CAPITAL BUDGETING

This unit introduces you to the concept of Capital Budgeting. A capital expenditure is an expenditure on fixed assets and other long-term infrastructure necessary for the implementation of projects. The assets bought for the project are expected to generate cash flows. The appraisal of capital projects is done in two steps. Firstly, we must determine the cash flows that are expected to be generated by the project. Secondly, we must estimate the cost of the funds (cost ofcapital ) that have been used in the project. Finally, we subject the expected cash flows to certain appraisal techniques.

Capital budgeting involves long-term decision making on the use of funds. This implies the evaluation of investment opportunities, the essence of which is the detailed consideration of expected future cash flows. A cash flow may be defined as the receipt or expenditure of cash during an interval of time. For the sake of simplicity, it is generally assumed that cash flows occur at the end of each interval of time (usually at the end of each year).

It is a budget that deals with Capital Expenditure, for example the purchase of plant and equipment, construction of a building .It is expenditure that is not easily reversible due to the values committed. There are different types of capital budgeting expenditures,e.g. Replacement projects and Expansion projects.

So a detailed risk analysis needs to be done before the investment introduction of new projects and regulatory projects. The investment in the project is prescribed by the regulatory board on the safety issues.

Importance of capital budgeting.

You recall that the goal of financial management is to maximize the wealth of shareholders by acquiring funds at the least possible cost and utilizing them to obtain the highest possible return for the shareholders.

Capital budgeting techniques are used to make an appraisal of the company’s investment projects, whereby assets are acquired in order to carry out approved investment projects. It is expected that a project will generate cash flows . In project appraisal, it is the project’s cash flows that are subjected to a series of tests in order to find out whether the wealth of

the shareholders is being maximized by embarking on a particular project. In making this appraisal, we utilize the discounted cash flow [DCF] techniques, which are based on thetime value of money concept, as well as other methods which are not based on the time value of money concept.

Incremental cash flows

In capital budgeting we should be careful not to include cash flows that are not relevant to the decision under consideration. The relevant cash flows for capital budgeting purposes are incremental cash flows. Cash flows for a project are defined as the difference between the cash flows of the firm without the project and the cash flows with the project :

Project CFt = CFt for the firm - CFt for the firm with project without project

Cash flow versus Accounting income.

Accounting income may differ from the cash flows that we consider under financial management. Income statements mix apples and oranges. For example, accountants deduct costs, which are cash outflows, from revenues, which may or may not be entirely cash inflows (some sales may be on credit). At the same time, they do not deduct capital outlays, which are cash outflows, but they also deduct depreciation, which is not a cash outflow. In capital budgeting, it is critical that we base our decisions strictly on cash flows, the actual dollars that flow into and out of the company during each time period.

Study the following example.

Example A company is considering investing in a project for which the following information has been generated:

$Initial capital outlay (600 000 )Profit/loss for the year: Year 1 100 000

Year 2 150 000Year 3 250 000Year 4 300 000

The capital outlay was on plant and machinery which is expected to have an economic life of four years with no scarp value.

In capital budgeting, we are concerned with "cash flows", not "profits or losses". To turn these into cash flows, we add back the depreciation, which is not a cash outlay. The profits are therefore adjusted as follows :

Annual depreciation = 600 000 / 4 = 150 000Annual cash flows = profit + depreciationYear 1 100 000 + 150 000 =250 000Year 2 150 000 + 150 000 =300 000Year 3 250 000 + 150 000 =350 000Year 4 300 000 + 150 000 =450 000

Sunk Costs

Sunk costs are not incremental costs, so they should not be included in capital budgeting. A sunk cost is an outlay that has already occurred (or been committed). Since the outlay has already occurred, it is not affected by the decision to accept or reject a project.

Opportunity Costs

An opportunity cost is the benefit lost or alternative foregone in making a decision. For example, the use of a factory building to implement a project may require that the other alternative uses to which the building could be put have to be foregone, for example rentals to other users of the building. Opportunity costs should be charged to the project as an additional cost.

Externalities

The effects of the project on other projects are called externalities. Externalities may be positive or negative. For example, a project may result in the production of a new product for the firm. If this results in the reduction of the demand for the firm’s other products as some of the existing customers shift to the new product ( this is known as “cannibalization”) , these reduced sales should be deducted from the new product sales [a negative externality]. On the other hand, the new product may create sales for other related existing products, and these should be attributed to the new product [ a positive externality].

Types of Project Cashflows

Project cash flows are dividedinto three categories : the initial investment; the annual cash flows; and the terminal cashflow.

The Initial investment [I0]

The initial investment is the net cash outlay on buying the capital, that is the plant andequipment, buildings, and other infrastructure for the project. The amount of the initialpayment and the way it is calculated is determined by whether the project is a newinvestment project or a replacement project.

A new investment is when a totally new project is being analysed , or the implementation ofthe project does not have any effect on the present cash flows of the firm. If a newinvestment is being considered the initial investment can be made up of the following:

1. Thecost of the assets and installation, and 2. Change in net working capital.

The cost of the assetsincluding installation costs is an outflow (-) including any other opportunity costs.

Change in net working capital caused by the implementation of the project. Normallyadditional inventories are required to support a new project, and the new sales will alsogenerate additional accounts receivable. At the same time, accounts payable and accrualsmay also spontaneously increase. If there is a net increase, this is anoutflow (-). At the end of the project’s life, the firm’s total working capital may revert toprior levels, the net increase in net working capital is therefore recovered and becomes aninflow (+).

