( ) ( )( ) ( ) 2 ( ) 3. n n = = =

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CAPITAL BUDGETING CAPITAL BUDGETING

Managerial Economics Managerial Economics

Mariusz Próchniak

Chair of Economics II

Warsaw School of Economics

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Future value (FV) Future value (FV)

r r –– annual interest rateannual interest rate

B B –– the amount of money held todaythe amount of money held today

Interest is compounded annually (annual capitalisation)Interest is compounded annually (annual capitalisation)

The today’s sum B will be worth:

( )( ) ( )

²

2 1 1 1

FV =B +r + =r B +r

( )

1 1

FV =B +r

( )

³

3 1

FV =B +r

(

¹

)

ⁿ

FVn=B +r

after one year:after one year:

after 2 years:after 2 years:

after 3 years:after 3 years:

after n years:after n years:

Future value

Future value –– an examplean example

B = 100 000 $ B = 100 000 $ r = 10% = 0.10 r = 10% = 0.10

( )

1 100 000 1 0.10 100 000 1.1 110 000

FV = ⋅ + = ⋅ =

( )

²

2 100 000 1 0.10 100 000 1.21 121 000

FV = ⋅ + = ⋅ =

( )

³

3 100 000 1 0.10 100 000 1.331 133100

FV = ⋅ + = ⋅ =

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Present value (PV) Present value (PV)

r r –– annual interest rateannual interest rate

Annual capitalisation of interestAnnual capitalisation of interest

The amount B received The amount B received one year from nowone year from nowis worth today:is worth today:

The amount B received The amount B received 2 years from now2 years from nowis worth today:is worth today:

The amount B received The amount B received 3 years from now 3 years from now is worth today:is worth today:

The amount B received The amount B received n years from now n years from now is worth today:is worth today:

(

¹^B

)

PV= r +

(

¹

)

²

PV B

= r +

(

¹

)

³

PV B

= r +

(

¹

)

ⁿ

PV B

= r +

Present value

Present value –– an examplean example

r = 10% = 0.10 r = 10% = 0.10

The amount 100 000 $ received

The amount 100 000 $ received one year from nowone year from nowis worth today:is worth today:

(

^{100 000}

)

^{100 000} 90 909.09 $ 1 0.10 1.1

PV= = =

+

The amount 100 000 $ received two years from now two years from now is worth today:is worth today:

The amount 100 000 $ received three years from nowthree years from nowis worth today:is worth today:

( )

²

( )

²

100 000 100 000 100 000

82 644.63 $ 1 0.10 1.1 1.21

PV= = = =

+

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Present value and discounting Present value and discounting

Interest rate used to calculate the present value of cash flows Interest rate used to calculate the present value of cash flows is known as

is known as the discount ratethe discount rate, and the process , and the process ––discountingdiscounting..

If If cash flows (CF) cash flows (CF) occur at different points in time, the present occur at different points in time, the present value of these future cash flows can be expressed as:

value of these future cash flows can be expressed as:

( )

0

1

T

n n n

PV CF

r

=

= ∑ +

Net present value (NPV) Net present value (NPV)

Net present value (NPV)Net present value (NPV)is equal to discounted cash inflows is equal to discounted cash inflows (operating profits from an investment) minus discounted cash (operating profits from an investment) minus discounted cash outflows (investment outlays).

outflows (investment outlays).

Net present value indicates the profitability of a given Net present value indicates the profitability of a given investment project.

investment project.

NPV is calculated according to the formula given on the previous NPV is calculated according to the formula given on the previous slide, but cash inflows (revenues or profits from investment) slide, but cash inflows (revenues or profits from investment) have positive sign whereas cash outflows (investment outlays) have positive sign whereas cash outflows (investment outlays) are included into the summation with a negative sign.

are included into the summation with a negative sign.

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Net present value

Net present value –– an examplean example

A

A textiletextile firmfirm isis consideringconsidering buildingbuilding aa newnew facilityfacility.. BuildingBuilding thethe plant

plant willwill requirerequire anan immediateimmediate capitalcapital outlayoutlay ofof500500 000000 $$andand willwill take

takeoneone yearyear.. TheThe firmfirm expectsexpects thatthat thethe plant,plant, whenwhen inin operation,operation, will

will generategenerate anan additionaddition toto thethe firm’sfirm’s operatingoperating profitprofit ofof200200 000000 $$ per

per yearyear forfor thethe nextnext 55 yearsyears beginningbeginning oneone yearyear fromfrom nownow.. TheThe annual

annual interestinterest raterate applicableapplicable overover thisthis timetime periodperiod isis 1212%%.. WhatWhat is

is thethe presentpresent valuevalue associatedassociated withwith buildingbuilding thethe plant?plant?

