CAPITAL BUDGETING AND CAPITAL RATIONING MODELS AS AN ANALYSIS FOR CAPITAL INVESTMENT DECISIONS

A COMPARATIVE STUDY OF

CAPITAL BUDGETING AND CAPITAL RATIONING MODELS AS AN ANALYSIS FOR CAPITAL INVESTMENT DECISIONS

A literature study done

by:

,Churiah Agustini Santoso

\9f199 ^\2.-1 Pls:rp

UNIVERSITAS KATOLIK PARAHYANGAN FAKULTAS ILMU SOSIAL DAN ILMU POLITIK

BAN DUNG

1984

9-1 <;;. .

[)'6

No. Klass ?~.~.:.(,(:') .. 9.:t.-: ... ;:;.

No. Induk.l?:\!~c~ ^Tgi .~~.:~.;~.

t-Iadioh/lleli ... .

page

ABSTRACT i

TABLE Of CONTENTS i i

CHAPTER

1

CHAPTER

2.

2.1 2.2 2.3 2 4 2.5 2.6

CHAPTER

3.

3.1 3.2 3 3 3.4 3.5 36

CHAPTER

4.

4.1 4.2 . 4.3

CHAPTER

5 5.1 5.2 5.3 5 4

CHAPTER

6.

6.1 6.2 6.3 64 6.5

CHAPTER 7

INTRODUCTION . . . 1

CAPITAL BUDGETING UNDER CERTAINTy • . . . • • . • • • . . . 4

Net Present Value . . .

4

Payback . . • • • . . • • • . . . • • • . . . • • • • • . . . • . • . • • . • . • . • . •

5

Average Return on Book Value . . .

6

Internal Rate of Return . . • . . . • . . . • . . • • . . • •

8

Profitability Index . . . 10

Application on The Investment Analys:'.s • . . . • • . . Program . . .

11

CAPITAL RATIONING UNDER CERTAINTy . . . . • . • • . • • . . .

16

Definition . . .

16

Structure . . . 17

Profitability Index. . . . . 17

Ma themetica 1 Programming Linea r . . . . • . . . • . . . Programming. . • . . • . . . • . . . • . . . • . . . . • . . . .. .. 19

Integer Programming . . . 29

Goal Programm:'.ng • . . . • . . . •

34

CAPITAL BUDGETING UNDER UNCERTAINTy . . • . . . • . . .

40

Sen~i~ivi~y Analysis/ Break-even Analysis . . .

40

DeCl.S10n frees . • . . .

42

Simulation . . .

5

CAPITAL RATIONING UNDER UNCERTAINTy . . . • . . . .

52

Stochastic Li.near Programmi.ng . . . •

52

Chance-contra:'.ned Programming . . .

53

Quadratic Programming . . .

53

An Example by Salazar and Sen . . .

54

DESCRIPTION

OF

THE SENSITIVITY ANALySIS; . . . . PROGRAM . . . ; • . . . • . . . . • . . • . . . 59

Assumptions . . . • . . . • . . . • . . . 59

In pu t D a t a . . • . . . • . . . • . . . • . • . . . • . . . . 59

Flowchart of The Program . . . • . . .

60

Output Data . . . . • . . . • . . . • . . • . . .

63

Example . . . • . . . •

63

SUHt·1ARy . . . 67

i(i'""""RFNC' .

--.c..

bS . . . • . . . • . . . 68

Ai)P[\DIX . . . . . • . . . • . . . 70 .

. .

1 1

ABSTRACT

lhe process of planning and evaluating is called capital

budget~ng.

Capital

proposals t6r

~nvestments

budgeting

dec~sions

are important because large 880unt ot money are committee tor long periods of time and because these type ot

decis~on

are often difficult or impossible to reverse once the funds have been committee.

