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Chapter 10
Decision Making in Finance: Capital Budgeting
Babita Goyal
Key words: Capital budgeting, discounting criteria, non-discounting criteria, urgency, payback
period, ARR, NPV, IRR, PI, and BCR, benefits, costs, depreciation, cash flows, simple and non-simple
investments, pure and mixed investments, ACC, capital rationing, divisible and indivisible projects,
linear programming.
Suggested readings:
1. Chandra P. (1970), Appraisal Implementation, Tata-McGraw Hill Publishing Company
Limited, New Delhi.
2. Hampton J.J. (1992), Financial Decision Making (4th edition), Prentice hall of India Private
Limited
3. Khan M.Y. and Jain P.K. (2004), Financial Management (4th edition), Tata-McGraw Hill
Publishing Company Limited.
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10.1 Introduction
Capital budgeting is the technique of making decisions regarding fixed or long-term assets. The time
horizon for these decisions is long, usually more than one year. In general, capital budgeting is a
technique to evaluate or appraise expenditure decisions regarding current outlay(s), which are expected
to produce benefits in future.
The inherent features of capital budgeting are, thus
(i) They involve potentially large anticipated benefits;
(ii) The risk involved with these decisions has a relatively high degree (than that of short-term
decisions); and
(iii) The time distance between initial outlay and benefits received is large.
Capital budgeting decisions are the key processes in management of a firm. These decisions are not
easy to make. The basic reason for these decisions to be difficult is the relatively large outlays
associated with these decisions. The main difficulties associated with these decisions are listed below:
(i) These decisions are made with an eye on the future, which is uncertain. With this uncertainty,
creeps in the element of risk. For example, if the time horizon of a decision is 15 years, then
we have to take into consideration the factors which may prevail till or at that time and which
may not even be present today but may arise in due course of time. A plant whose anticipated
life is 15 years is installed after taking into account the costs associated with the plant,
production, and marketing of the product. However, emergence of new technology may
altogether alter the potential market of the product. Thus the decision may not be easy to be
made.
(ii) Another difficulty associated with these decisions lies in the different time periods associated
with the cash flows involved with these decisions. Since money has time value, which does
not change at a constant or even predictable rate (at times price rise is higher than that at some
other times) so the net value of the flows may not be strictly comparable.
313
(iii) A third potential difficulty, which affects the capital budgeting decisions, is that a decision
may have several aspects (benefits and costs), which may not be quantifiable at all. Then it
may be difficult to evaluate a decision strictly in terms of financial aspects only.
In spite of these difficulties, the capital budgeting decisions are the integral part of the management of
the firm. Now we discuss the reasons why theses decisions are so important and why utmost care
should be practiced while making these decisions.
(i) Such decisions affect the profitability of the firm due to the fact that these are related to the
fixed assets of the firm, which are in fact the true earning assets of the firm. These include the
plant, the machinery, the inventory, and the long-term investments etc. So when huge capital
decisions are made, they may result in substantial departure from the past performance of the
firm. A strategic decision would significantly influence the firm's expected profits and risks,
which in turn would affect the firm's standing in the market. Thus a well-planned decision
may result in receipt of fortunate gains whereas an ill planned decision may force the firm to
be bankrupt.
(ii) Since the time period over which these decisions are to be implemented is very large and these
decisions involve huge outlays, so they have an impact over the cost structure of the company.
For example, if the decision is regarding initiation of a new product line, it means the costs to
be incurred on plant, machinery, labour, rent, insurance, marketing and so on. An
unsuccessful venture may heavily burden the cost structure of the firm.
(iii) Capital budgeting decisions are not easily reversible, and even if reversible, would put huge
financial restrain on the system.
(iv) In case of capital scarcity, as is the case with most of the firms, these decisions enable the firm
to choose more profitable or viable projects.
10.2 Types of capital budgeting decisions
The objective of any firm is to increase its profitability, which has two major factors, the optimal,
(minimum) costs and (maximum) revenue. While revenue generation is not under much control of the
314
management, the cost structure can be so organized so as to increase the profitability of the firm. A
key instrument of this task is to increase the efficiency of the operations of the firm. This includes
adoption of latest technology, replacement of old and worn out machinery, targeting new products and
making strategic investment decisions. Capital budgeting decisions help management in increasing
efficiency of the firm's operations.
Depending upon the two factors of the profitability of the firm, capital budgeting decisions can be of
two types.
(i) Revenue expanding investment decisions With an aim to increase the revenue of the firm,
these decisions are regarding the expansion of the present operations of the firm or initiation
and development of the new product lines.
(ii) Cost reducing investment decisions With an objective to reduce the costs of the firm,
these decisions are regarding replacement of the old and worn out machinery and assets,
acquisition of new technology and selection of the most suitable technology.
Cost reducing investment decisions are subject to less risk as compared to revenue expanding
investment decisions due to the fact that the former have the lesser element of risk than the latter.
Depending upon the type of the decision to be taken, different decision criteria can be prescribed.
(i) Acceptance-rejection criterion There can be several proposals before the management to
decide upon. Out of these proposals the ones, which meet some specific criterion (e.g., rate of
return exce4eding a specific limit or payback period less than a specific interval) are accepted.
All these proposals are independent in the sense that acceptance of one does not affect the
acceptance or rejection of the others.
(ii) One among several criterion All accepted decisions under acceptance-rejection
criterion may not be implemented and at times, we may need to select one project out of
several accepted projects. For example, if the decision is to buy a new machine, then several
brands may fulfill the acceptance-rejection criterion. However, all the machines are not to be
purchased and choice of one will eliminate the chances of the selection of the others. Such
projects are mutually exclusive projects.
315
(iii) Capital rationing Under acceptance-rejection criterion, several projects may be
selected and would be implemented if the firm had unlimited capital. But this is not the case
and every firm has limited capital. In such situation, the selected projects are ranked
according to some criterion (e.g., rate of return) and the projects on the top of ranking list are
selected if they meet the limited capital criterion also. More than one projects may be
undertaken if their joint capital requirement is meets the limited capital criterion.
On the basis of the costs and benefits defined for an investment project, the worthwhile ness of a
project can be evaluated. Various criteria have been suggested for this purpose, which can be
categorized into two broad categories:
(i) Non-discounting criteria This category of evaluation techniques consists of
(a) Urgency
(b) Payback period
(c) Accounting rate of return; and
(d) Debt service coverage ratio.
(ii) Discounting criteria This category of evaluation techniques consists of
(a) Net present value
(b) Benefit cost ratio
(c) Internal rate of return
(d) Terminal value; and
(e) Annual capital charge.
Now, we shall discuss these techniques in detail.
(i) Non-discounting criteria
(a) Urgency According to this criterion, projects, which are deemed to be most urgent,
get priority over the projects that are regarded as less urgent.
316
However, it may be difficult to assign the degree of urgency in general. In certain situations, this may
not be very difficult. For example, replacement of minor equipment may be immediate to ensure the
continuity of production. In such situations, detailed analysis delay decisions.
Urgency is a relative concept. In general, the reliable relative degree of urgency may be difficult to
determine. This is due to lack of an objective and quantifiable basis of assigning urgency levels to
different alternatives. In such situations, persuasiveness and presentability of project proposers may be
the determining criterion. But this is not a scientific basis of determining preferences and hence this is
not a preferred criterion except for some emergency situations.
(b) Payback period The payback period is the time required to recover the initial cost outlay
of the project. A project with a short payback period is considered to be a desirable project. The firms
using this criterion generally specify a maximum acceptable time period and the projects below this
time period are considered to be worth accepting. This criterion is simple to use and useful particularly
in those situations where the risk increases with time.
Example 1: The following table gives the cash flow streams for four alternative projects A, B; C
and D. find the desirability of the projects in terms of payback periods:
Table 10.1
Year A B C D
0 (200)* (300)* (210)* (320)*
1 40 40 80 200
2 35 35 50 50
3 35 35 80 -
4 35 35 60 -
5 35 35 80 -
6 20 20 50 -
7 20 15 40 -
8 20 15 40 -
9 20 15 40 400
10 20 15 40 300 * Initial outflow
317
Sol:
Table 10.2
Project Payback period
A 6
B 6
C 3
D 9
From payback period point of view, investment C is the best investment followed by investments A, B,
and D.
This method takes into account the face value of the cash flows without considering their time values,
thus violating the fundamental principal of financial accounting to appropriately discount the money.
Secondly, it ignores the cash flows beyond the payback period. For example, cash inflows of project D
are attractive ninth year onwards but this method is rejecting this project because of a large payback
period. Thus it is not measuring the project from profitability point of view.
(c) Accounting rate of return (ARR) Also known as average rate of return, it is the ratio of
income to the investment. Since, in accounting management, income and investments can be variously
defined, so there are various measures of ARR. We give below some of the common measures of ARR:
Example 2: The following financial information is available about a project:
Table 10.3
Year 1 2 3 4 5 6 7 8
Investment (Rs. In lacs) 24.0 21.0 18.0 15.0 12.0 9.0 6.0 3.0
Depreciation 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0
Income before interest and tax 6.0 6.5 7.0 7.0 7.0 6.5 6.0 5.0
318
Interest 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5
Income before tax 3.5 4.0 4.5 4.5 4.5 4.0 3.5 2.5
Tax - 1.0 2.5 2.5 2.5 2.2 1.9 1.4
Income after tax 3.5 3.0 2.0 2.0 2.0 1.8 1.6 1.1
For this data
Average income after tax 2.125A: 0.088 8.8%Initial investment 24
Average income after tax 2.125B: 0.1574 15.74%Average investment 13.5
Average income after tax bC:
= = =
= = =
ut before interest 4.625 0.193 19.3%Initial investment 24
Average income after tax but before interest 4.625D: 0.342 34.26%Average investment 13.5
Average income before tax aE:
= = =
= = =
nd interest 7.1875 0.299 29.9%Initial investment 24
= = =
Average income before tax and interest 7.1875F: 0.5324 53.24%Average investment 13.5
Total income afetr tax but before depriciation -Initial investment G: Initial investment × years
2
= = =
41 24 17 0.177 17.7%12 8 96−
= = = =×
From this example, it is clear that although this method is simple to use and interpret and it is based on
the entire cash flows throughput the life of the project, still the concept is somewhat confusing due to
various measures available for calculating it. Even if a particular measure is selected still there will be
different values of ARR as income and investments can be defined in more than one ways.
Another drawback of this method is that it is giving too much weight age to the cash flows occurring in
distant future due to the fact that it is not taking into account the time value of the money. Consider the
following two projects.
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Table 10.3
Particulars Project A Project B
Cost (Rs.) 50,000 50,000
Estimated life (years) 7 7
Annual estimated income (after depreciation
and tax) (Rs.): Year:
1
2
3
4
5
6
7
Total
3500
5500
7500
10000
13500
11500
9500
61000
13500
11500
10000
9500
7500
5500
3500
61000
Estimated salvage value (Rs.) 6000 6000
Average income (Rs.) Total incomen
⎛ ⎞⎜ ⎟⎝ ⎠
7625 7625
Average investment
(Rs.) 1Salvage value + (Cost - Salvage value)2
⎛ ⎞⎜ ⎟⎝ ⎠
33500 33500
ARR Average income 100Average investment
⎛×⎜ ⎟
⎝ ⎠
⎞ 22.76% 22.76%
Thus ARR of both the projects is the same. However, the nature of cash flows is indicating that project
B is more viable than project A.
