principles of money-time relationships. money medium of exchange -- means of payment for goods or...

PRINCIPLES OF MONEY-TIME

RELATIONSHIPS

PRINCIPLES OF MONEY-TIME

RELATIONSHIPS

MONEYMONEY• 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;

• 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;

CAPITALCAPITAL

Wealth in the form of money or property that can be used to

produce more wealth.

Wealth in the form of money or property that can be used to

produce more wealth.

CAPITALCAPITAL

• The majority of engineering economy studies involve commitment of capital for extended period of time, so the effect of time must be considered.

• A dollar today is worth more than a dollar one or more years from now because of the interest (or profit) it can earn. Therefore, money has a time value.

• The majority of engineering economy studies involve commitment of capital for extended period of time, so the effect of time must be considered.

• A dollar today is worth more than a dollar one or more years from now because of the interest (or profit) it can earn. Therefore, money has a time value.

KINDS OF CAPITAL

Capital in the form of money for the people, machines, materials, energy, and other things needed in the operation of an organization may be classified into two basic categories:

• 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. In return the lenders receive interest from the borrowers.

Financing Definition Instrument Description

• Debt financing

• Equity financing

• Borrow money

• Sell partial ownership of company;

• Bond

• Stock

• Promise to pay principle & interest;

• Exchange shares of stock for ownership of company;

Financing Definition Instrument Description

• Debt financing

• Equity financing

• Borrow money

• Sell partial ownership of company;

• Bond

• Stock

• Promise to pay principle & interest;

• Exchange shares of stock for ownership of company;

Exchange money for shares of stock as proof of partial ownership

INTERESTINTERESTThe 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.• Is the rate of gain received from an investment.

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.• Is the rate of gain received from an investment.

HOW INTEREST RATE IS DETERMINED

HOW INTEREST RATE IS DETERMINEDInterest

Rate

Quantity of Money



Rate

Quantity of Money

Money Demand



Rate

Quantity of Money

Money Demand

Money SupplyMS1



Rate

Quantity of Money

ieMoney Demand

Money SupplyMS1



Rate

Quantity of Money

ie

Money Demand

Money SupplyMS1 MS2

i2



Rate

Quantity of Money

ie

Money Demand

Money SupplyMS1 MS2

i2

MS3

i3

SIMPLE INTERESTSIMPLE 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

• 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 INTERESTCOMPOUND 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

• 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 EQUIVALENCEECONOMIC 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.

• 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 LOANECONOMIC 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)

• 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


• 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


• 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



REPAYMENT PLANS OF AN $8,000 LOAN


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


• 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



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 $02 $8,800 $880 $9,680 $0 $03 $9,680 $968 $10,648 $0 $04 $10,648 $1,065 $11,713 $8,000 $11,713Total interest paid ($3,713) is 10% of total dollar-years ($37,128)

• 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 $02 $8,800 $880 $9,680 $0 $03 $9,680 $968 $10,648 $0 $04 $10,648 $1,065 $11,713 $8,000 $11,713Total interest paid ($3,713) is 10% of total dollar-years ($37,128)

CASH FLOW DIAGRAMS / TABLE NOTATION

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-of-period 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

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-of-period 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

1 2 3 4 5 = N1

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.


1 2 3 4 5 = N1


P =$8,000 2

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


1 2 3 4 5 = N1


P =$8,000 2


A = $2,524 3

3 Annual income (cash inflow) of $2,524 for lender.


1 2 3 4 5 = N1


P =$8,000 2


A = $2,524 3


i = 10% per year4

4 Interest rate of loan.


1 2 3 4 5 = N1


P =$8,000 2


A = $2,524 3


i = 10% per year4

4 Interest rate of loan.

5

5 Dashed-arrow line indicates amount to be determined.

INTEREST FORMULAS FOR ALL OCCASIONSINTEREST 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;

• 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 OCCASIONSINTEREST 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 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


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

column 2 of Appendix C of text.

• 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

column 2 of Appendix C of text.P

0

N =

F = ?

• 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

column 3 of Appendix C of text;

• 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

column 3 of Appendix C of text;


FLOWS


FLOWS

P = ?