Th~s

stUdy surveys difterent models dnd techniques that are used to solve several types of

~~~i_a. bld~eting

problems These types are of simple capital budgeting and capital rationing; under conditions ot certainty and uncertainty

·lhe analysis helps on the difficult and critical deCisions management has to make

_ 1

in assisting most goals.

tus~~ess ~~~ms to achieve their various

Capital budgec~=~ ~s :~e

management which esta~~~shes

investing investment

resources projects

long

co~conly t~e

decision area in financial goals and criteria for term projects. Capital include land, buildings,

like. These assets are facilities, equipmeD~, and

extremely important ~o :~e

all of the firm's ?r0f~t

fir~ because, in general, are derived from the use

nearly of its capital investments; ~hese assets represent very large commitments of resources; and c~e funds ~ill usually remain invested over a long ?eriod of time. The future development investment of the firm hinges on the selection of capital

projects, and the decision to abandon previously

undertakin~s which turn out to be less attractive firm than was originally thought~

accepted to the

The benefits of capital projects are received Over some future period, and the time element lies at the core of capital budgeting. The firm must time the start of a project to take advantage of short-ter~ business conditions (construction costs for example vary ~ith the stage of business cycle) and financing of the project to capitalize on trends in the money markets (such as the pattern of short and long term interest r a ( 2 ) , In addition, the longevity of capital assets and the ~3rge outlays required for their acquisition suggest tha: :je estimates of income and cost aSSOCiated with the pro:e:: to be documented for the time

are

:.~ons

--::..ence

2

received or out Moreover, investment are al~ays bas~: JPon incomplete information using of fut~re reve=':~s and costs. It is known from the that such fC~~:25~s will always err on one side

~~e other, and the de;=~e of error may correlate "(but not

:'J.J;':;1"S) with the durat:::: of the projects. Short-term

(1 year or ::'''5S) generally display greater than long-ter::! =stimates (S years or more). The

:; dimly seen enta:":"~ risk, and any appraisal of a

(_~:. ~ "'_21 project ¹ therefo::;;;:. must necessarily comprehend some ,,"';'" ",ocsment of the risk =.c::ompanying the project. Finally,

.... ,~:;.tment

, " ^{decisions mus::.} be matched aga~nst some future

to ascertain t~e accur~cy of the forecast and _the

·/j;:.;",,:·.L~ty of evaluating c::~teria. In summary, the components 'if '.cpital budgeting

cost.s of

a=E~ysis involve n forecast of the

and project, discounting the funds

j fl 'f!-; .~ ted in the project 8,\:. an appropriate

project and

rate, assessing following up to

1 11 {- associated wit:: the

Ill" I~:~i.ne if the project pe~:orm cs expected.

The application of capital budgeting are many

'I ;; r : ^.~ rj •

In

^theory, a :arge n urn be r or problems

and lend

1 J I 1". /f, .'~ e 1 v e s to analysis by 'lar~ous methods. Analysis absorbs

! i I:, (:

I!! 1/ :~ !.

,,,

and ::noney, especiall J :he more ranking and risk management. The

sophisticated tec~niques cost of these approaches be justified by the ?erceived benefits~ Theory adapts costly

circumstances. Conceptually appealing but

! P ' :Jniques of analysis do not merit ac=oss-the-board

:\ !' jI j j r:: a t ion. Accordingly, in establishing a cutoff by .size

(~xpenditure; that i s , projects requiring an investment

,,\, ': [ a specified amount ..,il1 be subject to searching.

.; 1 I {j!. 1 n y ; below this amount, less costly _{criteria of}

1'1 l~ptahce will be applied.

This study is an attempt to survey dif£ere~t models

.\ 11.1 t~c.hniques tha~ are used to solve several _types

or

, 'li I 1..j 1 budgetiag problems. In the f i r s t place we make a

~~~~~~ction between conditions of certainty and _ - c : : certain t y . Another important distinction

conditions is bet;..reen budgeting problems (no capital restriction) and

~~o~tal rationing problems (capital restriction).

Chapter 2 summarizes some techniques to solve the

s~~ital budgeting problem under certainty. Five alternatives discussed net present value, payback period. average on book value, internal rate of return and

~r~f~tability index.

These simple techniques cannot be used in situations where the capital budget is limited. Chapter 3 investigates 30me techniques that can be used to solve the capital rationing problem under certainty. Simple techniques, like the ranking based on the profitability index proposed by Lorie-Savage, can be used when there is a restriction in ()nl'f one

be used:

period. Otherwise mathematical programming should this can be linear programming, integer programming or goal programming.

Chapters 4 and 5 handle conditions or uncertainty.

Techniques budgeting

;J n (] 1 y s i s ,

problem break

that can be used to -solve the capital under uncertainty are: sensitivity

even analysis, decision trees and

~imulation. These techniques are described in chapter 4.

Finally, uncertainty is

't" .lne capital rationing problem under

discussed in chapter 5~ He~e, a combination

llf mathematical programming and simulation can be used, like S.llazar and Sen proposed in their paper.

Most of the discussed techniques are illustrated with

t~xamples • For this purpose,programs available on the apple (.omputers of the department Industrieel 3eleid have been :lsed. The' existing capital budgeting package did not include

3 sensitivity analysis. That is why a new program was

jt~veloped • This program and some examples are discussed in

: ~1.) pte r 6.

Chapter 2.

CAPITAL

3CDGETI~G

UNDER CERTAINTY.

In order to be 301e to perform an economic evaluation of a project's desiraDil~ty. i t is necessary to understand the decision rule for accepting or rejecting investment projects. Some methods whic~ guide management in the acceptance or rejecc~on of proposed investments are explained in this chapter.

2.l.Net present value.

This measure is a direct application of the present value concept. Its computation requires the following steps:

first, choose an appropriate rate of interest4 Second, compute the present value of the substr~ction of the cash outl~ys from the invescment from the cash proceeds ex"ected from the inves~ment. This gives the present value

or

cash flows·. The present value of the cash flows minus initial investment is the net present value of investment.

The recommended accept or reject criterion is the the the

to accept all independent investments whose net present value is g,reater or equal to zero and reject all invest~ents ~hose

net present value is less than zero.

The formula for calculating PV and NPV can simply be written as:

P'I ~ C1/0+r) + C2!(l+r;: + ••. + Cn/(l+r)'.'I.

~iPV ~ - CO + PV

4

with :

C1.C2 •••• Cn - Cash flows:of the investment.

r - Interest rate.

CO - The initial investment.

