W w Gw/rM Ccsr \ Cosr WPlTE MIL-CAPACITY Cvr-Osv&/?Ao£ Cu T-Orr GAAPS' * O-1 yes Yes ST0P DETERMINAT/ON OE STRIPPING RAT/0 I + 1 V AVEKAG6 G/?AP£ » Ctsr-Of£(7f!ADE * J/3 MINCLIFEJOKS Resewes/fMru. CAPAC/rr' 3SO))i-.95 M/ue Ore r£ I DETER Mi'NAT/ON OFM/A//A/G COSTS \ MlMOPE&QT/AQ&Sr -Aur/£oe(ioe/./9-/3694 • < Af/MA/G RATS'. Mv*. CAPAC/TY ArfweC/iP/TAL COST = M/Mfi/Q RA T£*Arr/toO •* 7brAuC*v*r/u Cosr? S.C. 7Zx M/A/£CAfvrAiCosr > WRITE \Cur- OrrG/vtoE \Fletxi£crl'Fe DETERM/A/AT/0A/ Of CAS/-/ f/LOlA/ ' BEFORE TAX. \ M=0 O.O 1 ~ s CASHfcOIA/fA/) ~ -(rke/WOOe/cr/otf /o A/o • • OutReserves =Oee A}£S£A?f£S- •{350* M/HCAAAC/TY) 1 ANNUAL G#oss£si/£/ja£ r\ve/iA6£GfiAa£*M/u. CAAAC/ry*350 Pescavrs AUA/UAV. £XP£A/S£S = AMIWAJ. Expenses -((SrAIAA'HafiATIO ic *Jwe Off/fArwe COST)+ Kf/itO/>£*Ar//&Cosr) % 350 *AS/t* C*f*c/ry Afi/w/H.S/£r#eY£/Vl/e = 4/VAfPAt t p A W S & n w v r -AMMVAL £XA>JT/<S£S O&RfSSXVe-S ' O DETFRM/NAT/OA/ Of AFTER- TAX CASH FLO tA/ No CASUFLOIA/^) -ANNUAL A/erReveMG 10 CASHFLOW/M) ' BCTax. FfceAea&vcr/o/v t?as~r--FttaAeaauoTtoiY Ossr-8. C. fox ^/AWI/AL Msr * JO) - asooo A/a YES 3.C.7AX.* O PxgC.CA.-AA/MI/**. Yes • 7orA£ t/jfS/rAL <!?(>fr » 7OTA'4.L1'AA/T-AJ- ^SwW AfEAXoavano*/ £osr "O (7orrV£ OlP/rAi. Coert. 3) ADD - 7 o r A i &>p/r/i£ far *.3 PxeTAX- 3C.7AK XATee r/o/v Cosr-O The f l o w sheets f o r determining cash flows under United States t a x laws are on the f o l l o w i n g pages. The program i s i d e n t i c a l t o the preceeding ones up t o the p o i n t where mining costs are determined. Mine op e r a t i n g and c a p i t a l costs are found by usin g the same l o g - l o g f u n c t i o n s that were used i n the Canadian cases. A f t e r these costs are determined, ammortization per t o n of concentrator feed i s i n i t i a l l y set at 0. Then, pre-mining cost i s determined by adding m i l l c a p i t a l cost t o mine c a p i t a l cost and preproduction c o s t . The ammortization per ton of m i l l feed i s equal t o the pre-mining cost d i v i d e d by the ore reserve tonnage. This f i g u r e i s added t o m i l l i n g c o s t . The t o t a l c a p i t a l cost i s then defined as the sum of mine c a p i t a l cost and concentrator c a p i t a l c o s t . Next, a t e s t i s made to determine whether revenue per ton of m i l l feed w i l l exceed cost per t o n of m i l l feed. I f cost exceeds revenue, the c u t - o f f grade i s increased by 0.1%, then s t r i p p i n g r a t i o and costs are recomputed. I f an ope r a t i n g p r o f i t i s p o s s i b l e ; c u t - o f f grade, average grade, and p r o j e c t l i f e are p r i n t e d ; f o l l o w i n g which, cash flow before t a x i s determined. The f i r s t step i n c a l c u l a t i n g cash flow before t a x i s t o set the year s u b s c r i p t f o r the cash flow t o 0. This s u b s c r i p t i s then increased by 1. I f the s u b s c r i p t i s l e s s than 3 5 the cash flow f o r the year equals of pre-mining c o s t , and the cash flow f o r the next