In a replacement project, some assets are replaced, and this may result in the firm nolonger deriving any cash flows from the replaced assets but from the new assets. Thefollowing factors make up the initial investment in a replacement project:

The cost of the new project + installation costs + changes in net working capital. This isa cash outflow (-).

The disposal value of the old assets. The implementation of the new project results inthe old assets being disposed of. The funds that are received at the disposal of the oldassets are an inflow (+).

Tax effects. Selling a fixed asset can result in tax effects. The tax savings or taxpayable as a result of the disposal of the old assets must be incorporated in the initialinvestment calculation.

Annual Cash Flows.The annual cash flows are the result of revenues less expenses. Remember we always use net incremental, relevant, after-tax cash flows. We therefore need to incorporate tax effects in these cash flows. Let us do this now.

To convert before-tax cash flows into after-tax cash flows we use the following procedure:1. After-tax cash flow = Before-tax CF - Tax payable.2. Tax payable = Taxable × tax rate.3. Taxable CFs = Before-tax CF - Tax allowance.

In Zimbabwe, tax allowances e.g W&T and SIA are claimed in place of depreciation charges, which are notrecognized for tax purposes.

The terminal cash flow

The terminal cash flow is the net after tax amount received by the firm when a project is terminated. For a new investment the estimated terminal cash flow might include the following : the estimated salvage value of the new assets, the tax effects due to the disposal of the assets and the recovery of the net working capital.

The estimated salvage value of the new assets is the amount that is expected to be received when the assets are sold at the termination of the project. This is a cash inflow (+).

The tax effects due to the disposal of the assets. This depends on whether there is a net taxable recoupment (-) or scraping allowance (+)

Recovery of the net working capital. If there is a change in net working capital when the project is implemented, we expect an opposite change to occur at termination of the project. The change is usually an inflow (+).

Thus, the terminal cash flow will be equal to:Proceeds from sale of assets XXXXTax on recoupment (XXXX)Recovery of working capital (XXXX)Total (XXXX)

Practice Question

Thodes Bus Company is considering the replacement of one of its buses with a new one.It is estimated that the new bus will bring in extra revenues amounting to $12 000 000 per year as well as savings in maintenance costs amounting to $1 300 000 per year. The new bus is expected to cost $19 000 000 plus shipping costs of $1 200 000. The bus is expected to operate for five years and to have a salvage value of $5 000 000. There will be an increase of $100 000 per year in working capital resulting from the use of the new bus.

The old bus was bought four years ago and now has a market value of $300 000 and a zero book value.

The company elects to claim SIA on the bus using the current rates and has a tax rate of 30% per year.

Calculate: 1. The initial investment. 2. The annual cash flows. 3. The terminal cash flow of the project.

Classification of Capital Projects

The effects of a capital budgeting decision continue over many years and therefore, such a decision may not be easy to reverse. Capital budgeting decisions also define the strategic direction of the firm because a decision to move into new products, services or markets, for example, must be preceded by a capital budgeting expenditure.

A company usually considers more than one project at a time. Based on this notion, thereare basically two types of the projects: mutually exclusive or independent.

Mutually exclusive projects cannot be carried out at the same time. If project A and project B are mutually exclusive, for example, accepting one of them means the rejection of the other. On the other hand,

Independent projects, the acceptance of one may not necessarily mean the rejection of other projects that may be under consideration.

Another issue related to the classification of projects is the types of decisions. In investment appraisal there are two types of decision that we face.

1. We either have to accept or reject decisions, for mutually exclusive projects.

2. We have to rank independent projects. Projects which are more attractive according to the appraisal, are ranked higher than those which are less attractive.

Steps in Investment Appraisal

We have seen that the first step in the appraisal of a capital expenditure project is the determination of the project's net, after-tax cash flows. The next step after this, is to determine the cost of the funds used in the project. This is especially important if we are to use discounted cash flow techniques in our appraisal.

In estimating the cost of funds, we need to consider the sources of those funds, since the cost of funds is the return that is required by the suppliers of both debt and equity capital that has been used to finance the project, taking into account the risk of the project. In other words, it is the weighted average cost of debt and equity.

The following example clarifies this.

Example A firm requires $400m to finance its capital project. $300 of this will come from the issue of new shares on which the investors require a return of 20%. The balance will be borrowed at an interest rate of 18%.

Ignoring taxation, the weighted average cost of capital for this project would be:

[(300 / 400) x 0.20] + [(100 /400) x 0.18] = 0.195 = 19.5%.

Project appraisal techniques

After we have got our cash flows as well as the cost of the funds used in the project, we are now in a position to make an appraisal of the project.

There are various elements of the accounting system that are used including;

Payback period. Discounted payback period. Net Present Value [NPV] method Internal Rate of Return [IRR] method. Modified Internal Rate of Return [MIRR] method. Profitability Index [PI].

We discuss each of these, including the problems associated with them, and their merits and demerits in the following subsections.

The Payback method

The payback period tells us the number of years required to recover the initial cash outlay from the project’s expected net cash flows (The payback period, defined as the expected number of years required to recover the original investment). It is the ratio of the initial cash outlay to the annual net cash flows.