Example:

The payment for the construction of a building is on the following schedule:

a. 100.000 down payment now. The land. worth 50.000 must also be committed now.

b. 100.000 progress payment after 1 year.

c. A final payment of 100.000 when the building is ready for occupancy at the end of the second year.

d. Despite the delay the building will be worth 400.000 when completed.

All this yields a new set of cash-flow forecasts:

Period t==l

Land - 50.000

const:-uction payoif

Total

-100.000

CO--150.000

-100.000

C1--100,000

-100.000 +400.000 C2-+300,000

If the interest rate is 7 percent, then NPV is:

NPV CO + Cl/Cl+r) + C2/(l+r)'

- -150.000 - 100,000/1.07 + 300,000/(1.07)2 - 18,400

Since the NPV is positive,we should go ahead.

1 1 P . b ... --. 201 ack.

Companies frequently require that the initial olltlays

a:1Y project should be recoverable within some specified

cutoff period.

counting the number

:;eriod

-~"rs i t

- - 3 project is

~:'!kes before forecasted cash flows e~'~~--_ :~e init~21 investment.

Example:

Consider projects A and - -

Project

A B

Cash f::.~.:~llars

========;;:;;=====:;;;;====.=========

CO

-2000 -2000

Cl

+2000 +1000

C3

o

+5000

Payback

?eriod,years

1 2

6

found by cumulated

NPV at 10 :

-182 +3492

The NPV rule t e l l s project A, we take 1 ye~~

B we take 2 years.

uS to reject A and accept B. With

cO

recover our 2000, with project

If the firm use= the pavback rple with a cutoff period of 1 year,

it

woc~d accept only project A. If i t used the payback rule with a 2utoff period of 2 or more years i t would accept both

A

an&

3.

The reason for the difference is that payback gives equal weight to a l l cash flows before payback date and no weight at all to subsequent flows.

order to use the payback rule the firm has to decide on appropriate cutoff date.

2.3.Average retur~ on book value.

the

In

an

Some companies jud~= an investment project by looking at i t s book rate of retur~. To calculate book rate of return i t is necessary to divide :ie average forecasted profits of a project after deprec:~::on and taxes, by the average r~turn on book value of ::e investment. This ratio is then

measured against the book rate of ~etu~c the firm as a whole or against ^{some external} such as the average book rate of return for the ~nd~2:~7.

This criterion ignores the :ppor~~~~~! cost of money and is not based on the cash flovs o~ ;-:--eject, and the investment decision may be relatec to -~C ~rofitability of the firm's existing business.

Ex amp Ie:

The table below shows projected ir,==~e statements for project A over its 3-year life4 Its 2'~r3ge net income is 2000 per year. The required in~est~~~~ ~s 9000 at t=O.

_This amoun t is then depreciated a~ ~ constant rate of 3000 per year.

Cash ~:J~

ex

¹⁰⁰⁰⁾

--- ---

Project A Year 1 Year 2 Year3

---

Revenue

Out-of-pocket_ cost Cash flow

Depreci~tion

Net income

12 6 6 3 3

10 5 5 3 2

8 4 4 3 1

So the book value of the new investment will decline from 9000 in year 0 :0 zero in year 3

Yearl Year2 Year3 Year4

---

Gross book ^val'Je of investment 9000 9000 9000 9000 A"ccumula teed de:,J:eciation 0 3000 6000 9000 Net book value: of investment 9000 6000 3000

a

Average 12: book value=4500

8

The average net income is 2000, and the average net investment is 4500. Therefore the average book rate of return in 2000/4jOO = 0.44. Project A ~ould be undertaken if the firm's target book rate of return were less

44%.

2.4.Internal Rate of Return.

than

The internal rate of return is defined as the discount rate which makes

NPV=O.

This means that to find the internal rate of return for an investment lasting T years,

~e must solve for the

IRR

in the following expression

NPV = Co + Cl/(l+IRR)

+

C2/(1+IRR)'+ .•• +CT/(1+IRR)t = 0

Actual calculation of

IRR

usually involves trial and error. The easiest way to calculate IRR, if we have to do it by ^h uano, ^• is to plot three or four combi~ations of ~PV ana disr:ount

,

^.

.

... lne,ana.

rate on a graph, connect the points with a read off the discount rate at which NPV=O

smooch

The rule for Invest~ent decisions on tlle basis of IR~

is to aCcEpt an investmenL project if the opportunity cost of capital is less than the IRR. If the opportunity cost of capital is less than t!l.e IRR ^f then the project has a positive

NPV

~hen discounted at the opportunity cost of capital.

If

i t is equal to

TiR, tl.e

project ~as a zero

NPY.

And if it is greater than the IRR,the projec~ has a negative

NPV.

Therefore when we compare the opportunity cost of capital wirh the IRR .Qn our project, W~ are effectivel]

asking whether our pr?j~ct has a~·positive NPV.

2xample:

;\ neW' pr.oject has an af~~r-tax cost of 10,000 and result in after-tax cash inflows of 3,000 in year 1,

i~ year 2, and 6,000 in year 3.

will 5,000

NPV

=

^-10,000

^(l-;-IRR)

⁺

S,OOO/(I+IRR)"

₊

6,OOO/(1+IRR "

By trial and erra,," :::--- ~o find the

IRR

which gives result close to zero. Uo-- ~ ~ -::;

IRR,

the NPV is close to zero but negative (-38). T:=~ ,-::, for

IRR 16%

is positive

(146).

