Unformatted text preview: ENGINEERING ECONOMY, Sixth
Edition
by Blank and
Tarquin CHAPTER 7 Mc Gra
Hill
w Rate of Return
Analysis: Single
Alternative Chapter 7 Learning Objectives 1. Definition of Rate of Return (ROR)
2. ROR using PW and AW
3. Calculations about ROR
4. Multiple RORs
5. Composite ROR
6. ROR of Bonds 02/19/10 Authored by Don Smith, Texas A&M University 2004 2 Section 7.1
INTERPRETATION OF A RATE OF
RETURN VALUE
•The IRR method is one of the popular
timediscounted measures of investment
worth that is related to the NPV approach. DEFINITION follows 02/19/10 Authored by Don Smith, Texas A&M University 2004 3 7.1 INTERPRETATION OF A RATE
OF RETURN VALUE
DEFINITION ROR is either the interest rate paid on
the unpaid balance of a loan, or the
interest rate earned on the
unrecovered investment balance of an
investment such that the final
payment or receipt brings the
terminal value to equal “0”. 02/19/10 Authored by Don Smith, Texas A&M University 2004 4 7.1 INTERPRETATION OF A RATE
OF RETURN VALUE In rate of return problems you seek
an unknown interest rate (i*) that
satisfies the following:
PWi* (+ cash flows) – PWi* (  cash
flows)
=0
This means that the interest rate –
i*, is an unknown parameter and must
be solved or approximated.
02/19/10 Authored by Don Smith, Texas A&M University 2004 5 7.1 Unrecovered Investment
Balance ROR is the interest rate
earned/charged on the unrecovered
investment balance of a loan or
investment project
ROR is not the interest rate earned
on the original loan amount or
investment amount 02/19/10 Authored by Don Smith, Texas A&M University 2004 6 7.1 Unrecovered Investment
Balance Consider the following loan
You borrow $1000 at 10% per year
for 4 years
You are to make 4 equal end of year
payments to pay off this loan
Your payments are:
A=$1000(A/P,10%,4) = $315.47
02/19/10 Authored by Don Smith, Texas A&M University 2004 7 7.1 The Loan Schedule
Year BOY Bal Payment Interest
Amount Prin. Red.
Amount UnPaid
Balance 0 $1,000 0   $1,000 1 1,000 315.47 100.00 215.47 784.53 2 784.53 315.47 78.45 237.02 547.51 3 547.51 315.47 54.75 260.72 286.79 4 286.79 315.47 28.68 286.79 0 02/19/10 Authored by Don Smith, Texas A&M University 2004 8 7.1 Unrecovered Investment
Balance For this loan the unpaid loan
balances at the end of each year are: 0 $1,000 1 784.53 2 547.51 3
4
02/19/10 286.79
0 Unpaid loan
balance is now
“0” at the end of
the life of the
loan Authored by Don Smith, Texas A&M University 2004 9 7.1 Unrecovered Investment
Balance The URBt=4 is exactly 0 at a 10%
rate 0
1 784.53 2 547.51 3 286.79 4
02/19/10 $1,000 0 Authored by Don Smith, Texas A&M University 2004 10 7.1 Reconsider the following Assume you invest $1000 over 4
years
The investment generates
$315.47/year A=
Draw the315.47flow diagram
+ cash
0 1 2 3 4
P=$1,000 02/19/10 Authored by Don Smith, Texas A&M University 2004 11 7.1 Investment Problem What interest rate equates the
future positive cash flows to the
initial investment?
We can state: $1000= 315.47(P/A, i*,4)
Where i* is the unknown interest
rate that makes the PW(+) = PW() 02/19/10 Authored by Don Smith, Texas A&M University 2004 12 7.1 Investment Problem $1000= 315.47(P/A, i*,4)
Solve the above for the i* rate
(P/A,i*,4) = 1000/315.47 = 3.16987
Given n = 4 what value of i* yields a
P/A factor value = 3.16987?
