CHAPTER 3
MONEY-TIME
RELATIONSHIPS
AND EQUIVALENCE
INTEREST
“Hurt no one so that no one may hurt you. Remember
that you will indeed meet your LORD, and that HE will
indeed reckon your deeds. ALLAH has forbidden you to
take usury (interest), therefore all interest obligation
shall henceforth be waived”.
“Janganlah kamu sakiti sesiapapun agar orang lain tidak
menyakiti kamu pula. Ingatlah bahawa sesungguhnya kamu
akan menemui Tuhan kamu dan Dia pasti akan membuat
perhitungan atas segala amalan kamu. Allah telah
mengharamkan riba', oleh itu segala urusan yang melibatkan
riba' hendaklah dibatalkan mulai sekarang”
Muhammad (pbuh), The Farewell Sermon, Mount Arafat (0632)
MONEY
• Medium of Exchange -Means of payment for goods or services;
What sellers accept and buyers pay ;
• Store of Value -A way to transport buying power from one time
period to another;
• Unit of Account -A precise measurement of value or worth;
Allows for tabulating debits and credits;
CAPITAL
Wealth in the form of money or
property that can be used to
produce more wealth.
KINDS OF CAPITAL
• Equity capital is that owned by individuals who
have invested their money or property in a
business project or venture in the hope of
receiving a profit.
• Debt capital, often called borrowed capital, is
obtained from lenders (e.g., through the sale of
bonds) for investment.
Financing
Definition
Instrument Description
• Bond • Promise to
pay
principle &
interest;
• Debt
financing
• Borrow
money
• Equity
financing
Exchange
• Sell partial • Stock •Exchange
of
sharesfor
ownership of
money
stock for
company;
shares
of
of
ownership
stock
as
company;
proof of
partial
ownership
INTEREST
The fee that a borrower pays to a
lender for the use of his or her
money.
INTEREST RATE
The percentage of money being
borrowed that is paid to the lender
on some time basis.
HOW INTEREST RATE IS
DETERMINED
Interest
Rate
Quantity of Money
HOW INTEREST RATE IS
DETERMINED
Interest
Rate
Money Supply
MS3MS1 MS2
i3
ie
i2
Money Demand
Quantity of Money
SIMPLE INTEREST
• The total interest earned or charged is linearly
proportional to the initial amount of the loan
(principal), the interest rate and the number of
interest periods for which the principal is
committed.
• When applied, total interest “I” may be found by
I = ( P ) ( N ) ( i ), where
– P = principal amount lent or borrowed
– N = number of interest periods ( e.g., years )
– i = interest rate per interest period
COMPOUND INTEREST
• Whenever the interest charge for any interest
period is based on the remaining principal
amount plus any accumulated interest charges
up to the beginning of that period.
Period Amount Owed Interest Amount Amount Owed
Beginning of
for Period
at end of
period
( @ 10% )
period
1
$1,000
$100
$1,100
2
$1,100
$110
$1,210
3
$1,210
$121
$1,331
ECONOMIC EQUIVALENCE
• Established when we are indifferent between a
future payment, or a series of future payments,
and a present sum of money .
• Considers the comparison of alternative
options, or proposals, by reducing them to an
equivalent basis, depending on:
– interest rate;
– amounts of money involved;
– timing of the affected monetary receipts and/or
expenditures;
– manner in which the interest , or profit on invested
capital is paid and the initial capital is recovered.