The actual

IRR

is be!:-.~",-:: : 6% and 1

17%

and may be found using linear interpolatio a •

Example: NPV versuS - - -

Consider two projec-::oo ~ and B which are mutually exclusive.

The cost of capital '--'" -'.J%.

Project year

a

^{::ear 1 year 2 year 3}

---

A B

- 1, 000 -11, 000

The

NPVs and

the IRRs

IRR

50S 5,000

50S 5,000

the two projects are

NPV

---

A B

24%

17%

256 1,435

505 5,000

If

the firm uses the NPV criterion, project B will be chosen, However if the firm uses the

IRR

criterion project

A

will be preferred.

Now, ~e consider the incremental cash flow

Project year

a

^{year 1 year 2 year 3}

^IRR ---

3-A

-10,000

4,495

^{4,495 4,495 16,58%}

The IRR on this incremental cash flow is l6,58%, and given a 10% cost of capital, this represents a profitable

10

opportunity and shoul~ be ac=epted. This results in a total to the firm of A -;. (B-A) ~ B. Thus, the IRR rule

cash

when

flow

used properly (i.e.on incremental basis) leads the firm to prefer project

B,

but this is precisely the project which has the higher NPV.

2.S.Profitability index.

The profitability index is the present value

or

^the

forecasted cash flows divided by the initial investment

Profitability_index ~ present value/investment

=

^PV/CO

The profi ta bili ty i~dex rule t e l l s us to accept all projects with an index greater than 1. If the profitability index is greater than I, the present value is greater than the initial invest::nent, and so the project must have a positive NPV. The proritabi2..ity index! there:orc, leads to exactly the same decisions as NPV.

For many people the profitability index is illore intuitively appealing than t~e NPV criterion. The statement that a P?rticular investme~t has a NPV of say $20 is not sufficiently clear to man~ people who prefer a relative measure of profitability. B~ adding the inior3ation that the projectfs initial outlay (CO) is $lOO,the profitability index (120/100 ⁼ 1.2) provides a meaningful ~easure or the project's relative profitability in more readily understandable terms. It is then only a small step to convert the index of 1.2 to 20%.

However once again problems can arise ~hen mutually

excl~sive alternatives are considered. The profitability index may be useful f,or exposition, it should not be used as a measure of investment wor:i for projects of di££~ring size when mutually exclusive choices have to be made.

Example :

Consider the fol::"~:i t~o projects:

Cash flo·.'s. ~ : l l a r s

Project CO

PV at 10

%

?rofitabil:'ty index

NPV at 10

%

---

K 100 ::"0

L -10,000 ^-~

=

^JO

182 13,636

1 :-'?

-

.

^{- ' ....}

! .26

82 3.636

---

Both are 5 cod projects, as :~e profitability index correctly indicates.

exclusive. We shG~ld take L, the ?~oject ~it~ the higher NPV. Yet the prof:':abilit7 index gives

K

the higher ranking.

We can always solve such pro~lems by looking at the profitability index on the increment2~ investment.

But suppose tbe projects are mutually

Example:

Cash flows,dollars

Project Co Cl

PV at 10 %

Profitability index

NPV at 10

%

---

L-K

-9900 +14,800 13,454 1.36 3554

The profitability index on the additional is greater than 1, so L is the bette~ project.

investment

2.6.Application of tbe investme~t ana:ysis program.

At the department, a program has been

developed to solve capital budgeting under Certainty. The following criter::'a are!

problems considered:

net present value, profitability inte:-nal _{rate of} return and

12

(discounted) payback period.

Figure 2.6 is based on example 2.1 ~tax effects are also considered now) and illustrates input and output data.

The program allows several depreciation different

concerned,

types not

of revenues and expenses.

schemes and As output is only the investment criteria are given, but also a detailed picture of the periodic cash flows.

1. GENERAL DATA - - - -

PROJECT LIFETIME TAX RATE ^{( %)} INTEREST RATE ^(;.) INFLATION RATE ^(I.) ACTUALISATIoN RATE

= =

= =

(;. ) =

INPUT DATA

**********

2 50 7 0 7

2. INVESTMENT AND DEPRECIATION DATA 3. REVENUES

Tm Of jrIESTl!E~, --··AI()L~l

P£SJ WAL 'IAWE O£PP.ITlATlOW DOOATlM DEP!'£CrmOH XETHOD

4. EXPENSES

TYl'f

?IUlCE!! OR ll~R iXCR !

SPfC~FiC IKFlATlOH EllUl: fIRST ?SHOD

lM'D ~

o '

50000 '

o

^~

2 !

~ !

COlIS,R ! r ,

" .

o

! 100000 '

CD:/STRUCT '

o

~

100000 '

o '

2 ! V '

- - - -

TYPE

PROWIT OR ll)(SlR 1 XCR ' SPECIfiC lliFUTION REVElWE FJRSi i'EP.lOD

PERIODIC '1A!1JI.liU Rfl'ElllF.3

PERIOD ! I 2

PAYOFF !

o '

HIOOOO '

PAYOFF'

o

!

o '

PERIODIC DEPRECIATIONS

PERIOD' 1 1

LAAD! CCXSTRUCT!

o

!

o

!

o

!

o

!

