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Engineering Economics - EIT Review
Cash Flow Evaluation
Hugh Miller
Colorado School of Mines
Mining Engineering Department
Fall 2007
Introduction
As we have addressed the fundamental concepts associated with
engineering economics and cash flows, is now time to convert these
estimates into measures of desirability as a tool for investment
decisions.
We will use the following criteria:






Present & Future Value
Annual Value
Benefit / Cost Ratio
Payback period
Internal Rate of Return
Variations of IRR
Present Value
The Present value or present worth method of evaluating
projects is a widely used technique. The Present Value
represents an amount of money at time zero representing
the discounted cash flows for the project.
PV
T=0
+/- Cash Flows
Net Present Value (NPV)
The Net Present Value of an investment it is simply the
difference between cash outflows and cash inflows on a
present value basis.
In this context, the discount rate equals the minimum rate
of return for the investment
Where:
NPV = ∑ Present Value (Cash Benefits) - ∑ Present Value (Cash Costs)
Present Value Example






Initial Investment:
Project Life:
Salvage Value:
Annual Receipts:
Annual Disbursements:
Annual Discount Rate:
$100,000
10 years
$ 20,000
$ 40,000
$ 22,000
12%
What is the net present value for this project?
Is the project an acceptable investment?
Present Value Example Solution

Annual Receipts


$ 226,000
Salvage Value


$40,000(P/A, 12%, 10)
$20,000(P/F, 12%, 10)
$
6,440
Annual Disbursements
$22,000(P/A, 12%, 10)
-$124,000

Initial Investment (t=0)
-$100,000

Net Present Value
$ 8,140


Greater than zero, therefore acceptable project
Future Value
The future value method evaluates a project
based upon the basis of how much money will be
accumulated at some future point in time. This is
just the reverse of the present value concept.
FV
T=0
+/- Cash Flows
Future Value Example






Initial Investment:
Project Life:
Salvage Value:
Annual Receipts:
Annual Disbursements:
Annual Discount Rate:
$100,000
10 years
$ 20,000
$ 40,000
$ 22,000
12%
What is the net future value for this project?
Is the project an acceptable investment?
Future Value Example Solution

Annual Receipts


$ 20,000
$22,000(F/A, 12%, 10)
-$386,078
Initial Investment


$20,000(year 10)
Annual Disbursements


$ 701,960
Salvage Value


$40,000(F/A, 12%, 10)
$100,000(F/P, 12%, 10)
Net Future Value

-$310,600
$ 25,280
Positive value, therefore acceptable project
 Can be used to compare with future value of other projects
PV/FV Example
No theoretical difference if project is evaluated in
present or future value
PV of $ 25,282
$25,282(P/F, 12%, 10)
$ 8,140
FV of $8,140
$8,140(F/P, 12%, 10)
$ 25,280
Annual Value

Sometimes it is more convenient to evaluate a
project in terms of its annual value or cost. For
example it may be easier to evaluate specific
components of an investment or individual
pieces of equipment based upon their annual
costs as the data may be more readily available
for analysis.
Annual Analysis Example

A new piece of equipment is being evaluated for
purchase which will generate annual benefits in
the amount of $10,000 for a 10 year period, with
annual costs of $5,000. The initial cost of the
machine is $40,000 and the expected salvage is
$2,000 at the end of 10 years. What is the net
annual worth if interest on invested capital is
10%?
Annual Example Solution

Benefits:


$
125
$5,000 per year
-$ 5,000
Investment:


$2,000(P/F, 10%, 10)(A/P, 10%,10)
Costs:


$10,000
Salvage


$10,000 per year
$40,000(A/P, 10%, 10)
Net Annual Value
-$ 6,508
-$1,383
Since this is less than zero, the project is expected to
earn less than the acceptable rate of 10%, therefore
the project should be rejected.
Benefit/Cost Ratio

The benefit/cost ratio is also called the
profitability index and is defined as the ratio of
the sum of the present value of future benefits to
the sum of the present value of the future capital
expenditures and costs.
B/C Ratio Example