ECONOMIC EQUIVALENCE FOR FOUR
REPAYMENT PLANS OF AN $8,000 LOAN
• Plan #1: $2,000 of loan principal plus 10% of BOY
principal paid at the end of year; interest paid at the
end of each year is reduced by $200 (i.e., 10% of
remaining principal)
Year Amount Owed Interest Accrued Total Principal Total end
at beginning
for Year
Money Payment of Year
of Year
owed at
Payment
( BOY )
end of
Year
1
$8,000
$800
$8,800 $2,000 $2,800
2
$6,000
$600
$6,600 $2,000 $2,600
3
$4,000
$400
$4,400 $2,000 $2,400
4
$2,000
$200
$2,200 $2,000 $2,200
Total interest paid ($2,000) is 10% of total dollar-years ($20,000)
ECONOMIC EQUIVALENCE FOR FOUR
REPAYMENT PLANS OF AN $8,000 LOAN
• Plan #2: $0 of loan principal paid until end of fourth
year; $800 interest paid at the end of each year
Year Amount Owed Interest Accrued Total Principal Total end
at beginning
for Year
Money Payment of Year
of Year
owed at
Payment
( BOY )
end of
Year
1
$8,000
$800
$8,800 $0
$800
2
$8,000
$800
$8,800 $0
$800
3
$8,000
$800
$8,800 $0
$800
4
$8,000
$800
$8,800 $8,000 $8,800
Total interest paid ($3,200) is 10% of total dollar-years ($32,000)
ECONOMIC EQUIVALENCE FOR FOUR
REPAYMENT PLANS OF AN $8,000 LOAN
• Plan #3: $2,524 paid at the end of each year; interest
paid at the end of each year is 10% of amount owed
at the beginning of the year.
Year Amount Owed Interest Accrued Total Principal Total end
at beginning
for Year
Money Payment of Year
of Year
owed at
Payment
( BOY )
end of
Year
1
$8,000
$800
$8,800 $1,724 $2,524
2
$6,276
$628
$6,904 $1,896 $2,524
3
$4,380
$438
$4,818 $2,086 $2,524
4
$2,294
$230
$2,524 $2,294 $2,524
Total interest paid ($2,096) is 10% of total dollar-years ($20,950)
ECONOMIC EQUIVALENCE FOR FOUR
REPAYMENT PLANS OF AN $8,000 LOAN
• Plan #4: No interest and no principal paid for first
three years. At the end of the fourth year, the original
principal plus accumulated (compounded) interest is
paid.
Year Amount Owed Interest Accrued Total Principal Total end
at beginning
for Year
Money Payment of Year
of Year
owed at
Payment
( BOY )
end of
Year
1
$8,000
$800
$8,800 $0
$0
2
$8,800
$880
$9,680 $0
$0
3
$9,680
$968
$10,648 $0
$0
4
$10,648
$1,065
$11,713 $8,000 $11,713
Total interest paid ($3,713) is 10% of total dollar-years ($37,128)
CASH FLOW DIAGRAMS / TABLE
NOTATION
i = effective interest rate per interest period
N = number of compounding periods (e.g., years)
P = present sum of money; the equivalent value of one
or more cash flows at the present time reference point
F = future sum of money; the equivalent value of one or
more cash flows at a future time reference point
A = end-of-period cash flows (or equivalent end-ofperiod values ) in a uniform series continuing for a
specified number of periods, starting at the end of the
first period and continuing through the last period
G = uniform gradient amounts -- used if cash flows
increase by a constant amount in each period
CASH FLOW DIAGRAM NOTATION
A = $2,524 3
5
1
P =$8,000
1
2
2
3
4
4
5=N
i = 10% per year
1
Time scale with progression of time moving from left to
right; the numbers represent time periods (e.g., years,
months, quarters, etc...) and may be presented within a
time interval or at the end of a time interval.
2
Present expense (cash outflow) of $8,000 for lender.
3
Annual income (cash inflow) of $2,524 for lender.
4
Interest rate of loan.
5
Dashed-arrow line indicates
amount to be determined.
INTEREST FORMULAS FOR ALL OCCASIONS
• relating present and future values of single cash
flows;
• relating a uniform series (annuity) to present and
future equivalent values;
– for discrete compounding and discrete cash flows;
– for deferred annuities (uniform series);
• equivalence calculations involving multiple interest;
• relating a uniform gradient of cash flows to annual
and present equivalents;
• relating a geometric sequence of cash flows to
present and annual equivalents;
INTEREST FORMULAS FOR ALL OCCASIONS
• relating nominal and effective interest rates;
• relating to compounding more frequently than
once a year;
• relating to cash flows occurring less often than
compounding periods;
• for continuous compounding and discrete cash
flows;
• for continuous compounding and continuous
cash flows;
RELATING PRESENT AND FUTURE
EQUIVALENT VALUES OF SINGLE CASH
FLOWS
• Finding F when given P:
• Finding future value when given present value
• F = P ( 1+i ) N
– (1+i)N single payment compound amount factor
– functionally expressed as F = ( F / P, i%,N )
– predetermined values of this are presented in
Interest Factors Table for Discrete Compounding.