15

1. EVALUATION CRITERIA

PROJECT LIFETIME 2

TAX RATE ::.0

ACTUALISATrON RATE 7

NET ?'RESENT YALUE PROFITABILITY INDEX

INTERNAL RATE OF RETURN

PA)'8AQ( PERIOD

PERIODIC FLOWS

WITHOUT TAXES 18574 1.124 11.96 1.929

WITH TAxES

-~11S

.562 NEGATIVE

> 2

PO/II!

"""" """"

! [l'{11T!/E I !'O'm:TA-I fRHAI Tim ! If!U ru ' CASI! P J:w ~

''"''

"

^{o '}

^"

o ' ^{100000 ~ -l~!}

""'" .

^{1000») ! ;~~}

PERIODIC R£VENUES

.

^'

-,

4. PERIODIC EXPENSES

nlJOS ^PR!llT "",n

,

,

^'

,

^'

_" ,

^,

"

"

^-1@lQ1

^..,..., -- ^.

^-~'

,

^{' ~~ 1~1. l~~ l~~}

:""'-

^{11O"' ! rAIB ;,! EU2I}

or Wv~ , _ilJlI

I~I

.. _" " " ^,

^{' -l~'_ ~!}

"

^{j '}

^,

^{' !~!}

Chapter 3.

CAP~TAL

RATIONING UNDER CERTAINTY

3.l.DefinitiDn

The capital budgeting prDblem involves the allocatiDn of scarce capital resources amDng cDmpeting eCDnomically desirable projects, nDt all of which can be carried out due to a capital (or other) constraint. This problem is referred to as capital rationing.

is often that the restriction on the supply of b.ottlenecks capital reflects non capital constraints or

within the firm. For example, the supply of key personnel to carry out

restricting Similarly,

Df

the projects maybe severely limited, thereby the dollar amount of feasible investment.

consideration of management time may preclude the programs beyond some level. This should more adoption

properly be called lahor Dr management constraincs rat.her than capital cons~raints. These restrictions limit or

the maximum amau'nt of investment which can be undertaken by the firm.

Many firms capital constraints are soft. They ~eflect

no imperfection i.:1 the capital market. Instead they are provisional limits adopted by management as an aid to fin2ncial control. Soft rationing should never cost t~e firm If capital constraints become tigh~ enough to hu:-:, then

constraints ..

the firm raises mDre money and loosens But what if i t cannot raise more money-what

the if it :aces hard rationing?

Hard rationi~g implies market imperfect~on-a barrier be:'""'een the firm and capital markets. If that barrier also that the £:~mts shareholders lack free access to a wel:-runctioning ca?ital market, the very foundacion of net present value crumble.

16

3.2.Structure.

The entire discussion of methods of capital rests on the proposition tha t the wealth of

rationing a firmts shareholder is highest if the firm accepts every projects that has a positive net present value. Suppose, however,that there are limitations on the investment program that prevent the company to undertake all such projects. If this is the case, we need a method of selecting the package of projects that is within the company's resources yet gives the highest possible net present value.

We will survey several techniques which are applied to the capital budgeting problem setting, under conditions of certainty and under conditions of risk

- Capital rationing under conditions of certainty:

Lorie and Savage proposed profitabi~ity index techniques for the single period case and the multi period case. Weingartner formulates th-e capital ratloning problem

model.

Since advances

to the model.

first as an LP theh as an integer programming

these pioneering works, there have been many in the area of mathematical programming applied capital budgeting problem, e.g. goal programming

Capital rationing under condi~ions of uncertainty (see c.hapter 5).

3.3.Profitability Index.

Lorie and Savage proposed a solution to ~he capital rationing problem for the single period case and the multi period case, on the assumption that

18

l . The timing and magnitude of the cash flows of all

projects are known at: the outset with complet2 certainty.

2. The cost of capital is known (independent of the

inve~tment decisions under consideration) so that the present value of all the projects are also given.

3. All projects are strictly independent, i.e.the execution oi one project does not affect the costs and beneiits or some other project:.

Under these simplifying assumptions our problem becomes one of how to selec:: among projects which ha,"e positive net present values.

For a single period case, the solution is to rank all projects by the profitability index and then select from the top of the list until the budget is exhausted. The ?rocedure is simple and easily understood. However, it depends

set of very limiting and unrealistic assumptions:

on a name!.y that cash flows are known, the eost of capital is independent of the investment decision, mutually excl usi \'e projects are ruled out; and all outlays occur in a single period of time so that the budget constraint applies

single period.

Lorie and Savage drop the latter restriction by considering

limitation

the selection process in which occurs in more than one ?eriod~ Like

the also

to a

also budget many seminal articles in finance and economics it is the question they asked, and not the par~icular solution which they proposed, which is of lasting interest.

Example

Lorie and Savage consider the followi~g example:

Project

/

1 2 3 4 5 6 7 8 9

Suppose

NPV

14 17 17 15 40 12 14 10 12

the are selected,

Cash. outflow in period 1

12 54 6 6 30 6 48 36 18

budget is 100. This and the budget used

PI

1 • 167 0.314 2.83 2.5 1.33 2.0 0.292 0.277 0.666

means projects is 98.

