Chapter 01 - Engineering Economics 04-10-2010
Transcript of Chapter 01 - Engineering Economics 04-10-2010
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Interest Formula Derivations
I Makalanda
MSc(Fin. Maths), BSc (Eng.), DPM(UK), ACIM
Chapter 01
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Simple & Compound Interest
Simple interest
The interest owed upon repayment of a loan isproportional to the length of time the principal
sum has been borrowed. Let I represent the interest earned, P the
principal amount, n the interest period, and Ithe interest rate.
PniI !
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Example
Suppose that LKR 1,000 is borrowed at a
simple interest rate of 18% per annum. At the
end of one (01) year, the interest owed would
be,
180
18.011000
!
vv!
!
I
I
niI
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Describing cash flows over time
To aid in identifying & recording the economic
effects of investment alternatives, a graphical
description of each alternatives cash
transactions may be used.
This graphical descriptor, referred to as a cash
flow diagram, will provide the information
necessary for analyzing an investment
proposal.
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Describing cash flows over time
Borrower Lender
1000
0
1 2 3 4
160 160 160
1160
0
1 2 3 4
1000
160 160 160
1160
Net cash flow is the arithmetic sum of the receipts (+) and the
disbursements (-) that occur at the same point in time.
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Interest formulas (discrete compounding,
discrete payments)
i the annual interest rate
n the number of annual interest periods
P a present principal sum A a single payment, in a series of n equal
payments, made at the end of each annual
interest period. F a future sum, n annual interest period
hence.
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Interest formulas (discrete compounding,
discrete payments)
The end ofone (01) year is the beginning of
the next year.
Pis at the beginning of a year at a timeregarded as being the present.
Fis at the end of the nth year from a time
regarded as being present.
An Aoccurs at the end of each of the period
under consideration.
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Single - payment compound amount
factors
If an amount P is invested now and earns at
the rate i per year, how much principal and
interest are accumulated after n years?
1 2 3 .. n-1 n
P
F
0
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Single - payment compound amount
factors
Year Amount
beginning of year
Interest earned
during year
Compounded amount
at the end of year
1 P Pi P(1+i)1
2 P(1+i) P(1+i)i P(1+i)2
3 P(1+i)2 P(1+i)2i P(1+i)3
n P(1+i)n-1 P(1+i)n-1i P(1+i)n = F
(1+i) is known as the single-paymentcompounded amount
factor and is designated
),,/( niPF
),,/()( niPFPiiPF n !!
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Single - payment compound amount
factors
The equations for solving problems may be setup prior to looking up values of the factorsfrom the tables and inserting them in theparenthesis.
The source and the identity of values takenfrom the tables are maintained during thesolution.
),,/( niPF
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Single-payment present worth factor
1 2 3 .. n-1 n
P ?
F
0
!n
i
FP
1
1
),,/( niFP
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Equal-payment-series compound amount
factor
0 1 2 3 .. n-1 n
F??
A A A A A A
!
i
iAF
n1)1(
),,/( niAF
The resulting factor, [(1+i)n-1]/i, is known as the equal-payment-series
compound-amount factor & is designated
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Equal-payment-series sinking-fund
factor
0 1 2 3 .. n-1 n
F
A? A? A? A? A? A?
!
1)1(n
i
iFA
),,/( niFA
The resulting factor, [(1+i)n-1]/i, is known as the equal-payment-series
sinking-fund factor & is designated
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Equal-payment-series capital recovery
factor
1 2 3 .. n-1 n
P
A?
0
A?A?A?A?A?
!
1)1(
)1(n
n
i
iiPA
),,/( niPA
The resulting factor, i(1+i)n/[(1+i)n-1], is known as the equal-payment-
series sinking-fund factor & is designated
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Equal-payment-series present worth factor
1 2 3 .. n-1 n
P?
A
0
AAAAA
!
n
n
ii
iiAP
)1(
1)1(
),,/( ni
The resulting factor, [(1+i)n-1]/i(1+i)n, is known as the equal-payment-series sinking-fund factor & is designated
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Example 01
What interest rate is required to triple 1000/
in 10 years?
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Answer for Example 01
%6.11
11612.1)10986.0exp(%)1(
%)1ln(10986.0
%)1ln(100986.1
%)1ln(10)3ln(
%)1.(10003000
%)1(
10
!
!!
!
!
!
!
!
i
i
i
i
i
i
iPFn
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Or
11.6%i
ion,interpolatlinearBy
3220.0)10%,12,/(
rate,interest12%atSimilarly
3855.0)10%,10,/(
10%atthatfindcanwetablesUsing
10%iAssume,
3
1)10,,/(
),,/(
!
!
!
!
!
!
FP
FP
iFP
NiFPPF
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What annual year-end payment must be
made each year to have 20000 available 5
years from now? The compound annualinterest rate is 6%.
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A A A A A
F=20000
1 2 3 4 5
DR 6%
Year 0 1 2 3 4
DR 1 1.06 1.1236 1.1910161.262477
Payments A A A A A
FV 20000
3548
)63709.5(20000
262477.1191016.11236.106.1
!
!
vvvv!
A
A
AAAAAFV
.3548
)1774.0(20000
)5%,6,/(
!
!
!
A
A
FAFA
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Example
A machine needed for 3 years can be
purchased for 77662 and sold at the end of
the period for about 25,000. A comparable
machine can be leased for 30000 per year. If
the firm expect a return of 20% on their
investment, should it buy or lease the
machine?
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A A A
S S S
77662
25000
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For a New Product Development Project you are required capital of
LKR 5,000,000.00. Prepare the loan amortization plan showing the
yearly loan commitment using the following criterion.
1 Repayment period - 8 years
2 Installments per year 2
3 Interest charge 9% per 6 months
Clearly state assumptions you made as foot notes. ( 10 marks)
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