U 4 SSS Mine valuation 24-25 mine e.pptx

Mine Valuation

1. UNIT IV VALUATION & DEPRECIATION

2. S. No Title of Book Author   Publication   1 Mineral Economics R.K.Sinha& N.L. Sharma Lovely Prakashan , Dhanbad (our library ) 2 Mine Valuation Sparks.    On net 3 Mine Sampling and Valuation Sepulchre    On net 48 Mine Geology Arogyaswami. Oxford and IBH Publishing Co. New Delhi (our library ) 5 Mine Economics R.T. Deshmukh Lovely Prakashan , Dhanbad (our library ) 6 Mine and Mineral economics Subhash C. Ray and Indra N Sinha Asok K Ghosh PHI Learning Pvt Ltd Rimjhim House 111,Patpranj Industrial Estate Delhi 110092 market 7 Elements of Mineral exploration   Available on IBM website www.ibm.nic.in 8 Mines & Minerals (Development & Regulation), amendment Act 2015   Available on IBM website www.ibm.nic.in 9 Mineral Concession Rules, 2016   Available on IBM website www.ibm.nic.in 10 Mineral Conservation & Development Rules, 2017   Available on IBM website www.ibm.nic.in 11   Minerals (Evidence of Mineral Contents) Rules, 2015.   Available on IBM website www.ibm.nic.in 12 Mineral (Auction) Rules, 2015     Available on IBM website www.ibm.nic.in 13 Minor mineral concession Rules of respective states     On net Suggested learning material

3. Unit Marks as per Curriculum Max Marks [1.5 x Clo.-(2)] Que1 Que-2 Que-3 Que-4 Que-5 Que-6 Total Marks in a paper set Remarks IV 16 24 2 4 8 4 6 - 24   Q1 Define type of 2 marks Q2 Explain type of 4 marks or 6 marks  

4. UNIT IV Mine Valuation & Depreciation Describe the given method of mine property valuation. Describe the given method of debt redemption. Calculate depreciation using given method. 4.1 Valuation of mining property Mine valuation concept & Purpose S teps for valuation and cost estimation Factors considered in mine valuations, risk factors associated with valuation. Capital budgeting Discounted cash flow methods e.g. Net Present Value, Internal Rate of Return and Profitability index and Payback period 4.2 Debt Redemption Methods Diminishing annuity (EMI) Method. Sinking fund method. 4.3 Depreciation Concept of depreciation, its importance and causes. Calculation of Depreciation using: Straight line method, Declining or Reducing balance method. Sinking fund provision against depreciation. Calculation of Life of mine.

5. NOTE : The presentation is respect of various topics and units are only indicative and not exhaustive. Students are advised to keep themselves abreast and well informed about the subject by referring to the suggested reference books & latest information available from different available sources. IMP slides/definitions from exam view point are marked . Some lectures are available on YouTube: Basic mining by sharad sapkal in Hindi Rest of the material is for general information and subject understanding. Hoskols formula is not in the course but still questions are being asked on this topic they are marked

6. 4.1 Valuation of mining property Mine valuation concept & Purpose S teps for valuation and cost estimation Factors considered in mine valuations, risk factors associated with valuation. Capital budgeting Discounted cash flow methods e.g. Net Present Value, Internal Rate of Return and Profitability index and Payback period

7. Inventory Of metal input in concentrate Of metal input in Final metal Geological Some times saleable fraction is different from reserves which is recovered trough some physical process its grade is called as effective grade.

8. Circular flow model Of economy

9. Valuation of mining property The production capacity of a mine is generally pre-estimated by mining experts, based on reserves, extent of mechanisation, carrying capacity of haulage infrastructure and numerous other techno-economic variables of the mine. Initial and final phase of life of mine is generally non productive due to gestation period and reclamation and decommissioning activities respectively. Accordingly the financing, investing & operating activity schedule can also be planned during gestation, production and reclamation stage. Similarly, revenue, expenditure and incomes can also be projected. The future incomes or profits generated from sales after meeting out cost of production is expressed in terms of cashflows . These cashflows are used to work out (find out) the economic viability of a mine or a deposit and in turn its present purchase price .

10. Valuation of mining property cont.. Ore deposits are considered to have value when they are capable of yielding a profit on exploitation over a term of period, as calculated on the date of valuation.

11. Value (purchase price) of a mine Definition : The fair market value of a mine ( mineral deposit ) is that sum which the exploitation of mineral will return , together with a fair rate of interest ( return ) on it, besides paying operating and other expenses, taxes and loan repayments with interest within the period of investment . This price ( or the cash equivalent) is determined by economic experts. Period of investment :An investor invests an amount in purchase of a mine and expects the above return along with investment within the life f mine, or an investor invests an amount in mining company for a certain period of time and expects return along with investment within that expected time this time period of time is called as period of investment. The value of a mine is said to be Rs100 if the future activity in mine will yield Rs100 + all costs incurred + taxes and loan repayments with interest + expected rate of return on investment of Rs100. All factors and cashflow remaining constant the return on investment is deciding factor whether it can be sold to an individual or not. Rate of return is equivalent to compound interest earned on invested amount.

12. Valuation : Valuation is the process of estimating the fair market value of a business in monitory terms by the experts on the date of purchase. The exchange of hands should be in an open and unrestricted market between a willing buyer and a willing seller at “ arm’s length ”( ie as is where is basis), with each party acting knowledgeably, prudently and without compulsion . Valuations can be done on assets like mineral properties at any stage of development or on liabilities of a company. 1. In modern days the present fair market value of a mining property is calculated, based on its future earning capacity and is equal to the sum of present values of its future annual incomes (cash flows). 2.In monitory term it is the present worth or capital investment equivalent to future income to be received as annuity (fixed income annually) over a period of time or equivalent to a lump sum amount payable after a specified time.

13. This price ( or the cash equivalent) determined by economic experts is the amount for which the mineral asset should change hands on the valuation date in an open and unrestricted market between a willing buyer and a willing seller at an “ arm’s length ” with each party acting knowledgeably, prudently and without compulsion . Purpose of valuation : 1. The object of valuation of a working mine or mining project is to assess the selling or purchasing price of a mineral deposit or a mining enterprise. 2.To find out future earning potential and thereby its Profitability besides return of capital investment . 3. To carry out capital budgeting or investment analysis at the time of , merger and acquisition of firms , financial reporting, taxation calculations and in litigation cases .

14. V aluation of mineral property is different from Other Projects . Mine Valuation is the most crucial financial aspect of the mining industry. 1.The rate of annual income from the mine is cyclical & not uniform depending on various factors. 2.Unlike other industries which are perpetual in nature, in mining, the reserves get exhausted by continued working hence the original capital invested is to be recovered during earning period of mine. 3. Each ore body has unique features hence the v aluation of a mineral property is a time specific, site specific and mineral specific exercise.   4. Lengthy Gestation period is involved due to exploration and development . 5.High capital requirement during development period and requirement of specialized equipment . 6.Large Reclamation costs involved at the end of project 7.In other industry flow of raw material is almost perennial whereas in mining industry it depends of depleting resources. 8.Large environmental/social /infrastructure cost.

15. Methods of valuation: There are two methods for “Mine property Valuation” or capital budgeting. Valuation methods 1 . Static or Traditional methods: In these methods the present purchase value of a project/mine is calculated based on the face value of future earnings ( cashflows) without considering the time value of money. 2 . Dynamic or Discounted cash flow methods or modern methods: In these methods the present purchase value of a project/mine is based on future earning capacity of the mine duly considering the time value of money. In monitory terms the present purchase value is equal to the sum of present (discounted) values of future year wise cashflows (incomes) received during the life of mine . Future cashflows likely to be generated as per the capacity of mine is estimated year wise and they are discounted at the expected rate of return by the investor as on date of investment. The sum of discounted values gives the present value (or purchase value) acceptable to a particular investor . However the experts find present market value by adding discounted values of future cashflows at reasonable/fair rate of return .

16. (Minimum) O 1.Estimation of ore reserves

17. Profit from a mining business depends on revenue and cost incurred Cost is the amount of resources in monetary terms , used for the purpose of production of mineral. Various costs involved CAPITAL COST The capital cost will be in two parts and most of the capital cost is incurred in first few years . 1. Cost of the equipment 2. Cost of Development ( cost incurred on pre production work ) Working capital , is that portion of capital required to finance and maintain stockpiles, inventories (stores) , and the costs incurred during lag time between expending ( paying for) production costs and receiving accounts receivable. It is required to meet out day to day expenses . Unlike the capital spent for equipment , the investor gets working capital back when the operation closes, this is also required to be borrowed on interest.

18. Sources of funds : Capital market provides source of funds through public equity (shares) and loan (debenture). Cost is the amount of resources in monetary terms , used for the purpose of production of goods or rendering services . The costs of mine production include : i . Purchase, procurement and transportation costs of raw materials if any, other components and sub-assemblies . ii. Purchase cost of consumables such as oils, lubricants, tools of small value, fuel oil, machinery spares etc. iii. Wages and salaries paid to direct production workers , maintenance, inspection, stores staff, supervisors and other staff. iv. Costs paid to subcontractors for the orders placed on them. v. Cost incurred on line rejections, wastages, spoilage and rework. vi. Interest on working capital to the extent it relates to inventory. vii. Cost ( EMI) of procurement of capital assets like machinery & equipment. Viii : D epreciation of these capital assets & provisions for its replacement once worn out. ix. Indirect cost incurred on fulfilling social , environmental and legal obligations.

19. Fixed cost of production : The costs which do not change for a given period in spite of change in volume of production are called fixed costs . Eg. rent, taxes, salaries, depreciation, insurance etc. Fixed costs are normally expressed to be incurred in terms of time period e.g. per day, per annum etc. Average fixed cost per unit of production can be reduced by increasing production level. Variable cost of production : Variable cost varies with the output . Direct material cost and direct (piece rated) labour cost are generally variable cost. Variable cost includes cost of raw materials, running cost on machinery such as fuel , repairs , routine maintenance etc.

20. The break-even point for a production rate is the point ( or production rate ) where total revenue received equals the total costs associated with the sale of the product. Beyond this production rate mine gives profit . Profit margin Zone Cost revenue

21. Demand & supply curve :Demand & supply of quantity at a point of time determines the price Demanded/supplied

22. Valuation reports of a small mine The complete valuation report of a mineral property would involve not only attention and study of the mine and works, but also of the title deeds and account – books. Broadly, the following points should be covered 1) Property - legal rights or title without encumbrance , size, boundaries, ownership, royalties, taxation, rent. 2) Geographical position – access, climate, business center , markets . 3) Economic position – labour , water, energy, fuel, timber. 4) Geological position - type of deposit, possible genesis, country rock, mineralization. 5) Past history – past production and result, old working.

23. Present mine working – extent, equipment, mineral reserves, cost etc. Beneficiation – plant and equipment, recovery and cost. 8) Marketing - prices, freight, agencies. 9) General surface plant and equipment – building, water, land, sources, reservoir. 10) Valuation - revenue ,cost, profit, life of mine and the internal rate of return. 11) Finance requirement on improvement and reconstruction.

24. RISK IN VALUATION Valuation of the assets is being done with due care and correctness however it involves the following risks due to typical nature of the mining enterprise. 1) Cost of production varies – a) when the grade and reserves varies to that of computed figures. b) increase in the amount of gangue c)possibilities of contamination and dilution due to barren rock. Over and under estimation of the cost seriously effect mine economics. Unpredictable cost of lands, building, plant and machinery. Import policies and restrictions during procurement of the machineries. Existences of geological disturbances in the mine property, some times ore body may comprise of the several independent shoots separated by barren rocks. Extraction losses due to particular method of mining, premature collapse, faulted zone, safety restrictions.

25. 7 ). Sudden changes in various govt. taxes royalties and cess etc. 8). Risky nature of the mining due to unpredictable hazards in working i.e fall of roof, gas/coal and dust explosion and inundation. 9). High cost of working in the course of time as the workings are at a distant place and deeper . 10). Variation in the govt. policies towards labour and their wages affects the cost of productions. 11). In computation of the cost whether it is a pit mouth delivery or for destinations (free on rail ) should be taken care of before hand.

26. Capital budgeting is 1. Evaluation of different projects from the profitability ( irr ) point of view. 2.Consideration of those proposals giving higher returns than expected rate. 3.Selection, planning and phasing the budget for the most profitable one. OPPORTUNITY COST Opportunity cost is defined as the benefits lost by rejecting the best competing alternative to the chosen one. The benefit lost is usually the net earnings or profits that might have been earned from the rejected alternative.   Capital is the resource used to produce goods/service, it includes physical assets, equipment ,stock, bond, cash or say money invested in co. through share, loan etc.

27. Capital budgeting is evaluation of different projects from the profitability point of view and planning the budget for the most profitable one. The three most common approaches to project selection are payback period (PB), internal rate of return (IRR), and net present value (NPV). The payback period determines how long it would take a company to recover the original investment

28. Capital budgeting

29. Various methods of “Mine property Valuation” or capital budgeting. Valuation methods 1 . Static or Traditional methods: a. On mutually agreed terms: b. Book value approach. c. Payback period method d. Hoskold`s and Morkill`s valuation methods. 2 . Dynamic or Discounted cash flow methods or modern methods: a. Net Present Value, b. Internal Rate of Return c. Profitability index, d. Discounted payback period

30. Static or Traditional methods: Unlike modern methods these methods for transection/valuation of mineral property give same weighing (value) to the distant and near earnings & ignore the timing of cash flows. These methods consider money to have constant value ( purchasing power ) regardless of time of cash flow. In these methods all techno-economic parameters of mining business are considered but time value of money is ignored. Value of money means its utility or purchasing power. Due to rising inflation value of money reduces with the passage of time.

31. On mutually agreed terms: This is the most common approach, the prices are derived by comparison to similar nearby projects which have been sold in the recent past. It is also called as market approach. Since it is impossible to have two similar mining projects existing at any time, the method of valuation by comparison is not appropriate. Being without any scientific background the main disadvantages with this method, is, it may lead to under/over estimation of value. It does not take into account the future earning potential of the business.  

32. b . Book value approach ( Cost based approach) Book value of a company = Value of its tangible assets – liability of the company . Tangible assets= physical assets buildings, reserves, machinery, infrastructure facilities etc and not patent trademarks, copyrights, and   good will etc Machinery /equipment / buildings value is determined after due depreciation. Liability is considered on the day of assessment according to its "books" or  financial statements . Drawbacks :

33. Book value  refers to the total amount a company would be worth if it liquidates its assets and pays back all its liabilities. Book value represents the value of a particular  asset  on the company's  balance sheet  after taking accumulated  depreciation  into account. Book value literally means the value of the business according to its "books" or  financial statements . In this case, book value is  calculated from the balance sheet , and it’s the difference between a company's total assets and  total liabilities . On the company's balance sheet, book value is recorded as  shareholders' equity (contribution) . For example, if Company XYZ has total assets of $100 million and total liabilities of $80 million, the book value of the company is $20 million. In a very broad sense, this means that if the company sold off its assets and paid down its liabilities, the equity value or  net worth  of the business would be $20 million. Market Value Market value is the value of a company according to the  stock market . Market value is calculated by multiplying a company's  shares outstanding  by its current  market price . This is also known as  market capitalization .   If Company XYZ has 1 million shares outstanding, and each share trades for $50, then the company's market value is $50 million. Market value is most often the number analysts, newspapers and investors refer to when they mention the value of a company. Book value & Market value could be different based on company`s reputation in business.

34. c . Payback period method : The payback period is the time period required for the firm to recover its initial investment in a project, as calculated from its surplus cash inflows. If the project does not pay back itself within the maximum acceptable payback period to the purchaser, it is rejected. Lesser the period more is the security. Also called as payoff method it is a traditional, simplest and most widely employed quantitative method for apprising capital expenditure decisions. It is the indication of period of uncertainty connected with investment . Thus the pay-back method measures the number of years required for cash benefit to payback the original out lay . This method is based on the principle that every capital expenditure pays itself back within a certain period out of the additional earnings generated from the capital assets .

35. In the case of an annuity ( annual income ), the payback period can be found by dividing the initial investment by the surplus annual cash inflow. Cash Flow = profit= (Net )Cash flow/yr = Revenue earned (+ve) + Cost input (-ve) Payback period in years = N I = Initial investment I p = periodic Investment in subsequent years ( yearly ) A s = Annual gross surplus (annuity) r = simple rate of interest on investment Draw backs :

36. Eg . So the investment year is taken as 0 year Investment is made at the beginning of 1 st year. This is called as 0 year ,cash flows are considered at the end of each year so the investment amount is negative at the end of 0 year. End flow Cash flow Cumulative cash flow column is a must for getting payback period 3.29

37. Time value of money Rationale Value of money is the purchasing power of money .(How much commodity can be purchased with Rs 1) “ Time value of money ” is the value / worth or purchasing power of a unit of money (Rs 1)at different time Intervals ie today ,after one yr, after 2 yrs ---after n yrs and value of money decreases every subsequent year . If Govts ensures zero inflation money will have constant value or constant purchasing power . The value of the money received today is considered to be more than its value received at a later date . In other words, the value of money changes over a period of time ( maybe due to inflation or other uncertainties ). Since a rupee received today has more value, a prudent or wise investors would prefer current receipts over future receipts because he can use it in better & desired ventures. Some important factors for preferring current receipts over future receipts are: 1. Investment opportunities ie he may earn more by investing it somewhere else . 2.Preference for consumption ie he may like to enjoy it today. 3.Risk of loosing money. 4.Inflation  

38. Why does money have time value and not constant value ? Some of the reasons are:  Production  Money can be employed productively to generate income or real returns. For example, if we spend Rs. 500 on raw materials, Rs. 300 on labour and financial Management and Rs. 200 on other expenses and the finished product is sold for Rs. 1100, we can say that the investment of Rs. 1000 has fetched us a return of 10% in a particular business. Maybe in another business we could earn more than 10% also.  Inflation  During periods of inflation, a today rupee has higher purchasing power than a rupee in the future.  Risk and uncertainty  We all live under conditions of risk and uncertainty. As the future is characterized by uncertainty, individuals prefer current consumption over future consumption . Most people have subjective preference for present consumption either because of their current preferences or because of inflationary pressures. On the other hand money looses its value day by day in the market economy hence people deposit their surplus money with bank or invest it in business & expect return/interest against investment in order to take care of above factors.

