engeco chap 04 - the time value of money
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CHAPTER 4CHAPTER 4CHAPTER 4CHAPTER 4
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The term capital refers to financial resources that can be used to produce more
wealth.
It is one of the factors of production (inputs or resources to produce goods orservices)
Factors of production:
Land: comprising all naturally occurring resources; economically referred to asland or raw materials. Examples: agriculture, plants, coal, fossil fuels, rock, minerals, forestry,pasture, soils, water, ocean, lakes, river
Labor: the physical and mental capacity of humans to produce goods and
services
Capital: financial resource to operate and enterprise, man-made goods such asmachines to facilitate further production of goods and services
Entrepreneurship: the creative ability of individuals to seek profits by
combining land, labor, and capital to produce goods and services
Information: includes market forecasts, specialized knowledge of the human
resource, other economic data.
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Majority of the engineering economyMajority of the engineering economy
studies involve commitment of capital forstudies involve commitment of capital forextended periods of time.extended periods of time.
Therefore, the effect of time must beTherefore, the effect of time must beconsidered.considered.
Money changes in value due to changes inMoney changes in value due to changes in
the interest rate, inflation (or deflation),the interest rate, inflation (or deflation),and currency exchange ratesand currency exchange rates
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SIMPLE INTERESTSIMPLE INTERESTSIMPLE INTERESTSIMPLE INTEREST
The total interest earned or charged is linearlyThe total interest earned or charged is linearlyproportional to the initial amount of the loanproportional to the initial amount of the loan(principal), the interest rate and the number of(principal), the interest rate and the number of
interest periods for which the principal isinterest periods for which the principal is
committed.committed.
When applied, total interest I may be foundWhen applied, total interest I may be foundby I = ( P ) ( N ) ( i ), whereby I = ( P ) ( N ) ( i ), where
P = principal amount lent or borrowedP = principal amount lent or borrowed
N = number of interest periods ( e.g., years )N = number of interest periods ( e.g., years )
i = interest rate per interest periodi = interest rate per interest period
The total interest earned or charged is linearlyThe total interest earned or charged is linearly
proportional to the initial amount of the loanproportional to the initial amount of the loan
(principal), the interest rate and the number of(principal), the interest rate and the number of
interest periods for which the principal isinterest periods for which the principal is
committed.committed.
When applied, total interest I may be foundWhen applied, total interest I may be foundby I = ( P ) ( N ) ( i ), whereby I = ( P ) ( N ) ( i ), where
P = principal amount lent or borrowedP = principal amount lent or borrowed
N = number of interest periods ( e.g., years )N = number of interest periods ( e.g., years )
i = interest rate per interest periodi = interest rate per interest period
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COMPOUND INTERESTCOMPOUND INTERESTCOMPOUND INTERESTCOMPOUND INTEREST
Whenever the interest charge for any interestWhenever the interest charge for any interest
period is based on the remaining principalperiod is based on the remaining principalamount plus any accumulated interest chargesamount plus any accumulated interest charges
up to theup to the beginningbeginningof that period.of that period.
PeriodPeriod Amount OwedAmount Owed Interest AmountInterest Amount Amount OwedAmount OwedBeginning ofBeginning of for Periodfor Period at end ofat end of
periodperiod ( @ 10% )( @ 10% ) periodperiod
(1)(1) (2) = (1) * 10%(2) = (1) * 10% (3) = (1) + (2)(3) = (1) + (2)
11 $1,000$1,000 $100$100 $1,100$1,10022 $1,100$1,100 $110$110 $1,210$1,210
33 $1,210$1,210 $121$121 $1,331$1,331
Whenever the interest charge for any interestWhenever the interest charge for any interest
period is based on the remaining principalperiod is based on the remaining principalamount plus any accumulated interest chargesamount plus any accumulated interest charges
up to theup to the beginningbeginningof that period.of that period.
