# Financial Management ```Lecture 2 Question of the Week
wondering how much loan you have to borrow, whether you will be able to pay for
the loan each month and which financial institutions or banks you should borrow
the loan from?
The concept we are going to learn this week will help answer the above questions.
The question of this week concerns the last period of your life, your retirement.
You are saving for retirement. You plan to retire when you reach 65 years
old. To live comfortably during retirement, you plan to withdraw money
from your savings account \$48,000 each year. Suppose today is your 25th
birthday, and you decide, starting today and continuing on every year
up to your 64th birthday, that you will put the same amount into a savings
account. If the interest rate is 2.4% per annum, compounded annually,
how much must you set aside each year to make sure that you will have
enough money in the account on your 65th birthday in order to be able
to withdraw \$48,000 per year when you retire and you expect to live until
your are 90 years old (i.e., the first withdrawal is on your 65th birthday and
the last withdrawal occurs on your 89th birthday)?
By the end of the lecture, you would be able to answer this question.
0
Lecture 2 Time Value of Money and Loan Amortisation
Learning Objectives
• Explain how time value of money works and why it is important in
Finance
• Calculate the present value (PV) and future value (FV) of:
‒
‒
‒
‒
A lump sum
Annuity
Uneven cash flow stream
Perpetuity (only PV)
• Differentiate between annuity due and ordinary annuity
• Explain the difference between nominal, periodic, and effective
interest rates
‒ Understand how to compare alternative investments with different
compounding periods
• Understand loan amortisation and able to calculate the relevant
outputs (e.g. payments, principal outstanding).
1
AB1201: Financial
Management
Lecture 2: Time Value of Money
and Loan Amortisation
By: Chanika Charoenwong
2
What is time value of money? &gt; FV of lump sum &gt; PV of lump sum &gt; LL1 &gt; FV of ordinary annuity &gt; PV of ordinary annuity &gt; FV of
annuity due &gt; PV of annuity due &gt; PV of perpetuity &gt; LL2 &gt; PV of uneven CF stream &gt; Three types of interest rates &gt; Why EAR? &gt;
When is each rate used? &gt; LL3 &gt; Loan amortisation &gt; LL4 &gt; Conclusion
What is time value of money?
• Time value of money: Idea that money
available today is worth more than the same
amount in the future because you can invest
the money, for example
– Deposit the money in a bank to earn interest
– Invest in stocks and bonds
3
What is time value of money? &gt; FV of lump sum &gt; PV of lump sum &gt; LL1 &gt; FV of ordinary annuity &gt; PV of ordinary annuity &gt; FV of
annuity due &gt; PV of annuity due &gt; PV of perpetuity &gt; LL2 &gt; PV of uneven CF stream &gt; Three types of interest rates &gt; Why EAR? &gt;
When is each rate used? &gt; LL3 &gt; Loan amortisation &gt; LL4 &gt; Conclusion
What is time value of money?
• Time value of money is important in finance. Its
analysis is used in many ways such as
Valuing stocks
and bonds
Planning
retirement
making corporate
decisions regarding
investing in new
plant and equipment
Setting up loan
payment
schedule
4
What is time value of money? &gt; FV of lump sum &gt; PV of lump sum &gt; LL1 &gt; FV of ordinary annuity &gt; PV of ordinary annuity &gt; FV of
annuity due &gt; PV of annuity due &gt; PV of perpetuity &gt; LL2 &gt; PV of uneven CF stream &gt; Three types of interest rates &gt; Why EAR? &gt;
When is each rate used? &gt; LL3 &gt; Loan amortisation &gt; LL4 &gt; Conclusion
Time Lines
• Show the timing of cash flows.
• Tick marks occur at the end of periods, so
Time 0 is today, Time 1 is the end of the first
period (year, month, etc.) or the beginning
of the second period.
