EMSE 260 Clhapter 4 Time Value of Money
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Transcript of EMSE 260 Clhapter 4 Time Value of Money
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Copyright 2009 by Pearson Education, Inc.
Upper Saddle River, New Jersey 07458
All rights reserved.
Engineering Economy, Fourteenth Edition
By William G. Sullivan, Elin M. Wicks, and C. Patrick Koelling
Engineering Economy
Chapter 4: The Time Value of Money
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Key Concepts
Cash Flow Diagram: the financial description
(visual) of a project Time Value of Money: the value of money
changes with time
Money provides utility (value) when spent
Value of money grows if invested
Value of money decreases due to inflation Interest: used to move money through time for
comparisons
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Time Value of Money
Money has value because it gives us
utility. Generally, money is preferred now, as
opposed to later (same amount)One can spend it now and get utility
One can invest it and watch it grow withinterest for greater future utility
One can put it under the mattress and watchit losepurchasing power
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Copyright 2009 by Pearson Education, Inc.
Upper Saddle River, New Jersey 07458
All rights reserved.
Engineering Economy, Fourteenth Edition
By William G. Sullivan, Elin M. Wicks, and C. Patrick Koelling
Time Value of Money
To describe the same amount of money at
different periods of time requires the useof an interest rate. With a positive rate:
Money grows (compounds) into largerfuture sums in the future.
Money is smaller (discounted ) in the past.
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Copyright 2009 by Pearson Education, Inc.
Upper Saddle River, New Jersey 07458
All rights reserved.
Engineering Economy, Fourteenth Edition
By William G. Sullivan, Elin M. Wicks, and C. Patrick Koelling
Money has a time value. Capital refers to wealth in the form of
money or property that can be used toproduce more wealth.
Engineering economy studies involve the
commitment of capital for extended periods
of time.
A dollar today is worth more than a dollarone or more years from now (for several
reasons).
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Copyright 2009 by Pearson Education, Inc.
Upper Saddle River, New Jersey 07458
All rights reserved.
Engineering Economy, Fourteenth Edition
By William G. Sullivan, Elin M. Wicks, and C. Patrick Koelling
Return to capital in the form of interest and
profit is an essential ingredient ofengineering economy studies.
Interest and profit pay the providers of capital forforgoing its use during the time the capital is beingused.
Interest and profit are payments for the risktheinvestor takes in letting another use his or hercapital.
Any project or venture must provide a sufficientreturn to be financially attractive to the suppliersof money or property.
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Cash Flow
Movement of money in (out) of a project
Inflows: revenues or receipts
Outflows: expenses or disbursements
Net Cash Flow:
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Cash Flow Analysis
Given that any investment opportunity
can be drawn by a cash flow diagram,how can we select the best?
Transform all cash flow diagrams intosomething similar for comparison.
Use a Common Interest RateUse Time Value of Money Calculations
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Cash Flows
Discrete: Movement of cash to or from a
project at a specific point in time.
Continuous: Rate of cash moving from orto a project over some period of time.
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Cash Flow Diagram
Financial representation of a project.
Describes type, magnitude and timing ofcash flows over some horizon.
Can describe any investment opportunity. Typical investment:
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Cash Flow Diagram
Describes type, magnitude and timing of
cash flows over some horizon
0 1 2 3 4 5
500K
200K
50K100K
200K
500K
200K
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Cash Flow Diagram
Continuous cash flows define a rate of
movement of cash over time.
While good for analysis, not used often.
0 1 2 3 4 5
500K
200K200K
500K
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Cash Flow Diagram
Net Cash Flow Diagram
0 1 2 10
490M
3
98.0M 97.2M 96.4M 113M
9
89.5M
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Interest
Interest Rate comprised of many factors
Example: Home Mortgage: 7.5%
Prime Rate : (Banks borrow money at this rate fromthe Federal Reserve banks when needed) 5%
Risk Factor : 1%
Administration Fees : 0.5%
Profit : 1%
May inflate more for higher risk client.
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Interest
Cost of MoneyRental amount charged by lender for use of money
In any transaction, someone earns and someonepays interest
Savings Account: bank pays you;1.5% fee to depositor
Home/Auto Loan: borrower pays bank;7.5% fee to bank
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Copyright 2009 by Pearson Education, Inc.
Upper Saddle River, New Jersey 07458
All rights reserved.
Engineering Economy, Fourteenth Edition
By William G. Sullivan, Elin M. Wicks, and C. Patrick Koelling
Simple Interest: infrequently used
When the total interest earned or charged is linearlyproportional to the initial amount of the loan
(principal), the interest rate, and the number of
interest periods, the interest and interest rate are said
to be simple.