Debt service coverage ratio (DSCR) DSCR is the ability of the firm to meet the interest and the
principal repayment obligation of the firm.
Example 3: The projected profit after tax, depreciation, interest on loan, and loan repayment
installment for a company are given below.
320
Table 10.4 (Rs. in lacs)
Year Profit after tax (PAT) Depreciation (D) Interest on loan (I) Loan repayment installment (LRI)
1 -2.0 5.0 8.1 6.0
2 9.0 5.0 8.1 6.0
3 20.0 5.0 7.2 6.0
4 32.0 5.0 6.3 6.0
5 32.0 5.0 5.4 6.0
6 36.0 5.0 4.5 6.0
7 40.0 5.0 3.6 6.0
8 36.0 5.0 2.7 6.0
9 32.0 5.0 1.8 6.0
10 30.0 5.0 0.9 6.0
For a year
PAT D IDSCRI LRI+ +
=+
For the total period
( )
( )
PAT D IDSCR
I LRI+ +
=+
∑∑
Table 10.5
Year Numerator Denominator DSCR
1 11.1 2.1 1.37
2 22.1 14.6 1.51
3 32.2 13.2 2.44
4 43.3 12.3 2.52
5 42.4 11.4 3.52
6 45.5 10.5 3.72
7 48.6 9.6 4.33
8 43.7 8.7 5.06
321
9 38.8 7.8 5.02
10 35.9 6.9 4.97
Total 363.6 103.1 5.2
For the total period 363.6 3.53103.1
DSCR = =
The company is in a good position to repay its loans.
A high value of the ratio means that corresponding to repayment obligations, the firm is earning a
good income, and a project with the high value (2 or more) is considered as worth accepting.
One drawback of DSCR is its interpretation since it involves both pre-tax cash flow (interest) and post-
tax cash flow (profit and loan repayment installment).
A major drawback of techniques involving non-discounting criteria is that they do not take into account
the time value of money. So whereas payback period technique is just taking care of cash flows till the
initial investment is recovered, ARR takes into consideration all the cash flows situated even in distant
future without considering the actual worth of those cash flows. The results on the basis of these
techniques may be distorted if the value of money decreases significantly. To overcome this drawback,
we make use of discounting criteria.
(ii) Discounting criteria
The main disadvantage of non-discounting criteria is that they don’t consider the time value of money,
while evaluating a project in terms of costs and benefits. In such situations, the comparisons of cash
flows at different time periods become meaningless as we are comparing essentially non-homogeneous
quantities. Thus, in order to make valid and meaningful comparisons, it is essential to discount cash
flows at a certain and appropriate rate. The criteria, which take into account the time value of money,
are called the discounting criteria. The rate at which the cash flows are discounted is called the cost of
the capital. The cost of the capital is thus the discount rate at which the market value of the project
remains unchanged as the time progresses.
322
Another important feature of discounting criteria is that they take into account all the cash flows (costs
and benefits) occurring during the life of the project.
(a) Discounted payback period Discounted payback period is the period needed to recover
the initial outlay on the project when the estimated future flows are discounted at a rate equal to the
cost of the capital. In our example of pay back period if the future flows are discounted at a rate of
10%, then the discounted payback period can be calculated as follows.
Table 10.6
Year PV factor (k = 10%) A B C D
0 1 (200)* (300)* (210)* (320)*
1 0.909 36.36 36.36 72.72 181.80
2 0.826 28.91 28.91 41.30 41.30
3 0.751 26.29 26.29 60.08 -
4 0.683 23.91 23.91 40.98 -
5 0.621 21.74 21.74 49.68 -
6 0.564 11.28 11.28 28.20 -
7 0.513 10.26 7.70 20.52 -
8 0.467 9.34 7.01 18.68 -
9 0.424 8.48 6.36 16.96 169.60
10 0.386 7.72 5.79 15.44 115.80
Total discounted
flows
184.28 175.33 364.56 508.50
Payback period 4 (215.08) 18
2
(392.70)
Thus when the time value of money is taken into consideration, projects A and B are not capable of
recovering the initial outlay, for project C, this is approximately 4 years, whereas for project D, it is
approximately eight and half years.
323
(b) Net present value (NPV) The net present value of a project is the sum of the present
value of all the cash flows associated with the project, i.e.,
0 1 20 1 2
1NPV = ...
(1 ) (1 ) (1 ) (1 ) (1 )
where NPV = Net present value;
= cash flow occurring at the end of the period ;
= life of the
nn i
n ii
i
CF CF CFCF CFk k k k
CF i
n
=
+ + + + =+ + + + +∑
project; and
= cost of capital used as the discount rate.k
k
A cash inflow has a positive sign but a cash outflow has a negative sign. Typically, first cash flow is
an outflow.
The decision rule is to accept the project if the NPV is positive and reject the project if NPV is
negative. If NPV = zero, it is a matter of indifference.
However, NPV = 0 means that if the project is accepted, then only the original investment will be
recovered.
For mutually exclusive projects, a project with the highest NPV is selected.
There are two principal features of this method:
(i) This method is based on the assumption that the intermediate cash (in) flows of the project are
invested at a rate of return equal to the firm's cost of capital.
(ii) The NPV of a simple project (i.e. the one involving a single cash outflow, i.e., the initial
investment and subsequent outflows) monotonically decreases as the discount rate increases.
However the decrease in NPV is at a decreasing rate.
Mathematically, let the cash flows associated with the project be C0, C1… Cn. if k is the discount rate,
then
1 20 2 ...
(1 ) (1 ) (1 )n
n
CC CNPV C
k k k= − + + + +
+ + +
If k is a continuous variable and NPV a continuous function of k, then
324
1 2
2 3 1
2( ) ... 0 1(1 ) (1 ) (1 )
NPV is monotonically decreasing.
nn
nCC Cd NPV kdk k k k += − − + − < ∀ > −
+ + +
⇒
Also,
21 2
2 3 4 2
( 1)2 2.3.( ) ... 0 1(1 ) (1 ) (1 )
the rate of decrease is decreasing.
nn
n n CC Cd NPV kdk k k k +
+= + + + > ∀ >
+ + +
⇒
−
Rationale underlying NPV criterion NPV criterion can be very instrumental in making choice
between different investments/ consumptions. To illustrate the point consider a simple problem
concerning a choice between current consumption and future consumption.
A
Year 1
Lending – borrowing opportunity B
O
D
Suppose we have a cash inflow of OA no
point X. we assume that through the capita
borrowing. The opportunity for lending o
1/r, r is the annual rate of interest. If th
consumption in year 1 will be OA (1+r) =
the current consumption OB ,r+1
i.e., AC. A
X .
C Year 0
Fig. 10.1
w (year 0) and OB in a year from now (year1) as shown by
l market, wealth can be transferred across time by lending or
r borrowing is shown across the line CXD, which has slope
e present cash inflow of OA is lent at an interest rate r, the
OD. Similarly a borrowing against future inflow will make
ny point on CXD can be accessed by lending or borrowing.
325
However, investment opportunities are not limited to just lending or borrowing in capital market. Let
the second opportunity is real assets. The opportunity line for real assets investment is shown as
Year 1
Investment opportunity curve
Year 0O F
This is not a straight line, unlike the lending borrowing opportunity curve. Its slope is high to begin
with but declines progressively because the marginal return from additional investment in real assets
tends to decline.
If different amounts are to be invested in real assets and some amount in lending – borrowing, the
situation can be represented as
Year 1
Y
O
M
R
G
L E P
F
. X
. .F H
ig 10.3
32
D
6
ig. 10.2
S NQ
. Z
Year 0
DL is the line representing the lending-borrowing opportunity in the capital market.
Now we consider different investments in the real assets.
(i) Investment of DF A sacrifice of DF in year 0 brings a benefit of FX in year 1. As a result,
position of real asset investment opportunity curve is shifted from point D to point X from where one
can move along the line MN by availing of lending –borrowing opportunities (MN || DL, all lending-
borrowing lines are parallel to each other)
(ii) Investment of DG An investment of DG leads to point Y from D, thus setting the line PQ
to function in lending-borrowing opportunities.
(iii) Investment of DH (Amount less than DF) An investment of DH causes a
movement across the line RS.
Comparison of all the three cases leads us to the fact that investment of amount DF leads us to
consumption frontier MN that is higher than PQ (investment of DG) or RS (investment of DH). Thus
the first investment is the best.
Now, we show that this is also the investment that has the highest present value:
NPV of investment DF = present value of benefits – present value of costs
= NF – DF = DN
NPV of DG = QG – DG = DQ
NPV of DH = SH – DH = DS
Thus, the NPV maximizes at the highest consumption frontier. This is the rationale of NPV criterion.
The NPV method of choosing a project is instrumental in achieving the objective of the financial
management, viz. the maximization of the stockholders wealth. The phenomenon can be explained as
follows.
When NPV exceeds zero, the rate of return is more than the cost of capital employed (when the rate of
return is equal to the rate of the capital employed, then NPV is equal to zero). In such situations, the
327
actual return would exceed the expected return, which in turn will enhance the market price of shares
thus increasing the shareholders' wealth.
NPV is a criterion, which takes into account the time value of money and gives appropriate weight age
to the cash flows occurring in near future or distant future. Also it considers all the cash flows in their
entirety. Another feature of the NPV criterion is that NOPV of various projects can be measured as
NPV (A+B) = NPV (A) + NPV (B)
Also, the method works even if the discount rate changes at some point of time.
However, the ranking of projects on the basis of this criterion may be influenced by the discount rates,
i.e., a project, which appears to be more attractive for one discount rate, may become less attractive for
another discount rate. Consider the following two projects.
Table 10.7
Year Project A (Rs.) Project B (Rs.)
0 (3,00,000) (3,00,000)
1 60000 130000
2 100000 100000
3 120000 80000
4 125000 75000
For different discount rates, we have the following table of NPV's:
Table 10.8
Discount rate (%) NPV (A) (Rs.) NPV (B) (Rs.)
10 11,567.76 11,052.40
12 (1,656.96) 354.55
14 (13,521.98) (9,310.89)
15 (18,991.84) (13,790.75)
16 (24,176.09) (18,051.98)
328
Another limitation of this criterion is that it is an absolute measure so a businessman may be reluctant
to uses it that wants the results in terms of rate of return.
(c) Benefit-cost ratio (BCR) Also known as profitability index (PI), BCR is defined as
the benefit per rupee of cost. Mathematically, it can either be expressed in terms of present worth of
money or its net present value, i.e.
(i)
where, = Benefit cost ratio
PVBBCRI
BCR
=
= Present value of benefits; and
= Initial investment
- (ii)
PVB
I
NPVB PVB I PNBCRI I
= = = - 1
= Net benefit cost ratio
= Net present value of benefits
VBI
NBCR
NPVB
The decision rule is: Accept a project when BCR > 1 (or, NBCR > 0); reject it if BCR < 1 (or,
NBCR < 0); and it is indifferent if BCR = 1 (or, NBCR = 0).