PeriodPeriod Amount OwedAmount Owed Interest AmountInterest Amount Amount OwedAmount OwedBeginning ofBeginning of for Periodfor Period at end ofat end of
periodperiod ( @ 10% )( @ 10% ) periodperiod
(1)(1) (2) = (1) * 10%(2) = (1) * 10% (3) = (1) + (2)(3) = (1) + (2)
11 $1,000$1,000 $100$100 $1,100$1,10022 $1,100$1,100 $110$110 $1,210$1,210
33 $1,210$1,210 $121$121 $1,331$1,331
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ECONOMIC EQUIVALENCEECONOMIC EQUIVALENCEECONOMIC EQUIVALENCEECONOMIC EQUIVALENCE
The question:The question: How to compare the various alternatives (proposalsHow to compare the various alternatives (proposals
or options) involve in the economic analysis?or options) involve in the economic analysis?
Considers the comparison of alternative options, or proposals, byConsiders the comparison of alternative options, or proposals, byreducing them to an equivalent basis, depending on:reducing them to an equivalent basis, depending on:
interest rate;interest rate;
amounts of money involved;amounts of money involved;
timing of the affected monetary receipts and/or expenditures;timing of the affected monetary receipts and/or expenditures;
manner in which the interest , or profit on invested capital is paidmanner in which the interest , or profit on invested capital is paid
and the initial capital is recovered.and the initial capital is recovered.
Established when we areEstablished when we are indifferentindifferent between a future payment, or abetween a future payment, or aseries of future payments, and a present sum of money.series of future payments, and a present sum of money.
The question:The question: How to compare the various alternatives (proposalsHow to compare the various alternatives (proposals
or options) involve in the economic analysis?or options) involve in the economic analysis?
Considers the comparison of alternative options, or proposals, byConsiders the comparison of alternative options, or proposals, byreducing them to an equivalent basis, depending on:reducing them to an equivalent basis, depending on:
interest rate;interest rate; amounts of money involved;amounts of money involved;
timing of the affected monetary receipts and/or expenditures;timing of the affected monetary receipts and/or expenditures;
manner in which the interest , or profit on invested capital is paidmanner in which the interest , or profit on invested capital is paid
and the initial capital is recovered.and the initial capital is recovered.
Established when we areEstablished when we are indifferentindifferent between a future payment, or abetween a future payment, or aseries of future payments, and a present sum of money.series of future payments, and a present sum of money.
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CASH FLOW DIAGRAMS / TABLE NOTATIONCASH FLOW DIAGRAMS / TABLE NOTATIONCASH FLOW DIAGRAMS / TABLE NOTATIONCASH FLOW DIAGRAMS / TABLE NOTATION
Cash flow refers to the movement of cash into or out of a business orCash flow refers to the movement of cash into or out of a business ora project. It is usually measured during a specified, finite period ofa project. It is usually measured during a specified, finite period oftime.time.
A cash flow diagram is a tool used by accountants and engineers toA cash flow diagram is a tool used by accountants and engineers torepresent the transactions which will take place over the course of arepresent the transactions which will take place over the course of agiven project.given project.
i = effective interest rate per interest periodi = effective interest rate per interest period
N = number of compounding periods (e.g., years)N = number of compounding periods (e.g., years)
P = present sum of money; the equivalent value of one or more cash flows at theP = present sum of money; the equivalent value of one or more cash flows at thepresent time reference pointpresent time reference point
F = future sum of money; the equivalent value of one or more cash flows at aF = future sum of money; the equivalent value of one or more cash flows at afuture time reference pointfuture time reference point
A = end-of-period cash flows (or equivalent end-of-period values ) in a uniformA = end-of-period cash flows (or equivalent end-of-period values ) in a uniformseries continuing for a specified number of periods, starting at the end of theseries continuing for a specified number of periods, starting at the end of thefirst period and continuing through the last periodfirst period and continuing through the last period
G = uniform gradient amounts --- used if cash flows increase by a constantG = uniform gradient amounts --- used if cash flows increase by a constantamount in each periodamount in each period
Cash flow refers to the movement of cash into or out of a business orCash flow refers to the movement of cash into or out of a business ora project. It is usually measured during a specified, finite period ofa project. It is usually measured during a specified, finite period of
time.time.
