Word Problems
• Organize the Data Given:
•Determine the objective and
your strategy.
•Draw the Cash Flow Diagram.
•Write Equations and Solve.
• Reflect Back on What You
Learned – What does your
answer mean?
1
Reviewing…
• Complex Cash Flows – Must splitup and recombine at the SAME
point in time.
• Linear gradient cash flows always
begin at the end of year two.
• Strategy – comparisons must
occur at the same point in time.
• Periods for i and n must match!
2
Present Worth Analysis
Net Present Worth of initial and
future cash flows can be used to
select among alternative projects.
It is important to understand what
Net Present Worth means, especially
when the cash flows include both
revenue and expenses.
3
Present Worth Analysis
Project selection based on Present
Worth Analysis:
• If all expenses and revenues are
included, select the largest NPW
that is greater than zero.
•Doing nothing is an option
(leaving the $ to earn “safe” interest)
• If some or none of the revenues are
included, select the largest NPW.
•“Must Do situation”
4
Terminology
Salvage Value – the amount of
money you can expect to receive
by selling an asset when you are
done with it. What value does it
have when you are done with it?
MARR – Minimum Attractive Rate
of Return – I expect or need this
return in order to be willing to
invest my money.
5
Example Problem
Project A costs $10,000 and will last for 10 years.
Annual, end of the year revenues will be
$3,000, and expenses will be $1,000. There is
no salvage value.
Project B costs $10,000 and will also last for 10
years. Annual revenues will be $3,000 with
annual expenses of $1,500. Salvage value is
$5,000.
Conduct an economic analysis to select the
preferred project using a MARR of 10% per
year, compounded annually.
6
Example Problem
Project A costs $10,000 and will last for 10 years. Annual,
end of the year revenues will be $3,000, and expenses
will be $1,000. There is no salvage value.
GIVEN:
DIAGRAM:
LIFETIME = 10 YRS
NPWA ?
MARR = 10%/YR, CPD ANNUALLY
FIRST COST = $10 000
ANNUAL REVENUES = $3 000/YR
0
1
2
3
ANNUAL COSTS = $1 000/YR
SALVAGE VALUE = $0
$10 000
FIND NPWA:
NET ANNUAL = ANNUAL REVENUES – ANNUAL COSTS
= $3000/YR – $1000/YR = $2000/YR
$3 000
4
10
$1 000
NPWA = A(P|A,i,n) – 1ST COST
= $2 000(P|A,10%,10) – $10 000
= $2 000(6.1446) – $10 000 = $2 289
7
Example Problem
Project B costs $10,000 and will also last for 10 years.
Annual revenues will be $3,000 with annual expenses
of $1,500. Salvage value is $5,000.
GIVEN:
DIAGRAM:
LIFETIME = 10 YRS
NPWB ?
$5 000
MARR = 10%/YR, CPD ANNUALLY
$3 000
FIRST COST = $10 000
ANNUAL REVENUES = $3 000/YR
10
0
1
2
3
4
ANNUAL COSTS = $1 500/YR
$1 500
SALVAGE VALUE = $5 000
$10 000
FIND NPWB:
NET ANNUAL = ANNUAL REVENUES – ANNUAL COSTS
= $3 000/YR – $1 500/YR = $1 500/YR
NPWB = A(P|A,i,n) + SALVAGE(P|F,i,n) – 1ST COST
= $1 500(P|A,10%,10) + $5 000(P|F,10%,10) – $10 000
= $1 500(6.1446) + $5 000(0.3855) – $10 000 = $1 144 ►PREFER A
8
What does this mean?
NPWA = $2 289 NPWB = $1 144
We prefer project A over project B.
Does NOT mean a $2 289 profit!
Concept:
We favor Project A by $2 289 over
taking $10 000 and putting it in an
account earning 10%.
9
In other words…
With expenses and revenues known,
select the largest NPW > 0
 Select Project A
What does this mean?
At i = 10% (cpd yearly), $2 000 at the
end of each of the next 10 years is
worth, today, $2 289 more than the
initial cost of $10 000.
10
Further…
You would be willing to pay as much as
$10,000 + 2,289 = $12,289 for the project.
At that price and at i = 10%, you are indifferent
between:
1. Investing $12,289 for 10 years at i = 10%.
2. Obtaining $2000 at the end of each of the
next 10 years (and reinvesting each receipt
at 10%).
11
Illustrating…
F10 = (10,000 + 2,289) (F | P, 10%, 10)
= (10,000 + 2,289) (2.594)
= $31,875
F10 = 2000 (F | A, 10%, 10)
= 2000 (15.937)
= $31,875
12
Thus…
The project only costs $10 000, but
at i = 10% it is equivalent to
investing $10 000 + $2 289 for 10
years.
Since PW > 0, you are actually
earning more than 10% on
investment.
13
Present Worth Analysis
When applied correctly, NPW can
be used to select among various
alternative projects.
• The larger the NPW the better.
• Requires establishing MARR.
• MARR is used as the (i) in the
equations.
14
Plotting NPW vs. i
10 000
NPW ($)
IRR
2 289
0
10%
15.1%
i (%)
Why does NPW decrease as i increases?
15