Present value cash inflows
Project A
$500,000
Present value cash outflows
$300,000
Net Present Value
$200,000
Benefit/Cost Ratio
1.67
Project B
$100,000
$ 50,000
$ 50,000
2.0
Payback Period
This is one of the most common evaluation criteria used by
engineering and resource companies.
The Payback Period is simply the number of years required for the
cash income from a project to return the initial cash investment in the
project.
The investment decision criteria for this technique suggests that if the
calculated payback is less than some maximum value acceptable to
the company, the proposal is accepted.
The following example illustrates five investment proposals having
identical capital investment requirements but differing expected annual
cash flows and lives.
Payback Period
Example
Calculation of the payback period for a given investment proposal.
a) Prepare End of Year Cumulative Net Cash Flows
b) Find the First Non-Negative Year
c) Calculate How Much of that year is required to cover the previous period
negative balance
d) Add up Previous Negative Cash Flow Years
Initial
Investment
1
Alternative A
(45,000) 10,500
a
2
11,500
3
8
9
10
12,500 13,500 13,500 13,500 13,500 13,500 13,500 13,500
End of Year Cummulative Net Cash Flow
(45,000) (34,500) (23,000) (10,500)
Pay Back Period
Fraction of First Positive Year
Pay Back Period
Annual Net Cash Flows
4
5
6
7
3,000 16,500 30,000 43,500 57,000 70,500 84,000
b
0.78
3.78
c) 0.78 = 10,500/13,500
d) 3 + 0.78
Example:
Calculate the payback period for the following
investment proposal
Initial
Investment
1
2
3
Alternative A
(120)
10
10
50
Annual Net Cash Flows
4
5
6
7
9
10
50
50
50
50
50
50
End of Year Cummulative Net Cash Flow
(120) (110) (100) (50)
0
50
100
150
200
250
300
Pay Back Period
Fraction of First Positive Year
Pay Back Period
50
8
1.00
4.00
Example:
Calculate the payback period for the following
investment proposal
Initial
Investment
1
2
3
Alternative A
(120)
10
10
50
Annual Net Cash Flows
4
5
6
7
9
10
50
50
50
50
50
50
End of Year Cummulative Net Cash Flow
(120) (110) (100) (50)
0
50
100
150
200
250
300
Pay Back Period
Fraction of First Positive Year
Pay Back Period
50
8
1.00
4.00
Example:
Calculate the payback period for the following
investment proposal
Initial
Investment
1
2
3
Alternative A
(120)
10
10
50
Annual Net Cash Flows
4
5
6
7
9
10
50
50
50
50
50
50
End of Year Cummulative Net Cash Flow
(120) (110) (100) (50)
0
50
100
150
200
250
300
Pay Back Period
Fraction of First Positive Year
Pay Back Period
50
8
1.00
4.00
Example:
Calculate the payback period for the following
investment proposal
Initial
Investment
1
2
3
Alternative A
(250)
86
50
77
Annual Net Cash Flows
4
5
6
7
9
10
41
70
127
24
6
40
End of Year Cummulative Net Cash Flow
(250) (164) (115) (38) 14
55
124
252
276
282
322
Pay Back Period
Fraction of First Positive Year
Pay Back Period
52
8
0.73
3.73
Example:
Calculate the payback period for the following
investment proposal
Initial
Investment
1
2
3
Alternative A
(250)
86
50
77
Annual Net Cash Flows
4
5
6
7
9
10
41
70
127
24
6
40
End of Year Cummulative Net Cash Flow
(250) (164) (115) (38) 14
55
124
252
276
282
322
Pay Back Period
Fraction of First Positive Year
Pay Back Period
52
8
0.73
3.73
Example:
Calculate the payback period for the following
investment proposal
Initial
Investment
1
2
3
Alternative A
(250)
86
50
77
Annual Net Cash Flows
4
5
6
7
9
10
41
70
127
24
6
40
End of Year Cummulative Net Cash Flow
(250) (164) (115) (38) 14
55
124
252
276
282
322
Pay Back Period
Fraction of First Positive Year
Pay Back Period
52
8
0.73
3.73
Effect of Pre-production Period
When calculating the payback period for a new project we typically
have several years of negative cash flows (investment) prior to positive
cash flows.
Two Approaches: Total Payback and Payback After 1st Production
Effect of Pre-production Period
The total payback period is calculated from the start of the project and
represents the commitment of the investor throughout the pre-production
period, particularly the opportunity cost associated with the investment
during this period.
Disadvantages:
A)
B)
This criterion fails to consider cash flows after the payback period,
therefore, it can’t be regarded as a suitable measure of profitability.
It doesn’t consider the magnitude or timing of the of cash flows during the
payback interval.
Advantages
1.
Simple and easy to calculate, providing a simple number which
can be used as an index of proposal profitability.
2.
Prevents management from exposure to excessive risk
 The shorter the payback the less risk associated with the
investment
3. Measure of lost opportunity risk
 Projects with short payback will minimize opportunity risk since
early cash flows will be returned to the firm within a short span of
time. (Liquidity)
4.
Payback period represents a break even point.
 Projects with life greater than the payback period will
contribute profit to the firm
Engineering Economics
EIT Review
IRR & Discount Rates
Investment Rate of Return
In mineral evaluation, any reference to rate of return normally refers to
the discounted cash flow return on investment (DCF-ROI) or the
discounted cash flow rate of return (DCF-ROR)
These terms are special versions of the more generic term, Internal
Rate of Return (IRR) or sometimes called marginal efficiency of capital
Besides NPV, is probably the most common evaluation technique used
in the minerals industry
Internal Rate of Return