3.4.Mathematical Programming: Linear Programming.

Ranking

5 7 1 2 4 3 8 9 6

3,4,6,5,1,

By the linear programming(LP) the firm can examine very large-scale choice problem in which the number or alternatives is virtually unlimited in the relevant range.

It helps allocating and evaluating scarce resources and provides decision makers .... ith some very insightful information regarding the ~arginal value of resources.

The LP formulation of capital rationing is as follows:

fI

max NPV =

I

_{OJ x J} ⁽¹⁾

J>' II

s.t

I

^{C v ~}

^K,

t=l,2, . . . T ^{(2 )} Jt ^., J

J"

0 ~ ~J v -" 1 ( 3 )

XJ = percent of projec~ j that is accepted b

J =

NPV of project j over its usef~l life

.I

CJ~ = cash outflow requireC: by project

K

_t

=

budget available in year t

in year t

The following aspects should )e noted about problem formulation above:

20

the

1. The x

J ^{decision var~ables} are assumeQ to be continuous- that is, partial projects are ^a~lowed in the

LP

formulation.

2. The usual nonnegativity constant or L? is modified as shown in equation(3) to also show an ,oper limit for each project.

3. I t is assumed that all the input par=.neters - D_J ,C J ' .K~

,are known with certainty.

4.

The b_J parameter shows the NPV of project over i t s useful l i f e . where all cash flows are discounted at the cost of capital, which is kno~n with certainty.

The marginal value to the ^::c~ of the budget constraints are obtained by comparing the total NPV with and without an extra dollar of resources. In .L? such values are referred to as dual 'lariables

represent the opportunicy costs firm's resources~

or s~adow prices.

of us:~g a unit of

~. .. ney

~he

The dual LP formulation for the cap~:al ~3tioning proble~ is as follows

K II

Min

I

P~

Kt

₊

I ^Y

_J

t .. t

J.,

s. t

(a)

V

I ^P,

^{Cjt + \l j ~ ':J. j} :;;;lt2,.",~~

,.,

^{.; ( b )}

P"

^{IJ) ~ 0 t =1 2 , l' =1}

, _.N

_{( c )}

• ^•

. .

_•

_-

^j

.

^-

^{, .}

where:

P.,. = dual decision va~iable whic~ re?::-esent3 the cost.

associaced with resource of type t.

/ .-IJ

⁼ dual decision variable associated with project j.

The dual formulation has important implications for the financial manager. Namely, the dual L? and its optimal solution provide valuable information for both planning and control functions in the capital budgeting

This optimal solution gives the shadow accepted and rejected projects.

decision process.

prices for the

These values enable the decision maker to rank all projects according to their relative attr3c~iveness.

under

For accepted projects the slack variables (

S,,)

the shadow prices are found for the following constraint:

The shadow prices are computed using the following expression

= shadow price associated with accepted project (shown under the slack variable associated with project j)

= NPV for project j shown in the objective function.

p:

^{= shadow} price in the optimal solution associated with each resource t which is required to accept a project. (shown under the slack variable associated

with the corresponding resources).

C[\. = quantity of j .

resources of type t required by project

The shadow prices

Y

J

may very well give a ranking for the projects which di£fe~s from that given by any of the simple models as payback, NPV, IRR,or the profitability index. Such differences in ranking will exist because the latter models look at projects independently.

,/

22

The shadow pric.es show interrelationships among projects by means of the budget constraints. They evaluate the projects at cost of capital ( p~) that is implied by the optimal use of resources of type t .

For the rejected projects the shadow prices are computed in an analogous way as for accepted project.

llJ

=

T

I

^P'"

,

= shadow prices associated with rejected project j (shown in the objective function row in the column for x J ).

The llJ value shows the amount by which the objective funtion would decrease if the firm were forced to accept the unattractive

would mean

project that the

j. If such projects were accepted,this

scar~e capital budget dollars would be used in a suboptimal way, since the- opportunity cost

with the cash outflows

C-Ip" ,

c.,\:---)

.

exceeds

associated the

present value of the benefits generated by the projects.

It should be mentioned that the values must be zero for all projects that are accepted (including partially accepted projects) because the benefits of these projects must justify the cash outlays in the various periods of the planning horizon (C)~

)

when they are evaluated at the implied cost of capital ( p~) when the budgets each year are

"

llsed in an optimal way~

Example:

Lorie and Savage nine-project problem.

Project NPV

1 14

2 17

3 17

4 15

5 40

6 12

7 14

8 10

9 12

Cas.h outflow in period 1

12 54 6 6 30 6 48 36 18

Budget available

I c.

_J^x) ~50

The

LP

formulation is as follows :

Cash outflow in period 2

3 7 6 2 35 6 4 3 3

Max NPV ~ 14x, + 17x~ +17x~ +lSx~ +4Cx r +12xb

s • t •

, 1 " ...

T.l."' .... '3

12x, + 54x, +6x l +6x ... +30x s

+6x& +48x, +36x& +18x~ ~ SI ~ 50.

3x, + 7X, +6x, +2x.., +35x,.

+6x, +4x 1 +3x 8 +3x, + 5,

x, + S:; ^~ 1 x., + S , ^~ 1 x, +

S ...