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Copyright 2009 by Pearson Education, Inc.
Upper Saddle River, New Jersey 07458
All rights reserved.
Engineering Economy, Fourteenth Edition
By William G. Sullivan, Elin M. Wicks, and C. Patrick Koelling
Computation of simple interestThe total interest, I, earned or paid may be computed
using the formula below.
P = principal amount lent or borrowed
N= number of interest periods (e.g., years)
i = interest rate per interest period
The total amount repaid at the end ofNinterest
periods is P +I.
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Copyright 2009 by Pearson Education, Inc.
Upper Saddle River, New Jersey 07458
All rights reserved.
Engineering Economy, Fourteenth Edition
By William G. Sullivan, Elin M. Wicks, and C. Patrick Koelling
If $5,000 were loaned for five years at asimple interest rate of 7% per year, the
interest earned would be
So, the total amount repaid at the end
of five years would be the originalamount ($5,000) plus the interest
($1,750), or $6,750.
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Copyright 2009 by Pearson Education, Inc.
Upper Saddle River, New Jersey 07458
All rights reserved.
Engineering Economy, Fourteenth Edition
By William G. Sullivan, Elin M. Wicks, and C. Patrick Koelling
Compound interest reflects both the remaining principal
and any accumulated interest. For $1,000 at 10%
Period
(1)
Amount owedat beginning of
period
(2)=(1)x10%
Interestamount for
period
(3)=(1)+(2)
Amountowed at end
of period
1 $1,000 $100 $1,100
2 $1,100 $110 $1,210
3 $1,210 $121 $1,331
Compound interest is commonly used in personal and
professional financial transactions.
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Copyright 2009 by Pearson Education, Inc.
Upper Saddle River, New Jersey 07458
All rights reserved.
Engineering Economy, Fourteenth Edition
By William G. Sullivan, Elin M. Wicks, and C. Patrick Koelling
Figure 4-1
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Copyright 2009 by Pearson Education, Inc.
Upper Saddle River, New Jersey 07458
All rights reserved.
Engineering Economy, Fourteenth Edition
By William G. Sullivan, Elin M. Wicks, and C. Patrick Koelling
Economic equivalence allows us to
compare alternatives on a common basis.
Each alternative can be reduced to anequivalent basis dependent on
interest rate,
amount of money involved, andtiming of monetary receipts or expenses.
Using these elements we can move cashflows so that we can compare them at
particular points in time.
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Copyright 2009 by Pearson Education, Inc.
Upper Saddle River, New Jersey 07458
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Engineering Economy, Fourteenth Edition
By William G. Sullivan, Elin M. Wicks, and C. Patrick Koelling
We need some tools to find economic
equivalence. Notation used in formulas for compound interestcalculations.
i = effective interest rate per interest periodN = number of compounding (interest) periods
P = present sum of money; equivalentvalue of one or
more cash flows at a reference point in time; the present F = future sum of money; equivalentvalue of one or
more cash flows at a reference point in time; the future
A = end-of-period cash flows in a uniform seriescontinuing for a certain number of periods, starting atthe end of the first period and continuing through thelast
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Copyright 2009 by Pearson Education, Inc.
Upper Saddle River, New Jersey 07458
All rights reserved.
Engineering Economy, Fourteenth Edition
By William G. Sullivan, Elin M. Wicks, and C. Patrick Koelling
Figure 4-5
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Copyright 2009 by Pearson Education, Inc.
Upper Saddle River, New Jersey 07458
All rights reserved.
Engineering Economy, Fourteenth Edition
By William G. Sullivan, Elin M. Wicks, and C. Patrick Koelling
Figure 4-6
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Copyright 2009 by Pearson Education, Inc.
Upper Saddle River, New Jersey 07458
All rights reserved.
Engineering Economy, Fourteenth Edition
By William G. Sullivan, Elin M. Wicks, and C. Patrick Koelling
Cash flow tables are essential to
modeling engineering economy
problems in a spreadsheet
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Copyright 2009 by Pearson Education, Inc.
Upper Saddle River, New Jersey 07458
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Engineering Economy, Fourteenth Edition
By William G. Sullivan, Elin M. Wicks, and C. Patrick Koelling
We can apply compound interest
formulas to move cash flows along thecash flow diagram.
Using the standard notation, we find that apresent amount, P, can grow into a future
amount, F, inNtime periods at interest rate
i according to the formula below.
In a similar way we can find P given Fby
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Copyright 2009 by Pearson Education, Inc.
Upper Saddle River, New Jersey 07458
All rights reserved.