A cash flow diagram is a tool used by accountants and engineers toA cash flow diagram is a tool used by accountants and engineers torepresent the transactions which will take place over the course of arepresent the transactions which will take place over the course of agiven project.given project.
i = effective interest rate per interest periodi = effective interest rate per interest periodN = number of compounding periods (e.g., years)N = number of compounding periods (e.g., years)
P = present sum of money; the equivalent value of one or more cash flows at theP = present sum of money; the equivalent value of one or more cash flows at thepresent time reference pointpresent time reference point
F = future sum of money; the equivalent value of one or more cash flows at aF = future sum of money; the equivalent value of one or more cash flows at afuture time reference pointfuture time reference point
A = end-of-period cash flows (or equivalent end-of-period values ) in a uniformA = end-of-period cash flows (or equivalent end-of-period values ) in a uniformseries continuing for a specified number of periods, starting at the end of theseries continuing for a specified number of periods, starting at the end of thefirst period and continuing through the last periodfirst period and continuing through the last period
G = uniform gradient amounts --- used if cash flows increase by a constantG = uniform gradient amounts --- used if cash flows increase by a constantamount in each periodamount in each period
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CASH FLOW DIAGRAM NOTATION
1 2 3 4 5 = N1
1 Time scale with progression of time moving from left toright; the numbers represent time periods (e.g., years,months, quarters, etc...) and may be presented within atime interval or at the end of a time interval.
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CASH FLOW DIAGRAM NOTATION
1 2 3 4 5 = N1
1 Time scale with progression of time moving from left toright; the numbers represent time periods (e.g., years,months, quarters, etc...) and may be presented within atime interval or at the end of a time interval.
P =$8,000 2
2 Present expense (cash outflow) of $8,000 for lender.
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CASH FLOW DIAGRAM NOTATION
1 2 3 4 5 = N1
1 Time scale with progression of time moving from left toright; the numbers represent time periods (e.g., years,months, quarters, etc...) and may be presented within atime interval or at the end of a time interval.
P =$8,000 2
2 Present expense (cash outflow) of $8,000 for lender.
A = $2,524 3
3 Annual income (cash inflow) of $2,524 for lender.
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CASH FLOW DIAGRAM
1 2 3 4 5 = N
1
1Time scale with progression of time moving from left to right; thenumbers represent time periods (e.g., years, months, quarters, etc...)and may be presented within a time interval or at the end of a timeinterval.
P =$8,000 2
2 P = Present investment (cash outflow) of $10,000
A = $3,5003
3 A = Annual revenue (cash inflow) of $3,500
i = 10% per year4
4 i = Rate of return
0
5 N = number of compounding period
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CASH FLOW DIAGRAM NOTATION
1 2 3 4 5 = N1
1 Time scale with progression of time moving from left toright; the numbers represent time periods (e.g., years,months, quarters, etc...) and may be presented within atime interval or at the end of a time interval.
P =$8,000 2
2 Present expense (cash outflow) of $8,000 for lender.
A = $2,524 3
3 Annual income (cash inflow) of $2,524 for lender.
i = 10% per year4
4 Interest rate of loan.
5
5 Dashed-arrow line indicatesamount to be determined.
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RELATING PRESENT AND FUTURE EQUIVALENTRELATING PRESENT AND FUTURE EQUIVALENT
VALUES OFVALUES OF SINGLE CASH FLOWSSINGLE CASH FLOWS
RELATING PRESENT AND FUTURE EQUIVALENTRELATING PRESENT AND FUTURE EQUIVALENT
VALUES OFVALUES OF SINGLE CASH FLOWSSINGLE CASH FLOWS
Finding F when given P:Finding F when given P:
Finding future value when given present valueFinding future value when given present value
F = P ( 1+i )F = P ( 1+i ) NN (1+i)(1+i)NN single payment compound amount factorsingle payment compound amount factor
functionally expressed as F = ( F / P, i%,N )functionally expressed as F = ( F / P, i%,N )
predetermined values of this are presented inpredetermined values of this are presented incolumn 2 of Appendix C of text.column 2 of Appendix C of text.