This method is an improvement over the NPV method in the sense that NPV being an absolute measure
may not work reliably when evaluating projects with different initial outlays. For example, for a
project with initial outlay Rs. 50,000, the NPV is Rs. 56,000 and for another project with the initial
outlay Rs, 1,00,000, the NPV is Rs. 1,08,000. Then the NPV criterion will select the second project.
However the PI method in this situation would yield that
56000 of first project = 1.12;and50000
108000 of first project = 1.08100000
BCR
BCR
=
=
Thus second project should be selected.
329
Under the unconstrained conditions, BCR criterion is same as the NPV criterion. In case of limited
capital, this criterion may be a suitable one, as it will rank the projects in decreasing order of capital
efficiency. However this method is not suitable if the cash outflows occur beyond the current period.
Also, since BCRs of two different projects cannot be added up so an assessment of a project involving
several projects cannot be done.
( d) Internal rate of return (IRR) IRR is the discount rate which makes NPV equal to zero,
i.e., it is the discount rate in the equation
0 1 20 1 2
00 = ...
(1 ) (1 ) (1 ) (1 ) (1 )
where = cash flow occurring at the end of the period ;
= life of the project; and
= Internal ra
nn i
n ii
i
CF CF CFCF CFr r r r
CF i
n
r
=
+ + + + =+ + + + +∑
te of return.
r
On solving this equation for r, we get the internal rate of return.
Like the NPV criterion, IRR method also takes into account the appropriate discount rate. In case of
NPV, this discount rate is the cost of the capital, which is determined by the factors other than those,
involved in the project and hence is external to the project. In case of IRR, the rate is determined by the
factors of the project (cash flows and their timings). That is why the discount rate is referred to as the
internal rate of return.
IRR of a project can be calculated as follows:
(i) Find the expected future average annual cash inflow associated with the project.
(ii) Divide the initial outlay by the average annual cash inflow.
(iii) From the present value of annuity tables, find the discount rate that will make the present
value of an annuity of one rupee (running for a period equal to the life of the project) equal to
the figure obtained in step (ii). This is the starting discount rate.
Consider the following cash flows associated with a project:
330
Table 10.9
Year Cash flows (Rs.)
0 -1,00,000
1 30,000
2 30,000
3 40,000
4 45,000
For this project, we calculate the IRR.
(i) 1, 45,000Average annual cash inflow = Rs.36, 2504
=
(ii) Initial outlay 1,00,000 = 2.759Average annual cash inflow 36, 250
=
(iii) For n = 4, r corresponding to 2.759 is 16%. This is the starting discounting rate.
Now,
0 1 2 3
100000 30000 30000 40000 45000 (1 0.16) (1 0.16) (1 0.16) (1 0.16) (1 0.16)
100000 25862.069 22294.89 25626.37 24853.1
-1363.57
−+ + + +
+ + + + +
= − + + + +
=
4
So, we have to reduce the rate of discount to reach at the solution. Take r = 0.15, we have
0 1 2 3
100000 30000 30000 40000 45000 (1 0.15) (1 0.15) (1 0.15) (1 0.15) (1 0.15)
100000 26086.96 22684.31 26300.65 25728.9
800.82
−+ + + +
+ + + + +
= − + + + +
=
4
Thus the real IRR lies between r = 15% and r = 16%. By interpolation of values, we get r ≅ 15.36%
The decision rule is: Accept a project with maximum IRR.
Interpretation of IRR - Physical significance There are two possible interpretations of IRR:
(i) IRR is the rate of return on the unrecovered investment balance in the project.
331
(ii) IRR is the rate of return earned on the initial investment made in the project.
To understand the meanings of these interpretations, we consider the following project.
Table 10.10
Year Cash flows (Rs.)
0 -5,00,000
1 0
2 6,30,000
3 2,40,000
IRR for this project is 27.88%.
(i) First interpretation: The unrecovered investment balance is defined as
1 (1 )
where unrecovered investment balance at the end of year ; and
cash flow at the end of year .
t t t
t
t
F F r CF
F t
CF t
−= + +
=
=
For r = 27.88%, the balance schedule of the project is as follows.
Table10.11
Year (t) Unrecovered investment
balance (Rs.) at the
beginning of the period
(Ft-1)
Interest
(Rs.) for
the year t
(r)
Cash flow
(Rs.) at the end
of the year
(CFt)
Unrecovered investment
balance (Rs.) at the end of
the period
(Ft= Ft-1(1+r) + CFt)
1 -500000 -139400 0 -639400
2 -639400 -178265 630000 -187665
3 -187665 -52320.9 240000 14.36*
*This amount is due to approximation error.
(ii) Second interpretation If a project involving an initial outlay of amount I has an IRR of
r%, and a life of n years, the value of the benefits of the project at the end of n years is I (1+r)n.
For our project,
I = Rs. 5,00,000, r = 27.88% and n = 3.
332
If the intermediate flows of the project are reinvested at rate r, then the value of all benefits at the end
of three years is given as:
Table10.12
Year (t) Benefits (cash inflows) (Rs.) Compound value at the end of 3 years (Rs.)
1 0 -
2 6,30,000 8,05,644
3 2,40,000 3,06,912
11,12,556
However, it is not possible for the firm to reinvest the intermediate flows at a rate of return equal to the
project's IRR, the first interpretation seems more realistic, i.e., IRR is the rate of return on the time
varying unrecovered investment balance in the project and not the rate of return earned on a sustained
basis on the initial investment over the life of the project.
Multiple rates of return We have seen that for computing IRR, we have to solve the
equation
0
= 0 (1 )
ni
ii
CFr= +∑
10 1 (1 ) (1 ) ... 0 n n
nCF r CF r CF−⇒ + + + + + =
which is a polynomial of degree n in (1+r).
If the only cash outflow is the initial flow then CF0 is negative and the subsequent cash flows are cash
inflows so for every i, CFi is positive. Thus the polynomial expression has just one change of sign and
hence only one real root. Now, consider the following project, which needs investment more than
once.
Table 10.13
Year Cash flows (Rs.)
0 -6,000
1 25,000
2 -18000
3 -9,000
333
The IRR for this project is given by
0 1 2
3 2 2
6000 25000 18000 9000 0 (1 ) (1 ) (1 ) (1 )
or 6(1 ) 25(1 ) 18(1 ) 9 0
133%, 200%, 50%
r r r r
r r r
r
− − −= + + +
+ + + +
+ − + + + + =
⇒ = −
3
Negative r is discarded since it does not make any sense financially. Then r = 200% and r = 50%.
Consider the NPV of the project for various values of r.
Table 10.14
r (%) NPV (Rs.)
0 -8,000.00
10 -4,464.18
20 -2,395.83
30 -1,166.63
50 0.00
75 416.49
100 437.50
125 340.19
150 217.60
200 0.00
225 -84.31
250 -153.27
The graph of this data shows that
NPV (Rs.)
Discount rate (%)r = 300% r = 100%r = 50%
NPV curve
Fig 10.4
334
For r = 0, NPV = -8000. As r increases, NPV increases till it becomes 0 at r = 50% after which it
attains positive value. Maximum NPV occurs at r = 100% after which it starts declining and again
becomes 0 at r = 200%. After this point it is always negative. Thus this project has two rates of return.
A polynomial of degree n has n roots. If the polynomial has k (<n) changes in sign then exactly k roots
will be real. However out of those real roots we are interested in those which have magnitude more
than one since 1 + r < 1 does not make sense in financial decision making.
Note: The reciprocal of IRR is the payback period when the annual cash flow is constant over the
life of the project, which is fairly long as can be shown as follows.
For a project which has a life of n years, an initial outlay I, constant annual cash flow C and salvage
value S, the IRR is the value of r in the equation
0
= 0(1 )
ni
ii
CFr= +∑
1
(1 ) (1 )
ni
i ni
CF SIr r=
⇒ = ++ +∑
1
1Now, 1(1 )
1 1 (1
nni
ii
n
n
CF C Cr r rr
C C SIr r r r
=
⎛ ⎞= − ⎜ ⎟++ ⎝ ⎠
⎛ ⎞⇒ = − +⎜ ⎟+ +⎝ ⎠
∑
)
as
C nr
CrI
→ →
⇒
∞
The IRR is a preferred criterion as it takes into account the time value of the money and considers all
the cash flows in their entirety. Further it makes sense to businessmen who are interested in knowing
rates in place of absolute measures like NPV.
However, this criterion may not be uniquely defined. If the cash flow stream of the project has more
than one change in sign, then there is a possibility of more than one IRR, a situation that may be
difficult to interpret.
335
Another drawback of this measure is that it does not differentiate between lending and borrowing and
hence a large IRR may not mean a viable project. To illustrate this point consider the following
projects:
Table 10.15
Cash flows in years (Rs.) Projects
0 1
IRR (%)
A -400 600 50
B 400 -700 75
The option B means a borrowing at a rate of interest of 75% whereas option A means lending at a rate
50%. Thus option A is better than option B although IRR is suggesting something else.
A further limitation of the IRR method is that this criterion can be misleading when the projects are
mutually exclusive and have substantially different outlays. Consider the following two projects.
Table 10.16
Cash flows (Rs.)
Years
Projects
0 1
IRR (%)
NPV (Rs.) (k=10%)
A
B
-10,000
-50,000
20,000
75000
100
75
7,857
16,964
Both projects are good and since A has higher IRR than B, it seems more lucrative. However, B with a
higher NPV is adding more to the stockholders wealth. Thus IRR may not be a suitable criterion. In
such cases, we consider the rate of return on incremental cash flows. If we switch from the low outlay
project to higher outlay project, the incremental cash flows are
Table 10.17
Years 0 1
Incremental cash flows (Rs.) -40,000 55,000
336
IRR for these flows is 37.5%; much above the cost of the capital hence it is desirable to switch from
project A to project B.
Under IRR, it is assumed that all the intermediate cash (in) flows will be reinvested at the same rate as
IRR. While this may be difficult to justify, some times this assumption is ridiculous as well. Consider
the following projects.
Table 10.18
Cash flows (Rs.) Year
Project A Project B
0 -55,000 -55,000
1 10.000 25,000
2 15,000 22,000
3 18,000 18,000
4 22,000 15,000
5 25,000 10.000
IRR 16.24% 0.23%
This means that cash flows of project A can be invested at a rate equal to 16.24% whereas the cash
flows of project B can be invested at a rate equal to 0.23% or in other words, the nature of the cash
flows from different projects is different which is something ridiculous. Moreover all the intermediate
cash flows may not be reinvested as they may be used somewhere else in the firm.
(e) Terminal value method As the future is uncertain, so the assumption of IRR that
all the future flows will be reinvested at the same rate may be unrealistic. The rate of return of future
investments will depend upon the time of the investment. The terminal value method takes into
account this aspect when choosing a project.
Consider the following project
337
Table 10.19
Initial outlay Rs. 50,000
Life of the project 5 years
Year Cash inflow Rate of reinvestment (%)
1 10000 6
2 15000 6
3 15000 8
4 20000 8
Cash inflows
5 20000 10
We want to determine the present value of this project for cost of capital k = 10%.
Table 10.20
Year Cash flow (Rs.) Rate (%) Year of reinvestment
Compounding factor (k=10%)
Compounded sum (Rs.)
