Blank L., Tarquin A. Engineering Economy (McGraw-Hill Series in Industrial Engineering and Management)
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SECTION
7.5
Composite Rate
of
Return: Removing Multiple
i*
Values
7.5 COMPOSITE
RATE
OF RETURN: REMOVING
MULTIPLE
i*
VALUES
The
rates
of
return
we
have
computed
thus far
are
the rates that exactly balance
plus and minus cash flows with the time value
of
money considered.
Any
method which accounts for the time value
of
money can
be
used in calculating
this balancing rate, such as
PW, AW,
or
FW.
The
interest rate obtained from
these calculations
is
known
as the internal rate
af
return (IRR).
Simply
stated,
the internal rate
of
return
is
the rate
of
return on the unrecovered balance
of
an
investment, as defined earlier.
The
funds that remain unrecovered are still inside
the investment, hence the
name
internal rate afreturn.
The
general terms rate
of
return and interest rate usually imply internal rate
of
return.
The
interest rates
quoted
or
calculated
in
previous chapters are all internal rates.
The
concept
of
unrecovered balance
becomes
important
when positive
net
cash flows are generated (thrown
off)
before the
end
of
a project. A positive net
cash flow,
once
generated, becomes released as externalfunds ta the praject
and
is
not considered further in an internal rate
of
return calculation.
These
positive net
cash flows may
cause
a nonconventional cash flow series
and
multiple
i*
values to
develop. However, there
is
a method to explicitly
consider
these funds, as dis-
cussed below. Additionally, the
dilemma
of
multiple
i*
roots is elimjnated.
It
is important to
under
stand that the procedure detailed below is used to
Determine the
rate
of
return
for cash flow estimates when there
are
multiple
i*
values indicated by both the cash flow rule
of
signs
and
the cu-
mulative cash flow rule
of
signs,
and
net positive cash flows from the project
will
earn
at
a stated
rate
that
is
different from any
of
the multiple
i*
values.
For
example,
assume
a cash flow series has two
i*
values that balance the
ROR
equation-IO
% and
60%
per
year-and
any cash released by the pr
oject
is in-
vested by the
company
at a rate
of
return
of
25%
per
year
.
The
procedure
below
will find a single unique rate
of
return for the
cash
flow series.
However
,
if
it is
known that released
ca
sh will earn exactly 10%, the unique rate is 10%.
The
same
statement
can
be
made using the
60
% rate.
As before,
if
the
exact
rate
of
return for a project's cash flow estimates is not
needed, it is much
simpler
,
and
equally correct, to use a
PW
or
AW analysis at the
MARR
to
determine
if
the project is financially viable. This is the normal
mode
of
oper
ation
in
an engineering
economy
study.
Consider
the internal rate
of
return calculations for the following cash flows:
$10,000
is
invested at I = 0,
$8000
is
received in
year
2, and
$9000
is received
in
year 5. The
PW
equation to
determine
i*
is
o = - 10,
000
+ 8000(P /
F,i
,2) + 9000(P / F,i,5)
i* =
16
.815%
If this rate
is
used for the unrecovered balances, the investment will be recovered
exactly at the end
of
year 5.
The
procedure to verify this
is
identical to that used
in
Table 7- 1, which describes
how
the
ROR
works to exactly remove the unre-
:overed balance with the final cash flow.
255
256
CHAPTER
7 Rate
of
Return Analysis: Single Alternative
Unrecovered balance at end
of
year 2 immediately before $8000 receipt:
-lO,OOO(Fj P,16.81S%,2) = -10,000(1 +
0.1681Si
=
$-13,646
Unrecovered balance at end
of
year 2 immediately after $8000 receipt:
-13
,646 + 8000 =
$-S646
Unrecovered balance at end
of
year S immediately before $9000 receipt:
-S646(Fj
P,16.81S%,3) =
$-9000
Unrecovered balance at end
of
year S immediately after $9000 receipt:
$-9000
+ 9000 = $0
In this calculation, no consideration
is
given to the $8000 available after year 2.
What happens if funds released from a project are considered in calculating the
overall rate
of
return
of
a project? After all, something must be done with the re-
leased funds.
One possibility
is
to assume the money
is
reinvested at some stated
rate. The
ROR method assumes funds that are excess
to
a project earn at the
i*
rate,
but this may not be a realistic rate in everyday practice. Another approach
is
to
sim-
ply assume that reinvestment occurs at the MARR. In addition to accounting for
all
the money released during the project period and reinvested at a realistic rate,
th
e
approach discussed below has the advantage
of
converting a nonconventional cash
flow series (with multiple
i*
values) to a conventional series with one root, which
can be considered
the rate
of
return for making a decision about the project.