^{~ 1 x ,} 51 ^{~ 1}

X 3 +

S

5' = 1

x '" +

Sa

^{= 1}

::: 20~

x₁ +

S"

x& ^.;- S ^'\0 x , ^.;-

S"

x S ⁾ 0 i ::;; 1 ,2 t .. ,11.

J

^{, ~} •

j

::;; 1

^f

2 ,

,9.

=

=

=

budget constraint year 1

budget constraint year 2

1 Upper limits 1 on project 1 acceptance

Non negativity constra.int

The problem has been solved with a package on Apple II developed at the department.

24

5olu·hon

ttlttttttttttttttlllllllltltttttl'ttttt

lMAX PROBLEM TWO-PERIOD LP 1

tCONSTRAINTS 11 «30) L=ll G= 0 E= 0 i ,VARIABLES 20«80) D= 9 S=11 A= 0 1 tttfftil1itttllittltltlltlllttttttttt'l

OPTIMAL SOLUTION 70.2727273

---

VALUE OF Xl ^;:::;=;. 1 VALUE OF v.,

"~ ==.;. 0 VAl.UE OF v_

"" ^::=::>

VALUE OF X4 :::=> 1 VALUE OF X5 => 0

VALUE OF X6 ^==} .969697 VALUE OF X7 ^{= }} .. 045455 VALUE OF XS ==;- 0

VALUE OF X9

==>

-- CONSTRAINT ANALYSIS

CONSTR .. ACTIV ^? MARGINAL LEVEL. OF

NR VALUE SLACK Ui)

8UDYE-j YES .. 1:6364

--

⁽⁾

BUDYE-2 YES 1.863636 0

-tJPPER-l YES 6.77:727 ^(l

UPF'£R-2 NO 0

UPPER-3 YES 5 0

UPPER-4 YES 10.454546 0

UPPER-5 NO 0 1

UPPER-6 NO 0 .030303-

UPPER-? NO 0 .. 954545

UPPER-8 NO 0 1

UPPER-9 YES 3.954545 0

We

see that the basic variables, that _{i s} f the variables that are equal to a positive value in the _~pt=-mal solution are ! x." x~, x~, x" x_{1 ,} x~ (column value

or

^x)

and S."S1' S" S~, S,o (column level of slack).

Any of the variables in the problem which are equal to zero, are in fact, non basic va~iables in the opt:.mal solution. Thus, X 1. = xI" = xa = 0, which shows that these three projects should be completely rejected; in addition

S,=

S~

=

0, shows that the entire budget of 50 in year

1

and 20 in year 2 has been spent on the six projects that have been designated lor acceptance. Further-, S;

=

Ss-

=

S6

=

S"

=

0, since the project corresponding

100%

accepted.

to these slack

variabl~s have been

These projects require the use of the entire budget in both years and generate the maximuQ objective func~ion

value of 70,2727273, which is the NPV of the accepted projects.

The marginal value of const~aint ~umber 1 (the shadow price of

s,

^{) is O~136364} which means that _the object:"ve

func~ion would increase by this amount (b.136364) for each dollar of additional budget they could obtain.

constraint is active because any changes in tha cons~raint

can change the objective function.

The same explanation can conseraints.

be used

LP

models seem tailor-made :or

for the other

solving capi'L.al budgeting

they not

problems when resources are 1~3ited. Why then are universally accepted either in theory or in practice? .One reason is that these models are often not cheap to use. LP is considerably c~eaper in terms of computer time, but it cannot be used when large t indiviSible

projec'L.3 are involved.

As with any sophisticated long-~3nge planning too 1 there is the general problem of getting good data. It is not just worth applying costly, sophisticatad methods for poor

26

data. Furthermore, these models are based on the assumDtion that all future investment opportunities are known.

In

reality, process.

the discovery of investment ideas is an unfolding

Before we leave the area of linear programming a appropriate LP noteworthy controversy relative to the

formulation should be mentioned. Under capital rationing,the appropriate discount rate to use in determining the net present values of projects under consideration cannot be determined until the optimal set of projects is determined, so that

well as

the size of the capital budget is ascertained as the sources of the subsequent finanCing and hence the cost of capital (or the appropriate discount rate to use in calculating NPVs). This is a simultaneous problem wherein the firm· should concurrently determine through an iterative mathematical programming process bot~ the optimal set of capital projects and the optimal finanCing package, with its associated cost of capital to be used in the discounting

process.

Weingartne~ suggested an operational approach; his model assumes %hat a:1 shareholders have the same linear utility preferences for consumption~ And it makes the period- by-period .utilities iadependent of one another. He t.hen proposes

dividends

a more operational mode1 1 which maximizes the to be paid in a terminal ye~r, where throughout the planning horizon dividend are nondecreasing and can be required to achieve a specified annual growth rate. Over the past decade, several other authors have jumped into the c~n,troversy, each suggesting his own reformulation.

