Time value of money

Time is money? Why?

Discussion: Would you prefer to get the
scholarship now?
The school has decided to give you a
scholarship of $5,000 in 1 year’s time, but you
may request to be paid now. Would you prefer
to get the $5,000 now or a year later? Why?
Justify your answer with calculation.

https://www.youtube.com/watch?v=oqpqb5ljliY
Time value of money

● The time value of money means that

that a dollar received today is worth more than a dollar received
____________________________________________

● This is because a dollar received today can be invested

________ and earn
interest
________.

Today Future

Investment
Time value of money

► As money ______
grows over time, the additional amount reflects
the time value of money.
► We often use the concept of _______________
time value of money to make
financial decisions. Most calculations of _________
savings and
investment _________
returns are based on time value.

Today Future

Investment
Time value of money
Future value

Future value (FV) refers to the value of money in the
_______.
future

The concept of FV is used to estimate our future income
and help us make investment decisions.

https://www.youtube.com/watch?v=gyiiqUQgEeA (Self-learning video: simple vs compound

interest)
Time value of money
Compounding
● The process of calculating the future value over more
than one period is known as compounding.
● It is because the interest received in the first period will
become part of the principal for the second period, and
so on.
Time value of money
Compounding
● The process of calculating the future value over more
than one period is known as compounding.
● It is because the interest received in the first period will
become part of the principal for the second period, and
so on.

https://www.youtube.com/watch?v=weUPXHkf3O4
Problem Solving (Problem 1)
● Your mother has bought $4,500 Australian dollars and placed it in
the bank on a fixed deposit term of 1 year. If the deposit interest
rate is 5.19% per year, how much will she have at the maturity of
the deposit?

Problem Solving (Problem 2) – compound interest

● An investment product promises to pay an 8% return for each year,
how much will an initial investment of $100,000 become in 3 years
based on compound interest calculation?

Problem Solving (Problem 3) – compound interest

● Future Value of a Single Cash Flow Using Compound Interest

● Your aunt has made a 5-year fixed New Zealand dollar deposit
today. The principal amount is NZ$10,000 and interest rate is 6.75%
compounded yearly. How much would your aunt get back at the
maturity of her deposit?
Problem Solving (Problem 1)
● Your mother has bought $4,500 Australian dollars and placed it in
the bank on a fixed deposit term of 1 year. If the deposit interest
rate is 5.19% per year, how much will she have at the maturity of
the deposit?

 At the maturity of the deposit, your mother will have:

FV = PV  (1 + i)
= A$4,500  (1 + 0.0519)
= A$4,733.55
Problem Solving (Problem 2) – compound interest

● An investment product promises to pay an 8% return for each year,

how much will an initial investment of $100,000 become in 3 years
based on compound interest calculation?


FV = PV  (1 + i)n
= $100,000  (1 + 0.08)3
= $125,971.20
 Problem Solving (Problem 3) - compound interest
● Future Value of a Single Cash Flow Using Compound Interest

● Your aunt has made a 5-year fixed New Zealand dollar deposit
today. The principal amount is NZ$10,000 and interest rate is 6.75%
compounded yearly. How much would your aunt get back at the
maturity of her deposit?


FV = PV  (1 + i)n
= NZ$10,000  (1 + 0.0675)5
= NZ$13,862.43
 Time value of money
Calculate the future values of using (i) simple interest and
(ii) compound interest for a $1,000 principal at an interest
rate of 10% for 30 years.
Time value of money
The Difference between Simple and Compound Interest of 10% on
$1,000 for 30 Years is:
Simple
Simpleversus
versusCompound
CompoundInterest
Interest

20000
20000
Simple
Simple
Compound
Compound $17,449
15000
15000
Future Value
Future Value

10000
10000

5000
5000 $4,000
0
0
0 5 10 15 20 25 30
0 5 10 15 20 25 30
Number of Year
Number of Year
 Time value of money

Present value (PV) is the current/present value of
future money.

It is the amount needed today to grow in specific future
value in the future.

