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?