0 N = F

RELATING A UNIFORM SERIES (ORDINARY ANNUITY) TO PRESENT AND FUTURE EQUIVALENT

VALUES


VALUES

• Finding F given A:• Finding F given A:


VALUES


VALUES• Finding F given A:• Finding future equivalent income (inflow) value given

a series of uniform equal Payments

• Finding F given A:• Finding future equivalent income (inflow) value given

a series of uniform equal Payments


VALUES



a series of uniform equal Payments ( 1 + i ) N - 1

• F = A

i



• F = A

i


VALUES




• F = A

i– uniform series compound amount factor in [ ]



• F = A

i– uniform series compound amount factor in [ ]


VALUES




• F = A

i– uniform series compound amount factor in [ ]– functionally expressed as F = A ( F / A,i%,N )



• F = A

i– uniform series compound amount factor in [ ]– functionally expressed as F = A ( F / A,i%,N )


VALUES




• F = A

i– uniform series compound amount factor in [ ]– functionally expressed as F = A ( F / A,i%,N )– predetermined values are in column 4 of Appendix

C of text



• F = A

i– uniform series compound amount factor in [ ]– functionally expressed as F = A ( F / A,i%,N )– predetermined values are in column 4 of Appendix

C of text


VALUES


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 column 4 of Appendix C

of text

• 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 column 4 of Appendix C

of textF = ?

1 2 3 4 5 6 7 8 A =

( F / A,i%,N ) = (P / A,i,N ) ( F / P,i,N )

( F / A,i%,N ) = F / P,i,N-k )N

k = 1


VALUES


VALUES• Finding P given A:• Finding P given A:


VALUES


VALUES• Finding P given A:• Finding present equivalent value given a series of

uniform equal receipts

• Finding P given A:• Finding present equivalent value given a series of



VALUES




( 1 + i ) N - 1• P = A

i ( 1 + i ) N



( 1 + i ) N - 1• P = A

i ( 1 + i ) N


VALUES




( 1 + i ) N - 1• P = A

i ( 1 + i ) N

– uniform series present worth factor in [ ]



( 1 + i ) N - 1• P = A

i ( 1 + i ) N

– uniform series present worth factor in [ ]


VALUES




( 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 )



( 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 )


VALUES




( 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 Appendix

C of text



( 1 + i ) N - 1• P = A

i ( 1 + i ) N


C of text


VALUES




( 1 + i ) N - 1• P = A

i ( 1 + i ) N


C of text



( 1 + i ) N - 1• P = A

i ( 1 + i ) N


C of text

P = ?

1 2 3 4 5 6 7 8A =

( P / A,i%,N ) =P / F,i,k )N

k = 1


VALUES


VALUES• Finding A given F:• Finding A given F:


VALUES


VALUES• Finding A given F:• Finding amount A of a uniform series when given the

equivalent future value

• Finding A given F:• Finding amount A of a uniform series when given the



VALUES




i

A = F

( 1 + i ) N -1



i

A = F

( 1 + i ) N -1


VALUES




i

A = F

( 1 + i ) N -1– sinking fund factor in [ ]



i

A = F

( 1 + i ) N -1– sinking fund factor in [ ]


VALUES




i

A = F

( 1 + i ) N -1– sinking fund factor in [ ]– functionally expressed as A = F ( A / F,i%,N )



i

A = F

( 1 + i ) N -1– sinking fund factor in [ ]– functionally expressed as A = F ( A / F,i%,N )


VALUES




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 Appendix

C of text



i

A = F


C of text


VALUES




i

A = F


C of text



i

A = F


C of text F =

1 2 3 4 5 6 7 8 A =?

