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Internal Rate of Return

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					Internal Rate of Return
Andrew Jain and Ravinder Saidha
What We Will Cover
• What is Internal Rate of Return?
• Formula to calculate IRR for:
    •   Projects / Common Stocks
    •   Zero-Growth Models
    •   Constant Growth Models
    •   Multiple Growth Models
•   Crossover Rate
•   Independent & Mutually Exclusive Projects
•   Advantages and Disadvantages of IRR
•   Conclusion
What is Internal Rate of Return?
• Another way of making a capital budgeting decision
• Is calculated when the Net Present Value is set equal
  to Zero
• There are four model types we will cover:
   •   Projects / Common Stocks
   •   Zero Growth
   •   Constant Growth
   •   Multiple Growth
 IRR for Common Stocks
 • Formula
                CF1            CF2                  CFN
NPV  CF0                              ...               0
            (1  IRR ) 1
                           (1  IRR ) 2
                                                (1  IRR ) N



        N
                CFt
                       0
      t  0 (1  IRR )
                       t
  Sample Question
  Time Period:            0       1     2     3     4

  Cash Flows:            -1,000   500   400   300   100




PV of the
inflows         -1,000
discounted
at IRR


                 NPV = 0
     Sample Question Continued
     • Can only find IRR by trial and error
                   CF1        CF2                  CFN
   NPV  CF0                          ...               0
               (1  IRR ) (1  IRR )
                         1           2
                                               (1  IRR ) N



                500        400            300            100
0  1000                                       
            (1  IRR ) (1  IRR )
                      1           2
                                      (1  IRR ) 3
                                                     (1  IRR ) 4


     • IRR = 14.49%
Practice Question
Professor Stephen D'Arcy is planning to invest $500,000 in to his own
insurance company, but is unsure about the return he will gain on this
investment. He produces estimated cash flows for the following years:
•   Year 1: $200,000
•   Year 2: $250,000
•   Year 3: $300,000
How do you find his internal rate of return for this investment?

                             200 ,000    250 ,000     300 ,000
• A            500 ,000                           
                            (1  IRR )1 (1  IRR ) 2 (1  IRR )3
                               200 ,000    250 ,000     300 ,000
• B             500 ,000                           
                              (1  IRR )1 (1  IRR ) 2 (1  IRR )3

                               200 ,000    250 ,000     300 ,000
• C             500 ,000                           
                              (1  IRR )3 (1  IRR ) 2 (1  IRR )1

                             200 ,000    250 ,000     300 ,000
• D            500 ,000                           
                            (1  IRR )1 (1  IRR ) 2 (1  IRR )3

• E           This is a trick question
IRR for Zero Growth Models
• A zero growth model is when dividends per
  share remain the same for every year
• Formula:

         D1
   IRR 
         P
• Where:
   • D1 = Dividend paid
   • P = Current price of stock
Sample Question
• Andrew is prepared to pay his stockholders $8 for
  every share held. The current price
  that his stock is currently held for is $65.
  What is his internal rate of return?
         $8
  IRR 
        $65

• IRR = 12.3%
IRR for Constant Growth Models
• A constant growth model is when the
  dividend per share grows at the same rate
  every year
• Formula is similar to zero growth, except
  you have to add growth:
         D1
   IRR     g
         P
Sample Question
• Rav paid $1.80 in dividends last year. He
  has forecasted that his growth will be 5%
  per year in the future. The current share
  price for his company is $40.
  What is his IRR?
 What is D1?
    Do * (1 + Growth Rate)
    $1.80 * (1+5%) = $1.89

           $1.89
   IRR           0.05
            $40

 IRR = 9.72%
IRR for Multiple Growth Model
•   A multiple growth model is when dividends growth
    rate varies over time
•   The focus is now on a time in the future after which
    dividends are expected to grow at a constant rate g
•   Unfortunately, a convenient expression similar to the previous
    equations is not available for multiple-growth models.
    You need to know what the current price
    of the stock is to find IRR
•   Formula:           N
                              D                  D
                    P                 t
                                                                  t 1

