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      Mill Modification ROI, IRR and Payback Explanation  
                         
      1 Introduction:              
        Two calculators are available.            
        The first calculator is valid for cement mills with 2 compartments and for the modification of the internals only.    
        The second calculator is used for circuit modifications like pregrinding system or new separator.      
                         
      2 Definition:              
        In business, the purpose of the profitability of an investment is to measure, per period, rates of return on money invested in an   
        economic entity in order to decide whether or not to undertake an investment.          
        As a performance measure, it is used to evaluate the efficiency of an investment or to compare the efficiency of a number of  
        different investments.              
        In purely economic terms, it is one way of considering profits in relation to capital invested.        
        Three indicators are used to determine the profitability of an investment:           
        - Payback              
        - ROI              
        - IRR              
                         
      3 Payback Period:              
        - Payback period is the time in which the initial cash outflow of an investment is expected to be recovered from the cash inflows  
          generated by the investment. It is one of the simplest investment estimation techniques.        
        - The formula to calculate the payback period of a project depends on whether the cash flow per period from the project is even  
          or uneven.              
        - In case they are even, the formula to calculate the payback period is:          
         
     
               
                     
                     
        - When cash inflows are uneven, we need to calculate the cumulative net cash flow for each period and then use the following   
          formula for payback period:            
         
     
               
                     
                     
          Where:              
          A is the last period with a negative cumulative cash flow,            
          B is the absolute value of cumulative cash flow at the end of the period A and          
          C is the total cash flow during the period after A            
        - Example for Even Cash Flows:            
          A Cement Plant is planning to undertake a project requiring initial investment of US$ 10,5 million.      
          The project is expected to generate US$ 3,8 million per year for 8 years.          
          Calculate the payback period of the project.            
          Solution:              
          Payback Period = Initial Investment ÷ Annual Cash Flow = US$ 10,5M ÷ US$ 3,8M = 2,8 years        
        - Example for Uneven Cash Flows:            
          A Cement Plant is planning to undertake another project requiring initial investment of US$ 5 million and is    
          expected to generate US$ 1 million in Year 1, US$ 1,3 million in Year 2, US$ 1,6 million in year 3, US$ 1,9 million in Year 4  
          and US$ 2,2 million in Year 5.            
          Calculate the payback period of the project.            
          Solution:              
         
     
               
                     
                     
                     
                     
                     
                     
                     
                     
          Payback Period = 3 + ( abs(-1,1) ÷ 1,9 )            
          Payback Period = 3 + 0,58              
          Payback Period = 3,58 years            
        - Advantages of payback period are:            
          1. Payback period is very simple to calculate.            
          2. It can be a measure of risk inherent in a project.            
          3. For companies facing liquidity problems, it provides a good ranking of projects that would return money early.    
        - Disadvantages of payback period are:            
          1. Payback period does not take into account the time value of money (see Discounted Payback Period).      
          2. It does not take into account the cash flows that occur after the payback period.        
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      4 Discounted Payback Period:            
        - One of the major disadvantages of simple payback period is that it ignores the time value of money. To counter this limitation,  
          an alternative procedure called discounted payback period may be followed, which accounts for time value of money by   
          discounting the cash inflows of the project.            
        - The formula to calculate the discounted payback period is:            
         
     
               
                     
                     
          Where:              
           i is the discount rate and              
          n is the period to which the cash inflow relates.            
         
     
                 
                         
                         
          Where:              
          A is the last period with a negative discounted cumulative cash flow,          
          B is the absolute value of discounted cumulative cash flow at the end of the period A and        
          C is the discounted cash flow during the period after A.            
        - Example:              
          An initial investment of US$ 3.500.000 is expected to generate US$ 900.000 per year for 6 years.      
          Calculate the discounted payback period of the investment if the discount rate is 11%.        
          Solution:              
         
     
           
                 
                 
                 
                 
                 
                 
                 
                 
                 
                 
          Discounted Payback Period = 5 + (abs(-173 000) ÷ 481177 = 5,37 years          
        - Advantage of discounted payback period is:            
          Discounted payback period is more reliable than simple payback period since it accounts for time value of money.    
        - Disadvantage of discounted payback period is:            
          It ignores the cash inflows from project after the payback period.            
                         
      5 ROI (Return on Investment):            
        - ROI is the most common profitability ratio.            
        - ROI is a performance measure used to evaluate the efficiency of an investment.        
        - To calculate ROI, the benefit of an investment is divided by the cost of the investment.        
        - The return on investment formula:            
         
     
               
                     
                     
        - Example:              
          Gain of Investment = US$ 20 000            
          Cost of Investment = US$ 7 500            
          Solution:              
          Roi = (20 000 - 7 500) ÷ 7 500 = 167%            
                         
      6 IRR (Internal rate of return):            
        - Internal rate of return is used to evaluate the attractiveness of a project or investment. If the IRR of a new project exceeds  
          a company’s required rate of return, that project is desirable. If IRR falls below the required rate of return, the project    
          should be rejected.              
        - Internal rate of return (IRR) is the discount rate at which the net present value of an investment becomes zero.    
        - The formula is:              
         
     
           
                 
                 
          Where:              
          CF is the net cash inflow,              
          1...n are the years              
          r is the IRR              
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      7 Depreciation:              
        - Depreciation is an income tax deduction that allows a taxpayer to recover the cost of certain property.      
        - It is an annual allowance for the wear, deterioration, or obsolescence of the property.        
        - The two most well-known methods to calculate the depreciation of an industrial equipment are:      
          1. Straight-Line Depreciation            
          2. Declining-Balance Depreciation            
                         
      8 Straight-Line Depreciation Example:            
        - It is a system of depreciation in which one deducts the same amount every year.        
        - Example:              
          Cost of investment: 1 200 000 US$            
          Salvage (or residual value): 200 000 US$            
          Life: 5 years              
         
     
       
             
             
         
     
           
                 
                 
                 
                 
                 
                 
                 
                 
                         
      9 Declining-Balance Depreciation Example:            
        - Also called Double-Declining-Balance Depreciation            
        - The (double) declining balance method of depreciation is a common form of accelerated depreciation.      
        - In the calculator, two degressive depreciation rate are available:            
         
     
               
                     
                     
                     
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        - Example with Ddr1:              
          Cost of investment: 1 500 000 US$            
          Salvage (or residual value): 100 000 US$            
          Life: 6 years              
         
     
           
                 
                 
                 
          Comparizon between DDB and SL yearly depreciation:            
         
     
           
                 
                 
                 
                 
                 
                 
                 
                 
                 
          When the DDB depreciation is lower than the SL depreciation, SL depreciation replaces DDB depreciation.    
          Result:              
         
     
           
                 
                 
                 
                 
                 
                 
                 
                 
                 
                         
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