( A / F,i%,N ) = 1 / ( F / A,i%,N )

( A / F,i%,N ) = ( A / P,i%,N ) - i


VALUES


VALUES• Finding A given P:• Finding A given P:


VALUES


VALUES• Finding A given P:• Finding amount A of a uniform series when given the

equivalent present value

• Finding A given P:• Finding amount A of a uniform series when given the



VALUES




i ( 1+i )N

A = P

( 1 + i ) N -1



i ( 1+i )N

A = P

( 1 + i ) N -1


VALUES




i ( 1+i )N

A = P

( 1 + i ) N -1– capital recovery factor in [ ]



i ( 1+i )N

A = P

( 1 + i ) N -1– capital recovery factor in [ ]


VALUES




i ( 1+i )N

A = P

( 1 + i ) N -1– capital recovery factor in [ ]– functionally expressed as A = P ( A / P,i%,N )



i ( 1+i )N

A = P

( 1 + i ) N -1– capital recovery factor in [ ]– functionally expressed as A = P ( A / P,i%,N )


VALUES




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 Appendix

C of text



i ( 1+i )N

A = P


C of text


VALUES




i ( 1+i )N

A = P


C of text



i ( 1+i )N

A = P


C of text P =

1 2 3 4 5 6 7 8 A =?

( A / P,i%,N ) = 1 / ( P / A,i%,N )

RELATING A UNIFORM SERIES (DEFERRED ANNUITY) TO PRESENT / FUTURE

EQUIVALENT VALUES

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)

• 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

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

• 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 EXPENDITUREPRESENT 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 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.

• 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 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 F when given G:



• 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 + + ... +



(1+i)N-1 -1 (1+i)N-2 -1 (1+i) 1 -1• F = G + + ... +

i i i





(1+i)N-1 -1 (1+i)N-2 -1 (1+i) 1 -1• F = G + + ... +

• Functionally represented as (G/ i) (F/A,i%,N) - (NG/ i)



(1+i)N-1 -1 (1+i)N-2 -1 (1+i) 1 -1• F = G + + ... +

• Functionally represented as (G/ i) (F/A,i%,N) - (NG/ i)

i i i





(1+i)N-1 -1 (1+i)N-2 -1 (1+i) 1 -1• F = G + + ... +

• 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



(1+i)N-1 -1 (1+i)N-2 -1 (1+i) 1 -1• F = G + + ... +

• 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

i i i

Cash Flow Diagram for a Uniform Gradient Increasing by G Dollars per period

1 2 3 4 N-2 N-1 N

G

2G3G

(N-3)G

(N-2)G

(N-1)Gi = effective interest rate per period

End of Period

RELATING A UNIFORM GRADIENT OF CASH FLOWS TO ANNUAL AND PRESENT

EQUIVALENTS


EQUIVALENTS• Find A when given G:• Find A when given G:


EQUIVALENTS


EQUIVALENTS• Find A when given G:• Find the annual equivalent value when given the


• Find A when given G:• Find the annual equivalent value when given the



EQUIVALENTS




1 N• A = G -

i (1 + i ) N - 1



1 N• A = G -

i (1 + i ) N - 1


EQUIVALENTS




1 N• A = G -

i (1 + i ) N - 1• Functionally represented as A = G ( A / G, i%,N )



1 N• A = G -

i (1 + i ) N - 1• Functionally represented as A = G ( A / G, i%,N )


EQUIVALENTS




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 Appendix C (represented in the above parenthetical expression).



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 Appendix C (represented in the above parenthetical expression).


EQUIVALENTS


EQUIVALENTS• Find P when given G:• Find P when given G:


EQUIVALENTS


EQUIVALENTS• Find P when given G:• Find the present equivalent value when given the


• Find P when given G:• Find the present equivalent value when given the



EQUIVALENTS




1 (1 + i ) N-1 N• P = G -

i i (1 + i ) N (1 + i ) N



1 (1 + i ) N-1 N• P = G -

i i (1 + i ) N (1 + i ) N


EQUIVALENTS




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 )



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 )


EQUIVALENTS




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 Appendix C (represented in the above parenthetical expression).



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 Appendix C (represented in the above parenthetical expression).

