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Unformatted text preview: Engineering Economic Analysis — Newnan, Lavelle, and Eschenbach Chapter 3
Interest and Equivalence A. Interest is paid to the supplier of capital for the use of money. The interest rate, i, is established
based on the risk the supplier takes in making an investment. B. Simple Interest is used for bank loans, mortgage loans, coupon and registered bonds and other
investments where the interest is paid out and not reinvested at the end of each payment period. C. Compound Interest is used for most other investments than those listed in B. Any interest not paid
out at the end of a payment period is added to the capital investment (principal) to earn interest
during the succeeding period. D. Cash Flow (vs. time) Diagrams: Sign convention  Revenues are generally positive and costs are
generally negative in sign. Sometimes both revenues and expenditures are shown and sometimes the
yearly or monthly cash ﬂows are netted in order to calculate internal rates of return as discussed in
Chapter 7. Ifonly equivalence is being demonstrated, the cash ﬂows would all be of the same sign.
This is implied in the equations in the following lesson. E. Cash Flow Equivalence is illustrated in Tables 3.1 and 3.2 using various payment schedules for
repaying a $5,000 loan in ﬁve years with interest at 8%. The cash ﬂow diagrams are from the
borrower's point of View. 5,000
Plan 1:
Pay an amount equal to the
principal/n plus the interest due at
the end of each payment period sim le interest . ' 5
( p
0
Plan 1 is typical of bank loans.
1400 132° 124° 1160 1080
5,000 Plan 2: Pay interest due at the end of each
payment period and the principal at
the end of the loan term (simple
interest). Plan 2 is typical for coupon bonds, 400/year
registered bonds, international loans. 5,000 Plan 3: Pay equal amounts at the end of
each payment period for the term
of the loan (simple interest). Plan 3 is typical of house and
auto loans when the payment
periods are stated in months. Plan 4: Borrower pays principal and
accumulated interest in one
payment at the end of the loan
term (compound interest). Plan 4 is typical of bank
Certiﬁcates of Deposit (CD'S) and
IRA‘s. Equivalence 5,000 1 ,252/year 5.000 7.347 When we are indifferent as to whether we have a quantity of money now or the assurance of some
other sum of money in the future, or series of ﬁJture sums of money, we say that the present sum of
money is equivalent to the future sum or series of ﬁiture sums. If a ﬁrm believed 8% was an appropriate interest rate, it would have no particular preference
whether it received $5000 now or was repaid by Plan 1 of Table 31. Thus $5000 today is equivalent
to the series of ﬁve end ofyear payments. In the same fashion, the industrial ﬁrm would accept
repayment Plan 2 as equivalent to $5000 now. Logic tells us that if Plan 1 is equivalent to $5000
now and Plan 2 is also equivalent to $5000 now, it must follow that Plan 1 is equivalent to Plan 2. In
fact, all four repayment plans must be equivalent to each other and to $5 000 now. G. Single Payment Compound Interest Formulas
(see inside front cover and yellow section tables)
Note: All the formulas calculate absolute values based on equivalence. The sign convention
used in the cash ﬂow diagram depends on the viewpoint of the borrower or lender of money. 1. Single Pament Compound Amount Factor F
(Compound interest) (F/P,i,n) = (1+i)n = F/P
thus F = P(F/P,i,n) 2. Single Payment Present Worth Factor n
(Compound interest) (P/F,i,n) = 1/(1+i)" = P/F Notation: i = interest rate per payment period
n = number of payment periods
P = present value of a sum of money
Pa = Future value of a sum of money in year 11
A = Uniform endof—period payment in a uniform series of 11 payments , ﬁvemrlalples mustheiknown to solveﬁmevalue of money
‘4 ' " ‘ I‘ Engineering Economic Analysis  Newnan, Lavelle, and Eschenbach Chapter 4
More Interest Formulas A. Uniform Series Formulas
Conventions for uniform series payments:
a. A occurs at the end of each period.
b. P occurs one payment period before the ﬁrst A.