Interest Table search yields i*=10% 02/19/10 Authored by Don Smith, Texas A&M University 2004 13 7.1 Investment Problem i*=10% per year
Just like the loan problem, we can
calculate the unrecovered investment
balances that are similar to the
unpaid loan balances
See the next slide…. 02/19/10 Authored by Don Smith, Texas A&M University 2004 14 7.1 Unrecovered Investment
Balances (UIB) We set up the following table
t C.F(t) Future Value for 1 year UIBt 0 1,000  1,000 1 +315.47 1000(1.10)+315.47= 784.53 2 +315.47 784.53(1.10)+315.47= 547.51 3 +315.47 547.51(1.10)+315.47= 286.79 4 +315.47 286.79(1.10)+315.47= 0 02/19/10 Authored by Don Smith, Texas A&M University 2004 15 7.1 UIB’s for the Example See Figure 7.1
t C.F(t) UIBt 0 1,000 1,000 1 +315.47 784.53 2 +315.47 547.51 3 +315.47 286.79 4 +315.47 0 02/19/10 •The unrecovered
investment
balances have been
calculated at a 10%
interest rate.
•Note, the
investment is fully
recovered at the
end of year 4
•The UIB = 0 at a
10% interest rate Authored by Don Smith, Texas A&M University 2004 16 7.1 UIB’s for the Example See Figure 7.1
t C.F(t) UIBt 0 1,000 1,000 1 +315.47 784.53 2 +315.47 547.51 3 +315.47 286.79 4 +315.47 0 02/19/10 •The 10% rate is the
only interest rate
that will cause the
UIB at the end of
the project’s life to
equal exactly “0” Authored by Don Smith, Texas A&M University 2004 17 7.1 UIB’s for the Example See Figure 7.1
t C.F(t) UIBt 0 1,000 1,000 1 +315.47 784.53 2 +315.47 547.51 3 +315.47 286.79 4 +315.47 0 02/19/10 •Note, all of UIB’s
are negative at the
10% rate.
•This means that
the investment is
unrecovered
throughout the life. Authored by Don Smith, Texas A&M University 2004 18 7.1 Pure Investment The basic definition of ROR is the
interest rate that will cause the
investment balance at the end of the
project to exactly equal “0”
If there is only one such interest
rate that will cause this, the
investment is said to be a “PURE”
investment or, Conventional
investment
02/19/10 Authored by Don Smith, Texas A&M University 2004 19 7.1 UIB’s for the Example “Unrecovered” means that the
investment balance for a given year is
negative.
If a project’s UIB’s are all negative
(underrecovered) then that
investment will possess one unique
interest rate to cause the UIBt= to
n
equal “0”
02/19/10 Authored by Don Smith, Texas A&M University 2004 20 7.1 UIB’s for the Example See Figure 7.1
This diagram depicts the unpaid
loan balance
At the ROR, the URB will be exactly
“0” at the end of the life of the
project 02/19/10 Authored by Don Smith, Texas A&M University 2004 21 7.1 ROR  Explained ROR is the interest rate earned on
the unrecovered investment balances
throughout the life of the investment.
ROR is not the interest rate earned
on the original investment
ROR (i*) rate will also cause the
NPV(i*) of the cash flow to = “0”. 02/19/10 Authored by Don Smith, Texas A&M University 2004 22 Section 7.2
ROR using Present Worth
•PW definition of ROR
•PW(CF’s) = PW(+CF’s)
•PW(CF’s)  PW(+CF’s) =0 •AW definition of ROR
•AW(CF’s) = AW(+CF’s)
•AW(CF’s)  AW(+CF’s) = 0
02/19/10 Authored by Don Smith, Texas A&M University 2004 23 7.2 ROR using Present Worth
•Finding the ROR for most cash flows is a
trial and error effort.
•The interest rate, i*, is the unknown
•Solution is generally an approximation
effort
•May require numerical analysis
approaches
02/19/10 Authored by Don Smith, Texas A&M University 2004 24 7.2 ROR using Present Worth
+$1,500 •See Figure 7.2
0 1 +$500 2 3 4 5 •$1,000
Assume you invest $1,000 at t = 0: Receive
$500 @ t=3 and $1500 at t = 5. What is the
ROR of this project?