Engineering Economy, Fourteenth Edition
By William G. Sullivan, Elin M. Wicks, and C. Patrick Koelling
It is common to use standard notation for
interest factors.
This is also known as the single payment
compound amountfactor. The term on the
right is read Fgiven P at i% interest perperiod forNinterest periods.
is called the single payment present worth
factor.
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Copyright 2009 by Pearson Education, Inc.
Upper Saddle River, New Jersey 07458
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Engineering Economy, Fourteenth Edition
By William G. Sullivan, Elin M. Wicks, and C. Patrick Koelling
We can use these to find economically
equivalent values at different points in time.
$2,500 at time zero is equivalent to how much after sixyears if the interest rate is 8% per year?
$3,000 at the end of year seven is equivalent to how
much today (time zero) if the interest rate is 6% peryear?
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Copyright 2009 by Pearson Education, Inc.
Upper Saddle River, New Jersey 07458
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Engineering Economy, Fourteenth Edition
By William G. Sullivan, Elin M. Wicks, and C. Patrick Koelling
There are interest factors for a series of
end-of-period cash flows.
How much will you have in 40 years if you
save $3,000 each year and your account
earns 8% interest each year?
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Copyright 2009 by Pearson Education, Inc.
Upper Saddle River, New Jersey 07458
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Engineering Economy, Fourteenth Edition
By William G. Sullivan, Elin M. Wicks, and C. Patrick Koelling
Finding the present amount from a series
of end-of-period cash flows.
How much would is needed today to provide
an annual amount of $50,000 each year for 20
years, at 9% interest each year?
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Copyright 2009 by Pearson Education, Inc.
Upper Saddle River, New Jersey 07458
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Engineering Economy, Fourteenth Edition
By William G. Sullivan, Elin M. Wicks, and C. Patrick Koelling
Finding A when given F.
How much would you need to set aside eachyear for 25 years, at 10% interest, to have
accumulated $1,000,000 at the end of the 25
years?
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Copyright 2009 by Pearson Education, Inc.
Upper Saddle River, New Jersey 07458
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Engineering Economy, Fourteenth Edition
By William G. Sullivan, Elin M. Wicks, and C. Patrick Koelling
Finding A when given P.
If you had $500,000 today in an accountearning 10% each year, how much could you
withdraw each year for 25 years?
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Copyright 2009 by Pearson Education, Inc.
Upper Saddle River, New Jersey 07458
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Engineering Economy, Fourteenth Edition
By William G. Sullivan, Elin M. Wicks, and C. Patrick Koelling
It can be challenging to solve forNor i.
We may know P,A, and i and want to findN.
We may know P,A, andNand want to find
i.
These problems present special challenges
that are best handled on a spreadsheet.
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Copyright 2009 by Pearson Education, Inc.
Upper Saddle River, New Jersey 07458
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Engineering Economy, Fourteenth Edition
By William G. Sullivan, Elin M. Wicks, and C. Patrick Koelling
FindingN
Acme borrowed $100,000 from a local bank, whichcharges them an interest rate of 7% per year. If Acme
pays the bank $8,000 per year, now many years will it
take to pay off the loan?
So,
This can be solved by using the interest tables and
interpolation, but we generally resort to a computer
solution.
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Copyright 2009 by Pearson Education, Inc.
Upper Saddle River, New Jersey 07458
All rights reserved.
Engineering Economy, Fourteenth Edition
By William G. Sullivan, Elin M. Wicks, and C. Patrick Koelling
Finding i
Jill invested $1,000 each year for five years in a local
company and sold her interest after five years for
$8,000. What annual rate of return did Jill earn?
So,
Again, this can be solved using the interest tables
and interpolation, but we generally resort to a
computer solution.
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Copyright 2009 by Pearson Education, Inc.
Upper Saddle River, New Jersey 07458
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Engineering Economy, Fourteenth Edition
By William G. Sullivan, Elin M. Wicks, and C. Patrick Koelling
There are specific spreadsheet functions
to findNand i.The Excel function used to solve forNis
NPER(rate,pmt,pv), which will compute thenumber of payments of magnitudepmtrequired to
pay off a present amount (pv) at a fixed interest
rate (rate).One Excel function used to solve for i is
RATE(nper,pmt,pv,fv), which returns a fixedinterest rate for an annuity ofpmtthat lasts for nper
periods to either its present value (pv) or future value
(fv).
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Copyright 2009 by Pearson Education, Inc.
Upper Saddle River, New Jersey 07458
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Engineering Economy, Fourteenth Edition
By William G. Sullivan, Elin M. Wicks, and C. Patrick Koelling
We need to be able to handle
cash flows that do not occur untilsome time in the future.
