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
0)1(
...)1()1( 2
21
10
NN
IRR
CF
IRR
CF
IRR
CFCFNPV
N
tt
t
IRR
CF
0
0)1(
Sample Question
Time Period: 0 1 2 3 4
Cash Flows: -1,000 500 400 300 100
PV of theinflows discounted at IRR
-1,000
NPV = 0
Sample Question Continued
• Can only find IRR by trial and error
• IRR = 14.49%
0)1(
...)1()1( 2
21
10
NN
IRR
CF
IRR
CF
IRR
CFCFNPV
4321 )1(
100
)1(
300
)1(
400
)1(
50010000
IRRIRRIRRIRR
Practice QuestionProfessor 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?
• A
• B
• C
• D
• E This is a trick question
321 )1(
000,300
)1(
000,250
)1(
000,200000,500
IRRIRRIRR
321 )1(
000,300
)1(
000,250
)1(
000,200000,500
IRRIRRIRR
123 )1(
000,300
)1(
000,250
)1(
000,200000,500
IRRIRRIRR
321 )1(
000,300
)1(
000,250
)1(
000,200000,500
IRRIRRIRR
IRR for Zero Growth Models
• A zero growth model is when dividends per
share remain the same for every year
• Formula:
• Where:• D1 = Dividend paid
• P = Current price of stock
P
DIRR 1
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 $65.
What is his internal rate of return?
• IRR = 12.3%
65$
8$IRR
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:
gP
DIRR 1
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%) = $1.89
IRR = 9.72%
05.040$
89.1$IRR
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:
• 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
Tt
N
tt
t
IRRgIRR
D
IRR
DP
)1)(()1(1
1
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%) = $3.30
• IRR = 14.9%
221 )1)(1.0(
30.3$
)1(
3$
)1(
2$55
IRRIRRIRRIRR
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 $100, how do you find his internal rate of return?
• A
• B
• C
• D
• E I have no idea what you want me to do
4321 )1)(1.0(
11$
)1(
10$
)1(
10$
)1(
10$100
IRRIRRIRRIRRIRR
4321 )1)(1.0(
31.13$
)1(
1.12$
)1(
11$
)1(
10$100
IRRIRRIRRIRRIRR
3321 )1)(1.0(
11$
)1(
10$
)1(
10$
)1(
10$100
IRRIRRIRRIRRIRR
3321 )1)(1.0(
10$
)1(
10$
)1(
10$
)1(
10$100
IRRIRRIRRIRRIRR
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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