02/19/10 Authored by Don Smith, Texas A&M University 2004 25 7.2 ROR using Present Worth
•Write a present worth expression, set equal to “0”
and solve for the interest rate that satisfies the
formulation.
1000 = 500(P/F, i*,3) +1500(P/F, i*,5)
•Can you solve this directly for the value of
i*?
•NO!
•Resort to trial and error approaches
02/19/10 Authored by Don Smith, Texas A&M University 2004 26 7.2 ROR using Present Worth
1000 = 500(P/F, i*,3) +1500(P/F, i*,5)
•Guess at a rate and try it
•Adjust accordingly
•Bracket
•Interpolate
•i* approximately 16.9% per year on
the unrecovered investment
balances.
02/19/10 Authored by Don Smith, Texas A&M University 2004 27 7.2 Trial and Error Approach •Iterative procedures require an initial
guess for i*
•One makes an educated first guess and
calculates the resultant PV at the guess
rate. 02/19/10 Authored by Don Smith, Texas A&M University 2004 28 7.2 Trial and Error Approach
•If the NPV is not = 0 then another i* value is
evaluated…. Until NPV “close” to “0”
•The objective is to obtain a negative PV for
an i* guess value then.
•Adjust the i* value to obtain a positive PV
given the adjusted i* value
•Then interpolate between the two i* values
02/19/10 Authored by Don Smith, Texas A&M University 2004 29 7.2 Trial and Error Approach –
Bracket “0”
•If the NPV is not = 0 then another i* value is
evaluated
•A negative NPV generally indicates the i*
value is too high
•A positive NPV suggests that the i* value
was too low 02/19/10 Authored by Don Smith, Texas A&M University 2004 30 7.2ROR Criteria
•Determine the i* rate
•If i* => MARR, accept the project
•If i* < MARR, reject the project 02/19/10 Authored by Don Smith, Texas A&M University 2004 31 7.2 Spreadsheet Methods
Excel supports ROR analysis
•RATE(n,A,P,F) can be used when a time t =
0 investment (P) is made followed by “n”
equal, end of period cash flows (A)
•This is a special case for annuities only 02/19/10 Authored by Don Smith, Texas A&M University 2004 32 7.2 Example 7.3 – In Excel
MARR=
Life
Year
0
1
2
3
4
5
6
7
8
9
10 RORGuess
ROR
NPV=
02/19/10 10.00%
10
Cash Flow •Excel Setup for ROR $500,000
$10,000
$10,000
$10,000
$10,000
$10,000
$10,000
$10,000
$10,000
$10,000
$710,000
$300,000 =IRR(D6:D16,D19) D19 = Guess Value 0%
5.16%
$168,674.03
Authored by Don Smith, Texas A&M University 2004 33 7.2 Example 7.2 Investment
Balances
•Investment balances
at the i* rate
•The time t = 10
balance = 0 at i*
•As it should! 02/19/10 Authored by Don Smith, Texas A&M University 2004 34 ENGINEERING ECONOMY Fifth Edition
Blank and
Tarquin Mc Gra
Hill
w CHAPTER VII Section 7.3
Cautions when using the
ROR Method Section 7.3
Cautions when using the
ROR
Method •Important Cautions to
remember when using the ROR
method…… 02/19/10 Authored by Don Smith, Texas A&M University 2004 36 7.3 Cautions when using the
ROR Method No.1
•Many realworld cash flows may possess
multiple i* values
•More than one i* value that will satisfy the
definitions of ROR
•If multiple i*’s exist, which one, if any, is the
correct i*??? 02/19/10 Authored by Don Smith, Texas A&M University 2004 37 7.3 Cautions when using the
ROR Method. 2. Reinvestment
Assumptions •Reinvestment assumption for the ROR
method is not the same as the reinvestment
assumption for PW and AW
•PW and AW assume reinvestment at the
MARR rate
•ROR assumes reinvestment at the i* rate
•Can get conflicting rankings with ROR vs.
