180211 applications

8/7/2019 180211 Applications

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APPLICATIONS OF

MONEY-TIMERELATIONSHIPS



MINIMUM ATTRACTIVE RATE OFRETURN ( MARR )

• An interest rate used to convert cash flows into

equivalent worth at some point(s) in time• Usually a policy issue based on: - amount, source and cost of money available for

investment - number and purpose of good projects available for

investment - amount of perceived risk of investment

opportunities and estimated cost of administeringprojects over short and long run

- type of organization involved

• MARR is sometimes referred to as hurdle rate



CAPITAL RATIONING• MARR approach involving opportunity cost

viewpoint• Exists when management decides to restrict

the total amount of capital invested, by

desire or limit of available capital• Select only those projects which provide

annual rate of return in excess of MARR

• As amount of investment capital andopportunities available change over time, afirm’s MARR will also change



PRESENT WORTH METHOD ( PW )

• Based on concept of equivalent worth of all

cash flows relative to the present as a base• All cash inflows and outflows discounted to

present at interest -- generally MARR

• PW is a measure of how much money can beafforded for investment in excess of cost

• PW is positive if dollar amount received for

investment exceeds minimum required byinvestors



FINDING PRESENT WORTHFINDING PRESENT WORTH•Discount future amounts to the present by using the

interest rate over the appropriate study period•Discount future amounts to the present by using the

interest rate over the appropriate study period





PW = Σ Fk ( 1 + i ) - k

–i = effective interest rate, or MARR per

compounding period–k = index for each compounding period

–Fk = future cash flow at the end of period k

–N = number of compounding periods in studyperiod

•Discount future amounts to the present by using theinterest rate over the appropriate study period

PW = Σ Fk ( 1 + i ) - k

–i = effective interest rate, or MARR per compounding period

–k = index for each compounding period


–N = number of compounding periods in study

period

k = 0k = 0

NN





PW = Σ Fk ( 1 + i ) - k

–i = effective interest rate, or MARR per

compounding period–k = index for each compounding period


–N = number of compounding periods in studyperiod

•interest rate is assumed constant through project

•Discount future amounts to the present by using theinterest rate over the appropriate study period

PW = Σ Fk ( 1 + i ) - k

–i = effective interest rate, or MARR per compounding period



–N = number of compounding periods in study

period•interest rate is assumed constant through project

k = 0k = 0

NN



O S O



BOND AS EXAMPLE OFPRESENT WORTH

• The value of a bond, at any time, is the present

worth of future cash receipts from the bond• The bond owner receives two types of

payments from the borrower: -- periodic interest payments until the bond is

retired ( based on r ); -- redemption or disposal payment when the bond

is retired ( based on i );

• The present worth of the bond is the sum of thepresent values of these two payments at thebond’s yield rate



PRESENT WORTH OF A BOND• For a bond, let Z = face, or par value C = redemption or disposal price (usually Z ) r = bond rate (nominal interest) per interest period N = number of periods before redemption

i = bond yield (redemption ) rate per period

VN = value (price) of the bond N interest periods

prior to redemption -- PW measure of merit VN = C ( P / F, i%, N ) + rZ ( P / A, i %, N )

• Periodic interest payments to owner = rZ for N periods-- an annuity of N payments

• When bond is sold, receive single payment (C), based

on the price and the bond yield rate ( i )



FUTURE WORTH METHOD (FW )FUTURE WORTH METHOD (FW )•FW is based on the equivalent worth of all cash inflows

and outflows at the end of the planning horizon at aninterest rate that is generally MARR

•FW is based on the equivalent worth of all cash inflows






•The FW of a project is equivalent to PW

FW = PW ( F / P, i %, N )




FW = PW ( F / P, i %, N )






FW = PW ( F / P, i %, N )

•If FW > 0, it is economically justified




FW = PW ( F / P, i %, N )







FW = PW ( F / P, i %, N )


FW ( i % ) = Σ Fk ( 1 + i ) N - k




FW = PW ( F / P, i %, N )


