Post on 01-Oct-2015
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MS-291: Engineering Economy(3 Credit Hours)
Pervious Lecture!!!
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Single payment Factors
F/P FactorF= P(F/P, i%, n)
P/F FactorP= F(P/F, i%, n)
P is given
F = ?
n and i is given
F = given
n and i is given
P =?
Uniform Series Factors
P/A FactorP = A(P/A, i%, n)
A/P FactorA = P(A/P, i%, n)
F/A FactorF = A(F/A, i%, n)
A/F FactorA = F(A/F, i%, n)
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Athematic Gradient
PA = A(P/A, i%, n)
PG = G(P/G, i%, n)
FG = ?
F = PT(F/P, i%, n)
or
/ = 1 (1 + ) 1
Athematic Gradient
PA = A(P/A, i%, n)
PG = G(P/G, i%, n)
A = PT(A/P, i%, n)or
AT = AA + AG
= 1 (1 + ) 1AA = A (Annual Worth) &AG = G(A/G, i, n)
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Geometric Gradient
Pg = ?
= 1 1 +1 +for g i:
for g = i:= 1 +
Fg = ?
F = Pg(F/P, i%, n)Similarly
A = Pg(A/P, i%, n)
Engineering Economy
Chapter 3Combining Factorsand Spreadsheet
Functions
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This Chapter Objectives
1. Shifted uniform series
2. Shifted series and single cash flows
3. Shifted gradients
Example
0 62 3 4 51 7 8 9 10 11 12 13 Year
A = $50
P = ?
Use the P/F factor to find the present worth of each disbursement at year 0 and addthem.
Use the F/P factor to find the future worth of each disbursement in year 13, addthem, and then find the present worth of the total, using P/F= F( P/F, i ,13).
Use the F/A factor to find the future amount F/A =A( F/A, i ,10), and then computethe present worth, using P/F=F(P/F, i ,13).
Use the P/A factor to compute the present worth P3 =A( P/A , i ,10) (which will belocated in year 3, not year 0), and then find the present worth in year 0 by using the(P/F , i ,3) factor.
How can we get Present worth of this series ?
P3 = ? F
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Shifted Uniform Series Typically the last method is used for calculating the present worth
of a uniform series that does not begin at the end of period 1.
Note that a P value is always located 1 year or period prior to thebeginning of the first series amount. Why? Because the P/A factorwas derived with P in time period 0 and A beginning at the end ofperiod 1.
The most common mistake made in working problems of this typeis improper placement of P .
Remember: When using P/A or A/P factor, PA is always one year ahead of first A When using F/A or A/F factor, FA is in same year as last A The number of periods n in the P/A or F/A factor is equal to the number of uniform
series values
0 62 3 4 51 7 8 9 10 11 12 13 Year
A = $50
P3 = ?
0 62 3 4 51 7 8 9 10 11 12 13 Year
A = $50
F = ?
PA is always one year aheadof first A
FA is in same year as last A
The number of periods n in the P/A orF/A factor is equal to the number ofuniform series values
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Steps for applying factors toShifted Cash Flows
1. Draw a diagram of the positive and negative cash flows.2. Locate the present worth or future worth of each series on
the cash flow diagram.3. Determine n for each series by renumbering the cash flow
diagram.4. Draw another cash flow diagram representing the desired
equivalent cash flow. (Optional)5. Set up and solve the equations.
ExampleThe offshore design group at Bechtel just purchasedupgraded CAD software for $5000 now and annualpayments of $500 per year for 6 years starting 3 yearsfrom now for annual upgrades. What is the presentworth in year 0 of the payments if the interest rate is8% per year?