PV and AW
02/19/10 Authored by Don Smith, Texas A&M University 2004 38 7.3 Cautions when using the
ROR Method: 3. Computational
Difficulties •ROR method is computationally more
difficult than PW/AW
•Can become a numerical analysis problem
and the result is an approximation
•Conceptually more difficult to understand 02/19/10 Authored by Don Smith, Texas A&M University 2004 39 7.3 Cautions when using the
ROR Method: 4. Special Procedure
for Multiple Alternatives •For analysis of two or more alternatives
using ROR one must resort to a different
analysis approach as opposed to the PW/AW
methods
•For ROR analysis of multiple alternatives
must apply an incremental analysis
approach
02/19/10 Authored by Don Smith, Texas A&M University 2004 40 7.3 Cautions when using the
ROR Method: ROR is more difficult!
•ROR is computationally more difficult
•But is a popular method with financial
managers
•ROR is used internally by a substantial
number of firms
•Suggest using PW/AW methods where
possible
02/19/10 Authored by Don Smith, Texas A&M University 2004 41 7.3 Valid Ranges for usable i*
rates
•Mathematically, i* rates must be: −100% < i ≤ +∞
* • If an i* <= 100% this signals total and
complete loss of capital.
•i*’s < 100% are not feasible and not
considered
•One can have a negative i* value
(feasible) but not less than –100%!
02/19/10 Authored by Don Smith, Texas A&M University 2004 42 Section 7.4
Multiple Rates of Return
•A class of ROR problems exist that will
possess multiple i* values
•Capability to predict the potential for
multiple i* values
•Two tests can be applied 02/19/10 Authored by Don Smith, Texas A&M University 2004 43 7.4 Tests for Multiple i* values 1. Cash Flow Rule of Signs
2. Cumulative Cash Flow Rule of Signs
test
• Example follows 02/19/10 Authored by Don Smith, Texas A&M University 2004 44 7.4 Cash Flow Rule of Signs Test
•The total number of real values i*’s is
always less than or equal to the number of
sign changes in the original cash flow series.
•Follows from the analysis of a nth degree
polynomial
•A “0” value does not count as a sign change
•Example follows…
02/19/10 Authored by Don Smith, Texas A&M University 2004 45 7.4 C.F. Rule of Signs example
•Consider Example 7.4
Year
0
1
2
3 Cash Flow
$2,000
$500
$8,100
$6,800 +
+ Result: 2 sign changes in the Cash
Flow 02/19/10 Authored by Don Smith, Texas A&M University 2004 46 7.4 Results: CF Rule of Signs
Test •Two sign changes in this example
•Means we can have a maximum of 2
real potential i* values for this problem
•Beware: This test is fairly weak and
the second test must also be performed 02/19/10 Authored by Don Smith, Texas A&M University 2004 47 7.4 Norstrom’s Test •Norstrom’s Test1972 works with the
accumulated cash flow
•For the example problem the
accumulated cash flow (ACF) is…. 02/19/10 Authored by Don Smith, Texas A&M University 2004 48 7.4 Accumulated CF Sign Test
(ACF)
•For the problem form the accumulated cash
flow from the original cash flow.
Year
0
1
2
3 Cash Flow
$2,000
$500
$8,100
$6,800 Accum.
C.F
$2,000
$1,500
$6,600
$200 Count sign changes
here
02/19/10 Authored by Don Smith, Texas A&M University 2004 49 7.4 Accumulated CF signs Example
A.C.F
Year
0
1
2
3 Cash Flow
$2,000
$500
$8,100
$6,800 Accum.
C.F
$2,000
$1,500
$6,600
$200 +
+
+ 2 sign changes in the ACF. 02/19/10 Authored by Don Smith, Texas A&M University 2004 50 7.4 ACF Sign Test States: •A sufficient but not necessary condition
for a single positive i* value is:
•The ACF value at year “N” is > 0
•There is exactly one sign change in
the ACF 02/19/10 Authored by Don Smith, Texas A&M University 2004 51 7.4 ACF Test  continued •If the value of the ACF for year “N” is
“0” then an i* of 0% exists
•If the value of ACF for year “N” is > 0,
this suggests an i* > 0
•If ACF for year N is < 0 there may
exist one or more negative i* values –
but not always
02/19/10 Authored by Don Smith, Texas A&M University 2004 52 7.4 ACF Test  continued •Alternatively:
•If the ACF in year “0” < 0
•And….
