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Internal Rate of ReturnAndrew Jain and Ravinder Saidha

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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

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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

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IRR for Common Stocks

Formula

0)1(

...)1()1( 2

2

1

10

N

N

IRR

CF

IRR

CF

IRR

CFCFNPV

N

t

t

t

IRR

CF

0

0)1(

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Sample Question

Time Period: 0 1 2 3 4

Cash Flows: -1,000 500 400 300 100

PV of theinflowsdiscountedat IRR

-1,000

NPV = 0

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Sample Question Continued

Can only find IRR by trial and error

IRR = 14.49%

0)1(

...)1()1( 2

2

1

10

N

N

IRR

CF

IRR

CF

IRR

CFCFNPV

4321 )1(

100

)1(

300

)1(

400

)1(

50010000

IRRIRRIRRIRR

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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 thisinvestment. 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

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IRR for Zero Growth Models

A zero growth model is when dividends pershare remain the same for every year

Formula:

Where:

D1 = Dividend paid

P = Current price of stock

P

DIRR

1

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Sample Question

Andrew is prepared to pay his stockholders $8 forevery 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

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IRR for Constant Growth Models

A constant growth model is when thedividend per share grows at the same rate

every year

Formula is similar to zero growth, except

you have to add growth:

gP

DIRR 1

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Sample Question

Rav paid $1.80 in dividends last year. Hehas 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

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IRR for Multiple Growth Model

A multiple growth model is when dividends growthrate 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 = Currentprice of stock

T

tN

t

t

t

IRRgIRR

D

IRR

DP

)1)(()1(

1

1

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Sample Question

The University of Illinois paid dividends in the first andsecond year amounting to $2 and $3respectively. 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

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Practice Question

Professor Stephen D'Arcy is the CEO of a large insurancefirm, AIG. He is prepared to pay $10 in dividends for the first three years, inwhich 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

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Crossover Rate

The crossover rate is defined as the rate at which theNPVs of two projects are equal.

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

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Internal Rate of Return

Advantages

Doesnt require a discount rate to calculate

like NPV calculations

Disadvantages

Lending vs. Borrowing

Multiple IRRs

Mutually Exclusive projects.

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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 frontbut 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

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Disadvantages Continued

Multiple IRRs In certain situations, various rates will cause

NPV to equal zero, yielding multiple IRRs.

This occurs because of sign changes in the

associated cash flows. In a case where there are multiple IRRs,

you should choose the IRR that provides

the highest NPV at the appropriate discountrate.

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Disadvantages Continued

Mutually exclusive projects can be misrepresented by theIRR rule.

Example: Project C requires an initial investment of $10,000and 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 shouldchoose 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

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Conclusion

There are various types of models for calculating IRRincluding 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 evenreturn on book.

When applied correctly, IRR calculations yield the

same decisions that NPV calculations would.

In cases where IRR causes conflicts indecision-making, it is more useful to use NPV.

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Questions?