year i s c a l c u l a t e d . I f the mine i s not i n the c o n s t r u c t i o n phase, a t e s t i s made to determine whether ore reserves exceed annual mining c a p a c i t y . I f p l a n t c a p a c i t y i s greater than ore res e r v e s , production f o r the year equals the ore-reserve tonnage. Annual gross revenue i s c a l c u l a t e d as 20 times the smelter r e t u r n times the average grade times production f o r the year. S i m i l a r l y , annual expenses equals production f o r the year times the q u a n t i t y of co n c e n t r a t i n g costs plus the product of s t r i p p i n g r a t i o and mining co s t . T o t a l ammortization i s then equated to ammorti-z a t i o n per t o n times production f o r the year, and ore-reserve tonnage i s decreased by the year's p r o d u c t i o n . Annual net revenue i s obtained by deducting expenses from gross revenue. The next phase of the program i n v o l v e s the c a l c u l a t i o n of a f t e r -tax cash f l o w . The d e p l e t i o n allowance i s computed by t a k i n g 15% of annual net revenue. I f the d e p l e t i o n allowance exceeds 50% of net revenue, tax-able income i s equal t o 50% of net revenue. However, i s d e p l e t i o n allowance i s l e s s than 50% of net revenue, t a x a b l e income equals annual net revenue minus d e p l e t i o n allowance. State taxes are then c a l c u l a t e d as 5% of taxable income, and are then deducted from taxable income to give the f e d e r a l i n -come tax base. I f the f e d e r a l t a x base exceeds $25,000, cash flow i s equal t o : 18,750 + (.52 ( f e d e r a l income t a x base - 25,000)) + d e p l e t i o n allow-ance + t o t a l ammortization. Should the f e d e r a l t ax base be l e s s than $25,000, cash flow i s equal t o : (.75 ( f e d e r a l income tax base) + d e p l e t i o n allowance + t o t a l ammortization. A f t e r the cash flow f o r the year i s computed, the program increases the cash-flow year s u b s c r i p t and c a l c u l a t e s cash flow f o r the next year. F7ows/)eef *S Cast? F~/ow With Unifeci States Taxes START XTOA/S=(OX£&S£*I/£V STP/PPW&/?Ar/0~ 4-1 Af/A/£ mffiAf/A/MT/OA/Of M/A/WG COSTS aa9/JS»-/.e634Uoe[ j 0>P4a/TY)-Zo<?500o) • M/M/V(S/?Ar£-= •. /0,00o) = O Ax-A/w/ve der-Chaw Qsr/Arfmrfo/rAt Oxrt/keneoexrro/v £XT Y  CvA/rA£ L^hs-r « M//v£C/v>/rAt\ COST WRITE AvEKAOB (3-&AO0 CASHflOW 3£fV/?£rAX ± M - O Res&wes -( i6o AHWAL £x£>£rssfs = 0#E&ESE£tVes<WM/Ae 0/?E#£SEtfY£S = O AWIW. £/tV$Stf£V£M£= A i/EeQOE&eAVE * 35~C A/I/A/CM £~XPES/S&S •-* M/ttCxAJC/ry TorAiAM/tfoer/ZA r/o/v MI.L£WJ4C/TY X36>0 AMM/AL Mer/PE/EME At/AWA£ £Tx£>ejVSE$ AFT£/?