1 10000 6 4 1.262 12620
2 15000 6 3 1.191 17865
3 15000 8 2 1.166 17490
4 20000 8 1 1.080 21600
5 20000 10 0 1 20000
Total 89575
PV of the project 55626.08
The decision rule is that if the present value of the sum of the compounded reinvested cash inflows
(PVTS) is greater than the present value of the cash outflows (PVO), accept the project. If PVTS is
less than PVO, reject the project. In case of equality the firm is indifferent to the selection or
rejection of the project.
In the above example, the firm should go for the project.
The main difference between the above method and the NPV method is that in the later case the cash
flows are discounted whereas in the earlier case, the cash flows are compounded.
338
This method is easy to use and explain. It makes better sense than the NPV method since NPV method
does not consider the cash inflows in terms of interest earnings.
The major limitation of this method is that the future interest rates are difficult to determine.
10.3 Basic principals for measuring benefits and costs
A capital budget may be viewed as a stream of costs and benefits (negative and positive cash inflows).
Typically, the costs are incurred in the beginning year(s), which are then followed by benefits, which
extend over a fairly long interval of time. The investment decision process of a firm is a complex
process and does not depend upon just one criterion. Also, several proposals may have to be examined
at the same time, which may have entirely different structures. In order to examine such proposals,
more than one decision criterion may have to be employed. Sometimes application of more than one
criterion may lead to conflicting conclusions. In such situations, a further analysis is required to reach
at final decisions.
Since capital budgeting process uses costs and benefits as the raw material for the process, so one
should know how to measure the benefits and costs. Certain facts and rules should be taken into
account while measuring benefits and costs. Now, we will discuss some of such rules.
(i) Cash flow principal Costs and benefits must be measured in terms of cash flows, costs
as cash outflows and the benefits as cash inflows. The cash flows in aggregation represent the
purchasing power.
(ii) Post-tax principal Cash flows must be measured in post-tax terms because that
represents the net flow from the point of view of firm.
(iii) Incremental principal Cash flows must be measured in incremental terms. The
incremental principal says that changes in the cash flows of the firm, arising from the adoption of the
proposed project, alone are relevant.
In estimating incremental cash flows, following points are taken care of.
339
(a) Consider the incidental effects In addition to the direct cash flows of the project, all the
incidental effects it has on the rest of the firm must be considered. The project may enhance the
profitability of some lines of existing activities because they complement each other or it may detract
from the profitability of some lines of existing activities because of competitiveness.
(b) Ignore sunk cost Sunk costs are bygone. Hence they do not matter for the present
decision-making.
(c) Include opportunity costs If a project employs some resources available within the
firm, it should be changed into opportunity cost of these resources even if there are no explicit cash
outflows arising from the use of these resources.
(d) Allocation of overhead costs Normally, overhead costs are allocated to different
projects of a firm on some predetermined basis. So when a new project starts, it is expected to bear a
part of the total overhead costs, which may have hardly any relationship with the incremental overhead
cost of the project. For the purpose of investment appraisal, it is the incremental overhead cost that
matters and not the allocated overhead cost.
(iv) Long-term funds principal In capital investment appraisal, the principal focus is
usually on the profitability of long-term funds. Hence cash flows relating to long-term funds need to
be segregated.
(v) Interest- exclusion principal Interest on long-term debt should be excluded from the
computation of profit and taxes. This is so because the cost of capital used for appraising the cash
flows stream reflects the time value of money. Hence, interest cost, which represents the time cost of
debt, must be excluded from the cash flow estimation. Otherwise, double counting will occur.
10.4 Components of cash flow
Cash flows associated with a project may be divided into three parts
(i) Initial flows Initial flows are outlays made in the beginning of the project and are usually
negative. The initial cash flows consist of payments for the acquisition of land, infrastructure, and
machinery, preliminary and pre-operative expenses, and the build-up of working capital.
340
(ii) Operational cash flows These cash flows arise out of operations of the project and are
generally positive. For determining operational cash flows, projected profitability statement is the
starting point. The adjustments for non-cash charges, like depreciation, amortization of patent cost, and
write-off preliminary expenses are made in the profitability statement.
If the investment is measured by long-term funds committed to the project, which consists of share
capital (both equity and preference) and long-term borrowings, then the post-tax operational cash flows
are
Profit after tax + Interest on long-term borrowings (1- tax rate) + Depreciation
+ Other non-cash charges
In this expression, interest on long-term borrowings is added after adjustment for the tax factor since
we are considering post-tax measure of return.
The expression can be rewritten as
Profit before tax on long-term borrowings (1- tax rate) + Depreciation + Other non-cash charges
as
Profit after tax + Interest on long-term borrowings (1- tax rate)
= Profit before tax (1- tax rate) + Interest on long-term borrowings (1- tax rate)
= Profit before tax on long-term borrowings (1- tax rate)
(iii) Terminal cash flows These cash flows are result from the winding up of the operations
of the projects and are generally positive. Assuming that the measure of investment is long-term funds
committed to the project, the terminal cash flows are given by
Net salvage value of fixed assets +Net recovery of working capital margin
(a) Net salvage value of fixed assets: Usually, the sale price of fixed assets is higher than the
depreciated book value at the time of sale. Therefore, net salvage value of fixed assets is equal to
Sale value of fixed assets - Tax on profit arising from the sale.
341
If, loss is incurred on the sale, then the net salvage value of fixed assets is equal to
Sale value of fixed assets + Tax shield on loss from the sale.
(b) Net recovery of working capital margin Working capital is normally expected to be
liquidated at its face value. Hence no profit or gain is expected from its liquidation and net recovery of
working capital margin is equated with working capital margin provided at the outset.
10.5 Depreciation of fixed assets
With the passage of time, all assets (except land) lose their capacity to render service. Accordingly a
fraction of the cost of the asset is chargeable as an expense for the years the asset is in service. The
accounting process for this gradual conversion of capitalized cost of fixed assets into expenses is called
depreciation. The major factors to which the decline in the usefulness of any asset can be attributed
are: The deterioration of the asset and the obsolescence of the asset. Whereas deterioration is a
physical phenomenon associated with the wear and tear of the asset, obsolescence is on part of
improved equipment or processes, changes in style and other factors not linked with the physical
condition of the asset. Thus
“ Depreciation is the allocation of the depreciable amount of an asset over the estimated useful
life,” where “useful life is the period over which a depreciable asset is expected to be used by the
enterprise.”
10.5.1 Methods of depreciation
The amount of depreciation of a fixed asset is determined by taking into account the original cost of the
asset, the recoverable cost at the time of the retirement of the asset and the expected useful life. Out of
these factors, only the original cost is known with certainty and the other two factors can only be
estimated. The total amount to be depreciated is the difference between the original cost of the asset
and the recoverable amount at the time of its retirement and this difference is to be charged over the
useful life of the asset. There are four frequently used methods of computing depreciation.
342
(i) Straight line method;
(ii) Units of production method;
(iii) Diminishing balance method; and
(iv) Sum-of-the-years-digit method.
(i) Straight line method (SL Method)
This method is based on the assumption that the asset provides the same level of service throughout its
useful life and hence equal amount should be charged as the expense over the estimated life of the
asset. Consider the following example.
Example 4: The original cost of a machine is Rs. 15,000. After being used for 5 years, the
machine is expected to fetch Rs. 5,000. Calculate the depreciation charged per annum by the straight-
line method.
Sol: The annual depreciation can be calculated as follows
Original cost - Salvage valueAnnual depreciation = Useful life
15,000 5,000 5
. 2,000Rs
−=
=
The annual depreciation can also be calculated as a percentage on the net cost. The annual percentage
is the 100 divided by the number of years of useful life, i.e. 100/5 = 20 in this case. Then the annual
depreciation is given by
( )
( )
20Annual depreciation = Original cost - Salvage value100
20 15,000 5,000100
. 2,000Rs
×
= − ×
=
This method is a simple method of computing depreciation and it provides a uniform allocation of costs
to periodic revenues. Hence this is a widely used method.
343
(ii) Units of the production method (UP Method)
In this method, depreciation is charged on the basis of estimated productive capacity of the asset under
consideration. In this method, first of all the depreciation is calculated in an appropriate unit of
production such as hour, kilometer or the number of operations. Then annual depreciation is calculated
by multiplying the unit depreciation by the number of units in one year.
In our case, let the estimated life of the machine in hours is 10,000 hours. Then depreciation per hour
(if the number of units in a year is 3,000) is given by
Original cost - Salvage valueUnit depreciation = Useful life in units
15,000 5,000 10,000
Re. 1.00 per hour
Annual depreciation
−=
=
= Unit depreciation units per year
Re. 1.00 per hour 3000 hours
. 3,000Rs
×
= ×
=
(iii) Diminishing balance method (written-down value method) (DB Method)
This method results in a diminishing periodic depreciation charge over the estimated life of the asset.
In this method, each year, depreciation is charged by applying a rate to the net cost of the asset as at the
beginning of that year. Next book value at a particular point of time is the original cost minus total
depreciation accumulated up to that point of time. The rate to be applied is usually the double the
straight-line depreciation rate.
In our example, the straight-line depreciation rate is 20% and, therefore, the diminishing balance rate
would be 40%. This rate would be applied to the original cost of the asset for the first year and
thereafter to the net book value over the estimated life of the asset. The asset's residual value is not
taken into consideration for calculation of net book value. However, the asset is not to be depreciated
below its residual value in the last year. We have the following table to calculate the depreciated value
of the asset over different years:
344
Table 10.21 Year Net cost (Rs.) Rate (%) Depreciation for the year (Rs.)
1 10,000 5/15 3,333
2 10,000 4/15 2,667
3 10,000 3/15 2,000
4 10,000 2/15 1,333
5 10,000 1/15 667
Both diminishing balance method and the sum-of-the-year-digits method provide for a higher
depreciation charge in the first year of the use of the asset and a gradually declining periodic charge
thereafter. Hence they are referred to as accelerated methods of depreciation.
Comparison of different methods of depreciation Depending upon the different methods of
computing depreciation, the annual depreciation may be different and hence difference in the annual
profit. But overall depreciation charged and the overall profit will be same at the end of the project.
For our case, let the annual profits before depreciation be Rs. 30,000. The following table shows the
impact of different methods of depreciation on this annual profit and the overall profit of the project:
Table 10.22
Depreciation (Rs.) Profit after depreciation (Rs.) Year
Profit before
depreciation
(Rs.)
Straight-
line
method
Diminishing
balance
method
Sum-
of-the-
year-
digits
method
Straight-
line
method
Diminishing
balance
method
Sum-of-
the-
year-
digits
method
1 30,000 2,000 6,000 3,333 28,000 24,000 26,667
2 30,000 2,000 3,600 2,667 28,000 26,400 27,333
3 30,000 2,000 400 2,000 28,000 29,600 28,000
4 30,000 2,000 - 1,333 28,000 30,000 28,667
5 30,000 2,000 - 667 28,000 30,000 29,333
Total (Rs.) 1,50,000 10,000 10,000 10,000 1,40,000 1,40,000 1,40,000
345
Rs. (‘000)
5000 .
4000 .
2000 .
1000 .
3000 .
0 .
4
Sum- of- the- years -digit - method Straight-line method
Diminishing balance method
6000 .
Time
Example 5: A firm is i
equipment by a new one. T
same price. The remai
salvage value.