FW ( i % ) = Σ Fk ( 1 + i ) N - k

k = 0

N



FUTURE WORTH METHOD (FW )• FW is based on the equivalent worth of all cash

inflows and outflows at the end of the planninghorizon at an interest rate that is generally MARR

• The FW of a project is equivalent to PW FW = PW ( F / P, i %, N )

• If FW > 0, it is economically justified FW ( i % ) = Σ Fk ( 1 + i ) N - k

k = 0

N

–i = effective interest rate



–N = number of compounding periods in study period



ANNUAL WORTH METHOD ( AW )• AW is an equal annual series of dollar amounts, over

a stated period ( N ), equivalent to the cash inflowsand outflows at interest rate that is generally MARR

• AW is annual equivalent revenues ( R ) minus annualequivalent expenses ( E ), less the annual

equivalent capital recovery (CR) AW ( i % ) = R - E - CR ( i % )

• AW = PW ( A / P, i %, N )

• AW = FW ( A / F,i

%, N )• If AW > 0, project is economically attractive

• AW = 0 : annual return = MARR earned



CAPITAL RECOVERY ( CR )• CR is the equivalent uniform annual cost of the

capital invested• CR is an annual amount that covers:

– Loss in value of the asset

– Interest on invested capital ( i.e., at the MARR ) CR ( i % ) = I ( A / P, i %, N ) - S ( A / F, i %, N ) I = initial investment for the project

S = salvage ( market ) value at the end of thestudy period N = project study period



INTERNAL RATE OF RETURN METHOD ( IRR )• IRR solves for the interest rate that equates the

equivalent worth of an alternative’s cashinflows (receipts or savings) to the equivalentworth of cash outflows (expenditures)

• Also referred to as:

– investor’s method – discounted cash flow method

– profitability index

• IRR is positive for a single alternative only if: – both receipts and expenses are present in the

cash flow pattern

– the sum of receipts exceeds sum of cashoutflows



INTERNAL RATE OF RETURN METHOD ( IRR )INTERNAL RATE OF RETURN METHOD ( IRR )

•IRR is i ’ %, using the following PW formula:

Σ R k ( P / F, i ’ %, k ) = Σ E k ( P / F, i

’ %, k )


ΣR k ( P / F,

i ’ %, k ) =

ΣE k ( P / F,

i ’ %, k )

N N N N

k = 0k = 0k = 0k = 0





Σ R k ( P / F, i ’ %, k ) = Σ E k ( P / F, i

’ %, k )

R k = net revenues or savings for the kth year


ΣR k ( P / F,

i ’ %, k ) =

ΣE k ( P / F,

i ’ %, k )


N N N N

k = 0k = 0k = 0k = 0





Σ R k ( P / F, i ’ %, k ) = Σ E k ( P / F, i

’ %, k )


E k = net expenditures including investment

costs for the kth year


ΣR k ( P / F,

i ’ %, k ) =

ΣE k ( P / F,

i ’ %, k )



costs for the kth year

N N N N

k = 0k = 0k = 0k = 0





Σ R k ( P / F, i ’ %, k ) = Σ E k ( P / F, i

’ %, k )



costs for the kth year N = project life ( or study period )

•If i ’ > MARR, the alternative is acceptable


ΣR k ( P / F,

i ’ %, k ) =

ΣE k ( P / F,

i ’ %, k )



costs for the kth year N = project life ( or study period )

•If i ’ > MARR, the alternative is acceptable

N N N N

k = 0k = 0k = 0k = 0



INTERNAL RATE OF RETURN METHOD ( IRR )

• IRR is i ’ %, using the following PW formula:

ΣR k ( P / F,

i ’ %, k ) =

ΣE k ( P / F,

i ’ %, k )

R k = net revenues or savings for the kth

year

E k = net expenditures including investmentcosts for the kth year

N = project life ( or study period )