The rate
of
earnings used for the released funds
is
called the reinvestment rate
or external rate
of
return and
is
symbolized by
c.
This rate, established outside
(external to) the cash flow estimates being evaluated, depends upon the market
rate available for investments.
If
a company is making, say, 8% on its daily in-
vestments, then c
= 8%. It is common practice to set c equal to the MARR. The
one interest rate that now satisfies the rate
of
return equation is called the com-
posite rate
of
return (CRR) and
is
symbolized by
i'
. By definition
The composite rate
of
return i'
is
the unique
rate
of
return
for a project
that
assumes
that
net positive cash flows, which
represent
money not im-
mediately needed by the project,
are
reinvested
at
the reinvestment
rate
c.
The term composite
is
used here to describe tills rate
of
return because it is de-
rived using another interest rate, namely, the reinvestment rate
c.
If
c happens to
equal
anyone
of
the
i*
values, then the composite rate i' will equal that i* value.
The
CRR
is
also known by the term return on invested capital (RIC). Once the
unique
i'
is determined, it is compared to the
MARR
to decide on the project's
financial viability, as outlined
in
Section 7.2.
The
con
"ect procedure to determine
i'
is
called the net-investment proc
edur
e.
The
technique involves finding the future worth
of
the net investment amount
I year in the future. Find the project's net-investment value
F,
in year t from F
t-J
by
using the F j P factor for 1 year at the reinvestment rate c
if
the previous net
investment
F,
- J
is
positive (extra money generated by project), or at the
CRR
rate
i'
if
F'_I is negative (project used all available funds).
To
do this mathemat-
ically, for each year
t set up the relation
F, =
F,_I(l
+
i)
+
C,
[7.6
SECTION 7.5
Co
mpo
site Rate
of
Return: Removing Multiple i* Values
where t = 1, 2,
...
, n
n
= total years
in
project
C{
= net cash flow in year t
. _ {c
if
F{
_ ) > 0
l-
.f
l
if
F{-) < 0
(net positive investment)
(net negative investment)
Set
the net-investment relation for year n equal to zero
(F
Il
= 0) and solve for
if.
The
if value obtained is unique for a stated reinvestment rate
c.
The
development
of
F)
through F3 for the cash flow series below, which is
graphed
in
Figure 7-8a, is illustrated for a reinvestment rate
of
c = MARR =
15
%.
Year
Cash
Flow, $
a 50
I
-2
00
2 50
3 100
The
net investment for year t = 0 is
Fa = $50
which is positive, so
it
returns c =
15
% during the first year.
By
Equation
[7
.
6]
,
F)
is
F)
=
50(1
+
0.
15
) - 200 =
$-142.50
This result is shown
in
Figure 7
-8b.
Since the project net
in
vestment is now neg-
ative, the va
lu
e
F)
earns interest at the
co
mposite rate if for year 2. Therefore, for
year 2,
50
o
200
(a)
Figure 7-8
50
2
F2 = F)(l + if) + C
2
= - 142.500 +
if)
+ 50
100 100 100
50 50
3
o
2 3 o
2
3 o
1
42.50
14
2.50(1 + i
')
(iJ)
(e)
Cash flow
se
ri
es
for w
hi
ch
th
e
composi
te rate
of
retum
i' is
co
mput
ed: (a)
or
iginal form; eq
ui
valent form
in
(iJ)
year I. (e)
year
2,
and
(d)
year
3.
2
(d)
257
1
00
3
[14
2.50(
1 + i
')
-5
0l
(1 + i
')
258
CHAPTER 7 Rate
of
Return Analysis: Single Alternative
The
i' value is to be determined (Figure 7-8c). Since F2 will be negative for all
i' > 0, use i' to set up F3 as shown in Figure 7-8d.
F3 = F
2
0 +
i')
+ C
3
= [- 142.50(1 +
i')
+ 50](1 +
i')
+ 100 [7.7]
Setting Equation [7.7] equal to zero and solving for
i' will result in the unique
composite rate
of
return i'. The resulting values are 3.13% and
-168%,
since
Equation [7.7] is a quadratic relation (power 2 for
i')
.
The
value
of
i' = 3.13%
is
the correct
i*
in the range
-100
% to
00
.
The
procedure to find i' may be summa-
rized as follows:
I. Draw a cash flow diagram
of
the original net cash flow serie
s.
2. Develop the series
of
net investments using Equation [7.6] and the c valu
e.
The
result is the
FIl
expression in terms
of
i'.
3.
Set
FIl
= 0 and find the i' value to balance the equation.
Several comments are in order.