(Source

(Source:: Samuelson,Samuelson, Marks,Marks,ManagerialManagerial EconomicsEconomics))

Net present value –– an examplean example

Solution:

( )

²

( )

³

( )

⁴

( )

⁵

200 000 200 000 200 000 200 000 200 000 500 000

1.12 1.12 1.12 1.12 1.12

NPV= − + + + + +

Net present valueNet present value

500 000 720 955 NPV= − +

220 955 $ NPV=

NPV is positive, so the investment should be made.NPV is positive, so the investment should be made.

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Net present value –– an example (cont.)an example (cont.)

Suppose

Suppose thethe firmfirm payspays aa flatflat 3434%% taxtax raterate onon itsits taxabletaxable incomeincome..

The

The200200 000000 $$annualannual profitprofit flowsflows areare taxabletaxable withwith oneone exceptionexception..

The

The firmfirm isis allowedallowed aa deductiondeduction forfor thethe depreciationdepreciation ofof itsits production

production facilityfacility overover itsits lifetimelifetime.. TheThe firmfirm cancan depreciatedepreciate thethe building

building onon aa straightstraight--lineline basisbasis overover 55 yearsyears.. ThisThis meansmeans thethe firmfirm can

can taketake aa deductiondeduction ofof 100100 000000 $$ ((¹¹//₅₅ ofof thethe totaltotal capitalcapital cost)cost) against

against itsits annualannual incomeincome forfor eacheach ofof thethe nextnext55 yearsyears..

What

What isis thethe netnet presentpresent valuevalue ofof thisthis investmentinvestment project?project?

Net present value –– an example (cont.)an example (cont.)

Solution:

( )

²

( )

³

( )

⁴

( )

⁵

166 000 166 000 166 000 166 000 166 000 500 000

1.12 1.12 1.12 1.12 1.12

NPV= − + + + + +

Net present value (NPV)Net present value (NPV)

500 000 598 393 NPV= − +

98 393 $ NPV=

NPV is positive, so the investment should be made.NPV is positive, so the investment should be made.

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Annuities Annuities

Annuity

Annuity–– periodic cash flow of fixed amount.periodic cash flow of fixed amount.

During

Duringthethe nextnext 44 yearsyears,, beginningbeginning oneone yearyear fromfrom nownow,, youyou willwill bebe given

given aa fixedfixed amountamount ofof 50005000 $$annuallyannually.. TheThe interestinterest raterate isis 88%%..

What

What isis thethe presentpresent valuevalue ofof thesethese cashcash flows?flows?

Annuity

Annuity –– an examplean example

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This

This isisaa fourfour--yearyear annuityannuity..

Annuity

Annuity –– an examplean example

( ) (

²

) (

³

)

⁴

5000 5000 5000 5000 1.08 1.08 1.08 1.08

PV= + + +

16 561 $ PV = Solution:

Solution:

A perpetual annuity (a perpetuity)

A perpetual annuity (a perpetuity) –– an annuity that goes on an annuity that goes on forever.

forever.

Perpetual annuity (perpetuity) Perpetual annuity (perpetuity)

PV CF

= r

CF

CF –– the constant annual cash flow,the constant annual cash flow, r

r –– the discount rate.the discount rate.

The present value of a perpetuity is:

InIn thethe aboveabove formulaformula itit isis assumedassumed thatthat constantconstant annualannual cashcash flowsflows beginbegin oneone year

year fromfrom nownow (and(and gogo onon forever)forever);; ii..ee.. thethe firstfirst cashcash flowflow isis discounteddiscounted byby aa discount

discount factorfactor thatthat correspondscorresponds toto thethe firstfirst yearyear (division(division byby 11 ++ r)r)..

DerivationDerivation ofof thethe aboveabove formulaformula isis easyeasy (see(see thethe sumsum ofof thethe infiniteinfinite geometricgeometric sequence)

sequence)..

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Perpetual annuity

Perpetual annuity –– an examplean example

You

You willwill bebe givengiven aa fixedfixed annualannual amountamount ofof 50005000 $$ forever,forever, beginning

beginning oneone yearyear fromfrom nownow.. TheThe interestinterest raterate isis88%%.. WhatWhat isis thethe present

present valuevalue ofof thesethese cashcash flows?flows?