Internal Rate of Return refers to the interest rate that the investor
will receive on the investment principal

IRR is defined as that interest rate (r) which equates the sum of
the present value of cash inflows with the sum of the present
value of cash outflows for a project. This is the same as defining
the IRR as that rate which satisfies each of the following
expressions:
∑ PV cash inflows - ∑ PV cash outflows = 0
NPV = 0 for r
∑ PV cash inflows = ∑ PV cash outflows
In general, the calculation procedure involves a trial-and-error solution
unless the annual cash flows subsequent to the investment take the form of
an annuity. The following examples illustrate the calculation procedures for
determining the internal rate of return.
Example
Given an investment project having the following annual cash flows; find the IRR.
Year
Cash Flow
0
(30.0)
1
(1.0)
2
5.0
3
5.5
4
4.0
5
6
7
8
9
17.0
20.0
20.0
(2.0)
10.0
Solution:
Step 1. Pick an interest rate and solve for the NPV. Try r =15%
NPV
= -30(1.0) -1(P/F,1,15%) + 5(P/F,2,15) + 5.5(P/F,3,15) + 4(P/F,4,15)
+ 17(P/F,5,15) + 20(P/F,6,15) + 20(P/F,7,15) - 2(P/F,8,15) + 10(P/F,9,15)
= + $5.62
Since the NPV>0, 15% is not the IRR. It now becomes necessary to select a
higher interest rate in order to reduce the NPV value.
Step 2. If r =20% is used, the NPV = - $ 1.66 and therefore this rate is too high.
Step 3. By interpolation the correct value for the IRR is determined to be r =18.7%
IRR using Excel
Using Excel you should insert the following function in the
targeted cell C6:
Analysis
The acceptance or rejection of a project based on the IRR
criterion is made by comparing the calculated rate with the required
rate of return, or cutoff rate established by the firm. If the IRR
exceeds the required rate the project should be accepted; if not, it
should be rejected.
If the required rate of return is the return investors expect the
organization to earn on new projects, then accepting a project with an
IRR greater than the required rate should result in an increase of the
firms value.
Analysis
There are several reasons for the widespread popularity of the IRR as
an evaluation criterion:

Perhaps the primary advantage offered by the technique is
that it provides a single figure which can be used as a
measure of project value.