• If i ’ > MARR, the alternative is acceptable• To compute IRR for alternative, set net PW = 0 PW = Σ R k ( P / F, i ’ %, k ) - Σ E k ( P / F, i ’ %, k )

= 0

N N

k = 0k = 0

N N

k = 0k = 0

N N

k = 0k = 0

N N

k = 0k = 0

O O S



INTERNAL RATE OF RETURN PROBLEMS

• The IRR method assumes recovered funds, if

not consumed each time period, arereinvested at i ‘ %, rather than at MARR

• The computation of IRR may be

unmanageable• Multiple IRR’s may be calculated for the same

problem

• The IRR method must be carefully applied andinterpreted in the analysis of two or morealternatives, where only one is acceptable



THE EXTERNAL RATE OF RETURN METHOD( ERR )

• ERR directly takes into account theinterest rate ( ε ) external to a project atwhich net cash flows generated over the

project life can be reinvested (or borrowed ).

• If the external reinvestment rate, usually

the firm’s MARR, equals the IRR, thenERR method produces same results asIRR method



C C G O



CALCULATING EXTERNAL RATE OFRETURN ( ERR )


ΣEk ( P / F,

ε%, k )( F / P,

i

‘

%, N )=

Σ Rk ( F / P, ε %, N - k )

Σ Ek

( P / F, ε %, k )( F / P, i ‘ %, N )

=

Σ Rk ( F / P, ε %, N - k )

N N

k = 0k = 0

N N

k =k =

00

CALCULATING EXTERNAL RATE OF





ΣEk ( P / F,

ε%, k )( F / P,

i

‘

%, N )=

Σ Rk ( F / P, ε %, N - k )

Rk= excess of receipts over expenses in period k

Σ Ek

( P / F, ε %, k )( F / P, i ‘ %, N )

=

Σ Rk ( F / P, ε %, N - k )

Rk= excess of receipts over expenses in period k

N N

k = 0k = 0

N N

k =k =

00






ΣEk ( P / F,

ε%, k )( F / P,

i

‘

%, N )=

Σ Rk ( F / P, ε %, N - k )

Rk= excess of receipts over expenses in period kEk = excess of expenses over receipts in period k

Σ Ek

( P / F, ε %, k )( F / P, i ‘ %, N )

=

Σ Rk ( F / P, ε %, N - k )


N N

k = 0k = 0

N N

k =k =

00






ΣEk ( P / F,

ε%, k )( F / P,

i

‘

%, N )=

Σ Rk ( F / P, ε %, N - k )


N = project life or period of study

Σ Ek

( P / F, ε %, k )( F / P, i ‘ %, N )

=

Σ Rk ( F / P, ε %, N - k )



N N

k = 0k = 0

N N

k =k =

00






ΣEk ( P / F,

ε%, k )( F / P,

i

‘

%, N )=

Σ Rk ( F / P, ε %, N - k )



ε = external reinvestment rate per period

Σ Ek

( P / F, ε %, k )( F / P, i ‘ %, N )

=

Σ Rk ( F / P, ε %, N - k )




N N

k = 0k = 0

N N

k =k =

00





Σ Ek

( P / F, ε %, k )( F / P, i ‘ %, N )

=

Σ Rk ( F / P, ε %, N - k )




N N

k = 0k = 0

N N

k =k =

00

i ‘ %= ?

Time N

0

Σ Rk ( F / P, ε %, N - k ) N

k = 0

Σ Ek ( P / F, ε %, k )( F / P, i ‘ %, N )

N

k = 0



ERR ADVANTAGES

• ERR has two advantages over IRR:

1. It can usually be solved for directly, rather than by trial anderror.

2. It is not subject to multiple ratesof return.



PAYBACK PERIOD METHODPAYBACK PERIOD METHOD•Sometimes referred to as simple payout method•Sometimes referred to as simple payout method



PAYBACK PERIOD METHODPAYBACK PERIOD METHOD•Sometimes referred to as simple payout method