If
the reinvestment rate c
is
equal to the inter-
nal rate
of
return
i*
(or one
of
the
i*
values when there are multiple ones), the i'
that
is
calculated will be exactly the same as
i*
; that is, c =
i*
= i'.
The
closer
the c value
is
to i*, the smaller the difference between the composite and internal
rates. As mentioned earlier, it is correct to assume that c
=
MARR
, if all throw-
off
funds from the project can realistically earn at the
MARR
rat
e.
A summary
of
the relations between
c,
i', and i* follows, and the relations are
demonstrated in Example 7.6.
Relation
between
Reinvestment
Rate c
and
j*
c =
i*
c <
i*
c >
i*
Relation
between
CRR
j'
and
j*
i' =
i*
i'
<
i*
i' > i*
Remember: This entire net-investment procedure
is
used when multiple i* values
are indicated. Multiple
i*
values are present when a nonconventional cash flow
series does not have one positive root, as determined by Norstrom's criterion.
Additionally, none
of
the steps
in
this procedure are necessary if the present
worth
or
annual worth method
is
used to evaluate a project at the MARR.
The
net-investment procedure can also be applied when one internal rate
of
return (i*)
is
present, but the stated reinvestment rate (c) is significantly different
from
i*. The same relations between c, i*, and i' stated above remain correct for
this situation.
EXAMPLE
7.6
~c
Compute the composite rate
of
return for the Honda Motor Corp. engineering group
in
Example 7.4
if
the reinvestment rate
is
(a) 7.47% and (b) the corporate MARR
of
20%.
The multiple
t'
values are determined in Figure
7-6.
SECTION 7.5 Composite Rate
of
Return: Removing Multiple
i*
Values
Soluti
on
(a) Use the net-investment procedure to determine i' for c = 7.47%.
1.
Figure 7- 9 shows the original cash
flow.
2.
The first net-investment expression is
Fo
= $+2000. Since
Fo>
0, use
c
= 7.47% to write
FI
by
Equation
[7
.6
].
F\ = 2000(1.0747) - 500 = $1649.40
S.ince
F\
> 0, use c = 7.47% to determine F
2
•
F2
= 1649.40(1.0747) - 8100 =
$-6327.39
Figure 7-
10
shows the equivalent cash flow at this time. Since
F2
< 0, use
i' to express F
3
.
F3
= - 6327.39(1 +
i')
+ 6800
$6800
Figure
7-9
Original cash flow
(in thousands),
Examp
le 7.6.
$2000
o
0
1
$5
00
2
$8100
2
$6327.39
Year
3
$6800
Figure
7-10
Equivalent cash flow (in
thousands)
of
Figure 7- 9
with reinvestment at
c
= 7.47%.
Year
3
259
260
CHAPTER 7
Rate
of
Return Analysis: Single Alternative
3. Set
F3 = 0 and solve for i' directly.
-6327.39(1 +
i')
+ 6800 = 0
6800
I
+
i'
= = 1.0747
6327.39
i'
= 7.47%
The
CRR
is 7.47%, which
is
the same as c, the reinvestment rate, and the
if
value determined in Example 7.4, Figure 7-6. Note that 41.35
%,
which
is
the
second
i*
value, no longer balances the rate
of
return equation. The equiva-
lent future worth result for the cash flow
in
Figure 7-10,
if
i'
were 41.35%,
is
6327.39(F/P,41.35%,I) = $8943.77
'*
$6800
(b) For MARR = c = 20%, the net-investment series is
Fo
= +2000
FI
= 2000(1.20) - 500 = $1900
F2
= 1900(1.20) - 8100 = $- 5820
F3
= -5820(1 +
i')
+ 6800
Set F3 = 0 and solve for
i'
directly.
(Fo > 0, use c)
(FI > 0, use c)
(F2
< 0, use
i')
1 + ., = 6800 = I 1684
I 5820 .
i'
= 16.84%
The CRR
is
i'
= 16.84%
at
a reinvestment rate
of
20%, which is a marked in-
crease from
i'
= 7.47% at c = 7.47%.
Note that since i' <
MARR
= 20
%,
the project is not financially justified. This is
verified by calculating PW =
$-106
at 20% for the original cash flows.
EXAMPLE 7.7
~
';.j-
Determine the composite rate
of
return for the cash flows
in
Table
7-6
if the reinvest-
ment rate
is
the
MARR
of
15%
per year. Is the project justified?
Solution
A review
of
Table
7-6
indicates that the nonconventional cash flows have two sign
changes and the cumulative cash flow sequence does not start with a negative value.
There are a maximum
of
two
i*
values.
