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Engineering Economics
M. Abinanth
ME Industrial Engineering
PSG College of Technology
1. INTRODUCTION
CONTENTS
• Present economic policy
• Liberalization
• Privatization
• Globalization
• scope for industrial growth
• Interest and time value of money cash-flow diagram
• simple interest - compound interest
• single payments
• Uniform series payments
• Nominal and effective interest rates
• continuous compounding
• Uniform continuous payments
INDIAN ECONOMIC POLICY
 The main objectives of the India Economic Policy is to take care of the basic
parameters of the Indian Economy as mentioned below:
 Agriculture
 Industry
 Trade and Commerce
• The main policies are:
 Fiscal Policy
 Monetary Policy
Meaning
• Fiscal policy deals with the taxation and expenditure decisions of the government.
These include, tax policy, expenditure policy, investment or disinvestment strategies
and debt or surplus management.
- Kaushik Basu ( Former Chief Economic Adviser )
Objectives of fiscal policy:
1. To achieve desirable price level
2. To achieve desirable consumption level
3. To achieve desirable employment level
4. To achieve desirable income distribution
5. Increase in capital formation
6. Degree of inflation
Fiscal Policy
Fiscal Policy there are three possible positions
• A neutral stance of fiscal policy implies a balanced budget where G = T
(Government spending = Tax revenue).
• An expansionary stance of fiscal policy involves a net increase in
government spending (G > T) through rises in government spending, a fall in
taxation revenue, or a combination of the two..
• A contractionary fiscal policy (G < T) occurs when net government
spending is reduced either through higher taxation revenue, reduced
government spending, or a combination of the two.
Instruments of Fiscal Policy:
• Taxation Fiscal policy, especially tax policy, can be used to enhance growth, by
encouraging the efficient use of any given amount of scarce resources.
• Public expenditure embraces all the public sector spending including that of
central governments, state governments, local authorities and public corporations.
The pattern of public expenditure is influenced by interest groups and by
economic, political, demographic, sociological and technological factors. In
addition, international demonstration effect induces developing countries like
India to follow spending patterns of advanced countries.
Monetary Policy
• The term monetary policy refers to actions taken by central banks to affect
monetary magnitudes or other financial conditions.
• Monetary Policy operates on monetary magnitudes or variables such as
money supply, interest rates and availability of credit.
• Monetary Policy ultimately operates through its influence on expenditure
flows in the economy.
• In other words affects liquidity and by affecting liquidity, and thus credit, it
affects total demand in the economy.
Credit Policy
• Central Bank may directly affect the money supply to control its growth.
• Or it might act indirectly to affect cost and availability of credit in the economy.
• In modern times the bulk of money in developed economies consists of bank
deposits rather than currencies and coins.
• So central banks today guide monetary developments with instruments that control
over deposit creation and influence general financial conditions.
• Credit policy is concerned with changes in the supply of credit.
• Central Bank administers both the Credit and Monetary policy
Operation of Monetary Policy
Instruments
1. Discount Rate
(Bank Rate)
2.Reserve Ratios
3. Open Market
Operations
Operating
Target
• Monetary Base
• Bank Credit
• Interest Rates
Intermediate
Target
•Monetary
Aggregates(M3)
•Long term
interest rates
Ultimate
Goals
•Total Spending
• Price Stability
Etc.
What is New Economic Policy ?
 It refers to ongoing economic liberalisation or
relaxation started in 1991 of the countries
economic policies
 It was introduced with the goal of making the
economy more market-oriented and
expanding the role of the private and foreign
investment.
 Specific changes include the reduction in import tariffs, deregulation of markets,
reduction of taxes, and greater foreign investment.
 The liberalization has been credited by its proponents for the high economic growth
recorded by the country in the 1990s and 2000s.
 On the other hand, its opponents have blamed it for increased poverty, inequality and
economic degradation.