Finding F when given P:Finding F when given P:
Finding future value when given present valueFinding future value when given present value
F = P ( 1+i )F = P ( 1+i ) NN
(1+i)(1+i)NN single payment compound amount factorsingle payment compound amount factor
functionally expressed as F = ( F / P, i%,N )functionally expressed as F = ( F / P, i%,N )
predetermined values of this are presented inpredetermined values of this are presented incolumn 2 of Appendix C of text.column 2 of Appendix C of text.
P
0
N =
F = ?
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Suppose you borrow $8,000 now, and promiseSuppose you borrow $8,000 now, and promise
to pay back the loan principal plus accumulatedto pay back the loan principal plus accumulated
interest in four years at interest rate of 10% perinterest in four years at interest rate of 10% per
year. How much would you repay at the end ofyear. How much would you repay at the end of
the fourth year?the fourth year?
P = $8,000
F = ?0 4
i= 10% per year
RELATING PRESENT AND FUTURE
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Finding P when given F:Finding P when given F:
Finding present value when given future valueFinding present value when given future value
P = F [1 / (1 + i ) ]P = F [1 / (1 + i ) ]NN
(1+i)(1+i)-N-N single payment present worth factorsingle payment present worth factor
functionally expressed as P = F ( P / F, i%, N )functionally expressed as P = F ( P / F, i%, N )
predetermined values of this are presented inpredetermined values of this are presented in
column 3 of Appendix C of text;column 3 of Appendix C of text;
Finding P when given F:Finding P when given F:
Finding present value when given future valueFinding present value when given future value
P = F [1 / (1 + i ) ]P = F [1 / (1 + i ) ] NN
(1+i)(1+i)-N-N single payment present worth factorsingle payment present worth factor
functionally expressed as P = F ( P / F, i%, N )functionally expressed as P = F ( P / F, i%, N )
predetermined values of this are presented inpredetermined values of this are presented in
column 3 of Appendix C of text;column 3 of Appendix C of text;
RELATING PRESENT AND FUTUREEQUIVALENT VALUES OF SINGLE CASH
FLOWS
RELATING PRESENT AND FUTUREEQUIVALENT VALUES OF SINGLE CASH
FLOWS
P = ?
0 N =F
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An investor has an option to purchase a plot ofAn investor has an option to purchase a plot of
land that will be worth $25,000 in six years. Ifland that will be worth $25,000 in six years. If
the interest rate is 8% per year, how muchthe interest rate is 8% per year, how muchshould the investor be willing to pay now forshould the investor be willing to pay now for
this property?this property?
P = ?
F = $25,000
60
i= 10% per year
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RELATING A UNIFORM SERIES (ORDINARY ANNUITY)RELATING A UNIFORM SERIES (ORDINARY ANNUITY)
TO PRESENT AND FUTURE EQUIVALENT VALUESTO PRESENT AND FUTURE EQUIVALENT VALUES
RELATING A UNIFORM SERIES (ORDINARY ANNUITY)RELATING A UNIFORM SERIES (ORDINARY ANNUITY)
TO PRESENT AND FUTURE EQUIVALENT VALUESTO PRESENT AND FUTURE EQUIVALENT VALUES
Finding F given A:Finding F given A: Finding future equivalent income (inflow) value givenFinding future equivalent income (inflow) value given
a series of uniform equal Paymentsa series of uniform equal Payments
( 1 + i )( 1 + i ) NN - 1- 1
F = AF = Aii
uniform series compound amount factor in [ ]uniform series compound amount factor in [ ]
functionally expressed as F = A ( F / A,i%,N )functionally expressed as F = A ( F / A,i%,N ) predetermined values are in column 4 of Appendixpredetermined values are in column 4 of Appendix
C of textC of text
Finding F given A:Finding F given A: Finding future equivalent income (inflow) value givenFinding future equivalent income (inflow) value givena series of uniform equal Paymentsa series of uniform equal Payments
( 1 + i )( 1 + i ) NN - 1- 1
F = AF = Aii
uniform series compound amount factor in [ ]uniform series compound amount factor in [ ]
functionally expressed as F = A ( F / A,i%,N )functionally expressed as F = A ( F / A,i%,N ) predetermined values are in column 4 of Appendixpredetermined values are in column 4 of Appendix
C of textC of textF = ?
