We have compared cash flow options based on PW and EAW. These methods assumed we knew the value of i, and took no offense to whatever i was used.

In real world i fluctuates widely depending upon:

• Who we are. (individual, corporation, government agency)
• How good our credit is.
• How much money we have to invest.
• How long the money can be committed.

Terms

MARR = Minimum attractive rate of return

• Rate set by an organization to designate lowest level of 'i' that makes a cash flow option acceptable.
• The rate of return available to investor, should they choose to invest funds elsewhere rather than in the current proposal.
• Target ROR or Required ROR.

IRR = Internal rate of return = i*

• Interest rate at which the PW of cash flow equals 0.
• ROR of the cash flow under consideration.
• True rate of return.

Would like to have IRR>MARR for a profitable venture.

Objective in ROR Analysis

Find i* and compare it to MARR to see if the investment is profitable. Find i* where

PW (Benefits & Receipts) = PW (Costs & Disbursements)   

Hence PW(Benefits) - PW(Costs) = 0

Types of Cash Flows

Simple

Net cash flow changes signs only once over study period. Either   + to -   or    - to+

Non-simple

Net cash flow changes sign more than once over study period. This will result in multiple IRR values.

Types of Investments (In Non-simple Cash Flows)

Pure investment : 

Project cash flow balances are greater than 0 when evaluated at i* at any time period.
All simple investments are pure investments.

Mixed investment:

Project cash flow balances are both   +   and   -  
Balances are reinvested at MARR for analysis purposes in order to find i*.

Example: Let's look at the Cash flow balance method of a pure investment.
  
Find balance of cash at end of each year.

End Period     Cash Flow            Current Balance

     0                 -1000                                                 -1000.00
     1                263.80              -1000(1+i)¹+263.80  = -836.20    (Note: 1+i = 1+0.10 = 1.1)     
     2                263.80          -836.20(1.1)¹+263.80   = -656.02
     3                263.80          -656.02(1.1)¹+263.80   = -487.82
     4                263.80          -487.82(1.1)¹+263.80   = -239.80
     5                263.80          -239.80(1.1)¹+263.80   =       0.00

All balances are less than 0.
If i was not equal to i*, the final balance at the end of year 5 would not be equal to 0.
If we had used i = 5%  then CB(5) = 65.12 which is greater than 0. 

 Hence i =5% would not be i*, since at i* the cash balance at the last period must be 0.

Simple Investments

Assume MARR= 10%. Is this a good investment?

Recall that PW(Disbursements) = PW(Benefits)  and
     P = F(P/F,i,N) = F(1+i)^-N  sets up the problem to look like this :
1000 = 200(1+i*)^-1  + 200(1+i*)^-2  +  400(1+i*)^-3 + 600(1+i*)^-4   or
      0 = -1000 + 200(1+i*)^-1 + ............

Solve the problem by trial and error.

               200x            200x            400x              600x                    PW
   i*        (1+i*)^-1      (1+i*)^-2       (1+i*)^-3        (1+i*)^-4                   Total   

12%      178.57        159.44       284.71          381.31          $4.03 greater than 0

13%      176.99         156.63       277.22           367.99       -$21.17 lesser than 0

Since we need a PW Total = 0, solve for i*.

By interpolation :  i* = 12% + 1%    (     4.03    )    = 12.16%
                                                     [4.03-(-21.17)]

12.16%> MARR= 10%

Hence investment is acceptable compared to MARR.

Trial and error is great, but where do you start? Let's look at several ways to at least get close.

1) Sum the cash flows to determine if the gain is substantial, minor or negative.

            End Period             A              B            C             D
                 0                 -$100        -$100      -$100      -$100
                 1                      20             30            10           30
                 2                      30             60            30           30
                 3                      40             40            30           30
                 4                      20             70            20           20   
             Total                    $10          $100        -$10       $10
      Actual   i*                 3.85%      30.69%    -3.85%    7.71%

Use the magnitude of the gain to help with a starting point. If there is no gain, (Negative total), i* is negative and thus less than MARR.

2)  Approximate constant cash flows.

Using Option A or D above.

A(approx) = $110/4 = $27.50 / yr
A/P = $27.50/$100 = 0.2750  ---->(Tables for (A/P, i ,4))
Thus i = 4% for a starting point for PW calculation.

3)  Rule of 72.

An investment will double in size in N years at i% relative to the following relationship: 72 / N = i          
Hence if most of income is at yr N and about double the initial outlay   i = 72 / N
    

4)  If a trial results in (+)PW --- Try a  higher i on the next trial.
     If a trial results in (-)PW  --- Try a  lower i on the next trial.
     Go the way of the sign!

5)  Use standard cash flows and table amounts, whenever possible.
    
P = A(P/A, i, 6)
P/A = 1000/350 = (P/A, i, 6)
2.8571 = (P/A, i, 6)
(P/A, 25, 6) = 2.95142
(P/A, 30, 6) = 2.64275

Then by interpolation, i* = 25% + 5% (2.95142  -  2.8571)  = 26.53%
                                                          (2.95142 - 2.64275)

6)  Use a computer program or spread sheet.
Excel IRR function = IRR(A1: AXX) where A1:AXX represents the rows where you have entered the periodic cash flows. When doing so, year labels are not required.
                          

Non - Simple Investments

What do we do if the cash flow changes signs more than once?
Assume the following cash flow:

To find IRR
         PW = 0 = -A₀+A₁(1+i*)^-1+A₂(1+i*)^-2+.............+A_N(1+i*)^-N
         0 = -A₀+A₁((1+i*)^-1)¹+A₂((1+i*)^-1)²............+A_N((1+i*)^-1)^N

Factoring out the term   X= (1+i*)^-1 we get 0 = -A₀+A₁X+A₂X²+...........+A_NX^N 

Descarates rule of sign of Polynomials:  
A Polynomial can have as many real roots as sign changes to the A _{j 's}