NEW ECONOMIC
POLICY
LIBERALISATION PRIVATISATION GLOBALISATION
LIBERALISATION
The first aspect of new economic policy was
liberalisation
Liberalisation of an economy means removing or
relaxing government controls and restrictions on
economic activities
Relief for foreign invertors
Revaluation of Indian Currency
New Industrial Policy
New Trade Policy
Import Technology
Encouraging foreign tie-ups
Privatisation in Public Sector
POSITIVE EFFECTS
 Increase in foreign investment
 Increase in Production
 Technological advancement
 Increase in GDP growth rate
NEGATIVE EFFECTS
 Increase in Unemployment
 Decrease in Tax Receipt
Impacts of Liberalisation
Growth of GDP Post Liberalisation
 According to World Bank, “Privatisation is the
transfer of state owned enterprises to the private
sector by sale of going concerns or by sale of assets
following their liquidation “
 Increasing inefficiency on part of public sector led
to privatization
 Forms of Privatization :-
 Denationalisation
 Joint Venture
 Leasing
 Franchising
Privatisation
POSITIVE EFFECTS
 Private companies cut cost and be more
efficient
 Increased competition
 More Responsive to customer
complaints
NEGATIVE EFFECTS
 Public service
 Job loses
 Privatisation is expensive
Impacts of Privatisation
Globalisation
Globalisation means reduction or removal of
government restriction on the movement of goods and
service, capital, technology and talent across national
boundaries.
It is the increasing interdependence, integration and
interaction among people and cooperation in various
locations around the world.
POSITIVE EFFECTS
 Expansion of market
 Development of infrastructure
 Higher living standards
 International cooperation
NEGATIVE EFFECTS
 Cut throat competitions
 Rise in Monopoly
 Take over of Domestic Firms
 Increase in Inequalities
Impacts of Globalisation
Impact of NEP 1991 on Indian Economy
a) Increasing Competition
b) More Demanding Customers
c) Rapidly Changing Technological Environment
d) Necessity for Change
e) Need for Developing Human Resources
f) Market Orientation
g) Loss of Budgetary Support to Public Sector
h) Export a Matter of Survival
Time value of money
• The time value of money is important when one is interested either in investing
or borrowing the money
Definition:
Time value of money is the premise that an investor prefers to receive a payment
of a fixed amount of money today, rather than an equal amount in the future, all
else being equal.
TIME allows one the opportunity to postpone consumption and earn INTEREST.
Types of Interest
• Simple Interest
– Interest paid (earned) on only the original amount, or principal, borrowed
(lent).
SI = P0(i)(n)
SI: Simple Interest I :Interest Rate per Period
n : Number of Time Periods P0:Deposit today (t=0)
• Compound Interest
Interest paid (earned) on any previous interest earned, as well as on the
principal borrowed (lent).
Simple Interest Example
• Assume that you deposit $1,000 in an account earning 7% simple interest for 2
years. What is the accumulated interest at the end of the 2nd year?
SI = P0(i)(n)
= $1,000(.07)(2)
= $140
• What is the Future Value (FV) of the deposit?
FV = P0 + SI
= $1,000 + $140
= $1,140
• Future Value is the value at some future time of a present amount of money, or a
series of payments, evaluated at a given interest rate.
That is the value today!
• Present Value is the current value of a future amount of money, or a series of
payments, evaluated at a given interest rate.
Simple Interest (PV)
• What is the Present Value (PV) of the previous problem?
The Present Value is simply the $1,000 you originally deposited.
0
5000
10000
15000
20000
1st Year 10th Year 20th
Year
30th
Year
Future Value of a Single $1,000 Deposit
10% Simple Interest
7% Compound Interest
10% Compound Interest
Why Compound Interest?
FutureValue(U.S.Dollars)
Assume that you deposit $1,000 at a compound
interest rate of 7% for 2 years.
Future Value Single Deposit
0 1 2
$1,000
FV2
7%
FV1 = P0 (1+i)1
= $1,000 (1.07)
= $1,070
Compound Interest
You earned $70 interest on your $1,000 deposit over the first year.
This is the same amount of interest you would earn under simple interest.
FV2 = FV1 (1+i)1
= P0 (1+i)(1+i)
= $1,000(1.07)(1.07)
= P0 (1+i)2
= $1,000(1.07)2
= $1,144.90
You earned an EXTRA $4.90 in Year 2 with compound over simple interest.
Future Value Single Deposit
FV1 = P0(1+i)1
FV2 = P0(1+i)2
General Future Value Formula:
FVn = P0 (1+i)n
or FVn = P0 (FVIFi,n) -- See Table
General Future Value Formula
Period 6% 7% 8%
1 1.060 1.070 1.080
2 1.124 1.145 1.166
3 1.191 1.225 1.260
4 1.262 1.311 1.360
5 1.338 1.403 1.469
FVIFi,n is found on Table I
at the end of the book.
