ACCA Paper F9 Financial Management Study Materials F9FM Session04 d08

OVERVIEW

Objective

¾

To apply the time value of money to investment decisions.

INTEREST

DISCOUNTING SIMPLE

INTERNAL RATE OF RETURN (IRR) ¾ Single sum

¾ Annuities

¾ Effective Annual Interest Rates (EAIR)

¾ Procedure ¾ Meaning

¾ Cash budget pro forma ¾ Tabular layout

¾ Annuities ¾ Perpetuities

DISCOUNTED CASH FLOW (DCF)

TECHNIQUES COMPOUND

NET PRESENT VALUE (NPV)

¾ Definition and decision rule

¾ Perpetuities ¾ Annuities

¾ Uneven cash flows ¾ Unconventional cash

flows

NPV vs. IRR

1

SIMPLE INTEREST

¾

Interest accrues only on the initial amount invested.

Illustration 1

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

Year Amount on deposit Interest Amount on deposit

(year beginning) (year end)

1 $100 0.1 × 100 = 10 $110

2 $110 0.1 × 100 = 10 $120

3 $120 0.1 × 100 = 10 $130

¾

A single principal sum, P invested for n years at an annual rate of interest, r (as a decimal) will amount to a future value FV.

Where FV = P (1 + nr)

2

COMPOUND INTEREST

¾

Interest is reinvested alongside the principal.

2.1

Single sum

Illustration 2

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 In 3 years’ time, $121 would have grown by 10% to $133.10 Or

FV = P (1 + r) n where

P = initial principal

r = annual rate of interest (as a decimal)

Example 1

$500 is invested in a fund on 1.1.X1. 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) Simple interest FV = P (1 + nr) FV =

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

Example 2

$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. How much will be on deposit by 31.12.X2?

Solution

Date Amount

invested × Compound interest factor = Compounded cashflow

$ $

1.1.X0 1,000

1.1.X1 500

1.1.X2 700

_________

Amount on deposit =

2.2

Annuities

¾

Many saving schemes involve the same amount being invested annually.

¾

There are two formulae for the future value of an annuity. Which to use depends on whether the investment is made at the end of each year or at the start of each year. (i) first sum paid/received at the end of each year

(ii) first sum paid/received at the beginning of each year

(i) FV =

(

)

_

r = interest rate (interest payable annually in arrears) n = number of years annuity is paid/invested

Commentary

These formula will not be provided in the examination

Illustration 3

Andrew invests $3,000 at the start of each year in a high interest account offering 7% pa. How much will he have to spend after a fixed 5 year term?

Solution

2.3

Effective Annual Interest Rates (EAIR)

¾

Where interest is charged on a non-annual basis it is useful to know the effective annual rate.

¾

Foe 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.

Formula

1 + R = (1 + r) n R = annual rate

Illustration 4

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

Using FV = P (1 + r)n ⇒ = £100 × (1.02)12

= £100 × 1.2682 * = £126.82 EAIR is 26.82%

3

DISCOUNTING

3.1

“Compounding in reverse”

¾

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 5

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

or alternatively PV = CF × 1

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 PV = $251.94 × ₃ (1.08)

1 _{= $200}

3.2

Points to note

¾

_n

r) 1 (

1

+ is known as the “simple discount factor” and gives the present value of $1 receivable in n years at a discount rate, r.

¾

A present value table is provided in the exam

¾

The formula for simple discount factors is provided at the top of the present value table.

¾

For a cash flow arising now (at t0) the discount factor will always be 1.

¾

t1 is defined as apoint in timeexactly one year after t0.

¾

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

Example 3

Find the present value of

(a) 250 received or paid in 5 years time, r = 6% pa (b) 30,000 received or paid in 15 years time, r = 9% pa.

Solution

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

4

DISCOUNTED CASH FLOW (DCF) TECHNIQUES

4.1

Time value of money

¾

Investors prefer to receive $1 today rather than $1 in one year.

¾

This concept is referred to as the “time value of money”

¾

There are several possible causes:

‰ 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

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

4.2

DCF techniques

¾

DCF techniques can be used to evaluate business projects i.e. for investment appraisal.

¾

Two methods are available:

NET PRESENT

5

NET PRESENT VALUE (NPV)

5.1

Procedure

¾

Forecast the relevant cash flows from the project

¾

Estimate the required return of investors i.e. the discount rate. The required return of investors represents the company’s cost of finance, also referred to as its cost of capital.

¾

Discount each cash flow (receipt or payment) to its present value (PV).

¾

Sum present values to give the NPV of the project.