Example: payback period with equal annual cash flows . A firm is considering a project whose expected net, after-tax cash flows are as follows:Initial Investment = $ 18 000.00Annual cash flows = $ 5 600.00 per year for the next 5 years. What is the payback period?

SolutionThe payback period = 18 000 / 5 600 = 3.2 years.

Example: payback period with unequal annual cash flows .A project has the following expected annual net, after-tax cash flows :

Year Expected Net Cash flow Cumulative Cash Flow0 ($ 18 000) ($ 18 000)1 $ 4 000 ($ 14 000)2 $ 6 000 ($ 8 000)3 $ 6 000 ($ 2 000)4 $ 4 000 $ 2 0005 $ 4 000 $ 600

In this example, you can see that the payback period falls between 3 years and 4 years. Toget the actual payback period, we use the following formula:

Payback = Year before full recovery + [unrecovered cost at start of year / cash flowduring year]

Therefore the payback period = 3 + [ 2 000 / 4 000 ] = 3.5 years.

Decision criteria of Payback period

A company might have a standing policy that all projects must recover their full cost within a certain period of time. If the payback for a particular project falls within this stipulated period, then the project is acceptable.

If, for example, the company policy payback was 3 years, then the project would be accepted. In this sense therefore, the payback period may be said to be a rough measure of the risk associated with the project. The longer the payback, the greater the risk, therefore the less acceptable a project is.

In the case of mutually exclusive projects, those with longer paybacks are eliminated in favour of those with shorter paybacks. As for independent projects, those with shorter paybacks would be ranked higher than those with longer paybacks.

Criticism of payback period

The regular payback does not take into account the time value of money, assuming that cash flows received in the future are just as good as cash flows received today. In this sense it does not take into account the cost of capital. A project may be financed by both debt and equity and we need to factor in the cost of obtaining these funds, using an appropriate discount rate.

It suffers from “Fish-bait criteria” i.e. (the size of the fish matters, not just catching something). It focuses only on the covering the initial investment than profitability.

it ignores cash flows beyond the payback period, as is evident from the above examples.

Discounted Payback Method

It is a refinement of the payback method and seeks to overcome the problems of the time value of money before calculated.

To overcome the problems of the regular payback, the expected cash flows are discounted at the project’s cost of capital. The discounted payback period is therefore the number of years required to recover the investment from discounted cash flows .

Example

A project has the following net after tax cash flows;

Year 0 1 2 3 4Cash flows -1000 500 400 300 100

Calculate the discounted payback period if the cost of capital is 10%.

Solution

Year 0 1 2 3 4Cash flows -1000 500 400 300 100Discounted Cash flows

-1000 5001.1=454.55

400(1.1)2

=330.5

300(1.1)3

=225.39

100(1.1)4

=68.30Cumulative cash flow

-1000 -545.45 -214.89 10.5 78.8

Discounted Payback Period=2 years + 214.89/225.39

= 2.95 years

The discounted payback shows the break-even year after covering the cost of debt and equity.

General critique of payback methods

The payback provides information on how long funds will be tied up in a project.Therefore, the shorter the payback, the greater the project’s liquidity. Since cash flows expected in the distant future are generally riskier than near-term cash flows, the payback can be used as a crude

measure of risk. The company that is cash poor may find the method useful in gauging the early recovery of funds invested.

The proper measure of risk, however, is the standard deviation of expected cash flows, which takes into account the dispersion of the possible cash flows. The payback measures only the magnitude and timing of the expected cash flows relative to the original investment. It cannot therefore, be considered an adequate indicator of risk. It is more appropriately treated as a constraint to be satisfied rather than a measure of profitability.

The payback discriminates against longer term projects, which may turn out to be more profitable for the shareholders, by ignoring the cash flows after the payback period. This is demonstrated for you in the following example.

Example

The cash flows for projects X and Y are as follows :

Project X Project YYear 0 (10 000) (10 000)Year 1 1 000 5 000Year 2 2 000 3 000Year 3 3 000 2 000Year 4 4 000 1 000Year 5 8 000 500

The payback period for project X is 4 years whereas that for Y is 3 years. If these projects were mutually exclusive, Y would be accepted and X rejected. Project Y is, however, a shorter-term project than X in that the cash flows of X show a rising trend and those for Y are decline drastically after the payback period.

The Net Present Value (NPV)

It is the difference between the present value of the initial cash outlay and the present value of the cash inflows of project discounted at the cost of the capital.

The steps to be followed in evaluating a project using the NPV method are as follows :

Find the present value [PV] of each period’s cash flow, including both inflows and outflows, discounted at the project’s cost of capital,

Sum the PVs to find the NPV. If the NPV is positive, accept the project and if the NPV is negative,

reject the project. For two or more projects, accept the project with the highest NPV, if

the projects are mutually exclusive. If the projects are independent, accept the project with the highest NPV first and rank them accordingly.

The NPV is found by the following formula :

NPV=∑t=1

n CF((1+t)t)

– I0

Where I0=Initial cash outlay

t=time of periods 1,2,3,4 ……n

n=number of periods

Example

Using the information from the example on discounted payback period calculate the Net Present Value of the project.