How much would you be willing to lend to Thomson with

a current interest rate of 3%, if he promises to pay you
$1,000 in (i) 1 year time? What if Thomson pays you (ii)
3 years later?

https://www.ifec.org.hk/web/en/tools/calculator-widget/index.page#how-much-will-i-get-in-the-future

https://www.youtube.com/watch?v=G-xFadv9KE4&t=1s
Present value - Discounting
 A given future value is used to find the present value. This process
is known as discounting.
 If the number of periods is more than one, the process is called
discounting.
Present Value of a Single Cash Flow
How much would you be willing to lend to Thomson
with a current interest rate of 3%, if he promises to
pay you $1,000 in 1 year time? What if Thomson
pays you 3For
years later?
1 year:
PV = FV / (1 + r)
= $1,000 / 1.03
= $970.87
 For 3 years:
PV = FV / (1 + r)n
= $1,000 / 1.033
= $915.14
Time value of money
Present value or Future value
1.

2.
Time value of money
Compoundi
ng

Present value Future value

Discounti
ng
 Concept check
Are the following statements true or false? Explain.
When we deposit money in banks, we usually earn compound
(a)

interest.
Compounding is the process of finding the present value, which
(b)
is the current value of a future sum of money.
If a person deposited $1,500 into a bank account a year ago
(c)
and the amount is $2,000 now, the $2,000 is the present value.
 Concept check

Are the following statements true or false? Explain.

(a) When we deposit money in banks, we usually earn compound interest.
True
(b) Compounding is the process of finding the present value, which is the current value
of a future sum of money.
False
The process of finding the present value is called discounting
() If a person deposited $1,500 into a bank account a year ago and the amount is
$2,000 now, the $2,000 is the present value.
 False
 The $2,000 is the future value.
 Concept check

Find (to one decimal place) the future values (FVs) in the following
scenarios.
(a)Invest $5,000 today at a 6% interest rate per annum for three years,

compounded annually.
$5,000 × (1 + 0.06)3 = $5,955.1
(b)Make a three-year deposit of $1,000 today. The deposit can earn a 4%

interest rate per annum, compounded annually.

$1,000 × (1 + 0.04)3 = $1,124.9
(c)Deposit $3,500 today at an annual interest rate of 8% for three years,

compounded annually.
$3,500 × (1 + 0.08)3 = $4,409.0
(d)Put $4,500 into a bank today for two years at an annual interest rate of

10%, compounded annually.

$4,500 × (1 + 0.1)2 = $5,445
Concept check
Concept check
 Concept check

Assume you want to buy a $5,000 smartphone and are offered two
payment schemes. Which scheme will you choose?
•Scheme 1: Pay $5,000 now
•Scheme 2: Pay $5,700 two years later
Assume the discount rate is 4% per annum. Which scheme would you choose?

FV of Scheme 1:
$5,000 × (1 + 0.04)2
= $5,408
FV of Scheme 2:
$5,700
Since the FV of Scheme 1 is lower,
Scheme 1 should be chosen.
Using net present value to make investment decisions
 Net present value (NPV) is the present value of a series of cash flows, less the
initial investment.
 The concept of NPV is used to determine whether an investment project is
acceptable.
 The calculations of NPV refers to finding the difference between the present value
of series of future income flows and the initial investment.

Calculation of Net present (Note p.19)

PV of all annual PV of the asset Initial cost of

NPV= cash flows + in the last year
of use (scrap
- the asset
(can be +ve/-ve)
value)

If NPV >1 : Benefit outweighs cost

If NPV <1 : Benefit does not outweigh
cost
Using net present value to make
Capital investment
decisions

Case1 Mr Kwok is considering purchasing a new computer for his business,

which will cost him $40,000.
•He expects the computer will be used for three years, and that it will be sold
for $5,000 at the end of Year 3.
•He estimates that the future cash inflows generated by this computer will be
$15,000 each year for the next three years.
•The cost of capital is 10%.

Should Mr Kwok buy the computer?