                           t 1   (1  IRR ) t       ( IRR  g )(1  IRR )T
•   Where:
     •   Dt = Dividend payments before dividends are made constant
     •   Dt+1 = Dividend payment after dividends are set to a constant rate
     •   t = time dividends are paid at
     •   T = time that dividends are made constant
     •   P = Current price of stock
Sample Question
•   The University of Illinois paid dividends in the first and
    second year amounting to $2 and $3 respectively. It then
    announced that dividends would be paid at a constant rate of 10%. The
    current price of the stock is $55.
•   We know:
     •   D1 = $2
     •   D2 = $3
     •   P = 55
     •   T = 2 (as after second year, dividends become constant)
•   We need to find D3:
     • $3 * (1+10%) = $3.30

               $2          $3               $3.30
    55                           
           (1  IRR )1 (1  IRR ) 2 ( IRR  0.1)(1  IRR ) 2

•   IRR = 14.9%
Practice Question
•   Professor Stephen D'Arcy is the CEO of a large insurance
    firm, AIG. He is prepared to pay $10 in dividends for the first three years, in
    which after the third year, the growth rate in dividends will be 10%. If the
    stock currently sells for $100,
    how do you find his internal rate of return?
                         $10         $10          $10              $11
• A         100                                       
                     (1  IRR )1 (1  IRR ) 2 (1  IRR )3 ( IRR  0.1)(1  IRR ) 4
                         $10         $11         $12 .1           $13 .31
• B          100                                      
                     (1  IRR )1 (1  IRR ) 2 (1  IRR )3 ( IRR  0.1)(1  IRR ) 4
                         $10         $10          $10              $11
• C          100                                      
                     (1  IRR )1 (1  IRR ) 2 (1  IRR )3 ( IRR  0.1)(1  IRR )3
                         $10         $10          $10              $10
• D          100                                      
                     (1  IRR )1 (1  IRR ) 2 (1  IRR )3 ( IRR  0.1)(1  IRR )3

• E          I have no idea what you want me to do
Crossover Rate
• The crossover rate is defined as the rate at which the
  NPV’s of two projects are equal.




    Source: http://people.sauder.ubc.ca/phd/barnea/documents/lecture%202%20-%202004.pdf
Internal Rate of Return

• Advantages
  • Doesn’t require a discount rate to calculate
    like NPV calculations


• Disadvantages
  • Lending vs. Borrowing
  • Multiple IRRs
  • Mutually Exclusive projects.
Disadvantages
• Lending vs. Borrowing

• Example: Suppose you have the choice between projects A
  and B. Project A requires an investment of $1,000 and pays
  you $1,500 one year later. Project B pays you $1,000 up front
  but requires you to pay $1,500 one year later.




Project     C_0           C_1         IRR         NPV at 10%


A           -1,000        +1,500      +50%        +364
B           +1,000        -1,500      +50%        -364
Disadvantages Continued
• Multiple IRR’s
  • In certain situations, various rates will cause
    NPV to equal zero, yielding multiple IRR’s.
  • This occurs because of sign changes in the
    associated cash flows.
  • In a case where there are multiple IRR’s,
    you should choose the IRR that provides
    the highest NPV at the appropriate discount
    rate.
Disadvantages Continued
• Mutually exclusive projects can be misrepresented by the
  IRR rule.
• Example: Project C requires an initial investment of $10,000
  and yields a inflow of $20,000 one year later. Project D
  requires an initial investment of $20,000 and yields an inflow
  of $35,000 one year later. It would appear that we should
  choose project C due to its higher IRR. Project D, however,
  has the higher NPV.

   Project      C_0         C_1          IRR (%)     NPV at 10%


   C            -10,000     +20,000      100         +8,182

   D            -20,000     +35,000      75          +11,818
Conclusion
• There are various types of models for calculating IRR
  including common stock, zero growth, constant
  growth, and multiple growth.
• Despite the disadvantages covered, IRR is still a much
  better measure than the payback method or even
  return on book.
• When applied correctly, IRR calculations yield the
  same decisions that NPV calculations would.
• In cases where IRR causes conflicts in
  decision-making, it is more useful to use NPV.
Questions?

				
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posted:1/24/2013
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