Investment Appraisal - Capital Budgeting

Compounding & Discounting

Compounding

A sum invested today will earn interest. Compounding calculates the future or terminal value of a given sum invested today for a number of years. To compound a sum, the figure is increased by the amount of interest it would earn over the period. To speed up the process, we can use a formula to calculate the future value of a sum invested now. The formula is:

Example:

Suppose that a business has $100 to invest and wants to earn a return of 10%.

What is the future value at the end of each year using compound interest?

Solution:

Year 1 - 100 x 1.10 = $110

Year 2 - 110 x 1.10 = $121 or 100 x (1.10) ^ 2

Year 3 - 121 x 1.10 = $133 or 100 x (1.10) ^ 3

This future value can be calculated as:

FV = PV (1 + r) n

Where,

FV is the future value of the investment with interest

PV is the initial or ‘present’ value of the investment

r is the compound annual rate of return or rate of interest expressed as a proportion

n is the number of years

e.g., $100 x 1.1 ^3 = $133

Discounting

Discounting is compounding in reverse.

It starts with a future amount of cash and converts it into a present value.

A present value is the amount that would need to be invested now to earn the future cash flow, if the money is invested at the ‘cost of capital’.

Hence, when looking at whether we should invest in something we will be looking at future cash flows coming in. We want to know what these future cash flows are worth now, in today’s money ideally.

Discount factor = (1 + r) - n

Where r = discount/interest rate and n = time period of cash flow

PV = FV × DF

PV = FV × (1 + r)–n

Example:

A business is to receive $100 in one year’s time and the interest rate/discount rate is 10%. What is the PV of that money?

Solution:

PV = 100 /1.10 ^ 1

PV = $90.9

Discount Rate

The present value can also be calculated using a discount factor (saving all the dividing by 1.1 etc.)

The discount factor can be calculated as:

1/ (1+r) ^ n

r - rate of interest

n - number of time periods

So, the discount factor for 10% in 3 years is:

1/1.1 ^ 3 = 0.751

There are also tables that give you a list of these ‘discount factors’ – a copy of these tables is included at the end of these notes.

Hence, to calculate a present value for a future cash flow, you simply multiply the future cash flow by the appropriate discount factor.

Conventions Used in DCF

• Time 0 is today, it is usual to assume that time 0 is the first day of a project, ie the start of its first year.

• Time 1 is the last day of the first period (normally a year).

• A cash flow which occurs during the course of a time period is assumed to occur all at once at the end of the time period (at the end of the year).

• A cash flow which occurs at the start of a time period is taken to occur at the end of the previous time period e.g. a cash outlay of $5,000 at the start of time period 2 is taken to occur at the end of time period 1.

Annuities

An annuity is a constant annual cash flow for a number of years. If a project involves equal annual cash flows (or annuities) then each future cash flow can be discounted separately back to a present value, but it is quicker to use a single discount factor (called an annuity factor or a cumulative discount factor).

PV = Annual Cash flow × AF

AF = {1- (1+r) – n} ÷ r

Annuity Tables

You will be provided with an Annuity Table (sometimes referred to as a Cumulative Present Value Table), which provides pre-calculated annuity factors for various different discount rates over various periods. So again, you have a choice. For example, for a three-year annuity at 10%:

There might be a small difference due to rounding. The tables should not be used as a substitute for knowing how to use the formula. Remember, the tables only cover a small range of discount rates and time periods, and you may be required to calculate a discount factor or annuity factor for variables outside of this range.

Example:

A firm has arranged a 10-year lease at an annual rent of $17,264. Each rental payment is to be made at the start of the year.

Required:

What is the present value of the lease at 12%? (Give your answer to the nearest $.)

