ACCA paper f9 financial management study materials F9FM session04 d08

SESSION 04 – DISCOUNTED CASH FLOW TECHNIQUES

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

INTEREST

SIMPLE

COMPOUND

DISCOUNTING

Single sum
Annuities
Effective Annual Interest Rates
(EAIR)

“Compounding in
reverse”
Points to note

DISCOUNTED
CASH FLOW (DCF)
TECHNIQUES

Procedure
Meaning
Cash budget pro forma
Tabular layout

Annuities
Perpetuities
Definition and decision
rule
Perpetuities
Annuities
Uneven cash flows
Unconventional cash
flows

Time value of money
DCF techniques
NET PRESENT
VALUE (NPV)

INTERNAL RATE
OF RETURN (IRR)

NPV vs. IRR
Comparison

0401


SESSION 04 – DISCOUNTED CASH FLOW TECHNIQUES

1

SIMPLE INTEREST
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
(year beginning)
$100
$110
$120

1
2
3

Interest
0.1 × 100 = 10
0.1 × 100 = 10
0.1 × 100 = 10

Amount on deposit
(year end)
$110
$120
$130

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
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)
n
=
number of years for which the principal is invested

0402


SESSION 04 – DISCOUNTED CASH FLOW TECHNIQUES

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
$

1.1.X0
1.1.X1
1.1.X2

1,000
500
700

Compound
interest factor

Amount on deposit

=

Compounded
cashflow
$

_________
=

_________

0403


SESSION 04 – DISCOUNTED CASH FLOW TECHNIQUES

2.2

Annuities
Many saving schemes involve the same amount being invested annually.
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

 (1 + r )n − 1 

(i) FV = a 

r


where

a
r
n


  (1 + r )n + 1 − 1 
 − 1
(ii) FV = a  



r




= annuity (i.e. annual sum)
= interest rate (interest payable annually in arrears)
= 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

  (1.07 )6 − 1 
 − 1  = $3,000 × 6.153 = $18,460
FV = $3,000 ×  




0.07




2.3

Effective Annual Interest Rates (EAIR)
Where interest is charged on a non-annual basis it is useful to know the effective annual
rate.
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
r = rate per period (month/quarter)
n = number of periods in year
0404


SESSION 04 – DISCOUNTED CASH FLOW TECHNIQUES

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”
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 ×
where

PV
r
n

1
(1 + r ) n

= the present value of a future cash flow (CF)
= annual rate of interest/discount rate.
= number of years before the cash flow arises

In this case PV = $251.94 ×

1
= $200
(1.08) 3

The present value of $251.94 receivable in three years time is $200.

0405


SESSION 04 – DISCOUNTED CASH FLOW TECHNIQUES

3.2

Points to note
1
is known as the “simple discount factor” and gives the present value of $1
(1 + r) n
receivable in n years at a discount rate, r.
A present value table is provided in the exam
The formula for simple discount factors is provided at the top of the present value
table.
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.
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 =
(b) From the tables: r = 9%, n = 15, discount factor =

Present value =

0406

SESSION 04 – DISCOUNTED CASH FLOW TECHNIQUES

4

DISCOUNTED CASH FLOW (DCF) TECHNIQUES

4.1

Time value of money
Investors prefer to receive $1 today rather than $1 in one year.
This concept is referred to as the “time value of money”
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
DCF techniques can be used to evaluate business projects i.e. for investment appraisal.
Two methods are available:

NET PRESENT

VALUE

INTERNAL RATE
OF RETURN

0407


SESSION 04 – DISCOUNTED CASH FLOW TECHNIQUES

5

NET PRESENT VALUE (NPV)

5.1

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

5.2

Meaning
NPV shows the theoretical change in the $ value of the company due to the project.
It therefore shows the change in shareholders’ 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.

5.3

Cash budget pro forma
Time

0
$000

1
$000

2
$000

3
$000

Capital expenditure
Cash from sales
Materials
Labour
Overheads
Advertising
Grant

(X)
–

(X)
–
–
(X)
–
___

–
X
(X)
(X)
(X)
–
X
___

–
X
(X)
(X)
(X)
(X)
–
___

X
X
–
(X)
(X)

–
–
___

Net cash flow

(X)
___

X
___

X
___

X
___

1

1
1+ r

1
(1 + r ) 2

1
(1 + r )3

(X)