We will use the “Rule-of-72”.
Double Your Money!!!
Quick! How long does it take to double $5,000 at a compound rate of
12% per year (approx.)?
Approx. Years to Double = 72 / i%
72 / 12% = 6 Years
[Actual Time is 6.12 Years]
Cash flow diagram:
• The graphical representation of the cash flows i.e. both cash outflows and cash inflows with respect
to a time scale is generally referred as cash flow diagram.
• The cash outflows (i.e. costs or expense) are generally represented by vertically downward arrows
whereas the cash inflows (i.e. revenue or income) are represented by vertically upward arrows.
• In the cash flow diagram, number of interest periods is shown on the time scale.
• The cash outflows are Rs.100000, Rs.15000 and Rs.25000 occurring at end of year (EOY) „0‟ i.e. at
the beginning, EOY „4‟ and EOY „7‟ respectively. Similarly the cash inflows Rs.35000, Rs.80000
and Rs.45000 are occurring at EOY „3‟, EOY „6‟ and EOY „10‟ respectively .
Compound interest factors:
• P = Present worth or present value
• F = Future worth or future sum
• A = Uniform annual worth or equivalent uniform annual worth of a uniform
series continuing over a specified number of interest periods
• n = number of interest periods (years or months)
• i = rate of interest per interest period i.e. % per year or % per month
In this figure the present worth, P is at the beginning and the uniform annual series with annual value
“A” is from end of year 1 till end of year 5. Both “P” and “A” are cash outflows. It may be noted that the
uniform annual series with annual value “A” may be also continued throughout the entire interest
periods i.e. from beginning till end of year 10 or for some intermediate interest periods like commencing
from end of year 3 till end of year 8
The future worth “F” is occurring at end
of year 4 (cash outflow), at end of year 6
(cash inflow) and at the end of year 10 (cash inflow)
Single payment compound amount factor (SPCAF)
• The single payment compound amount factor is used to compute the future worth (F)
accumulated after “n” years from the known present worth (P) at a given interest rate ‘i’
per interest period.
• It is assumed that the interest period is in years and the interest is compounded once per
interest period.
• The known present worth (P), unknown future worth (F) and the total interest period “n”
years are shown in Fig below (Cash flow diagram for ‘known P’ and ‘unknown F’ )
• The future worth (F1) accumulated at the end of year 1 i.e. 1st year is given by;
• The future worth accumulated at the end of year 2 i.e. F2 will be equal to the amount that was
accumulated at the end of 1st year i.e. F1 plus the amount of interest accumulated from end of
1st year to the end of 2nd year on F1 and is given by;
• Putting the value of F1 from equation (1) in equation (2), the value of F2 is given by
• Similarly, the future worth accumulated at the end of year 3 i.e. F3 is equal to the amount that
was accumulated at the end of 2nd year i.e. F2 plus the amount of interest accumulated from
end of 2nd year to the end of 3rd year on F2 and is given by
• Putting the value of F2 from equation (3) in equation (4), the value of F3 is given by
• Similarly, the future worth accumulated at the end of year 4 i.e. F4 is given by
• Thus the generalized formula for the future worth at the end of „n‟ years is given by;
The factor is known as the single payment compound amount factor (SPCAF).
1
2
4
5
6
3
7
Single payment present worth factor (SPPWF):
• The single payment present worth factor is used to determine the present worth of a known future
worth (F) at the end of “n” years at a given interest rate ‘i’per interest period.
• The present worth (P), future worth (F) and the total interest period n years are shown in Fig. 1.8.
• The factor in above equation is known as single payment present worth factor
(SPPWF).
• Thus if future worth (F) at the end of n years is known, the present worth (P) at interest rate
of i (per year) can be calculated by multiplying the future worth with the single payment
present worth factor.
8
Uniform series present worth factor (USPWF)
• The uniform-series present worth factor is used to determine the present worth of a
known uniform series.
• Let A be the uniform annual amount at the end of each year, beginning from end of
year 1 till end of year n.
• The known A, unknown P, and the total interest period n years are shown in Fig. 1.9.
This cash flow diagram refers to the case; if a person wants to get the known uniform
amount of return every year, how much he has to invest now.
