mer439- design of thermal fluid systems engineering economics lecture 2- using factors professor...
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MER439- Design of Thermal Fluid Systems
Engineering Economics Lecture 2- Using Factors
Professor AndersonSpring 2012
Some Definitions
Capital: Invested money and resources
Interest: The return on capital
Nominal IR: the interest rate per year without adjusting for the number of compounding periods
Effective IR: the interest rate per year adjusting for the number of compounding periods
Different sums of money at different times can be equal in economic value.
i.e. $100 today with i = 6% is equivalent to $106 in one year.
Equivalence depends on the interest rate!
Equivalence occurs when different cash flows at different times are equal in economic value at a given interest rate.
Equivalence
Cash Flow Diagrams: An Important Tool
Income
time
Initial Capital Cost
ReplacementCosts
Operating &Maintenance
Costs
Salvage“Costs”
- Arrows up represent “income” or “profits” or “payoffs”- Arrows down represent “costs” or “investments” or “loans”- The “x axis” represents time, most typically in years
Time Value of Money
If $4500 is invested today for 12 years at 15% interest rate, determine the accumulated amount. Draw this.
F = P(1+i)n
P =Present Value (in dollars)
F = Future Value (in dollars)
$4500
Ft=0
t=12
n = 12, i = 15%
Factors
Single Payment Compound Amount Factor (future worth)
(F/P, i%, n) :
Single Payment Present Worth Factor (P/F, i%, n):
neffi
PF
)1(
neffiF
P
)1(
1
n is in years if the ieff is used.
Example - Factors
How much inheritance to be received 20 years from now is equivalent to receiving $10,000 now? The interest rate is 8% per year compounded each 6-months.
Uniform Series (Annuity)
An Annuity is a series of equal amount money transactions occurring at equal time periods
Ordinary Annuity - one that occurs at the end of each time period
neffeff
neff
ii
i
AP
)1(
1)1(
1)1(
)1(
neff
neffeff
i
ii
PA
Uniform Series Present Worth
Factor
Capital Recovery
Factor
Annuities
Can Relate an Annuity to a future value:
eff
neff
i
i
AF 1)1(
1)1(
n
eff
eff
i
i
FA
Uniform Series Compound
Amount Factor
Uniform Series Sinking Fund
Factor
Annuity Example
How much money can you borrow now if you agree to repay the loan in 10 end of year payments of $3000, starting one year from now at an interest rate of 18% per year?
Factors
Fortunately these factors are tabulated…
And Excel has nice built in functions to calculate them too….
Spreadsheet Function
P = PV(i,N,A,F,Type)F = FV(i,N,A,P,Type)i = RATE(N,A,P,F,Type,guess)Where, i = interest rate, N = number of
interest periods, A = uniform amount, P = present sum of money, F = future sum of money, Type = 0 means end-of-period cash payments, Type = 1 means beginning-of-period payments, guess is a guess value of the interest rate
Gradient Factors
Engineering Economic problems frequently involve disbursements or receipts that increase or decrease each year (i.e. equipment maintenance)
If the increase is the same every year this is called a uniform arithmetic gradient.
Gradient Factors
Present Value @ time zero
The Uniformamount of increaseeach period is thegradient amount
The amount in the initial year is calleda baseamount, and itdoesn’t need toequal the gradientamount
Gradient Factors
To get the Gradient Factors we subtract off the base amount, and start things in year (period) 2:
PG = Present worth of thegradient starting in year 2…This is what is calculated byP/G factor.
PT (total) = PG+PA
PA comes from using the P/Afactor on an annuity equal to the base amount.
PG/G and AG/G
n
n
ii
ininiGP
)1(
1)1(),,/(
2
1)1(
1),,/(
ni
n
iniGA
P/G = factor to convert a gradient series to a present worth.
A/G = factor to convert a gradient series to an equivalent uniform annual series.
Gradient Example
Find the PW of an income series with a cash flow in Year 1 of $1200 which increases by $300 per year through year 11. Use i = 15%
Review of Factors
Using the tables..
Single Payment factors (P/F), (F/P)
Uniform Series factors (P/A), (F/A)
Gradients (A/G), (P/G)
Unknown Interest Rates and Years
Unknown Interest rate:
-i.e. F = $20K, P = $10K, n = 9 i = ?
-Or A = $1770, n = 10, P = $10K i =?
Unknown Years – sometimes want to determine the number of years it will take for an investment to pay off ( n is unknown)
-A = $100, P = $2000, i = 2% n = ?
Unknown interest example
If you would like to retire with $1million 30 years from now, and you plan to save $6000 per year every year until then, what interest rate must your savings earn in order to get you that million?
Use of Multiple Factors
Many cash flow situations do not fit the single factor equations.
It is often necessary to combine equationsExample? What is P for a series of $100
payments starting 4 years from now?
1 2 3 4 5 6 7 8 9 10 11 12 13
$100
P = ?
years
Use of Multiple Factors
Several Methods:1. Use P/F of each payment2. F/P of each and then multiply by P/F3. Get F =A (F/A, i,10), then P = F (F/P,i,13)4. Get P3 = A(P/A,I,10) and P0 = P3(P/F,i,3)
1 2 3 4 5 6 7 8 9 10 11 12 13
$100
P = ?
years
Use of Multiple Factors
Step for solving problems like this:1. Draw Cash Flow Diagram.2. Locate P or F on the diagram.3. Determine n by renumbering if necessary.4. use factors to convert all cash flows to
equivalent values at P or F.
Multiple Factors: Example
A woman deposited $700 per year for 8 years. Starting in the ninth year she increased her deposits to $1200 per year for 5 more years. How much money did she have in her account immediately after she made her last deposit?
Eng Econ Practice Problems
Check Website for Practice Problems…
Remember you ALL have a quiz on
Engineering Econ on Monday, not just the
economists!
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