Solution

Given that ; NPV=∑t=1

n CF((1+t)t)

– I0

NPV=∑(( 500(1.1 )1

+ 400(1.1)2

+ 300(1.1)3

+ 100(1.1)4

)-1000)

=$1078.82-$1000

=$78.82

This investment has a total value or present value of future cash flows of $1078.82. Since the investment is acquired at a cost of $1000(the initial outlay). The investment company is giving up $1000 of its wealth in exchange of an investment worth $1078.82. This means the investors wealth has increased by a margin of $78.82.

It can be safely said that: the NPV is the amount by which the investors wealth increases or decreases as a result of an investment. The decision rule is:

1. Invest: if NPV >0,2. Do not invest: if NPV <0

It should be noted that positive NPV investments are wealth increasing whilst negative NPV investments are wealth decreasing.

Example: Calculating NPV with Constant Cash flowsA project is expected to generate net cash flows of $600.00 per year for the next three years. The initial investment in the project is $ 1000.00 and the cost of capital is 10%.

Solution

Since this is an annuity, the NPV will be found as follows :NPV = [ 600.00 ( PVIFA10%,3years ) ] - 1 000.00

= [ 600.00 ( 2.487 ) ] - 1 000.00.= $ 492.00.

Rationale for the NPV

An NPV of zero signifies that the project’s cash flows are exactly sufficient to repay the invested capital and to provide the required rate of return demanded by the providers of the capital used on the project [both equity and debt ].

If the NPV is positive, the project’s cash flows are generating more than the required rate of return [RRR]. Since the return to bond holders [the providers of debt capital] is fixed [ the interest on debt], the extra return accrues solely to the firm’s stock holders [the providers of equity capital,] who receive their return [dividend] only after the bond holder have been paid their fixed amount. It therefore follows that if a firm takes on a zero-NPV project, the position of the shareholders remains unchanged. The firm only becomes larger to the extent of the size of the project, but the wealth of the shareholders, that is the price of the company’s shares, remains constant.

If the firm takes on a positive-NPV project, the position of the shareholders is improved by the amount of the NPV. Thus, if the firm takes on the

project, the wealth of the shareholders is increased directly by the NPV amount, thus enhancing the share price of the firm.

Practice QuestionA project has an initial cash outlay of $800000. The life of the project is 4 years; residual value of the asset in 4 years is $90000. Expected revenue per year is $650000. $450000 of the capital required is borrowed at 15% after tax amount. The balance is raised through the issue of new shares at a required rate 18%. Evaluate if the project is worth investing in or not. (-$55622.09) NPV=Reject the project

Internal Rate of Return (IRR)

The IRR is the yield or rate of return generated by the project’s internal cash flows. It is an internally generated rate of return. It can also be defined as a discount rate that makes the present value of the future after tax cash flows of a project equal to the initial outlay. It yields an NPV of zero (NPV=0).

NPV=∑t=1

n CFt(1+ IRR )t

−I 0=0

∑t=1

n CFt(1+ IRR )t

=I

At the IRR,NPV=0

Therefore IRR is a rate of return generated by the internal cash flows of a project.

Decision Rules for IRR

It is such that: Accept the project: if IRR >Required Rate of Return

: Reject the project: if IRR < Required Rate of Return

The Required Rate of Return is equivalent to the Weighted Average Cost of Capital.

How do we find the IRR?

We find the IRR using the Iterative Methods- specifically Estimation by Interpolation

Example

An investment with an initial outlay of $12000 returns a constant after tax cash flows of $24000 per annum for 10 years. What is the IRR for the project?

Solution

Given that ; ∑t=1

n CFt(1+ IRR )t

=I

I= CF 1(1+ IRR)1

+ CF 2(1+ IRR)2

+ CF 3(1+ IRR)3

+............ CF 10(1+ IRR)10

120000= 24000(1+ IRR)1

+ 24000(1+ IRR)2

+ 24000(1+ IRR)3

+............ 24000(1+ IRR)10

Let’s try to make IRR the subject of the formulae and see if we can win.[we can’t ]….lol

Also notice that the pattern of the payments is in the form of an annuity.

Let’s employ our knowledge of annuities then.

120000= R×PVIFAn,%

=24000× (1+ IRR )10−1IRR (1+ IRR)10

Even so , we cannot make IRR subject by mere algebraic manipulation. So we use trial and error then we interpolate.

Formulas for estimation of the value of IRR

If Z is to the left of both X and Y we use:

IRR =a-( z−xx− y )(b-a)

If Z is between X and Y we use:

IRR =a+( x−zx− y )(b-a)

If Z is to the right of both X and Y we use:

IRR =b+( x−zx− y )(b-a)

Where: Z=initial outlay or investment initial outlay (PV)

a=is the lower discount rate

b=is the higher discount rate

x=is the Present Value associated with the lower discount rate

y= is the Present Value associated with the higher discount rate

The relationship between the Present Value and a Discount Rate;

To reduce the Present Value you increase the discount rate (an inverse relationship) and vice versa.

Finding IRR by trial and error then Interpolation

You can just try any percentage and calculate the PVs so that you compare with the I0.

For the problem above;

Try with 10%

PV (10%) =10%=( (1.10 )10−10.1(1.10)10 )*24000

=147469.61

This is not the same with Z of $120 000 ,it is greater than Z so let’s increase the discount rate to get a r PV that is slightly lower or the same as Z

Try with 18%

PV (18%) =18%=( (1.18 )10−10.18(1.18)10 )*24000

=107858.07

This is also not equal to Z of $120000. We have to look for another discount rate which will equalize the PV to Z,but this may take us forever so we resort to interpolation to find an estimate of this IRR.