39. Present & Future Value concept Time preference rate or required rate of return or interest rate expected on investment: The time preference for money is generally expressed by an interest rate, which remains positive even in the absence of any risk. It is called the risk free rate (as in govt bonds). For example, if an individual's time preference is 8%, it implies that he is willing to forego Rs. 100 today to receive Rs. 108 after a period of one year. Thus he considers today`s Rs. 100 and Rs. 108 after one year respectively as equivalent. In reality though this is not the only factor he considers. He requires another rate for compensating him for the amount of risk involved in such an investment. This risk is called the risk premium .   Required rate of return = Risk free rate + Risk premium

40. Say somebody wants to invest his amount P with a company called as principal. Pv = Present Value of principle P , Term = n = No of years for which principal P is invested/ locked. r = Rate of interest or return in % age. There are two methods by which the time value of money can be calculated : A. Compounding technique for finding future value of present amount . B . Discounting technique for finding present value of future amount . Present and future value Basic Concepts : Interest is defined as the cost of borrowing money. Interest can be calculated in two ways,  simple interest and compound interest. Some Important terms Principal = P =Original amount deposited in some investment scheme like Fixed deposits in bank or invested in a company Or lent to a company/individual with an expectation of return or interest r . ( loan for one is deposit for another ) Pv = present value= present purchasing power of an amount of present time P , Fv =future value =future purchasing power of an amount of future time S.

41. Future value Fv of a present amount P , is an amount S of future time having purchasing power equal to the purchasing power of present amount P as on today . Present value Pv of a future amount S , is an amount P of present time having purchasing power equal to the purchasing power of future amount S . Say if P = Rs.100 then Today`s value of principle P= Rs.100 Say this P becomes S after a period of n years due to accumulated interest (or return) whether SI or CI. Then this S = Sum accrued (accumulated)in n years will have certain value at that time (at the end of n yrs ) say it is S = Rs 500 Future value of Rs 500 is purchasing power of Rs 500 n years hence. As the same P today`s P ie Rs.100 becomes the then S, Rs.500 by lending on interest or investing, in a venture, this S is also called as future value of P and it is considered that the purchasing power or value of today`s P ie Rs.100 and the value of then S Rs.500 is same. Or the quantity of goods available today for Rs 100 will be available for Rs 500 after n years. Or an investor is presumed to be ready to part with Rs.100 today in the hope of getting Rs.500 after n years.

42. Note : Interest is always compound interest unless mentioned otherwise. ( considering no intervening (in between ) repayment of interest or principal is made ). Thus S the then Rs.500 is called as Future Value Fv of P todays Rs.100 Note: Future Value Fv of todays Rs 100 is not value of Rs 100 after n years Simple interest  is calculated only on the principle P, or original amount of a loan. Here the interest due is paid regularly at the end of every year/period on borrowed money for a pre-decided period called as Term . The principle, or original loan amount is paid back at the end of term to the lender. 1.Simple Interest paid every year If principal is P & Interest Rate =r % per year then Simple Interest to be paid every year = P.r And Amount to be returned at the end of term =P

43. Thus, if simple interest is charged at 5% on a Rs. 10,000 loan that is taken for a three-year period, . Interest paid at end of every yr = 10000x0.05=500 (for 3 yr ) After 3 yrs original loan amount 10000 is also returned. This Simple Interest method is still adopted in commercial world. 2.Simple Interest not paid every year The formula for calculating simple interest for n years is: Simple Interest = Principal x Term of the loan x Interest Rate = P.n.r S= Sum accrued or Future valve of P = Fv of P = P + Pnr Thus, if simple interest is charged at 5% on a Rs. 10,000 loan that is taken for a three-year period, the total amount of interest payable by the borrower after 3 yrs is calculated as  Simple Interest= Rs 10,000 x 0.05 x 3 = Rs, 1,500. Total amount to be paid by borrower after 3 years is =10000 Principal + 1500 interest Fv of principal P 10000 after 3 yrs= 11500 as simple interest is being charged. ( This cumulative Simple Interest policy is never adopted in commercial world .)

44. Compounding technique / Compound interest: If interest due is not paid at the end of every year on borrowed money, interest and interest on gets accumulated here generally compound interest is charged every year on the principal amount and also on the accumulated interest of previous periods, and can thus be regarded as “ interest on interest .” Generally compound interest is taken into account in business transactions. Compound Interest (CI)= Say Principle = P = having Present Value = Pv is invested or lent for n years S = sum accrued (accumulated) after the deposit term period of n years is over. It is also called as Future Value of P . Or S= Fv of P n = Term of deposit = No of compounding periods = generally No. of years for which P is deposited. Note : If compounding (interest calculation) is t times a year then No of compounding periods = tn ) r = Rate of Compound interest in %age per year unless stated otherwise. Compound Interest CI = Total amount of Principal and Interest in future less the Principal amount. CI = S-P = Future Value of P - Present Value of P = Fv -Pv In Compound Interest the lump sum Money is received by lender after the term of n yrs is over, the lump sum amount includes principal + Compound interest ( As there is no pay back in the intervening period ) .

45. Year = Principle Amount in the beginning of year having Present value Interest accrued during year= Principle X rate of interest Amount accumulated at the end of the year = Principle + Interest accrued during the year =S 1 P PX r = Pr S sum = P+ Pr =P(1+r) = P(1+r) 1 2 P(1+r) 1 P(1+r) 1 X r S sum = P(1+r) 1 + P(1+r) 1 Xr =P(1+r) 1 (1+r)= P[1+r] 2 3 P(1+r) 2 P(1+r) 2 X r S sum = P(1+r) 2 + P(1+r) 2 Xr =P(1+r) 2 [1+r]= P(1+r) 3 n  P(1+r) n-1  P(1+r) n-1 X r   S sum = P(1+r) n-1 + P(1+r) n-1 X r =P(1+r) n-1 [1+r]= P(1+r) n Derivation Compound Interest : P=Principle at beginning of term having present value Pv , invested for n yrs becomes S @ CI r at the end of term n Then S or Fv of P after n yr compounding = P(1+r) n OR S = Fv of Pv = Pv( 1+r) n

46. Compounding technique or Compound Interest The compounding of interest can be calculated by the following equation: 1.Future value of a single flow (lump sum) The process of calculating future value will become very cumbersome if they have to be calculated over long maturity periods of 10 or 20 years. A generalized procedure of calculating the future value of a single cash flow compounded annually is as follows. S = Fv = Pv (1+r) n 1  Where, S = Amount accumulated at the end of the period n years & having future value as Fv , called as future value of P or P becomes S after n yrs Pv = Present value of Principle P at the beginning of the period r = Rate of interest /year n = Number of years or periods for which investment is made Net Compound Interest = I = S – P = P(1+r) n - P I = P[(1+r) n -1] ---- 2

47. Example Mr. A invests Rs. 1,000 for 3 yrs in a bank which offers him 5% interest compounded annually . Substituting the actual figures for the investment or Rs. 1000 in the formula Interest compounded annually @ 5% interest Principle at Beginning of year Interest Earned @ 5% Amount at end of yr 1 st yr 1000 50 1050 2 nd yr 1050 52.50 1102.5 3 rd yr 1102.5 55.13 1157.63 Mr. A has Rs. 1050 in his account at the end of the first year. The total of the interest and principal amount Rs. 1050 constitutes the principal for the next year. He thus earns Rs. 1102.50 for the second year. This becomes the principal for the third year. This compounding procedure will continue for an indefinite number of years. Amount at the end of year 1 = Rs. 1000 (1+0.05) == Rs. 1050 Amount at the end of year 2 = Rs. 1050 (1+0.05) == Rs. 1102.50 Amount at the end of year 3 = Rs. 1102.50 (1+0.05) == Rs. 1157.63 The amount at the end of the second year can be ascertained by substituting Rs.1000 (1+0.05) for Rs.1050, that is, Rs.1000 (1+0.05) (1+0.05)=Rs.1102.50 Similarly, the amount at the end of the third year can be ascertained by substituting Rs.1000 (1+0.05) for Rs.1102.50, that is, Rs.1000 (1+0.05) (1+0.05) (1+0.05)=Rs.1157.63

48. Thus, if compound interest is charged at 5% on Rs. 10,000 loan that is taken for a three-year period, I = P[(1+r) n -1] it would be: Rs. 10,000 [(1 + 0.05) 3  – 1] = Rs. 10,000 [1.157625 – 1] = Rs. 1,576.25. Or S = Fv of 10000= Pv (1+r) n = 10000(1+0.05) 3 = Rs. 11,576.25 While the total interest payable over the three-year period of this loan is Rs. 1,576.25, Whereas Simple Interest= Pnr = Rs 10,000 x 0.05 x 3 = Rs, 1,500. Total amount to be paid by borrower after 3 years is =10000 Principal + 1500 interest or Fv of principal P 10000 after 3 yrs = 11500 as simple interest is being charged. With CI 76.25 is required to be paid more In CI the interest amount is not the same for all three years because compound interest also takes into consideration accumulated interest of previous periods.

49. Q. what will be the sum of Rs.50/- in 10 years at 10% compound interest? A: S = 50 (1 + 0.1) 10 = Rs. 129.68/- Only Q: what will be the amount of C. I. accrued on Rs. 50 in 10 years at 10% A: I = P [(1+r) n – 1] = 50 [(1+0.1) 10 -1] = 50[(1.1) 10 -1] = 50X1.5937or I = 79.68/- Q . What will be the status of Rs. 10, 000 after three years, if it is invested at this point of time @ 11% ? Solution Fv = Pv (1+r) n = 10, 000 (1+0.11) 3 = Rs.13, 680 The status of Rs. 10, 000 after three years, if it is invested at this point of time, would be Rs.13, 680.

50.   Sinking fund; & its formula : A sinking fund is an account that is used for regularly (generally annually) depositing a fixed amount, so that it multiplies at a fair rate of compound interest and at the end of term the sum accumulated in the sinking fund is received. A sinking fund is nothing but cumulative future value of an amount regularly deposited at the end of each period. ( period could be annually, monthly, quarterly, half yearly etc ). It is also called as immediate annuity as amount is regularly & annually deposited at the end of each year. Sinking fund is created in order to 1.R ecover the invested amount in a business by depositing part of income in SF . 2.R epay a long term debt & prevent defaulting. 3.Replace a wasting capital asset. ( by keeping aside depreciation amount ) 4.Get a lump sum amount in the future for specific purpose like purchases of large fixed asset in the future etc.

51. 2. Future value of a series of equal amount deposited at the end of every year for n years in a Recurring Deposit scheme or Sinking fund. Derivation for Sinking fund Say If every year we get surplus Rs A from a venture or take out from our savings and deposit it @ annual CI r for n years in sinking fund. ( Say R = 1+r ) Then Rs A deposited at end of 1 st yr will remain in Sinking fund for n-1 year and will become = A(1+ r) n-1 = A(R) n-1 at end of n th year Rs A deposited at end of 2 nd yr will remain in Sinking fund for n-2 year and will become = A(1+ r) n-2 = A(R) n-2 at end of n th year Rs A deposited at end of ( n-1) th yr will remain in Sinking fund for n- (n-1) year or 1 yr and will become = A(1+ r) = AR at end of n th year Rs A deposited at end of n th yr will remain in Sinking fund for n- n year or 0 yr and will become = A(1+ r) n-n = A(R) n-n = A at end of n th year ( no interest ) S A = Sum of the future values of all A deposited during the period of n years Thus S A = A (R) n-1 + A(R) n-2 ------ + AR 2 +AR + A ----- Eq1 R. S A = R [ A(R) n-1 + A(R) n-2 ------ +AR + A ]- ---- multiplying both sides by R R. S A = A(R) n + A(R) n-1 ------ AR 2 + AR ---- Eq 2 R. S A - S A = [ A(R) n + A(R) n-1 ---- AR 2 + AR ]- [A(R) n-1 + A(R) n-2 --- AR 2 + AR + A ] {Eq 2 - Eq 1 }

52. R. S A - S A = A(R) n –A or S A [R-1] =A [ (R) n -1] S A [R-1] =A [ (R) n -1] This is Sinking fund formula S A = A[ R n -1] [R-1] S A = A[ R n -1] [r+1-1] put R= r+1 S A = A[ R n -1] = A [(1+r) n -1] r r S A is the total amount accumulated in n yr from Rs A (annuity) deposited at the end of every yr @ CI r in a sinking fund /RD In place of A if Rs 1 is deposited at the end of every yr for “n” years @ CI r t hen S the Sum received after n years = S 1 S 1 = 1[R n -1] = [ ( r+1) n -1] r r S A = A[R n -1 ] = A. S 1 if amount deposited is A in place of Rs 1 r Hence R n -1 = S 1 ---3 r

53. Say investment is Rs 1 which we want after the term This is called sinking fund factor = amount required to be deposited at the end of every year for n years to get Rs.1 after n years when CI rate is r % Say r is 8% ie 0.08 and period is 10 yrs 0.08/1.08^10-1=0.069 0.069 if deposited at the end of every year for 10 years @ 8% it will get you Rs.1after 10 yrs To get Rs.10000 after 10 yrs. yearly deposit will be 10000x0.069=690 Check 690 [1.08^10 -1]/0.08 = 10000 = sinking fund factor

54. Hoskold’s formula

55. Hoskold’s formula for mine valuation Introduction : The first scientific approach was made by H.D. Hoskold (1877)for mine valuation prior to the advent of DCF ( discounted cash flow) analysis and corporate income taxes. Where a mining enterprise does not invest in increasing the reserves by acquiring further properties then the normal practice is to make a periodical payment to the investors (mine owner or other investors in co.) in the form of dividend or profit , which would not only cover the risk on the capital but provides additional funds which can be invested by him at safe rate of compound interest which will help the investor get back the capital invested, by the time the mine is worked out. Additional payment is actually for recovery or Redemption of capital invested No inflation is taken into account . Although corporations today do not generally use the Hoskold formula to determine mineral property values, governments sometimes use it to determine value for the purposes of taxation .

56. In Hoskold formula case we can have Two types of investors : 1.An investor invests capital (money) P from his own pocket for a period of n years or purchases a mine having life n years and expects a constant minimum dividend d/year. This d should be equal to his speculative rate of interest on investment P + an additional amount , such that by depositing this additional amount in some safe RD like fund ( sinking fund ) annually he can recover back his original investment P by the end of the term ( n years) of investment. The speculative rate of interest covers his risk on investment hence it is always higher than bank deposit rates and as per his own expectations , this is simple interest as he is getting it regularly at the end of every year, where as the interest that he will get from sinking fund is lower and at compound interest also called as safe interest . 2. An investor borrows capital (money) P from bank for n years on certain rate of interest and buys a mine having life more than n years . He also expects same constant minimum dividend d/yr. This d should be equal to rate of interest on his borrowing from bank + an additional amount, such that by depositing this additional amount in some safe RD like fund (sinking fund) annually he can recover back his original investment P by the end of the term ( n years) of loan to pay back the bank loan. Once the bank loan is paid back he can enjoy dividend d for rest of mine life.