X

X

X

r% discount factor
Present value
NPV = X

0408


SESSION 04 – DISCOUNTED CASH FLOW TECHNIQUES

5.4

Tabular layout
Time
0
1–10
0–9
1–10
0
2
1
10

Cash flow

$000

Discount
factor
@ r%

Present
value
$000

(X)
X
(X)
(X)
(X)
(X)
X
X

1
x
x
x
x
x
x
x

(X)
X

(X)
(X)
(X)
(X)
X
X
___

CAPEX
Cash from sales
Materials
Labour and overheads
Advertising
Advertising
Grant
Scrap value

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
$

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

8% DF

PV
$

_____
NPV =

_____

0409


SESSION 04 – DISCOUNTED CASH FLOW TECHNIQUES

5.5

Annuities
An annuity is a stream of identical cash flows arising each year for a finite period of
time.
The present value of an annuity is given as
CF ×

1
1 

1 −
r  (1 + r) n 

where CF is the cash flow received each year commencing at t1.

1
1 
 is known as the “annuity factor” or “cumulative discount factor”. It is
1 −
r  (1 + r) n 
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.

Illustration 6
Calculate the present value of $1,000 receivable each year for 3 years if interest
rates are 10%.
Time

Description

t1–3

Annuity

Cash flow
$
1,000

10% Annuity factor
1 
1 
1−
= 2.486
0.1 
1.1 3 

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

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

0410

PV
$
2,486


SESSION 04 – DISCOUNTED CASH FLOW TECHNIQUES

Solution

5.6

Perpetuities
A perpetuity is a stream of identical cash flows arising each year to infinity.
As n → ∞
(1 + r)n → ∞
1
→0
(1 + r) n
1
1
1 −

r
(1 + r ) n

 1
→


r


1
is known as the “perpetuity factor”.
r
The present value of a perpetuity is given as CF ×

1
r

where CF is the cash flow received each year.
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

t1–∞

Perpetuity

Cash flow
$

1,000

10% Annuity factor
1
= 10
01
.

PV
$
10,000

0411


SESSION 04 – DISCOUNTED CASH FLOW TECHNIQUES

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

IRR is the discount rate where NPV = 0
IRR represents the average annual % return from a project.
It therefore shows the highest finance cost that can be accepted for the project.
If IRR > cost of capital, accept project.
If IRR < cost of capital, reject project.

6.2

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

Annual cash inflows
× 100%
Initial investment

Illustration 8
An investment of $1,000 gives income of $140 per annum indefinitely, the
return on the investment is given by
IRR = 140/1000 × 100% = 14%

0412


SESSION 04 – DISCOUNTED CASH FLOW TECHNIQUES

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
To give an NPV of zero, the present value of the cash inflows must equal the initial cash
outflow.
i.e. annual ash inflow × Annuity factor = Cash outflow
Annuity factor =

Cash outflow
Cash inflow

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
0
1-4

Description
Initial investment
Annuity

NPV

AF1-4 years =

CF
(6,340)
2,000

DF
1
AF1-4 years

PV
(6,340)
6,340
_____
Nil
_____

6 ,340
= 3.17
2 ,000

From the annuity table, the rate with a four year annuity factor closest to 3.17 is 10% and this
is therefore the approximate IRR for this investment.

0413


SESSION 04 – DISCOUNTED CASH FLOW TECHNIQUES

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
0
1-15

Description
Investment
Annuity

Cash flow
$

Discount factor

(10,000)
1,000

PV
$
______
______

6.4

Uneven cash flows

Method
Calculate the NPV of the project at a chosen discount rate.
If NPV is positive, recalculate NPV at a higher discount rate (i.e. to get closer to IRR).
If NPV is negative, recalculate at a lower discount rate.
The IRR can be estimated using the formula:
IRR ~ A +

Where

A
B
NA
NB

NA
(B − A)
NA − NB

=
=
=
=

Lower discount rate
Higher discount rate
NPV at rate A

NPV at rate B

This method is known as “linear interpolation”.

0414


SESSION 04 – DISCOUNTED CASH FLOW TECHNIQUES

Illustration 10
The NPVs of a project with uneven cash flows are as follows.

Discount rate

NPV
£

10%
20%

64,237
(5,213)

Estimate the IRR of the investment.

Solution
IRR ~ A +

NA
(B – A)

NA − NB

IRR ~ 10% +

64 ,237
(20 – 10)%
64 ,237 − ( −5,213)

IRR ~ 19%

Graphically
NPV

IRR using formula
(interpolated)

NA

A
NB
Actual IRR

B

Discount rate
Actual NPV as
discount rate varies

0415

SESSION 04 – DISCOUNTED CASH FLOW TECHNIQUES

Example 9
An investment opportunity with uneven cash flows has the following net
present values
$
At 10%
At 15%

Required:
Estimate the IRR of the investment.