Deferred annuities are uniform series that
do not begin until some time in the future. If the annuity is deferredJperiods then the
first payment (cash flow) begins at the end
of periodJ+1.
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Copyright 2009 by Pearson Education, Inc.
Upper Saddle River, New Jersey 07458
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Engineering Economy, Fourteenth Edition
By William G. Sullivan, Elin M. Wicks, and C. Patrick Koelling
Finding the value at time 0 of a
deferred annuity is a two-stepprocess.
1. Use (P/A, i%,N-J) find the value of thedeferred annuity at the end of periodJ
(where there areN-Jcash flows in theannuity).
2. Use (P/F, i%,J) to find the value of the
deferred annuity at time zero.
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Copyright 2009 by Pearson Education, Inc.
Upper Saddle River, New Jersey 07458
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Engineering Economy, Fourteenth Edition
By William G. Sullivan, Elin M. Wicks, and C. Patrick Koelling
Sometimes cash flows change by a
constant amount each period.We can model these situations as a uniform
gradientof cash flows. The table belowshows such a gradient.
End of Period Cash Flows1 0
2 G
3 2G
: :
N (N-1)G
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Copyright 2009 by Pearson Education, Inc.
Upper Saddle River, New Jersey 07458
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Engineering Economy, Fourteenth Edition
By William G. Sullivan, Elin M. Wicks, and C. Patrick Koelling
It is easy to find the present value
of a uniform gradient series.
Similar to the other types of cash flows, there is aformula (albeit quite complicated) we can use to find
the present value, and a set of factors developed for
interest tables.
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Copyright 2009 by Pearson Education, Inc.
Upper Saddle River, New Jersey 07458
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Engineering Economy, Fourteenth Edition
By William G. Sullivan, Elin M. Wicks, and C. Patrick Koelling
We can also findA or F
equivalent to a uniform gradient
series.
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Copyright 2009 by Pearson Education, Inc.
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Engineering Economy, Fourteenth Edition
By William G. Sullivan, Elin M. Wicks, and C. Patrick Koelling
End of Year Cash Flows ($)
1 2,0002 3,000
3 4,000
4 5,000
The annual equivalent of
this series of cash flows can
be found by considering an
annuity portion of the cash
flows and a gradient
portion.
End of Year Annuity ($) Gradient ($)
1 2,000 02 2,000 1,000
3 2,000 2,000
4 2,000 3,000
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Copyright 2009 by Pearson Education, Inc.
Upper Saddle River, New Jersey 07458
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Engineering Economy, Fourteenth Edition
By William G. Sullivan, Elin M. Wicks, and C. Patrick Koelling
Sometimes cash flows change bya constant rate, ,each period--this
is a geometric gradient series.
End of Year Cash Flows ($)
1 1,000
2 1,200
3 1,440
4 1,728
This table presents ageometric gradient series. It
begins at the end of year 1
and has a rate of growth, ,of 20%.
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Copyright 2009 by Pearson Education, Inc.
Upper Saddle River, New Jersey 07458
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Engineering Economy, Fourteenth Edition
By William G. Sullivan, Elin M. Wicks, and C. Patrick Koelling
We can find the present value of a
geometric series by using the appropriateformula below.
Where is the initial cash flow in the series.
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Copyright 2009 by Pearson Education, Inc.
Upper Saddle River, New Jersey 07458
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Engineering Economy, Fourteenth Edition
By William G. Sullivan, Elin M. Wicks, and C. Patrick Koelling
When interest rates vary with
time different procedures are
necessary. Interest rates often change with time (e.g., a
variable rate mortgage). We often must resort to moving cash flows
one period at a time, reflecting the interest
rate for that single period.
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Copyright 2009 by Pearson Education, Inc.
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Engineering Economy, Fourteenth Edition
By William G. Sullivan, Elin M. Wicks, and C. Patrick Koelling
The present equivalent of a cash flow occurring at
the end of periodNcan be computed with theequation below, where ikis the interest rate for the
kthperiod.
If F4 = $2,500 and i1=8%, i2=10%, and i3=11%, then
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Copyright 2009 by Pearson Education, Inc.
Upper Saddle River, New Jersey 07458
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Engineering Economy, Fourteenth Edition
By William G. Sullivan, Elin M. Wicks, and C. Patrick Koelling
Nominal and effective interest rates.
More often than not, the time between successive
compounding, or the interest period, is less than
one year (e.g., daily, monthly, quarterly). The annual rate is known as a nominal rate.