57. Hoskold considers that a mining investor, would invest amount Pv for purchase of mine for n years if he gets constant annual dividend /income d on his investment for n years. After n years he is no more connected with the mine. This d should be equal to his speculative rate of interest /return on investment Pv + additional amount to recover investment capital Pv after n years from a sinking fund. Say Pv = P =investment =purchase value of mine at the time of investment. Annual Dividend /income/return / interest on Pv = d r 1 % = Compound rate of Interest per year from sinking fund for recovery of capital Pv invested in n years. r 2 % = expected speculative or risk rate of interest per year on capital invested Note : r 2 simple interest > r 1 Compound interest Simple interest amount or return per year on investment Pv = Pv.r 2 d - Pv.r 2 is surplus amount left for depositing annually in sinking fund. We have from sinking fund formula if any amount say A is deposited annually at Compound interest r for n yrs it becomes S A = A [(1+r) n -1] r

58. If this ( d - Pv.r 2 ) is deposited annually at Compound interest r 1 for n years it becomes as per sinking fund formula : Sum S = ( d - Pv.r 2 ) [(1+ r 1 ) n -1 ] say R= 1+ r 1 r 1 S = ( d - Pv.r 2 ) R n -1 but S should be equal to P or Pv or present purchase value of mine r 1 Hence Pv = ( d - Pv.r 2 ) R n -1 replacing S with Pv r 1 Pv. r 1 = ( d - Pv.r 2 ) R n -1 Or Pv. r 1 = d (R n -1) - (Pv.r 2 ) R n -1 Pv. r 1 + Pv r 2 (R n -1) = d (R n -1) Or Pv [ r 1 + r 2 (R n -1)] = d (R n -1) Pv = d (R n -1) r 1 + r 2 (R n -1) Pv = d r 1 + r 2 R n – 1 Where Pv is the present purchase value of the mine as per HOSKOLD formula

59. HOSKOLD FORMULA A= constant earning/year during the life of the mine (fixed annuity or dividend ) r 1 =safe rate of Compound interest from sinking fund, which is in %.  r 2 =speculative rate of simple interest, which is in %. r 2 = simple interest > r 1 Compound interest  Pv= Present value of the property. If A, n , r 1 and r 2 are known Pv can be found out  Say R= (1+ r 1 ), f= Hoskold’s factor. Then Pv= A.f

60. HOSKOLD FORMULA in brief Hoskold considers that a mining investor, would invest amount Pv for purchase of mine if he gets constant annual dividend /income d on his investment . This d should be equal to his expected rate of simple interest every year on investment Pv + an additional amount to set aside in a sinking fund which can get him back his invested capital Pv after n years. A= constant earning/year during the life of the mine (fixed annuity or dividend ) r 1 =safe rate of Compound interest from sinking fund, which is in %.  r 2 =speculative rate of simple interest, which is in %. r 2 = simple interest > r 1 Compound interest, d= r 1 + r 2 & Say R= (1+ r 1 ) Pv = Present purchase value of the property. n = life of mine or period of investment in mine If A, n , r 1 and r 2 are known Pv can be found out

61. Gestation period The annuity (annual dividend) d does not become available during gestation period for some yrs after purchase of mine. gestation period m is period in years before start of income If the gestation period is m yrs then the present value at the date of purchase is found by discounting the present value of mine at the beginning of production period ( end of gestation period) at the bank interest rate or expected interest rate applicable. Say interest rate during m year gestation period is r 3 % present value at the beginning of production period = PV then Present value on the day of purchase = PV p PV p = PV/(1+ r 3 ) m    

62. Or A- r 2 = r 1 R n – 1 = dividend on capital invested - rate of speculative interest on capital .

63. To calculate dividend required to get back investment in given mine life period Say Rs 1 is invested then to get it back after the term 20 yrs , dividend - speculative interest on Rs 1 will be invested in SF which will yield Rs 1 after the term 20 yrs Say dividend = Rs d , speculative interest on Rs 1 = Rs 0.15 then extra income d-0.15 should be invested in SF which will yield Rs 1 after the term 20 yrs or S= A [(1+r) n -1] r 1= [ d-0.15]x [(1+0.1)^20 -1]/0.10 or d-0.15 = 0.10/ (1.1)^20 -1 from above d-0.15 = 0.10/[(1.1)^20 -1] Or d =0.1674=dividend Check 0.1674 -0.15= 0.0174 invested for 20 yr in SF will yield 0.0174[(1.1)^20 -1]/0.1= Rs 1 Calculate dividend.

64. Pv = A r 1 + r 2 ( 1+ r 1 ) n –1 A= constant earning/year during the life of the mine (fixed annuity or dividend ) r 1 =safe rate of Compound interest from sinking fund, which is in %.  r 2 =speculative rate of simple interest, which is in %. r 2 = simple interest > r 1 Compound interest  Pv= Present purchase value of property. If the earning A is t times / yr & interest is also compounded t times / yr and interest rates are per year then Pv = A r 1/ t + r 2/ t ( 1+ r 1/ t ) t n –1 If the earning A is once/ yr & interest is also compounded once / yr then

65. No of compoundings S = Fv = Pv (1+r) n Here total time period of investment/deposit is denoted by n = no. of years and interest compounding is once in a year means the interest in 2 nd year will be on principle P + interest on P for 1 st year = r(P + Pr ) = P r (1+r) In total Period of n years no. of compounding is n times If interest r is compounded 2 times in a year the interest rate will be r/2 ( as r is annual compounding rate) and no. of compounding periods in n years will be 2n If interest is compounded t times in a year the interest rate will be r/t (as r is annual compounding rate) and no. of compounding periods in n years will be tn If the Interest is compounded t times in a year (quarterly, half yearly, monthly etc ) S = Pv (1+r/t) tn Higher the value of t more is beneficial to investor For deposit of Rs 10000 for 5 years @ 8% CI, Investor gets Rs =10000(1.08)^5= 14693 by annual compounding and 10000 (1+0.08/2) 2x 5 = 14802 by half yearly compounding. 10000 (1+0.08/4) 4x 5 = 14859 by quarterly compounding. 10000 (1+0.08/12) 4x 5 = 14898 by monthly compounding

66. A. Sinking fund amount if interest is compounded t times in a year but annuity deposited once in a year S A = A[R n -1] [R-1] S A = A[(1+r) n -1] compounded 1 time /yr [(1+r)-1] S A = A[(1+r/t) tn -1] compounded t times /yr 4 [(1+r/t) t -1] Sinking fund amount of annuity ( ie deposited annually ) or Rs 500 if interest is compounded quarterly in 10 years at 8% annual CI S A = A [(1+r) n -1] = 500 [(1+0.08) 10 -1] = 7243 compounded 1 time /yr r 0.08 S A = A[(1+r/t) tn -1] compounded t times/ yr as R=1+r [(1+r/t) t -1] S A = 500[(1+0.08/4) 4x10 -1] compounded 4 times / yr = 7323 [(1+0.08/4) 4 -1] = 7327 It is more than 7243

67. B. Sinking fund amount if interest is compounded t times in a year & annuity is also paid t times in a year for n years In sinking fund if amount A is deposited at the end of every year for n years final amount after n years becomes: S A = A[R n -1] [R-1] S A = A[(1+r) n -1] compounded 1 time /yr [(1+r)-1] S A = A[(1+r) n -1] compounded 1 time /yr r S A = A[(1+r/t) tn -1] compounded t times /yr r/t The periodic contribution to SF is denoted by A. Annualized rate of interest of the fund denoted by r Frequency of the periodic payment/yr denoted by t Then the periodic interest rate is = r / t Number of years is denoted by n. Total number of periods = n * t

68. Sinking fund amount of annuity or Rs 500 if interest is compounded quarterly in 10 years at 8% annual CI if annuity is also quarterly. compounded 1 time / yr S A = A [(1+r) n -1] = 500 [(1+0.08) 10 -1] = 7243 compounded once a yr r 0.08 S A = A[(1+r/t) tn -1] compounded t times and annuity quarterly as R=1+r r/t S A = 500[(1+0.08/4) 4x10 -1] compounded 4 times / yr 0.08/4 = 30200

69. Time required to get back investment amount Say investment amount is Rs 1 = S Dividend from a mine = return/income from a mine = Rs D % per yr Bank interest rate on loan if investment money is borrowed or speculative interest rate expected if investment money is spent from pocket simple interest = r 2 % Sinking fund CI rate r 1 % Annual contribution to SF = D-r 2 = A ( income – interest paid will be saved in SF) Time required to recover investment amount of Rs 100 from SF= n yrs Time required to get amount S S = A [(1+r 1 ) n -1] r 1 Say Rs 1 is invested for n yrs D , r 1, r 2 will be in decimal ie 12%=0.12 1 = D-r 2 [(1+ r 1 ) n -1] r 1 D-r2 = r 1 [(1+r 1 ) n -1] R n – 1 = r 1 or R n = r 1 +1 D- r 2 D- r 2 n= log r 1 +1 Log R D- r 2

70. E.g. it is proposed to buy a mine which is expected to yield a 12% dividend on capital invested. If the rate of interest on capital is 7% & the contribution to the sinking fund is at the rate of 5% then how many years will it take to repay the investment. R n = r 1 +1 Answer: D- r 2 D= 0.12, r 1 = 0.05, r 2 = 0.07, ( 12% dividend means, Rs 0.12 earning on Rs 1 investment) D- r 2 = 0.12 -0.07 = 0.05 R n =0.05/0.05 +1 =2 (1.05) n = 2 or n = log 2/log 1.05 n = 14.2 years.    

71. Re exam S 22 Production from mine starts after 3 yrs from purchase with profit10 L annual Mine Life 20 yrs speculative rate desired 15% and safe rate 8%, rate desired during gestation period 10% on investment. Find purchase value. A = Annuity =10L r 1 =8%, r 2 =15%, r 3 =10%, n=20 yrs Pv after 3 yr =10L/[0.08/(1.08)^20 -1]+0.15 =581895 Pv now = 581895 /(1.1)^3 =437186

72. Re exam S 22 Production from mine starts after 3 yrs from purchase with profit10 L quarterly Mine Life 25 yrs speculative rate desired 15% and safe rate 8%, rate desired during gestation period 10% on investment. Find purchase value. A = dividend Annuity / Qr =10L r 1 =8%, r 2 =15%, r 3 =10%, m=3yrs n=25 yrs t=4 Pv = A r 1/ t + r 2/ t ( 1+ r 1/ t ) t n –1 = 10L 0.08 /4 + 0.15 / 4 ( 1+ 0.08 / 4 ) 4x25 –1 Pv at begining of 25 yr life =245.7 L , Present Pv =245.7/(1+0.10) 3 = 184.6 L

73. Limitations : Hoskold formula doesn`t assure safety of mining investment. The factors upon which it mainly depends are the life of the mine and b) the interest rate & dividends. Any variation in these estimates will profoundly affect the resulting figure for present value. Moreover the mining entrepreneur never keeps the money aside in sinking fund for procuring a replacement deposit but always likes to reinvest in the business. Mining companies invest in projects that offer the highest return ; they do not invest portions of cash flow at a riskless return to provide funds to replace depleted deposits. Therefore, the Hoskold formula uses a lower discount rate than a corporation would actually use, which results in an overvaluing of the project. It also doesn’t take into account the time value of money.

74. Application ( amortization/ recovery of capital ) From 2.18 , n is obtained on basis of r & r` A and P. From n and a, S is known from S ,depth C is found P = present value a = capacity/ yr r = CI rate A= Earning/ yr

75. R^n –1 = r 1 d – r 2 .

76. E.g. it is proposed to buy a mine which is expected to yield a 12% dividend on capital invested. If the rate of interest on capital is 7% & the contribution to the sinking fund is at the rate of 5% then how many years it will take to repay the investment . R n = r 1 +1 Answer: d- r 2 Here the entrepreneur is buying a mine he borrows money from bank for which he has to pay simple interest per year 7% for whole term and pay back original borrowed amount at the end of term n. Say Rs 100 is borrowed and invested then d= 12, r 1 = 5, r 2 = 7, d- r 2 = 12 -7 = 5 R = 1+0.05 =1.05 (R =1+ r 1 ) R n =5/5 +1 =2 (1.05) n = 2 or n = log 2/log 1.05 n = 14.2 years.     For 14.2 years he will pay Rs 7 /yr to bank simple interest . After 14.2 years Rs 100 collected in sinking fund will be returned to bank, there after entire dividend 7% received is his profit till end of mine life.

77. E.g. it is proposed to buy a mine which is expected to yield a 13% dividend on capital invested. If the rate of interest on capital is 7% & the contribution to the sinking fund is at the rate of 4% then how many years it will take to repay the investment. R n = r 1 +1 Answer: D- r 2 Here the entrepreneur is buying a mine he borrows money from bank for which he has to pay simple interest for whole term and pay back original borrowed amount at the end of term. Say Rs 100 invested D= 13, r 1 = 4, r 2 = 7, D- r 2 = 13 -7 = 6 R = 1+0.04 =1.04 R n =4/6 +1 =1.67 (1.04) n = 1.67 or n = log 1.67/log 1.04 n = 13 years.   Check 6[(  1.04)^13 -1)/0.04 = 100 app

78. A mine has reserve for 20Years what remunerative interest (or dividend ) is expected on investment if speculative rate of int is 15% and current safe rate for capital redemption is 10% If A is deposited /yr after n yr it becomes A[ (r+1)^n -1]/r Say Rs 1 is invested then A is deposited / yr it should yield Rs 1 after n yrs Say Rs 1 is invested, dividend on Rs 1 = d Rs d-0.15 should be invested in SF @ CI 10% which will yield Rs 1 after the term of 20 yrs 1= d-0.15 [ (1+0.1)^20 -1] 0.10 or d-0.15 = 0.10/[(1.1)^20 -1] Or d =0.1674= or dividend =16.74% Now 0.1674 -0.15= 0.0174 will be invested for 20 yr in SF will yield Rs 1 check 0.0174[(1.1)^20 -1]/0.1= Rs 1 Find dividend required to get back investment within given mine life period

79. Simple cash flow ( no loan, no inflation, no DF ) : Income = Revenue - Capital cost– Operating cost – tax Year=n Capital cost-a Revenue-b Operating cost-c Tax Payment-d Cash Flow-E=( b- a-c-d ) Cumulative CF=F F n = F n-1 + E n 1 5,000 -5000 -5000 2 7,000 -7000 -12000 3 6,000 2,000 4000 -8000 4 6,000 2,000 4000 -4000 5 6,000 2,000 200 3800 -200 6 6,000 2,000 600 3400 3200 7 6,000 2,000 1,000 3000 6200 8 6,000 2,000 1,600 2400 8600 9 6,000 2,000 1,900 2100 10700 10 6,000 2,000 1,900 2100 12800 net gain in 10 yrs from venture. Here the value of money is considered constant and cashflows are taken at their face values

80. Modern Mine property valuation methods:

81. Present value concept: Future value Fv = Future purchasing value / purchasing capacity of an amount S Present value Pv = Present purchasing value/ purchasing capacity of an amount P Future value Fv of P, is future equivalent of P Present value Pv of S is present equivalent of S Eg. In the year 2023 Present value of Rs 100 is Rs 100 which can buy X gms of gold. Say someone deposits /invests Rs 100 to day and gets @ r (compound interest) rate of return Rs 500 in 2030 after 7 years , the then Rs 500 is said to be the future value of todays Rs 100 ( though in the year 2030 the then Present value of Rs 500 will be Rs 500 only) and taking inflation into account it is presumed that in 2030 Rs 500 will be required to buy same X gms of gold considered earlier. Present value= Pv Rs 100 Future value = Fv = Rs 500 or Future value /Fv of Rs 100 after 7 yrs = Rs.500 Fv of Pv 100 = Rs.500 =100(1+r)^n= Pv(1+r)^n Fv=Pv(1+r)^n this r is called compounding rate Compounding rate is used to calculate future value of any present value Conversely Pv = Fv/ (1+r)^n this r is called discounting rate Same Compounding rate when used to calculate present value of any future value is called discounting rate and generally denoted by i

82. Present values of a single cashflow Pv = Present value or Present purchase power of Principle P at the beginning of the period say P=Rs100 r = Rate of bank compound interest /year = say 10% n = Number of years or period of investment say 5 yrs S = Amount accumulated at the end of the period n years & having future value or future purchase power Fv ,called as future value of P S = Fv = Pv (1+r) n = 100 (1+0.10) 5 = 161 P = Pv = Fv/ (1+r) n = 161/ (1+0.10) 5 = 100 Rs 100 if kept in bank for 5 yrs @ 10 compound interest you will get Rs 161 after 5 yrs It is presumed that today`s 100= Rs 161 received after 5 yrs or what can be purchased today for Rs 100 will cost Rs 161 after 5 yrs

83. Solved Problem If Ms. Sapan expects to have an amount of Rs. 1000 after one year what should be the amount she has to invest today, if the bank is offering 10% interest rate? Solution Fv = Pv (1+i) n  Pv= Fv /(1+i) n = 1000/(1+0.10) 1 = Rs. 909.09 ie The amount to be invested today to have an amount of Rs, 1000 after one year is Rs. 909. Q . An investor wants to find out the present value of an amount of Rs. 100000 to be received after 15 years. The interest offered by bank is 9%. Calculate the Pv of this amount. Solution Pv= Fv /(1+i) n or 100000/ (1+0.09) 15 = 100000*0.275 = Rs. 27500 The PV of Rs. 100, 000 is Rs. 27,500.

84. Eg Rs 500 worth todays deposited @ 8% CI for 6 yrs becomes 500 (1+r) n = 500 (1+0.08) 6 = 793 Or the Fv of 500 is 793 if compounded @ 8% for 6 yrs Similarly Rs 793 likely to be received after 6 yrs if discounted @ 8% is worth Pv = Fv / (1+r) n = 793/ (1+0.08) 6 = Rs 500 today. Q. what is the present value of Rs.50/- due in 10 years @ 10% compound interest? Answer: Pv or V P = S/ (1+r) n = 50/ (1.10) 10 = 50/2.5937 = Rs.19.277/- only   Mr. A requires Rs.1050 at the end of the first year. Given the rate of interest as 5%,find out how much Mr. A would invest today to earn this amount. Solution If P is the unknown amount, then P (1+0.05) =1050 P=1050/ (1+0.05) =Rs.1000 Thus Rs. 1000 would be the required principal investment to get Rs . 1050 at the end of the first year at 5% interest rate. The present value of the money is the reciprocal of the compounded future value .

85. Minimum expectation of investor : The ore reserves intended for mining must ultimately pay for the development of the deposit; environmental protection expenses before, during and at the closer of mining ; direct costs of operation, transportation , refining (if necessary ); and indirect costs including overhead charges; tax impositions ; interest payments on working capital & investment capital and royalties . AND Return the invested capital to the investor along with acceptable rate of return ( profit ) in percentage on his investment . Every individual has an expectations in the form of return from the venture, he puts money in. Minimum expectation of investors in the form of return will be different for different investors , if the calculated returns are as per the expectations of an investor then only the purchase deal can materialize.