P
N=
0
F=?
Example:
• Given: i = 10%, N = 8 years, P = 2000
• F = 2000(1+0.10)8 = 4,287.18
RELATING PRESENT AND FUTURE
EQUIVALENT VALUES OF SINGLE CASH
FLOWS
• Finding P when given F:
• Finding present value when given future value
• P = F [1 / (1 + i ) ] N
– (1+i)-N single payment present worth factor
– functionally expressed as P = F ( P / F, i%, N )
– predetermined values of this are presented in
Interest Factors Table for Discrete Compounding
0
N=
P=?
F
Example:
• Given: i = 12%, N = 5 years, F = 1000
• P = 1000(1+0.12)-5 = 567.40
RELATING A UNIFORM SERIES (ORDINARY
ANNUITY) TO PRESENT AND FUTURE EQUIVALENT
VALUES
• Finding F given A:
• Finding future equivalent income (inflow) value given
a series of uniform equal Payments
(1+i)N-1
• F=A
i
– uniform series compound amount factor in [ ]
– functionally expressed as F = A ( F / A,i%,N )
– predetermined values are in Interest Factors Table
for Discrete Compounding
F=?
1 2 3 4 5 6 7 8
A=
Example:
• Given: i = 6%, N = 5 years, A = 5000
• F = 5000 (F/A, 6%, 5)
•
= 5000 (5.6371)
= 28,185.46
RELATING A UNIFORM SERIES (ORDINARY
ANNUITY) 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
A= 1 2 3 4 5 6 7 8
• P=A
P=?
N
i(1+i)
– uniform series present worth factor in [ ]
– functionally expressed as P = A ( P / A,i%,N )
– predetermined values are in Interest Factors Table
for Discrete Compounding
Example:
• Lump Sum $104 m or 7.92 m annually
in 25 years total 198 m
• Given: i = 8%, N = 25 years, A = 7.92m
• P = 7.92 (P/A, 8%, 25) m
•
=7.92(10.6748)
= 85.54 m
RELATING A UNIFORM SERIES (ORDINARY
ANNUITY) TO PRESENT AND FUTURE EQUIVALENT
VALUES
• Finding A given F:
• Finding amount A of a uniform series when given the
equivalent future value
i
F=
A=F
1 2 3 4 5 6 7 8
A =?
( 1 + i ) N -1
– sinking fund factor in [ ]
– functionally expressed as A = F ( A / F,i%,N )
– predetermined values are in Interest Factors Table
for Discrete Compounding
Example:
• Given: i = 7%, N = 8 years, F = 100,000
• A = 100,000 (A/F, 7%, 8)
•
= 100000 (0.09747)
= 9,747
RELATING A UNIFORM SERIES (ORDINARY
ANNUITY) 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
P=
A=P
1 2 3 4 5 6 7 8
N
A =?
( 1 + i ) -1
– capital recovery factor in [ ]
– functionally expressed as A = P ( A / P,i%,N )
– predetermined values are in Interest Factors Table
for Discrete Compounding
Example:
You borrowed RM21,061.82 to finance the educational
expenses for your diploma course. The loan carries an
interest rate of 6% per year and is to be repaid in equal
annual installments over the next five years
• i = 6%, N = 5 years, P = 21,061.82
• A = P (A/P, 6%, 5)
•
= 21,061.82 (0.2374)
= 5,000
RELATING A UNIFORM SERIES (DEFERRED
ANNUITY) TO PRESENT / FUTURE
EQUIVALENT VALUES
• If an annuity is deferred j periods, where j < N
• And finding P given A for an ordinary annuity
is expressed by:
P = A ( P / A, i%,N )
• This is expressed for a deferred annuity by:
A ( P / A, i%,N - j ) at end of period j
• This is expressed for a deferred annuity by:
A ( P / A, i%,N - j ) ( P / F, i%, j )
as of time 0 (time present)
EQUIVALENCE CALCULATIONS INVOLVING
MULTIPLE INTEREST
• All compounding of interest takes place once per time
period (e.g., a year), and to this point cash flows also
occur once per time period.
• Consider an example where a series of cash outflows
occur over a number of years.
