Issues for the IRR Rule
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The IRR Rule has many shortcomings.

 First, it ignores the time value of money: $100 today has the same value
  as $100 in two years. The first problem could be overcome using the 
  Present Value Payback Rule, which  applies the present value of all 
   future cash flows instead of their face values, see one of the 
  end-of-chapter problems. Second, it ignores the future cash flows
  after the payback period. The first case, as an illustration, 
  is shown below. The cash flows in years 5 and 6 are ignored. 

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Monotone function
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A monotone function which is a function where its value is either entirely non-increasing or non-decreasing for a given input value. Binary search is a classic algorithm used to find a specific element in a sorted array or sequence. The algorithm operates by repeatedly dividing the search interval in half. If the value of the search key is less than the item in the middle of the interval, the algorithm narrows the interval to the lower half. Otherwise, it narrows it to the upper half. This process continues until the search key is found or the interval is empty. Binary search is highly efficient, with a time complexity of O(log n), making it much faster than linear search algorithms, especially for large datasets. It is a fundamental technique used in computer science for problems involving sorted data and is the backbone of more complex search and optimization algorithms. 

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inner-,left-,right-,all-join
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Assume that we have two data sets A and B. 

    inner-join: if an item exists in both A and B. 

    left-join: if an item exists in A. 

    right-join: if an item exists in B. 

    out-join: if an item exists in eight A or B. 

 Note: out-join is also called all-join

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fv annuity         [new]
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Definition of annuity:
   1) the same cash flows
   2) at the same interval,
   3) for n periods
      If we add the follwoing condition: 
         The 1st cash flow happends at the end of the first period
      then, we have the following formula
                    c
    fv(annuity) =  --  ( (1+r)^n  -1 )
                    r

    see .ice(28) for one example. 

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