Capital Investment Appraisal

1 Introduction

‘Capital investment’ refers to the purchase of non-current assets. These purchases are often known as capital expenditure. Non-current assets are used by the business to make profits over a number of accounting periods. Examples include: factories, machinery, office equipment and motor vehicles.

Capital investment therefore typically involves spending now to enjoy increased income in the future. Income and expenditure are therefore not matched in a profit and loss account and it is important to have techniques that help businesses to decide if the expenditure is justified.

2 introduction to discounted cash flow

As stated above, a key feature of capital expenditure is that expenditure occurs now and income is in the future. The cash outflows (expenditure) and inflows (income) occur in different time periods and so cannot be directly compared.

For example, here is a simple choice. Would you rather have:

1. $1,000 now, or

2. $1,000 in one year’s time?

Most people would prefer option 1 because it is:

• Safer
  

• Gives immediate enjoyment
  

• Even if you do not need it now, you can invest it for a year.
  

The value of investment can be evaluated by looking at interest rates. If the money could be placed on deposit at 6% for a year, then the two options really are:

1. $1,000 x 1.06 = $1,060 in 1 year, or

2. $1,000 in one year.

So earlier flows are worth more than later ones.

Instead of projecting amounts into the future (which calculates the terminal values), it is more normal to bring all amounts back to the present. So, for option 2, we want to find out how much would need to be received now to become (be identical to) $1,000 in one year.

So, if p is received now and invested at 6% to become $1,000, the following must be true:

    p x 1.06 = 1,000

    p = 1,000/1.06 = 943.40.

$943.40 is known as the present value of receiving $1,000 in 1 year at a discount rate of 6%.

This means $943.40 is equivalent to receiving $1,000 in 1 year if interest rates are 6%. [Check $943.40 invested at 6% becomes $943.40 x 1.06 = $1,000]

So the two original options can be rephrased as:

Would you rather have:

1. $1,000 now, or

2. $943.40 now?

Obviously, option 1 is preferable.

3 Discounted cash flow tables

The above examples should help you to realise that:

or Cash flow × (1 + r)⁻ⁿ

where:

r = discount rate as a decimal

  and

n = number of years in the future the flow occurs.

You can see this in previous example 1 where the present value of $4,000 received in 4 years at an interest rate (discount rate) of 7% is:

0.763 is known as the discount factor for 4 years at 7%.

Instead of having to work this out manually, discount tables are routinely provided. They are at the start of this set of notes, but here is an excerpt:

Present Value Table — present value of 1, i.e. (1 + r)⁻ⁿ, where r = discount rate and n = number of periods until payment.

You will see that with a discount rate of 7% and a period of 4 years, the discount factor is as was previously calculated, 0.763.

It is common in investment appraisal to have a number of years with equal flows: this would be termed an ‘annuity’. For example, the same rent might have to be paid for several years.

So, if a payment of $1,500 had to be made after one, two, and three years, with a discount rate of 9%, the calculation could be set out as:

However, a faster way of working this out would be to add up the discount factors and multiply their total by $1,500:

This would normally be written out as:

Once again, this type of calculation is made easier by the use of tables known as Annuity Tables or Cumulative Discount Tables. Here’s an excerpt:

Annuity Table — present value of an annuity of 1, i.e. (1 − (1 + r)⁻ⁿ) ÷ r, where r = discount rate and n = number of periods.

You will see that at a discount rate of 9% and for three periods (n = 3), the annuity discount factor from this table is 2.531, as was calculated above.

It is important to realise that if you are using discount factors straight from annuity tables, then the cash flows must start after 1 year and occur every year up to the stated number of periods.

Therefore, if any other pattern of flows occurs the discount factor will have to be adjusted.

So, if $200 is to be received in years 4 – 10 with a 5% discount rate, the required calculation would be:

Therefore, the present value of the cash received is $200 x 4.999 = $999.8

Note: the cumulative factor for years 1 – 3 has to be removed to leave 4 – 10. The commonest error made is to remove the cumulative factor for 1 – 4, but that would only leave a factor for years 5 –10.

4 The cumulative discount factor formula and perpetuities

You will see a formula at the top of the annuity tables:

Some people prefer to write this as:

Annuity factor = (1 ÷ r) × [ 1 − 1 ÷ (1 + r)ⁿ ]

This will give the figures in the tables (a pointless exercise in itself!) but might need to be used if the discount rate or the number of periods is not on the tables.

An annuity is an annual cash flow. As the annuity becomes longer (ie more years of flow) the annuity becomes close to a perpetuity, which is an equal cash flow for ever ie in perpetuity. Obviously no set of flows is for ever, but after 20 years or so, at moderate discount rates, discount factors are so small that further flows can be ignored (for example, the 20 year 10% factor is 0.149) and for convenience the flows can be treated as a perpetuity.

