ENGINEERING 161

1
ENGINEERING ECONOMICS
Lecture 10
Freshman Engineering Clinic II Engineering
Economics Part I Dr. R. Polikar
2
Engineering Economics

• Engineers make economic decisions, and they need
  bases for making these decisions
• Engineering economic decisions consider
• Money moved through time
• Economic indications, such as inflation, are
  considered through the rate of return
• Financial analysis of investments, depreciation,
  etc. through annuities
• Comparisons of alternative engineering projects
  can be based on engineering economics
• The Engineering In Training (EIT) examination has
  questions on engineering economics - many
  engineers will not take a formal course in
  engineering economics

3
Terminology

• P principal, (present sum), present worth,
• F (future sum), future worth,
• r interest rate or rate of return, used as a
  decimal per year given as per year
• N years, number of yearly periods in the
  project, may be a fraction of a year

4
Simple Interest

• F1 P r P P (1 r) the future worth, F1
  after one year is the principal , P plus the
  interest, rP earned (interest is removed)
• F2 F1 r P P (1 2 r) future worth after
  two years
• FN P (1 N r) future worth is the present
  worth plus the interest earned, interest does not
  earn additional interest
• This is not a useful method for engineering
  calculations

5
Compound Interest
Compound interest includes interest on accrued
interest year beginning of year (BOY) end of
year (EOY) 1 P P r P P (1 r) 2 P(1
r) P(1r)P(1r)r P (1 r)2 3 P(1r)2 P(1
r)2P(1r)2 r P (1 r)3 after N years
N P(1r)N-1 P(1r)N-1 P(1r)N-1 r P (1
r)N F Future worth F in terms of present worth
P is F P (1 r)N C P
C compound interest factor C (1 r)N P
C-1 F P (1 r)-N F
6
Cash-Flow Diagram
1. The horizontal line is a time scale.
Normally, years are given as the interval of time
with a breakdown for compounding at finer
increments 2. The vertical arrows signify cash
flow.
Generally we will want to convert a series of
payments into a future worth. In this example we
want to convert two revenue payments (P1 and P3)
and one expedtidure (P2) into a future worth F -
important note we cannot add the payments and
receipts UNLESS they are at the same point in
time, otherwise the values must be moved through
time with the compound interest factor
7
Example
Compound Interest Example
If one deposits 2 000.00 today at 7 interest
compounded annually, what will be the balance in
3 years?
F ?
P
0
3
1
2
year
N 3 r 0.07
P 2 000.00 C (1 r)N (1 0.07)3
1.225 043 F CP 2 450.09 (answer given to the
nearest 0.01) As money moves through time, into
the future, it increases in value
8
Example
If 4 000.00 is needed in 3 years, how much money
should be deposited today, if the money earns at
a 7 yearly interest rate?
F
P ?
0
3
1
2
year
N 3 r 0.07
9
Compound Interest on a Monthly Basis
Monthly compounded interest is interest on
monthly accrued interest period beginning of
month (BOM) end of month (EOM)
r
1 mo P P(1 )
12
r
r
2 P(1 ) P(1 ) P(1 )
P(1 )2
12
12
r
r
3 P(1 )2 P(1 )3
12
12
. . .
r
r
12 P(1 )11 P(1 )12
after 1 year
12
12
r
r
12N P(1 )12N-1 P(1
)12N after N years
12
12
10
Periodic Compound Interest
F P (1 )nN C P C
compound interest factor C º (1
)nN new definition for C P C-1 F P
(1 ) -nN F
r
n
r
n
r
n
n is the number of times compounded annually
(period, e.g. n4 for quarterly) N is the number
of years in the project (for annual compounding,
n1. For monthly compounding, n12)
11
Compounding For Different Periods
consider r 0.07 N 1 calculate the
compound interest factor n 1 C (1
r) 1.07 n 2 C (1 )2
1.0712 n 4 C (1 )4
1.0719 n 12 C (1 )12
1.0722 n 365 C (1 )365
1.07250
Continuous compounding
r
n
r
C
e




lim
(
)
.
1
1
07251
n
n


e 2.7182818
12
Compound Interest Example
If 4 000.00 is needed in 3 years, how much money
should be deposited today, if the money earns at
a 7 yearly interest rate but is compounded
quarterly?
F
P ?
0
3
1
2
year
n 4 N 3 r 0.07
r
0.07
F 4 000.00 C (1 )4N (1
)12 1.231 439 315 P 3 248.23
4
4
F
C
Note that this value is smaller than annual
compounding example for the same interest rate
13
Compound Interest Summary
Given P, n, N, r solve for F, the future
worth Given F, n, N, r solve for P, the present
worth P present worth, F future worth,
n times compounded per year N number of
years r interest rate as a decimal
n N
r
æ
ö
C


1

ç

è
ø
n
14
Other Compounding Problems
Given F, P, n, N solve for r, the interest rate
C
æ
ö
ln(
)
F
r
n
e
C

-


1
nN
ç

P
è
ø
Given F, P, n, r solve for N, the length of
project in years
æ
ö
ç

C
F
ln(
)
ç

N
C



r
ç

P
æ
ö
n

ln
1
ç

ç

è
ø
n
è
ø
15
Example
If 4 000.00 is needed in 3 years and you
currently have 3 500.00 to invest, what rate of
return should you seek, if the money is
compounded monthly?
F
n 12 N 3 r ?
P
0
1
2
3
year
F 4 000.00 P 3 5000.00
F
4000
C



1
1428571
.
P
3500
æ
ö
ln(
.
)
1
1428571
C
æ
ö
ln(
)
ç

(
)
12
3
r
n
e
e
r

-

-


.

.
1
12
1
0
0446
4
46
nN
ç

ç

è
ø
è
ø
16
Example
If 4 000.00 is needed and you currently have 3
500.00 to invest, how long would you have to
invest it at a rate of 7, if the money is
compounded monthly?
F
n 12 r 0.07 N ?
P
N
0
1
2
3
year
F 4 000.00 P 3 5000.00
F
4000
C



1
1428571
.
P
3500
æ
ö
æ
ö
ç

ç

C
ln(
)
ln(
.
)
1
1428571
ç

ç

N
years
or
about
months



.
1
913
23
r
ç

ç

.
07
æ
ö
æ
ö
n


ln
ln
1
12
1
ç

ç

ç

ç

è
ø
è
ø
n
è
ø
è
ø
12
17
Example
You have 3 500.00 to invest now for 3 years, 2
500.00 more at the end of one year, but you
withdraw 1500.00 at the end of 18 months. The
investment earns at a rate of 7 and the money is
compounded monthly, what is the future worth of
the investments? the present worth of the
investments?
F
P2 2500
P1 3500
r 0.07 n 12
1
2
3
0
year
P3 - 1500
N3
N2
N1
F1 P1 C1 C1 (1 )nN1 N1 3
C1 1.2329256 F1 4315.24
F2 P2 C2 C2 (1 )nN2 N2 2
C2 1.149806 F2 2874.52
F3 P3 C3 C3 (1 )nN3 N3 1.5
C3 1.1103718 F3 -1665.56
F F1 F2 F3 P 4480.56
F 5524.20

18
Hints
1. Draw proper cash-flow diagram (yearly,
quarterly, etc. compounding) 2. Do not add values
unless they are located at the same point on the
time-line 3. Make sure your final answer is
rounded to nearest 0.01 (no significant digits)
- keep all significant digits the calculator
holds in any intermediate steps in the
calculation 4. Make sure answer is reasonable 5.
The compound interest relations are F C P