• The present worth (P) of the uniform series can be calculated by considering each A
of the uniform series as the future worth.
• Then by using the formula the present worth of these future worth can be calculated
and finally taking the sum of these present worth values.
Cash flow diagram for ‘known A’ and ‘unknown P’
USPWF
• The present worth (P) of the uniform series is given by
The expression in the bracket is a geometric sequence with first term equal to (1+i)-1 and common
ratio equal to (1+i)-1. Then the present worth (P) is calculated by taking the sum of the first n terms
of the geometric sequence (at i ≠ 0) and is given by,
The simplification of equation
The factor within the bracket in above equation is known as uniform series present worth factor
(USPWF)
9
10
11
13
12
Capital recovery factor (CRF):
• The capital recovery factor is generally used to find out the uniform annual amount A of a
uniform series from the known present worth at a given interest rate ‘i’per interest period.
• The cash flow diagram is shown in Fig below. This cash flow diagram indicates, if a person
invests a certain amount now, how much he will get as return by an equal amount each year.
• From equation (12), the expression for the uniform annual amount (A) can be written as follows
• The factor within bracket in equation above is known as the capital recovery factor (CRF)
Cash flow diagram for ‘known P’ and ‘unknown A’
Uniform series compound amount factor:
• The uniform series compound amount factor is used to determine the future sum (F) of a known
uniform annual series with uniform amount A.
• The cash flow diagram is shown in Fig. This cash flow diagram states that, if a person invests a
uniform amount at the end of each year continued for n years at interest rate of i per year, how
much he will get at the end of n years.
• Putting the value of present worth (P) from equation (8) in equation (13) results in the following:
• The factor within bracket in equation (19) is
known as uniform series compound amount
factor (USCAF).
• Hence the future worth „F‟ can be computed by
multiplying the uniform annual amount „A‟
with the uniform series compound amount
factor.
Sinking fund factor
• The sinking fund factor is used to calculate the annual amount A of a uniform series from the
known future sum F.
• The cash flow diagram is shown in Fig. 1.12. This cash flow diagram indicates that, if a person
wants to get a known future sum at the end of n years at interest rate of i per year, how much he
has to invest every year by an equal amount.
• From equation (19), the expression for the uniform annual amount (A) can be written as follows
• The factor within bracket in equation (20) is known as sinking fund factor (SFF). Thus one can
find out the annual amount A of a uniform series by multiplying the future worth F with the
sinking fund factor.
Interest factors (Discrete compounding) with rate of
interest ‘i’ (%) and number of interest periods ‘n’
Cash flow involving arithmetic gradient payments
or receipts
• Some cash flows involve the payments or receipts in gradients by same amount.
• In other words, the expenditure or the income increases or decreases by same amount.
• The cash flow involving such payments or receipts is known as uniform gradient series.
• For example, if the cost of repair and maintenance of a piece of equipment increases by same
amount every year till end of its useful life, it represents a cash flow involving positive uniform
gradient
• The cash flow diagrams for positive gradient and negative gradient are shown in Fig. 1.3 and Fig.
1.4 respectively
... continue
• The present worth, future worth and the equivalent uniform annual worth of the uniform gradient
can be derived using the compound interest factors.
• The generalized cash flow diagram involving a positive uniform gradient with base value B and the
gradient G is shown in Fig. 1.15a.
• The cash flow shown in Fig. 1.15a can be split into two cash flows; one having the uniform series
with amount B and the other having the gradient series with values in multiples of gradient amount
G and is shown in Fig. 1.15b.
• This gradient series is also know as the arithmetic gradient series as the expense or the income
increases by the uniform arithmetic amount G every year.
=
...continue
• The future worth of the cash flow shown in Fig. 1.15a can be obtained by finding the out the
individual future worth of the cash flows shown in Fig. 1.15b and then taking their sum.
• As already stated, the future worth of the cash flow involving an uniform series can be determined
by multiplying the uniform annual amount „B‟ with the uniform series compound amount factor.
• The future worth (F) of the gradient series shown in Fig 1.15b can be determined by finding out the
individual future worth of the gradient values (i.e. in multiples of gradient amount G) at the end of
different years at interest rate of i per year and then taking the sum of these individual futures
values.