The next step is to compare the relationship of X, Y and Z ,so that we knw which formula to use for our interpolation.

If we compare the 2values (a= 10%) ⇛X=147469.61 and (b=18 %) ⇛( Y= 107858.07) with Z= 120 000 ,we can see that our Z figure lies between the two figures X and Y so we construct a table.

a% IRR b%X Z Y10% IRR 18%147469.61 120000 107858.07

So our Z is between X and Y so the formulae to use is the formulae #2

Graphical line representation

Y Z X

107 858.120 000 147 469.61

So IRR =a+ ( x−zx− y )(b-a)

= 0.1+ ( 147469.61−120000147469.61−107858.07 )(0.18-0.10)

= 0.1+ ( 27469.6139611.54 )(0.08)

=0.1+0.693474931*0.08

=15.5%

Example

An investment has an initial capital outflow of $600000 and is going to generate the following successive yearly cash flows of $50000, $70000,

$150000, $200000, $250000 and $300000. If theinvestor RRR is 14%. Determine whether the project is worthwhile using the IRR method.

Solution

Use the same methods above.

Example

A company is trying to decide whether to buy a machine for $ 80 000.00. The machine will save the company costs of $ 20 000 per year for five years and have a resale value of $ 10 000 at the end of that period. What is the IRR? (Ignoring tax effects ).

Solution

The IRR is found by trial and error [interpolation] if one is not using a financial calculator.We apply different discount rates to the cash flows until we get one which produces an NPV of zero.

Try 9%:

Year Cash flow x PVIF = present value0 (80000) x 1.000 = (80000)1 –5 20 000 x 3.890= 77 8005 10 000 x 0.650 = 6 500

NPV = 4 300

Since this is a positive NPV, we try a higher discount rate, say 12% , to get a negative NPV:

NPV = [20 000 (3605) + 10 000 (0.567) ] - 80 000 = - 2 230.

Since the NPV is negative, the required discount rate lies somewhere between 12 % and 9%, as shown in the following diagram, known as the NPV profile of the project :

NPV Profile for the project

NPV ($)IRR = 10.98%

$ 4 300

0 9% 10.98% 12% Discount Rate (%)

-$ 2 230

The IRR is the discount rate that will result in a zero NPV, therefore lies somewherebetween 9% and 12%. It is found by the following formula :

IRR = A + [(a / a + b)][(B – A)]

Where A = the lower discount rate which produces a positive NPV,a = the NPV resulting from a discount rate of A%,B = the higher discount rate which produces a negative NPV,b = the NPV resulting from a discount rate of B%.

NB. We treat the –NPV as a positive in the formulae.

Thus, in this example, the IRR is equal to 10.98%, which is approximately 11%.

The decision rule for the IRR is that we should accept the project if the IRR is greater thanthe RRR, that is the cost of capital. If the IRR is less than the RRR, then the project shouldbe rejected. If we are comparing two mutually exclusive projects, we would take the onewith the higher IRR.

Rationale for the IRR

The IRR is the return that is expected to be generated by the project's net cash flows, assuming that all the cash flows are reinvested into the project. If the IRR exceeds the cost of the funds used to finance the project [the RRR], a surplus remains after paying for the funds, and this surplus belongs to the firm’s shareholders. Therefore, taking on a project whose RRR exceeds the cost of capital increases the shareholders’ wealth.

The decision criterion is therefore that if the project's internal rate of return exceeds the required rate of return, the project should be accepted.

The decision criteria for independent projects is to take those projects with a higher IRR first. If the projects are mutually exclusive, we take on those projects with a higher IRR and reject those with a lower IRR.

Conflict between the NPV and the IRR

There may be a conflict between the decision resulting from an NPV analysis and that resulting from an IRR analysis. This conflict arises when the projects are mutuallyexclusive rather than independent.

For independent projects, the NPV and the IRR always lead to the same accept / rejectdecision, that is if NPV says accept, IRR also says accept. The criterion for acceptance isthat the project’s cost of capital [RRR] is less than [to the left of] the IRR. Whenever theproject’s cost of capital is less than the IRR, its NPV is positive, therefore the project isacceptable using both methods. Both methods reject the project if the cost of capital isgreater than the IRR.

If IRR >Cost of Capital(RRR), then NPV >0

If two projects, A and B are mutually exclusive, we can choose either A or B, or we canreject both, but we cannot accept both projects.

Let us suppose we had two projects, A and B.

PROJECT A PROJECT BNPV $100 000 $150 000IRR 20% 18%

You can see that project A has a lower NPV than project B, but it has a higher IRR. Thisconflict between A and B is illustrated as below :

Conflict between projects

NPV($)Cross-over rate

IRRbIRRa

NPVb

NPVa

RRR Discount Rate (%)

NPV profile (A) NPV profile (B)

According to the NPV method, project B should be accepted and A rejected. But accordingto the IRR project A should be accepted and B rejected.As long as the cost of capital [RRR] is greater than the cross-over rate, the NPV of projectA is greater than the NPV of project B and the IRR for project A is greater than the IRRfor project B.

Therefore for IRR greater than the cross-over rate, the two methods lead tothe selection of the same project. However, if the cost of capital [RRR] is less than thecross-over rate, a conflict arises. The NPV ranks project B higher but the IRR ranks Ahigher.

