F9-04 Discounted Cash Flow Techniques PDF

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Session 4

Discounted Cash Flow Techniques FOCUS This session covers the following content from the ACCA Study Guide. D. Investment Appraisal 1. Investment appraisal process techniques d) Calculate net present value and discuss its usefulness as an investment appraisal method. e) Calculate internal rate of return and discuss its usefulness as an investment appraisal method. f) Discuss the superiority of discounted cash flow (DCF) methods over nonDCF methods. g) Discuss the relative merits of NPV and IRR.

Session 4 Guidance Work through Illustrations 1–4 and Examples 1–3. Read section 3 on discounted cash flow (DCF) techniques. Understand the steps required to find a project's net present value (s.4.1) and how to interpret a project's NPV (s.4.2). Attempt Example 4 on the basics of NPV calculations.

(continued on next page) F9 Financial Management

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VISUAL OVERVIEW Objective: To apply the time value of money to investment decisions.

INTEREST • Simple Interest • Compound Interest • Effective Annual Interest Rates

DISCOUNTING • Compounding in Reverse • Simple Discount Factor

DISCOUNTED CASH FLOW (DCF) TECHNIQUES • Time Value of Money • Investment Appraisal • Limitations NET PRESENT VALUE (NPV)

INTERNAL RATE OF RETURN (IRR)

• Procedure

• Perpetuities

• Meaning

• Annuities

• Cash Budget Pro Forma

• Uneven Cash Flows • Unconventional Cash Flows

• Tabular Layout • Annuities • Perpetuities

COMPARISON OF NPV AND IRR

Session 4 Guidance Work through sections 4.5 and 4.6 on finding the NPV of annuities and perpetuities, including Illustrations 5–6 and Examples 5–6. Read section 6 on the internal rate of return (IRR), including applying IRR to annuities, perpetuities, and projects with uneven cash flows. Understand the issue that arises when trying to find the IRR for a project with unconventional cash flows (s.5.4). Use section 6 to review your understanding of both NPV and IRR.

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4-1

Session 4 • Discounted Cash Flow Techniques

1

Interest

1.1

Simple Interest

F9 Financial Management

Interest accrues only on the initial amount invested.

Illustration 1 Simple Interest If $100 is invested at 10% per annum (pa) simple interest:

Year

Amount on deposit (year beginning)

Interest

Amount on deposit (year end)

1

$100

0.1 x $100 = $10

$110

2

$110

0.1 x $100 = $10

$120

3

$120

0.1 x $100 = $10

$130

For simple interest, a single principal sum, P, invested for n years at an annual rate of interest, r (expressed as a decimal), will amount to a future value (FV). Where: FV = P (1 + nr)

1.2

Compound Interest

Interest is reinvested alongside the principal.

When a single sum is invested to earn compound interest, the formula to find the FV is: FV = P (1 + r)n where: P = initial principal r = annual rate of interest (as a decimal) n = number of years for which the principal is invested

4-2

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Illustration 2

Session 4 • Discounted Cash Flow Techniques

Compound Inter

If Zarosa placed $100 in the bank today (t0) earning 10% interest per annum, what would this sum amount to in three years' time?

Solution In 1 year's time, $100 would have increased by 10% to $110 In 2 years' time, $110 would have grown by 10% to $121

*Conversely, the present value of $133.10 to be received in three years' time is $100.

In 3 years' time, $121 would have grown by 10% to $133.10*

Example 1 7% Simple and Compound Intere $500 is invested in a fund on 1.1.X1.

Required: Calculate the amount on deposit by 31.12.X4 if the interest rate is: (a)

7% per annum simple

(b)

7% per annum compound.

Solution The $500 is invested for a total of 4 years (a)

7% Simple interest

FV = P (1 + nr) FV =

(b) 7% Compound interest FV = P (1 + r)n FV =

Example 2 5% Compound Interest $1,000 is invested in a fund earning 5% per annum on 1.1.X0. $500 is added to this fund on 1.1.X1 and a further $700 is added on 1.1.X2.

Required: Calculate the amount on deposit by 31.12.X2 if the interest rate is 5% per annum compound.

Solution Date

Amount Invested

×

Compound Interest Factor

=

$

Compounded Cash Flow $

1.1.X0

1,000

×

1.1.X1

500

×

1.1.X2

700

×

Amount on deposit

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=

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Session 4 • Discounted Cash Flow Techniques

1.3

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Effective Annual Interest Rates (EAIR)

< Where interest is charged on a non-annual basis, it is useful to know the effective annual interest rate (EAIR).

< For example, interest on bank overdrafts (and credit cards) is often charged on a monthly basis. To compare the cost of finance to other sources, it is necessary to know the EAIR (also called the annual percentage rate or APR), which is calculated as follows: 1 + R = (1 + r)n where:

R = annual rate (EAIR) r = interest rate per period (month/quarter) n = number of periods in year

Illustration 3 EAIR Sarah borrows $100 at a cost of 2% per month. How much (principal + interest) will be owed after a year?

