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FM – Investment Appraisal

Contents

Capital Rationing .................................................................................................................... 2

CONSTRAINTS AND RATIONING ......................................................................................... 2

Replacement Analysis ............................................................................................................ 7

EQUIVALENT ANNUAL COSTS ............................................................................................. 7

EXAMPLE ............................................................................................................................. 7

SOLUTION ........................................................................................................................... 7

Lease vs. Buy .......................................................................................................................... 9

LEASING VERSUS BUYING THE ASSET ................................................................................. 9

Risk - One .............................................................................................................................. 14

RISK AND UNCERTAINTY ................................................................................................... 14

EXPECTED VALUES ............................................................................................................ 14

RISK ADJUSTED DISCOUNT RATE ...................................................................................... 15

PAYBACK PERIOD .............................................................................................................. 15

SIMULATIONS ................................................................................................................... 15

Sensitivity Analysis and Certainty Equivalents ..................................................................... 16

SENSITIVITY ANALYSIS ...................................................................................................... 16

CERTAINTY EQUIVALENTS ................................................................................................ 18

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Capital Rationing

CONSTRAINTS AND RATIONING

Let us consider how to deal with questions that involve a capital constraint, known as

‘capital rationing’ problems. We may have loads of good ideas for projects to do all with

positive NPVs, but not enough capital to invest. In such events we need to decide which

projects to do.

Firstly, let us consider why we have a constraint in the first place.

The capital constraint can be described as ‘hard’ which means it’s externally imposed.

This means the money isn’t there and it can’t be obtained in time.

It could also potentially be described as ‘soft’ which means it’s internally imposed, this

generally means the money is there, but self-discipline means that the company doesn’t

want to spend it all. It may, for example, want to put some aside in case of difficulties.

Often, businesses put together a capital expenditure budget to limit their spending on

capital expenditure. This isn’t to say that if they spent a little more that the business would

be in financial difficulty, it’s just that the company may not want to spend every last penny it

has on new projects.

To be honest whether the capital constraint is hard or soft doesn’t change the numbers.

Either the money is not there, or it’s there and the company doesn’t want to spend it.

EXAMPLE

Let’s work with an example.

Suppose the company has identified four projects with the following profiles:

Project Initial investment

($’000)

NPV

($’000)

A 100 50

B 150 100

C 125 100

D 200 130

Total 575

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Now suppose it only has $300,000 dollars to spend. It cannot do all the projects as this

would cost a total of $575,000. Assuming that the projects couldn’t be delayed and all

avenues for finance been explored let’s consider how will decide what our investment plan

will be.

Firstly, we need to know if the projects are divisible or indivisible.

A divisible project means that you can do, for example 40% of the project for 40% of

the investment and get 40% of the NPV. For example, if you were buying a chain of

shops, you could buy the whole chain or potentially 40% of it.

With indivisible projects it’s either all or nothing. For example, investing in a machine to

create a new product, you either invest in the machine or you don’t, 40% of a machine

is not a machine, it is just a scrap metal!

The question will guide you as to whether the projects are divisible or not.

Divisible projects

Suppose in our example the projects are divisible. The approach we take to solving this

problem is very similar to limiting factors that you may recall from earlier studies.

We want to direct our investment when we have limited capital available to ensure that we

maximise the NPV per dollar invested. So our key calculation is to divide the NPV by the

initial investment (sometime known as the profitability index) to help us rank our projects

from most favourable to least favourable.

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For example, for project A the profitability index is 50/100 = 0.5; and for the other projects:

Project Initial investment

($’000)

NPV

($’000)

Profitability index

A 100 50 50/100 = 0.50

B 150 100 100/150 = 0.67

C 125 100 100/125 = 0.80

D 200 130 130/200 = 0.65

Total 575

We then rank the projects from the highest profitability index down to the lowest as

follows:

Project Initial investment

($’000)

NPV

($’000)

Profitability

index Rank

A 100 50 50/100 = 0.50 4

B 150 100 100/150 = 0.67 2

C 125 100 100/125 = 0.80 1

D 200 130 130/200 = 0.65 3

Total 575

As we can see, project C is our preferred starting point. It has the highest profitability index

because as we can see it gives us the greatest NPV per dollar invested which is an important

consideration when you don’t have that many dollars available to invest.

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Let’s set up a pro forma for our investment plan, we will start off at the top of the rankings

and work our way down until we’ve run out of money. In total, we have $300,000 to spend,

let’s put that at the bottom of our working:

Rank Project Investment Capital Spent

NPV Explanation

% ($’000) ($’000)

1 C 100 125 100 We can afford to do 100% of project C as we have $300,000 dollars to spend and the full project C only takes $125,000. That leaves us with: $300,000 – $125,000 = $175,000.

