Chapter 9 Engineering Economic Analysis
Copyright R.Turton and J. Shaeiwitz - 2012
Chemical Engineering Department
West Virginia University
1
Definitions
• P – Principal or Present Value (of an investment)
• Fn – Future Value (of an investment)
• n – Years (or other time unit) between P and F
• i – Interest Rate (based on time interval of n) per anum
Basic premise: Money when invested earns money
$1 today is worth more than $1 in the future
Copyright R.Turton and J. Shaeiwitz - 2012 2
Interest
• Simple Interest – Annual Basis
– Interest paid in any year = Pis
• Pis – Fraction of investment paid as interest per year
– After n years total interest paid = Pisn
– Total investment is worth = P + Pisn
– Total investment after 1 year (n = 1) = P (1+is)
– What is the drawback of simple interest?
Copyright R.Turton and J. Shaeiwitz - 2012
We can earn interest on earned interest
3
Interest
• Compound Interest
At time 0 we have P
At the end of Year 1, we have F1 = P (1 + i )
At the end of Year 2, we have F2 = P (1 + i )2
At the end of Year n, we have Fn = P (1 + i )n
or P = Fn / (1 + i )n
Copyright R.Turton and J. Shaeiwitz - 2012 4
Example
• How much would I need to invest at 8 % p.a. to yield $5000 in 10 years?
Copyright R.Turton and J. Shaeiwitz - 2012
10
10
0.08
10
5000
5000$2315.97
1 0.08
i
n
F
P
5
What if Interest Rate Changes with Time?
Copyright R.Turton and J. Shaeiwitz - 2012
1 21(1 ) 1 1 ....... 1
n
n j nj
F P i P i i i
Eq. (7.7)
6
Different Time Basis for Interest Calculations
• Relates to statement “Your loan is 6 % p.a., compounded monthly”
• Define actual interest rate per compounding period as r
– inom = Nominal annual interest rate
– m = Number of compounding periods per year (12)
Copyright R.Turton and J. Shaeiwitz - 2012 7
Different Time Basis for Interest Calculations cont.
– ieff = effective annual interest rate
• Look at condition after 1 year
Copyright R.Turton and J. Shaeiwitz - 2012
nomir
m
1 1 1
mnom
eff
iF P i P
m
1 1
mnom
eff
ii
m
8
Example
• I invest $1000 at 10 % p.a. compounded monthly. How much do I have in 1 year, 10 years?
Copyright R.Turton and J. Shaeiwitz - 2012
12
1
12
10
10
0.101 1000 1 $1104.71
12
0.101 1 0.1047
12
1 $2707.04
mnom
eff
eff
iF P
m
i
F P i
9
Example cont.
• As m decreases ieff increases
• Is there a limit as m goes to infinity
– Yes – continuously compounded interest
– Derivation – pp. 265-266
– ieff (continuous) = e inom – 1
Copyright R.Turton and J. Shaeiwitz - 2012 10
Cash Flow Diagram (CFD)
• Represent timings and approximate magnitude of investment on a cfd
– x-axis is time and y-axis is magnitude
– both positive and negative investments are possible.
• In order to determine direction (sign) of cash flows, we must define what system is being considered.
Copyright R.Turton and J. Shaeiwitz - 2012 11
Consider a Discrete Cash Flow Diagram
• Discrete refers to individual CFDs that are plotted
Copyright R.Turton and J. Shaeiwitz - 2012 12
Example
• I borrow $20 K for a car and repay as a $400 monthly payment for 5 years.
Copyright R.Turton and J. Shaeiwitz - 2012
For Bank For Me
1 2 3
$400
60 3 2 1
$20,000
60
$400
$20,000
13
Cumulative CFD
Copyright R.Turton and J. Shaeiwitz - 2012
Cumulative CFD
14
Annuities
Copyright R.Turton and J. Shaeiwitz - 2012
1 2 3 n
Uniform series of equally spaced, equal value cash flows Note: The first payment is at the beginning of year 1 not
at t = 0
15
Annuities
• What is future value Fn?