The expression in the bracket is a geometric sequence with first term equal to 1/(1+i) and common ratio
equal to 1/(1+i). Thus equation (26) is rewritten by taking sum of the first „n‟ terms of the geometric
sequence (at i ≠ 0) and is given by;

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Engineering Economics Introduction

  • 1. Engineering Economics M. Abinanth ME Industrial Engineering PSG College of Technology
  • 2. 1. INTRODUCTION CONTENTS • Present economic policy • Liberalization • Privatization • Globalization • scope for industrial growth • Interest and time value of money cash-flow diagram • simple interest - compound interest • single payments • Uniform series payments • Nominal and effective interest rates • continuous compounding • Uniform continuous payments
  • 3. INDIAN ECONOMIC POLICY  The main objectives of the India Economic Policy is to take care of the basic parameters of the Indian Economy as mentioned below:  Agriculture  Industry  Trade and Commerce • The main policies are:  Fiscal Policy  Monetary Policy
  • 4. Meaning • Fiscal policy deals with the taxation and expenditure decisions of the government. These include, tax policy, expenditure policy, investment or disinvestment strategies and debt or surplus management. - Kaushik Basu ( Former Chief Economic Adviser ) Objectives of fiscal policy: 1. To achieve desirable price level 2. To achieve desirable consumption level 3. To achieve desirable employment level 4. To achieve desirable income distribution 5. Increase in capital formation 6. Degree of inflation Fiscal Policy
  • 5. Fiscal Policy there are three possible positions • A neutral stance of fiscal policy implies a balanced budget where G = T (Government spending = Tax revenue). • An expansionary stance of fiscal policy involves a net increase in government spending (G > T) through rises in government spending, a fall in taxation revenue, or a combination of the two.. • A contractionary fiscal policy (G < T) occurs when net government spending is reduced either through higher taxation revenue, reduced government spending, or a combination of the two.
  • 6. Instruments of Fiscal Policy: • Taxation Fiscal policy, especially tax policy, can be used to enhance growth, by encouraging the efficient use of any given amount of scarce resources. • Public expenditure embraces all the public sector spending including that of central governments, state governments, local authorities and public corporations. The pattern of public expenditure is influenced by interest groups and by economic, political, demographic, sociological and technological factors. In addition, international demonstration effect induces developing countries like India to follow spending patterns of advanced countries.
  • 7. Monetary Policy • The term monetary policy refers to actions taken by central banks to affect monetary magnitudes or other financial conditions. • Monetary Policy operates on monetary magnitudes or variables such as money supply, interest rates and availability of credit. • Monetary Policy ultimately operates through its influence on expenditure flows in the economy. • In other words affects liquidity and by affecting liquidity, and thus credit, it affects total demand in the economy.
  • 8. Credit Policy • Central Bank may directly affect the money supply to control its growth. • Or it might act indirectly to affect cost and availability of credit in the economy. • In modern times the bulk of money in developed economies consists of bank deposits rather than currencies and coins. • So central banks today guide monetary developments with instruments that control over deposit creation and influence general financial conditions. • Credit policy is concerned with changes in the supply of credit. • Central Bank administers both the Credit and Monetary policy
  • 9. Operation of Monetary Policy Instruments 1. Discount Rate (Bank Rate) 2.Reserve Ratios 3. Open Market Operations Operating Target • Monetary Base • Bank Credit • Interest Rates Intermediate Target •Monetary Aggregates(M3) •Long term interest rates Ultimate Goals •Total Spending • Price Stability Etc.
  • 10. What is New Economic Policy ?  It refers to ongoing economic liberalisation or relaxation started in 1991 of the countries economic policies  It was introduced with the goal of making the economy more market-oriented and expanding the role of the private and foreign investment.
  • 11.  Specific changes include the reduction in import tariffs, deregulation of markets, reduction of taxes, and greater foreign investment.  The liberalization has been credited by its proponents for the high economic growth recorded by the country in the 1990s and 2000s.  On the other hand, its opponents have blamed it for increased poverty, inequality and economic degradation.