0
1
2
3
CF1
CF2
CF3
I%
CF0
5
What is time value of money? &gt; FV of lump sum &gt; PV of lump sum &gt; LL1 &gt; FV of ordinary annuity &gt; PV of ordinary annuity &gt; FV of
annuity due &gt; PV of annuity due &gt; PV of perpetuity &gt; LL2 &gt; PV of uneven CF stream &gt; Three types of interest rates &gt; Why EAR? &gt;
When is each rate used? &gt; LL3 &gt; Loan amortisation &gt; LL4 &gt; Conclusion
Drawing Time Lines
\$100 lump sum due in 2 years
0
I%
1
2
100
Uneven cash flow stream
0
I%
1
2
100
200
3
50
6
What is time value of money? &gt; FV of lump sum &gt; PV of lump sum &gt; LL1 &gt; FV of ordinary annuity &gt; PV of ordinary annuity &gt; FV of
annuity due &gt; PV of annuity due &gt; PV of perpetuity &gt; LL2 &gt; PV of uneven CF stream &gt; Three types of interest rates &gt; Why EAR? &gt;
When is each rate used? &gt; LL3 &gt; Loan amortisation &gt; LL4 &gt; Conclusion
What is the future value (FV) of an initial
\$100 after three years, if I/YR = 10%?
• Finding the FV of a cash flow or series of cash flows
is called compounding.
• FV can be solved by using the step-by-step,
0
1
2
3
10%
100
FV = ?
7
What is time value of money? &gt; FV of lump sum &gt; PV of lump sum &gt; LL1 &gt; FV of ordinary annuity &gt; PV of ordinary annuity &gt; FV of
annuity due &gt; PV of annuity due &gt; PV of perpetuity &gt; LL2 &gt; PV of uneven CF stream &gt; Three types of interest rates &gt; Why EAR? &gt;
When is each rate used? &gt; LL3 &gt; Loan amortisation &gt; LL4 &gt; Conclusion
Solving for FV:
The Step-by-Step and Formula Methods
• After 1 year:
– FV1 = PV(1 + I) = \$100(1.10) = \$110.00
• After 2 years:
– FV2 = PV(1 + I)2 = \$100(1.10)2 = \$121.00
• After 3 years:
– FV3 = PV(1 + I)3 = \$100(1.10)3 = \$133.10
• After N years (general case):
FVN
= PV(1 + I)N
8
What is time value of money? &gt; FV of lump sum &gt; PV of lump sum &gt; LL1 &gt; FV of ordinary annuity &gt; PV of ordinary annuity &gt; FV of
annuity due &gt; PV of annuity due &gt; PV of perpetuity &gt; LL2 &gt; PV of uneven CF stream &gt; Three types of interest rates &gt; Why EAR? &gt;
When is each rate used? &gt; LL3 &gt; Loan amortisation &gt; LL4 &gt; Conclusion
Solving for FV:
The Calculator Method
• Solves the general FV equation.
• Requires four inputs into calculator, and will
solve for the fifth. (Set to P/YR = 1 and END
mode.)
INPUTS
OUTPUT
3
10
-100
0
N
I/YR
PV
PMT
FV
133.10
9
What is time value of money? &gt; FV of lump sum &gt; PV of lump sum &gt; LL1 &gt; FV of ordinary annuity &gt; PV of ordinary annuity &gt; FV of
annuity due &gt; PV of annuity due &gt; PV of perpetuity &gt; LL2 &gt; PV of uneven CF stream &gt; Three types of interest rates &gt; Why EAR? &gt;
When is each rate used? &gt; LL3 &gt; Loan amortisation &gt; LL4 &gt; Conclusion
What is the present value (PV) of \$133.10
due in three years, if I/YR = 10%?
• Finding the PV of a cash flow or series of
cash flows is called discounting (the reverse
of compounding).
2
0
1
3
&gt; Calculating FV of lump sum &gt;
10%
133.10
• PV – how much is a stream of future cash
flows worth today?
PV = ?
10
What is time value of money? &gt; FV of lump sum &gt; PV of lump sum &gt; LL1 &gt; FV of ordinary annuity &gt; PV of ordinary annuity &gt; FV of
annuity due &gt; PV of annuity due &gt; PV of perpetuity &gt; LL2 &gt; PV of uneven CF stream &gt; Three types of interest rates &gt; Why EAR? &gt;
When is each rate used? &gt; LL3 &gt; Loan amortisation &gt; LL4 &gt; Conclusion
Solving for PV:
The Formula Method
• Solve the general FV equation for PV:
– PV = FVN /(1 + I)N
– PV = FV3 /(1 + I)3
= \$133.10/(1.10)3
= \$100
11
What is time value of money? &gt; FV of lump sum &gt; PV of lump sum &gt; LL1 &gt; FV of ordinary annuity &gt; PV of ordinary annuity &gt; FV of
annuity due &gt; PV of annuity due &gt; PV of perpetuity &gt; LL2 &gt; PV of uneven CF stream &gt; Three types of interest rates &gt; Why EAR? &gt;
When is each rate used? &gt; LL3 &gt; Loan amortisation &gt; LL4 &gt; Conclusion
Solving for PV:
The Calculator Method
• Exactly like solving for FV, except we have
different input information and are solving
for a different variable.