86. Discount rate Def :The minimum acceptable rate of return in % age, from a business is called as Discount rate. This discount rate is used by investor to discount the future cash flows receivable from different businesses for finding their present values and comparing their profitability. Discount rate / minimum acceptable rate of return =Risk free rate ( cost of capital) + Risk premium. Higher the perceived risk, higher is risk-adjusted discount rate . This can be compared with the compound interest he gets from bank but at more %age as he is putting money in a risky venture . This Discount rate is used by the investor to find out the purchase value of a mine with given series of cashflow.

87. 1.Discounting or finding present value of a single cashflow at a point of time in future by an investor in a business Investors expectation in terms of CI is obviously more than bank rates offered, due to risk involved in business this is as per individual`s risk appetite. Say ‘ S ’ is sum offered after a period of ‘ n’ years by a business. Then present value of S for the investor = P = Pv = S /(1+i) n where i is investors expected rate of return/discount factor for an investment of Rs P This i is investor`s minimum expected rate of return in % age or discount rate it could be different for different investor. If discount factor of Mr X is 10 % then he is ready to part with Rs 100 today if he gets Rs 161 after 5 yrs because = 100 = 161 /(1+0.1) 5 Whereas Mr Y is ready to part with Rs 100 today if he gets Rs 201 after 5 yrs because his minimum expected rate of return in 15% and 100= 201 /(1+0.15) 5 Note : Interest rate r is decided by bank or the company raising loan whereas discount rate i is investor`s choice.

88. 2. Present value of series of uneven incomes to be received in future. ( This actually happens in industry ) Say C1,C2 ------- & Cn as future incomes at end of 1st, 2nd and nth year from now, for a given investment P in a venture. Present value = Pv of any Fv = Fv / (1+i)^n & i is rate of return expected or discount rate. The present value of above cash flow = A1/(1+i) + A2/(1+i)^2 +--- -+ An/(+ i )^n The year wise incomes generated from the mine can be compared with a series of cashflows as above and investment P can be compared with investment in a mine. Thus the present value of cashflows generated from the mine can give its present purchase price.

89. Modern Mine property valuation methods: Modern Mine property valuation methods are based on Dynamic or Discounted cash flow analysis . These are most objective methods of Mine property valuation They don’t give same weight (value) to the distant and near earnings & take in to consideration the timing of cash flows . They make fair estimation of mining property if reasonable rate of discount is applied . Modern Definition of value of mine : The present fair market value of a mine is based on its future earning capacity and is equal to sum of the present values of its future annual incomes (cash flows). To arrive at present value of the future annual incomes, the incomes are discounted as on date of purchase at the expected rate of return by individual purchaser or by economic experts. The Modern Mine property valuation methods are : a . Net Present Value Method. b. Internal Rate of Return Method c. Profitability index Method d. Discounted payback period method.

90. Main three Business activities are financing, operating and investing . Financing = Collecting money for investment from investors, banks, lenders or financial institutions. ( This money ideally goes for investing when spent in acquiring assets /equipment for production and meeting other cost of production . ) Operating =Day to day direct/indirect activity to produce and sale minerals to generate revenue. Operating cost is incurred while doing this. Investing = Using company resources or financed money to acquire assets for operating activity to generate revenue. Sources of funds/capital To run a business capital is one of the prerequisites. Capital market provides source of funds through public equity ( shares ) and loan (debt ) in the form of bonds and debentures . Shares are like investments : On shares, share of profit ie dividend is given to share holders by the industrial company . Dividend is profit or surplus got after meeting all costs of production, taxes, payment of loan installments , loan interest and investment etc. Theoretically Profit or Dividend doesn’t come into picture unless initial investment is fully recovered . Bonds and debentures are issued against loan raised: S imple or compound interest is paid to lender by the industrial company to lender irrespective of profit or loss .

91. Types of financing : In case of investment An investor invests his capital in an industrial company instead of keeping his money in bank with a hope to earn more returns /profit /dividend on it. If invested in a company ,company`s liability is only to pay dividend (share of profit) proportionate share capital to investor, every year so long as the individual is owner of shares. Nothing else apart from this is the liability of company towards share holder or business promotors. B. In case of borrowing by Co. Lender expects pre-decided interest at more rate than risk free banks rates along with principal, whether co. makes profit or loss. lender can be paid back in following ways: Lender gets simple interest on his lent amount at the end of every year and also gets back the original amount at the end of term. At the end of term amortization (co. liability) is complete.

92. 2. Lender gets a lumpsum amount equal to future value P (1+r) n @ compound interest r on his lending amount P after the end of term n years . At the end of term the amortization is complete with payment of S= P (1+r) n . 3.Lender can be payed an amount comprising of equal instalments of principal along with interest on balance loan amount Eg. say Loan Rs.500 for 5 yrs @ 10% he will pay 100+50 int=150 at the end of 1 st year 100+40=140at the end of 2 nd year ---- and 110 at the end of 5 th yr ( This is common practice here amount paid changes every year) 4. Lender can be payed a n annuity A = [ P X r (1+r) n /[(1+r) n -1] at the end of every year till the end of term like EYI( equal yearly installments, consisting of part of principal and interest on balance principal ). At end of term the amortization is complete. (here amount paid remains constant every year)

93. Important terms Capital or capital equivalent is resources used to produce goods/services, it includes physical assets, equipment ,stock, bond, cash or say money invested in co. through share, loan etc. For producer In a year Revenue = What co. gets by selling mineral or what capital comes into the Co. Cost = Cost of producing mineral ie Resources in monetary terms , used for the purpose of mineral production this includes Direct cost of prod = (cost of prod +wages +fuel+ freight etc) it is directly proportional to production + Indirect /overhead cost of prod = ( office rent +wages +interest +administration + depreciation + some taxes ) & all expenses and losses at mines and administrative level.) it is independent of production. Profit = return=dividend investor gets on his investment = Net profit or Return = Revenue ( Sale price x Production) – total Cost ( direct + indirect /overhead cost of prod) Income is actual amount earned by co in a year Income from business = Net profit – dividend If dividend is not paid then income = net profit For our purpose per year , Profit / Return/ income = Revenue – Total cost of prod

94. The production capacity of a mine is generally pre-estimated by mining experts, based on reserves, extent of mechanisation, carrying capacity of haulage infrastructure and numerous other techno-economic variables of the mine. Based on the production capacity of a mine and market economics the annual cash inflows and out flows in the business can be determined. Cash flow means net cashflow or income or profit on given investment. Cash flow determines position of cash balance in account at a point of time. Annual cashflow in a particular year = Annual cash inflow into the business – Annual cash outflow from the business. Annual cash inflow in a business is = Revenue generated + loan taken in the year Annual cash outflow is = Capital Investment + Operating expenses + working capital spent in that yr + Royalty + Corporate Tax + interest on loan + Loan repayment [ Operating expenses includes direct and indirect cost ( overheads) ] Period/term/period of investment is period in years for which money is invested in the business with expectation to get back investment along with return on it.

95. Net Cash flow or cash flow in a year = Revenue + loan taken if any – capital Investment – Operating expenses – working capital spent in that yr – Royalty – Corporate Tax – interest on loan – Loan repayment If tax component is involved : Year wise Taxable income = Revenue – Actual Cost incurred on production = ( Revenue – Operating expenses – working capital spent in that yr – Royalty – Depreciation – Interest paid on borrowings) ( NO loan raised or loan repayment or capital investment made is considered for calculating Taxable Income ) b. Corporate Tax= taxable income X tax rate ] Cash flow could be even negative in a year if expenditure or cash out flow is more than cash inflow. Accordingly Cash flow will be +ve or – ve as case may be. The date of investment is termed as O year, this date is beginning of 1 st year. Transactions made during any year are accounted for at the end of that year.

96. Return submitted to IBM Sl.No . Item C ost of production in MCDR returns Cost per unit (`) (i) Direct Cost (a) Exploration (b) Mining (c) Beneficiation(Mechanical Only) (ii) Over-head cost (iii) Depreciation on machinery (iv) Interest on borrowing (v) Royalty (vi) Payments made to DMF (vii) Payments made to NMET (viii) Taxes (ix) Dead Rent (x) Others (specify) Total

97. Purchaser of a mine is ready to invest money in a mine because the mine can generate a given income or cash flow in future against his investment. These incomes received in future , have certain future value. Def: Year wise Cashflow from a mine is sum of series of projected cashflows at the end of every year, right from the 0 year ( ie date of purchase or date of investment) till the end of period. For eg if a mine is purchased on 1 st Jan 2025 investing Rs 5 Cr. 01.01.25 to 31.12.25 will be 1 st year . 01.01.26 to 31.12.26 will be 2 nd year and so on. Say revenue in 1 st year is Rs 30 L and cost incurred 26 L, revenue in 2 nd year is Rs 25 L and cost incurred 10 L, ----- revenue in n th year is Rs 40 L and cost incurred 20 L, Then cash flow in 0 yr is -5cr in 1 st year is 30-26=4L, cash flow in 2 nd year is 25-10=15 L cash flow in n th year is 40-20=20L Year wise Cashflow from the mine = - 5 cr + 4L+15 L +----+ 20L At the time of investment ie in year cash flow is invariably – ve due to investment expenditure. In gestation and reclamation period also it could be – ve . In the midst of mine life also it could be – ve if any heavy investment for expansion is made and revenue in that year is relatively less

98. In modern days the present value/worth of a mine (purchase value for an investor as on today ) Or worth capital investment is that sum/amount which is equivalent to future income to be received as annuity (fixed income annually) for a specified period of time or equivalent to a lump sum amount payable after that specified time. These future incomes will of course be acceptable to investor only if he gets fair rate of return (according to compound interest) on his today`s capital investment. Thus he can estimate the future earnings based on capacity of the mine and discount those incomes to present time and figure out the present purchase value of the mine. Discounting rate depends on his minimum expected rate of return.

99. For example: Someone is said to be ready to invest Rs 100 for 6 years in a venture, if he gets 12% return on it . It means he expects either 100 (1+0.12) ^6= Rs 197 ( like future value of a single flow by compound interest ) after 6 yrs OR he expects annuity ( A = P X r (1+r) n /[(1+r) n -1] ) =100x0.12(1.12)^6/(1.12)^6-1= Rs 24.3 per year. ( annuity of Rs 24.3 per year for 6 years like in Diminishing Annuity method or EYI) These are equivalent future values of his today`s investment Rs 100. Because present value of Rs 197 received after 6 yrs or present value of annuity of Rs 24.3 per year for next 6 years is Rs 100. However in business we don’t get same incomes every year hence we have to discount ( find present values ) of all future incomes as on today and add them to arrive at present value of entire cashflow . As the expected rate of return will be different for different individual investors and so will be the purchase value in their views. In modern times this present value concept is preferred.

100. Investors whether business promotor or share holders expect certain minimum return or profit or income on their invested amount in terms of % age in the form of compound interest , below this rate he is not willing to part with his money. An entrepreneur investing money in a mining project would either borrow money from market/banks and put it in the mining venture, obviously his expected rate of return ( on compound interest basis )from the mining business would be fairly more than the interest rate to be paid on borrowings. Or he may put money from his own pocket in the mining business and expect rate of return from the mining business that is fairly more than the interest rates that he could have fetched on fixed deposits from banks /bonds Or from an alternative business . So he will discount a particular mining project cash flows with his expected rate of return and find out the present purchase value. He can also asses the actual rate of return based on sale price and ensuing cash flows from mining project . In short the Minimum expectation of an investor is the desired profit in %age on his investment after meeting all expenses & recovery of his invested amount.

101. Pv Present value of a mine as per Hoskold A = constant earning or dividend /year during the life of the mine (fixed annuity ) r 1 =safe rate of Compound interest from govt sinking fund, which is in % . R= (1+ r 1 ),  r 2 =speculative rate of simple interest, expected by investor. (r 2 = > r ) Present Purchase value of a mine as per OLD method

102. 4. Calculate Net present value NPV= present value - Initial or todays investment required

103. Steps or factors involved in Mine Valuation (modern methods) Step-I (Technical appraisal) 1 .Estimation of ore reserves : By assessing Geological Resources & Grades, Mineable reserves & Grades, Mining losses, effective grades, cut off grade ,minimum mining grade etc. 2. Determination of grade wise production rate of mine & life of mine. 3. Estimation of capital cost, (investment in fixed assets), working capital, other investment, and its phasing. 4. Estimation of all direct Operating costs, - including drilling, blasting, loading ,hauling labour, store, energy etc. and 5. Estimation of all indirect costs like overhead costs of plant infrastructure, equipment, utilities, mine management, loan and interest payments, taxes in view of depreciation, environmental & social cost in view of, government policies etc. 6. Projecting tentative Revenue for life of the mine based on market survey and forecast . Step-II ( Financial Appraisal ) Calculation of ensuing annual cash inflows (+ ve ), cash outflows(- ve ) and cashflow ( or net Cashflow(+ ve or – ve as the case may be). Calculate the present values of ensuing cashflows based on cash flows, applying the expected rate of return (discount rate) Find NPV,IRR and discounted payback period if it suits to your expectations.

104. Valuation of mining property Valuation : Valuation is the process of estimating the fair market value of a business in monitory terms by the experts considering its future earning capacity. The exchange of hands should be in an open and unrestricted market between a willing buyer and a willing seller at “ arm’s length ” (as is where is basis), with each party acting knowledgeably, prudently and without compulsion .

105. Discounted cash flow methods for project valuation . Net Present Value and Internal Rate of Return are the two basic methods for assessing project feasibility using Discounted cash flow analysis. NPV is a measure of value of a stock of wealth, whereas IRR is a measure of the efficiency of capital & gives the rate of accumulation of wealth. A. Net Present Value method for project valuation This method considers that the present fair market value of a property depends on its future earning capacity and time value of money. Say P is sale price declared by owner of a mine then, P = investment required to be made by buyer today in order to materialize the deal. Cashflow/ net cashflow is series of the actual cashflows received at the end of every year from 0 year till the end of term like C0, C1,C2 ------- & Cn depending on production capacity of mine, cost and revenue, where C0 or P is invariably negative as money goes out against investment. r = Acceptable rate of return or discount factor DF for buyer for given investment and period (R=1+r)

106. The cash flow from the end of 1 st year C1,C2 ------- & Cn must be generating some rate of return for given investment of Rs P . This rate of return is called internal rate of return irr from business on investment P An investor will purchase above mine only if the above internal rate of return irr agrees with his expected rate of return r or is more than that. To confirm this he will find out the sum of Present values of this cashflow by discounting C1,C2 ------- & Cn at his expected rate of return r and add them. (R= 1+r) C1/R^1 +C2/R^2+C3/R^3---------+ Cn/ R^n = Say P1 If P1 is more or equal to P then only he will purchase the mine.

107. Present value or Net Present Value of a cash flow = Present value or Net Present Value of a mine because mine generates cash flow that buyer is interested in. 1.Def. PV or P resent value or Purchase value of a mine (in the eyes of investor) is sum of the present (or discounted ) values of all future cash flows from mine @ investor`s acceptable rate of return r from the end of 1sr year till the end of period of investment. PV = C1/R^1 +C2/R^2+C3/R^3---------+ Cn/ R^n (R= 1+r) Where C1,C2 ----& Cn are expected cash flows from the mine at the end of1st , 2 nd ----& nth year . 2 . Def : NPV / Net Present Value is the sum of the present (discounted) values of all future cash flows from the yr ( from beginning of 1 st year ) till the end of period of investment. NPV = C0/R^0 +C1/R^1 +C2/R^2+C3/R^3---------Cn/ R^n ( where C0=C0/R^0 = - P) Or Net Present Value : NPV is equal to the sum of the present values of all future cashflows from end of 1 st year minus investment. NPV = P resent value – investment required ( P ) = PV- P

108. Q. An investor from purchase of mine will receive Rs. 10000, Rs. 15000, Rs. 8000, Rs. 11000 and Rs. 4000 respectively at the end of each of the ensuing five years. Find out the present value of this stream of uneven cash flows, if the investors discount rate/ expected interest rate is 8%. Solution Cashflow/Future values= 10000 + 15000 + 8000+ 11000 + 4000 = Rs.48000 Present values = 10000/ (1+0.08) + 15000/ (1+0.08) 2 + 8000/ (1+0.08) 3 + 11000/ (1+0.08) 4 + 4000/ (1+0.08) 5 =9260+12680+6351+8085+2722= app Rs.39276 Investor`s offer price is Rs 39276 or he can purchase the mine for maximum Rs 39276 & have his expected return 8%. In case the sale price of mine declared by mine owner is also Rs 39276 then the irr of mine is also 8% at Rs 39276 price for given cash flow & NPV will be Present values – investment = Rs.39276- Rs.39276= 0 , If he gets it in less than Rs.39276 say for Rs 35000 he is likely to get rate of return about 13% ie more than 8% & NPV =39276 – 35000= 4276 positive In above case if he is purchasing mine for Rs 40000, NPV =39276 – 40000= - 724 this NPV is –ve , he is in loss situation because the actual rate of return or IRR is < 8% its about 7% Deal wont get through.