  • 13. LIBERALISATION The first aspect of new economic policy was liberalisation Liberalisation of an economy means removing or relaxing government controls and restrictions on economic activities Relief for foreign invertors Revaluation of Indian Currency New Industrial Policy New Trade Policy Import Technology Encouraging foreign tie-ups Privatisation in Public Sector
  • 14. POSITIVE EFFECTS  Increase in foreign investment  Increase in Production  Technological advancement  Increase in GDP growth rate NEGATIVE EFFECTS  Increase in Unemployment  Decrease in Tax Receipt Impacts of Liberalisation
  • 15. Growth of GDP Post Liberalisation
  • 16.  According to World Bank, “Privatisation is the transfer of state owned enterprises to the private sector by sale of going concerns or by sale of assets following their liquidation “  Increasing inefficiency on part of public sector led to privatization  Forms of Privatization :-  Denationalisation  Joint Venture  Leasing  Franchising Privatisation
  • 17. POSITIVE EFFECTS  Private companies cut cost and be more efficient  Increased competition  More Responsive to customer complaints NEGATIVE EFFECTS  Public service  Job loses  Privatisation is expensive Impacts of Privatisation
  • 18. Globalisation Globalisation means reduction or removal of government restriction on the movement of goods and service, capital, technology and talent across national boundaries. It is the increasing interdependence, integration and interaction among people and cooperation in various locations around the world.
  • 19. POSITIVE EFFECTS  Expansion of market  Development of infrastructure  Higher living standards  International cooperation NEGATIVE EFFECTS  Cut throat competitions  Rise in Monopoly  Take over of Domestic Firms  Increase in Inequalities Impacts of Globalisation
  • 20. Impact of NEP 1991 on Indian Economy a) Increasing Competition b) More Demanding Customers c) Rapidly Changing Technological Environment d) Necessity for Change e) Need for Developing Human Resources f) Market Orientation g) Loss of Budgetary Support to Public Sector h) Export a Matter of Survival
  • 21. Time value of money • The time value of money is important when one is interested either in investing or borrowing the money Definition: Time value of money is the premise that an investor prefers to receive a payment of a fixed amount of money today, rather than an equal amount in the future, all else being equal. TIME allows one the opportunity to postpone consumption and earn INTEREST.
  • 22. Types of Interest • Simple Interest – Interest paid (earned) on only the original amount, or principal, borrowed (lent). SI = P0(i)(n) SI: Simple Interest I :Interest Rate per Period n : Number of Time Periods P0:Deposit today (t=0) • Compound Interest Interest paid (earned) on any previous interest earned, as well as on the principal borrowed (lent).
  • 23. Simple Interest Example • Assume that you deposit $1,000 in an account earning 7% simple interest for 2 years. What is the accumulated interest at the end of the 2nd year? SI = P0(i)(n) = $1,000(.07)(2) = $140 • What is the Future Value (FV) of the deposit? FV = P0 + SI = $1,000 + $140 = $1,140 • Future Value is the value at some future time of a present amount of money, or a series of payments, evaluated at a given interest rate.
  • 24. That is the value today! • Present Value is the current value of a future amount of money, or a series of payments, evaluated at a given interest rate. Simple Interest (PV) • What is the Present Value (PV) of the previous problem? The Present Value is simply the $1,000 you originally deposited.
  • 25. 0 5000 10000 15000 20000 1st Year 10th Year 20th Year 30th Year Future Value of a Single $1,000 Deposit 10% Simple Interest 7% Compound Interest 10% Compound Interest Why Compound Interest? FutureValue(U.S.Dollars)
  • 26. Assume that you deposit $1,000 at a compound interest rate of 7% for 2 years. Future Value Single Deposit 0 1 2 $1,000 FV2 7% FV1 = P0 (1+i)1 = $1,000 (1.07) = $1,070
  • 27. Compound Interest You earned $70 interest on your $1,000 deposit over the first year. This is the same amount of interest you would earn under simple interest. FV2 = FV1 (1+i)1 = P0 (1+i)(1+i) = $1,000(1.07)(1.07) = P0 (1+i)2 = $1,000(1.07)2 = $1,144.90 You earned an EXTRA $4.90 in Year 2 with compound over simple interest. Future Value Single Deposit
  • 28. FV1 = P0(1+i)1 FV2 = P0(1+i)2 General Future Value Formula: FVn = P0 (1+i)n or FVn = P0 (FVIFi,n) -- See Table General Future Value Formula Period 6% 7% 8% 1 1.060 1.070 1.080 2 1.124 1.145 1.166 3 1.191 1.225 1.260 4 1.262 1.311 1.360 5 1.338 1.403 1.469 FVIFi,n is found on Table I at the end of the book.