~ TAX CASH FLOW 4- Os-ACET/aMAuaMHa;* YES GePisnavA^iotvAWCE" . S*AwuAi. AJerAkrMc* TAXABLE//vaoAfE -AvA/VAtA/er&vEfScie -DEPiEr/oAS AUOVAWCE TAXABLE A i w < f = STATE 7* x A r t h u r Andersen and Co. "Organization, Management, A d m i n i s t r a t i o n and Accounting f o r a Mining Venture." Short course at the Colorado School of Mines, 1966. Mimeographed notes. Baker, W i l l i a m D. "'Present Value' A n a l y s i s of a Mining P r o j e c t . " Unpublished r e p o r t . B l o n d e l F., and S.G. Lasky. "Mineral Reserves and M i n e r a l Resources," Economic Geology, L I 1956, pp 686 - 697 Bucovetsky, M.A. The Taxation of M i n e r a l E x t r a c t i o n , Studies of the Royal Commission on Taxation,' No. 8 Ottawa: Queen's P r i n t e r , 196T7 Callaway, H.M. "Basic Break-even Formulas Devised to S i m p l i f y Mine E v a l u a t i o n , " Engineering and Mining J o u r n a l , CLV (November, 1954), pp. 90 - 92. Canada, Department of Mines and T e c h n i c a l Surveys, M i n e r a l Resources D i v i s i o n . Summary Review of F e d e r a l Taxation and L e g i s l a t i o n  A f f e c t i n g the Canadian M i n e r a l Industry. .Mineral Information B u l l e t i n MR 82. Ottawa: Queen's P r i n t e r , 1966. C a r l i s l e , D. "The Economics of a Fund Resource w i t h P a r t i c u l a r Reference to Mining, "American Economic Review, XLIV (195^), pp 609 - 612. "Maximum T o t a l Recovery Through Mining High-grade and Low-grade ore together i s Economically Sound," Canadian I n s t i t u t e of Mining and M e t a l l u r g y B u l l e t i n , XLVI (January 19537"^  Chandler, John W. "Economic E v a l u a t i o n of Proposed Mining Ventures --Mine Development and Mine Operating Costs," Mining Congress  J o u r n a l XLV (November, 1959) pp. 4-5 - 51. C o l l i n s , J.E. "An Evaluation and Comparison of Identical Hypothetical Copper Properties i n B r i t i s h Columbia and i n the Northwestern United States." An unpublished report. Conrad, Gordon R., and Irving H. P l o t k i n . "Risk/Return: U.S. Industry Patterns, " Harvard Business Review, XLVI (March-April, 1 9 6 8 ) , pp. 9 0 - 9 9 -C o r l e t t , A.V. "Valuation Formulae -- Parts 1 and 1 1 , "Canadian Mining Journal, LXXX (August, 1 9 6 7 ) , pp. 6 5 - 6 8 , and (November, 1 9 6 7 ) , pp. 5 9 - 6 2 . Evans, J.B. "Evaluation of Mining Properties -- a Graphical Approach," Canadian In s t i t u t e of Mining and Metallurgy, Transactions, LXIII ( I 9 6 0 ) , pp. 5 4 7 - 5 5 3 . "Mining Property Evaluation and i t s Influence on Project Submissions." An unpublished report. Graham, E.P. "Exploiting a Small Orebody," Canadian In s t i t u t e of Mining and Metallurgy B u l l e t i n , LVII ( A p r i l 1967), pp.~~4~09 - 412 Henning, U l f . "Calculation of Cut-off Grade," Canadian Mining Journal LXXXIV (March, 1 9 6 3 ) , pp. 54 - 5 7 -Hichens, Antony. "Minimum Standards for Capital Investment," Mining  Magazine, CXVI ( A p r i l , 1 9 6 7 ) pp. 2 7 7 - 2 8 6 . Just, Evan "Economic Evaluation of Proposed Mining Ventures," Mining  Congress Journal, XLV (November, 1 9 5 9 ) King, H.K. 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"Marginal A n a l y s i s : , i t s A p p l i c a t i o n i n Determining Cut-off Grade," Mining Engineering, X I I I (June, 196l), pp 579 - 582. 

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