•One sign change in the ACF series then
•Have a unique i* value!
02/19/10 Authored by Don Smith, Texas A&M University 2004 53 7.4 ACF Test  continued •If the number of sign changes in the
ACF is 2 or greater this strongly suggest
that multiple rates of return exist. 02/19/10 Authored by Don Smith, Texas A&M University 2004 54 7.4 Example 7.4 – ACF Sign Test
Year
0
1
2
3 Cash Flow
$2,000
$500
$8,100
$6,800 Accum.
C.F
$2,000
$1,500
$6,600
$200
2 Sign Changes
here •Strong evidence that we have multiple i*
values
•ACF(t=3) = $200 > 0 suggests positive i* (s)
02/19/10 Authored by Don Smith, Texas A&M University 2004 55 7.4 Excel Analysis of Ex. 7.4
Year
0
1
2
3
Sum Cash Flow
$2,000
$500
$8,100
$6,800
$200 RORGuess
ROR
NPV= 0%
7.47%
$200.00 RORGuess
ROR 30%
41.35% 02/19/10 •We find two i* values:
•{7.47%,41.35%} First i* using a
guess value of 0%
Second i* value
using guess value
of 30% Authored by Don Smith, Texas A&M University 2004 56 7.4 Investment (Project) Balance
•Examine the Investment or Project
balances at each i* rate
Inv Bal
irate
Year
0
1
2
3 02/19/10 7.47%
Project Balances
@ i* Rate
$2,000.00
$1,649.36
$6,327.47
$0.00 Inv Bal
irate
Year
0
1
2
3 41.35%
Project Balances
@ i* Rate
$2,000.00
$2,327.04
$4,810.69
$0.00 Authored by Don Smith, Texas A&M University 2004 57 7.4 Investment Balances at both
i*’s
Inv Bal
irate
Year
0
1
2
3 7.47%
Project Balances
@ i* Rate
$2,000.00
$1,649.36
$6,327.47
$0.00 •Terminal IB(7.47%) =
0
•Terminal
IB(41.35%)=0
02/19/10 Inv Bal
irate
Year
0
1
2
3 41.35%
Project Balances
@ i* Rate
$2,000.00
$2,327.04
$4,810.69
$0.00 •Both i*’s yield
terminal IB’s equal to
0! Authored by Don Smith, Texas A&M University 2004 58 7.4 Investment Balances at both
i*’s
•Important Observations
•The IB’s for the terminal year (3) both
equal 0
•Means that the two i* values are valid
ROR’s for this problem
•Note: The IB amounts are not all the
same for the two i* values.
•IB amounts are a function of the
interest rate used to calculate the
investment balances.
02/19/10 Authored by Don Smith, Texas A&M University 2004 59 7.4 PV Plot of 7.4
250.000
200.000 i* = 7.47%
i*=41.35% PV  $$ 150.000
100.000
50.000
0.000 0.00
50.000 0.20 0.40 Interest
Rate 0.60 100.000
150.000 02/19/10 Authored by Don Smith, Texas A&M University 2004 60 7.4PV Plot  continued
250.000
200.000 PV < 0 150.000
100.000
50.000
0.000 0.00
50.000 0.20 0.40 0.60 If the
MARR is
between
the two i*
values this
investmen
t would be
rejected! 100.000
150.000 02/19/10 Authored by Don Smith, Texas A&M University 2004 61 7.4 Comments on ROR
•Multiple i* values lead to
interpretation problems
•If multiple i*’s – which one, if any is
the “correct” one to use in an
analysis?