2 B 100 150 100 So, we move on project B which was ranked number two on the list. We can also afford to do 100% of project B as we have $175,000 left to spend and the full project B only takes $150,000. This leaves us with:

$175,000 – $150,000 = $25,000.

3 D 12.5 25 16.25 Next on the rankings is Project D. We cannot afford to do a whole project D as that will take $200,000 and we only have $25,000 left. However we can afford to do: 25/200 = 12.5% of a project D.

Remember these projects are divisible. This means it will cost us:

12.5% x $200,000 = $25,000.

We assume we will generate an additional NPV of:

12.5% x $130,000 = $16,250.

Total 300 216.25

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This means therefore that the highest NPV we can generate if we have a capital constraint

of $300,000 dollars is $216,250, this is the total of the NPV column.

In summary, if the projects are divisible then we would want to do 100% of project C, 100%

percent of project B, and 12.5% of project D

Indivisible projects

If the projects are indivisible, then the only approach we can use is to look at all possible

combinations of positive

NPV projects trying to spend as much money as possible, and see which combination gives

us the highest NPV.

For example here, we could do:

A + B = 150 NPV,

A + C= 150 NPV,

B + C = 200 NPV,

A + D = 180 NPV.

So, the combination of projects B and C give the highest NPV when the project are

indivisible and this NPV equals $200,000.

NOTE: We have considered here the impact of having capital constraints and how company

can decide what to invest in to maximum effect. The company should make sure that it is

definitely constrained as far as the amount of capital available is concerned before turning

away positive NPV projects, it could consider for example delaying a project, leasing the

assets or trying to raise more finance.

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Replacement Analysis

EQUIVALENT ANNUAL COSTS

The concept of equivalent annual cost is useful in making asset replacement decisions. The

equivalent annual cost method involves the following steps:

Step 1 – Calculate the net present value (NPV) of the cost of each replacement cycle.

Step 2 – For each replacement cycle, an equivalent annual cost is calculated.

Step 3 – Choose the replacement cycle with the lowest equivalent annual cost.

EXAMPLE

A delivery van costs $10,000 and the company is trying to decide whether to replace the van

every 2 or every 3 years. It costs $1,000 to maintain in the first year, $2,000 in the second

year and $2,500 in its third year. If the van is sold after 2 years, the scrap value is $4,000. If

the van is sold after 3 years, the scrap value is $2,500.

The cost of capital is 10% per annum. Decide whether the van should be replaced every 2

years or 3 years.

SOLUTION

Two Year Replacement Cycle

T0 T1 T2

Initial cost 10,000

Maintenance 1,000 2,000

Scrap value (inflow) -4,000

10,000 1,000 -2,000

Discount factor @ 10% 1 0.909 0.826

10,000 909 -1,652

NPV Cost of $9,257

Equivalent annual cost = $9,257 / 1.736 = $5,332 a year

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Three Year Replacement Cycle

T0 T1 T2 T3

Initial cost 10,000

Maintenance 1,000 2,000 2,500

Scrap value (inflow) -2,500

10,000 1,000 2,000 -

Discount factor @ 10% 1 0.909 0.826 0.751

10,000 909 1,652 -

NPV Cost of $12,561

Equivalent annual cost = $12,561 / 2.487= $5,051 a year

Given that the equivalent annual cost of 3 year replacement cycle is lower, the van should

be replaced every 3 years.

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Lease vs. Buy

LEASING VERSUS BUYING THE ASSET

Suppose you’ve already done your NPV analysis assuming that you’re going to borrow the

money and buy the asset involved in the project. We are now asking the question ‘could we

improve the NPV of that project by leasing as opposed to buying the asset?’

It’s important to note that we’ve already decided to go ahead with the project, so the actual

operational cash flows the project generates is not relevant here. We are only considering

the incremental cash flows of comparing leasing versus buying the asset.

You need to read the question to find out exactly what the differences are between leasing

and buying in the jurisdiction of the question. Typically, it will include the following:

If you lease the asset you pay rentals. These rental payments are incremental to the

leasing option, as is the fact that you will receive a tax deduction when paying them.

They will be allowable for corporation tax purposes.

If you lease the asset, you won’t have to pay the initial cost of buying it, but you will also

lose the scrap proceeds as you won’t have an asset to sell at the end of the lease.

Finally, if you lease the asset, you won’t own it so it’s unlikely you’ll be able to claim

capital allowances.