c. F occurs at the same time as the last A, and N periods aﬁer P. 1. Uniform Series Compound
Amount Factor (Compound interest) (F/A,i,n) = [(1+i)“ 1]/i
= F/A 2. Uniform Series Sinking Fund
Factor (Compound interest)
(A/F,i,n) = i/[(1+i)“'—1] = NF 3. Uniform Series Capital Recoveﬂ Factor
(Simple interest) (A/P,i,n) = [i(1+i)“]/[(1+i)“ 1]
= A/P =
A/F*F/P 4. Uniform Series Present Worth Factor
(Simple interest) (P/A,i,n) = [(1+i)n —1]/[i(l+i)“] = P/A
Note that: (A/P,i,n) = (A/F,i,n) B. Deferred Annuities (Uniform Series)  If the number of annuity payments is less than the
number of analysis periods, the equivalent worth at the beginning or end of the annuity can
be calculated and that value moved to a different location on the cash ﬂow diagram. For example, to ﬁnd P0
given A at the end of years 59, P4 = A(P/A,i,5) = F4 P0 = F4(P/F,i,4) = A(P/A,i,5)(P/F,i,4) 1’4 C. Situations where N is unknown in uniform series calculations
Rearrange the equation to the form (1+i) = ( ).
Take the logarithm of both sides and solve for n. D. Situations where i is unknown in uniform series calculations
To determine i, given P, A, and n, (or F, A, and n), rearrange the appropriate equation to
the following forms and iterate. i= [A/P][((1+i)n  1)/(l+i)“] i= (Fi/A + 1)“n  I
assume an i and iterate assume an i and iterate 0 The spreadsheet ﬁnancial function @IRATE(Term,Pmt,PV) solves this situation
efﬁciently. E. Arithmetic Gradient Series (n1)G 1. Arithmetic Gradient Present
Worth Factor (P/G,i,n) = [(1+i)“  in  1]/[i2(1+i)“] = P/G 2. Arithmetic Gradient Uniform
Series (A/G,i,n) = [(1+i)“ — in 1]/[i(1+i)“  i]
= A/G note that (P/G,i,n) = [(P/A,i,n) 
n(P/F,i,n)]/i Geometric Gradients (very complex unless programmed on a computer) will not be used
as part of the normal course material, but may be appropriate for some study projects,
especially those involving inﬂation. Geometric gradients are easily handled in spreadsheet
analysis. G. Nominal and Effective Interest Rates Nominal Interest Rate per year, r, is the annual interest rate without considering the
eﬂ‘ect of more ﬁequent compounding. The nominal rate is sometimes called the "stated
rate.” Effective Interest Rate per year, ieff , is the annual interest rate taking into account more
frequent compounding (also called "yield"). Note: The interest rate per payment
subperiod, i or im, is an effective interest rate per subperiod.
ieff = (1+r/m)m 1 m = number of compounding subperiods/year
i = im = r/m = interest rate per compounding subperiod "Annual Percentage Rate" (APR) is a term which the truthinlending laws require banks
and other installment lenders to report to the consumer. In advertising, APR sometimes
refers to nominal interest rates and sometimes refers to eﬁective interest rates, depending
on whose point of view is favored. Table 41, p125 compares nominal and eﬁective interest rates for several compounding
subperiods. Nominal Effective interest rate per year, ia,
interest when nominal rate is compounded
rate per year
r Yearly Semiannually Monthly Daily Continuously
1% 1.0000% 1.0025% 1.0046% 1.0050% 1.0050%
2 2.0000 2.0100 2.0184 2.0201 2.0201
3 3.0000 3.0225 3.0416 3.0453 3.0455
4 4.0000 4.0400 4.0742 4.0809 4.0811
5 5.0000 5.0625 5.1162 5.1268 5.1271
6 6.0000 6.0900 6.1678 6.1831 6.1837
8 8.0000 8.1600 8.3000 8.3278 8.3287
10 10.000 10.2500 10.4713 10.5156 10.5171
15 15.000 15.5625 16.0755 16.1798 16.1834
25 25.000 26.5625 28.0732 28.3916 28.4025 _,__._—_—__.__—_—_——_———————_—— H. Interest problems with uniform cash ﬂows less often than compounding periods. Either
a) convert the uniform cash ﬂow to an equivalent more frequent uniform cash ﬂow, or
b) use the effective interest rate between cash ﬂows as the basis for compounding. I. Interest problems with uniform cash ﬂows occurring more often than the compounding
periods: Use simple interest between compounding periods. J. Continuous Compounding
Continuous compounding formulas are given on pp129—136. They will not be emphasized in
most Engineering Economics courses as daily compounding is generally the most frequent
compounding used by the banking system, i.e., checks and deposits are cleared once per day.
ATM machines could be programmed to compound continuously, but the cash ﬂows are
almost never uniform, so the equations are of little use. ...
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This note was uploaded on 07/29/2011 for the course EGN 3612 taught by Professor Staff during the Fall '09 term at UNF.
 Fall '09
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