109. Steps : Calculate and tabulate the actual year wise cashflows from 0 year till the last year. Discount the cash flows as on investment date applying DF and tabulate. 3.Add the discounted values of entire cashflow from end of 1 st year to last year .The sum of discounted values gives the present (purchase)value of the mine. If the sum is equal to or more than price declared by seller ( PV = or > P ) , the buyer can go with the deal. 4.Deduct initial investment P ( price declared by seller ) from PV ( Present value calculated above ) to get Net Present Value . If PV – P is + ve deal can be finalised if not, it can`t go through. or NPV = – P +C1/R^1 +C2/R^2+C3/R^3---------Cn/ R^n ( Where, – P , C1,C2 –Cn are cashflows at the end of 0 yr,1st, 2 nd ----nth year ( R =1+r where r is investor`s expected rate return or profit in % . (P is – ve because it is investment and – P = C0/R^0 )

110. 1.The economic experts determine sale price P of the mine based on reasonable and fair rate of return taking into consideration risk profile of the particular mining industry . This rate of return considered to arrive at sale price for given cashflow stream is called as irr .(internal rate of return) . 2. The present values PV estimated by individuals differ as per their expected rate of returns ( Df ) 3. Investment is supposed to be made in the beginning of 1 st year which is called as year and this is always – ve cash flow. ( Investment is considered as Money going out P=C0=C0/(R)^0 ) 4. Present purchase value or sale value can`t be arrived at, without applying some discount factor to future earnings because Present value means discounted value. 5. In a cashflow statement date of investment = beginning of 1 st year = 0 year all cash flows are indicated at the end of every year like end of O yr , end of 1 st yr -----end of n th year etc.

111. 6 . Internal rate of return ( IRR ) : In any business, arranging funds and its investment is the very first activity. Unless and until investment is there , no question of returns arises, because returns are expected only on investment in terms of % age. Every cashflow generates an inherent rate of return for the given sale price P it is called as irr . The cashflow remaining constant, different irr will give different sale prices. Sale price = C1/(1+irr)^1+C2/(1+irr)^2+C3/(1+irr)^3-------Cn/(1+irr)^n = P Or C1/(1+irr)^1+C2/(1+irr)^2+C3/(1+irr)^3-------Cn/(1+irr)^n – P =0 7. The sale price of a mine or deposit is declared by owner of mine( in consultation with experts ) based on its future earning capacity and his own profit margin. If this sale price is equal or less than present value acceptable to buyer at his discount factor the deal can be finalised. Sale price declared by owner = Initial investment required to be made by buyer in case the deal materializes. 8. Buyer`s Present value can be different from sale price declared by owner because of difference in rates of returns projected by owner and that acceptable to buyer.

112. 9. Actual Cash flows generated are based on not only the production capacity of mine but the efficiency of financing, investing and operating activities carried out during the respective years also. Revenue generated and cost incurred depends on proposed production rates management skill and market parameters. Productively used investment can recovers its value and generate profit more than expectations. 10. The discount rate applied to cash flow and investment cost, makes NPV positive or negative. Cash flows in the beginning years are generally negative due to more investments compared to revenue ,in subsequent years also it could be negative if out flow is more than inflow due to reinvestment etc . 11. Cashflow in any year = cash inflow in the business in that year - cash outflow from the business in that year . Revenue earned is cash inflow or + ve & Cost incurred and investment made is – ve or cash out flow. Cash flow in a year = Revenue + loan taken if any – capital Investment – Operating expenses – working capital spent in that yr – Royalty – Corporate Tax – interest on loan – Loan repayment

113. 1 . If NPV or sum total of the series is positive or > 0 at expected discount rate it means PV is > investment required & the project irr gives more than expected rate of return, the project is economically viable & can be accepted by investing Rs P today. H ere irr is > r . Present value acceptable to buyer > sale price. 2.If IF NPV < 0 or – ve at expected discount rate then irr < expected discount rate r of buyer or the project irr gives less than expected rate of return & Present value acceptable to buyer < sale price declared by seller. It is economically not viable & can not be accepted and he may put his money in bank or in other business where he can get at least expected rate of return. here irr is < r . 3.IF NPV = 0 , at expected discount rate, it means the project irr is just equal to buyer`s expected rate of return r. Present value of buyer = sale price of seller & investor may or may not accept the deal. NPV= C1/(1+r)^1+C2/(1+r)^2+C3/(1+r)^3-------Cn/(1+r)^n – investment (P) =0 Also C1/(1+irr)^1+C2/(1+irr)^2+C3/(1+irr)^3-------Cn/(1+irr)^n = Price P Or C1/(1+irr)^1+C2/(1+irr)^2+C3/(1+irr)^3-------Cn/(1+irr)^n – P =0 Hence irr is = r .

114. Indicators NPV or The sum of the discounted yearly cash flows gives the present value of the entire income stream right in the beginning of the time period ie on the date of investment, thereby helps in deciding about the viability of project . NPV criterion measures economic consequences, of time , by converting the all future cash flow amounts to equivalent amounts at the present time and sum of the discounted yearly cash flows determine net loss or gain . However, it is more difficult to calculate NPV by hand, hence computer iteration , graphic or excel format is used to find out NPV. Indicators : The larger the NPV, the richer the investor becomes by undertaking the project. The net present value is higher if the income amounts are larger , or if they come sooner , or if the discount rate or expectation is lower .  The net present value is lower if the income amounts are smaller , or if they come later , or if the discount rate is higher . The word " net " in "net present value" indicates that our calculation includes the initial and subsequent investments made ( shown negative ) as well as the subsequent profits earned (shown positive).

115. Project valuation in modern times . Say P is the initial investment required to be done today ie at the end of 0 year and A 1 , A 2 , ----- A n all are future uneven cash flows (inflows- out flow ) at end of 1 st 2 nd ---and N th year from today, r is discount rate expected by buyer on given investment P. Then NPV = - P + A 1 /(1+r) + A 2 /(1+r) 2 +-----+ A n /(1+r) n Today or at the end of 0 yr cashflow is –P because here there is no revenue and only investment is P. If NPV or sum total of the series is positive at expected discount rate the project is economically viable & can be accepted by investing Rs P today. Or the project is worth investing Rs P. If NPV or sum total of the series is negative it is rejected and he may put his money in bank or in other business where he can get expected rate of return Advantage : NPV or The sum of the discounted yearly cash flows gives the present value of the entire income stream right in the beginning of the time period ie on the date of investment, thereby helps in deciding about the viability of project . NPV criterion measures economic consequences, of time , by converting the all future cash flow amounts to equivalent amounts at the present time and sum of the discounted yearly cash flows determine net loss or gain

116. Year =n Amount Fv Rs. =Future Value Present Value factor pvf at DR 10% or 0.1= 1/(1+0.1)^ n Present Value = Fv X pvf. NPV= ∑ Pv 1.1.16 - 3000 cost 1 - 3000 + 790 1 31.12.16 1000 income .909 909 2. 31.12.17 1000 income .826 826 3. 31.12.18 1000 income .751 751 4. 31.12.19 1000 income .683 683 5. 31.12.20 1000 income .621 621 Discounted cash flow: The discount rate converts the then value /amounts ( without inflation taking into account ) to present value / amounts just according to expected rate of return. . NPV = -3000 +909+826+751+683+621=790 ie The Project will yield money which is equivalent to Rs 790 in terms of today after recovering 3000 investment in 5 yrs. Investment date is say 1.1.16 this date itself is investment year or 0 yr , work started on 1.1.16 then 1.1.16 to 31.12.16 is 1 st year and 31.12.16 is end of 1 st yr 1.1.17 to 31.12.17 is 2 nd yr and so on ---& 1.1.20 to 31.12.20 is 5 th year If NPV is positive or more than 1 as in this case it is +790 at the expected rate of return (discounting rate 10% ) project is viable & acceptable or else it would be rejected

117. Discounted cash flow Example Schedule of net annual Income from an investment Project is given in column 2 of the table given below Initial Investment is Rs 160000 , discount factor or expected rate is 10 % .salvage value expected at the end of 5 yr project life is Rs 20000 . It is required to find if the project is Profitable. Year Net annual Income Rs Discount factor at 10% Present Worth 1 2 3 4 1 50000 0.9091 45455 2 45000 0.8264 37188 3 35000 0.7513 26296 4 30000 0.6830 20490 5 20000 0.6209 24836 + Sv 20000 Total 154265 NPV = 154265 - 160000 (Investment) = - 5735 The Project is not Profitable as NPV< 0

118. Internal rate of return

119. b. INTERNAL RATE OF RETURN : The second discounted cash flow method for adjudging the viability of project or for capital budgeting is the internal rate of return method. Every cashflow generates an inherent rate of return for the given sale price P it is called as irr . The cashflow remaining constant, different irr will give different sale prices & different prices will give different irr . Sale price = C1/(1+irr)^1+C2/(1+irr)^2+C3/(1+irr)^3-------Cn/(1+irr)^n = P Def : IRR is the actual compound annual rate of return (interest )that the firm will generate for its investment, with the given actual cash flows. Mathematically IRR is the discount rate at which NPV is equal to zero . IRR shows the efficiency at which capital grows. Say r is DF for a purchaser for a mine having C1, C2, C3--- Cn as cashflows and purchase price P If NPV= C1/(1+r)^1+C2/(1+r)^2+C3/(1+r)^3----Cn/(1+r)^n – investment (P) = We have C1/(1+irr)^1+C2/(1+irr)^2+C3/(1+irr)^3-------Cn/(1+irr)^n = Price P Or C1/(1+irr)^1+C2/(1+irr)^2+C3/(1+irr)^3-------Cn/(1+irr)^n – P =0 Hence irr is = r .

120. Thus if the cashflow is discounted at rate of irr it will give present value = investment . And NPV will be IRR is internally determined with actual net cash flow and not with discounted cash flows hence it is independent of discount factor. Cash flow for a mine is constant based on its capacity. As the discount rate (expectations) increases for a specific cash flow, the NPV of the cash flow necessarily decreases and the purchase offer of buyer lowers down. As the discount rate (expectations) decreases for a specific cash flow, the NPV of the cash flow necessarily increases and the purchase offer of buyer goes up. At a given DF the NPV becomes .At zero NPV expectations of buyer and seller agree with each other. ( It means if the expectations of an investor is equal to the IRR then for him NPV is 0 ,if it`s > IRR ,NPV is – ve and if < IRR , NPV is + ve ) .At irr =r , investors expectation is just fulfilled. IRR is calculated from the end of 0 year (date of investment or beginning of 1 st year) till the end of mine life or period of investment on actual year wise cash flow (not discounted one). IRR is also called discounted cash flow rate of return.

121. The higher the IRR, the more profitable the project is in terms of return on given invested capital. The (IRR) is widely used technique. However, it is more difficult than NPV to calculate by hand, hence computer iteration , graphic or excel format is used. When IRR is used to make accept–reject decisions , criteria is : • If the IRR is greater than the cost of capital ( interest rate at which the capital or loan is available from market )or is sufficiently higher than banking interest rate or higher than minimum desired /expected rate of return ( for mining it is 18 % due to risks involved) the project is accepted. If the IRR is less than the cost of capital or minimum desired rate, the project is rejected. The difference between the discount rate and IRR : The investor chooses the discount rate , whereas the characteristics of the actual cash flow for given investment determines the IRR . Consequently, IRR is determined internally on actual cash flows hence called as the internal rate of return, Where as the discount rate for NPV, is determined externally as per buyer`s expectations .

122. The formula to calculate IRR = r for an investment proposal is : This equation can give value of r ie irr where CF CF 1 CF 2 CF n are actual net cash flows at end of 0 , 1 st ,2 nd ---nth year Inflation not accounted for. CF is the investment and CF 1, CF 1 ,CF 2 are incomes. That right red dot is between the 0.05 and 0.06 marks on the  r  axis, so the internal rate of return is between 0.05 and 0.06. The actual internal rate of return is about 0.0547 or irr = 5.47%   = NPV Say investment is 1000 and income 200/ yr for 6 yrs

123. Indicators : IRR is the discount rate at which NPV equals zero . NPV at higher Discount Rate is negative NPV at lower Discount Rate is positive Hence the IRR which corresponds to zero NPV must be between higher & lower discount rates. IRR also helps you in making a decision when comparing different projects. Formula for IRR from NPVs : IRR= Lower Discount Rate + ( difference between two Discount Rates X NPV at lower Discount Rate )/ absolute difference between NPV at two discount rates . (Absolute difference between NPV at two discount rates = sum two NPVs irrespective of + ve or – ve sign.) Lower & higher Discount Rates are arbitrarily chosen. Link for understanding NPV and IRR in a simple way https://youtu.be/qAhV3xG0i8s

124. Internal Rate of Return Finding IRR by Calculation of NPV at various discount factors Year Cash Flow Discounted Cash Flow at 0% 5% 10% -1000 -1000 -1000 -1000 1 100 100 95.2 80.9 2 150 150 136.0 123.9 3 200 200 172.8 150.2 4 250 250 205.7 170.7 5 300 300 235.2 136.3 6 350 350 261.1 197.4 Tolal NPV 350 106 -80.6 IRR corresponding to zero NPV must be between 5% and 10 % IRR= lower DR + difference between two DR x NPV at lower DR/absolute difference between NPV at two discount rates . IRR = 5+5*106/186.6 =7.84 = say 8%

125. a ) Year n b )(The then Net Cash Flow at the end of yr ) c) Present Value or discounted CF. at 10% c=b /(1+0.1) ^n d )Present Value or discounted CF at 20% d =b/(1+0.2) ^n IRR corresponding to zero NPV must be between 10% and 20 % IRR = lower DR + difference between two DR x NPV at lower DR /absolute difference between NPV at two discount rates = 10+10x17.3/22.2 = 17.8% -100 -100 -100 1 40 36.40 33.30 2 35 28.90 24.30 3 30 22.50 17.40 4 25 17.30 12.10 5 20 12.40 8.0 NPV = (+)17.30 NPV =(-)4.90 17.80% = IRR In this example, the negative income amount Rs.100 in year 0 represents the cost of buying and installing the machine.

126. T A B C D=B-A-C E =D (T)+ E (T-1) F=D/(1+0.1)^T G= F (T)+ G (T-1) Yr Investment Revenue Operating Expenses Year wise Cash flow Cumulative cash flow Dis.cash flow @ 10% Cum. Dis C cash flow 34767866 -34767866 -34767866 - 34767866 -34767866 1 17930853 40000000 12058823 10010323 -24757543 9100293 -25667573 2 10189120 61500000 18088235 33222644 8465101 27456730 1789157 3 102397500 30060000 72337500 80802601 54348234 56137391 4 102397500 30060000 72337500 153140101 49407485 105544876 5 102397500 30060000 72337500 225477601 44915896 150460772 6 102397500 30060000 72337500 297815101 40832632 191293404 7 1127510 330994 796516 298611617 408738 19,17,02,142         298611617   19,17,02,142 NPV  

127. Net present value (Rs.) at 10% irr 19,17,02,142 Internal rate of return (%) 91% at NPV =0 Payback period (years)=1+ -25667573/ 27456730 1.93

128. C. PROFITABILITY INDEX METHOD : It is also a time adjusted method(time vale of money) for evaluating the investment proposals . Profitability index is also called benefit : cost ratio or desirability factor. It is the relationship between present value of cash inflows and the present values of cash outflows. ( which takes place in the life of mine right from the day of investment or 0 year. ) Profitability Index= Sum of present values of all cash inflows / Sum of present values of all cash out flows   It can be worked out either in unitary return per Rs of investment or in percentage terms. P.I . = (Present value of cash inflows) / (Present value of Cash outflows) x 100 % The discounted value ( Present value) of future cash flows can be determined by the following formula: Say cashflow from a mine is :C0,C1,C2,C3,C4,C5------Cn C0 is invariably – ve because it is investment, say C2 and C4 are also – ve then Cashflow becomes: – C0,+C1, – C2,+C3, – C4,+C5,------+Cn

129. Then Profitability Index , P.I . = (Present value of cash inflows) / (Present value of Cash outflows) x 100 % PI =[C1/(1+r)^1+C3/(1+r)^3+C5/(1+r)^Cn/(1+r)^n] / [ C0/(1+r)^0+C2/(1+r)^2+C4/(1+r)^4 ] / x 100 % Selection of an appropriate discount rate is a major factor, higher the discount rate, lower is the present value & so is the viability of the project. A ratio of 1.0 is logically the lowest acceptable measure ( i ) on the index. Any value lower than 1.0 indicates the project's present value is less than the initial investment. So discard it. Higher the ratio or index better the project. In the given sum PI = [ 19,17,02,142- 34767866] /34767866=4.5

130. D. Discounted Payback period method : The payback period is the time period required for the firm to recover its initial investment in a project, as calculated from surplus cashflows. Here the payback period is calculated on the basis of discounted future cash-flows which tells how long it would take a company to recover the original investment from its profit. Pay back period is = E+ B/C years Year of final recovery is the year in which first time cumulative DCF becomes + ve E is serial number of year preceding the year of final recovery in cumulative DCF column , B is cumulative (– ve ) DCF in year E, C is discounted cash flow in the year of final recovery . If Year of final recovery is 4 th year, E will be 3 rd year & Pay back period =3+ cumulative (– ve ) discounted cash flow in 3 rd year/ discounted cash flow in the 4 th year In the given sum Payback period (years)=1+ -25667573/ 27456730=1.93

131. Link for understanding NPV and IRR in a simple way https://youtu.be/qAhV3xG0i8s

132. Difference between Traditional and modern methods of mine valuation. 1 .In traditional methods present purchase value of a project/mine is calculated based on the face value of future earnings without considering the time value of money thus gives same weighing (value) to the distant and near earnings whereas in modern methods it is based on future earning capacity of the mine duly considering the time value of money. 2. Traditional methods are not based on any scientific background and future earning potential of the mine is not taken into consideration whereas these aspects are taken care of in modern methods. 3.Traditional methods may lead to under or over estimation of value of mine and true profitability of the project is not ascertained.