A nominal rate of 12%, compounded monthly,means an interest of 1% (12%/12) would accrue
each month, and the annual rate would be
effectively somewhat greater than 12%. The more frequent the compounding the greater
the effective interest.
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Copyright 2009 by Pearson Education, Inc.
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Engineering Economy, Fourteenth Edition
By William G. Sullivan, Elin M. Wicks, and C. Patrick Koelling
The effect of more frequent
compounding can be easilydetermined.
Let rbe the nominal, annual interest rate andMthe
number of compounding periods per year. We can
find, i, the effective interest by using the formulabelow.
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Copyright 2009 by Pearson Education, Inc.
Upper Saddle River, New Jersey 07458
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Engineering Economy, Fourteenth Edition
By William G. Sullivan, Elin M. Wicks, and C. Patrick Koelling
Finding effective interest rates.For an 18% nominal rate, compounded quarterly, the
effective interest is.
For a 7% nominal rate, compounded monthly, the
effective interest is.
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Engineering Economy, Fourteenth Edition
By William G. Sullivan, Elin M. Wicks, and C. Patrick Koelling
Effects of Compounding
18%Compoundedlessthen1yearn Ratepern Effectivei PV FV5years1 0.18000 18.000% $ 10,000 $22,877.58
2 0.09000 18.810% $ 10,000 $23,673.64
4 0.04500 19.252% $ 10,000 $24,117.14
12 0.01500 19.562% $ 10,000 $24,432.20
360 0.00050 19.716% $ 10,000 $24,590.50
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Copyright 2009 by Pearson Education, Inc.
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Engineering Economy, Fourteenth Edition
By William G. Sullivan, Elin M. Wicks, and C. Patrick Koelling
Figure 4-18
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Copyright 2009 by Pearson Education, Inc.
Upper Saddle River, New Jersey 07458
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Engineering Economy, Fourteenth Edition
By William G. Sullivan, Elin M. Wicks, and C. Patrick Koelling
Interest can be compounded
continuously. Interest is typically compounded at the end
of discrete periods. In most companies cash is always flowing,
and should be immediately put to use. We can allow compounding to occur
continuously throughout the period.
The effect of this compared to discrete
compounding is small in most cases.
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Copyright 2009 by Pearson Education, Inc.
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Engineering Economy, Fourteenth Edition
By William G. Sullivan, Elin M. Wicks, and C. Patrick Koelling
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Copyright 2009 by Pearson Education, Inc.
Upper Saddle River, New Jersey 07458
All rights reserved.
Engineering Economy, Fourteenth Edition
By William G. Sullivan, Elin M. Wicks, and C. Patrick Koelling
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Copyright 2009 by Pearson Education, Inc.
Upper Saddle River, New Jersey 07458
All rights reserved.
Engineering Economy, Fourteenth Edition
By William G. Sullivan, Elin M. Wicks, and C. Patrick Koelling
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Copyright 2009 by Pearson Education, Inc.
Upper Saddle River, New Jersey 07458
All rights reserved.
Engineering Economy, Fourteenth Edition
By William G. Sullivan, Elin M. Wicks, and C. Patrick Koelling
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Copyright 2009 by Pearson Education, Inc.
Upper Saddle River, New Jersey 07458
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Engineering Economy, Fourteenth Edition
By William G. Sullivan, Elin M. Wicks, and C. Patrick Koelling
InterpolationFind 6.5% for 5 years from
the table.Find the incremental
difference and add the
appropriate amount tothe low number
Or
For 6.5 add the factor for 7and 6 then divide by 2!
(.7473 +.713)/2 = .73015
% N Factor
6% 5 0.7473
7% 5 0.713
Subtract7%Factorfrom6%factor0.0343
Divideby10 0.00343
Multiplyby5 0.01715
AddtothelowerFactorfor6%=6.5% 0.73015
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Engineering Economy, Fourteenth Edition
By William G. Sullivan, Elin M. Wicks, and C. Patrick Koelling
Interpolation
What about 6.9%?
% N Factor
6% 5 0.7473
7% 5 0.713
Subtract7%Factorfrom6%factor0.0343
Divideby10 0.00343
Multiplyby9 0.01372
AddtothelowerFactorfor6%=6.9% 0.73358
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Copyright 2009 by Pearson Education, Inc.E i i E F t th Editi
Names
Engineering Economy
Present Worth (PW)
Future Worth (FW)
Annual Worth (AW)
Internal Rate of Return (IRR)
Minimum Acceptable Rate of
Return (MARR)
Finance
Net Present Value (NPV) Present Value (PV)
Future Value (FV)
Annuity (A)
Internal Rate of Return (IRR) Hurdle Rate
Discount Rate