•Serves to illustrate the
computational difficulties associated
with ROR analysis
•Section 7.5 provides an alternative
ROR approach – Composite ROR (C02/19/10
Authored by Don Smith, Texas A&M University 2004
ROR) 62 Section 7.5
Composite ROR Approach
•Consider the following investment
+$9,000
+$8,000 0
5 $10,000 02/19/10 1 2 3 4 Determine the ROR as…. Authored by Don Smith, Texas A&M University 2004 63 7.5 Composite ROR Approach
•Analysis reveals:
Year
0
1
2
3
4
5
Sum
RORGuess
ROR 02/19/10 Cash Flow
$10,000
$0
$8,000
$0
$0
$9,000
$7,000 i* = 16.82%/year on
the unrecovered
investment balances
over 5 years 0%
16.82% Authored by Don Smith, Texas A&M University 2004 64 7.5 Composite ROR Approach:
IB’s
• The Investment Balances ar i* are:
Year
0
1
2
3
4
5
Sum
RORGuess
ROR 02/19/10 Cash Flow
$10,000
$0
$8,000
$0
$0
$9,000
$7,000
0%
16.82% Project Balances
@ i* Rate
$10,000.00
$11,681.59
$5,645.95
$6,595.36
$7,704.43
$0.00 •All IB’s are
negative for t
= 0 –4: IB(5) =
0
•Conventional
(pure)
investment Authored by Don Smith, Texas A&M University 2004 65 7.5 Composite ROR Approach
• i* = 16.82% +$9,000 +$8,000 0
5 1 2 3 4 Question: Is it reasonable to
$10,000 assume that the +8,000 can be
invested forward at 16.82%?
02/19/10 Authored by Don Smith, Texas A&M University 2004 66 7.5 Composite ROR Approach
• Remember …….
•ROR assumes reinvestment at
the calculated i* rate
•What if it is not practical for the
+$8,000 to be reinvested forward
one year at 16.82%? 02/19/10 Authored by Don Smith, Texas A&M University 2004 67 7.5 Reinvestment Rates
• Most firms can reinvest surplus
funds at some conservative market
rate of interest in effect at the time
the surplus funds become available.
•Often, the current market rate is less
than a calculated ROR value
•What then is the firm to do with the
+$8,000 when it comes in to the firm?
02/19/10 Authored by Don Smith, Texas A&M University 2004 68 7.5 Reinvesting
• Surplus funds must be put to good
use by the firm
•These funds belong to the owners –
not to the firm!
•Owners expect such funds to be put
to work for benefit of future wealth of
the owners 02/19/10 Authored by Don Smith, Texas A&M University 2004 69 7.5 Composite ROR Approach
• Consider the following
representation.
Invested Funds The
Firm Project
Returns
back 02/19/10 Authored by Don Smith, Texas A&M University 2004 70 7.5 Composite ROR Approach
• Or, put in another context…. The
Firm Project borrows from the
firm Project
Project Lends back to the
firm 02/19/10 Authored by Don Smith, Texas A&M University 2004 71 7.5 Composite ROR Approach
• Or, put in another context….
Project borrows from the
firm The
Firm At the i* rate (16.82%) Project Project Lends back to the
firm
But can the firm reinvest
these funds at 16.82%?
Probably not!
02/19/10 Authored by Don Smith, Texas A&M University 2004 72 7.5 Composite ROR Approach
• So, we may have to consider a
reinvestment rate that is closer to the
current market rate for reinvestment
of the $8,000 for the next time
period(s)
•Assume a reasonable market rate is
say, 8% per year.
•Call this rate an external rate  c
02/19/10 Authored by Don Smith, Texas A&M University 2004 73 7.5 The external rate  c
• The external interest rate – c, is a
rate that the firm can reinvest surplus
funds for at least one time period at a
time.
•c is often set to equal the firm’s
current MARR rate 02/19/10 Authored by Don Smith, Texas A&M University 2004 74 7.5 Composite ROR Approach
• Thus, a procedure has been
developed that will determine the
following:
•Find i* given c – if multiple ROR’s
exist.