•Indicates liquidity (riskiness) rather than profitability

•Sometimes referred to as simple payout method





•Indicates liquidity (riskiness) rather than profitability•Calculates smallest number of years ( Θ ) needed for

cash inflows to equal cash outflows -- break-even life









•Θ ignores the time value of money and all cash flowswhich occur after Θ











Σ ( Rk -Ek) - I > 0



•Calculates smallest number of years ( Θ ) needed for cash inflows to equal cash outflows -- break-even life


Σ ( Rk -Ek) - I > 0

k = 1k = 1

Θ



PAYBACK PERIOD METHOD• Sometimes referred to as simple payout method

• Indicates liquidity (riskiness) rather than profitability• Calculates smallest number of years ( Θ ) needed for

cash inflows to equal cash outflows -- break-evenlife

• Θ ignores the time value of money and all cash flowswhich occur after Θ

Σ ( Rk -Ek) - I > 0

• If Θ is calculated to include some fraction of a year, itis rounded to the next highest year

k = 1k = 1

Θ

PAYBACK PERIOD METHODPAYBACK PERIOD METHOD



PAYBACK PERIOD METHODPAYBACK PERIOD METHOD•The payback period can produce misleading results,

and should only be used with one of the other methods of determining profitability

•The payback period can produce misleading results,and should only be used with one of the other methods of determining profitability






•A discounted payback period Θ ‘ ( where Θ ‘ < N )may be calculated so that the time value of money is

considered



considered







considered



considered

Σ ( Rk - Ek) ( P / F, i %, k ) - I > 0k = 1k = 1

Θ







considered

i ‘ is the MARR

•The payback period can produce misleading results,and should only be used with one of the other

methods of determining profitability


considered

i ‘ is the MARR

Σ ( Rk - Ek) ( P / F, i %, k ) - I > 0k = 1k = 1

Θ







considered

i ‘ is the MARR

I is the capital investment made at the present time




considered

i ‘ is the MARR


Σ ( Rk - Ek) ( P / F, i %, k ) - I > 0k = 1k = 1

Θ







considered

i ‘ is the MARR


( k = 0 ) is the present time




considered

i ‘ is the MARR


( k = 0 ) is the present time

Σ ( Rk - Ek) ( P / F, i %, k ) - I > 0k = 1k = 1

Θ

PAYBACK PERIOD METHOD



PAYBACK PERIOD METHOD• The payback period can produce misleading results,

and should only be used with one of the other

methods of determining profitability• A discounted payback period Θ ‘ ( where Θ ‘ < N )

may be calculated so that the time value of money

is considered

i ‘ is the MARR I is the capital investment made at the present time ( k = 0 ) is the present time Θ ‘ is the smallest value that satisfies the equation

Σ ( Rk - Ek) ( P / F, i %, k ) - I > 0k = 1k = 1

Θ ’



INVESTMENT-BALANCE

DIAGRAM

Describes how much money istied up in a project and how therecovery of funds behaves over

its estimated life.

INTERPRETING IRR USING



INTERPRETING IRR USINGINVESTMENT-BALANCE DIAGRAM

• downward arrows represent annual returns (Rk - Ek) : 1 < k < N

• dashed lines represent opportunity cost of interest, or intereston BOY investment balance

• IRR is value i ‘ that causes unrecovered investment balance toequal 0 at the end of the investment period.

0 1 2 3 N

$0

Unrecovered

InvestmentBalance, $

1 + i‘

1 + i‘1 + i‘

1 + i‘

P (1 + i‘)[ P (1 + i‘) - (R 1 - E1) ] (1 +i‘)

(R1 - E1)

(R2 - E2)(R3 - E3)

(RN-1 - EN-1 ) (RN - EN)

Initial investment

= P

INVESTMENT BALANCE



INVESTMENT-BALANCEDIAGRAM EXAMPLE

• Capital Investment ( I ) = $10,000

• Uniform annual revenue = $5,310

• Annual expenses = $3,000• Salvage value = $2,000

• MARR = 5% per year

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