The Excel function NPER

Capital Budgeting: the key numerical techniques

Worked Examples with Explanations

The purpose behind capital budgeting is to assist managers of organisations make better informed decisions on acquiring and disposing of assets. For example, how and when would you know, without some form of detailed analysis, whether and when to buy a new machine for your factory; or a new vehicle to deliver your goods; or even new land on which to build an extension to your showrooms?

Although there are many aspects to capital budgeting other than the numerical aspects, it is these aspects with which this page is solely concerned. The output of any of the following techniques tells us whether a project is viable - in financial terms only.

Additionally, although I have said that capital budgeting is concerned with acquiring and disposing of assets, I will not be giving examples relating to disposals in this page. The techniques can easily be applied to such situations; and if you are interested in following through to disposals either let me know or look for financial management texts in a good library which contain examples of disposals.

The key numerical techniques to be covered in this page are:

  • The Net Present Value Technique (NPV)
  • The Internal Rate of Return Technique (IRR)
  • The Payback Period Technique (PB)
  • The Accounting Rate of Return Technique (ARR)
  • The Profitability Index (PI)

I will deal with each one in turn.

Net Present Value

The idea behind the NPV technique is that it DISCOUNTS the cash flows generated by an asset back to the present day: thus the NPV technique is concerned with the time value of money. The result we are faced with is usually in the form:

Year Cash
Flows
()
Discount
Factors
(15%)
Present
Values
()
0 -25,000 1.0000 -25,000.00
1 20,000 0.8696 17,391.30
2 25,000 0.7561 18,903.59
3 12,500 0.6575 8,218.95
4 9,000 0.5718 5,145.78
Net Present Value     24,659.63

The residual value is taken to be zero.

The key points to notice here are that we are dealing with the NET present value which is the net of the initial (original) cost and the present value of all other cash flows. This is as opposed to the present value of the cash flows which would simply be the sum of 17,319.30 + 18,903.59 ... + 5,145.78 = 49,659.63

Thus we are dealing with the value, in terms of today's prices, of an asset for which we are expecting to pay 25,000 today. A positive NPV of 24,659.63 says that we are being asked to pay 25,000 for an asset worth 49,659.63: a bargain!! Had the NPV been negative - let's say MINUS 24,659.63 - then we would not be facing such a bargain. In this example, a negative NPV of the value just given would say we were being asked to pay 25,000 for an asset worth only 340.37: definitely not a bargain.

The above example is a case of a CONVENTIONAL investment. A conventional investment is one where an initial outflow of cash (the original capital cost) is followed by positive inflows. A non conventional investment would behave differently: for example, it could have several negative initial outflows followed by some positive and some negative inflows:

YearConventional
Investment
Non conventional
Investment
0 Negative Negative
1 Positive Negative
2 Positive Positive
3 Positive Positive
4 Positive Negative
5 Positive Positive
6 Positive Positive

The fact that an investment is non conventional does not alter the way the NPV technique works: it is merely another fact for you to impress your friends with!!

Internal Rate of Return

In some senses this is the simplest of the techniques to understand. However, it is the most difficult to cope with mathematically. The best way of viewing the IRR of a project is to consider it in the form of a graph: the NET PRESENT VALUE PROFILE:

Construct a Net Present Value Profitle for yourself!

Reading from the point where the NPV profile itself cuts the horizontal (interest rate) axis gives the value of the IRR - in this case it is 66.15%.

By appreciating that this is how to find the value of the IRR, you can, in fact, define the term yourself: it is the rate of interest applying to a project at which its net present value is precisely zero. The usefulness of this knowledge is that if the IRR is known (and it is relatively simple to discover it for any project with either a good calculator or a computer) then, for any rate of interest which the company has to bear, the management will know whether the project under review is a good one, a risky one or a safe one.

In the example above, then, if the current interest rate being borne by that company is 15% on average, then it knows, with an IRR of 66.15% that interest rates will have to rise a long way before this project becomes non viable. It follows from this that, in general, if the rate of interest being borne or considered is LESS THAN the IRR, the net present value of the project is sure to be positive; and similarly, if the rate of interest is GREATER THAN the IRR, the NPV is sure to be negative.

Confirm the last few statements by inspecting the NPV profile and considering the numerical aspects of that example for yourself.

The Payback Period

In spite of what I said above about the IRR technique being the simplest to understand of all the techniques being presented here, in fact, PB outshines them all for simplicity - once you have got used to it!!

The payback period is measures the length of time it takes a project to repay its initial capital cost. For example, if I buy a machine for 10,000 and it earns me a cash flow of 10,000 for the whole of the first year of its life, I can see immediately that the cash flows have repaid the initial capital cost and therefore that the payback period is exactly one year.