¾

If NPV is positive then accept the project as it provides a higher return than required by investors.

5.2

Meaning

¾

NPV shows the theoretical change in the $ value of the company due to the project.

¾

It therefore shows the change in shareholders’ wealth due to the project.

¾

The assumed key objective of financial management is to maximise shareholder wealth.

¾

Therefore NPV must be considered the key technique in business decision making.

5.3

Cash budget pro forma

Time 0 1 2 3

$000 $000 $000 $000

Capital expenditure (X) – – X

Cash from sales – X X X

Materials (X) (X) (X) –

Labour – (X) (X) (X)

Overheads – (X) (X) (X)

Advertising (X) – (X) –

Grant _{___} – _{___} X _{___} – _{___} –

Net cash flow _{___} (X) _{___} X _{___} X _{___} X

r% discount factor 1 1

1+r

1 1 2

( +r)

1 1 3

( +r)

Present value (X) X X X

5.4

Tabular layout

Discount Present

Time Cash flow factor value

$000 @ r% $000

0 CAPEX (X) 1 (X)

1–10 Cash from sales X x X

0–9 Materials (X) x (X)

1–10 Labour and overheads (X) x (X)

0 Advertising (X) x (X)

2 Advertising (X) x (X)

1 Grant X x X

10 Scrap value X x _{___} X

Net present value _{___} X

Example 4

Elgar has $10,000 to invest for a five-year period. He could deposit it in a bank earning 8% pa 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.

Required:

Calculate the net present value of the project.

Solution

Time Description Cash flow 8% DF PV

$ $

0 Investment (10,000)

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 _{_____}

5.5

Annuities

¾

An annuity is a stream of identical cash flows arising each year for a finite period of time.

¾

The present value of an annuity is given as

CF ×

1

_{is known as the “annuity factor” or “cumulative discount factor”. It is}

simply the sum of a geometric progression.

¾

The formula is given in the exam as 1 - (1+ r)

r

−n

¾

Annuity factor tables are also provided in the exam

¾

Remember that the formula and tables are based on the assumption that the cash flow starts after one year.

Illustration 6

Calculate the present value of $1,000 receivable each year for 3 years if interest rates are 10%.

Time Description Cash flow 10% Annuity factor PV

$ $

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

Example 5

Solution

5.6

Perpetuities

¾

A perpetuity is a stream of identical cash flows arising each year to infinity.

¾

As n → ∞

1 is known as the “perpetuity factor”.

The present value of a perpetuity is given as CF ×

r 1

where CF is the cash flow received each year.

¾

The formula is based on the assumption that the cash flow starts after one year.

Illustration 7

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 1,000 1

Example 6

Calculate the present value of $2,000 receivable in perpetuity commencing in 10 years time. Assume interest at 7%.

Solution

6

INTERNAL RATE OF RETURN (IRR)

6.1

Definition and decision rule

¾

IRR is the discount rate where NPV = 0

¾

IRR represents the average annual % return from a project.

¾

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

¾

If IRR > cost of capital, accept project.

¾

If IRR < cost of capital, reject project.

6.2

Perpetuities

¾

If a project has equal annual cash flows receivable in perpetuity then

IRR =

investment Initial

inflows cash

Annual _{× 100%}

Illustration 8

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

Example 7

An investment of $15,000 now will provide $2,400 each year to perpetuity.

Required:

Calculate the return inherent in the investment.

Solution

6.3

Annuities

¾

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

¾

i.e. annual ash inflow × Annuity factor = Cash outflow

Annuity factor =

inflow Cash

outflow Cash

¾

Once the annuity factor is known the discount rate can be established from the appropriate table.

Illustration 9

An investment of $6,340 will yield an income of $2,000 for four years. Calculate the internal rate of return of the investment.

Solution

Year Description CF DF PV

0 Initial investment (6,340) 1 (6,340) 1-4 Annuity 2,000 AF1-4 years 6,340 _{_____}

NPV Nil _{_____}

AF1-4 years = 000 , 2

340 ,

6 _{= 3.17}

Example 8

An immediate investment of $10,000 will give an annuity of $1,000 for the next 15 years.

Required:

Calculate the internal rate of return of the investment.

Solution

Time Description Cash flow Discount factor PV

$ $

0 Investment (10,000)

1-15 Annuity 1,000 _{______}

______

6.4

Uneven cash flows

Method

¾

Calculate the NPV of the project at a chosen discount rate.

¾

If NPV is positive, recalculate NPV at a higher discount rate (i.e. to get closer to IRR).

¾

If NPV is negative, recalculate at a lower discount rate.