• Geometric progression
Copyright R.Turton and J. Shaeiwitz - 2012
1 2
1 1 .....n n
nF A i A i A
1 1n
n n
iF S A
i
16
Discount Factors
• Just a shorthand symbol for a formula in i and n
Copyright R.Turton and J. Shaeiwitz - 2012
1, ,
1 1
1, ,
1
1 1, ,
1
n n
n
n
n
F PP i n
Fi i
PP F i n F
F i
iPA P i n
A i i
See Table 9.1
17
Discount Factors
n
n
n
n
ii
ini
A
P
ini
F
P
i
ini
A
F
1
11,,
therefore
1
1,,
11,,
Copyright R.Turton and J. Shaeiwitz - 2012
Table 9.1 has six versions of these Three are reciprocals of the other three
18
Calculations with Cash Flow Diagrams
• Invest 5K, 1K, 2K at end of Years 0, 1, 3, and take 3K at end of Year 4
• Note that annuity payments are all at the end of the year
Copyright R.Turton and J. Shaeiwitz - 2012
0
$3,000
$2,000 $1,000
$5,000
3
7 4
1
19
Example 1
• How much in account at end of Year 7 if i = 8% p.a.?
• What would investment be at Year 0 to get this amount at Year 7?
Copyright R.Turton and J. Shaeiwitz - 2012
7 6 47
3
7
5,000 1 0.08 1000 1 0.08 2000 1 0.08
3000 1 0.08
$9097.84
F
F
7
9097.845308.50
1.08P
20
Example 2
• What should my annual monthly car payment be if interest rate is 8% p.a. compounded monthly?
Copyright R.Turton and J. Shaeiwitz - 2012
$20,000
A
21
Example 2 (cont’d) • Compare at n = 60
Copyright R.Turton and J. Shaeiwitz - 2012
60
60
60
60
0.081 1
1273.47
0.08
12
0.0820,000 1 29,796.90
12
73.47 29,796.90 0
$405.53
F A A
F
A
A
Interest paid = $4,331.20
22
Example 2 (cont’d)
• Another method
20.4331$000,20)52.405(60 paidinterest
52.405$)020276.0(000,2060,08.0,
020276.0
112
08.01
12
08.01
12
08.0
11
1
60,08.0,60
60
P
APA
m
i
m
i
m
i
P
Amn
mn
Copyright R.Turton and J. Shaeiwitz - 2012 23
Example 3
• You buy a house where you finance $200 K at 6% p.a. interest, compounded monthly. What is your monthly payment, and how much interest do you pay over the lifetime of the loan for a 15-year and a 30-year mortgage in current dollars?
Copyright R.Turton and J. Shaeiwitz - 2012 24
Example 3 (cont’d)
30,12mortgageyear -30
15,12mortgageyear -15
11
1
1)1(
)1(
nm
nm
m
i
m
i
m
i
i
ii
P
Amn
mn
n
n
Copyright R.Turton and J. Shaeiwitz - 2012 25
Example 3 (cont’d)
• For 15-year mortgage – $1687.71/month
– total of $303,788 paid
– $103,788 interest
• For 30-year mortgage – $1199.10/month
– total of $431,676 paid
– $231,676 interest
Copyright R.Turton and J. Shaeiwitz - 2012 26
Example 4
• You invest $5000/year (the maximum, for now) in a Roth IRA, starting at age 25 for 40 years. Assuming a return of 8% p.a., how much will you have at age 65 in future dollars?
Copyright R.Turton and J. Shaeiwitz - 2012 27
Example 4 (cont’d)
283,295,1$
5000
06.259
08.0
1)08.01(
1)1(
40
F
A
A
F
A
F
i
i
A
F n
Copyright R.Turton and J. Shaeiwitz - 2012 28
Example 5
• Repeat the previous calculation, assuming that you do not start investing until age 35 or age 45.