If the same machine gave rise to 5,000 cash flow in the first year and 5,000 cash flow in the second year then the payback period has become two years since that is how long it has taken cumulative cash flows to equal the initial capital cost. Developing that idea more generally now, let's go back to our original example above:

Year Cash
Flows
Cumulative
Cash
Flows
0 -25,000 -25,000
1 20,000 - 5,000
2 25,000 20,000
3 12,500 32,500
4 9,000 41,500

When, in the above example, does the cumulative cash flow equal 0? It isn't immediately obvious but we can see that it is somewhere between year one and year two: at the end of year one cumulative cash flows equal -5,000; and at the end of year two they equal 20,000; therefore somewhere between the two they have been equal to 0. But where? We can find out exactly where providing we assume that all cash flows accrue evenly throughout the year: in year two, 20,000 has been received by the company and we assume that 20,000/365 was the daily cash flow.

The payback period calculation is:

Number of years
immediately prior to the year
in which the payback
period occurs
PLUS The cash flow received
during the year to take
cumulative cash flow to zero
The total cash flow during the
year during which the payback period occurs

In our example, this is:

1 year + 5,000 years = 1.25 years
20,000

This will seem a very awkward way of dealing with a fairly simple calculation; and once you have practiced it you'll find it is a simple technique to use.

The payback period technique is the single most widely used technique of all of the techniques currently reported to be in use virtually anywhere in the world! It is so widely used for two major reasons:

  • it is the simplest method available
  • it acts as a proxy for risk.

The first reason should be self explanatory. The second reason may need some explanation. The method is a proxy for risk in that most people are risk averse - they do not like taking risks - and thus they prefer to minimise or offset risk altogether.

Risk arises in capital budgeting in that most of the data on which decisions are based are estimated - especially the data derived for the later years of a project - the further away from today a value for cash flow is, the less reliable it is (that is, the more risky it would be to believe it and act on it). The beauty of the payback period technique in this respect is that it tells management how quickly its cash inflows cover its cash outflows: the quicker the better. Hence, a decision will be favourable on a project with a lower value for the payback period when a manager is risk averse.

In reality, a value of between 3 and 5 seems to be the sort of value which most British managers wish to see: an average British manager is risk averse!

Accounting Rate of Return

The only method of the five we are discussing which relies on PROFIT rather than cash flows. Nevertheless, its calculations re not too onerous. Again, we'll work through the same example we've been considering so far for the other techniques.

Initial
Cost
Total
Cash
Flows
DepreciationNet
Profit
Annual
Average
Profit
ARR
%
25,00066,50025,00041,50010,37541.50

Note: depreciation is equal to the initial cost of the asset because we were told that there is to be a zero residual value at the end of the life of the asset: otherwise, the depreciation would have been initial cost less residual value.

The result of 41.50% is the average annual rate of profit earned by this asset or project and it can be compared with other projects: the higher the rate of profit earned the better.

The Profitability Index

Of itself, this method is nice and straightforward; BUT, to calculate the PI, you have first to calculate the NPV - then it's easy!! From our continuing example, the PI is:

(i) Net Present Value
Initial Capital Cost

Alternatively, you can calculate it as:

(i) Present Value
Initial Capital Cost

Although the values derived from each calculation are different, their interpretations are the same:

(i) 24,659.63 = 0.9864
25,000

(ii) 49,659.63 = 1.9864
25,000

The PI is relative view of how well a project is to perform: it compares NPV or PV (depending on whether you're using equation (i) or (ii)) with the initial capital cost. There is no single value which will tell you whether the project is a good one: rather, two or more projects may be being considered and the PI which is highest belongs to the optimal project to adopt.

The above has given an insight into the five methods you are most likely to face in theory as well as in practice. One thing we have not done here is to consider the advantages and disadvantages of these methods: a later page will describe these.

Exercises

You might care to work through the following exercises to confirm your understanding of the above techniques (except the IRR technique). I will give you the result for the IRR for each exercise.

Q1

YearCash
Flows
0-50,000
113,000
212,000
35,000
45,000
53,000

The rate of interest is to be 12%

Q2

YearCash
Flows
0-45,000
113,000
213,000
313,000
413,000

There is estimated to be a residual value of 10,000 receivable at the end of year 5; and the rate of interest is to be taken as 14%.

For exercise 1, the IRR is 23.03%; and for exercise 2 it is 12.81%.

Reference

Williamson Duncan (1996)
Cost and Management Accounting
Prentice Hall
ISBN 0-13-205923-1

Duncan Williamson
12 October 2001 revised 8 May 2003

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