133. Net Cash flow = Revenue + loan taken if any – capital Investment – Operating expenses – working capital spent in that yr – Royalty – Corporate Tax – interest on loan – Loan repayment Case 1 DCF = CF/(1+0.05)^ Year All data in million year Irr npv In CF 1 Sum =702 and sum of DCF or NPV=361

134. Case 2 Net Cash flow = Revenue + loan taken if any – capital Investment – Operating expenses – working capital spent in that yr – Royalty – Corporate Tax – interest on loan – Loan repayment

135. Net Cash flow = Revenue + loan taken if any – capital Investment – Operating expenses – working capital spent in that yr – Royalty – Corporate Tax – interest on loan – Loan repayment

136. Net Cash flow = Revenue + loan taken if any – capital Investment – Operating expenses – working capital spent in that yr – Royalty – Corporate Tax – interest on loan – Loan repayment Case 4 Cash Flows is shared among company , bank , C.govt , st govt , & re-investment. 702= 288.5 + 51+ 192.3 + 48.6+ 121.5 Taxation 40% Royalty 3% Depreciation 270 for 10 year remaining for 5 years Depr on 13.5 0.2 ie 2.7 inv in 5 th yr 2.7*2=5.4 in 6 th yr & so on Reinvestment 5% of 270 from 5 th yr to 5 yr before the end or 20% of mobile equipment) with loan the Return on Equity is higher than the IRR ( 15% compared to 11% in case 3 ) because instead of investing 391.5 the company invests only 229,5 . (leverage effect) LR = loan repay RL=Remaining loan Loan Royalty 3%, Tax 40% DCF 127 in place of 361 of case 1 pay back period = 9.5 yrs

137. Case 5 rest like case 1 Delayed investment In case 1 it was 361 pay back period = + 8 yrs

138. Case 6 rest like case 4 Net Cash flow = Revenue + loan taken if any – capital Investment – Operating expenses – working capital spent in that yr – Royalty – Corporate Tax – interest on loan – Loan repayment Cash Flows is shared among company, bank , C.govt , st govt, & re-investment . 702 =284+ 52.23 + 191.9 + 48.6 + 121.5 DCF =104 in place of 361 pay back period = 12 yrs

139. company 1 2 3 4 5 6 Case no Delay inv in 4 Delay inv in 1

140. 4.2 Debt Redemption Methods Diminishing annuity (EMI) Method. Sinking fund method.

141. In case of borrowing by Co. Lender expects pre-decided interest at more rate than risk free banks rates along with principal, whether co. makes profit or not. lender can be paid back in following ways: Lender gets simple interest on his lent amount at the end of every year and also gets back the original amount at the end of term. At the end of term amortization (co. liability) is complete. As in case of Hoskold`s formula 2. Lender gets a lumpsum amount equal to future value S= P (1+r) n @ compound interest r on his lending amount P which can be paid at the end of term n years . 3.Lender can be payed an amount comprising of equal yearly instalments of principal along with interest on balance loan amount Eg. say Loan Rs.500 for 5 yrs @ 10% .Then he will pay 100 instalment+50 int on Rs 500 for 1 yr =150 at the end of 1 st year. 100+40=140at the end of 2 nd year ---- and 110 at the end of 5 th yr ( It’s a common practice here amount paid reduces every successive year) .( case 4 ) 4. Lender is payed an annuity (A = P X r (1+r) n /[(1+r) n -1]) at the end of every year (from 1 st yr ) till the end of term like EYI(equal yearly installments, consisting of part of principal and interest on balance principal). At end of term the amortization is complete. Here amount paid remains constant every year

142. Net Cash flow = Revenue + loan taken if any – capital Investment – Operating expenses – working capital spent in that yr – Royalty – Corporate Tax – interest on loan – Loan repayment Case 4 Cash Flows is shared among company , bank , C.govt , st govt , & re-investment. 702= 288.5 + 51+ 192.3 + 48.6+ 121.5 Taxation 40% Royalty 3% Depreciation 270 for 10 year remaining for 5 years Depr on 13.5 0.2 ie 2.7 inv in 5 th yr 2.7*2=5.4 in 6 th yr & so on Reinvestment 5% of 270 from 5 th yr to 5 yr before the end or 20% of mobile equipment) with loan the Return on Equity is higher than the IRR ( 15% compared to 11% in case 3 ) because instead of investing 391.5 the company invests only 229,5 . (leverage effect) LR = loan repay RL=Remaining loan Loan Royalty 3%, Tax 40% DCF 127 in place of 361 of case 1 pay back period = 9.5 yrs

143. Def : An annuity is a series of equal receipts or payments or deposits made at equal interval of time . Calculations are based on compound interest basis The time interval also called as period may be weekly, monthly, quarterly, yearly, or at any other regular interval of time . Unless specified the interval can be taken as annual . Examples : Regular deposits into a RD account, EMI, home mortgage payments, insurance and pension payments or house rent payments etc. Term is total duration of time in years for which annuity is paid. The time between two successive instalment dates of an annuity is called its payment period. The payment of each single annuity is called an Installment . No of periods is generally equal to no of years = n = No of instalments If r is annual CI rate & t = no of times payments made in a year (t =12 in case of monthly 2 in case of half yearly etc) then n = t*no of years and interest rate will be r/t If annuity payments are once in a year No of instalments= No of periods = no of years.

144. Amount or sum is future sum /future value of annuities deposited by annuitant. The present value or the present worth of a given annuity is the sum of the present values of its future instalments . An annuitant deposits certain amount A regularly in a fund either to get its amount or future value S at the end of the term or Borrows an amount P today (having present value P) and pays back an amount A regularly for the term. A person can even deposit deposits certain amount A regularly in a fund for certain period and after maturity can receive regularly an amount B for the life (like pensioner) for specified time. Annuities can be classified on the basis of payment interval and timing of payments. 1.Immediate annuity : If the payments of annuity is made at the end of every period (end of every month, end of every year etc) it is called Immediate annuity. Like salary, Mortgage payments, EMI, pension, interest on bonds. Eg. A person deposits amount S on 1.1.2025 he will get 1 st instalment A of annuity immediately at the end of 1 st term on 31 st Dec 2025, and so on.

145. 2.Annuity due : If the payment of annuity is made at the beginning of every period (beginning of every month, beginning of every year etc) it is called annuity due . (payment is due at the beginning of year/period ) like house rent & deposits in RD made in advance . 3.Perpetual annuity is an annuity whose payments continue forever. Eg. LIC premium paid by the person till he is alive against some mishap . 4.Deferred annuity : If the first instalment or annuity payments is deferred for certain number of periods it is called deferred annuity. Eg. A person raised loan S but fails to pay first instalment or annuity on due date due to some reason.

146. Types of annuity 1. Annuity immediate or ordinary annuity or Immediate Annuity. Here the payments of an annuity are made at the end of every payment periods. 1,2, ---- n are periods (years) annuity payments made end of every period/year. 0 is today 1 st annuity payment It is also called as Diminishing Annuity 1 st year Say A = amount of annuity, S= Amount = Sum or future value of annuities at end of the n th year , r = annual rate of compound interest, n = no or periods S = A[ (1+r)^n -1] by sinking fund formula r ( 0 is today or say dt of borrowing money or dt of deposit planning)

147. If P =present value of S ,P= S/(1+r)^n. P is also called as present value of annuities or P = A[ (1+r)^n -1] r (1+r)^n We can say if annuitant deposits annuity A for n years in a fund it will get him sum S after n years S = A[ (1+r)^n -1] ( Used to get lumpsum amount after n yrs ) r And If annuitant borrows amount P today and pay back annuity for n years he will have to pay. P = A[ (1+r)^n -1]/r (1+r)^n or A = P . r (1+r)^n/ [ (1+r)^n -1] (Used to redeem the loan)

148. Example : Find The present value of a 5 - year annuity with a nominal annual interest rate of 12% and monthly payments of $100 is : No of periods 5*12=60 =n r = 0.12/12 monthly interest rate S = A[ (1+r)^n -1]/r & P= S/(1+r)^n S = 100[ (1+0.12/12)^60 -1] =8167 0.12/12 P= 8167/(1+0.12/12)^60 =4495 Thus he can deposit $100 at the end of every month for 5 yrs and get amount 8167 after the term of 5 yrs Or can borrow today 4495 and pay $100 at the end of every month for 5 yrs and get rid of the loan. A = P .r (1+r)^n/ [ (1+r)^n -1] =4495x0.12/12 (1+0.12/12)^60/ (1+0.12/12)^60-1 =4495x0.01x1.81/0.81=100

149. Future value of a series of equal amount paid in the beginning of every year ( or period ) for n years is Called as annuity due Eg. Recurring Deposit scheme or Sinking funds Derivation : Rs A deposited in beginning of 1 st yr will remain in Sinking fund for n year and will become = A(1+ r) n = A(R) n at end of n th year Rs A deposited at beginning of 2 nd yr will remain in Sinking fund for n-1 year and will become = A(1+ r) n-1 = A(R) n-1 at end of n th year Rs A deposited in beginning of n th yr will remain in Sinking fund for 1 year and will become = A(1+ r) = A(R) at end of n th year ( 1 yr interest ) S A = Sum of the future values of all A deposited during the period of n years

150. Thus S A = A(R) n + A(R) n-1 ------+ A(R) --- 1 S A R= A(R) n+1 + A(R) n +A(R) n-1 ------+ A(R) 2 --2 S A R - S A = A(R) n+1 + A(R) n +A(R) n-1 ------+ A(R) 2 - [A(R) n + A(R) n-1 ------+ A(R) ] Eqn 2 – Eqn 1 S A [R – 1] = A(R) n+1 + A(R) n +A(R) n-1 ------+ A(R) 2 - A(R) n -A(R) n-1 ------ -A(R) S A [R – 1] = A(R) n+1 + A(R) n +A(R) n-1 ------+ A(R) 2 - A(R) n -A(R) n-1 ------ -A(R) S A [R – 1] = A(R) n+1 - A(R) S A [R – 1] = AR[(R) n – 1] S A = A [(R) n – 1] R [R – 1] S A = A [(1+r) n -1] (1+r) r S A = A[R n -1] (R) = A [(1+r) n -1] (1+r) r r

151. 2. Annuity-due In this case the payments of an annuity are made at the beginning of every payment periods . Deposits in savings , rent or lease payments , and insurance premiums are examples of annuities due . 1,2, ---- n are periods (years) annuity payments made at beginning of every period/year. 0 is today here 1 st annuity payment is made today itself ie at beginning of 1 st year. S = A[ (1+r)^n -1] (1+r) as P= S/(1+r)^n P= A[ (1+r)^n -1] (1+r) r r(1+r)^n S= Sum or future values of annuities ,r = annual rate of compound interest, n = no or years P =present value of annuities or present value of S 1 st year.

152. We can say if annuitant deposits annuity for n years in a fund it will get him sum S after n years S = A[ (1+r)^n -1] (1+r) ( Used to get lumpsum amount after n yrs ) r And If annuitant borrows amount P today and pays back 1 st annuity A (1 st instalment) today itself and subsequently at beginning of every period for n years he will have to pay. P = A[ (1+r)^n -1] (1+r) r (1+r)^n A= P. r (1+r)^n [ (1+r)^n -1] (1+r) (Used to redeem the loan)

153. Example : Find The final value ( amount ) of a 7 - year annuity - due with a nominal annual interest rate of 9% and monthly payments of $100   S = A[ (1+r)^n -1]x (1+r)/r S= 100[(1+0.09/12)^84-1] (1+0.09/12) = 11730 0.09/12 P= 11730/ (1+r)^n = 11730/ [(1+0.09/12)^84 = 6262 Thus he can deposit $100 at the beginning of every month for 7 yrs and get amount 11730 after the term of 7 yrs Or can borrow today 6262 and pay $100 at the end of every month for 7 yrs and get rid of the loan.

154. Redemption or Amortization It is a process to get rid of debt raised against purchase of any equipment or to pay back the loan amount to lender or to recover the investment made in a business if money is spent from pocket. Investor invests Rs. P, principle for n years in a company which can be paid back /redeemed by borrower company in two ways 1. By paying uniform instalments @ Rs. A /year for a term of n years directly to the lender. Or 2. By depositing same uniform instalments @ Rs. A /year in a sinking fund on behalf of lender and payback the sinking fund maturity amount S to lender at the end of the term.

155. Diminishing Annuity method: Many times loan repayments are done on EMI basis. Diminishing Annuity method is used to calculate ( EMI) equal monthly or ( EYI) equal annual instalments  (annuity) to be paid over a specified number of years in order to pay off the loan in full including interest . N.B. For ease we are considering period as year however the period could be month or quarter or half year and thus annuity could be monthly half yearly & yearly etc . Say in order to purchase a machine company raises loan P at the beginning of an year and to redeem it, a fixed amount of instalment A is paid at the end of every year right from the end of 1 st year till the term is over. It is called EYI equal yearly instalments  and this type of annuity is called Diminishing annuity . This is also called as Annuity immediate or Ordinary annuity or Immediate Annuity. This annuity is the ‘ Annual Owning Cost” of a machine or any item. Def. Diminishing Annuity is the series of fixed amount to be paid at the end of every year for a certain period to redeem a loan . The constant amount paid consists of part of principal + interest on the unpaid or remaining principal.  

156. This method is reverse of sinking fund method here the lender is directly paid at the end of every year instead of depositing same amount in sinking fund. Subsequently every year from the annuity A , payment towards interest part gets reduced (as the principle is getting reduced due to its part payment ) and correspondingly the payment against principle part increases . Eventually by the end of term the loan with interest is fully paid back. This A = P.CRF Where P is the principle, A the annuity and CRF is called capital recovery factor. Capital recovery means amortization or redemption of loan P .

157. Year a Loan at beginning of year= b EYI = c loan part app d = c-e int part app e =b*0.1 Loan at end of year f =b-d 100 100 1 100 18.8 8.8 10 91.2 2 91.2 18.8 9.68 9.12 82.08 3 82.08 18.8 10.56 8.21 71.49 4 71.49 18.8 11.65 7.15 59.84 5 59.84 18.8 12.8 6.0 47 6 47 18.8 14 4.7 33 7 33 18.8 15.5 3.3 17.5 8 17.5 18.8 17.05 1.75 Loan=100 r=10% period 8 year EYI = P X r (1+r) n /[(1+r) n -1] = 100*0.1*1.1^8/1.1^8-1 =10*2.14 /1.14=18.8

158. Derivation Diminishing annuity method of amortization or EMI method Say P = Principal /loan taken for n years @ compound interest r%. S= sum or Future value of P after the term of n years = P (1+r) n If A = annuity of equal amount paid yearly for n yrs at the end of each yr Then as per sinking fund formula Future value of constant annuity A = S A = A [(1+r) n – 1] r The capital recovery factor is the ratio of a constant  annuity  to the loan P or present value of S   or CRF = A/P CRF = A/P or P = A / CRF , or constant annuity A = P.CRF ,

159. as S = S A = A [(1+r) n – 1] r Present value of S = P = S /(1+r) n P = A [(1+r) n – 1] putting value of S r (1+r) n A= P X r (1+r) n [(1+r) n – 1] A = P X r (1+r) n /[(1+r) n -1] = Annual payment or EYI for n years(beginning from the end of 1 st year) to redeem loan P with interest r% It is called diminishing annuity CRF = ( A/P) = P X r (1+r) n = r (1+r) n / [(1+r) n -1] [(1+r) n – 1] P   Capital recovery factor  CRF = r (1+r) n /[(1+r) n -1] A = P X CRF ( factor being constant, the annuity depends on loan amount P)

160. For finding the n or No of years required to pay the loan in case of diminishing annuity method of amortization or EMI method S A = A [(1+r) n – 1] by sinking fund formula r P (1+r) n = S= S A = A [(1+r) n – 1] r P.r. (1+r) n = A [(1+r) n – 1] [P.(1+r) n . r - A (1+r) n ] = - A A (1+r) n - P.(1+r) n . r = A (1+r) n [ A – P.r ] = A (1+r) n = A / [ A – P.r ]

161. A co. takes a loan of 20000 at 5% C I for extending mine working, by this investment additional saving of 2000 per year is expected for paying up the loan. What is the period required to pay up the loan? Sol : Saving will be used for loan amortization (1+r) n = A / [ A – P.r ] (1+0.05) n = 2000/(2000-20000*0.05) (1.05) n = 2 or n =14 years

162. Sol . (1+r) n = A / [ A – P.r ] A= 5000, P =50000, r =0.07 ie 7% (1+ 0.07) n = 5000 / [ 5000 – 50000X 0.07] Ans : n = 18 years A co. takes a loan of 50000 at 7% C I for renewal of existing machine to raise production, by this investment additional earning of 5000 per year is expected for paying up the loan. What is the period required to pay up the loan?

163. A machine is purchased for Rs . 30,00,000. Amount is to be paid in 5 installment over 5 year period. Interest rate is 10%. What will be the annuity or annual payment for this machine?   A = Pv X r (1+r) n /[(1+r) n -1] A = (Pv x r x R n )/(R n -1) P = 3000000, r = 10% or 0.1,R = 1+r = 1.1 A = [3000000 x 0.10 x (1.10) 5 ] / [(1.10) 5 -1)] = 7,91,400 per year What will be the hourly cost of owning if it works for 1000 hours per year. Hourly cost of operations = 791400/1000 = 791.40 Rs /hr.

