# Internal Rate of Return by dfhdhdhdhjr

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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
•   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
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

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

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