The new equipment will cos
be needed. The new equipm
by written down method an
replacement project.
. 1
nterested in
he book va
ning usefu
t Rs. 4,00,0
ent will sa
d applicab
. 2
Fig
assessing t
lue of the o
l life of the
00. It can b
ve Rs. 1,00
le tax rate i
3
.3
. 10.5he cash flow
ld equipment
equipment
e sold for Rs
,000 annually
s 50%, deter
46
.
s associated
is Rs. 90,00
is 5 years a
. 2,50,000 a
. If the dep
mine the cas
. 5
with the replacement of old
0 and it can be sold for the
fter which it will have no
fter 5 years when it will not
reciation is at a rate of 10%
h flows resulting from the
Sol:
Table 10.23: Cash flows from the replacement project (Rs.)
Year 0
1
2
3
4
5
A. Net investment (3,10,000)
B. Savings in operations 1,00,000 1,00,000 1,00,000 1,00,000 1,00,000
C. Depreciation on old equipment
9,000 8,100 7,290 6,561 5,905
D. Depreciation on new equipment
40,000 36,000 32,400 29,160 26,244
E. Incremental depreciation on new equipment (D-C)
31,000 27,900 25,110 22,599 20,339
F. Incremental taxable profit (B-E)
69,000 72,100 74,890 77,401 79,661
G. Incremental tax 34,500 36,050 37,445 38,700 39,830
H. Incremental profit after tax
34,500 36,050 37,445 38,700 39,830
I. Incremental net salvage 2,16,526*
J. Initial flow (A) (3,10,000)
K Operating flow (H+E) 65,500 63,950 62,555 61,299 60,169
L. Terminal flow (I) 2,16,526
M. Net cash flow (J+K+L) (3,10,000) 65,500 63,950 62,555 61,299 2,76,695
* Table 10.24: Table to calculate salvage value
Particulars Value (Rs.)
Salvage value of the new equipment after 5 years 2,50,000
Book value of the new equipment after 5 years 2,36,196
Profit on sale 13,804
Tax on profit 6,902
347
Salvage value of the old equipment after 5 years 0
Book value of the old equipment after 5 years 53,144
Loss on sale 53,144
Tax shield on loss 26,572
Incremental tax payable after 5 years if new equipment is bought 33,474
Net incremental salvage value 2,16,526
Example 6: The following information is available for a capital project:
Table 10.25
Particulars Value
Life of the project (years) 15
Initial outlay (Rs. in lacs)
Plant and machinery
Working capital
180
120
Financing (Rs. in lacs)
Equity
Long-term loans
Trade credit
Commercial banks
100
104
36
60
Expected annual sale (Rs. in lacs)
(Including depreciation but excluding interest)
350
Cost of sales (Rs. in lacs)
(Including depreciation but excluding interest)
190
Tax rate (%) 60
Salvage value (Rs. in lacs)
Plant and machinery
Working capital
180
120
Depreciation (written down method) 15%
348
Further, working capital will be fully recovered at the end of the project. The long-term debt carries an
interest rate of 14% and is payable in eight equal annual installments. Short-term advance from the
commercial banks will be maintained at Rs. 60 lakh and will have an interest rate of 8% per annum. It
will be fully liquidated at the end of the project. The level of trade credit will remain at Rs. 36 lacs and
will be fully paid at the end of the project. Calculate the cash flow stream associated with the
following measures of investment.
(a) Total funds; (b) Long-term funds; and (c) Equity.
Sol: Table 10.26: Net cash flows relative to equity
Years Particulars
1 2
3
4
5 6 7
A Sales 350 350 350 350 350 350 350
A' Depreciation 27 22.95 19.51 16.58 14.09 11.98 10.18
B
Operating cost
(Including depreciation but
excluding interest)
190
190
190 190 190 190 190
C Interest on short-term bank
borrowings
2.88 2.88 2.88 2.88 2.88 2.88 2.88
D Interest on term-loans 14.56 14.56 14.56 12.74 10.92 9.1 7.28
E Profit before tax 142.56 142.56 142.56 144.38 146.21 147.02 149.84
F Tax 85.536 85.536 85.536 86.628 87.72 88.81 89.9
G Profit after tax 57.024 57.024 57.024 57.752 58.48 59.39 59.97
H Net salvage value of fixed
assets
I Net salvage value of current
assets
J Repayment of long-term loans 13 13 13 13 13 13
K Repayment of short-term loans
L Repayment of trade credit
349
M Net cash flows related to equity
investors (G+H+I-J-K-L+ A')
84.024 66.974 63.534 61.332 59.57 58.37 57.15
N Net cash flows related to long-
term funds
(G+A'+D (1-.60)+H+I-K-L)
84.024 79.974 63.534 61.332 59.57 58.37 57.55
O Net cash flows related to total
funds
(G+A'+C(1-.6)+1.6D+H+I)
90.976 86.926 83.51 80.58 78.09 76.162 74.834
8 9 10
11 12 13 14 15
350 350 350 350 350 350 350 350
8.65 7.36 6.25 5.32 4.52 3.84 3.26 2.77
190
190
190
190 190 190 190 190
2.88
2.88
2.88
2.88
2.88
2.88
2.88
2.88
5.46 3.64 1.82 - 14.56 14.56 14.56 14.56
151.66 153.48 155.3 157.12 157.12 157.12 157.12 157.12
91 92.09 93.18 93.18 93.18 93.18 93.18 93.18
60.66 61.39 62.12 63.94 63.94 63.94 63.94 63.94
13.10
120.00
13 13 - - - - - -
350
60
36
56.31 55.75 68.37 69.26 68.46 67.78 67.2 103.81
39.01 55.75 55.37 69.26 68.46 67.78 67.7 103.81
72.646 71.358 70.25 70.412 69.612 68.932 68.352 200.96
10.6 IRR and return on invested capital
IRR, though a popular approach to appraise a project, is not always an unambiguous approach to do so.
It is meaningful to a certain type of projects.
Types of projects and relevance of IRR Investment proposals can be classified on the basis of two
types of factors, (i) the number of sign changes in the net cash flow stream; and (ii) the form of the
unrecovered investment stream.
On the basis of the number of sign changes in the net cash flow stream we can define the following
type of investments:
(i) Simple investment A simple investment is one, which has cash outflow(s) followed by
cash inflow(s). Thus there is only one sign change in the cash flow stream
(ii) Non-simple investment A non-simple investment is one in which one or more cash
outflows are interspersed with one or more cash inflows. There are at least two sign changes in a non-
simple investment.
Simple and non-simple investment are demonstrated in the following table
Table 10.27 Sign of cash flow at time Type of investment
0 1 2 3 4 5
Simple _ + + + + +
Simple _ _ + + + +
Non-simple _ _ + + - +
Non-simple _ + + - + -
351
The unrecovered balance at any time t during the life of the project is given by the expression
1
0 1( ) (1 ) (1 ) ...
Cash flow at time ;
to the project.
t tt t
t
F r CF r CF r C
CF t
r IRR
−= + + + + +
=
=
F
On the basis of the form of the unrecovered stream, we can define the following type of investments:
(i) Pure investment A pure investment is one for which the unrecovered investment
balance is either zero or negative throughout the life of the project, i.e.,
10 1
10 1
( ) (1 ) (1 ) ... 0; 1,2,2..., 1
and ( ) (1 ) (1 ) ... = 0;
is the life of the project.
t tt t
n nn n
F r CF r CF r CF t n
F r CF r CF r CF
n
−
−
= + + + + + ≤ = −
= + + + + +
A pure investment is one from which the firm does not borrow at any time during the life of the
investment but recovers fully its investment at the end of the project. Hence the internal rate of return
in case of a pure investment is the return earned on the funds invested in the project by the firm.
(ii) Mixed investment A mixed investment is the investment for which ( ) tF r is greater
than zero for some t and less than or equal to zero for some other t. Thus a mixed investment contains
unrecovered investment balance ( ) and over-recovered investment balance ( ). In
other words, there are periods when the firm lends to the project and there are periods when the firm
borrows from the project.
( ) < 0tF r ( ) > 0tF r
Two-fold classification On the basis of the above types of investments, we can classify investments
as follows.
(i) Simple investments (Simple investments are always pure investments);
(ii) Non-simple, pure investments; and
(iii) Non-simple, mixed investments.
Now, we shall study the relevance of IRR for such investments.
352
Consider the following information relating to the four projects viz. A, B, C, and D:
Table 10.28
Cash flows (Rs.) Year
A B C D
0 -5,000 -2,000 -50 -1,600
1 1,000 600 200 10,000
2 1,000 500 -200 -10,000
3 1,000 40 - -
4 1,833 -378 - -
5 5,000 1,800 - -
IRR (%) 20 12 100 25, 400
We, now, calculate the unrecovered investment balance for each of these IRR
Table 10.29
Unrecovered investment balance (Rs.)
D
Year
A B C
r = 25% r = 400%
0 -5000 -2000 -50 -1600 -1600
1 -5000 -1640 100 8000 2000
2 -5000 -1337 0 0 0
3 -5000 -1097 - - -
4 -4167 -1607 - - -
5 0 0 - - -
Type of investment Simple Non-simple, pure Mixed Mixed Mixed
If we interpret the above facts, we see that throughput its life, project A is a net borrower although the
investment is fully recovered at the end. The IRR in this case is internal to the project. Again, project
B is a non-simple, pure investment and IRR is internal to the project.
For project C, the unrecovered balance is negative in the beginning, positive at the end of first year and
zero at the end of the project. This implies that the firm has been a lender to the project from period 0
353
to 1 and a borrower from the project from period 1 to 2. The investment is a mixed investment and it is
both a user of and a source to the funds of the firm. In such cases, there is a problem of interpretation
of the IRR. If the rate of return imputed to the funds borrowed from the project is same as the rate of
interest on the funds lent to the project, then the rate of return can be viewed as internal to the project.
This is not so in general and in general, the rate of interest on the funds lent to the project is different
form the rate of return imputed to the funds borrowed from the project. Thus in case of mixed
investment the rate of return cannot be considered as internal to the project.
For project D, besides interpretation there is problem of multiple rates of return. Then IRR is not very
meaningful for such projects.
To sum up, IRR is meaningful for pure investments but not for mixed investments.
Decision-making in case of mixed investments Mixed investments are to be analyzed differently
as the usual computation of IRR in these cases is not very meaningful. Then we need to have some
criterion for decision making in case of mixed investments. First of all, we see how to identify a mixed
investment.
Test for pure and mixed investments To test whether an investment is a pure or mixed, we
compute , the minimum interest rate that makes investment balance mini ( ) 0; 0,1,2... 1tF r t n≤ = − .
Such a value always exists since . 0 ( ) < 0F r
Since ( )min 0 0,1,2... 1tF i t≤ ∀ = −n
( ) min 0 and 0,1,2... 1tF i i i t⇒ ≤ ∀ > = n− (As i increases, Ft decreases)
Then the test procedure is as follows
(i) Find the value of for the investment. mini
(ii) Find i* which satisfy ( *) 0nF i =
(iii) Compare i* with . If , the investment is pure, otherwise it is a mixed investment. mini *mini i>
Criterion for mixed investment- Return of invested capital In case of a mixed investment;
the firm commits funds to the project part of the time and borrows funds from the project rest of the
354
time. Define r* as the return on the invested capital and k as the cost of the funds borrowed from the
project.