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




• A k = (A k-1) ( 1 +f ),2 < k < N






• A k = (A k-1) ( 1 +f ),2 < k < N

• AN = A1 ( 1 + f ) N-1




• A k = (A k-1) ( 1 +f ),2 < k < N

• AN = A1 ( 1 + f ) N-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




• 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






• 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

• convenience rate = i cr = [ ( 1 + i ) / ( 1 + f ) ] - 1= ( i - f ) / ( 1 + f )




• 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

• convenience rate = i cr = [ ( 1 + i ) / ( 1 + f ) ] - 1= ( i - f ) / ( 1 + f )

0 1 2 3 4 N

A1

A2 =A1(1+f )

A3 =A1(1+f )2

AN =A1(1+f )N - 1

End of Period

Cash-flow diagram for a Geometric Sequence of Cash Flows

RELATING A GEOMETRIC SEQUENCE OF CASH FLOWS TO ANNUAL AND PRESENT

EQUIVALENTS


EQUIVALENTS• Find P when given A:• Find P when given A:


EQUIVALENTS


EQUIVALENTS• Find P when given A:• Find the present equivalent value when given the

annual equivalent value ( i = f )and ( icr = 0 )

• Find P when given A:• Find the present equivalent value when given the



EQUIVALENTS




A1

P = ( P / A, iCR%, N )

( 1 + f )



A1

P = ( P / A, iCR%, N )

( 1 + f )


EQUIVALENTS




A1

P = ( P / A, iCR%, N )

1 + f• Functionally represented as A = P (A / P, i%,N )



A1

P = ( P / A, iCR%, N )



EQUIVALENTS




A1

P = ( P / A, iCR%, N )

1 + f• 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 )



A1

P = ( P / A, iCR%, N )




EQUIVALENTS




A1

P = ( P / A, iCR%, N )



• The future equivalent of this geometric gradient is F = P ( F / P, i%, N )



A1

P = ( P / A, iCR%, N )





EQUIVALENTS


EQUIVALENTS• Find P when given A:• Find P when given A:


EQUIVALENTS







EQUIVALENTS




NA1

P =

1 + f



NA1

P =

1 + f


EQUIVALENTS




NA1

P =




NA1

P =



EQUIVALENTS




NA1

P =


constant rate f% is A0 = P ( A / P, iCR%, N )



NA1

P =


constant rate f% is A0 = P ( A / P, iCR%, N )


EQUIVALENTS




NA1

P =






NA1

P =




INTEREST RATES THAT VARY WITH TIME


• Find P given F and interest rates that vary over N





• Find the present equivalent value given a future value and a varying interest rate over the period of the loan







• FN P = -----------------

N (1 + ik)



• FN P = -----------------

N (1 + ik)k + 1k + 1

NOMINAL AND EFFECTIVE INTEREST RATESNOMINAL 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

• 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

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.

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.

CASH FLOWS LESS OFTEN THAN COMPOUNDING PERIODS

CASH FLOWS LESS OFTEN THAN COMPOUNDING PERIODS

• Find A, given i, k and X, where:– i is the effective interest rate per interest period– k is the period at the end of which cash flow occurs– X is the uniform cash flow amount

Use: A = X (A / F,i%, k )

• Find A, given i, k and X, where: – i is the effective interest rate per interest period– k is the period at the beginning of which cash flow

occurs– X is the uniform cash flow amount

Use: A = X ( A / P, i%, k )

• Find A, given i, k and X, where:– i is the effective interest rate per interest period– k is the period at the end of which cash flow occurs– X is the uniform cash flow amount

Use: A = X (A / F,i%, k )

• Find A, given i, k and X, where: – i is the effective interest rate per interest period– k is the period at the beginning of which cash flow

occurs– X is the uniform cash flow amount

Use: A = X ( A / P, i%, k )

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 M / r = p, [ 1 + (r / M ) ] M = [1 + (1/p) ] rp





• Given M / r = p, [ 1 + (r / M ) ] M = [1 + (1/p) ] rp

• Given lim [ 1 + (1 / p) ] p = e1 = 2.71828



• Given M / r = p, [ 1 + (r / M ) ] M = [1 + (1/p) ] rp

• Given lim [ 1 + (1 / p) ] p = e1 = 2.71828p





• Given M / r = p, [ 1 + (r / M ) ] M = [1 + (1/p) ] rp

• Given lim [ 1 + (1 / p) ] p = e1 = 2.71828 • ( F / P, r%, N ) = e rN



• Given M / r = p, [ 1 + (r / M ) ] M = [1 + (1/p) ] rp

• Given lim [ 1 + (1 / p) ] p = e1 = 2.71828 • ( F / P, r%, N ) = e rNp





• Given M / r = p, [ 1 + (r / M ) ] M = [1 + (1/p) ] rp


• i = e r - 1



• Given M / r = p, [ 1 + (r / M ) ] M = [1 + (1/p) ] rp


• i = e r - 1

p


Single Cash Flow


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 column 2 of appendix D of text