The Weingartner ¹ s basic horizon model is as follows

Max Z =

-; _11,

_x

J

^{i- v,}

-

^w_

- ^,

s • t • J

.1

_{a 'J}

J

Xl ^i- v,

-

^{w, 1.}

- ^D,

^(a)

~

a'l ^XJ - (l+r)v_t ^•. ₁ + v .. + (l+r)w •. , - ^WI. 1. D"

t = 2, •• o,T ( b )

j:::;: l , .... ,n ( c )

t = 1, .•• ,T ( d )

where:

~ a value of all cash flows of project j subsequent to J =

the horizon discounted to the horizon at the market rate of interest, r •

x_J = fraction of project j accepted.

T = horizon year.

v_{t =} amount available for lending in period t . w ... ^{= amount} borrowed in period t.

a 'J = cash flow in period t from project j.(pos-expenditure!outflow, neg-revenue/inflow)

D_t - anticipated cash t~row-off in period t.

The dual of the basic horizon model is as follows:

T n

Min

p.

^{D-t, +}

s. t

2.

J' ,

J.., ,. ^p.

^j =l, . . . ,n ..

(1)

~ 1 ( 2)

). -1

" ⁽³⁾

p. .. , -

^{O+r)p. ,. 0} _• ⁽⁴⁾

;;. 0

• ^t -2 ••••• T ( 5 )

( 6 )

28

The following derivations help to understand the meaning of the dual variables

(2)and(3)

Po'"

^{= 1 (7)}

(4)and(5)

p:- ,

_(1+r)

= (8)

f\

(7)and(8) P+..~

=

^(1+r)

f: ^~~

(1

^{+r) ,}

..

= ft.

^>.

=

(l+r) ^{i - l f~} t + r-t.

=

^(l+r)r-~

So, one can see that

f.

is the compounded rate of interest which expresses the yield at the horizon of an additional dollar in year t .

When a project is •

corresponding restriction

fully accept.ed ,- that: is, x_ = 1

J

of the form of (a) becomes

t,P~a"J .-

=

~ ^{a J}

,. T_ t

+

I

(-a,,)(1+r)

=

~ a J

t: "'-.

This is the value of the projec~ a~ time T. This the same as saying that the NPV of the ?roject should positive.

...

,the

is be

For the rejected project we know that xJ ⁼ O,and also

• ~ 11 ~

x J ' i , and therefore ,oJ = 0.

This is the same as saying that the NPV is smaller than O.

3.S.Integer programming .applied problem.

to the capital rationing

The main mocivations for the use of IP in the capital rationing problem setting are the following

1. Difficulties imposed by the acceptance of partial projects in LP are eliminated.

2. All the project interdependencies can be formally included in the constraints of ILP, while the same is not true for LP due to the possibility of accepting partial projects.

In using the simple capital budgeting models (NPV,IRR,PI.etc) i t is assumed that all the investment projects are independent each other (i.e.that project cash flows are not related to each other and do not influence or one another if various projects are accepted). In change

using ILP, virtually any project dependencies (mut:lally exclusive, prerequisite, complementary) can be incorporated on the model.

The general ILP formulation for :he capital rationing:

N

Max NPV =

l

. j-1

st

t=1,2, ... T

j=1.2 •••• N

There are three types of project dependencies:

a. MutuallY exclusive projects.

A

set of projeccs wherein the acceptance of one project in the set includes the simultaneous acceptance or any other project in the set. It is incorporated in the model by the following constraint:

J

l..

^x ~ 1

J.;:.J

~ set of mutually consideracion

30

exclusive projects under

jb:J ~ chat project j is an element of the set of mutually exclusive project J.

This constraint states that at most one project from set J can be accepted; this means that the firm could choose not to accept any project from set J. On the other hand, if i t was necessary to select: one project from the set, the above contraint would appear as a strict equality

2..

^X' ~ 1 jE.J J

b. Prerequisite/contingent projects.

Two or more projects wherein the acceptance of one project (A) necessitates the prior acceptance of some other project(s) (Z):

x""

~ X '1..

If project A cannot be accepted unless project Z is accepted, we could say that project Z is a prerequisite project for acceptance of- project"-A; alternatively, we could

upon

say that the accept:ance of project A is contingent the acceptance of project Z. However project Z can be accepted on its own and project A rejected.

c. Complementary projects.

Wherein the acceptance of one ~rojec~ enhances the cash flows of one or more other projects.

There are ^t~o complementary projects,7 and 8. Either of these projec=s can be accepted in isolation. However, i f both are accepted simultaneously : the COSt will be

r~duced by say 10%. The net cash inflow will be increased by 15%. To handle the problem, a ne~ project (project 78) would be constracted. The constraint below is needed to preclude acceptance of both projects 7 and 8 as well as

78; because the latter is the consisting of the t~o former projects.

xl + x 8 + x r8

::i

1

The shortcomings of the ILP:

composite project

a. ILP is probably only feasible for small to medium size capital budgeting problems (25 constraints and 100 projects).

be paid

We have to think whether the price that has to in moving to ILP from LP is worth the benefit gained.

b. 'leaningful shadow prices (which show the marginal change in the value of the objective function for an incremental change in the right hand side of various constraints) are not available in ILP. That is, many of the constraints on ILP problem which are not binding on the optimal

of zero,

integer solution whic!) indicates

will be assigned shadow that these resources are

prices

"free goods^fT •

In

reality this is not true since the objective function would clearly decrease if the availability of such resources were decreased.

Example:

Consider the following 15 projects