• Consider that the value of the outflows is unique for
each of a number (i.e., first three) years.
• Consider that the value of outflows is the same for
the last four years.
• Find a) the present equivalent expenditure; b) the
future equivalent expenditure; and c) the annual
equivalent expenditure
PRESENT EQUIVALENT EXPENDITURE
• Use P0 = F( P / F, i%, N ) for each of the unique years:
-- F is a series of unique outflow for year 1 through year 3;
-- i is common interest for each calculation;
-- N is the year in which the outflow occurred;
-- Multiply the outflow times the associated table value;
-- Add the three products together;
• Use A ( P / A,i%,N - j ) ( P / F, i%, j ) -- deferred annuity -- for
the remaining (common outflow) years:
-- A is common for years 4 through 7;
-- i remains the same;
-- N is the final year;
-- j is the last year a unique outflow occurred;
-- multiply the common outflow value times table values;
-- add this to the previous total for the present equivalent
expenditure.
RELATING A UNIFORM GRADIENT OF CASH
FLOWS TO FUTURE EQUIVALENTS
• Find F when given G:
• Find the future equivalent value when given the
uniform gradient amount
(1+i)N-1 -1 (1+i)N-2 -1
(1+i) 1 -1
• F=G
+
+ ... +
i
i
i
• Functionally represented as (G/ i) (F/A,i%,N) - (NG/ i)
• Usually more practical to deal with annual and
present equivalents, rather than future equivalent
values
Cash Flow Diagram for a Uniform Gradient
Increasing by G Dollars per period
(N-1)G
i = effective interest rate
per period
(N-2)G
(N-3)G
3G
2G
G
1
2
3
4
End of Period
N-2 N-1 N
RELATING A UNIFORM GRADIENT OF CASH
FLOWS TO ANNUAL AND PRESENT
EQUIVALENTS
• Find A when given G:
• Find the annual equivalent value when given the
uniform gradient amount
1
N
• A=G
i
(1 + i ) N - 1
• Functionally represented as A = G ( A / G, i%,N )
• The value shown in [ ] is the gradient to uniform series
conversion factor and is presented in column 9 of
Compound Interest Factor Tables
RELATING A UNIFORM GRADIENT OF CASH
FLOWS TO ANNUAL AND PRESENT
EQUIVALENTS
• Find P when given G:
• Find the present equivalent value when given the
uniform gradient amount
1
(1 + i ) N-1
N
• P=G
i
i (1 + i ) N
(1 + i ) N
• Functionally represented as P = G ( P / G, i%,N )
• The value shown in{ } is the gradient to present
equivalent conversion factor and is presented in
column 8 of Compound Interest Factor Tables
RELATING GE0METRIC SEQUENCE OF CASH
FLOWS TO PRESENT AND ANNUAL EQUIVALENTS
• Projected cash flow patterns changing at an average
rate of f each period;
• Resultant end-of-period cash-flow pattern is referred
to as a geometric gradient series;
• A1 is cash flow at end of period 1
• A k = (A k-1) ( 1 +f ),2 < k < N
• AN = A1 ( 1 + f ) N-1
• f = (A k - A k-1 ) / A k-1
• f may be either positive or negative
AN =A1(1+f )N - 1
A3 =A1(1+f )2
A2 =A1(1+f )
A1
0
1
2
3
4
End of Period
N
Cash-flow diagram for a Geometric Sequence of
Cash Flows
RELATING A GEOMETRIC SEQUENCE OF
CASH FLOWS TO ANNUAL AND PRESENT
EQUIVALENTS
• Find P when given A:
• Find the present equivalent value when given the
annual equivalent value ( i = f )
A1
1+i
P=
( P / A,
-1, N )
(1+f)
1+f
RELATING A GEOMETRIC SEQUENCE OF
CASH FLOWS TO ANNUAL AND PRESENT
EQUIVALENTS
• Find P when given A:
• Find the present equivalent value when given the
annual equivalent value ( i = f )
A1[1 – (1+i)–N (1+f)N]
P=
1-f
which may also be written as
A1[1 - (P/F,i%,N) (F/P,f%,N)]
P=
i-f
RELATING A GEOMETRIC SEQUENCE OF
CASH FLOWS TO ANNUAL AND PRESENT
EQUIVALENTS