EXAMPLE

Let’s work through a question to see how we tackle the situation. Suppose we are

considering borrowing at an after tax cost of 10% to buy an asset at a cost of $10,000 for a

three year project. At the end of the three years the asset will have a scrap value of $4,000.

25% writing down allowances are available on a reducing balance basis with a balancing

allowance or balancing charge for the final year.

Alternatively, you could lease the asset at a cost of $2,600 a year payable at the end of each

year. This $2,600 is tax-deductible. Corporation taxes is payable at a rate of 30% per annum,

payable in the same year as the cash flows that are being taxed.

Should the company lease or buy the asset?

There are two alternative approaches to dealing with a question like this:

1. You could prepare two NPV tables, one for leasing, and one for buying, and compare

the two to see which is cheapest.

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2. Alternatively, you can prepare one table which in one go compares ‘leasing as opposed

to buying’. We’ll have a look at both methods here, generally either are acceptable in the

exam.

Two table approach

First of all, let’s have a look at the two table approach.

We’ll start off by putting together an NPV table showing the cost of buying the asset:

NPV – Cost of buying asset ($)

Description T0 T1 T2 T3

Initial cost (10,000)

Scrap value 4,000

Writing down allowance (WDA) – Working 1 750 563 488

Net cash flow (10,000) 750 563 4,488

Discount factor @ 10% 1 0.909 0.826 0.751

Present value (10,000) 682 465 3,370

NPV = (10,000) + 682 + 465 + 3,370 = (5,483)

Firstly, don’t be put off by the fact that this is negative– We are only considering the cost of

ownership.

Secondly, you’ll notice we don’t put the loan that we take out as a cash inflow. We don’t put

the loan repayment as a cash outflow, and we don’t even put the interest cost as a cash

outflow. This is because it is all accounted for in the discount factor. That’s 10% represents

the cost of financing this asset.

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Working 1 – WDA

(We calculate the value of the tax saved as a result of claiming writing down allowances in

exactly the same way as we’ve seen already.)

Time Written down value

(WDV) Tax 30% Timing

T0 10,000

1st WDA (10,000 x 25%) (2,500) 750 T1

7,500

2nd WDA (7,500 x 25%) (1,875) 563 T2

5,625

Scrap (4,000)

Balancing allowance 1,625 488 T3

Let’s now have a look at the relevant cash flows associated with leasing the asset:

NPV – Leasing the asset ($)

Description T0 T1 T2 T3

Rental cost (2,600) (2,600) (2,600)

Tax saved (@30%) 780 780 780

Net cash flow (1,820) (1,820) (1,820)

Discount factor @ 10% 0.909 0.826 0.751

Present value (1,654) (1,503) (1,367)

NPV = (1,654) + (1,503) + (1,367) = (4,524)

So, the net present cost associated with buying the asset was $5,483, and the net present

cost associated with leasing asset was $4,524. Leasing is therefore (5483 – 4524 =) $959

cheaper. All else being equal we should therefore lease the asset.

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One table approach

As mentioned earlier, this can be done through one table. We will construct the table from

the point of view of saying ‘if we lease as opposed to buy the asset’. Let’s set up a pro forma

first, after that account for:

Save the initial outlay;

Lose the scrap proceeds;

Lose the capital allowances;

Pay annual rentals and receive a tax deduction for doing so;

We then add up all the relevant cash flows and multiply by the discount factor to give us

present values;

Which we then add up to give us the NPV of leasing as opposed to buying.

NPV – Cost of buying asset ($)

Description T0 T1 T2 T3

Initial investment saved 10,000

Scrap proceeds lost (4,000)

Lost capital allowance (750) (563) (488)

Annual rental (2,600) (2,600) (2,600)

Tax saved on annual rental (@ 30%) 780 780 780

Net cash flow 10,000 (2,570) (2,383) (6,308)

Discount factor @ 10% 1 0.909 0.826 0.751

Present value 10,000 (2,336) (1,968) (4,737)

NPV = 10,000 + (2,336) + (1,968) + (4,737) = $959

You’ll notice that the answer is the same as before, preparing a table that asks ‘should we

lease as opposed to buy’ has a positive NPV of $959. Meaning that leasing as opposed to

buying is a good thing to do.

It is worth considering how you could possibly be better off to lease as opposed to buy,

when presumably the company that leases you the asset is also making some money. There

are several possible reasons for this:

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The leasing company may buy in bulk and therefore be able to buy the assets at a

discount

The leasing company may have access to cheaper finance, for example if it is a bank

The leasing company may be prepared to take a small loss on the lease itself to

encourage the sale to happen in the first place.