1 2 3 4 5 6 7 8 A =
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( F / A,i%,N ) = (P / A,i,N ) ( F / P,i,N )
( F / A,i%,N ) = ( F / P,i,N-k )
N
k = 1
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RELATING A UNIFORM SERIES (ORDINARYANNUITY) TO PRESENT AND FUTURE EQUIVALENT
VALUES
RELATING A UNIFORM SERIES (ORDINARYANNUITY) TO PRESENT AND FUTURE EQUIVALENT
VALUES
Finding P given A: Finding present equivalent value given a series of
uniform equal receipts
( 1 + i )
N
- 1 P = A
i ( 1 + i ) N
uniform series present worth factor in [ ]
functionally expressed as P = A ( P / A,i%,N )
predetermined values are in column 5 of AppendixC of text
Finding P given A: Finding present equivalent value given a series of
uniform equal receipts
( 1 + i ) N - 1
P = A
i ( 1 + i ) N
uniform series present worth factor in [ ]
functionally expressed as P = A ( P / A,i%,N )
predetermined values are in column 5 of AppendixC of text
P = ?
1 2 3 4 5 6 7 8A =
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( P / A,i%,N ) = ( P / F,i,k )
N
k = 1
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RELATING A UNIFORM SERIES (ORDINARYANNUITY) TO PRESENT AND FUTURE EQUIVALENT
VALUES
RELATING A UNIFORM SERIES (ORDINARYANNUITY) TO PRESENT AND FUTURE EQUIVALENT
VALUES
Finding A given F: Finding amount A of a uniform series when given the
equivalent future value
iA = F
( 1 + i ) N -1
sinking fund factor in [ ]
functionally expressed as A = F ( A / F,i%,N )
predetermined values are in column 6 of AppendixC of text
Finding A given F: Finding amount A of a uniform series when given the
equivalent future value
i
A = F
( 1 + i ) N -1
sinking fund factor in [ ]
functionally expressed as A = F ( A / F,i%,N )
predetermined values are in column 6 of AppendixC of text F =
1 2 3 4 5 6 7 8 A =?
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( A / F,i%,N ) = 1 / ( F / A,i%,N )
( A / F,i%,N ) = ( A / P,i%,N ) - i
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RELATING A UNIFORM SERIES (ORDINARYANNUITY) TO PRESENT AND FUTURE EQUIVALENT
VALUES
RELATING A UNIFORM SERIES (ORDINARYANNUITY) TO PRESENT AND FUTURE EQUIVALENT
VALUES
Finding A given P: Finding amount A of a uniform series when given the
equivalent present value
i ( 1+i )
N
A = P
( 1 + i ) N -1
capital recovery factor in [ ]
functionally expressed as A = P ( A / P,i%,N )
predetermined values are in column 7 of AppendixC of text
Finding A given P: Finding amount A of a uniform series when given the
equivalent present value
i ( 1+i )N
A = P
( 1 + i ) N -1
capital recovery factor in [ ]
functionally expressed as A = P ( A / P,i%,N )
predetermined values are in column 7 of AppendixC of text P =
1 2 3 4 5 6 7 8 A =?
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( A / P,i%,N ) = 1 / ( P / A,i%,N )
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CONCEPT OF EQUIVALENCECONCEPT OF EQUIVALENCE
A sum of $10,000 is borrowed by a refining oilcompany. The capital and interest have to be repaid
within four (4) years at an interest rate of five (5)
percent per year.
Q1. Proposed four (4) different equivalent plans of
payments
Q2. Construct the cash flow diagram for each plan of
payment.
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Proposal of the four (4) plansProposal of the four (4) plans
Year Investment
($)
Plan I
($)
Plan II
($)
Plan III
($)
Plan IV
($)
0 10,000
1 500 2,500+500 2,820 0
2 500 2,500+375 2,820 0
3 500 2,500+250 2,820 0
4 10,500 2,500+125 2,820 12,155
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