When the unrecovered balance from the project is negative (i.e. the firm has funds committed to the
project) it is compounded at r*; When the unrecovered balance from the project is positive (i.e. the firm
has overdrawn funds from the project) it is compounded at k. Thus the unrecovered balance is a
function of both r* and k. Let we denote this by ( )* ,tF r k . As the life of the project is n years, so
( )* ,nF r k = 0 is the condition for realizing a return r*on invested capital. To obtain r*, we proceed as
follows.
(i) Determine whether the investment is a pure investment or a mixed investment. For a pure
investment r* = k.
(ii) For a mixed investment, calculate ( )*,tF r k as follows:
( )
( ) ( )
( )
( ) ( )
( )
( ) ( )
*0 0
* *1 0 1 0
0 1 0
* *1 1
1 1
* *1 1
,
, 1 if 0
1 if 0
, 1 if 0
1 if 0
, 1 if 0
t t t t
t t t
n n n n
F r k CF
F r k F r CF F
F k CF F
F r k F r CF F
F k CF F
F r k F r CF F
− −
− −
− −
=
= + + <
= + + >
= + + <
= + + >
= + + <
M
M
( )1 1 1 if 0n n nF k CF F− −= + + >
(iii) Obtain r* by solving ( )* *
min, 0 subject to 0 nF r k r i= ≤ ≤
=
For project D
(i) . So the project is a mixed project. min 25%, 400% and 525%r i=
(ii)
355
( )
( ) ( )
*0 0
* *1 0
*
, -1600
, 1600 1 10,000 ( 0)
8400 1600
F r k CF
F r k r F
r
= =
= − + + <
= − −
Since * *
min 5.25 ( ) 1600 < 8400r i r= ⇒
i.e., ( )*1 , 0F r k ≥ and it is compounded at k in the next step.
( ) ( )( )* *2 , 8400 1600 1 10000F r k r k= − + −
(iii)
( )
( )( )
*2
*
*
, 0
8400 1600 1 10000 0
6.25 5.251
F r k
r k
rk
=
⇒ − + − =
⇒ = −+
r* can be obtained for different values of k. Following points can be observed:
(i) r*, the return on the invested capital increases with k, the imputed cost of surplus funds;
(ii) asymptotically; and *min r i→
(iii) If k = r*, then k = r* = i*.
6.25 5.25 1
(1 ) 5.25(1 ) 6.25
0.25, 4.00
kk
k k k
k
= −+
⇒ + = + −
⇔ =
10.7 Conflict between NPV and IRR methods
Basically NPV and IRR methods are similar methods in the sense that they use the similar procedures
of discounting the future cash flows. In most of the cases, they would lead to the similar conclusions.
For example, for independent projects, both will either accept or reject a proposal. Also for simple
(conventional) investment, the two methods will yield the similar results. The reason for this similarity
of behavior lies in the structure of these methods. According to NPV criterion, a project is accepted if
the NPV of the project is positive. In case of IRR criterion, a project is accepted if its IRR is more than
356
a predetermined required rate of return (the cost of capital). The projects with positive NPV have IRR
greater than the cost of capital (k).
Consider the following figure, which depicts the relationship between NPV and the discount rate:
If k = 0, NPV is the highest. However, this situation is very unlikely. NPV is inversely proportional to
the figure, NPV is zero when k = 20%. This is the IRR (r) of the project. Now, let k = 10%. In this
Again, for k = 25%, NPV is negative (equal to OB) so when NPV is negative, IRR (20%) is less than
Thus the acceptance and rejection criteria of both the methods remain the same.
The situation will remain the same if the projects to be examined are independent. However if the
k and decreases as k increases. The point where NPV is zero is the IRR. After this point, NPV will be
negative and the project will be unacceptable.
B
IRR
Discount rate (%) . . 0 0
. 5 5 2
. 11
. 25
NPV (Rs.)
A
O
Fig. 10.6
In
case, NPV is positive (equal to OA). For positive NPV, IRR (20%) is more than the cost of the capital
(10%).
the cost of the capital (25%).
projects are mutually exclusive, this situation may change. The exclusiveness of the projects may be on
two accounts.
357
(i) Technical exclusiveness Technical exclusiveness refers to the situation when the
alternatives to be examined have different profitability and the most profitable alternative is to be
chosen. For example buy or manufacture decisions, purchase or lease decisions etc.
(ii) Financial exclusiveness Financial exclusiveness refers to the situation when the alternatives
are subject to the financial constraints. The most profitable alternatives are to be selected from a give
(profitable) set of alternatives. Due to limited funds all alternatives cannot be selected. This situation
is also referred to as capital rationing.
Consider the following projects:
Table 10.30
Project Initial outlay
(Rs.)
Annual cash flows
(Rs)
Life of the
project (years)
NPV (@ 10%)
(Rs.)
IRR (%)
A 14,000 2745 20 8158 19
B 19,000 3550 20 10203 18
The two methods are giving conflicting results. If the projects were independent or the firm had
sufficient funds, both could have been selected. But if only one project is to be selected, then the
decision maker is in a dilemma.
The conflicting ranking by the NPV and the IRR method is attributed to some of the following
situations.
(i) Size disparity – When the projects involve different cash outlays;
(ii) Time disparity – When the timings and the pattern of cash flows are different.; and
(iii) Life disparity - When the projects involve different lives
(i) Size disparity If the initial outlay of mutually exclusive projects under consideration are
different, then the NPV and IRR criteria are likely to provide conflicting rankings.
358
Consider the following projects.
Example 7: Two assembling systems manufactured by two different companies are being
considered by a manufacturing firm. The system A would cost Rs. 50,000, has expected life of 5 years
and would generate cash inflows of order Rs. 17,500 per year. The system B would cost Rs. 30,000,
has expected life of 5 years, and would generate cash inflows of order Rs. 11,500 per year. Rank the
two projects on the basis of NPV and IRR.
Table 10.31
Particulars Project A Project B
Initial outlay (Rs.) 50,000 30,000
Life of the project (years) 5 5
Annual cash flows (Rs.) 17500 11500
NPV (Rs.) 14853 12358
IRR (%) 22.11 26.50
Thus NPV criterion is leading to selection of project A whereas the IRR criterion is leading to selection
of project B. Hence the two methods are giving conflicting results.
Reason for conflict The reason for the conflict lies in the fact that NPV method assumed that the
intermediate cash (in) flows are reinvested at a rate equal to the cost of the capital which is a fairly
reasonable assumption ensuring the minimum opportunity rate on the intermediate flows. However,
IRR method assumed that the intermediate flows are reinvested at a rate equal to the IRR of the project.
This expectation is on the higher side of what actually is the rate of return. Liquid cash cannot generate
such a high rate of return. Thus the results of IRR method are upward biased.
Resolving the conflict In such situations, NPV method is invariably better than the IRR method.
The reason being that the objective of the NPV method is maximization of the shareholders' wealth,
which is ensured by a project earning the highest NPV. The conflict can be resolved by modifying IRR.
The approach to modify IRR is called the incremental approach.
359
According to this approach, in case of mutually exclusive projects with different outlays, if the IRR of
both the projects exceed some predetermined required rate, calculate IRR on the difference of the
outlays of the two projects. This difference of outlays of the two projects is called the incremental
outlay. If the IRR on this incremental outlay is more than the required rate of return, accept the project
with the bigger outlay.
In our case,
Table 10.32
Particulars Values
Incremental outlay (Rs.) 20,000
Incremental cash flows (Rs.) 6,000
NPV (Rs.) 2,495.20
IRR (%) 15.24
Since incremental IRR (15.24%) is more than the cost of the capital (10%), project A should be
selected, which the decision reached at by NPV method also.
The justification of this approach lies in the fact that if IRR of the incremental outlay exceeds the cost
of the capital, then the firm is earning profits of the smaller project and an additional profit on the
incremental outlay.
The incremental outlay approach would always yield results identical to those obtained by the NPV
method.
(ii) Time disparity This situation arises when the mutually exclusive projects are exhibiting
different cash flow patterns although their initial outlays may be the same.
Consider the following projects. Table 10.33
Particulars Project A Project B
Initial outlay (Rs.) 16800 16800
Life of the project (years) 3 3
360
Cash flows (Rs.) Years
1 2 3
14000 7000 1400
1400 8400
15500
NPV (Rs.) 2,512.94 2,782.05
IRR (%) 22.79 17.59
Under such situations, the cost of capital is the key determinant in the ranking of projects. The
following table gives the NPV of the projects for different values of k.
Table 10.34
k (%) Project A (Rs.)
Project B (Rs.)
0.05 3,897.06 5,277.97
0.08 3,033.06 3,705.87
0.1 2,512.94 2,782.05
0.175851 0.01
0.2 448.30 (691.74)
0.2279 0.41
0.25 (322.56) (1,894.40)
0.3 (962.71) (2,844.30)
0.5 (2,627.16) (5,027.16)
0.6 (3,108.64) (5,537.23)
Following is the graph of the NPVs of the two projects. Any discount rate till NPV of project A is
equal to the NPV of project B would support project B and any other discount rate would support
project A.
361
Fig. 10.7
. 0.05
. 0.1
. 0.2
. 0.3
. 0.4
. 0.5
. 0.6
4000 .
3000 .
2000 .
1000 .
-1000 .
-2000 .
-3000 .
Discount rate
5000 .
IRR (A) IRR (B)
NPV (Rs.) Project B
Project A
(iii) Life disparity – projects with unequal lives Consider the following proposals
A person has been allotted a piece of land for which he would get possession at the end of the year. He
has to deposit an amount of Rs. 10 lacs today in order to own land at the end of the year. Now, real
estate is a flourishing business and he has been offered a premium of Rs. 8 lacs if he sells the land as
soon as he gets its possession. However a well -wisher of him suggested that if he retains the plot for 5
years, he would be able to earn a premium of Rs. 28 lacs. If the cost of capital is 10%, what should be
his decision?
In this case we calculate the NPV and IRR of both the projects
362
Table 10.35
Particulars Project A Project B
Initial outlay (Rs. in lacs) 10 10
Life of the project (years) 1 5
Cash flows (Rs.)
Years
1
2
3
4
5
18
-
-
-
-
-
-
-
-
38
NPV (Rs.) 5.79 12.36
IRR (%) 80 31
Again the results by the two methods are different.
Resolving the conflict There are two ways of resolving this conflict.
(i) Common time horizon approach
In this approach, the projects are made to have equal lives by repeating them a number of times so that
their multiple life periods become equal. In our case, this can be done as follow:
Table 10.36
Project A Project B Years
Cash flows
(Rs. in lacs)
PV
factor
Total present value
(Rs. in lacs)
Cash flows
(Rs. in lacs)
PV
factor
Total present value
(Rs. in lacs)
0 -10 1.000 -10 -10 1.000 -10
1 18 0.909 16.362 -
2 -10 0.909* -9.09 -
3 18 0.826 14.868 -
4 -10 0.826* -8.26 -
5 18 0.751 13.518 38 22.36
17.4 12.36
363
*The cash inflow will be reinvested at the same time period.
