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RATE OF RETURN ANALYSIS

1

INTERNAL RATE OF RETURN (IRR)The Internal Rate of Return (IRR) method solves for the interest rate that

equates the equivalent worth of a project's cash outflows (expenditures)to the equivalent worth of cash inflows (receipts or savings).

2

INTERNAL RATE OF RETURN (IRR)By definition:

Given a cash flow stream, IRR is the interest rate i at which thebenefits are equivalent to the costs or the NPW=0

The Internal Rate of Return is the rate of return that yields a Net PresentValue of zero.

NPW=0

PW of benefits - PW of costs =0

PW of benefits = PW of costs

PW of benefits / PW of costs=1

EUAB-EUAC=0

3

INTERNAL RATE OF RETURN (IRR)

In other words, the IRR is the interest rate that makes the PW, AW, and FWof a project's estimated cash flows equal to zero. That is, PW(i') of cashinflow = PW(i') of cash outflow.

We commonly denote the IRR by i'.

PW(i' %) = 0

AW(i' %) = 0

FW(i' %) = 0

4

Internal Rate of Return

-$1,000

$350

years1 2 3 4

$350$350 $350

15.00000,1

1

350

1

350

1

350

1

350432

IRRIRRIRRIRRIRR

$350

years1 2 3 4

$350$350 $350

$1,063

$1,00015%

12%

Internal Rate of Return

• Example: A company invests $10,000 in a computer and resultsin equivalent annual labor savings of $4,021 over 3 years. Thecompany is said to earn a return of 10% on its investment of$10,000.

PROJECT BALANCE CALCULATION:

0 1 2 3

Beginningproject balance

Return oninvested capital

Paymentreceived

Ending projectbalance

-$10,000 -$6,979 3,656

-$1,000 -$697 -$365

-$10,000 +$4,021 +$4,021 +$4,021

-$10,000 -$6,979 -$3,656 0

The firm earns a 10% rate of return on funds that remain internallyinvested in the project. Since the return is internal to the project, we callit internal rate of return.

INTERNAL RATE OF RETURN (IRR)

8

700 = 100/(1+i) + 175/(1+i)2 + 250/(1+i)3 + 325/(1+i)4.

It turns out that i = 6.09 %.

Suppose you have the following cash flow stream. Youinvest $700, and then receive $100, $175, $250, and $325at the end of years 1, 2, 3 and 4 respectively. What is the IRRfor your investment?

0 1

$700

$100

time2 3 4

$175$250

$325

How to calculate IRR?

COMPUTATION OF IRR

Direct Solution

Trial and Error Solution

Computer Solution

9

Internal Rate of Return MethodInternal Rate of Return MethodInternal Rate of Return Method

Management is evaluating a proposal toacquire equipment costing $97,360. The

equipment is expected to provide annual netcash flows of $20,000 per year for seven

years.

Management is evaluating a proposal toacquire equipment costing $97,360. The

equipment is expected to provide annual netcash flows of $20,000 per year for seven

years.

$97,360

$20,000= 4.868

Determine the table valueusing the present value foran annuity of $1 table.


Amount to be invested

Equal annual cash flow

Present Value of an Annuity of $1Present Value of an Annuity of $1Present Value of an Annuity of $1

1 0.943 0.909 0.893 0.870

2 1.833 1.736 1.690 1.626

3 2.673 2.487 2.402 2.283

4 3.465 3.170 3.037 2.855

5 4.212 3.791 3.605 3.353

6 4.917 4.355 4.111 3.785

7 5.582 4.868 4.564 4.160

Year 6% 10% 12% 15%


Find the seven year line on thetable. Then, go across theseven-year line until the closestamount to 4.868 is located.


1 0.943 0.909 0.893 0.870

2 1.833 1.736 1.690 1.626

3 2.673 2.487 2.402 2.283

4 3.465 3.170 3.037 2.855

5 4.212 3.791 3.605 3.353

6 4.917 4.355 4.111 3.785

7 5.582 4.868 4.564 4.160

Year 6% 10% 12% 15%

4.8684.868



Move vertically to the top of thetable to determine the interestrate.


