اقتص4اد هندسي
TRANSCRIPT
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RATE OF RETURN ANALYSIS
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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).
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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
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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
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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%
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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.
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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.
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INTERNAL RATE OF RETURN (IRR)
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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?
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COMPUTATION OF IRR
Direct Solution
Trial and Error Solution
Computer Solution
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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.
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$97,360
$20,000= 4.868
Determine the table valueusing the present value foran annuity of $1 table.
Internal Rate of Return MethodInternal Rate of Return MethodInternal Rate of Return Method
Amount to be invested
Equal annual cash flow
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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%
Internal Rate of Return MethodInternal Rate of Return MethodInternal Rate of Return Method
Find the seven year line on thetable. Then, go across theseven-year line until the closestamount to 4.868 is located.
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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%
4.8684.868
Internal Rate of Return MethodInternal Rate of Return MethodInternal Rate of Return Method
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Internal Rate of Return MethodInternal Rate of Return MethodInternal Rate of Return Method
Move vertically to the top of thetable to determine the interestrate.
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%
4.8684.868
10%10%
10%
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DIRECT SOLUTION
Example
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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
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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
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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%
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PW (30%) = -4,684.24 < 0, tells us that i'% < 30%
25% < i' < 30%
Use linear interpolation to estimate i'%.
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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%
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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
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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
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Internal Rate of Return versus NPVExample:
-$10,000
$20,000Project A
-$20,000
$35,000Project B
000,101
000,20
rrNPVA
year1 year1
000,201
000,35
rrNPVB
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NPV(A) and NPV(B) as function of the discount rate
Example:
ProjectCash flows ($)
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%
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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
-$9,000
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-4000
-2000
0
2000
4000
6000
8000
10000
0 10 20 30 40 50
Project C
Project D
Another example:
ProjectCash flows ($)
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%
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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.
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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
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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
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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
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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?
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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!
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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
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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).
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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.
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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.
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NPW
Interest Rate,%
0
i*B2-
B1=15%
Select B2
Select B1
PW(i)B2 > PW(i)B1
B2
B1
NPW Profiles
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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!
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Example
• Option B has:
– Slightly lower rate of return,
• but
– Higher initial investment
• Present worth of benefit may be greaterthan option A!
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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*)
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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
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Example
• Option A has IRR 18%
– (B - A) has IRR 16.6%
• If i* = 20%, then:
– Neither option A nor option B is good
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Example
• Option A has IRR 18%
– (B - A) has IRR 16.6%
• If i* = 20%, then:
– Neither option A nor option B is good
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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
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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.
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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
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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,
/ /
![Page 47: اقتص4اد هندسي](https://reader031.vdocuments.mx/reader031/viewer/2022011722/577ccf031a28ab9e788ea75a/html5/thumbnails/47.jpg)
PW Plot for a Nonsimple Investment with MultipleRates of Return