Why conflicts arise

Conflicts arise for two reasons : When project size [scale] differences exist, that is when the initial

capital outlay on oneproject is greater than that on the other. When timing differences exist, that is the timing of the cash flows

from the twoprojects differs such that most cash flows from one project come in the early years andmost of the cash flows from the other project come in the later years.

You may be wondering ‘How can I deal with such a conflict ?” Let us study the next example for the answer.

Example :Resolving conflict between NPV and IRR

The following two projects aremutually exclusive:Project A Project B

Year 0 (10200) (35250)Year 1 6 000 18 000Year 2 5 000 15 000

Year 3 3 000 15 000

The company’s cost of capital is 16%.

At this rate of discount the NPV for A is $610.00and the NPV for B is $ 1 026. The IRR for A is 20% and the IRR for B is 18%. Therefore,using the NPV method, B is preferred to A but using the IRR, A is preferred to B.

In actual fact, B is better if we consider the differential [orincremental] cash flows thatwould occur from the adoption of B rather than A. If we discount these incremental cashflows at 16% we find that the present value of the incremental benefits from project Bexceed the present value of the incremental costs, that is, the NPV of the differential cashflow is positive.

Therefore, it is worth spending the extra capital on project B and also theIRR of the differential cash flows exceed the cost of capital [16%].

Year Project A Project B Difference PVIF (16%) PV of Differential CF0 (10 200) (35 250) (25 050) 1.000 (25 050)1 6 000 18 000 12 000 0.862 10 3442 5 000 15 000 10 000 0.743 7 4303 3 000 15 000 12 000 0.641 7 692

NPV = 416

The IRR of the differential cash flows is 18%.

Why the NPV is regarded as superior to the IRR

We have seen that the cost of capital is the weighted average between the cost of debt andthe cost of equity. When a project generates cash flows, they must be paid out to the debtholders and equity holders, who on average require a return which is equal to the cost ofcapital [the discount rate, or the RRR].

We have also seen that there are two sources of funds for the firm, debt and equity.However, in the case of equity, we have internally generated equity, that is retainedearnings, and new equity. Thus, the cash flows of the project can be paid out as dividendsafter we have paid interest on debt. Alternatively, the cash flows can be retained and usedas a substitute for outside sources of funds, after paying out the interest on debt capital.

Now, suppose that the cost of capital ( RRR, or WACC ) is 18%. This means that byretaining all the cash flows, we are saving the firm the cost of obtaining outside sources offunds at a cost of 18%. We can, therefore say that the value of the cash flows to the firm is18%. As you can see, this value is an opportunity cost. In other words, it is the requiredrate of return, which the shareholders would obtain on alternative investments of similarrisk if we had paid them their dividend instead of retaining the cash flows.

The NPV is regarded as superior to the IRR due to the assumptions they both make aboutthe treatment of the project’s cash flows. The IRR assumes that the cash flows arereinvested at the IRR itself. Obviously, the IRR will vary from project to project,depending on the cash flows of each particular project. As we have seen, a project is onlyacceptable if the IRR is greater than the RRR.

Now, if a firm can get funds from outside at the same cost as the RRR, which is also thediscount rate that is applied to projects, the appropriate reinvestment rate would be theopportunity cost of capital, that is the RRR. The NPV assumes that the cash flows from theproject are reinvested at the RRR, which is built into the NPV method. Thus, whenevaluating mutually exclusive projects, especially those with timing and scale differences,the NPV should be used rather than the IRR.

Problems with IRR

The use of IRR in appraising projects is fraught with problems. Firstly, the IRR ignores therelative size of the projects, as shown in the following example:

Project A Project BYear 0 (350 000) (35 000)Year 1 – 6 100 000 10 000

Project A is 10 times bigger than project B, therefore more profitable, but both have thesame IRR of 18%.

Secondly, the IRR is not effective when it comes to unconventional cash flows.

Let’s study thefollowingexampleinvolving two projects.

PROJECT A PROJECT BCash flows ($) Cash flows ($)

YEAR0 (100 000) (120000)1 (50000) 50 0002 60 000 80 0003 90 000 (50000)4 80 000 20 000

Project B is not a conventional project. As you can see, we have an outflow in year 0 whichis followed by two inflows before we get another outflow in year 3, which is followed byanother outflow in year 4. If the cash flows from the project are not conventional [out flowsfollowed by inflows, as in project A], there may be more than one IRR.

The equation for the IRR is a polynomial of n degrees, therefore it has n different roots orsolutions. All except one of the roots are imaginary numbers when the cash flows arenormal, therefore in the normal case only one value of IRR appears. For non-conventionalcash flows, there are multiple real roots, hence multiple IRRs. In the case of project B,there would be two internal rates of return.

Thus, the IRR is inferior to the NPV method.

The Modified Internal rate of return [MIRR]

In spite of the strong academic preference for the NPV over the IRR, surveys have shown that the IRR is by far the preferred method of investment appraisal. The IRR is easier to understand as it looks at percentages rather than absolute figures. In order to take into account the objections regarding the re-investment assumption, we can modify the IRR by assuming that the cash flows are re invested at the required rate of return before calculating the IRR.