1. Draw a diagram of the positive and negative cash flows.Solution
0 62 3 4 51 7 8
P0 = $5000
A = $500
PT = ? 2. Locate the presentworth or future worth ofeach series on the cashflow diagram.PA = ?PA = ? i= 8% per year
3. Determine n for eachseries by renumbering thecash flow diagram.
0 62 3 4 51 nYear
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P' A = $500( P /A ,8%,6)PA = P' A ( P /F ,8%, 2)
P T = P 0 + P A=5000 + 500( P /A ,8%,6)( P / F ,8%,2)=5000 +500(4.6229)(0.8573)$6981.60
5. Set upand solvetheequations.
PA = $500( P /A ,8%,6) ( P /F ,8%, 2)
Class Practice 5 Minutes Time
A = $10,000
0 1 2 3 4 5 6
i = 10%
Calculate the present worth of the cash flow shown below at i = 10%
Actual year
10% Single Payments Uniform Series Factorsn Compoun
d Amount(F/P)
PresentWorth(P/F)
SinkingFund(A/F)
CompoundAmount(F/A)
CapitalRecovery(A/P)
PresentWorth(P/A)
1 1.1000 0.9091 1.00000 1.0000 1.10000 0.90915 1.6105 0.6209 0.16380 6.1051 0.26380 3.7908
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Class Practice 5 Minutes Time
A = $10,000
PT = ?
0 1 2 3 4 5 6
i = 10%
Calculate the present worth of the cash flow shown below at i = 10%
PT = A(P/A,10%, 5) (P/F,10%,1)
$34462
0 1 2 3 4 5
PA = ?Actual yearSeries year
(2) Use P/F factor with n = 1 to move PA back for PT in year 0
= 10,000(3.7908)(0.9091)
(1) Use P/A factor with n = 5 (for 5 arrows) to get PA in year 1Solution
A(P/A,10%, 5)----(P/F,10%, 1)----
Shifted Series and RandomSingle Amounts
For cash flows that include uniform series and randomly placedsingle amounts:
Uniform series procedures are applied to the series amounts
Single amount formulas are applied to the one-time cash flows
The resulting values are then combined per the problem statement
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Example
Find the present worth in year 0 for the cash flows shown using an interestrate of 10% per year.
0 1 2 3 4 5 6 7 8 9 10
A = $5000
i = 10%
Find the cash flows both positive and negatives
$2000
0 1 2 3 4 5 6 7 8 9 10
PT = ?
A = $5000
i = 10%
$2000
0 1 2 3 4 5 6 7 8
Solution:
Actual year
Series year
Locate the present worth/ future worth Determine the n by re-numbering the cash flows series Uniform series procedures are applied to the series amounts. Single amount
formulas are applied to the one-time cash flows The resulting values are then combined per the problem statement
Use P/A to get PA in year 2: PA = 5000(P/A,10%,8) = 5000(5.3349) = $26,675
Move PA back to year 0 using P/F: P0 = 26,675(P/F,10%,2) = 26,675(0.8264) = $22,044Move $2000 single amount back to year 0: P2000 = 2000(P/F,10%,8) = 2000(0.4665) = $933Now, add P0 and P2000 to get PT: PT = 22,044 + 933 = $22,977
Example:
PA = ?
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Class Practice: 8 MinutesAn engineering company lease the mineral rights to a miningcompany on its land. The engineering company makes aproposal to the mining company that it pay $20,000 per year for20 years beginning 1 year from now, plus $10,000 six years fromnow and $15,000 sixteen years from now. If the mining companywants to pay off its lease immediately, how much should it paynow if the investment is to make 16% per year?
16% Single Payments Uniform Series Factorsn Compound
Amount (F/P)Present Worth(P/F)
CapitalRecovery (A/P)
Present Worth(P/A)
6 2.4364 0.4104 0.27139 3.68477 2.8262 0.3538 0.24761 4.038616 10.7480 0.0930 0.17641 5.668517 12.4677 0.0802 0.17395 5.748720 19.4608 0.0514 0.16867 5.9228
A =$20,000P = ?
0 1 2 3 4 5 6 7 16 17 18 19 20
$10,000 $15,000
P = 20,000(P/A ,16%,20)+
= $124,075
10,000( P /F ,16%,6) +15,000(P/F,16%,16)
P = $20,000(5.9288)+ $ 10,000( 0.4104) + $ 15,000(0.0930)
Solution
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Shifted Gradient Series
We already learnt how to get P (Presentvalue) or A ( Annuity or a Uniform series)from a Gradient Series
We will now discuss how to calculate P orA from Shifted Gradient Series agradient series not starting from year 1.