Then we find the NPV of the new (multiple) project A is higher than the NPV of project B. Then
project A should be selected.
The implicit assumption of this approach is that the investment which is being replaced will have the
similar pattern of cash flows in future as it had in past.
This approach works when the projects have short lives. For large projects having lives say 15 and 20
years, the common life period would have been 60 years, which is very large, and estimates over this
period may have very large errors.
(ii) Annual capital charge (ACC) This drawback of common life horizon approach can be
overcome by annual capital charge approach. Annual capital charge of an investment is the cost on an
annual basis of the initial outlay and operating costs associated with that investment, the time value of
money taken into account. Annual capital charge is also referred to as the equivalent annual cost.
The annual capital charge can be determined as follows:
(i) Find the present value of the initial outlay and operating costs.
(ii) Apply suitable capital recovery factor to convert this present value into the annual capital
charge.
Consider the following project:
Table 10.37 Particulars Values
Initial outlay (Rs.)
Operating costs (Rs.)
Years
1
2
3
4
5
10,00,000
2,00,000
2,50,000
3,00,000
3,50,000
4,00,000 If the cost of capital is 10%, we want to determine the ACC.
364
Sol: (i) The present value of costs =
1 2 3 4
2,00,000 2,50,000 3,00,000 3,50,000 4,00,000 10,00,000 (1 0.10) (1 0.10) (1 0.10) (1 0.10) (1 0.10)
Rs. 21,01,220
+ + + + ++ + + + +
=
5
(ii) The capital recovery factor (inverse of PVIFA (n=5, k= 10%)) = 1 0.26383.7908
=
ACC 21,01, 220 0.2638
Rs. 5,54,302
⇒ = ×
=
This method is quite useful in the area of public price regulation of utilities. For example, the
construction cost and the operating cost of a power station may be converted into an annual capital
charge, which serves as basis for the determination of the tariff structure. The tariff structure may be so
determined as to recover the annual capital charge.
Example 8: A firm is examining following two proposals for installing an electronic security
system:
Table 10.38 Particulars System A System B
Initial cost (Rs.) 50,000 75,000
Expected life (years) 6 10
Running costs (Rs.) 15,000 12,000
Salvage value (Rs.) 3000 10,000
The depreciation is charged on straight-line basis. If the tax rate is 35%, which system should be
installed?
Sol: To calculate the annual capital charge, first of all we calculate the present value of the cash
flows associated with the two projects.
365
Table 10.39 Costs (Rs.) Adjusted PV (Rs.) Particulars
Machine A Machine B
PV factor
(@10%) Machine A Machine B
Initial cost (Rs.) 50,000 75,000 1.000 50,000 75,000
Operating costs:
1-6 years (A)
1-10 years (B)
7008.33
5525
4.355
6.145
1,83,127.66
3,39,511.25
Less: salvage value
6th year (A)
10th year (B)
3,000
10,000
0.564
0.386
1692
3860
Present value of the total costs 231435
410651.25
Capital recovery factor
A
B
4.355
6.145
ACC 53,142.36 66,826.89
The operating costs can be determined as follows:
Table 10.40
Particulars Machine A Machine B
Running cost (Rs.)
Less: Tax (@35%)
Less: Tax shield on depreciation :
Initial cost - Salvage value 0.35Life of the machine
×
Effective operating cost (Rs.)
15,000
5250
2741.67
7008.33
12000
4200
2275
5525
Since machine A has lower ACC than machine B, so machine A should be opted.
366
Net present value (NPV) versus profitability index (PI) The NPV method and the PI method in
general will provide the similar results. If NPV is positive, PI will be greater than one. However, in
evaluating mutually exclusive projects, the two may give differing results.
Consider the following projects
Table 10.41
Project A Project B Year Present value
factor (k=0.10) Cash flow (Rs.) Present value (Rs.) Cash flow (Rs.) Present value (Rs.)
0
1
2
3
1.00
0.909
0.826
0.751
(60,000)
42,000
50,000
50,000
(60,000)
38178
41300
37550
(40,000)
35,000
32,000
30,000
(40,000)
31815
26432
22530
NPV 57028 40777
PI 1.95 2.02
In this case, the project selected by the NPV method must be chosen for the reasons already discussed.
Capital rationing
Capital rationing is the process of choosing investment proposals when there are financial constraints
in the form of limited capital expenditure budget.
There may be a large number of proposals before the management of a firm, all of which could have
been selected if the firm had unlimited funds or sufficient funds to have undertaken all the proposals.
But this is not the case in reality; and in reality, firms have limited funds available for capital
expenditure. Then from the given set of acceptable proposals, the firm has to choose the most
profitable combination of proposals. Capital rationing aims at selection of such combinations.
. Thus the process of capital rationing involves two steps.
(i) Identification of acceptable proposals; and
(ii) Selection of the most profitable combination of proposals.
367
(i) Identification of acceptable proposals We have discussed several criteria that can be
used to identify acceptable projects. Generally IRR or PI is used for this purpose.
(ii) Selection of the most profitable combination of proposals At this stage, we ought to
choose those combinations of proposals, which have the highest NPV.
At this stage, the proposals may be accepted/ rejected partially or in their entirety. In fact, on the basis
of whether or not the projects are divisible, the projects may be classified into two categories:
(a) Divisible projects Those projects, which can be accepted or rejected in parts (e.g.,
investment in mutual funds), are called the divisible projects.
(b) Indivisible projects Those projects, which are to be accepted or rejected in their
entirety (e.g., installation of new machinery), are called the indivisible projects.
Example 9: A firm has following proposals before it, all of which are divisible. The firm has a
limited fund of Rs. 10.5 lacs. What should be the decision of the management?
Table 10.42
Particulars Project A Project B Project C Project D
Initial investment (Rs. in lacs) 4.5 3.0 2.6 9.0
NPV (Rs. in lacs) 0.90 0.75 2.25 2.70
PI 1.20 1.25 1.60 1.30
Sol: Stage (i): Rank the projects in descending order of PI
Table 10.43
Rank Project PI Initial investment (Rs. in lacs) NPV (Rs. in lacs)
1
2
3
4
C
D
B
A
1.60
1.30
1.25
1.20
2.6
9.0
3.0
4.5
2.25
2.70
0.75
0.90
368
Stage (ii) NPV (Full C + part of D) = 3.565
NPV (Full D + part of C) = 4.47
Project D should be selected in its entirety along with a part of C.
Project selection is a bit simpler in case of indivisible projects. Consider the following example.
Example 10: A firm is considering the following proposals. It has a limited fund of Rs. 75 crores.
If the projects under consideration are indivisible, what should be the decision of the firm?
Table 10.44
Projects Initial outlay (Rs. in crores) NPV (Rs. in crores)
A 10 7
B 24 17
C 35 25
D 16 20
E 15 18
F 8 15
Sol: We arrange the projects in descending order of NPV
Table 10.45
Projects Initial outlay (Rs. crores) Cumulative outlay (Rs. crores) NPV (Rs. crores)
C 35 35 25
D 16 51 20
E 15 66 18
B 24 90 17
F 8 98 15
A 10 108 7
With the given financial restraint, NPV will be maximized by undertaking projects C, D, E and F. Rs.
one crores will be left unutilized which can be used elsewhere.
369
If the firm adopts IRR method, then the optimal investment policy suggests the acceptance of all
projects till the IRR is equal to the marginal cost of capital. Graphically this situation may be
represented as follows
IRR/ cost of capital
IRR k
Acceptance limit
Budget
Fig. 10.8
The firms cannot raise unlimited capital at a constant rate. After a certain level of borrowing, the cost
of capital (k) will rise. In such situations, some marginal projects with IRR close to the increased cost
of capital may no longer be acceptable. We arrange the projects in descending order of IRR and till
IRR is equal to the cost of capital, the projects are accepted.
Consider the following example.
Example 11: For the following investment proposals, what should be the decision of a firm if the
minimum required rate of return is 10%? The projects are mutually independent.
Table 10.46
Project Investment (Rs.) Life of the project (years) Net cash flow (per year) (Rs.)
A -12000 5 4281
B -10000 5 4184
C -17000 10 5802
370
Sol: We form all possible combinations in which projects can be selected:
Table 10.47
Combination Project Total investment (Rs.) Net cash flow (per year) (Rs.)
1 A 12000 4281 (1-5 years)
2 B 10000 4184 (1-5 years)
3 C 17000 5802 (1-10 years)
4 AB 22000 8465 (1-5 years)
5 AC 29000 10083 (1-5 years)
5802 (6-10 years)
6 BC 27000 9986 (1-5 years)
5802 (6-10 years)
7 ABC 39000 14267 (1-5 years)
5802 (6-10 years)
In order of the size of the investments, the combinations are rearranged as follows:
Table 10.48
Combination Project Total investment (Rs.) Net cash flow (per year) (Rs.)
2 B 10000 4184 (1-5 years)
1 A 12000 4281 (1-5 years)
3 C 17000 5802 (1-10 years)
4 AB 22000 8465 (1-5 years)
6 BC 27000 9986 (1-5 years)
5802 (6-10 years)
5 AC 29000 10083 (1-5 years)
5802 (6-10 years)
7 ABC 39000 14267 (1-5 years)
5802 (6-10 years)
371
Now, we calculate the incremental rate of return.
Table 10.49
Project Incremental investment (Rs.)
Incremental flow (Rs.) IRR (%) NPV (Rs.) (k =15%)
None
B 10,000 4184 31 3500
A-B 2,000 97 < 0 < 0
C-A 5,000 1521 16 85
2663 (1-5 years) AB-C 5,000
-5802 (6-10 years)
< 0 < 0
1521(1-5 years) BC-AB 5,000
5802 (6-10 years)
42 8494
97(1-5 years) AC-BC 2,000
0 (6-10 years)
< 0 < 0
4281(1-5 years) ABC-AC 10,000
0 (6-10 years)
31 3608
If the budget allows then the combination of project B and C is the best.
Example 12: The capital expenditure budget of a company is Rs. 25 lacs. The management has
found the following (independent) projects to be feasible.
Table 10.50
Project Life of the
project (years)
Initial outlay (Rs.)
Present value of the cash flows occurring from
the project (Rs.)
A
B
C
D
5
5
5
5
10,00,000
7,50,000
8,75,000
7,50,000
12,50,000
12,50,000
14,25,000
15,00,000
The cost of capital to the company is 19%. Any unutilized amount can be invested to earn an interest
of 6%, which is risk-free. If the projects are indivisible, what should be the decision of the firm?
372
Sol: We calculate the NPV of the feasible projects.
Table 10.51
Project Initial outlay (Rs.) Present value of the cash flows occurring from
the project (Rs.)
NPV (Rs.)
A
B
C
D
10,00,000
7,50,000
8,75,000
7,50,000
12,50,000
12,50,000
14,25,000
15,00,000
2,50,000
5,00,000
5,50,000
7,50,000
At the most three projects can be chosen under given budgetary constraint. Combinations ABC and
ACD are not feasible. We analyze remaining two projects.
Table 10.52
Combination Total outlay (Rs.) Unutilized sum (Rs.) NPV (Rs.)
ABD 25,00,000 0 15,00,000
BCD 23,75,000 1,25,000 19,00,912.5*
* NPV of this combination is equal to the NPV of the projects + NPV of the unutilized sum which can
be calculated as follows.