The modified internal rate of return (MIRR) is given by the following formula:

MIRR = PV Costs = TV / (1 + MIRR)

Where: TV = the future value of the inflows reinvested at the cost of capital [known as the terminal value ]

MIRR=the discount rate that forces the present value of the terminal value to equal the present value of the costs is the.

The following example illustrates this.

Example: Calculating the MIRR

Suppose we have the following time line for a particular project with a cost of capital of 10%

Timeline: Modified Internal Rate of Retrun

0 1 2 3 4

(1000) 500 400 300 100 100.00330.00484.00665.501 579.50

PV[TV]= 1000[I0] @ MIRR = 12.1%

To calculate the MIRR you take the following steps.

Step 1: Find the terminal value [TV] . This is the total of the future values of all the cash flows reinvested, that is compounded, at the cost of capital [RRR], 10%. The TV =$ 1 579.50.

Step 2:Find the discount rate that will give a present value of $1 000 on a future value of$1 579.50, the IRR. This is found by trial and error in the normal way. This discount rate is12.1%, which is the MIRR for the project.

Since the MIRR is greater than the cost of capital, 10%, the project is acceptable. Theconventional IRR for this project would have been 14.5%.

Revising NPV Analysis for Inflation.

In the absence of inflation, the real rate is equal to the nominal [quoted] rate of interest.The cost of capital, that is the RRR, includes an inflation premium because investorsalways try to protect themselves against a decline in their purchasing power by includingan adjustment for inflation in the required rate of return. The cost of capital is a nominalrate in that it takes into account the expected rate of inflation. On the other hand, cash

flowsare usually real cash flows in that they do not include inflation effects.

As inflation increases so does the minimum return required by an investor. It is, thereforeimportant to adjust the cash flows before we discount them at the nominal rate, whichalready contains an inflation premium. Alternatively, we should convert the nominal rateinto a real rate before we discount the cash flows.

To convert a nominal interest rate to a real rate, we use the following formula :

(1 + nominal rate) = (1 + real rate) (1 + inflation rate)

Let us say for example, the required [quoted ] rate of return is 20% under the current andanticipated conditions and the rate of inflation is currently running at 10% per year, we cansolve for R [ the real rate ] as follows :

(1 + 0.20 ) = ( 1 + R ) ( 1 + 0.10 )

Therefore R = 0.091 or 9.1%.

This means that a return of 20% will protect the shareholders against the loss of purchasingpower and also provide a real return of 9.1%.

In dealing with inflation, we should always:

discount nominal cash flows using a nominal discount rate; and discount real cash flows using a real discount rate.

If the information is mixed, that is real cash flows with nominal rates, or nominal cashflows with real rates, we should :

change nominal cash flows to real cash flows, or change real cash flows to nominal cash flows, change nominal rates to real rates, or change real rates to nominal rates,

,so that nominal cash flows are discounted at nominal rates or real cash flows are discountedat real rates.

Example: Discounting real cash flows at a real rate of return

The following nominal cash flows must first be converted to real cash flows before beingdiscounted at the real rate of 9.1%.

Year Nominal CF Real CF0 ( 15 000 ) ( 15 000 )1 9 000 9 000 / ( 1.10 ) = 8 1822 8 000 8 000 / ( 1.10 )2 = 6 6123 7 000 7 000 / ( 1.10 )3 = 5 259

Discounted cash flows at the real rate of 9.1%:Year Discounted cash flow0 (15000)1 8 182 / ( 1.091 ) = 7 5002 6 612 / ( 1.091 )2= 5 5553 7 000 / ( 1.091 )3= 4 050

NPV = 2 105This is the NPV of the project after taking into account the effects of inflation.

The Accounting Rate of Return

The accounting rate of return (ARR) is based on the return on investment ratio (ROI). The ROI is the ratio between the operating income for the year (from the income statement) and the total assets employed (from the balance sheet). It is a measure of the effectiveness with which the company is utilizing its assets to generate profits. Since management is often evaluated on the basis of this ratio, it is also an appropriate measure of the likely performance of a project. When applied to projects, the ratio is known as the ARR. Using this ratio, a project can be evaluated on the basis of an appropriate ARR. A project with a good ARR will in turn contribute positively to the firm’s overall ROI.

ARR=(Average incremental net income )

Averageinvestment

Where:1. Average incremental net income is the expected annual average increase innet income if the project is accepted. This is equal to

Total net Income from the project/Project economic life

However, as we have discussed before, when we are analyzing project cash flows, we must first deduct depreciation and other non-cash flow items from the netincome.2. Average investment = Cost of the investment

Example A firm is considering a project with an expected economic life of fiveyears. The following data also relate to the project:Capital outlay: $ 200 000Residual value: nil (assuming straight line depreciation)Expected Annual Net Income :

Year Expected Net Income ($)1 60 0002 80 0003 76 0004 58 000Total 274 000

The project’s ARR is calculated as follows :1. Total net income = 274 000.2. Average net income = 274 000 / 5 years = 54 800 per year.3. Depreciation per year = 200 000 / 5 year = 40 000 per year4. Average investment = 200 000 / 2 = 100 0005. Average net cash flow = 54 800 – $ 40 000 = 14 800

ARR = Average net cash flow/Average investment

= 14 800 / 100 000 = 0.148, or 14.8%.

The ARR can alternatively be calculated on the basis of the total investment rather thanthe average investment as follows:

ARR = 14 800 / 200 000 = 0.074, or 7.4%.