Perpetual annuity

Perpetual annuity –– an examplean example

This

This isisaa perpetualperpetual annuityannuity (a(a perpetuity)perpetuity)..

( ) (

²

) (

³

)

⁴

5000 5000 5000 5000 1.08 1.08 1.08 1.08

PV= + + + +K

62 500 $ PV= Solution:

Solution:

5000 PV= 0.08

Possible

Possible interpretationinterpretation::

You

You putput aa principalprincipal ofof 6262 500500 $$ inin aa bankbank accountaccount andand leaveleave itit therethere

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The present value of a growing perpetuity is:

Growing perpetuity Growing perpetuity

An annuity pays an amount of An annuity pays an amount of CF after the first yearCF after the first yearand the and the payment rises by

payment rises by g%g%each year thereafter. each year thereafter.

This is This is a growing perpetuitya growing perpetuity..

( )

( ) ( )

( )

2 3

2 3 4

1 1 1

1 1 1 1

CF g CF g CF g

PV CF

r r r r

+ + +

= + + + +

+ + + + K

PV CF

r g

= −

Inflation

Inflation often causes cash flows to grow.often causes cash flows to grow.

or:

There are two equivalent ways to compute present values while There are two equivalent ways to compute present values while properly accounting for inflation:

properly accounting for inflation:

Nominal and real cash flows and interest rates Nominal and real cash flows and interest rates

Listing cash flows in Listing cash flows in nominalnominalterms and discounting by a terms and discounting by a nominalnominal interest rate.

interest rate.

Listing cash flows in Listing cash flows in realrealterms and discounting by a terms and discounting by a realrealinterest interest rate.

rate.

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The

The internalinternal raterate ofof returnreturn (IRR)(IRR) ofof anan investmentinvestment isis thethe discountdiscount rate

rate atat whichwhich thethe project’sproject’s cashcash flowsflows havehave aazerozero presentpresent valuevalue..

Internal rate of return (IRR) Internal rate of return (IRR)

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1.

1. A singleA single--investment decision.investment decision.

2.

2. Mutually exclusive investments.Mutually exclusive investments.

3.

3. Making investment decisions with constrained resources.Making investment decisions with constrained resources.

MAKING INVESTMENT DECISIONS MAKING INVESTMENT DECISIONS

The firm should undertake the project if and only if the project’s net present value is positive.

1. A single

1. A single--investment decisioninvestment decision

The firm should undertake an investment project if and only if the project’s internal rate of return is greater than the discount rate.

In other words:

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2. Mutually exclusive investments 2. Mutually exclusive investments

In a choice among a number of mutually exclusive investment alternatives, the manager should choose the one that offers the greatest present value.

A firm has

A firm has 1 million $1 million $to invest and faces the following potential to invest and faces the following potential investment projects:

investment projects:

3. Making investment decisions with constrained resources 3. Making investment decisions with constrained resources

Project Initial investment ($) NPV ($) NPV per 1 $ invested

A 1 000 000 2 000 000 2.0

B 400 000 1 400 000 3.5

C 300 000 1 200 000 4.0

D 100 000 600 000 6.0

E 200 000 500 000 2.5

F 200 000 300 000 1.5

G 100 000 50 000 0.5

If the firm were not constrained, it would undertake If the firm were not constrained, it would undertake all the programs because NPV > 0.

all the programs because NPV > 0.

Given constrained resources the firm should choose Given constrained resources the firm should choose

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The discount rate should correspond to the rate of return The discount rate should correspond to the rate of return for projects in a comparable risk class.

for projects in a comparable risk class.

One of the methods of determining the discount rate:One of the methods of determining the discount rate:

Weighted average cost of capital (WACC) Weighted average cost of capital (WACC)

How to determine the discount rate?

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This method assumes that the considered investment project This method assumes that the considered investment project has the same risk characteristics as does the firm in general.

has the same risk characteristics as does the firm in general.

The weighted average cost of capitalThe weighted average cost of capitalis the average of is the average of the rate the rate of return on the firm’s debt

of return on the firm’s debt and and the rate of return on its equitythe rate of return on its equity..

Weighted average cost of capital (WACC) Weighted average cost of capital (WACC)

Weighted average cost of capital (WACC)

Weighted average cost of capital (WACC) –– an examplean example

The firm has 40% debt and 60% equity.The firm has 40% debt and 60% equity.

