CHAPTER 4 MONEY-TIME RELATIONSHIPS AND EQUIVALENCE.

CHAPTER 4CHAPTER 4

MONEY-TIME RELATIONSHIPS

AND EQUIVALENCE

MONEY-TIME RELATIONSHIPS

AND EQUIVALENCE

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.

KINDS OF CAPITALKINDS 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.

• 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

• 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

INTERESTINTEREST

The fee that a borrower pays to a lender for the use of his or her money.

INTEREST RATEThe percentage of money being borrowed that is paid to the

lender on some time basis.

The fee that a borrower pays to a lender for the use of his or her money.

INTEREST RATEThe percentage of money being borrowed that is paid to the

lender on some time basis.

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.

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.

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 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

• f may be either positive or negative




• 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

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 )

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



EQUIVALENTS




A1 1 + i

P = ( P / A, -1, N )

( 1 + f ) 1 + f



A1 1 + i

P = ( P / A, -1, N )

( 1 + f ) 1 + f


EQUIVALENTS




A1[1 – (1+i)–N (1+f)N]

P =

i - f

which may also be written as

A1[1 - (P/F,i%,N) (F/P,f%,N)]

P =

i - f



A1[1 – (1+i)–N (1+f)N]

P =

i - f

which may also be written as

A1[1 - (P/F,i%,N) (F/P,f%,N)]

P =

i - f


EQUIVALENTS


EQUIVALENTS• Note that the foregoing is mathematically equivalent

to the following (i = f ):

A1 1 + i

P = ( P / A -1, N ) `

1 + f 1 + f

• Note that the foregoing is mathematically equivalent to the following (i = f ):

A1 1 + i

P = ( P / A -1, N ) `

1 + f 1 + f


EQUIVALENTS


EQUIVALENTS

• The foregoing may be 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 )

• The foregoing may be 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 )



EQUIVALENTS


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


EQUIVALENTS




P = A1N (1+i)-1 which may be written as



P = A1N (1+i)-1 which may be written as


EQUIVALENTS




P = A1N (P/F,i%,1)

Functionally represented as A = P (A / P, i%,N )



P = A1N (P/F,i%,1)

Functionally represented as A = P (A / P, i%,N )


EQUIVALENTS




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 )




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 )


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

CHAPTER 4 MONEY-TIME RELATIONSHIPS AND EQUIVALENCE.

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Transcript of CHAPTER 4 MONEY-TIME RELATIONSHIPS AND EQUIVALENCE.