ACCA paper f9 financial management study materials F9FM session04 d08

Illustration 1 If $100 is invested at 10% per annum pa simple interest: Year Amount on deposit Interest Amount on deposit Ü A single principal sum, P invested for n years at an annual

NET PRESENT VALUE (NPV)

1 SIMPLE INTEREST

Ü 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

Ü 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

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

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

2.2 Annuities

Ü Many saving schemes involve the same amount being invested annually

Ü 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

r

r n

a

where a = annuity (i.e annual sum)

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?

.0

107

= $3,000 × 6.153 = $18,460

2.3 Effective Annual Interest Rates (EAIR)

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

rate

Ü 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

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%

3.1 “Compounding in reverse”

Ü 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?

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

Ü A present value table is provided in the exam

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

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

Ü t1 is defined as a point in time exactly one year after t0

Ü 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

4 DISCOUNTED CASH FLOW (DCF) TECHNIQUES

4.1 Time value of money

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

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

Ü 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

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

Ü Two methods are available:

NET PRESENT

5 NET PRESENT VALUE (NPV)

5.1 Procedure

Ü Forecast the relevant cash flows from the project

Ü 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

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

Ü Sum present values to give the NPV of the project

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

5.2 Meaning

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

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

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

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

5.3 Cash budget pro forma

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

1 r

1

r

1 is known as the “annuity factor” or “cumulative discount factor” It is

simply the sum of a geometric progression

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

r

−n

Ü Annuity factor tables are also provided in the exam

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

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%

(

11

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

Ü 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%

Example 6

Calculate the present value of $2,000 receivable in perpetuity commencing in

10 years time Assume interest at 7%

Solution

6.1 Definition and decision rule

Ü IRR is the discount rate where NPV = 0

Ü IRR represents the average annual % return from a project

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

Ü If IRR > cost of capital, accept project

Ü If IRR < cost of capital, reject project

6.2 Perpetuities

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

IRR =

investmentInitial

inflowscash

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%

outflowCash

Ü 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

340,

6 = 3.17

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

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

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

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

Ü The IRR can be estimated using the formula:

A)(BNN

NA

~

IRR

B A

−+

Where A = Lower discount rate

B = Higher discount rate

NA = NPV at rate A

NB = NPV at rate B

Ü This method is known as “linear interpolation”

NN

N

− (B – A)

IRR ~ 10% +

)213,5(237,64

237,64

NN

N

− (B – A) IRR ~

Graphically

6.5 Unconventional cash flows

Ü 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

IRR2

Ü The project appears to have two different IRR’s – in this case IRR is not a reliable

method of decision making

Ü However NPV is reliable, even for unconventional projects

7.1 Comparison

Ü An absolute measure ($) Ü A relative measure (%)

Ü If NPV ≥ 0 ,accept Ü If IRR ≥ target %, accept

Ü Always reliable for decision making

Ü If IRR ≤ target %, reject

Ü Does not show absolute change in wealth

Ü May be a multiple solution

Ü Not always reliable

Key points

ÐDiscounted cash flow techniques are arguably the most important

methods used in financial management

ÐDCF 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

ÐNPV 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:

Ü explain the difference between simple and compound interest rate and

calculate future values;

Ü calculate future values including the application of annuity formulae;

Ü calculate effective interest rates;

Ü explain what is meant by discounting and calculate present values;

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

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

Ü explain what is meant by, and estimate the internal rate of return, using a graphical and interpolation approach, and interpret the results;

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

Ü compare NPV and IRR as decision making tools

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

invested × interest factor = cashflow

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

Solution 4 — Net present value

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WORKING

Cdf10-∞ @ 7% = CDF1-∞ @ 7% - CDF1-9 @ 7%

=

07

Solution 8 — IRR (annuity)

Time Description Cash flow Discount factor PV

Solution 9 — IRR (uneven cash flows)

A

NN

,71

530,

71 (15 – 10)

IRR ~ 10 + 5.325

say 15.4% (rounded up)

IRR usingformula(extrapolated)

£

Actual

NPV