1 0.943 0.909 0.893 0.870

2 1.833 1.736 1.690 1.626

3 2.673 2.487 2.402 2.283

4 3.465 3.170 3.037 2.855

5 4.212 3.791 3.605 3.353

6 4.917 4.355 4.111 3.785

7 5.582 4.868 4.564 4.160

Year 6% 10% 12% 15%

4.8684.868

10%10%

10%

DIRECT SOLUTION

Example

15

n Cash Flow

1 -$2,000

2 1300

3 1500

0)1(

1500$

)1(

1300$000,2$)(

2

iiiPW

)1(

1

ixAssume

01500$1300$2000$)( 2 xxiPW

)(%160667.1

%25%258.0

667.18.0*

cesignificaneconomicnoix

iix

orxSolving

TRIAL AND ERROR METHOD

Aiming for i that makes PW(i)=0

Guess a value of i*

Compute the PW of net cash flows

Observe if PW is +, -, or zero

PW(i) is negative, lower the interestrate

PW(i) is positive, raise the interest rate

Continue until PW(i) is approximatelyzero

16

4 Example Continued – IRR method

Find i'% such that the PW(i'%) = 0.

0 = -$50,000 + $17,500(P|A, i'%,5) + $10,000(P|F, i'%,5)

PW (20%) = 6354.50 tells us that i' > 20%

PW (25%) = 339.75 > 0, tells us that i'% > 25%

17

PW (30%) = -4,684.24 < 0, tells us that i'% < 30%

25% < i' < 30%

Use linear interpolation to estimate i'%.

18

19

20

Example

0

1 2 3 4 5 6 7 8

Years

(Units in millions)

$10

$1.8 1.8 1.8 1.8 1.8 1.8 1.8

$2.8

After tax net cash flows

Guess i=8%

PW(8%) = -$10+$1.8(P/A, 8%, 8)+$1(P/F, 8%, 8) = $0.88

Sale value = $1

Since PW is positive, raise the interest rate

Assume i=12%

PW(12%) = -$10+$1.8(P/A, 12%, 8) + $1(P/F, 12%, 8) = -$0.65

Use interpolation

%3.10)65.0(88.0

088.0%)8%12(%8*

i

Check PW(i) with this i*, iterate if necessary. Computer value = 10.18%

Trial

interest

rates

NPW

0 $50.00

5 $26.46

10 $9.24

15 ($3.49)

20 ($12.97)

25 ($20.06)

30 ($25.37)

35 ($29.36)

40 ($32.34)

45 ($34.54)

50 ($36.16)($50.00)

($40.00)

($30.00)

($20.00)

($10.00)

$0.00

$10.00

$20.00

$30.00

$40.00

$50.00

$60.00

0 5 10 15 20 25 30 35 40 45 50

Ne

tP

res

en

tV

alu

e

Year Cash flow

0 ($100.00)

1 $20.00

2 $30.00

3 $20.00

4 $40.00

5 $40.00

21

EXAMPLE2 :GRAPHIC SOLUTION

PW of costs = PW of benefits

100=20/(1+i)+30/(1+i)2+20/(1+i)3+40/(1+i)4+40/(1+i)5

i=13.5%

NPW=-100+20/(1+i)+30/(1+i)2+20/(1+i)3+40/(1+i)4+40/(1+i)5

Internal Rate of Return versus NPVExample:

-$10,000

$20,000Project A

-$20,000

$35,000Project B

000,101

000,20

rrNPVA

year1 year1

000,201

000,35

rrNPVB

NPV(A) and NPV(B) as function of the discount rate

Example:

ProjectCash flows ($)

IRRNPV at

10%t=0 t=1

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

B -20,000 +35,000 75 +11,818

Internal Rate of Return versus NPV

IRR

NPV at 10%

Internal Rate of Return versus NPVAnother example:

-$9,000

$3,500

Project C

$6,000

Project D

000,91

000,4

1

000,5

1

000,632

rrrrNPVD

year1

000,91

500,35

1

n

nCr

rNPV

$3,500 $3,500 $3,500 $3,500

2 3 4 5

$5,000 $4,000

-$9,000

-4000

-2000

0

2000

4000

6000

8000

10000

0 10 20 30 40 50

Project C

Project D

Another example:

ProjectCash flows ($)

IRRNPV at

10%t=0 t=1 t=2 t=3 t=4 t=5

C -9,000 +6,000 +5,000 +4,000 0 0 33 +3,592

D -9,000 +3,500 +3,500 +3,500 +3,500 +3,500 27 +4,268

NPV(C) and NPV(D) as function of the discount rate

IRRNPV at 10%

26

Rate of Return AnalysisExample statements about a project:1. The net present worth of the project is $32,000.2. The equivalent uniform annual benefit is $2,800.3. The project will produce a 23% rate of return

The third statement is perhaps most widely understood.

Rate of return analysis is probably the most frequently used analysistechnique in industry.Its major advantage is that it provides a figure of merit that is readilyunderstood.

27

Motivating Example.

Banks 1 and 2 offer you the following Deals 1 and 2 respectively:

Deal 1.Invest $2,000 today. At the end of years 1, 2, and 3 get $100,$100, and $500 in interest; at the end of year 4, get $2,200in principal and interest.

Deal 2:Invest $2,000 today. At the end of years 1, 2, and 3 get $100,$100, and $100 in interest; at the end of year 4, get $2,000 inprincipal only.

Question. Which deal is the best?

Rate of Return Analysis

28

Deal 1:Find out the implicit interest rate you would be receiving;that is, solve for

2000 = 100/(1+i)1 + 100/(1+i)2 + 500/(1+i)3 + 2200/(1+i)4

IRR: i = 10.7844 %.

This is the interest rate for the PV of your payments to be $2,000.

Deal 2:We find i for which

2000 = 100/(1+i)1 + 100/(1+i)2 + 100/(1+i)3 + 2000/(1+i)4

IRR: i = 3.8194%.

Which deal would you prefer?

Rate of Return Analysis

29

Judging proposed investments

• IRR gets more complicated whencomparing multiple alternatives

– (Rather than evaluating a single project)

• Why?

– Desirability depends on both

• IRR

and

• size of initial investment

30

Example

• Consider two alternatives:

– Invest $1 at an IRR of 100%

– Invest $1,000,000 at an IRR of 20%

• Which investment would you prefer?

31

Example

• Consider two alternatives:

– Invest $1 at an IRR of 100%

– Invest $1,000,000 at an IRR of 20%

• The more expensive project has:

– Smaller IRR

but

– Larger present worth!

32

Judging proposed investments

• If you are going to pick only one alternativefrom several,

– Need to compare them against each other!

• (based on differences in cost)

– not only against the base rate of return i*

• Need to evaluate each incrementalinvestment to see if it is worthwhile

33

CFS AnalysisWe have two CFS’s.

1. Number them CFS1 and CFS2, with CFS1 having the largest year 0cost (in absolute value).

2. Compute CFS = CFS1 – CFS2. (It’s year 0 entry must be negative.)3. Find the IRR for CFS, say IRR .4. If IRR MARR, choose CFS1. If not, choose CFS2.

Example: there are two cash flows: (-20,28) and (-10,15). MARR = 6%.

1. CFS1= (-20,28), CFS2= (-10,15)

2. CFS = CFS1-CFS2 =(-10,13)

3. IRR = 30%.

4. IRR > MARR => we choose CFS1 = (-20,28).

34

Why we use ΔIRR in IRR analysis

Years01

A-1015

B-2028

B-A-1013

ΔIRRB-A 30% MARR < ΔIRRB-A Select B

MARR=6%

IRR 50% 40%

NPV 3.92 6.05 Select B

Select A

Although the rate of return of A is higher than B, B got $8 return fromthe $20 investment and A only got $5 return from $10 investment.