Solution
Formula
IRR ~ A +

NA
(B – A)
NA − NB

IRR ~

Graphically

0416

71,530
4,370

SESSION 04 – DISCOUNTED CASH FLOW TECHNIQUES

6.5

Unconventional cash flows
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

Actual NPV as
discount rate varies

IRR2

IRR1

Discount rate

Actual IRR

The project appears to have two different IRR’s – in this case IRR is not a reliable
method of decision making.
However NPV is reliable, even for unconventional projects.

7

NPV vs. IRR

7.1

Comparison
NPV

IRR

An absolute measure ($)

A relative measure (%)

If NPV ≥ 0 ,accept

If IRR ≥ target %, accept

If NPV ≤ 0, reject

If IRR ≤ target %, reject

Shows $ change in value of
company/wealth of shareholders

Does not show absolute change in
wealth

A unique solution i.e. a project has only
one NPV

May be a multiple solution

Always reliable for decision making

Not always reliable

0417


SESSION 04 – DISCOUNTED CASH FLOW TECHNIQUES

Key points
Discounted cash flow techniques are arguably the most important
methods used in financial management.
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.
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:
explain the difference between simple and compound interest rate and
calculate future values;
calculate future values including the application of annuity formulae;
calculate effective interest rates;
explain what is meant by discounting and calculate present values;
apply discounting principles to calculate the net present value of an investment project
and interpret the results;
calculate present values including the application of annuity and perpetuity formulae;

explain what is meant by, and estimate the internal rate of return, using a graphical and
interpolation approach, and interpret the results;
identify and discuss the situation where there is conflict between these two methods of
investment appraisal;
compare NPV and IRR as decision making tools.

0418


SESSION 04 – DISCOUNTED CASH FLOW TECHNIQUES

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

1.1.X0
1.1.X1
1.1.X2

Amount
invested
$

×

1,000
500
700

Compound
interest factor

=

(1 + 0.05)3
(1 + 0.05)2
(1 + 0.05)1
Amount on deposit

Compounded
cashflow
$
1,157.63
551.25

735.00
_________

=

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
(b) From the tables: r = 9%, n = 15, discount factor = 0.275

Present value = 30,000 × 0.275 = $8,250

0419


SESSION 04 – DISCOUNTED CASH FLOW TECHNIQUES

Solution 4 — Net present value
Time

Description

Cash flow
$

8% DF

PV
$

0

Investment

(10,000)

1

(10,000)

1

Net inflow

3,000

1
(1.08)

2,778

2

Net inflow

3,000

1
(1.08) 2

2,572

3

Net inflow

3,000

1
(1.08) 3

2,381

4

Net inflow

5,000

1
(1.08) 4

3,675

5

Net inflow

1,000

1
(1.08) 5

681

NPV =

_____
2,087
_____

Solution 5 — Annuity
Time

Description

t3-12

Annuity

Cash flow
$

7% Annuity factor

PV

$

2,000

6.135 (W)

12,270

WORKING
Cdf3-12 @ 7%

=

CDF1-12 @ 7% – CDF1-2 @ 7%

=

7.943 – 1.808 (per tables)

=

6.135

Solution 6 — Perpetuity

0420

Time

Description

Cash flow
$

7% Annuity factor

PV
$

t10-∞

Perpetuity

2,000

7.771 (W)

15,542


SESSION 04 – DISCOUNTED CASH FLOW TECHNIQUES
WORKING
=

Cdf10-∞ @ 7%

=

CDF1-∞ @ 7% - CDF1-9 @ 7%
1

– 6.515 (per tables)
0.07
14.286 – 6.515

=

7.771

=

Solution 7 — IRR (perpetuity)
IRR =

2,400
× 100 = 16%
15,000

Solution 8 — IRR (annuity)
Time

Description

0
1-15

Investment
Annuity

Cash flow
$

(10,000)
1,000

Discount factor

PV
$

1
Cdf1-15 = 10 (βal)

(10,000)
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.
IRR ~ A +

NA
(B – A)
NA − NB

71,530


IRR ~ 10 + 
 (15 – 10)
 71,530 − 4 ,370 
IRR ~ 10 + 5.325
say 15.4% (rounded up)

0421


SESSION 04 – DISCOUNTED CASH FLOW TECHNIQUES

Graphically
NPV
£

Actual
NPV

71,530
Actual
IRR
4,370
10

15
IRR using

formula
(extrapolated)

0422

Discount rate
(%)