To
find the one i' value, develop the net-
investment series
Fo
through
FlO
using Equation [7.6] and c = 15%.
Fo
= 0
FI
= $200
F2 = 200(1.15) + 100 = $330
F3 = 330(1.15) + 50 = $429.50
F4 = 429.50(1.15) - 1800 = $- 1306.08
Fs = - 1306.08(1 +
i')
+ 600
(F
I
> 0, use c)
(F2
> 0, use c)
(F3
> 0, use c)
(F4
< 0, use
i')
SECTION 7.6 Rate
of
Return of a Bond Investment
TABLE
7-6
Cash Flow and Cumulative Cash Flow Sequences,
Example 7.7
Cash Flow, $ Cash Flow, $
Year
Net
Cumulative Year
Net
Cumulative
0 0 0 6 500
-350
1 200
+200
7 400
+50
2
JOO
+300
8 300
+350
3 50
+350
9 200
+550
4
- J800
1450
10
100
+650
5 600
-850
Since we do not know if
F5
is
greater than zero or less than zero,
all
remaining expres-
sions use
i'.
F6
=
Fs(J
+
i')
+ 500 =
[-1306.08(1
+
i')
+ 600](1 +
i')
+ 500
F7
=
F6(1
+
i')
+ 400
Fg
= F
7
(l
+
i')
+ 300
F9
=
Fg(1
+
i')
+ 200
FlO
=
F9(1
+
i')
+ 100
To
find
i'
, the expression
FlO
= 0 is solved by trial and error. Solution determines that
i'
= 21.24%. Since
i'
> MARR, the project
is
justified.
In
order to work more with this
exercise and the net-investment procedure, do the case study in this chapter.
Comment
The two rates
wh
ich balance the ROR equation are
if
= 28.71% and
if
= 48.25%.
If
we rework this problem at either reinvestment rate, the i' value will be the same
as
this
reinvestment rate; that is, if c
= 28.
71
%,
then
i'
=
28
.
71
%.
There
is a spreadsheet function called
MIRR
(modified IRR) which deter-
mines a unique interest rate when you input a reinvestment rate c for positive
cash flows.
However
, the function does not
implement
the net-investment proce-
dure
for nonconventional cash flow series as discussed here, and the function re-
quires that a finance rate f
or
the funds used as the initial investment
be
supplied.
So
the formulas for
MIRR
and
CRR
computation are not the same.
The
MIRR
will not
produce
exactly the
same
answer
as Equation [7.6] unless all the rates
happen to
be
the
same
and this value is
one
of
the roots
of
the
ROR
relation.
7.6
RATE
OF RETURN OF A
BOND
INVESTMENT
In
Chapter
5
we
learned the terminology
of
bonds
and
how
to calculate the
PW
of
a bond investment.
The
cash flow series for a
bond
investment
is conventional
and has
one
unique
i*
, which is best determined by solving a PW-based rate
of
261
262
CHAPTER 7 Rate
of
Return Analysis: Single Alternative
return equation
in
the form
of
Equation
[7
.
1]
. Examples 7.8 and 7.9 illustrate the
procedure.
EXAMPLE
7.8
~
_
Allied Materials needs $3 million in debt capital for expanded composites manufactur-
ing. It
is
offering small-denomination bonds at a discount plice
of
$800 for a 4% $ 1000
bond that matures
in
20 years with interest payable semiannually. What nominal and
effective interest rates per year, compounded semiannually, will Allied Materials pay
an
investor?
Solution
The income that a purchaser will receive from the bond purchase
is
the bond interest
J = $20 every 6 months plus the face value
in
20 years. The PW-based equation for cal-
culating the rate
of
return
0=
-800
+ 20(P/A,i*,40) + 1000(P/F,i*,40)
Solve
by
computer (IRR function) or
by
hand
to
obtain
i*
= 2.87% semiannually. The
nominal interest rate per year
is
computed
by
multiplying
i*
by
2.
Nominal
i = 2.87%(2) = 5.74% per year, compounded semiannually
Using Equation [4.5], the effective annual rate
is
ia = (1.0287)2 - 1 = 5.82%
EXAMPLE
7.9
'
Gerry
is
an
entry-level engineer at Boeing Aerospace
in
California. He took a financial risk
and bought a bond from a different corporation that had defaulted on its interest payments.
He paid
$4240 for
an
8% $10,000 bond with interest payable quarterly. The bond paid no
interest for the first 3 years after Gerry bought
it.
If
interest was paid for the next 7 years,
and then Gerry was able to resell the bond for
$11,000, what rate
of
return did he make on
the investment? Assume the bond
is
scheduled to mature
18
years after he bought
it.