Solution:

$109,247

When discounting we are assuming that cash flows arise the end of the year, so the payments at

the start of years 1-10 can be viewed as a payment at time 0 and then nine payments at the end

of each fo the years 1-9. So, when we use the annuity table, we are looking at time periods 1-9.

r = 12%

n = 9 (1st payment now, 10th payment at the end of time 9)

Annuity factor = 5.328 for time periods 1-9 at 12%.

PV = 17,264 (the first payment is not discounted because it is paid in advance) + (5.328 × 17,264) = 109,247

Perpetuities

A perpetuity is an annuity that continues indefinitely into the foreseeable future, with no end date for the cash flows.

Cash Flow Analysis:

• Perpetuities can be evaluated using a single discount factor.

• This approach requires the use of a specific formula, which should be memorized for calculations.

PV = Cash flow ÷ r Or PV = Cash flow × (1÷ r)

1 ÷ r is known as the perpetuity factor.

The PV of a growing perpetuity is found using the formula:

PV = Cash flow at T1 × 1÷ (r – g)

1 ÷ (r – g) is known as the perpetuity factor with growth.

Example:

If a project involved the outlay of $20,000 today and provided a definite return of $3,000 per year

for the foreseeable future.

Required

Would you accept the project?

(Again, assume that you could get a return of 6% on investments of similar risk.)

Solution

The perpetuity factor here is: 1/0.06

So, the present value of the future cash flows is $3,000 × 1/0.06 =$50,000

And the present value of the inflows exceeds the cost of the project, so the project is acceptable.

Advanced and Delayed Annuities and Perpetuities

The use of annuity factors and perpetuity factors both assume that the first cash flow will be occurring in one year's time. Annuity or perpetuity factors will therefore discount the cash flows back to give the value one year before the first cash flow arose. For standard annuities and perpetuities, this gives the present (T0) value since the first cash flow started at T1.

Be careful: if this is not the case, you will need to adjust your calculation.

Advanced Annuities and Perpetuities

• Some regular cash flows may start now (at T0) rather than in one year's time (T1).

• Calculate the PV by ignoring the payment at T0 when considering the number of cash flows and then adding one to the annuity or perpetuity factor.

Advanced Annuities: An Illustration

A 5-year $600 annuity is starting today. Interest rates are 10%. Find the PV of the annuity.

Solution

This is essentially a standard 4-year annuity with an additional payment at T0. The PV could be calculated as follows:

PV = 600 × (1 + 3.17) = 600 × 4.17 = $2502

Advanced Perpetuities: An Illustration

A perpetuity of $2,000 is due to commence immediately. The interest rate is 9%. What is the PV?

Solution

This is essentially a standard perpetuity with an additional payment at T0. The PV could be calculated as follows:

PV =2,000 × (1 + 1÷ 0.09) = 2,000 × 12.11 = $24,222

Delayed Annuities and Perpetuities

Some regular cash flows may start later than T1.

These are dealt with by:

• Applying the appropriate factor to the cash flow as normal

• Discounting your answer back to T0

Delayed Annuities – An illustration

A seven-year annuity of $450 starting in five years’ time. Interest rates are 11 %. Find the present value of the cash flow.

A seven-year annuity of $450 starting in five years’ time.

Step 1: Look up the 7-year AF: AF = 4.712

Step 2: Discount the annuity as usual: 450 × 4.712 = $2,120.40

Note that this gives the value of the annuity at T4

Step 3: Discount the answer back to T0: 2,120.40 × 0.659 = $1,397.34

Delayed Perpetuities – An illustration

A perpetuity of $33,000 commencing at T2. Interest rates are 22%. Find the present value of the cash flow.

A perpetuity of $33,000 commencing at T2.

Step 1: Calculate the perpetuity factor: 1/0.22 = 4.545

Step 2: Discount the perpetuity as usual: 33,000 × 4.545 = $149,985

Note that this gives the value of the perpetuity at T1

Step 3: Calculate the PV: 149,985 × (1/1.22) = $122,939

The Net Present Value (NPV)

To appraise the overall impact of a project using DCF techniques involves discounting all the relevant cash flows associated with the project back to their PV.