164. Another method of Redemption or Amortization of loan is 2. Sinking Fund Method Say borrower borrows Rs P from lender, instead of paying installments (EMI) directly to lender, borrower deposits Rs . A in a Sinking Fund of bank or Govt. security at the end of every year and gets compound Interest @ r % for n years on behalf of lender & pays to lender S or Sinking Fund money after the entire period of n yrs is over. This S A or Sinking Fund money is equivalent to future value of P ie S Or S A or S = P (1+r) n S A = A [(1+r) n – 1] = maturity value of annuity A as per sinking fund formula r   A= S A X r/(1+r) n -1 --------- * Since S = S A = P (1+r) n A = P. r (1+r) n /[(1+r) n -1] putting value of S in above equation * Or A = P X r (1+r) n /[(1+r) n -1] This annuity A is same as determined by EMI or diminishing annuity method Hence A is same in both cases

165. Find the annuity for paying back a loan of Rs 10000 in 5 yrs borrowed @ 9% compound interest as per diminishing Annuity method and sinking fund method? The loan to redeem is Rs. 10000 . Amount is to be paid in 5 installment at the end of every year. Interest rate is 9%. The annuity A or annual payment :   A = Pv X r (1+r) n /[(1+r) n -1] (Pv is present value of loan & R=1+r) A = (Pv x r x R n )/(R n -1) P = 10000, r = 9% or 0.09,R = 1+r = 1.09 A = [10000 x 0.09 x (1.09) 5 ] / [(1.09) 5 -1)] = 2571 per year will be paid to lender. By sinking fund method also if Rs.10000 =Amount taken on loan future value of10000= 10000*1.09^5= 15387 By sinking fund 15387 = A [(1+0.09) 5 – 1] 0.09 A= 2571 per year will be paid in sinking fund and final amount 15387 will be handed over to lender.

166. 4.3 Depreciation Concept of depreciation, its importance and causes. Calculation of Depreciation using: Straight line method, Declining or Reducing balance method. Sinking fund provision against depreciation. Calculation of Life of mine .

167. DEPRECIATION   In any industry equipment, machines, buildings ( except land ) are subjected to wear & tear due to use & ageing, hence their value gets lowered, this loss in the value is called depreciation. Def : Depreciation is the reduction /loss in the value (book value) of capital asset during its economic life due to wear and tear or ageing. It is also called capital cost allowance or write down cost. The accounting process of gradual conversion of capital asset into expenditure is called depreciation. Since all these assets forms the part of capital the value of capital decreases with the passage of time hence a suitable deduction has to be made for this purpose from the book value of capital & after a certain period, an asset has to be completely written off ( discarded ) from the books & its value ultimately becomes zero or reaches salvage value (scrape value). This however need not always be true even though the book value be zero but the asset may yet have a salvage value or scrap value. An asset though subjected to depreciation can at the same time appreciate some value if it is properly maintained. The real cost of using an asset is depreciation plus repair expenses. Total cost of using an asset is depreciation plus repair expenses plus interest on

168. Hence while the depreciation is deducted from the original value of the asset, cost of maintenance appreciates the asset & adds to its cost. Indirectly depreciation is cost incurred on using equipment hence depreciation amount is allowed to be deducted from total income to arrive at taxable income thereby reducing the tax liability . Causes for Depreciation – Physical Causes Wear and tear due to use. Deterioration and decay with passage of time ( Even if the property/capital is not used ) Damage or destruction due to accident like fire ,flood earthquake etc Action of chemical elements on the component parts. Lack of maintenance . 2) Functional Causes Uselessness of the present purpose. Obsolescence of technology i.e due to innovation , invention , change in technique

169. Objectives of Depreciation 1) To Calculate the Correct Profits : The depreciation shows the diminishing value of assets or cost ( expenditure ) incurred due to use/owning an equipment. It must be deducted from the profits, so as to calculate the correct profits. 2) To Show the Assets at its Real Value - Real value of the property can only be shown after the charging depreciation every year. 3) To Replace the assets - Every Year the depreciated amount is set aside. It gives the management sufficient funds to replace the machinery /assets. 4) To Reduce Tax Liability – Depreciation is chargeable under expenses under Income tax Act and thus the taxable profits are reduced. It thus reduces the tax burden of the Company

170.   Methods of Calculating Depreciation The methods of calculating depreciation are as follows: Straight line method Declining/Reducing balance method Sinking fund method to provide for finance against depreciation.

171. 1)Straight line method It is most common method and based on assumption that wear & tear of the capital asset is uniform throughout its normal useful life. With the straight line depreciation method, the value of an asset is reduced uniformly over each period until it reaches its  salvage value. Yearly depreciation is calculated on life of machinery Depreciation per year = cost of machine - scrape value = (C-S)/n Useful life of asset C = Cost of the asset  or the purchase price of the asset. If installation cost is incurred separately it has to be added to Cost of the asset  . S = scrape value/ Salvage value  is the value of the asset at the end of its useful life. n= Useful life of asset  represents the number of years/periods in which the asset is expected to be used by the company.

172. A machine Costing Rs 20000/ was purchased and was installed . Installation charges are Rs 5000.If Scrap value is 5000/ and the useful life is 10 years , calculate the yearly depreciation by straight line method Total Cost is Rs 20000 / +5000 =25000 So the depreciation charges = 25000 – 5000 10 = 2000/yearly

173. 2 . Declining/ Reducing balance method : In the declining balance method the depreciation rate is pre-decided and the depreciation in any year is calculated on the written down or (left out) book value of the asset at the beginning of that year. Thus if depreciation rate 12% , every year 12% depreciation will be deducted from the value at the year beginning. Book value at the end of yr = book value at the beginning of yr - depreciation Taking the case of the machine valued at Rs . 5000/-. If 10% is its depreciation rate, the depreciation in 1 st yr = 5000 X 0.1 = 500 Depreciated cost (or book value after depreciation ) at the end of 1 st yr will be Rs. 5000- Rs.500= 4500. Similarly depreciation in 2 nd year will also be 10% of book value 4500 at the beginning of 2 nd year ie Rs 450 and book value at the end of 2 nd year will be 4500-450= 4050 and so on. This method is especially suitable to assets with long life , e.g., plant and machinery, furniture, motor car etc. .

174. Depreciation lowers net profit as it is accounted as an expense in income statement.  Actually the real cost of using an asset is depreciation plus repair expenses . so this method gives better results because here in early years when repair expenses are less the depreciation is more . As the asset gets older repair charges increases and amount of depreciation decreases . A heavier depreciation charge is born in the earlier days when repairs are lighter and increasing repair cost in later years with lower depreciation in later years is counter balanced . Thus the real cost of owning a machine remains constant & the combined effect of both these costs remain almost constant on the profit and loss of each year. This method is used by income tax authorities for granting depreciation allowance to assesses. Here assets hold their book values for an inordinate length of time and financial accounts and cost accounts get tipsy.

175. Reducing balance method This method is to make provision for depreciation by means of period charges , each of which is a constant proportion of the balance remaining after deducting from the cost of the asset depreciated, the aggregate of the amounts provided previously. Cost of machine Rs 50000 To be written down at the rate of 20% per annum Year Cost and Balance Brought forward Rs Beginning of yr Depreciation @20% Rs Balance carried down Rs End of yr Year 1 50000 10000 40000 Year 2 40000 8000 32000 Year 3 32000 6400 and so on 25600

176. For finding the rate of depreciation, the formula in the declining balance method is. r = 1 - (S/C) 1/n Where : r = Rate of depreciation = depreciation factor, n = Estimated useful life of asset. S = Residual value after the expiry of useful life. C = Original cost of asset It is a good system where the rate of depreciation can not be ascertained accurately. Eg. If n = 3 years, S = 64,000 and C = 1,000,000 calculate rate of depreciation . r = 1 - (64,000/1,000,000) 1/3 = 1 - 40/100 = 60/100 = 60%

177. Cost of machine 32000 ,scrape value 8000 ,life of mc 10 yrs find rate of depreciation. r = 1 - (S/C) 1/n r = 1 - (8000/32000) 1/10 = 1 - (1/4) 1/10 = 1 - (0.25) 1/10 r = 12.94%

178. Depreciation fund method  or  sinking fund method for finance against depreciation. Sinking fund method is a depreciation technique used to finance the replacement of an asset at the end of its life. The amount of money deposited in each year depends on life of equipment, cost to be incurred to replace equipment and rate of return on investment. A fund is created with the amount equivalent to annual depreciation of machine so as to make the accumulated fund available when the machine is worn out. This constant amount is invested each year in some securities ( sinking fund ) for n years or life of the asset outside the business throughout the life of the asset. This annuity accumulates at compound interest. At the end of machine life a sum equal to the original purchase cost of the asset is accumulated in the fund. Then the new asset is acquired with the sinking fund amount. Sinking fund method is specially applicable to costly machines in large scale industries .

179. Sinking fund formula : Amount of Sinking fund or maturity value received at the end of period is S = A [(1+r) n – 1] Or A = S X r r [(1+r) n -1 S is money required to be accumulated in SF = C or C-P if there is any scrape value P C =cost of m/c, P = scrape value, r =rate of return in SF, n =useful life of m/c ,A= Annual contribution to SF A= Annual contribution to SF = Annual rate of depreciation = r (C –P) (1+r) n – 1 r is called sinking fund factor [(1+r) n -1] A= (C –P) x sinking fund factor Def : Sinking fund method is a depreciation technique used to finance the replacement of an asset at the end of its life and annual rate of depreciation is given by Annual rate of depreciation = r (C –P)/ (1+r) n – 1 Whare C =cost of m/c, P = scrape value, r =rate of return in SF, n =useful life of m/c ,A= Annual contribution to Sinking fund . The sinking fund amount S can also be taken higher at fair rate of inflation , as the future value of new machine won`t be the same as present cost & annuity A can be increased accordingly.

180. Say cost of mc is 16000 scrape value is 8000 r=4% life of mc 4 yrs So we have to get 16000 - 8000 = 8000 form SF Annual contribution to SF = 0.04 (16000 –8000) = 1883= Annual rate of depreciation (1+0.04) 4 – 1 Or If we deposit 1883 every year in SF we get after 4 yrs S = A [(1+r) n – 1] r = 1883[ (1.04)^4 -1]/0.04 = 8000 Money required to buy new m/c Eg. What is the annual contribution to the sinking fund at 5% compound interest required for amortization (repayment or recovery ) of Rs. 10,000/- in 10 years? A = S X r [(1+r) n -1] A= S X r/ [(1+r) n -1] = 10,000 X 0.05 / (1.05) 10 – 1 = 500/ 1.6289 – 1 = Rs. 795/-

181. Life of a mine. Life of Mine is the number of years during which the mineable reserves can be profitably worked out. It depends on the degree of mechanization, modernization and the investment potential, For example with higher mechanization and higher capacity of machines the quarriable limit increases and we can mine at greater depth. This increases the life of the mine.

182. Life of the Mine The life of the mine (LOM), the average production rate (APR) and the total mineable reserve or the 'expected ore tonnage' (EOT) are related as follows: Or LOM = EOT/APR As a mine develops it is usual for further mineable reserves to be discovered. Thus APR and LOM will be modified as time passes. Taylor addressed this problem and derived an empirical guide which he called 'Taylor's Rule'. This is as follows: Life = 0.2 (Reserves)  0.25 Production rate = 5.0 (Reserves)  0.75 Life is in years Reserves are in metric tonnes Production rate is in metric tonnes per year

183. Thanks

184. Asked questions 2022 /23 What is depreciation. 2 Define “ Hoskold Formula” of mine valuation. 2 Define annuity 2 State Hoskold Formula 2 Define “Valuation”. 2 What is annuity explain diminishing method of annuity. 4 What is mine valuation compare between Hoskold and Morkills method of valuation. 4 Calculate the amount of annuity of Rs. 50,000/- per annum for 10 years @ 10% compound interest. 4 Find by straight line method of depreciation a machine costing Rs. 20,000/- was purchased and was installed. The installation charges are Rs. 5,000/- if the scrap value is Rs. 5,000/- and useful life is 10 years. Calculate the yearly depreciation. 4 Calculate the amount of annuity of Rs. 50,000/- per annum for 10 years @10% compared interest. 4 Find by straight line method of depreciation, a machine costing Rs. 20,000.00 was purchased and installed. The installation charges are Rs. 5,000/-. If the scrap value is Rs. 5,000/- and the useful life is 10 years. Calculate the yearly depreciation. 4 Classify the method of depreciation. 4 Classify various mine valuation methods and explain in brief different discounted cash flow methods. 4 For investment of Rs. 300/- cash flow in next 5 years are 50, 100, 125, 150 and 100 find NPV at DF 10%. Also find IRR. 6 Describe why is discounted cash flow method preferred compared to traditional methods. Explain net present value, internal rate of return, profitability index and payback period.  

185. Define value of a mine Definition: The fair market value of a mine (mineral deposit) is that sum which the exploitation of mineral will return, together with a fair rate of interest (return) on it, besides paying operating and other expenses, taxes and loan repayments with interest within the period of investment. This price ( or the cash equivalent) is determined by economic experts. In monitory terms the present fair market value of a mine is based on its future earning capacity and is equal to sum of the present values of its future annual incomes (cash flows). Define Valuation of a mine Valuation: Valuation is the process of estimating the fair market value of a business in monitory terms by the experts on the date of purchase. The exchange of hands should be in an open and unrestricted market between a willing buyer and a willing seller at “arm’s length” ( ie as is where is basis), with each party acting knowledgeably, prudently and without compulsion. Valuations can be done on assets like mineral properties at any stage of development or on liabilities of a company.

186. Define time value of money. Time value of money” is the value or purchasing power of a unit of money (Rs 1) at different time intervals ie today, after one yr , after 2 yrs ---after n yrs etc and value of money decreases every subsequent year due to inflation and other uncertainties. To compensate it interest or return is expected by investor on today`s investment/deposits. State the purpose of valuation. Ore deposits are considered to have value when they are capable of yielding a profit on exploitation over a term of period. Purpose of valuation : 1. The object of valuation of a working mine or mining project is to assess the selling or purchasing price of a mineral deposit or a mining enterprise. 2.To find out future earning potential and thereby its Profitability besides return of capital investment . 3. To carry out capital budgeting or investment analysis at the time of, merger and acquisition of firms, financial reporting, taxation calculations and in litigation cases.

187. Explain how the valuation of mineral property is different from Other Projects. Mine Valuation is the most crucial financial aspect of the mining industry. 1.The rate of annual income from the mine is cyclical & not uniform depending on various factors. 2.Unlike other industries which are perpetual in nature, in mining, the reserves get exhausted by continued working hence the original capital invested is to be recovered during earning period of mine. 3. Each ore body has unique features hence the v aluation of a mineral property is a time specific, site specific and mineral specific exercise.   4. Lengthy Gestation period is involved due to exploration and development. 5.High capital requirement during development period and requirement of specialized equipment. 6.Large Reclamation costs involved at the end of project 7.In other industry flow of raw material is almost perennial whereas in mining industry it depends of depleting resources. 8.Large environmental/social /infrastructure cost.

188. Explain the Methods of valuation Valuation methods Static or Traditional methods: In these methods the present purchase value of a project/mine is calculated based on the face value of future earnings without considering the time value of money. They are: a. On mutually agreed terms: b. Book value approach. c. Payback period method d. Hoskold`s and Morkill`s valuation methods. 2. Dynamic or modern methods: In these methods the present purchase value of a project/mine is based on future earning capacity of the mine duly considering the time value of money. In monitory terms the present purchase value is equal to the sum of present (discounted) values of future year wise cashflows (incomes) received during the life of mine. Future cashflows likely to be generated as per the capacity of mine is estimated year wise and they are discounted at the expected rate of return by the investor as on date of investment. The sum of discounted values gives the present value (or purchase value) acceptable to a particular investor.  

189. Modern methods are based on a. Net Present Value, b. Internal Rate of Return c. Profitability index, d. Discounted payback period Explain difference between traditional and modern methods of mine valuation. 1 .In traditional methods present purchase value of a project/mine is calculated based on the face value of future earnings without considering the time value of money thus gives same weighing (value) to the distant and near earnings whereas in modern methods it is based on future earning capacity of the mine duly considering the time value of money. 2. Traditional methods are not based on any scientific background and future earning potential of the mine is not taken into consideration whereas these aspects are taken care of in modern methods. 3.Traditional methods may lead to under or over estimation of value of mine and true profitability of the project is not ascertained.

190. Explain the important steps involved in mine valuation? Steps or factors involved in Mine Valuation (modern methods) Step-I (Technical appraisal) 1 .Estimation of ore reserves: By assessing Geological Resources & Grades, Mineable reserves & Grades, Mining losses, effective grades, cut off grade ,minimum mining grade etc. 2. Determination of grade wise production rate of mine & life of mine. 3.Estimation of capital cost, (investment in fixed assets), working capital, other investment, and its phasing. 4.Estimation of all direct Operating costs, - including drilling, blasting, loading ,hauling labour , store, energy etc. and 5.Estimation of all indirect costs like overhead costs of plant infrastructure, equipment, utilities, mine management, loan and interest payments, taxes in view of depreciation, environmental & social cost in view of, government policies etc. 6.Projecting tentative Revenue for life of the mine based on market survey and forecast. Step-II (Financial Appraisal) Calculation of ensuing annual cash inflows (+ ve ), cash outflows(- ve ) and cashflow ( or net Cashflow(+ ve or – ve as the case may be). Calculate the present values of ensuing cashflows based on cash flows, applying the expected rate of return (discount rate) Find IRR and discounted payback period if it suits to your expectations.