Furthermore, IRR is expressed as a percentage value.
Most managers and engineers prefer to think of economic
decisions in terms of percentages as compared with
absolute values provided by present, future, and annual
value calculations.
Analysis
Another advantage offered by the IRR method is related to
the calculation procedure itself:
As its name suggests, the IRR is determined
internally for each project and is a function of the
magnitude and timing of the cash flows.
Some evaluators find this superior to selecting a
rate prior to calculation of the criterion, such as in
the profitability index and the present, future, and
annual value determinations. In other words, the
IRR eliminates the need to have an external interest
rate supplied for calculation purposes.
Multiple Roots Case
One of the disconcerting aspects associated with the internal rate of return is
that more than one interest rate may satisfy the calculation. The solution
procedure for IRR is essentially the solution for an nth degree polynomial of the
form:
NPV = 0 = A0 + A1X + A2X2 + A3X3 + .... + AnXn
where X = 1/(1 + r)
For a polynomial of this type there may be n different real roots, or values of
r, which satisfy the equation. Multiple positive rates of return may occur when
the annual cash flows have more than one change in sign.
Multiple Roots Case
The following example illustrates the possibility of multiple rates
which satisfy the definition of IRR:
Year
0
Existing
1
2
3
4
5
6
7
8
10.0
10.0
10.0
10.0
10.0
10.0
10.0
10.0
Proposed
(15.0)
16.0
16.0
16.0
16.0
15.0
15.0
Difference
(15.0)
6.0
6.0
6.0
6.0
5.0
5.0
(10.0)
(10.0)
Suppose a mining operation has a remaining life of eight years,
but an investment is considered to increase the production rate. This
will result in depleting the deposit in six years. Assuming the following
cash flows, is the investment justified?
Multiple Roots Case
Graphically this appears as shown in the following figure:
Rate
NPV
0%
4%
5%
(1.00)
(0.07)
0.06
10%
0.05
12%
0.03
15%
(0.25)
Multiple Roots Case
Should the firm invest in the project or not?
If both rates were above the firm's required rate of return there
would be no problem and the firm would accept the project.
However, what if the required rate of return is 10%? Which of the
calculated IRR values is correct? The answers to these questions are
that they are both mathematically correct, but they are meaningless
from an economic standpoint.
Neither of these rates can be considered an adequate measure of
the project's rate of return because a project can not earn more than
one rate of return over its life. Therefore, the calculation of an IRR
value(s) does not always enable the decision-maker to make
accept/reject decisions on investment proposals.
Multiple Roots Case
How often this problem of multiple rates actually occurs?
The possibility of multiple-rate occurrences is perhaps more prevalent in
the case of new mining ventures than in most other industries. The negative
cash flows are typically the result of anticipated periods of reduced market
prices, major capital expenditures for equipment replacement, expansion
programs, and/or major environmental expenditures, particularly at the end
of project life.
Because of the possibility of multiple rates and the reinvestment
assumption when using-the IRR to rank projects, the evaluator must carefully
consider the exclusive use of this technique for decision-making.
Selecting a Discount Rate
“There is nothing so disastrous as a rational investment
policy in an irrational world” John Maynard Keynes
We have discussed the time value of money and illustrated
several examples of its use. In all cases an interest rate or
“discount rate” is used to bring the future cash flows to the
present (NPV - Net Present Value)
The selection of the appropriate discount rate has been the
source of considerable debate and much disagreement. In
most companies, the selection of the discount rate is
determined by the accounting department or the board of
directors and the engineer just uses the number provided to
him, but short of just being provided with a rate, what is the
correct or appropriate rate to use?
Example
What is the impact of the discount rate on the investment?
Cash
Flow Yr 0
Cash
Flow Yr 1
Cash
Flow Yr 2
Cash
Flow Yr 3
Cash
Flow Yr 4
Cash
Flow Yr 5
-500
-500
+750
+600
+800
+1000
IRR
ROR
NPV
2%
1,941
6%
1,581
10%
1,283
15%
981
20%
739
47.82%
0
What is in the Discount Rate?
According to our text book and several authors, the
discount rate has to cover the following items:
 Opportunity Costs
 Transaction Costs
 Compensate for Risk
 Cover anticipated Inflation
Some of these items can be accounted for in other
financial analysis methods and do not have to be
address in the discount rate itself.
Financial Cost of Capital
The financial cost of capital is based on the assumption
that financing is unlimited and the company can always
pay off loans or buy stock back, so the financial cost of
capital rate of return is the average cost of debt after tax
(remember interest is tax deductible) and the cost of
equity (what the share holders desired return is using the
capital asset pricing model CAPM)
Marginal Weighted Average Cost of Capital
The cost of capital is the minimum rate of return that a
firm needs to earn on new investments to maintain the
existing value of it’s shares of common stock. To
determine the cost of capital a weighted average of all
sources of capital must be evaluated. The weighted
average should include a mix of debt and equity on an
after tax basis.
Hurdle Rate
The hurdle rate is a common term used by companies as
an expression of their rate of return used for financial
analysis.
This is generally a higher number than the FCC rate as
they add an imposed “economic hurdle” for the project to
overcome. This helps companies express that a project
that just achieves a FCC rate of return does not add real
value to the company.
Opportunity Cost of Capital
The opportunity cost of capital is the most
common method of establishing the investor’s
minimum rate of return.
This is based upon the expected returns that the
company will generate in the next 1 to 15 years.
It is the average return that investors expect to
make over the next few years expressed as a
compound interest.