A major criticism of the ARR is that it ignores the time value of money and is thereforeinferior to other methods which are based on discounting techniques. If the ARR iscalculated purely on the basis of accounting income, without adjusting for depreciation andother non-cash flow items, it becomes even less suitable to apply to project appraisal.

Capital rationing.

Capital rationing is a situation in which the firm does not have adequate funds for all itsviable projects.

Single period capital rationing is where capital is limited for the current period only butwill be freely available in the future.

Multi-period capital rationing is where capital will belimited for several periods.

Divisible projects are those that can be undertaken completely or in fractions. This alsomeans that the NPV is divisible. Indivisible projects are those which must be undertakencompletely or not at all because it is not possible to invest in a fraction of the project.

Profitability index (PI)

The profitability index is the ratio between the present value and the cash outlay on theproject. The PI is a measure of the present value per dollar of capital invested. A projectshould be accepted if the PI is greater than 1. If we have capital rationing, projects shouldbe ranked according to their PIs.

Formula:

Profitability Index = Present Value of Future Cash Flows

Initial Investment Required

= 1 + Net Present ValueInitial Investment Required

Profitability index is actually a modification of the net present value method. While present value is an absolute measure (i.e. it gives as the total dollar figure for a project), the profibality index is a relative measure (i.e. it gives as the figure as a ratio).

The Decision Rule is to Accept a project if the profitability index is greater than 1, stay indifferent if the profitability index is zero and don't accept a project if the profitability index is below 1.

Profitability index is sometimes called benefit-cost ratio too and is useful in capital rationing since it helps in ranking projects based on their per dollar return.

Example: Capital rationing with divisible projectsA company has available $250 000 for investment in the forthcoming period at a cost ofcapital of 25%. The following six projects are under consideration.

PROJECT REQUIRED INVESTMENT PV @ 25%1 $50 000 $90 0002 $70 000 $100 0003 $40 000 $70 0004 $100 000 $160 0005 $90 000 $120 0006 $80 000 $95 000

The first step is to rank the projects according to the PI.

PROJECT PROFITABILITY INDEX RANKING1 1.8 12 1.4286 43 1.75 24 1.6 35 1.33 56 1.1875 6

The next step is to allocate the available funds according to the PI.

PROJECT INVESTMENT CUMULATIVE INVESTMENT1 50 000 50 0003 40 000 90 0004 100 000 190 0005 90 000 280 0006 80 000 360 000

The company should take projects 1, 3, and 4. These would require $190 000. The balanceof$60 000 (i.e. $250 000 - $190 000), will then be invested in a fraction of project 5. Thisfraction will be 67% (i.e. $60 000 / $90 000). Project 6 cannot be undertaken this year.

If the projects are not divisible, then both 5 and 6 will be shelved for this year and thefunds left over, $60 000, should be placed in other investments, such as money marketsecurities.

Practice Questions1. Discuss the conflict between NPV and IRR.2. Distinguish between single and multi-period capital rationing.3. Explain how the profitability index (PI) is used in appraising projects.

Sensitivity analysis

Sensitivity analysis is a way of determining how sensitive the project's NPV is to the manyvariables that may affect it. We know that the NPV is calculated on the basis of estimates,such as expected revenues and expected expenses. Expected revenues are in turn based onexpected sales levels in units and expected selling price per unit. On the other hand,expected expenses also depend on expected variable costs per unit, such as the cost of laborand materials.

Sensitivity analysis is based on asking "what if " questions regarding these input variables.For example, we could calculate the NPV using different estimates of material cost per unitto measure how sensitive the NPV would be to estimation errors regarding this variable. Ifour estimate of the expected cost of materials is wrong, then our estimate of the project'scash flows will also be inaccurate.

In sensitivity analysis we would want to find out what would happen to the estimate of theNPV under different assumptions regarding each input variable. We then determine theinput variables to which the NPV is most sensitive and put more resources and effort inestimating those input variables. We then put more attention to controlling those variablesonce the project has been implemented.

Some of the drawbacks of sensitivity analysis are as follows:

1. It assumes that we can alter each input variable in isolation from others. However, mostinput variables are interrelated. For example, changing the selling price per unit mayalso affect the number of units sold.

2. It ignores the probability distribution of each input variable. Since each variable isbeing estimated, it would have its own probability distribution.

The following example will show you how the concept of sensitivity analysis can beapplied:

Example A project requires an initial investment of $570 000 and provides annual net cash flows of$220 000 each for a period of 5 years. The cost of capital is 24%.

1. Calculate the NPV of the project.

SolutionNPV = R×PVIFA – I0NPV = ( 220 000 x 2.7454 ) - 570 000 = $33 988.

2. Suppose the variable cost per unit increases by 10%, resulting in the decrease of20% in the estimated net annual cash flows. Will the project still be acceptable?

SolutionNew cash flow estimate = 220 000 ( 1 - 0.20 ) = $176 000.

NPV = ( 176 000 x 2.7454 ) - 570 000 = -$86 809.60.

3. By how much should the annual cash flows decline before the NPV falls to zero?

Solution

Let the annual cash flow be equal to a, therefore:0 = 2.7454a - 220 000

220 000 = 2.7454aa = 80 134.04.

This means that annual cash flows should fall to $80 134.04 for the NPV tofall to zero, that is by 63.58% from the current estimate.