  • 29. We will use the “Rule-of-72”. Double Your Money!!! Quick! How long does it take to double $5,000 at a compound rate of 12% per year (approx.)? Approx. Years to Double = 72 / i% 72 / 12% = 6 Years [Actual Time is 6.12 Years]
  • 30. Cash flow diagram: • The graphical representation of the cash flows i.e. both cash outflows and cash inflows with respect to a time scale is generally referred as cash flow diagram. • The cash outflows (i.e. costs or expense) are generally represented by vertically downward arrows whereas the cash inflows (i.e. revenue or income) are represented by vertically upward arrows. • In the cash flow diagram, number of interest periods is shown on the time scale. • The cash outflows are Rs.100000, Rs.15000 and Rs.25000 occurring at end of year (EOY) „0‟ i.e. at the beginning, EOY „4‟ and EOY „7‟ respectively. Similarly the cash inflows Rs.35000, Rs.80000 and Rs.45000 are occurring at EOY „3‟, EOY „6‟ and EOY „10‟ respectively .
  • 31. Compound interest factors: • P = Present worth or present value • F = Future worth or future sum • A = Uniform annual worth or equivalent uniform annual worth of a uniform series continuing over a specified number of interest periods • n = number of interest periods (years or months) • i = rate of interest per interest period i.e. % per year or % per month In this figure the present worth, P is at the beginning and the uniform annual series with annual value “A” is from end of year 1 till end of year 5. Both “P” and “A” are cash outflows. It may be noted that the uniform annual series with annual value “A” may be also continued throughout the entire interest periods i.e. from beginning till end of year 10 or for some intermediate interest periods like commencing from end of year 3 till end of year 8 The future worth “F” is occurring at end of year 4 (cash outflow), at end of year 6 (cash inflow) and at the end of year 10 (cash inflow)
  • 32. Single payment compound amount factor (SPCAF) • The single payment compound amount factor is used to compute the future worth (F) accumulated after “n” years from the known present worth (P) at a given interest rate ‘i’ per interest period. • It is assumed that the interest period is in years and the interest is compounded once per interest period. • The known present worth (P), unknown future worth (F) and the total interest period “n” years are shown in Fig below (Cash flow diagram for ‘known P’ and ‘unknown F’ )
  • 33. • The future worth (F1) accumulated at the end of year 1 i.e. 1st year is given by; • The future worth accumulated at the end of year 2 i.e. F2 will be equal to the amount that was accumulated at the end of 1st year i.e. F1 plus the amount of interest accumulated from end of 1st year to the end of 2nd year on F1 and is given by; • Putting the value of F1 from equation (1) in equation (2), the value of F2 is given by • Similarly, the future worth accumulated at the end of year 3 i.e. F3 is equal to the amount that was accumulated at the end of 2nd year i.e. F2 plus the amount of interest accumulated from end of 2nd year to the end of 3rd year on F2 and is given by • Putting the value of F2 from equation (3) in equation (4), the value of F3 is given by • Similarly, the future worth accumulated at the end of year 4 i.e. F4 is given by • Thus the generalized formula for the future worth at the end of „n‟ years is given by; The factor is known as the single payment compound amount factor (SPCAF). 1 2 4 5 6 3 7
  • 34. Single payment present worth factor (SPPWF): • The single payment present worth factor is used to determine the present worth of a known future worth (F) at the end of “n” years at a given interest rate ‘i’per interest period. • The present worth (P), future worth (F) and the total interest period n years are shown in Fig. 1.8. • The factor in above equation is known as single payment present worth factor (SPPWF). • Thus if future worth (F) at the end of n years is known, the present worth (P) at interest rate of i (per year) can be calculated by multiplying the future worth with the single payment present worth factor. 8
  • 35. Uniform series present worth factor (USPWF) • The uniform-series present worth factor is used to determine the present worth of a known uniform series. • Let A be the uniform annual amount at the end of each year, beginning from end of year 1 till end of year n. • The known A, unknown P, and the total interest period n years are shown in Fig. 1.9. This cash flow diagram refers to the case; if a person wants to get the known uniform amount of return every year, how much he has to invest now. • The present worth (P) of the uniform series can be calculated by considering each A of the uniform series as the future worth. • Then by using the formula the present worth of these future worth can be calculated and finally taking the sum of these present worth values. Cash flow diagram for ‘known A’ and ‘unknown P’