• Note that the foregoing is mathematically equivalent
to the following (i = f ):
A1
P=
1+i
(P/A
1+f
-1, N )
1+f
`
RELATING A GEOMETRIC SEQUENCE OF
CASH FLOWS TO ANNUAL AND PRESENT
EQUIVALENTS
• The foregoing may be functionally represented as
= P (A / P, i%,N )
• The year zero “base” of annuity, increasing at
constant rate f % is A0 = P ( A / P, f %, N )
• The future equivalent of this geometric gradient is
F = P ( F / P, i%, N )
A
RELATING A GEOMETRIC SEQUENCE OF
CASH FLOWS TO ANNUAL AND PRESENT
EQUIVALENTS
• Find P when given A:
• Find the present equivalent value when given the
annual equivalent value ( i = f )
P = A1N (1+i)-1 which may be written as
RELATING A GEOMETRIC SEQUENCE OF
CASH FLOWS TO ANNUAL AND PRESENT
EQUIVALENTS
• Find P when given A:
• Find the present equivalent value when given the
annual equivalent value ( i = f )
P = A1N (P/F,i%,1)
Functionally represented as A = P (A / P, i%,N )
RELATING A GEOMETRIC SEQUENCE OF
CASH FLOWS TO ANNUAL AND PRESENT
EQUIVALENTS
• Find P when given A:
• Find the present equivalent value when given the
annual equivalent value ( i = f )
P = A1N (i+i)-1 which may be written as
P = A1N (P/F,i%,1)
Functionally represented as A = P (A / P, i%,N )
• The year zero “base” of annuity, increasing at
constant rate f % is A0 = P ( A / P, f %, N )
• The future equivalent of this geometric gradient is
F = P ( F / P, i%, N )
NOMINAL AND EFFECTIVE INTEREST RATES
• Nominal Interest Rate - r - For rates
compounded more frequently than one year,
the stated annual interest rate.
• Effective Interest Rate - i - For rates
compounded more frequently than one year,
the actual amount of interest paid.
• i = ( 1 + r / M )M - 1 = ( F / P, r / M, M ) -1
– M - the number of compounding periods per year
• Annual Percentage Rate - APR - percentage
rate per period times number of periods.
– APR = r x M
COMPOUNDING MORE OFTEN THAN ONCE A
YEAR
Single Amounts
• Given nominal interest rate and total number of
compounding periods, P, F or A can be determined
by
F = P ( F / P, i%, N )
i% = ( 1 + r / M ) M - 1
Uniform and / or Gradient Series
• Given nominal interest rate, total number of
compounding periods, and existence of a cash flow
at the end of each period, P, F or A may be
determined by the formulas and tables for uniform
annual series and uniform gradient series.
CONTINUOUS COMPOUNDING AND DISCRETE
CASH FLOWS
• Continuous compounding assumes cash flows occur
at discrete intervals, but compounding is continuous
throughout the interval.
• Given nominal per year interest rate -- r,
compounding per year -- M
one unit of principal = [ 1 + (r / M ) ] M
• Given M / r = p,
[ 1 + (r / M ) ] M = [1 + (1/p) ] rp
• Given lim [ 1 + (1 / p) ] p = e1 = 2.71828
pv
• ( F / P, r%, N ) = e rN
• i=er-1
•
•
•
•
•
•
CONTINUOUS COMPOUNDING AND
DISCRETE CASH FLOWS
Single Cash Flow
Finding F given P
Finding future equivalent value given present
value
F = P (e rN)
Functionally expressed as ( F / P, r%, N )
e rN is continuous compounding compound
amount
Predetermined values are in Continuous
Compounding Table
CONTINUOUS COMPOUNDING AND
DISCRETE CASH FLOWS
Single Cash Flow
• Finding P given F
• Finding present equivalent value given future
value
• P = F (e -rN)
• Functionally expressed as ( P / F, r%, N )
• e -rN is continuous compounding present
equivalent
• Predetermined values are in Continuous
Compounding Table
CONTINUOUS COMPOUNDING AND
DISCRETE CASH FLOWS
Uniform Series
• Finding F given A
• Finding future equivalent value given a series
of uniform equal receipts