Per-
form hand and computer analysis.
Soluti
on
by Hand
The bond interest received in years 4 through
10
was
J = (10,000)(0.08) = $200 per quarter
4
The effective rate
of
return per quarter can
be
determined
by
solving the PW equation
developed on a per quarter basis, since this basis makes
PP =
CPo
0 = - 4240 + 200(P/A,i* perql1arter,28)(P/ F,i* perql1arter,l2)
+ I I ,
OOO(P
/ F,
i*
per quarter,40)
The equation
is
con'ect for
i*
=
4.1
% per quarter, which
is
a nominal 16.4% per year, com-
pounded quarterly.
CHAPTER SUMMARY
Solution by
Computer
The solution
is
shown
in
Figure
7-11.
The spreadsheet is set up to directly calculate an
annual interest rate
of
16.41 % in cell
El
.
The
quarterly bond interest receipts
of
$200 are
converted to equivalent annual receipts
of
$724.24 using the PV function in cell E6. A
quarterly rate could be determined initially on the spreadsheet,
but
this approach would
require four times as many entries
of
$200 each, compared to the six times $724.24 is
entered here. (A circular reference may be indicated by Excel between cells
El,
E6, and
B6. However, clicking on
OK
will override and the solution
i*
= 16.41% is displayed. A
circular reference is avoided if all
40 quarters
of
$0 and $200 are entered in column B with
necessary changes
in
the column E relations to find the quarterly rate.)
Bond face value
Bond interest rate
Bond inte
re
st/quarter
PW
of bond interest/year
Figure 7-11
Spreadsheet solution
of
r:'
for a bond investment, Example 7.9.
CHAPTER SUMMARY
Rate
of
return, or interest rate,
is
a te
rm
used and understood by almost every-
body. Most people, however, can have considerable difficulty in calculating a
rate
of
return
i*
correctly for any cash flow series. For some types
of
series, more
263
~
E-Solve
!!Wl!11 ....
iWi'*f@:r'
264
CHAPTER 7
Rate
of
Return Analysis: Single Alternative
than one
ROR
possibility exists. The maximum number
of
i*
values
is
equal to
the number
of
changes in the signs
of
the net cash flow series (Descartes' rule
of
signs). Also, a single positive rate can be found
if
the cumulative net cash flow
series starts negatively and has only one sign change (Norstrom's criterion).
For all cash flow series where there
is
an indication
of
multiple roots, a decision
must be made about whether to calculate the multiple
i*
internal rates, or the one
composite rate
of
return using an externally-determined reinvestment rate. This
rate is commonly set at the MARR. While the internal rate is usually easier to cal-
culate, the composite rate is the correct approach with two advantages: multiple
rates
of
return are eliminated, and released project net cash flows are accounted
for using a realistic reinvestment rate. However, the calculation
of
multiple
i*
rates, or the composite rate
of
return, is often computationally involved.
If
an exact
ROR
is not necessary, it is strongly recommended that the
PW
or
AW
method at the
MARR
be used to
judge
economic justification.
PROBLEMS
Understanding
ROR
7.1 What does a rate
of
return
of
- 100% mean?
7.2 A
$10,000 loan amortized over 5 years at
an interest rate
of
10% per year would re-
quire payments
of
$2638 to completely
repay the loan when interest is charged on
the unrecovered balance.
If
interest is
charged on the principal instead
of
the
unrecovered balance, what will be the
balance after 5 years
if
the same $2638
payments are made each year?
7.3 A-I Mortgage makes loans with the inter-
est paid on the loan principal rather than
on the unpaid balance.
For
a 4-year loan
of
$10,000 at 10% per year, what annual
payment would be required to repay the
loan
in
4 years
if
interest
is
charged on
(a) the principal and (b) the unrecovered
balance?
7.4 A small industrial contractor purchased a
warehouse building for storing equipment
and materials that are not immediately
needed at construction
job
sites. The cost
of
the building was $100,000, and the con-
tractor made an agreement with the seller
to finance the purchase over a 5-year pe-
riod. The agreement stated that monthly
payments would be made based on a
30-
year amortization, but the balance owed at
the end
of
year 5 would be paid in a lump-
sum balloon payment. What was the size
of
the balloon payment
if
the interest rate
on the loan was 6% per year, compounded
monthly?
Determination
of
ROR
7.5 What rate
of
return per month will
an
entrepreneur make over a
2Yz-year
pro-
ject
period
if
he invested $150,000 to
produce portable 12-volt air compressors?
His estimated monthly costs are
$27,000
with income
of
$33,000 per month.
7.6 The Camino Real Landfill was required to
install a plastic liner to prevent leachate