engeco chap 04 - the time value of money

7/31/2019 Engeco Chap 04 - The Time Value of Money

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SEMESTER JANUARY 2010SEMESTER JANUARY 2010 11

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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1 2 3 4 5 = N1


P =$8,000 2

2 Present expense (cash outflow) of $8,000 for lender.


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1 2 3 4 5 = N1


P =$8,000 2


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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1 2 3 4 5 = N1


P =$8,000 2


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


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 )


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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VALUES


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 )


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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VALUES


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 )


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

engeco chap 04 - the time value of money

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