Notice what happens to the annuity factor formula as n, the number of years of flow, becomes very large

Annuity factor = (1 ÷ r) × [ 1 − 1 ÷ (1 + r)ⁿ ]

As n becomes large this term becomes very small and the formula can be simplified to

So, the present value of receiving $1,000 from time 1 to infinity at a discount rate of 10% is:

Note that as with annuities, perpetuities start at time 1 and occur every year. So, if the pattern is something else, the perpetuity factor will have to be adjusted.

5 Using present values in investment appraisal – net present value approach.

It was shown above that flows of cash occurring at different times cannot be directly compared. For comparison all flows are discounted to their present values.

For investment appraisal the cash outflows (usually the initial investment) and inflows (usually the future income and sale of the non-current asset at the end of its life) are all discounted to their present values. Outflows are negative, inflows are positive and when added up the net total is known as the net present value (NPV).

If the NPV > 0 the present value of the inflows exceeds the present value of the outflows, so the business would be richer. The investment would be worthwhile and should be accepted.

If the NPV< 0 the present value of the outflows exceeds the present value of the inflows and this means that the business would be poorer if it invested. The investment should be rejected

6 Net present value approach – care needed

To carry out NPV you have to identify relevant cash flows. The rules are similar to what you met earlier with relevant costing. Look for cash flows that are future incremental costs or income caused by the investment decision.

In particular:

• Cash flows are what you need, so look at income before depreciation.
  

• Reapportionment of existing fixed costs is irrelevant.
  

• Sunk or past costs are irrelevant.
  

• Opportunity costs are relevant ie cash flows forgone because of a decision
  

• Ignore all interest and other finance flows. These are taken account of by discounting.
  

7 Discounted cash flow – internal rate of return (IRR)

The internal rate of return (IRR) is defined as:

IRR = discount rate where NPV = 0

It is sometimes referred to as a breakeven discount rate.

If the discount rate to be used is greater than the IRR, then the project is not worthwhile. That would be like borrowing at 10% and investing in a project which generated 5%: obviously you would lose out.

If the discount rate to be used is less than the IRR, then the project is worthwhile. That would be like borrowing at 5% and investing in a project which generated 10%: obviously you would make money.

The IRR will tell you nothing about accepting/rejecting a project that the NPV has not already told you.

Estimating the IRR requires you to work out the NPV of the project at two discount rates. There is nothing special about the two rates chosen, but many people try 10% and 15%, or 5% and 15% or 10% and 20%.

For example:

NPV at 5% = $1,000

NPV at 15% = -$1,300

These results can be shown on a graph:

At a low discount rate the future positive cash flows stay relatively large after discounting so the NPV is likely to be positive.

At a high discount rate the future positive cash flows are considerably reduced after discounting so the NPV is likely to be negative

The IRR must be between 5% and 15% and this has to be estimated. However, the line joining the two discount rates and NPVs is usually curved and this makes estimation more difficult. To simplify matters it is assumed that the line is straight and this will result in an estimation of the IRR rather than a precise figure.

Simplifying, and drawing some triangles:

Our aim is to work out the distance BF as the IRR will then be estimated as 5+BF

The triangles ABF and ACD are similar, meaning that they have the same proportions. Therefore, for each, the base/height must be equal:

BF/AB   = CD/AC

BF   = AB x CD/AC

  = 1000 x (15 – 5) /(1,000 + 1,300) = 4.3

Therefore, point F must be 5 + 4.3 = 9.3%, and this is the estimate of the IRR.

Note that if someone had calculated the IRR using the NPVs at, say, 4% and 16%, the IRR estimate would be slightly different.

Some people are happy sketching diagrams such as those above. Others prefer to use a formula (not given in the exam):

Where

  A% = lower discount rate tried

  B% = higher discount rate tried

  N_A = NPV at A%

  N_B = NPV at B%

So, in the above example

A% = 5, B% = 15, N_A = 1,000, N_B = -1,300

Note the bottom line carefully: N_A – N_B;

N_A = 1,000 and N_B = –1,300.

So, N_A – N_B =   (1,000 – (- 1,300)) = 2,300   [the two negative signs make a plus]

8 Pay back period

The payback period measures how long it takes for a business to recoup its cash outlay.

It uses the same figures as an NPV calculation.

Companies set pay back targets and if the actual payback period is shorter than the target, the project will be accepted; if longer than the target, payback will be rejected.

The main faults with payback are:

• No account is taken of the time value of money: flows from each year are simply added together.
  

• Payback loses interest after the target has either been met or missed. So if the payback target in the above example had been 2 years the project would be rejected. That decision would be the same even if, say, the inflow in year 3 was forecast to be $60,000, not $6,000. Obviously, payback can’t be used just on its own.
  

9 Discounted payback

Here, future inflows are discounted to their present values. So, if the discount rate were 10%, the above illustration would be:

You can see here that the discounted payback period is almost 3 years

(strictly 2 + 4,234/4,506 = 2.94 years)