191. Explain the risk factors considered for valuation of a mineral property. RISK IN VALUATION . Valuation of the assets is being done with due care and correctness however it involves the following risks due to typical nature of the mining enterprise. . 1) Cost of production varies – a) when the grade and reserves varies to that of computed figures. b) increase in the amount of gangue c) possibilities of contamination and dilution due to barren rock. 2) Over and under estimation of the cost seriously effect mine economics. 3) Unpredictable cost of lands, building, plant and machinery. 4)Import policies and restrictions during procurement of the machineries. 5) Existences of geological disturbances in the mine property, sometimes ore body may comprise of the several independent chutes, separated by barren rocks. 6) Extraction losses due to particular method of mining, premature collapse, faulted zone, safety restrictions. 7). Imposition of various govt. taxes royalties and cess etc. 8). Risky nature of the mining due to unpredictable hazards in working i.e fall of roof, gas/coal and dust explosion and inundation. 9). High cost of working in courses or time as the workings are at a distant place and deeper. 10). Variation in the govt. policies towards labour and their wages affects the cost of productions 11). In computation of the cost whether it is a pit mouth delivery or for destinations (free on rail ) should be taken care of beforehand.

192. Enumerate the important contents of valuation report of a small working mine. Valuation reports of a small mine The complete valuation report of a mineral property would involve not only attention and study of the mine and works, but also of the title deeds and account – books. the following points should be covered 1)Property - legal rights or title without encumbrance, size, boundaries, ownership, royalties, taxation, rent. 2)Geographical position – access, climate, business center, markets. 3)Economic position – labour , water, energy, fuel, timber. 4)Geological position - type of deposit, possible genesis, country rock, mineralization. 5)Past history – past production and result, old working. 6)Present mine working – extent, equipment, mineral reserves, cost etc. 7)Beneficiation – plant and equipment, recovery and cost. 8) Marketing - prices, freight, agencies. 9) General surface plant and equipment – building, water, land, sources, reservoir. 10) Valuation - revenue cost, profit, life of mine and the internal rate of return. 11) Finance requirement on improvement and reconstruction.

193. Explain the sinking fund its purpose and its formula A sinking fund is an account that is used for regularly (generally annually) depositing a fixed amount, so that it multiplies at a fair rate of compound interest and at the end of term the sum accumulated in the sinking fund is received. A sinking fund is nothing but cumulative future value of an amount regularly deposited at the end of each period. Sinking fund is created in order to 1.R ecover the invested amount in a business by depositing part of income in SF. 2.R epay a long-term debt & prevent defaulting. 3.Replace a wasting capital asset. (By keeping aside depreciation amount) 4.Get a lump sum amount in the future for specific purpose like purchases of large fixed asset in the future etc. Future value of a series of equal amount deposited at the end of every year for n years in a Recurring Deposit scheme or Sinking fund. = S A = A[R n -1] = A [(1+r) n -1] r r where Rs A is deposited at the end of every year for n years where r is rate of compound interest of sinking fund & Say R = 1+r

194. Explain in brief Hoskolds formula Hoskold considers that a mining investor, would invest amount Pv for purchase of mine if he gets constant annual dividend /income d on his investment. This d should be equal to his expected rate of simple interest every year on investment Pv + an additional amount to set aside in a sinking fund which can get him back his invested capital Pv after n years. A= constant earning/year during the life of the mine (fixed annuity or dividend ) r 1 =safe rate of Compound interest from sinking fund, which is in %.  r 2 =speculative rate of simple interest, which is in %. r 2 = simple interest > r 1 Compound interest, d= r 1 + r 2 & Say R= (1+ r 1 ) Pv = Present purchase value of the property. n = life of mine or period of investment in mine If A, n, r 1 and r 2 are known Pv can be found out as follows :

195. Explain present value if there is Gestation period   If the gestation period is m yrs then the present value at the date of purchase is found by discounting the present value of mine at the beginning of production period at the bank interest rate or expected interest rate applicable. Say interest rate during m year gestation period is r 3 % present value at the beginning of production period = PV then Present value on the day of purchase = PV p PV p = PV/(1+ r 3 ) m  

196. Define Cashflow, Discount rate, discounted Cashflow, present value and net present value of a mine. Def : Cashflow means net cashflow or income or profit (year wise revenue – cost of production). Cashflow is a series of year wise incomes coming from a mine from the end of 0 year to the end of term and is equal to. Def : The minimum acceptable rate of return in % age, from a business is called as Discount rate. It includes safe bank rate + risk premium according to risk specific to the business. It is used by the investor to find out the purchase value of a mine with given series of cashflow.   Def. PV or Present value or Purchase value of a mine is sum of the present (or discounted) values of all future cash flows from mine from the end of 1sr year till the end of period of investment @ investor`s acceptable rate of return r. PV=C1/R^1+C2/R^2+C3/R^3---------+Cn/ R^n (R= 1+r) Where C1,C2 ----& Cn are expected cash flows from the mine at the end of1st , 2 nd ----& nth year .

197. Def : NPV /Net Present Value is the sum of the present (discounted) values of all future cash flows from the 0 yr ( from beginning of 1 st year ) till the end of period of investment. NPV = C0/R^0 +C1/R^1 +C2/R^2+C3/R^3---------Cn/ R^n Or Net Present Value : NPV is equal to the sum of the present values of all future cashflows from end of 1 st year till the end of period of investment minus investment. NPV = Present value – investment required ( P ) = PV- P If NPV is + ve the deal is profitable for purchaser if it`s - ve it is a loss proposal. Explain indicators of NPV The larger the NPV, the richer the investor becomes by undertaking the project. The net present value is higher if the income amounts are larger , or if they come sooner , or if the discount rate or expectation is lower .  The net present value is lower if the income amounts are smaller , or if they come later , or if the discount rate is higher .

198. Describe why is discounted cash flow method preferred compared to traditional methods. Explain net present value, internal rate of return, profitability index and payback period. These are most objective methods of Mine property valuation They don’t give same weight (value) to the distant and near earnings & take in to consideration the timing of cash flows . They make fair estimation of mining property by applying reasonable rate of discount.

199. The Modern Mine property valuation methods are: a . Net Present Value Method. b. Internal Rate of Return Method c. Profitability index Method d. Discounted payback period method. a . Project valuation by Net Present Value Method. Say P is the initial investment required to be done today ie at the end of 0 year and A 1 , A 2 , ----- A n all are future uneven cash flows (inflows- out flow ) at end of 1 st 2 nd ---and N th year from today, r is discount rate expected by buyer on given investment P. Then NPV = – P + A 1 /(1+r) + A 2 /(1+r) 2 +-----+ A n /(1+r) n Today or at the end of 0 yr cashflow is – P because here there is no revenue and investment is P. If NPV or sum total of the series is positive at expected discount rate the project is economically viable & can be accepted by investing Rs P today. Or the project is worth investing Rs P. If NPV or sum total of the series is negative it is rejected and he may put his money in bank or in other business where he can get expected rate of return

200. Advantage : NPV or the sum of the discounted yearly cash flows gives the present value of the entire income stream right in the beginning of the time period ie on the date of investment, thereby helps in deciding about the viability of project. NPV criterion measures economic consequences, of time, by converting the all-future cash flow amounts to equivalent amounts at the present time and sum of the discounted yearly cash flows determine net loss or gain Define Internal Rate of Return Def : IRR is the actual compound rate of return (interest) that the firm will generate for its investment with the given actual cash flows. Mathematically it is the discount rate at which NPV is equal to zero . Internal Rate of Return Method IRR is the actual compound rate of return (interest) that the firm will generate for its investment with the given actual cash flows Or It is the discount rate at which NPV is equal to zero . Say r is DF for a purchaser for a mine having C1,C2,C3--- Cn as cashflows and purchase price P Then NPV = C1/(1+r)^1+C2/(1+r)^2+C3/(1+r)^3----Cn/(1+r)^n – P If NPV = C1/(1+r)^1+C2/(1+r)^2+C3/(1+r)^3----Cn/(1+r)^n – P = then r is = Internal Rate of Return From above value of r to be found out mathematically or by graph or excel format value of r will be = irr

201. Formula for IRR from NPVs : NPV at higher Discount Rate is negative NPV at lower Discount Rate is positive Hence the IRR which corresponds to zero NPV must be between higher & lower discount rates. IRR= Lower Discount Rate + (difference between two Discount Rates X NPV at lower Discount Rate)/absolute difference between NPV at two discount rates. (Absolute difference between NPV at two discount rates= sum two NPVs irrespective of + ve or – ve sign.) Lower & higher Discount Rates are arbitrarily chosen.   IRR shows the efficiency at which capital grows. The higher the IRR, the more profitable the project is in terms of return on given invested capital. IRR also helps in making a decision when comparing different projects.

202. Profitability index method Profitability index is defined as ratio of Sum of the present values of all cash inflows and Sum of the present values of all cash out flows P.I . Profitability Index = (Present value of cash inflows) / (Present value of Cash outflows) x 100 % Say Cashflows from mine are: – C0,+C1 ,– C2,+C3, – C4,+C5,------+Cn and r is discount factor Then PI=[C1/(1+r)^1+C3/(1+r)^3+C5/(1+r)^Cn/(1+r)^n]/[C0/(1+r)^0+C2/(1+r)^2+C4/(1+r)^4 ] / x 100 % Higher the discount rate, lower is the present value & so is the viability of the project. A Profitability Index of 1.0 is logically the lowest acceptable index. Higher the ratio or index better the project.

203. D. Discounted Payback period method: Define Payback period and explain its significance: Def: The payback period is the time period required for the firm to recover its initial investment from surplus discounted cashflows. Payback period is =E+ B/C years Year of final recovery is the year in which first time cumulative discounted cashflow becomes + ve E is serial number of year preceding the year of final recovery in cumulative discounted cashflow column, B is cumulative (– ve ) discounted cashflow in year E, C is discounted cash flow in the year of final recovery. If the project does not pay back itself within the maximum acceptable payback period to the purchaser, it is rejected. Lesser the period more is the security.

204. Define annuity Def : An annuity is a series of receipts or payments or deposits of equal amount made at equal interval of time. Explain types of annuities . 1.Immediate annuity : If the payments of annuity is made at the end of every period (end of every month, end of every year etc) it is called Immediate annuity. Like salary, EMI, pension etc. 2.Annuity due : If the payment of annuity is made at the beginning of every period (beginning of every month, beginning of every year etc) it is called Immediate annuity. like house rent & deposits in RD made in advance. 3.Perpetual annuity is an annuity whose payments continue forever. Eg. LIC premium paid by the person till he is alive against some mishap. 4.Deferred annuity : If the first instalment or annuity payments is deferred for certain number of periods it is called deferred annuity. Eg. A person raised loan S but fails to pay first instalment or annuity on due date due to some reason.  

205. Define Diminishing Annuity Def. Diminishing Annuity is the series of fixed amount to be paid at the end of every year for a certain period to repay/redeem a loan . The constant amount paid consists of part of principal + interest on the unpaid or remaining principal. Explain debt redemption or amortization processes Redemption or Amortization is a process to get rid of debt against purchase of any equipment or to pay back the loan amount to lender or investor. This is done by two methods 1.Diminishing Annuity method: Paying uniform instalments A for a term of a period directly to the lender or 2.Sinking Fund Method: Depositing same uniform instalments A in a sinking fund on behalf of lender and payback the sinking fund amount S to lender at the end of the term.

206. 1. Diminishing Annuity method in brief : This method is used to calculate (EMI) equal monthly or equal annual instalments (annuity) to be paid over a specified number of years in order to pay off the loan in full including interest. Lender is directly paid at the end of every year constant amount A consisting of a part of principal + interest on the unpaid or remaining part of principal. Gradually the interest part gets reduced and principal part increases. Eventually by the end of term the loan with interest is fully paid back. Annual payment or annuity A = Pv X r (1+r)n /[(1+r)n -1]

207. 2.Sinking Fund Method Say borrower borrows Rs P from lender, instead of paying installments (EMI) directly to lender, borrower deposits Rs. A in a Sinking Fund of bank or Govt. security @ r % compound Interest for n years on behalf of lender & pay to lender S or Sinking Fund money after the entire period of n yrs is over. This S or Sinking Fund money is equivalent to future value of P Or S = P (1+r)^n But S = A [(1+r)^n – 1]/r = maturity value of annuity A as per sinking fund formula Or A= S X r/(1+r)^n -1 Since S = P (1+r)^n A = P X r (1+r)n /[(1+r)n -1] This annuity is same as by EMI or diminishing annuity method.

208. Explain diminishing method of annuity. 4 This method is used to calculate (EMI) equal monthly or annual instalments (annuity) to be paid over a specified number of years in order to pay off the loan in full including interest. Say P = Loan taken for n years @ compound interest r%. S= sum or Future value of P after the term of n years S = P (1+r) n If A = annuity of equal amount paid yearly for n yrs at the end of each yr Then as per sinking fund formula Future value of constant annuity A = S A = A [(1+r) n – 1] r Present value of S = P = S/(1+r) n P = A [(1+r) n – 1] putting value of S r(1+r) n

209. A= P X r (1+r) n (1+r) n -1 A = P X r (1+r) n /[(1+r) n -1] = Annual payment or EYI for n years (beginning from the end of 1 st year) to redeem loan P with interest r% Annuity A is called diminishing annuity This method of loan redemption is called diminishing method of annuity

210. Def: Depreciation is the reduction /loss in the value (book value) of capital asset during its economic life due to wear and tear or ageing.   In any industry equipment, machines, buildings (except land) are subjected to wear & tear due to use & ageing, hence their value gets lowered, this loss in the value is called depreciation.   Explain causes and the purpose of calculating depreciation? Def: Depreciation is the loss in the value of capital asset during its economic life due to wear and tear or ageing.

211. Causes for Depreciation – Physical Causes Wear and tear due to use. Deterioration and decay with passage of time ( Even if the property/capital is not used ) Damage or destruction due to accident like fire ,flood earthquake etc Action of chemical elements on the component parts. Lack of maintenance . 2) Functional Causes Uselessness of the present purpose. Obsolescence of technology i.e due to innovation, change in technique. Objectives of Depreciation 1) To Calculate the Correct Profits: The depreciation shows the diminishing value of assets or cost ( expenditure ) incurred due to use/owning an equipment. It must be deducted from the profits, so as to calculate the correct profits. 2) To Show the Assets at its Real Value - Real value of the property can only be shown after the charging depreciation every year. 3) To Replace the assets - Every Year the depreciated amount is set aside. It gives the management sufficient funds to replace the machinery /assets. 4) To Reduce Tax Liability – Depreciation is chargeable under expenses under Income tax Act and thus the taxable profits are reduced. It thus reduces the tax burden of the Company

212. Name the methods for calculating depreciation. Explain anyone method for calculating depreciation by taking an example. Methods of Calculating Depreciation The methods of calculating depreciation are as follows: 1) Straight line method 2) Declining/Reducing balance method 3) Depreciation fund method or sinking fund method Straight line method It is most common method and based on assumption that wear & tear of the capital asset is uniform throughout its normal useful life. Depreciation per year = (C-S)/n C = Cost of the asset or the purchase price of the asset including installation cost if any S = scrape value/ Salvage value is the value of the asset at the end of its useful life. n= Useful life of asset in years  

213. Declining/Reducing balance method : In the declining balance method the depreciation rate is pre-decided and the depreciation in any year is calculated on the book value of the asset at the beginning of that year. Taking the case of the machine valued at Rs. 5000/-. If 10% is its depreciation rate the depreciated cost at the end of 1st yr will be Rs. 5000- Rs.500= 4500. Similarly depreciation in 2nd year will be 10% of book value 4500 at the beginning of 2nd year ie Rs 450 and book value at the end of 2nd year will be 4500-450= 4050 and so on. This method gives real cost of using a machine as depreciation in earlier years is more when repair charges are lighter. It is especially suitable to assets with long life, e.g., plant and machinery, motor car etc. For finding the rate of depreciation the formula used in is. r = 1 - (S/C)1/n Where: r = pre-decided rate of depreciation = depreciation factor, n = Estimated useful life of asset. S = Residual value after the expiry of useful life. C = Original cost of asset  

214. Depreciation fund method or sinking fund method Sinking fund. Def: Sinking fund method is a depreciation technique used to finance the replacement of an asset at the end of its life. A sinking fund is created by depositing a constant amount equal to annual depreciation for the period equal to life of the asset so that the fund accumulated at compound interest is available for purchase of a new one when old machine is worn out. Sinking fund method is especially applicable to costly machines in large scale industries. Annual rate of depreciation A= r (C –P)/ (1+r) n – 1 Whare C =cost of m/c, P = scrape value, r =rate of return in SF, n =useful life of m/c in years, A= Annual contribution to sinking fund. Suitable inflation rate can also be taken into account to arrive at Annual rate of depreciation

215. Explain life of a mine . Life of Mine is the number of years during which the mineable reserves can be profitably worked out. It depends on the degree of mechanization, modernization and the investment potential, For example with higher mechanization and higher capacity of machines the quarriable limit increases and we can mine at greater depth. This increases the life of the mine. Conversely if reserves are limited the life of the mine reduces with higher mechanization and higher capacity of machines The life of the mine (LOM) ='expected ore tonnage' (EOT)/ the average production rate (APR)  

216. Thanks