The firm pays 10% on its debt.The firm pays 10% on its debt.

The firm returns 19.5% on its stock.The firm returns 19.5% on its stock.

0.4 10 0.6 19.5 15.7%

WACC = ⋅ + ⋅ =

The rate of return on the firm’s debt The rate of return on the firm’s debt is e.g. the (afteris e.g. the (after--tax) interest tax) interest rate (the rate of return) of the bonds offered by the firm.

rate (the rate of return) of the bonds offered by the firm.

The rate of return on equity: The rate of return on equity: we could calculate the internal rate we could calculate the internal rate of return of an investment in shares, say, 5 years ago if cashed in of return of an investment in shares, say, 5 years ago if cashed in

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A

A moremore preciseprecise measuremeasure ofof thethe raterate ofof returnreturn onon equityequity::

The rate of return on equity The rate of return on equity

s f p

r = +r r

rr_s_s thethe raterate ofof returnreturn onon thethe stockstock rr_ff thethe riskrisk--freefree raterate ofof returnreturn

rr_p_p thethe riskrisk premiumpremium ofof aa firm’sfirm’s stockstock

The riskThe risk--free rate of return free rate of return is typically the current rate of return is typically the current rate of return on short

on short--term bonds issued by the federal government.term bonds issued by the federal government.

The risk premiumThe risk premiumcan be estimated using can be estimated using the capital asset the capital asset pricing model (CAPM)

pricing model (CAPM)..

A firm’s risk (and, therefore, its expected rate of return) A firm’s risk (and, therefore, its expected rate of return)

depends on its correlation with movements in the stock market depends on its correlation with movements in the stock market as a whole.

as a whole.

BetaBetameasures the relationship between the individual stock’s measures the relationship between the individual stock’s return and the return of the stock market.

return and the return of the stock market.

E.g. if E.g. if ββββββββ= 1, the systematic risk of the stock is the same = 1, the systematic risk of the stock is the same as the market, meaning that the stock should have the same as the market, meaning that the stock should have the same risk premium as the market.

risk premium as the market.

Capital asset pricing model (CAPM) Capital asset pricing model (CAPM)

( )

p m f

r =β r −r

rr_p_p thethe riskrisk premiumpremium ofof aa firm’sfirm’s stockstock

ββββββββ betabeta

rr_m_m–– rr_ff thethe marketmarket riskrisk premiumpremium

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In the U.S., over the last 50 years, the annual rate of return In the U.S., over the last 50 years, the annual rate of return on stocks has averaged 12% and the return on risk

on stocks has averaged 12% and the return on risk--free, free, short

short--term treasury securities has averaged 3.5%. term treasury securities has averaged 3.5%.

Thus, the risk premium Thus, the risk premium on the stock marketon the stock market (r(r_m_m–– rr_ff) is: ) is:

12%

12% –– 3.5% = 8.5%.3.5% = 8.5%.

The beta on the firm’s stock is 1.4.The beta on the firm’s stock is 1.4.

ShortShort--term government bonds currently have a return of 7.6%.term government bonds currently have a return of 7.6%.

The rate of return on equity

The rate of return on equity –– an examplean example

( )

s f m f

r = + r β r − r

The firm’s rate of return on equity (the firm’s cost of equity):

( )

7.6 1.4 12 3.5 7.6 1.4 8.5 7.6 11.9 19.5%

rs= + − = + ⋅ = + =

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Appendix to Chapter 19: Present Value Tables Appendix to Chapter 19: Present Value Tables

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( ) ( )( ) ( ) 2 ( ) 3. n n = = =

CAPITAL BUDGETING CAPITAL BUDGETING

Mariusz Próchniak

Mariusz Próchniak

Chair of Economics II

Chair of Economics II

Warsaw School of Economics

Warsaw School of Economics

( )( ) ( )

( )

( )

(

)

( )

( )

( )

(

)

(

)

(

)

(

)

(

)

( )

( )

( )

1

PV CF

r

= ∑ +

( )

( )

( )

( )

( )

( )

( )

( )

( ) (

) (

)

PV CF

= r

( ) (

) (

)

( )

( ) ( )

( ) ( )

( )

PV CF

r g

= −

0.4 10 0.6 19.5 15.7%

WACC = ⋅ + ⋅ =

( )

( )

r = + r β r − r

( )

Thank you for Thank you for the attention!!!

the attention!!!