If we treat outflows of the project as negative and inflows as positive, the NPV of the project is the sum of the PVs of all flows that arise as a result of doing the project.

The NPV represents the surplus funds (after funding the investment) earned on the project, therefore:

• If the NPV is positive – the project is financially viable

• If the NPV is zero – the project breaks even (just returning enough money to cover the funding costs)

• If the NPV is negative – the project is not financially viable

• If the company has two or more mutually exclusive projects under consideration it should choose the one with the highest NPV

• The NPV gives the impact of the project on shareholder wealth.

Assumptions used in discounting

Unless the examiner tells you otherwise, the following assumptions are made about cash flows when calculating the net present value:

• All cash flows occur at the start or end of a year

• Although in practice many cash flows accrue throughout the year, for discounting purposes they are all treated as occurring at the start or end of a year. Note also that if today (T0) is 01/01/20X0, the dates 31/12/20X1 and 01/01/20X2, although technically separate days, can be treated for discounting as occurring at the same point in time, i.e. at T1

• Initial investments occur at T0 other cash flows start one year after that (T1).

• In project appraisal, the investment needs to be made before the cash flows can accrue. Therefore, unless the examiner specifies otherwise, it is assumed that investments occur in advance. The first cash flows associated with running the project are therefore assumed to occur one year after the project begins, i.e. at T1

Also, note you should never include interest payments as cash flows within an NPV calculation as these are taken account of by the cost of capital.

EXAMPLE: An organisation is considering a capital investment in new equipment. The estimated cash flows are as follows.

Year

                                          Cash flow

                                                $

0                                       (240,000)

1                                         80,000

2                                        120,000

3                                         70,000

4                                         40,000

5                                         20,000

The company’s cost of capital is 9%. Calculate the NPV of the project to assess whether it should be undertaken.

Year             Cash flow           DF at 9%            PV

0                  (240,000)             1.000              (240,000)

1                     80,000                0.917               73,360

2                    120,000               0.842             101,040

3                     70,000                0.772              54,040

4                     40,000                0.708             28,320

5                     20,000                0.650             13,000

NPV

29,760

–––––––

The PV of cash inflows exceeds the PV of cash outflows by $29,760, which means that the project will earn a DCF return in excess of 9%, i.e. it will earn a surplus of $29,760 after paying the cost of financing. It should therefore be undertaken.

Present Value Tables

(1 + r)–n is called the discount factor (DF). In the exam, you will be provided with a Present Value table that gives the discount factors for various different discount rates over various time period. So, to find the DF, for example if r = 10% and n = 5, you can

Advantages of NPV

When appraising projects or investments, NPV is considered to be superior (in theory) to most other methods. This is because it:

• Considers the time value of money – discounting cash flows to PV takes account of the impact of interest, inflation and risk over time. These significant issues are ignored by the basic methods of payback and annual rate of return (ARR or ROCE)

• Is an absolute measure of return – the NPV of an investment represents the actual surplus raised by the project. This allows a business to plan more effectively. Neither ARR nor payback is an absolute measure.

• Is based on cash flows rather than profits – the subjectivity of profits makes them less reliable than cash flows and therefore less appropriate for decision making.

• Considers the whole life of the project – methods such as payback only consider the earlier cash flows associated with the project. NPV takes account of all relevant flows associated with the project. Discounting the flows takes account of the fact that later flows are less reliable, which ARR ignores.

• Should lead to maximisation of shareholder wealth. If the cost of capital reflects the investors’ (i.e. Shareholders’) required return, then the NPV reflects the theoretical increase in their wealth. For a company, this is considered to be the primary objective of the business.

Disadvantages of NPV

• It is difficult to explain to managers. To understand the meaning of the NPV calculated requires an understanding of discounting. The method is not as intuitive as techniques such as payback.