The “/” is read “given”
•For multiple i*’s in a problem, the i.e.,
i* given a
analysis determines a single value for “c” c
i* given •Denoted i*/c or, i’
•i’ is called the composite University 2004i*/c 75
rate =
02/19/10
Authored by Don Smith, Texas A&M 7.5 Composite ROR Approach
• Finding i’ is a much more involved
process
•Prior to digital computers, only very
small (N <= say 45 time periods)
could be manually evaluated
•Requires a recursive analysis best
left to a computer program and
spreadsheet
•Example 7.6 illustrates a manual
02/19/10
approach Authored by Don Smith, Texas A&M University 2004 76 7.5 Example 7.6
• Cash Flow is :
Year
0
1
2
3 02/19/10 Cash Flow
$2,000
$500
$8,100
$6,800 Authored by Don Smith, Texas A&M University 2004 77 7.5 Ex. 75 Multiple i*’s
• i*1 = 7.468; i*2 = 41.35%
Year
0
1
2
3 02/19/10 Cash Flow RORGuess
ROR
NPV= $2,000
$500
$8,100 RORGuess
ROR
$6,800 0%
7.468%
$200.00
30%
41.35% Authored by Don Smith, Texas A&M University 2004 78 7.5 Investment Balances are:
• IB(7.468%)
Inv Bal
irate
Year
0
1
2
3 02/19/10 7.468%
Project Balances
@ i* Rate
$2,000.00
$1,649.36
$6,327.47
$0.00 • IB(41.352%)
Inv Bal
irate
Year
0
1
2
3 41.352%
Project Balances
@ i* Rate
$2,000.00
$2,327.04
$4,810.69
$0.00 Authored by Don Smith, Texas A&M University 2004 79 7.5 Project Balances @ 41.35% PB's  $ Project Balances
$3,000
$2,000
$1,000
$0
$1,000
$2,000
$3,000
$4,000
$5,000
$6,000 0 1 2 Years 02/19/10 Overrecovered 3 Underrecovered Authored by Don Smith, Texas A&M University 2004 80 7.5 Project Balances @ 7.468%%
Overrecovered Project Balances
$4,000 PB's  $ $2,000
$0
$2,000 0 1 2 3 $4,000
$6,000
$8,000 Years 02/19/10 Underrecovered Authored by Don Smith, Texas A&M University 2004 81 7.5 Assume you Reinvest at 20%
• c = 20%
•Positive IB’s are reinvested at 20% •Not at the computed i* rate
•Now, what is the modified ROR – i’?
•Must perform a recursive analysis 02/19/10 Authored by Don Smith, Texas A&M University 2004 82 7.5 Recursive IB’s are….
•Let Ibj = Fj
•F0 = +2000 (> 0; invest at 20%)
•F1 = 2000(1.20) – 500 = +1900
•+1900 > 0; invest at 20% Underrecovered •F2 = 1900(1.20) – 8100 = 5820
•5820 < 0; invest at i’ rate
02/19/10 Authored by Don Smith, Texas A&M University 2004 83 7.5 Recursive IB’s are….
Repeated from the
previous slide for
clarity •F2 = 1900(1.20) – 8100 = 5820
•5820 < 0; invest at i’ rate
• F3 =5820(1+i’) + 6800
•Since N = 3, F3 = 0
•Solve; 5820(1+i’) + 6800 = 0
02/19/10 Authored by Don Smith, Texas A&M University 2004 84 7.5 Single conditional ROR value
• 5820(1+i) + 6800 = 0
• i’ = [6800/5820] –1
• i’ = 16.84% given c = 20%
•If MARR = 20% and i’ = 16.84% this
investment would be rejected
•Reduced the problem to a single
conditional ROR value.
02/19/10 Authored by Don Smith, Texas A&M University 2004 85 7.5 What if c = i*1 (7.47%)?
• Here, we examine what happens IF
we assume the reinvestment rate, c,
equals one of the computed i* values.
•This approach assumes that the
reinvestment rate, c, is one of the i*
rates
•Recursive calculations follow…. 02/19/10 Authored by Don Smith, Texas A&M University 2004 86 7.5 Recursive IB’s are….