Solution Using FV

= P (1 + r)n = $100 x (1.02)12 = $100 x 1.2682 = $126.82

EAIR is 26.82%, which can also be found using the formula: 1+R R

4-4

= (1 + r)n = (1 + 0.02)12 = 1.2682 = 0.2682 = 26.82%

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F9 Financial Management

Session 4 • Discounted Cash Flow Techniques

2

Discounting

2.1

Compounding in Reverse

Discounting calculates the sum which must be invested now (at a fixed interest rate) in order to receive a given sum in the future.

Illustration 4 Discounting If Zarosa needed to receive $251.94 in three years' time (t3), what sum would she have to invest today (t0) at an interest rate of 8% per annum?

Solution The formula for compounding is: FV = P (1 + r)n Rearranging this: P = FV × (1 +1 r)n 1 Alternatively, PV = CF × (1 + r)n where: PV = the present value of a future cash flow (CF) r

= annual rate of interest/discount rate

n

= number of years before the cash flow arises

In this case: 1 = $200 PV = $251.94 × (1.08)3 The present value of $251.94 receivable in three years' time is $200.

2.2

Simple Discount Factor

< The formula

1 (1 + r)n

is known as the "simple discount factor" and it gives the present value of $1 receivable at the end of n years at a discount rate, r. < For a cash flow arising now (at t0) the discount factor will always be 1. < t1 is defined as a point in time exactly one year after t0.

Example 3 Present Value

The formula for simple discount factors is provided at the top of the present value table in the exam as (1 + r)-n.

See Tables in the Formulae and Tables section of the Introduction. Find the present value of: (a) $250 received or paid in 5 years' time, r = 6% per year. (b) $30,000 received or paid in 15 years' time, r = 9% per year.

Solution (a) From the tables: r = 6%, n = 5, discount factor =

In the exam, always assume that cash flows arise at the end of the year to which they relate (unless told otherwise).

Present value = (b) From the tables: r = 9%, n = 15, discount factor = Present value =

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Session 4 • Discounted Cash Flow Techniques

F9 Financial Management

3

Discounted Cash Flow (DCF) Techniques

3.1

Time Value of Money

The "time value of money concept" is based on the assumption that investors prefer to receive $1 today rather than $1 in one year. There are several possible reasons that underlie this assumption:

< Liquidity preference: if money is received today it can either be spent or reinvested to earn more in future. Hence, investors have a preference for having cash/liquidity today. < Risk: cash received today is safe, future cash receipts may be uncertain. < Inflation: cash today can be spent at today's prices but the value of future cash flows may be eroded by inflation.

3.2

Discounted cash flow techniques take account of the time value of money by restating each future cash flow in terms of its equivalent value today.

Investment Appraisal

Discounted cash flow (DCF) techniques can be used to evaluate business projects (i.e. for investment appraisal). Two methods are available: 1. Net present value (NPV); and 2. Internal rate of return (IRR).

3.3

Limitations of DCF Techniques

Despite the theoretical superiority of DCF techniques, it appears that in practice many company managers prefer to use nonDCF methods of appraisal such as payback or return on capital employed (see Session 3). Possible reasons for this reluctance to use DCF methods include: The potentially complex and time-consuming process of calculating NPV and/or IRR. Difficulty in explaining DCF techniques to non-financial managers. Complexity of estimating an appropriate discount rate, particularly for unquoted firms. Managers may feel little connection between DCF techniques and their own reported performance and bonus systems.

4 4.1

Net Present Value (NPV) Procedure

There are several steps required to use NPV to determine whether a project should be undertaken. 1. Forecast the relevant cash flows from the project. 2. Estimate the required return of investors (i.e. the discount rate). For a project that has risk equal to the company's risk, the required return of investors represents the company's overall cost of finance (also referred to as its cost of capital).

4-6

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F9 Financial Management

Session 4 • Discounted Cash Flow Techniques

3. Discount each cash flow (receipt or payment) to its present value (PV). 4. Sum present values to give the NPV of the project. 5. If NPV is positive then accept the project as it provides a higher return than required by investors.

4.2

Meaning

A project's NPV shows the theoretical change in the dollar value of the company due to the project. It therefore shows the change in shareholder wealth due to the project.

< The assumed key objective of financial management is to maximise shareholder wealth. < Therefore, NPV must be considered the key technique in business decision-making.