Project B: you put $20 in project B to get a return $8.Project A: you put $10 in project A (and $10 in your pocket) to get a

return $5.

35

Example:

n B1 B2 B2-B10 -$3,000 -$12,000 -$9,000

1 1,350 4,200 2,850

2 1,800 6,225 4,425

3 1,500 6,330 4,830

IRR 25% 17.43%

MARR=10%

0i,3),$4,830(P/F

i,2),$4,425(P/Fi,1),$2,850(P/F$9,000

Solve and obtain i*B2-B1= 15% (simple investment)

Since IRRB2-B1 > MARR, we select B2

Alternatively could have measured for B1 and B2 the NPW at MARR and accepted thelargest NPW in excess of zero.

36

NPW

Interest Rate,%

0

i*B2-

B1=15%

Select B2

Select B1

PW(i)B2 > PW(i)B1

B2

B1

NPW Profiles

37

Example

• Compare options A and B:

A: First cost = $1420

– Annual benefit = $256/year for 40 years

– Rate of return = 18%

B: First cost = $1684

– Annual benefit = $300/year for 40 years

– Rate of return = 17.8%

• You can only do one of these!

38

Example

• Option B has:

– Slightly lower rate of return,

• but

– Higher initial investment

• Present worth of benefit may be greaterthan option A!

39

Example

• Need to evaluate the incremental investmentto see if it is worthwhile:

– Delta first cost = $1684 - 1420 = $264

– Delta annual benefit = $300 - 256 = $44

(for 40 years)

– Rate of return = 16.6%

• Is option B worthwhile?

– (Depends on i*)

40

Example

• Option A has IRR 18%, first cost $1420

– (B - A) has IRR 16.6%, first cost $264

• If i* = 15%, then:

– Option A is worthwhile

– The delta for option B is also worthwhile

• If i* = 17%, then:

– Option A is worthwhile, but not B

41

Example

• Option A has IRR 18%

– (B - A) has IRR 16.6%

• If i* = 20%, then:

– Neither option A nor option B is good

42

Example

• Option A has IRR 18%

– (B - A) has IRR 16.6%

• If i* = 20%, then:

– Neither option A nor option B is good

Investment Classification

Simple Investment

• Def: Initial cash flowsare negative, and onlyone sign change occursin the net cash flowsseries.

• Example: -$100, $250,$300 (-, +, +)

• ROR: A unique ROR

Nonsimple Investment

• Def: Initial cash flowsare negative, but morethan one sign changes inthe remaining cash flowseries.

• Example: -$100, $300, -$120 (-, +, -)

• ROR: A possibility ofmultiple RORs

Period (N)

Project A Project B ProjectC

0 -$1,000 -$1,000 +$1,000

1 -500 3,900 -450

2 800 -5,030 -450

3 1,500 2,145 -450

4 2,000

Project A is a simple investment.Project B is a nonsimple investment.Project C is a simple borrowing.

Example 7.6 Multiple Rates of Return Problem

• Find the rate(s) of return:

2

$2,300 $1,320( ) $1,000

1 (1 )

0

PW ii i

$1,000

$2,300

$1,320

L et T h en ,

S o lv in g fo r yie ld s ,

o r

S o lv in g fo r yie ld s

o r 2 0 %

xi

P W ii i

x x

x

x x

i

i

1

1

0 0 03 0 0

1

3 2 0

1

0 0 0 3 0 0 3 2 0

0

1 0 1 1 1 0 1 2

1 0 %

2

2

.

( ) $ 1,$ 2 ,

( )

$ 1,

( )

$ 1, $ 2 , $ 1,

/ /

PW Plot for a Nonsimple Investment with MultipleRates of Return

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