  • 36. USPWF • The present worth (P) of the uniform series is given by The expression in the bracket is a geometric sequence with first term equal to (1+i)-1 and common ratio equal to (1+i)-1. Then the present worth (P) is calculated by taking the sum of the first n terms of the geometric sequence (at i ≠ 0) and is given by, The simplification of equation The factor within the bracket in above equation is known as uniform series present worth factor (USPWF) 9 10 11 13 12
  • 37. Capital recovery factor (CRF): • The capital recovery factor is generally used to find out the uniform annual amount A of a uniform series from the known present worth at a given interest rate ‘i’per interest period. • The cash flow diagram is shown in Fig below. This cash flow diagram indicates, if a person invests a certain amount now, how much he will get as return by an equal amount each year. • From equation (12), the expression for the uniform annual amount (A) can be written as follows • The factor within bracket in equation above is known as the capital recovery factor (CRF) Cash flow diagram for ‘known P’ and ‘unknown A’
  • 38. Uniform series compound amount factor: • The uniform series compound amount factor is used to determine the future sum (F) of a known uniform annual series with uniform amount A. • The cash flow diagram is shown in Fig. This cash flow diagram states that, if a person invests a uniform amount at the end of each year continued for n years at interest rate of i per year, how much he will get at the end of n years. • Putting the value of present worth (P) from equation (8) in equation (13) results in the following: • The factor within bracket in equation (19) is known as uniform series compound amount factor (USCAF). • Hence the future worth „F‟ can be computed by multiplying the uniform annual amount „A‟ with the uniform series compound amount factor.
  • 39. Sinking fund factor • The sinking fund factor is used to calculate the annual amount A of a uniform series from the known future sum F. • The cash flow diagram is shown in Fig. 1.12. This cash flow diagram indicates that, if a person wants to get a known future sum at the end of n years at interest rate of i per year, how much he has to invest every year by an equal amount. • From equation (19), the expression for the uniform annual amount (A) can be written as follows • The factor within bracket in equation (20) is known as sinking fund factor (SFF). Thus one can find out the annual amount A of a uniform series by multiplying the future worth F with the sinking fund factor.
  • 40. Interest factors (Discrete compounding) with rate of interest ‘i’ (%) and number of interest periods ‘n’
  • 41. Cash flow involving arithmetic gradient payments or receipts • Some cash flows involve the payments or receipts in gradients by same amount. • In other words, the expenditure or the income increases or decreases by same amount. • The cash flow involving such payments or receipts is known as uniform gradient series. • For example, if the cost of repair and maintenance of a piece of equipment increases by same amount every year till end of its useful life, it represents a cash flow involving positive uniform gradient • The cash flow diagrams for positive gradient and negative gradient are shown in Fig. 1.3 and Fig. 1.4 respectively
  • 42. ... continue • The present worth, future worth and the equivalent uniform annual worth of the uniform gradient can be derived using the compound interest factors. • The generalized cash flow diagram involving a positive uniform gradient with base value B and the gradient G is shown in Fig. 1.15a. • The cash flow shown in Fig. 1.15a can be split into two cash flows; one having the uniform series with amount B and the other having the gradient series with values in multiples of gradient amount G and is shown in Fig. 1.15b. • This gradient series is also know as the arithmetic gradient series as the expense or the income increases by the uniform arithmetic amount G every year. =
  • 43. ...continue • The future worth of the cash flow shown in Fig. 1.15a can be obtained by finding the out the individual future worth of the cash flows shown in Fig. 1.15b and then taking their sum. • As already stated, the future worth of the cash flow involving an uniform series can be determined by multiplying the uniform annual amount „B‟ with the uniform series compound amount factor. • The future worth (F) of the gradient series shown in Fig 1.15b can be determined by finding out the individual future worth of the gradient values (i.e. in multiples of gradient amount G) at the end of different years at interest rate of i per year and then taking the sum of these individual futures values.
  • 44. The expression in the bracket is a geometric sequence with first term equal to 1/(1+i) and common ratio equal to 1/(1+i). Thus equation (26) is rewritten by taking sum of the first „n‟ terms of the geometric sequence (at i ≠ 0) and is given by;