PA = A(P/A, i%, n)PG = G(P/G, i%, n)
P from Shifted Gradient Series
Shifted gradient begins at atime other than between periods1 and 2
Present worth PG is located 2periods before gradient starts
Must use multiple factors to find PTin actual year 0
P T =P A +P G=100(P/A , i ,8) + 50(P/G, i ,5)(P/F, i ,3)
Shifted Arithmetic gradient Series
PG = ?
PT = ?
PA = ?
PG = ?
P from Normal Arithmetic Gradient Series
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What will be the procedure for calculating P fromShifted Geometric Gradient Series?
Lets discuss it directly from a Numerical Example
Example:Shifted Geometric Gradient
Weirton Steel signed a 5-year contract to purchase water treatment chemicals from alocal distributor for $7000 per year. When the contract ends, the cost of the chemicals isexpected to increase by 12% per year for the next 8 years. If an initial investment instorage tanks is $35,000, determine the equivalent present worth in year 0 of all of thecash flows at i = 15% per year.
0 2 31 4 5
$7000
Year
$7840
6 7 8 9 10 11 12 13
$1733112% increaseper year
$35000
i =15% per year
GeometricGradient n
0 1 2 3 4 5 6 7 8
Pg = ?PT = ?
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P from Shifted Gradient Series
PT = 35,000 + A ( P /A ,15%, 4) + A1 ( P/A ,12%,15%,9) (P/F ,15%,4)
$83,232
PT = 35,000 + 7000 ( 2.8550) + 7000. .. . (0.5718)
A from Shifted Gradient SeriesShifted gradient Series
0 62 3 4 51 7 8
To calculate A for shifted Gradient Series (Arithmetic orGeometric), there are several possibilities.
The easiest way (and recommended also) is to get the P ofshifted Gradient Series first (procedure just explained in pervious slides) thenuse A/P factor to get A for the shifted gradient series
A, for above example will be: A = PT (A/PT, i%, n), where PT refers to the present value of the shifted gradient series that procedure is
already explained on pervious slide.
A (Annuity or Uniform Series)
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Important Points for P and Aof Shifted Gradient Series
Must use multiple factors to find P in actual year 0, for shiftedgradient series
The present worth (P) of an arithmetic gradient will always belocated two periods before the gradient starts.
To find the equivalent A series of a shifted gradient throughall the n periods, first find the present worth of the gradient atactual time 0, then apply the (A/P, i, n) factor.
F from gradient series can also be find by first calculating Pand then using F/P factor
Example: Shifted ArithmeticGradient
Solution:
John Deere expects the cost of a tractor part to increase by $5 per year beginning 4 years fromnow. If the cost in years 1-3 is $60, determine the present worth in year 0 of the cost through year10 at an interest rate of 12% per year.
i = 12%
First find P2 for G = $5 and base amount ($60) in actual year 2P2 = 60(P/A,12%,8) + 5(P/G,12%,8) = $370.41
Next, move P2 back to year 0 P0 = P2(P/F,12%,2) = $295.29
Next, find PA for the $60 amounts of years 1 and 2 PA = 60(P/A,12%,2) = $101.41
Finally, add P0 and PA to get PT in year 0 PT = P0 + PA = $396.70
60 60 6065
7095G = 5
0 1 2 3 104 5 Actual years0 1 2 3 8 Gradient years
PT = ? P2 = ?
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Class Question.. 5 minutesFor the cash flows shown, find the future value in year 7 at i = 10% per year
F = ?
0 1 2 3 4 5 6 7
700 650
500 450550600
G = $-50Solution:
Actual years0 1 2 3 4 5 6 Gradient years
PG is located in gradient year 0 (actual year 1); base amount of $700 is in gradient years 1-6
F = PG(F/P,10%,6) = 2565(1.7716) = $4544
i = 10%PGPG = ?
PG = A(P/A,10%,6) G(P/G,10%,6)PG = 700(P/A,10%,6) 50(P/G,10%,6) = 700(4.3553) 50(9.6842) = $2565
PG= PG(F/P,10%,1) = 2565(0.9091) = $2331.84
F = PG(F/P,10%,7) = 2331.84(1.9487) = $4544
Set up theequationsonly
Method 1
Method 2
Using Single Amount factors (Correctbut not Standard methods)
Method 3
Method 4
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THANK YOU