• It requires knowledge of the cost of capital. The calculation of the cost of capital is, in practice, more complex than identifying interest rates. It involves gathering data and making a number of calculations based on that data and some estimates. The process may be deemed too protracted for the appraisal to be carried out.

• It is relatively complex. For the reasons explained above, NPV may be rejected in favour of simpler techniques.

Internal Rate of Return

The IRR is essentially the discount rate where the initial cash out (the investment) is equal to the PV of the cash in.

So, it is the discount rate where the NPV = 0

It is actual return on the investment (%).

Consequently, to work out the IRR we need to do trial and error NPV calculations, using different discount rates, to try and find the discount rate where the NPV = 0.

The good news is you only need to do 2 NPV calculations and then apply this formula:

Where,

L = Lower discount rate

H = Higher discount rate

NPV L = NPV @ lower rate

NPV H = NPV @ higher rate

If the IRR is higher than the cost of capital, the project should be accepted.

Note:

IRR is greater than the required return (cost of capital)

Return from the investment is above that which is required

(undertake project)

IRR is less than the required

return (cost of capital)

Return from investment is below that which is required

(don’t undertake project)

IRR is equal to the required return

(cost of capital)

Return from investment is the same as cost of

Example:

A project has a positive NPV of $15,000 when discounted at 6% and a negative NPV of $3,000 when discounted at 12%.

Required:

Calculate the internal rate of return.

Solution:

11%

IRR = 6 + (15/(15 + 3) × 6) = 11%

Little Tricks

• If all the cashflows are the same

This is an annuity - simply take the Initial Cost / annual inflow - this gives you the cumulative discount factor (annuity factor).

• Then go to the annuity table and look for this figure (in the row for the number of years the project is for) - the column in which the figure is found is the IRR!

• If the cashflows are the same and go on forever.

• This is a perpetuity - simply take the Annual inflow / Initial cost and turn it into a percentage. That’s the IRR! Done.

Advantages of IRR

• Considers the time value of money

• Easily understood percentage

• Uses cash do not profit

• Considers whole life of project

• Increases shareholders’ wealth

Disadvantages of IRR

• Does not produce an absolute figure (percentage only)

• Interpolation of the formula means it is only an estimate

• Fairly complicated to calculate

• Non-conventional cashflows can produce multiple IRRs

Interpreting the IRR

• The IRR provides a decision rule for investment appraisal, but also provides information about the riskiness of a project – i.e., the sensitivity of its returns.

• The project will only continue to have a positive NPV whilst the firm’s cost of capital is lower than the IRR.

• A project with a positive NPV at 14% but an IRR of 15% for example, is clearly sensitive to:

- an increase in the cost of finance

- an increase in investors’ perception of the potential risks

- any alteration to the estimates used in the NPV appraisal.

IRR of a perpetuity

To calculate the IRR of a perpetuity:

1. IRR is the discount rate where NPV = 0.

2. Perpetuity formula: PV=Annual Inflow / r

3. At NPV = 0:
   Initial Investment = Annual Inflow / r

4. Rearrange to find IRR:
   r = Annual Inflow / Initial Investment

Example: For an initial investment of $30,000 and an annual inflow of $4,000:

r = 4,000 / 30,000 = 13.33%

Computer based exam method

In a computer-based exam you can use the =IRR function to calculate the project’s IRR. Note. The undiscounted cash flows are used for the IRR calculation. Also, the spreadsheet IRR formula does not work if the cash flows are set up as annuities.

NPV v IRR

• The net present value (NPV) method has several important advantages over the internal rate of return (IRR) method.

NPV is often simpler to use. As mentioned earlier, IRR may require hunting for the discount rate that results in a net present value of zero.

This can be a very laborious trial-and-error process, although it can be automated to some degree using a computer spreadsheet.

A key assumption made by IRR is questionable. Both methods assume that cash flows generated by a project during its useful life are immediately reinvested elsewhere.