Finally, sometimes leasing an asset may be the only way a business can finance the

acquisition, for example borrowing may not be possible if it has little to offer by way of

security.

So, here we have compared leasing versus buying as two financing alternatives, remember

the decision to go ahead with the project has already taken place so there is no need to

bring in the project cash flows, we are only considering how the cash flows change if we

were to lease as opposed to buy.

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Risk - One

RISK AND UNCERTAINTY

Risk and uncertainty are often used interchangeably. However, they are not the same.

Risk is quantifiable. The possibilities and probabilities can be predicted. An example of risk

would be tossing a coin. Although the exact outcome is not known, there is a 50% chance of

getting each side.

Uncertainty is not quantifiable. The possibilities and probabilities cannot be predicted.

Risk and uncertainty can be incorporated in the decision making with the help of following

techniques:

Expected values

Risk-adjusted discount factor

Payback

Simulations

Sensitivity analysis

Certainty equivalents

EXPECTED VALUES

An expected value is simply an average. Suppose a business is not certain about the next

year revenue. However, it thinks that there is a 25% chance of earning $100,000 and a 75%

chance of earning $200,000. The expected value for revenue next year will be:

(25% x $100,000) + (75% x $200,000) = $175,000.

The expected values assume risk-neutral attitude to decision making. It is important to note

that expected values are a long run average and are not suitable for one-off decisions. They

are also a highly dependent on the estimate of probabilities.

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RISK ADJUSTED DISCOUNT RATE

The cost of capital (discount rate) compensates the investors for: interest, inflation and risk.

If the risk is higher than normal, the discount rate can be increased to reflect it. Suppose a

business has cost of capital of 10%. However, it is undertaking a new project with higher

than normal risk. It will use a discount rate of, let’s say, 12% to evaluate the project. The

calculations are explained later in the course.

PAYBACK PERIOD

Payback period is also a measure of risk. A project with a short payback period is considered

to be less risky as the near future is more knowable than the distant future. A project with

long payback period is more risky as the conditions can change over the longer term.

SIMULATIONS

Simulation can be used to assess the impact of changes in multiple variables on the NPV.

The cash flows used to calculate NPV are point estimates. For example, if we estimate the

sales in year 1 to be $100 and year 2 to be $200, we are simply putting the best estimate to

the calculation and the actual values in year 1 and 2 may be different. To deal with the

problem, we can replace these point estimates with range of values, for example probability

distribution. If some of the point estimates are replaced with probability distributions, the

natural conclusion of this is that there won’t be a single number at the end of the

calculation for NPV; the NPV itself will be a distribution.

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Sensitivity Analysis and Certainty Equivalents

SENSITIVITY ANALYSIS

Once we have calculated net present value, sensitivity analysis turns to the estimates we

have made in the calculation and for any one estimate asks the question ‘how wrong could I

be before the decision changes?’ In other words, how much different from that estimate will

the number need to be before it turns out that I made a mistake in my initial conclusion. If

the answer is it has to be very different we are probably feeling reasonably comfortable. If it

only has to be slightly different before the decision changes, we are very sensitive to that

variable.

Example

For example, let’s suppose we have calculated the NPV for a project to be a positive

$50,000. Within the calculation there was an estimate for the initial investment of $1

million. Our initial conclusion would be to go ahead with the project given it has a positive

NPV. In fact, we would be happy to go ahead with the project, provided the NPV is not zero

or negative.

For every one dollar increase in the cost of the initial investment, the NPV will fall by one

dollar. In other words, initial investment could increase to $1,050,000 and the project would

still be viable. At $1,050,000, that initial investment cost has increased by $50,000 and has

therefore eliminated the positive NPV, the NPV would be zero.

We would typically express the sensitivity with the following calculation:

50,000 / 1,000,000 = 5% sensitivity.

A lower percentage sensitivity means that only a relatively small movement in the value of

that variable will change the decision. In other words, a lower percentage means we are

very sensitive to that variable.

In more general terms we can calculate the percentage sensitivity as follows:

Sensitivity =

NPV of the project

NPV of the cash flows affected by the variable

Now, suppose we are interested in the sensitivity of our project’s NPV to sales price.

Revenues are, let’s assume, $250,000 each year for five years. Corporation tax is at 30%. The

cost of capital is 10%. As previously mentioned, the NPV of the project is estimated to be

$50,000. Let’s now consider the sensitivity of our decision to sales price.

The net present value of the cash flows affected by sales price is as follows:

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$250,000 before tax is worth ($250,000 x [1 – 30%] =) $175,000 after tax each year for 5

years as a discount rate of 10%.