 Using net present value to make investment decisions
NPV = (PVs of all annual net cash inflows from using the computer
+ PV of the computer’s market value in Year 3)
– Initial cost/outlay of the computer
NPV = PV of annual net cash inflows + PV of the computer’s market value at Yr 3 – Initial outlay

=($15,000 /1.1) + ($15,000 /1.12) + ($15,000 /1.13) + 5,000 /1.13 - $40,000

= $1,059.35
Decision rules:
(i) If the NPV >0 , should buy / invest
(ii) If the NPV < 0, should NOT buy /
invest
(iii) If the NPV = 0, Indifferent
As the NPV is positive, Mr Kwok should purchase the
computer
 Interpreting the
NPV
Scenario Implication Decision
The financial benefit in doing so outweighs
Investment’s rate of the cost
NPV > 0 return  Accept the
> Cost of capital investment
The financial benefit in doing so does not
outweighs the cost
Investment’s rate of
NPV < 0 return  Reject the investment
< Cost of capital
One earns exactly the required rate of
return, which is enough to justify the
investment cost
Investment’s rate of
 Accept the investment
NPV = 0 return
= Cost of capital
Using net present value to make investment decisions
Capital investment => investment in non-current asset
e.g. machinery, equipment

For example, should a restaurant buy a dishwasher to replace

labour.

Dishwasher

Benefits > Costs :BUY

Benefits < Costs :NOT Buy
Using net present value to make capital investment decisions

Variables for Amount of Timing of the How to calculate

calculating NPV cashflow? cashflow? the PV of the
(Yr0,Yr1,Yr2,Y
cashflow?
r3)
What is the initial
investment cost?
How much cash inflows
at the end of year 1?

How much cash inflows

at the end of year 2?

How much cash inflows

at the end of year 3?

What is the estimated

market value of the
computer in Year 3?
Using net present value to make capital investment decisions

Variables for Amount of Timing of the How to calculate

calculating NPV cashflow? cashflow? the PV of the
(Yr0, Yr1, cashflow?
Yr2,Yr3)
What is the initial $40,000 Yr 0 $40,000
investment cost?
How much cash inflows $15,000/(1+10%)
at the end of year 1? $15,000 Yr 1 =$13,636.36

How much cash inflows

at the end of year 2? $15,000 Yr 2 $15,000
/(1+10%)2
=$12,396.69
How much cash inflows
at the end of year 3? $15,000 Yr 3 $15,000 /(1+10%)
3

=$11,269.72
What is the estimated
market value of the $5,000 Yr 3 $5,000 /(1+10%) 3
computer in Year 3? =$4,545.46
Q1: DSE 2021 P1A Q4
Mandy is considering investing in a 3-year project with the
following details:

The scrap value at the end of the project is $20 000. The discount rate is
5% per annum.
(a)NPV =

(b) State the variables of the NPV formula which will affect the accuracy of the NPV?
Q1:DSE 2021 P1A Q4
Mandy is considering investing in a 3-year project with the
following details:

The scrap value at the end of the project is $20 000. The discount rate is 5% per
annum.
(a)NPV =

(b) State the variables of the NPV formula which will affect the accuracy of the NPV?
 the initial investment cost
 amount of the future cash inflows
 timing of receiving the cash inflows
 scrap/residual/disposal value of the asset/project
 the interest rate/discount rate
Useful life of the project /machine
 Q2: DSE 2023 P1A Q4
A bank offers an investment plan which requires an initial
deposit of $180,000. An investor can get a $2,000 cash rebate
upon joining the plan. Cash inflows of $70,000 and $150,000
will be generated at the beginning of Year 2 and Year 3
respectively. The discount rate is 5% per annum.
(a) NPV =

(b) What are the variables of the NPV formula affecting the calculation of an
accuracy NPV?
 Q2: DSE 2023 P1A Q4
A bank offers an investment plan which requires an
initial deposit of $180,000. An investor can get a $2,000
cash rebate upon joining the plan. Cash inflows of
$70,000 and $150,000 will be generated at the beginning
of Year 2 and Year 3 respectively. The discount rate is
5% per annum.
(a) NPV =

(b) Which NPV variable(s) require(s) special attention in order to calculate the
NPV correctly?
 Cash rebate upon joining the plan = received at Year 0 => PV
 Timing of cash inflows (Beginning of Year 2 = End of Year 1)
Using net present value to make capital investment decisions

Q2. Mr Wong wants to purchase a dishwasher for his

restaurant. He is considering Brand A and Brand B:
• Brand A costs $10,000 and Brand B costs $12,000.
• Both of them can be used for three years and can be sold for
$1,000 at the end of Year 3.
• Estimated net cash inflows generated at the end of the next three
years:

Year
Year 11 Year
Year 22 Year
Year 33
Brand
Brand AA $5,000
$5,000 $4,000
$4,000 $3,000
$3,000
Brand
Brand BB $6,000
$6,000 $5,000
$5,000 $4,000
$4,000
If the cost of Mr Wong’s capital is 8%, which
brand should he buy?
Using net present value to make investment
decisions
Compare the NPVs of the two dishwashers:

Brand A Brand B
($5,000 ÷ 1.08) + ($6,000 ÷ 1.08) +
Total PV of annual net ($4,000 ÷ 1.082) + ($5,000 ÷ 1.082) +
cash inflows ($3,000 ÷ 1.083) ($4,000 ÷ 1.083)
= $10,440.48 = $13,017.58
PV of its market value $1,000 ÷ 1.083 $1,000 ÷ 1.083
at the end of Year 3 = $793.83 = $793.83
$10,440.48 + $793.83 $13,017.58 + $793.83
Total PVs
= $11,234.31 = $13,811.41
$11,234.31 – $10,000 $13,811.41 – $12,000
NPV
= $1,234.3 = $1,811.4

As the NPV of Brand B is higher, Mr Wong should buy Brand B.

 Using net present value to make investment
decisions
Compare the NPVs of the two dishwashers:
Total PV of annual Scrap / residual/disposal value
net cash inflows (PV of its market value at the
end of Year 3)
NPV (A): = ($5,000 ÷ 1.08) + ($4,000 ÷ 1.082) + ($3,000 ÷ 1.083) + ($1,000 ÷ 1.083) – $10,000
= $10,440.48-$10,000
= $1,234.3
NPV = ($6,000 ÷ 1.08) + ($5,000 ÷ 1.082) + ($4,000 ÷ 1.083) + ($1,000 ÷ 1.083) – $12,000
(B): = $13,017.58-$12,000
= $1,811.4

As the NPV of Brand B is higher, Mr Wong should buy Brand B.

The higher the NPV, the more one can gain from the investment.
higher
•If we have to choose one out of two or more investment projects  Choose the one with the ________ NPV
•If we can accept more than one project
largeror equal to zero
 Accept all projects with an NPV ________than
DSE 2020 Q8 & DSE 2018 Q19

51
%

66
%
Nominal versus effective rate of return
Not consider the
frequency of
compounding

nominal rate

Plan A: $100 x (1+1/12%)12

Plan B: $100 x (1+6/2%)2
Plan C: $100 x (1+12%)1

quoted
Nominal versus effective rate of return

Effective rate of return

●
Interest may be compounded more than
once during a year.
●
The frequency of compounding represents
how often the interest is compounded.
●
If the interest is compounded more than
once during a year, the nominal interest rate
cannot reflect the actual return on an
investment. The effective rate of return
(ERR) will be used.
Nominal versus effective rate of return
Effective rate of return
The effective rate of return is the rate of return that
includes the interest reinvested during a year.
The effective rate of return can be computed as follows:
 Nominal versus effective rate of return

Textbook p. 21 Check Your Progress (P1-7)

Calculate the ERR per year for each of the following cases.
Frequency of
Interest rate ERR
compounding

(a) 10% p.a. Annually 10%

(b) 8% p.a. Semi-annually

8.16
%
(c) 8% p.a. Quarterly
8.24
%
(d) 9% p.a. Twice a year 9.20
%

Chapter 7 Accounting Assumptions, Principles and Conventions

 Nominal versus effective rate of return

Textbook p. 21 Check Your Progress (P1-8)

Circle the correct answer for each of the following statements.
(a) If the compounding frequency of an investment plan increases,
its nominal / effective rate of return will also increase.
(b) The effective rate of return of an investment plan can be
greater / smaller than its nominal rate of return.

Home work:
Textbook p. 25-26 MCQs 1.13-1.15, p.27 1.21

42
Chapter 7 Accounting Assumptions, Principles and Conventions
Nominal versus effective rate of return

Effect of changing the frequency of compounding

● The more frequently interest is compounded, the ______ the final amount will be.
● Whenever the frequency of compounding differs, we should compare different investment plans based on their
larger
_____________________ instead of their _____________________.

effective rates of return

nominal rates of return
DSE Questions

38
%
DSE Questions
TB p.27 Q1.23