4.3

Cash Budget Pro Forma

Time

0

1

2

3

$000

$000

$000

$000

(x)

–

–

x

–

x

x

x

(x)

(x)

(x)

–

Labour

–

(x)

(x)

(x)

Overheads

–

(x)

(x)

(x)

Advertising

(x)

–

(x)

–

–

x

–

–

(x)

x

x

x

1

1 1+ r

1 (1 + r) 2

1 (1 + r)3

(x)

x

x

x

Capital expenditure Cash from sales Materials

Grant Net cash flow r% discount factor Present value NPV = ∑X = (sum of the Xs)

4.4

Tabular Layout

Time

0

Capital expenditure

1–10 Cash from sales 0–9

Materials

1–10 Labour and overheads

Cash Flow

Discount Factor

Present Value

$000

@ r%

$000

(x)

1

(x)

x

x

x

(x)

x

(x)

(x)

x

(x)

0

Advertising

(x)

x

(x)

2

Advertising

(x)

x

(x)

1

Grant

x

x

x

Scrap value

x

x

x

10

Net present value

x

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Session 4 • Discounted Cash Flow Techniques

F9 Financial Management

Example 4 Net Present Value Elgar has $10,000 to invest for a five-year period. He could deposit it in a bank earning 8% per year compound interest. He has been offered an alternative: investment in a low-risk project that is expected to produce net cash inflows of $3,000 for each of the first three years, $5,000 in the fourth year and $1,000 in the fifth year.

Required: Calculate the net present value of the project.

Solution Time

Description Cash Flow

0

Investment

1

Net inflow

3,000

2

Net inflow

3,000

3

Net inflow

3,000

4

Net inflow

5,000

5

Net inflow

1,000

8% DF

PV

$

$

(10,000)

NPV =

4.5

Annuities

Annuity—a stream of identical cash flows arising each year for a finite period of time.

< The present value of an annuity is given as: 1 1 − 1   CF × r  (1+r)n   < Where: CF is the cash flow received each year commencing at t1. 1  1 < r 1 − (1+r)n  is known as the "annuity factor" or   "cumulative discount factor". It is simply the sum of a geometric progression.

< The formula is given in the exam as

1 – (1 + r)–n r

< Annuity factor tables are also provided in the exam. < Remember that the formula and tables are based on the assumption that the cash flow starts after one year.

4-8

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F9 Financial Management

Session 4 • Discounted Cash Flow Techniques

Illustration 5 PV of Annuity* Calculate the present value of $1,000 receivable each year for three years if interest rates are 10%. Time

Description

Cash flow

10% Annuity Factor

PV

$ t1–3

Annuity

*An annuity received for the next three years is written as t1–3.

$ 1  1  1 −  = 2.487 0.1  1.13 

1,000

2,487

Example 5 Annuity Calculate the present value of $2,000 receivable for each of 10 years commencing three years from now. Assume interest at 7%.

Solution

4.6

Perpetuities

Perpetuity—a stream of identical cash flows arising each year to infinity. , where CF The present value of a perpetuity is given as CF × 1 r is the cash flow received each year.

< The

1 input is the result of the following: r

As n  ∞ (1 + r)n  ∞ 1 (1 + r)n  0 1  1  1 −  1 r  ( − r) n 



1 (1-0) r



1 r

1 is known as the "perpetuity factor". r

The present value of a perpetuity formula is based on the assumption that the cash flow starts after one year.

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Session 4 • Discounted Cash Flow Techniques

F9 Financial Management

Illustration 6 PV of Perpetuit Calculate the present value of $1,000 receivable each year in perpetuity if interest rates are 10%.

Solution Time

Description

Cash Flow

10% Annuity Factor

PV

$ t1–∞

Perpetuity

$ 1

1,000

0.1

= 10

10,000

Example 6 Perpetuity See Tables in the Formulae and Tables section of the Introduction. Calculate the present value of $2,000 receivable in perpetuity commencing in 10 years' time. Assume interest at 7%.

Solution

5

Internal Rate of Return (IRR)

Internal rate of return is the discount rate where NPV = 0. Internal rate of return represents the average annual percentage return from a project.*

*IRR is a "break-even" interest rate.

It therefore shows the highest finance cost that can be accepted for the project.

4-10

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F9 Financial Management

Session 4 • Discounted Cash Flow Techniques

The decision rule for IRR is: • If IRR > cost of capital, accept project. • If IRR < cost of capital, reject project.

Perpetuities

5.1

If a project has equal annual cash flows receivable in perpetuity, then: IRR =

Annual cash inflows Initial investment

× 100

Illustration 7 Perpetuity IRR An investment of $1,000 gives income of $140 per annum indefinitely, the return on the investment is given by

IRR =

Annual cash inflows Initial investment

=

$140 $1,000

=14%

Example 7 IRR (Perpetuity) An investment of $15,000 today will provide $2,400 each year to perpetuity.

Required: Calculate the internal rate of return inherent for the investment.

Solution IRR=

5.2

Annuities

To give an NPV of zero, the present value of the cash inflows must equal the initial cash outflow.

< That is: Annual cash inflow × Annuity factor = Cash outflow. This equation can be rearranged to solve for the ann...