800,228$76.4520if
400,566$28.11330if
5000
08.0
1)08.01(
1)1(
FA
Fn
FA
Fn
A
A
F
i
i
A
F
n
n
Copyright R.Turton and J. Shaeiwitz - 2012 29
Depreciation
• Total Capital Investment = Fixed Capital + Working Capital
– Fixed Capital – All costs associated with new construction, but Land cannot be depreciated
– Working Capital – Float of material to start operations cannot depreciate
Copyright R.Turton and J. Shaeiwitz - 2012
LTCI FCI Land WC
30
Definitions
• Salvage Value, S – Value of FCIL at end of project
– Often = 0
• Life of Equipment – n – set by IRS
• Not related to actual equipment life
• Total Capital for Depreciation – FCIL - S
Copyright R.Turton and J. Shaeiwitz - 2012 31
4 Basic Methods for Depreciation
• Straight Line
• Sum of Years Digits (SOYD)
• Double Declining Balance (DDB)
• Modified Accelerated Cost Recovery System (MACRS)
Copyright R.Turton and J. Shaeiwitz - 2012 32
Straight Line
Copyright R.Turton and J. Shaeiwitz - 2012
SL Lk
FCI Sd
n
n = # of years over which depreciation is taken
33
Sum of Years Digits (SOYD)
Copyright R.Turton and J. Shaeiwitz - 2012
1
11
2
LSOYDk
n k FCI Sd
n n
SOYD
34
Double Declining Balance (DDB)
Copyright R.Turton and J. Shaeiwitz - 2012
1
0
2k
DDBk L j
j
d FCI dn
35
MACRS
• Current IRS-approved method
Copyright R.Turton and J. Shaeiwitz - 2012
Year
1
2
3
4
5
6
Depreciation Percentage
20.00
32.00
19.20
11.52
11.52
5.76
See Chapter 9
Based on combination of DDB and SL
36
Example 9.21
Copyright R.Turton and J. Shaeiwitz - 2012
6
6
$150 10
$10 10
7
1 ?
L
stSL
FCI
S
n
Year d
Same for Years 1-7
150 1020
7SLd
37
Example 9.21 (cont’d)
Copyright R.Turton and J. Shaeiwitz - 2012
,1
,2
7 1 1 7150 10 150 10 35
1 287 82
7 1 2 6150 10 150 10 30
1 287 82
SOYD
SOYD
d
d
Sum of Year’s Digits
38
Example 9.21 (cont’d)
Copyright R.Turton and J. Shaeiwitz - 2012
,1
,2
2150 42.9
7
2150 42.9 30.6
7
DDB
DDB
d
d
Double Declining Balance
39
Example 9.21 (cont’d)
64.8)150(0576.0
28.17)150(1152.0
28.17)150(1152.0
8.28)150(192.0
48)150(32.0
30)150(20.0
6,
5,
4,
3,
2,
1,
yrMACRS
yrMACRS
yrMACRS
yrMACRS
yrMACRS
yrMACRS
d
d
d
d
d
d
Copyright R.Turton and J. Shaeiwitz - 2012
MACRS
40
Taxation, Cash Flow, and Profit
• Tables 9.3 – 9.4
• Expenses = COMd + dk
• Income Tax = (R – COMd - dk)t
• After Tax (net)Profit =
(R – COMd –dk)(1 – t)
• After Tax Cash Flow =
(R – COMd – dk)(1 – t) + dk (+ other cash flows)
• Other cash flows might include working capital return, salvage value, etc.
Copyright R.Turton and J. Shaeiwitz - 2012 41
Inflation
• $ Net Worth Now vs. $ Next Year
• f = Average inflation rate between years j and n
Copyright R.Turton and J. Shaeiwitz - 2012
1n
CEPCI j n f CEPCI j
42
Inflation
• Example
Copyright R.Turton and J. Shaeiwitz - 2012
10
0.1
1993 359
2003 402
4021
359
4021 0.0114 or 1.14%
359
CEPCI
CEPCI
f
f
43
Inflation
• What is inflation rate since 2003? The current CEPCI is 600 (2011).
%13.5
0513.014925.1
4925.1402
600)1(
125.0
8
f
f
Copyright R.Turton and J. Shaeiwitz - 2012 44
Inflation • Effect of inflation on interest rate
f affects the purchasing power of the $
• Look at the purchasing power of future worth, then
• If this future worth was obtained by investing at a rate i, then the inflation adjusted interest rate, i ’ is given by
Copyright R.Turton and J. Shaeiwitz - 2012
'
(1 )n
FF
f
' '(1 ) 1(1 )
1(1 ) (1 )
nnn
n n
F i iF P P P i
ff f
' 11
1
ii i f
f
45