Future value of . 1, 25,000 after 5 years @ 6% interest
1,25,000 6 5 = . 1, 25,000100
= . 1,62,500
Present value of . 1,62,500 @
Rs
Rs
Rs
Rs
× ×+
15% = . 1,62,500 0.621
= . 1,00,912.5
of the combination = ( ) ( ) ( ) of the unutilized sum
. 18,0
k Rs
Rs
NPV NPV B NPV C NPV D PV
Rs
= ×
⇒ + + +
= 0,000 . 1,00,912.5
. 19,00,912.5
Rs
Rs
+
=
The firm should select combination BCD.
373
Linear programming model for capital rationing Linear programming model (particularly integer
programming models) can be efficiently used for allocating funds to different feasible projects in case
of budgetary constraints. In this case the objective function is to maximize the net present value of
different combinations of projects subject to constraints of limited budgets, mutually exclusive
alternatives and project divisibility. A general formulation of the problem can be shown as
( )
( )
1 0
1
1
(1 )
subject to
; (Budgetary constraint)
m; 0 1 (Project divisibility constraint)
m nt
tj jj t
m
tj j tj
m
j jj
Max Z CF k X
IO X B
X X
−
= =
=
=
= +
≤
≤ ≤ ≤
∑∑
∑
∑
where
Cash flows of the project at the period;
Number of projects under consideration;
life of the project;
Cost of capital;
th thtj
th
CF j t
m
n j
k
=
=
=
=
Decision variable corresponding to the divisibilty of the project;
Initial outlay the project at time ; and
Total budget resources at time .
thj
thtj
t
X j
IO j t
B t
=
=
=
374
Problems
1. What is capital budgeting? Discuss its importance to the management of a firm.
2. On the basis of accounting rate of return, which project does you find more suitable:
Table 10.53
Project X
Project Y
Year
Book value (Rs.)
Depreciat
ion (Rs.)
Profit after tax
(Rs.)
Cash flow (Rs.)
Book value (Rs.)
Deprecia
tion (Rs.)
Profit after tax
(Rs.)
Cash flow (Rs.)
0 1,00,000 0 0 0 (1,00,000) 0 0 (1,00,000)
1 75,000 25,000 40,000 65,000 70,000 25,000 10,000 35,000
2 50,000 25,000 30,000 55,000 50,000 25,000 20,000 45,000
3 25,000 25,000 20,000 50,000 20,000 25,000 30,000 55,000
4 0 25,000 10,000 35,000 0 25,000 40,000 65,000
3. A company is considering replacement of a manual operation by a mechanized one. The cost
of this change is Rs. 50,000. The new system has an expected useful life of 5 years and no
salvage value. The estimated cash flows before depreciation and tax (CFBT) from the
proposal are as follows:
Table 10.54 Year CFBT (Rs.)
1 10,000
2 11,260
3 12,500
4 19,000
5 23,255
Assume the tax rate to be 35% and straight-line depreciation method (depreciation equally
spread over the useful life of the system) and no salvage value, calculate
(i) Payback period;
375
(ii) ARR;
(iii) NPV (k=10%)
(iv) IRR; and
(v) PI (k=10%).
4. In the above exercise, if the cash flows after tax and depreciation are reinvested at rates 6% for
the first two years, 8% for the third year and 10% for the remaining years, calculate the
present value of the system.
5. If an equipment costs Rs. 5,00,000 and lasts for eight years, what should be the minimum
annual cash inflow before it is worth to purchase the equipment. Assume the cost of capital to
be 10%.
6. How much can be paid for a machine which brings an annual cash inflow of Rs. 25,000 for 12
years at a discount rate of 15%?
7. A company is considering the purchasing and installation of a new welding machine and is
evaluating the following proposals:
(a) The company can buy a second hand machine for Rs. 1,00,000 which has an
expected life of 5 years and salvage value of Rs. 25,000. After its useful life the
machine can be replaced by another second hand machine having the expected cost
Rs. 1,25,000, working life of 5 years and salvage value Rs. 40,000.
(b) A new machine can be bought for Rs. 3,00,000 which would have an expected
working life of 10 years and salvage value Rs. 50,000.
Both the proposals would render the same service. If the cost of capital is 10%, which proposal
should be accepted?
8. The cash flow stream of a project is given below:
376
Table 10.55 Year 0 1 2 3 4
Cash flow (Rs. Lacs) -10 5 5 3.08 1.2
Calculate the internal rate of return for this stream of cash flows. Define the unrecovered
investment balance at the end of each year.
9. The management of a firm is considering the following two projects:
Table 10.56
Particulars Project A
Project B
Initial outlay (Rs.) 1,20,000 1,80,000
Expected life (years) 5 5
Cash flows (Rs.) 40,000 per annum 58,000 per annum
Cost of capital (%) 10 10
Which proposal should be acceptable under:
(a) NPV criterion;
(b) IRR criterion? Comment upon the dilemma you observe.
10. An electronic goods firm needs a special chip for which it has two options:
(a) The chip can be manufactured by the firm, for which the firm has to buy a machine at
a cost of Rs. 4,00,000 and the expected life of the machine is 5 years after which it
will have no salvage value. The manufacturing costs will be Rs. 6 lacs, Rs. 7 lacs,
Rs. 8 lacs, Rs. 10 lacs, and Rs. 12 lacs respectively.
(b) The chip can be purchased from outside at costs Rs. 9 lacs, Rs. 10 lacs, Rs. 11 lacs,
Rs. 14 lacs, and Rs. 17 lacs respectively.
377
An additional aspect of the proposals is that the machine of the first proposal would occupy floor
space at no cost, which in case of the second proposal can generate net cash inflows of Rs.
2,50,000 per annum. If the cost of the capital is 10%, which proposal should be accepted?
11. Consider the following projects
Table 10.57 Project Life of the project (years) Initial investment (Rs.) Annual cash flow (Rs.)
A
B
C
D
E
2
5
3
9
10
3,00,000
2,00,000
2,00,000
1,00,000
3,00,000
1,87,600
66,000
1,00,000
20,000
66,000
The capital expenditure budget of the firm is Rs. 5,00,000 and the cost of the capital is 12%.
Determine the optimal investment package. What would be your decision if the projects were
divisible? Make decisions if the budget is Rs. 4,00,000.
12. A firm is considering setting up a new production unit. It has two options before it
(a) To install six small machines at a unit cost of Rs. 25,000. The useful life of the
machines will be 6 years after which they would be able to fetch Rs. 10,000 (total).
(b) To install one large machine at a cost of Rs. 2,00,000. The useful life of this machine
will be 10 years after which it would be able to fetch Rs. 15,000.
The operating costs associated with the two alternatives are given below
Table 10.58
Years Alternative (i) (Rs.) Alternative (ii) (Rs.)
1 10,000 9,000
2 10,000 9,000
3 10,000 9,000
4 10,000 11,000
378
5 15,000 11,000
6 15,000 11,000
7 13,000
8 13,000
9 13,000
10 13,000
If the opportunity cost of funds is 10%, which alternative should be chosen by the firm if the
firm uses
(ii) NPV method;
(iii) IRR method for decision-making.
13 A company is planning to replace the machine, which is operational at present. It has option
of two machines either of which can be used for replacement purpose. The following
information is available about the cash flows associated with the machines
Table 10.59
Particulars Current machine Alternative I Alternative II
Book value
Resale value (now)
Purchase price
Annual fixed cost (including
depreciation)
Variable cost per unit
Units produced per hour
Life (years)
Salvage value
Rs. 1,00,000
Rs. 1,10,000
Rs. 90,000
Rs. 3
10
5
Rs. 10,000
Rs. 2,00,000
Rs. 1,10,000
Rs. 1.50
10
5
Rs. 15,000
Rs. 2,25,000
Rs. 1,30,000
Rs. 2.50
15
5
Rs. 18,000
Further information about the product is given below
379
Table 10.60 Particulars Value
Selling price
Cost of material
Annual operating hours
20
Rs.8
3,000
Existing machine has been in use for 2 years now. If the depreciation is by straight-line
method, tax rate is 40%, and the cost of capital is 10%, which alternative should be chosen by
the firm?
14 Consider the following information
Table 10.61
Alternatives Particulars
A B C D
Initial outlay (Rs. lakh)
PV of the future cash flows (Rs. lakh)
Life (years)
12
15
5
9
15
5
10.5
18
5
9]
17
5
The company wishes to invest Rs. 25,00,000. If the risk free rate of interest is 5% and the cost
of capital to the company is 10%, which investment(s) should be opted for if
(i) The projects are divisible?
(ii) The projects are indivisible?
15 Consider the following projects
Table 10.62 Project cash flows (Rs.’000) Years
A B C D E
1
2
3
4
(105)
(50)
75
30
(60)
(90)
105
120
(75)
(90)
105
120
(90)
30
60
75
-
(135)
100
110
380
All the projects are divisible. A project cannot be undertaken more than once. The amount
available for investment in the first year is Rs. 1,80,000 and there is no limit on the investment
second year onwards. If the cost of capital is 12%, what is the optimum investment package?
16 Consider the following projects
Table 10.63 Alternatives Particulars
A B C D E
Initial outlay (Rs. lakh)
Year end cash flows (Rs. lakh)
Life (years)
3
1.90
3
2
0.65
5
2
1
3
1.5
0.25
8
3
0.70
10
If the company has to invest Rs. 6,00,000, what should be the optimal choice according to
(i) NPV method;
(ii) PI method?
What is your final choice? It is given that the projects are divisible.
17 Consider the following information
Table 10.64 Alternatives Particulars
A B C D E F
Initial outlay (Rs. lakh)
PI
6
1.22
3
0.95
7
1.20
9
1.18
4
1.19
8
1.08
All the projects are indivisible and the investment capital is restricted to Rs. 15,00,000, which
projects should be undertaken?
18 Find the missing figures in the following table
381
Table 10.65 Particulars Value
Annual cost saving (Rs.)
Useful life (years)
IRR (%)
NPV (Rs.)
PI
Cost of capital (%)
Cost of the project
Pay back period (years)
Salvage value
40,000
4
14
7309
?
?
?
?
Nil
19 Consider the following information
Table 10.66
Particulars Project A Project B
Initial outlay (Rs.) 40,000 30,000
Life of the project (years) 5 5
Cash flows (Rs.)
Years
1
2
3
4
5
11,000
12,000
13,000
14,000
15,000
11,000
11,000
11,000
11,000
11,000
NPV (Rs.) (@6%) 14270 12120
NPV (Rs.) (@12%) 6050 6050
PV (Rs.) (@6%) 54270 42120
For the cost of capital (i) 10%; (ii) 12%; and (iii) 15%, which project should be selected?
382
20 Consider the following projects
Table 10.67 Project Initial outlay (Rs. lakh)
A
B
C
2
3
3
The projects A and B are not independent but are independent of C. If project A in
undertaken but project B is not undertaken, the NPV of A is Rs. 2,50,000. If project B in
undertaken but project A is not undertaken, the NPV of B is Rs. 2,90,000. If both A and B are
undertaken, their total NPV is Rs. 4,00,000. NPV of the project C is Rs. 1,80,000. The firm
has to invest Rs. 5,00,000. Which projects should be undertaken?
383
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