• 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



Single Cash Flow


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


• 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



Uniform Series


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



• Finding future equivalent value given a series of uniform equal receipts

• F = A (e rN- 1)/(e r- 1)


• (e rN- 1)/(e r- 1) is continuous compounding compound amount



Uniform Series


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


• 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)


• (e rN- 1) / (e rN ) (e r- 1) is continuous compounding present equivalent



Uniform Series


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



• Finding a uniform series given a future value

• A = F (e r- 1) / (e rN - 1)


• (e r- 1) / (e rN - 1) is continuous compounding sinking fund



Uniform Series


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



• Finding a series of uniform equal receipts given present equivalent value

• A = P [e rN (e r- 1) / (e rN - 1) ]


• [e rN (e r- 1) / (e rN - 1) ] is continuous compounding capital recovery


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



[ 1 + (r / p ) ] p - 1

P = ------------------------------

r [ 1 + ( r / p ) ] p





[ 1 + (r / p ) ] p - 1

P = ------------------------------

r [ 1 + ( r / p ) ] p • Given Lim [ 1 + ( r / p ) ] p = e r



[ 1 + (r / p ) ] p - 1

P = ------------------------------


p p --> oo --> oo





[ 1 + (r / p ) ] p - 1

P = ------------------------------


• For one year ( P / A, r%, 1 ) = ( e r - 1 ) / re r



[ 1 + (r / p ) ] p - 1

P = ------------------------------


• For one year ( P / A, r%, 1 ) = ( e r - 1 ) / re r

p p --> oo --> oo



• Finding F given A• Finding F given A




• Finding the future equivalent given the continuous funds flow







• F = A [ ( erN - 1 ) / r ]



• F = A [ ( erN - 1 ) / r ]





• F = A [ ( erN - 1 ) / r ]




• F = A [ ( erN - 1 ) / r ]






• F = A [ ( erN - 1 ) / r ]


• ( erN - 1 ) / r is continuous compounding compound amount



• F = A [ ( erN - 1 ) / r ]







• F = A [ ( erN - 1 ) / r ]



• Predetermined values are found in column 6 of appendix D of text.



• F = A [ ( erN - 1 ) / r ]






• Finding P given A• Finding P given A




• Finding the present equivalent given the continuous funds flow







• P = A [ ( erN - 1 ) / rerN ]



• P = A [ ( erN - 1 ) / rerN ]





• P = A [ ( erN - 1 ) / rerN ]




• P = A [ ( erN - 1 ) / rerN ]






• P = A [ ( erN - 1 ) / rerN ]


• ( erN - 1 ) / rerN is continuous compounding present equivalent



• P = A [ ( erN - 1 ) / rerN ]







• P = A [ ( erN - 1 ) / rerN ]






• P = A [ ( erN - 1 ) / rerN ]






• Finding A given F• Finding A given F




• Finding the continuous funds flow given the future equivalent







• A = F [ r / ( erN - 1 )]



• A = F [ r / ( erN - 1 )]





• A = F [ r / ( erN - 1 )]




• A = F [ r / ( erN - 1 )]






• A = F [ r / ( erN - 1 )]


• r / ( erN - 1 ) is continuous compounding sinking fund



• A = F [ r / ( erN - 1 )]


• r / ( erN - 1 ) is continuous compounding sinking fund



• Finding A given P• Finding A given P




• Finding the continuous funds flow given the present equivalent







• A = P [ r / ( erN - 1 )]



• A = P [ r / ( erN - 1 )]





• A = P [ r / ( erN - 1 )]




• A = P [ r / ( erN - 1 )]






• A = F [ rerN / ( erN - 1 )]


• rerN / ( erN - 1 ) is continuous compounding capital recovery



• A = F [ rerN / ( erN - 1 )]


• rerN / ( erN - 1 ) is continuous compounding capital recovery

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