• F = A (e rN- 1)/(e r- 1)
• Functionally expressed as ( F / A, r%, N )
• (e rN- 1)/(e r- 1) is continuous compounding
compound amount
• Predetermined values are in Continuous
Compounding Table
•
•
•
•
•
•
CONTINUOUS COMPOUNDING AND
DISCRETE CASH FLOWS
Uniform Series
Finding P given A
Finding present equivalent value given a series
of uniform equal receipts
P = A (e rN- 1) / (e rN ) (e r- 1)
Functionally expressed as ( P / A, r%, N )
(e rN- 1) / (e rN ) (e r- 1) is continuous
compounding present equivalent
Predetermined values are in Continuous
Compounding Table
CONTINUOUS COMPOUNDING AND
DISCRETE CASH FLOWS
Uniform Series
•
•
•
•
•
Finding A given F
Finding a uniform series given a future value
A = F (e r- 1) / (e rN - 1)
Functionally expressed as ( A / F, r%, N )
(e r- 1) / (e rN - 1) is continuous compounding
sinking fund
• Predetermined values are in Continuous
Compounding Table
•
•
•
•
•
•
CONTINUOUS COMPOUNDING AND
DISCRETE CASH FLOWS
Uniform Series
Finding A given P
Finding a series of uniform equal receipts given
present equivalent value
A = P [e rN (e r- 1) / (e rN - 1) ]
Functionally expressed as ( A / P, r%, N )
[e rN (e r- 1) / (e rN - 1) ] is continuous
compounding capital recovery
Predetermined values are in Continuous
Compounding Table
CONTINUOUS COMPOUNDING AND
CONTINUOUS CASH FLOWS
• Continuous flow of funds suggests a series of cash
flows occurring at infinitesimally short intervals of
time
• Given:
– a nominal interest rate or r
– p is payments per year
[ 1 + (r / p ) ] p - 1
P = -----------------------------r[1+(r/p)]p
p --> oo
• Given Lim [ 1 + ( r / p ) ] p = e r
• For one year ( P / A, r%, 1 ) = ( e r - 1 ) / re r
CONTINUOUS COMPOUNDING AND
CONTINUOUS CASH FLOWS
• Finding F given A
• Finding the future equivalent given the
continuous funds flow
• F = A [ ( erN - 1 ) / r ]
• Functionally expressed as ( F / A, r%, N )
• ( erN - 1 ) / r is continuous compounding
compound amount
• Predetermined values are found in Continuous
Compounding Table
CONTINUOUS COMPOUNDING AND
CONTINUOUS CASH FLOWS
• Finding P given A
• Finding the present equivalent given the
continuous funds flow
• P = A [ ( erN - 1 ) / rerN ]
• Functionally expressed as ( P / A, r%, N )
• ( erN - 1 ) / rerN is continuous compounding
present equivalent
• Predetermined values are found in Continuous
Compounding Table
CONTINUOUS COMPOUNDING AND
CONTINUOUS CASH FLOWS
• Finding A given F
• Finding the continuous funds flow given the
future equivalent
• A = F [ r / ( erN - 1 )]
• Functionally expressed as ( A / F, r%, N )
• r / ( erN - 1 ) is continuous compounding
sinking fund
CONTINUOUS COMPOUNDING AND
CONTINUOUS CASH FLOWS
• Finding A given P
• Finding the continuous funds flow given the
present equivalent
• A = P [ r / ( erN - 1 )]
• Functionally expressed as ( A / P, r%, N )
CONTINUOUS COMPOUNDING AND
CONTINUOUS CASH FLOWS
• Finding A given P
• Finding the continuous funds flow given the
present equivalent
• A = F [ rerN / ( erN - 1 )]
• Functionally expressed as ( A / P, r%, N )
• rerN / ( erN - 1 ) is continuous compounding
capital recovery
QUIZ 2
1. George obtained a new credit facilities from ABC
Bank, with a stated rate 8% compounded quarterly.
For a RM12,000 balance at the beginning of the
year, find the effective annual rate and total owned to
ABC Bank after a year, provided no payments are
made during the year.
2. Tasha plans to place money in a Koperasi Unimap
capital fund that currently returns 24% per year,
compounded daily. What effective rate is this (a)
yearly and (b) quarterly