However, the two methods make different assumptions concerning the rate of return that is earned on those cash flow.

• NPV assumes the rate of return is the discount rate

• IRR assumes the rate of return is the internal rate of return on the project.

• So, if the IRR is high, this assumption may not be realistic. It is more realistic to assume that cash can be reinvested at the discount rate - particularly if the discount rate is the company’s cost of capital. For example, by paying off the company’s creditors

• In short, when NPV and IRR do not agree, it is best to go with NPV. Of the two methods, it makes the more realistic assumption about the rate of return that can be earned on cash flows from the project.

Absolute V Percentage Figure

IRR has several weaknesses as a method of appraising capital investments. Since it is a relative measurement of investment worth, it does not measure the absolute increase in company value (and therefore shareholder wealth), which can be found using the net present value (NPV) method.

Mutually Exclusive Projects

• There is a potential conflict between IRR and NPV in the evaluation of mutually exclusive projects, where the two methods can offer conflicting advice as which of two projects is preferable.

For example, a small project may have a higher IRR but a lower NPV than a very big project.

Where there is conflict, NPV always offers the correct investment advice.

Assessment of DCF Methods of Appraisal

Discounted Cash Flow (DCF) methods of appraisal are widely used in investment analysis and project evaluation due to their numerous advantages:

• Cash Flow-Based: DCF methods rely on actual cash flows rather than accounting profits, offering a more accurate representation of a project's value.

• Consideration of Time Value of Money: DCF incorporates the principle that a dollar today is worth more than a dollar in the future, providing a realistic evaluation of cash flows.

• Comprehensive Cash Flow Analysis: All cash inflows and outflows throughout the project's life are included, ensuring a holistic appraisal.

• Flexibility in Timing of Cash Flows: DCF methods account for the exact timing of cash flows, which is critical for projects with uneven cash flow patterns.

• Universally Accepted Metrics: The Net Present Value (NPV) and Internal Rate of Return (IRR) are standard measures in financial analysis, providing clarity and consistency.

Assumptions of DCF Methods

• NPV Assumption: It is assumed that cash inflows are reinvested at the project’s discount rate, typically the cost of capital (e.g., 10%).

• IRR Assumption: It is assumed that cash inflows are reinvested at the internal rate of return (IRR).

However, these assumptions may not hold true in real-world scenarios, introducing potential inaccuracies.

Potential Problems with DCF Methods

• Forecasting Challenges: Predicting future cash flows accurately can be difficult, especially for projects with significant uncertainty.

• Capital Rationing: The NPV decision rule—accepting all projects with positive NPVs—does not apply when investment capital is limited.

• Estimating the Discount Rate: Determining the appropriate cost of capital can be complex and subjective, and it may vary over the project's lifetime.

• Changing Cost of Capital: Economic conditions or company-specific factors might cause fluctuations in the discount rate, complicating the evaluation.

• Stakeholder Conflict: The NPV method assumes the primary goal is shareholder wealth maximization, which may conflict with the interests of other stakeholders, such as employees or the community.

Conventional Cash Flows:

• Definition: A single outflow (investment) followed by multiple inflows (returns).

• Example: Initial investment of $10,000 (outflow) followed by yearly inflows of $2,000, $3,000, $5,000.

• Cash Flow Pattern: −,+,+,+

• Key Feature: Only one sign change in cash flows (negative to positive).

Non-Conventional Cash Flows:

• Definition: Cash flows with more than one sign change (outflows and inflows alternate).

• Example: $10,000 investment, followed by $5,000 inflow, then $3,000 maintenance cost (outflow), and $4,000 inflow.

• Cash Flow Pattern: −,+,−,+

• Key Feature: Multiple sign changes in cash flows.

Implications:

• Conventional Flows: Easier to analyze; results in a single IRR.

• Non-Conventional Flows: Can yield multiple IRRs, making analysis more complex.