The present value of after tax revenue is therefore:

$175,000 x AF1-510% = $175,000 x 3.791 = $663,425.

If, for example, sales price halved, NPV would reduce by (0.5 x 663,425 =) $331,713.

So, we can afford to lose:

50,000 / 663,425 = 7.5% of sales revenue before we reduce our NPV down to zero.

In other words, if sales price falls by 7.5% we will eliminate NPV.

We are therefore 7.5% sensitive to sales price.

The same approach can be taken with the vast majority of cash flows that have gone into

the NPV calculation. The decision maker can then focus in on the variables they are more

sensitive to, to make sure that their estimates are reasonable.

Extensions to sensitivity analysis

There are a couple of exceptions to this though:

1. Sensitivity to the cost of capital:

Firstly, let’s consider sensitivity to the cost of capital. With our project, the NPV at 10% is

$50,000. Internal rate of return is that discount rate that yields than NPV equal to 0.

Suppose for our project the IRR is 14%. This means if our estimate for the cost of capital of

10% turns out to be wrong, it will change our decision if the cost of capital turns out to be

14% or higher because at that point the NPV = 0.

In other words, we would need to be ([14 – 10] / 10 =) 40% wrong. We are 40% sensitive to

our estimate for the cost of capital.

2. Sensitivity to the life of the project:

Finally with sensitivity analysis, let’s consider sensitivity to the life of the project. Our five

year project has an NPV of $50,000 positive. Suppose we rerun the calculation as if the

project was a four-year project and the NPV turns out to be approximately zero if it only

lasts for four years.

The percentage sensitivity to the life of the project would be calculated as: ([5 – 4] / 5 =)

20%.

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Although sensitivity analysis can be a useful guide to decision makers to focus their

attention on the key variables in a decision, it does have its drawbacks:

It only looks at one variable changing at a time, in reality they’ll tend to change

together; a Simulation may be more appropriate with more than one variable changing.

It does not come with a decision rule, does a sensitivity of 6% mean that we don’t go

ahead? That is down to the judgement of the decision maker, there are no rules to help

make the decision.

Even though sensitivity analysis tells us how wrong we would need to be in order for the

decision to change, nowhere in our analysis does it include how likely we are to be

wrong. For example, on the face of it a 1% sensitivity to price might be worrying to the

decision maker. However, if they have a signed contract in place guaranteeing what the

price will be should they go ahead, there is in fact no need to be concerned.

CERTAINTY EQUIVALENTS

Finally, let’s consider certainty equivalents. This is something we tend to do naturally,

consider the receipt of $1,000 we may be receiving in a month’s time:

If you are absolutely certain it will arrive, we may include it in our forecasts as $1,000.

If you are reasonably certain but not absolutely certain it will arrive, we might include it

at say $900.

If we think it could possibly arrive but we are not at all certain, we might include it at,

say, $250.

In effect, what we are doing here is restating the possible $1,000 as a certainty equivalent.

For example, we are saying a possible $1,000 is worth the same as a certain $900 for

example.

Example

Let’s consider how this might be examined.

Consider the following estimated forecast cash flows for revenue in a project:

T1 $100,000

T2 $200,000 T3 $250,000

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Relevant certainty equivalent factors are as follows:

T1 0.9

T2 0.8 T3 0.7

These certainty equivalent factors would need to be given to you in the question, or at least

you would have to be given enough information to be able to work them out.

Suppose the cost of capital is 14% and the risk-free rate of return is 10%. The risk-free rate is

the rate of return required to compensate investors for interest and inflation only, i.e. no

premium is included for risk.

The present value using certainty equivalents will be calculated as follows:

Start off by writing out the estimated cash flows.

Multiply the estimated cash flows by the certainty equivalent factors. This strips out the

risk from the cash flows and turns them into risk-free certainty equivalents.

The certainty equivalents should then be discounted at the risk-free rate. There is no

risk any more in the cash flows, so we do not need a risk premium in the discount rate.

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Details T1 T2 T3

Estimated cash flows 100,000 200,000 250,000

Certainty equivalent factor 0.9 0.8 0.7

Certainty equivalent cash flows 90,000 160,000 175,000

Risk free discount factor @ 10% 0.909 0.826 0.751

Present value 81,810 132,160 131,425

Disadvantages

There is nothing wrong with this calculation in principle, but in practice the certainty equivalent

factors cannot consistently be derived from anything objective, they will in effect be subjective,

‘made up’ numbers which potentially undermines the accuracy of the overall answer.

NOTE: Investment appraisal questions require a methodical approach, they are often quite

numerically intensive, but with plenty of question practice and an orderly approach they are

eminently passable.