¾

The IRR can be estimated using the formula:

A) (B N N

N A

~ IRR

B A

A ₋

− +

Where A = Lower discount rate B = Higher discount rate NA = NPV at rate A NB = NPV at rate B

Illustration 10

The NPVs of a project with uneven cash flows are as follows.

Discount rate NPV

£

10% 64,237

20% (5,213)

Estimate the IRR of the investment.

Solution

IRR using formula (interpolated)

B A

Actual IRR NA

NB Actual NPV as

Example 9

An investment opportunity with uneven cash flows has the following net present values

$

At 10% 71,530

At 15% 4,370

Required:

Estimate the IRR of the investment.

Solution

Formula

IRR ~ A +

B A

A N N

N

− (B – A) IRR ~

6.5

Unconventional cash flows

¾

If there are cash outflows, followed by inflows are then more outflows (e.g. suppose at the end of the project a site had to be decontaminated), the situation of “multiple yields” may arise – i.e. more than one IRR.

NPV

Discount rate

IRR1

Actual IRR

Actual NPV as discount rate varies

IRR2

¾

The project appears to have two different IRR’s – in this case IRR is not a reliable method of decision making.

¾

However NPV is reliable, even for unconventional projects.

7

NPV

vs.

IRR

7.1

Comparison

NPV

IRR

¾

An absolute measure ($)

¾

A relative measure (%)

¾

If NPV ≥ 0 ,accept

¾

If IRR ≥ target %, accept

¾

If NPV ≤ 0, reject

¾

Shows $ change in value of company/wealth of shareholders

¾

A unique solution i.e. a project has only

one NPV

¾

Always reliable for decision making

¾

If IRR ≤ target %, reject

¾

Does not show absolute change in wealth

¾

May be a multiple solution

Key points

³

Discounted cash flow techniques are arguably the most important methods used in financial management.

³

DCF techniques have two major advantages (i) they focus on cash flow, which is more relevant than the accounting concept of profit (ii) they take into account the time value of money.

³

NPV must be considered a superior decision-making technique to IRR as it is an absolute measure which tells management the change in

shareholders’ wealth expected from a project.

FOCUS

You should now be able to:

¾

explain the difference between simple and compound interest rate and calculate future values;

¾

calculate future values including the application of annuity formulae;

¾

calculate effective interest rates;

¾

explain what is meant by discounting and calculate present values;

¾

apply discounting principles to calculate the net present value of an investment project and interpret the results;

¾

calculate present values including the application of annuity and perpetuity formulae;

¾

explain what is meant by, and estimate the internal rate of return, using a graphical and

interpolation approach, and interpret the results;

¾

identify and discuss the situation where there is conflict between these two methods of investment appraisal;

EXAMPLE SOLUTION

Solution 1 — 7% simple and compound interest

The $500 is invested for a total of 4 years

(a) Simple interest FV = P (1 + nr)

FV = 500 (1 + 4 × 0.07) = 500 × 1.28 = $640 (b) Compound interest FV = P (1 + r)n

FV = 500 (1 + 0.07)4 = 500 _× 1.3108 = $655.40

Solution 2 — 5% compound interest

Date Amount Compound Compounded

invested × interest factor = cashflow

$ $

1.1.X0 1,000 (1 + 0.05)3 _1,157.63

1.1.X1 500 (1 + 0.05)2 551.25

1.1.X2 700 (1 + 0.05)1 _735.00

_________ Amount on deposit = 2,443.88 _________

Solution 3 — Present value

(a) From the tables: r = 6%, n = 5, discount factor = 0.747 Present value = 250 × 0.747 = $186.75

Solution 4 — Net present value

Solution 5 — Annuity

Time Description Cash flow 7% Annuity factor PV

$ $

Solution 6 — Perpetuity

Time Description Cash flow 7% Annuity factor PV

$ $

WORKING

Solution 7 — IRR (perpetuity)

IRR =

Time Description Cash flow Discount factor PV

$ $

0 Investment (10,000) 1 (10,000)

1-15 Annuity 1,000 Cdf1-15 = 10 (βal) _{______} 10,000

Nil ______

From the annuity table the rate with a 15 year annuity factor of 10 lies between 5% and 6%. Thus if $10,000 could be otherwise invested for a return of 6% or more, this annuity is not worthwhile.

Solution 9 — IRR (uneven cash flows)

Formula

Commentary

The formula always works but take care with + and – signs.

Graphically

4,370 71,530

10 15

NPV

Discount rate (%) Actual

IRR

IRR using formula (extrapolated) £