(@7.47%)
•Let IBj = Fj
•F0 = +2000 (> 0; invest at 7.47%)
•F1 = 2000(1.0747) – 500 = +1649.40
•>0: Overrecovered balance
•F2 = +1649.40(1.074) – 8100 =
6327.39
•Now, underrecovered balance
Authored by Don Smith, Texas A&M University 2004 02/19/10 87 7.5 Recursive IB’s are….
•F2 = +1649.40(1.074) – 8100 =
6327.39
•F3 =6327(1+i’) + 6800
•Since N = 3, F3 must = 0 (terminal IB
condition)
•Solve; 6327(1+i’) + 6800 = 0
02/19/10 Authored by Don Smith, Texas A&M University 2004 88 7.5 Single conditional ROR value
•Must solve:
• 6327.39(1+i’) + 6800 = 0 •(1+i’) = 6800/6327.39
•i’ = 0.0747 = 7.47%
•Given c = one of the i*’s, the i’ will
equal the i* used as the
reinvestment rate!
02/19/10 Authored by Don Smith, Texas A&M University 2004 89 Section 7.6
Rate of Return on a Bond
Investment
• Review Chapter 5 Section on Bonds
•Bond problems represent a classical
ROR type problem
•Bond problems represent a
conventional investment with a
unique ROR
•Example Follows:
02/19/10 Authored by Don Smith, Texas A&M University 2004 90 7.6 Example 7.8
• Purchase Price: $800/bond
•4%, $1000 bond
•Life = 20 years
•Interest paid semiannually
•Question: If you pay $800 per bond
what is the ROR (yield) on this
investment?
02/19/10 Authored by Don Smith, Texas A&M University 2004 91 7.6 Ex. 7.8: Cash Flow Diagram
F40 =
$1000
A = +$20/6 months
0
39 1
40 2 3 4 ….
…. …. $800
$I/6 mos = $1000(0.04/2) = +$20.00 (every 6
months)
02/19/10 Authored by Don Smith, Texas A&M University 2004 92 7.6 Ex. 78: Closed Form Setup
Setup is:
•0 = $800 +20(P/A,i*,40 + $1000(P/F,i*,40)
•Solve for i*
•Manual or computer solution yields:
i*=2.87%/6 months
•Nominal ROR/year = (2.87%)(2) = 5.74%/yr,
C.S.A.
•Effective ROR/year = (1.0287)2 – 1 =
5.82%/yr
02/19/10 Authored by Don Smith, Texas A&M University 2004 93 Chapter 7 Summary
• ROR is a popular analysis method
•Understood in part by practitioners
•Presents computational difficulties
•No ROR may exist
•Multiple ROR’s may exist 02/19/10 Authored by Don Smith, Texas A&M University 2004 94 Chapter 7 Summary cont.
• For cash flows where multiple i*
values exist, one must decide to:
•Go with one of the i* rates or,
•Calculate the composite rate, i’
•If an exact ROR is not required, it is
suggested to go with PW or AW 02/19/10 Authored by Don Smith, Texas A&M University 2004 95 Chapter 7 Summary cont.
• For cash flows where multiple i*
values exist, one must decide to:
•Go with one of the i* rates or,
•Calculate the composite rate, i’
•If an exact ROR is not required, it is
suggested to go with PW or AW 02/19/10 Authored by Don Smith, Texas A&M University 2004 96 Chapter 7 Summary cont.
• To have a possible i* value with a
given cash flow, there must be at
least one negative CF amount in the
series
•IROR can be demanding and is best
accomplished using a computer
algorithm
•IROR is often misunderstood by
practitioners but is a popular method
of analysis
02/19/10
Authored by Don Smith, Texas A&M University 2004
97 Chapter 7 Summary cont.
• In real world applications one may
encounter a cash flow that possess no
usable i* value!
•These types of problems are difficult
to explain to uninformed decision
makers. 02/19/10 Authored by Don Smith, Texas A&M University 2004 98 ENGINEERING ECONOMY Sixth
Edition
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 Time Value Of Money, Don Smith

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