INPUTS
OUTPUT
3
10
N
I/YR
PV
0
133.10
PMT
FV
-100
12
What is time value of money? &gt; FV of lump sum &gt; PV of lump sum &gt; LL1 &gt; FV of ordinary annuity &gt; PV of ordinary annuity &gt; FV of
annuity due &gt; PV of annuity due &gt; PV of perpetuity &gt; LL2 &gt; PV of uneven CF stream &gt; Three types of interest rates &gt; Why EAR? &gt;
When is each rate used? &gt; LL3 &gt; Loan amortisation &gt; LL4 &gt; Conclusion
Lessons Learnt 1
• Time value of money: Idea that money available today is
worth more than the same amount in the future because
you can invest the money
• Time value of money is important in finance. Its analysis is
used to
– value the stocks, bonds and capital budgeting projects
– plan for retirements
– set up loan payment schedules, etc.
• Finding the FV of a cash flow or series of cash flows is
called compounding.
• Finding the PV of a cash flow or series of cash flows is
called discounting (the reverse of compounding).
13
What is time value of money? &gt; FV of lump sum &gt; PV of lump sum &gt; LL1 &gt; FV of ordinary annuity &gt; PV of ordinary annuity &gt; FV of
annuity due &gt; PV of annuity due &gt; PV of perpetuity &gt; LL2 &gt; PV of uneven CF stream &gt; Three types of interest rates &gt; Why EAR? &gt;
When is each rate used? &gt; LL3 &gt; Loan amortisation &gt; LL4 &gt; Conclusion
What is the difference between an
Annuity – series
of equal cash
flows an
at fixed
intervals fordue?
a specified
ordinary
annuity
and
annuity
no. of periods
Ordinary Annuity
Ordinary
Annuity:
Cash flows
occur at
end of period
0
I%
1
2
3
PMT
PMT
PMT
1
2
3
PMT
PMT
Annuity Due
Annuity Due:
Cash flows
occur at
beginning of
period
0
PMT
I%
14
What is time value of money? &gt; FV of lump sum &gt; PV of lump sum &gt; LL1 &gt; FV of ordinary annuity &gt; PV of ordinary annuity &gt; FV of
annuity due &gt; PV of annuity due &gt; PV of perpetuity &gt; LL2 &gt; PV of uneven CF stream &gt; Three types of interest rates &gt; Why EAR? &gt;
When is each rate used? &gt; LL3 &gt; Loan amortisation &gt; LL4 &gt; Conclusion
Ordinary Annuity, Solving for FV:
3-Year Ordinary Annuity of \$100 at 10%
• \$100 payments occur at the end of each
period
15
What is time value of money? &gt; FV of lump sum &gt; PV of lump sum &gt; LL1 &gt; FV of ordinary annuity &gt; PV of ordinary annuity &gt; FV of
annuity due &gt; PV of annuity due &gt; PV of perpetuity &gt; LL2 &gt; PV of uneven CF stream &gt; Three types of interest rates &gt; Why EAR? &gt;
When is each rate used? &gt; LL3 &gt; Loan amortisation &gt; LL4 &gt; Conclusion
Ordinary Annuity, Solving for FV:
3-Year Ordinary Annuity of \$100 at 10%
• \$100 payments occur at the end of each
period, but there is no PV.


PMT
N
1  I   1
FV 
I
INPUTS
OUTPUT


100
3
1 0.10 1
FV 
0.10
3
10
0
-100
N
I/YR
PV
PMT
FV
331
16
What is time value of money? &gt; FV of lump sum &gt; PV of lump sum &gt; LL1 &gt; FV of ordinary annuity &gt; PV of ordinary annuity &gt; FV of
annuity due &gt; PV of annuity due &gt; PV of perpetuity &gt; LL2 &gt; PV of uneven CF stream &gt; Three types of interest rates &gt; Why EAR? &gt;
When is each rate used? &gt; LL3 &gt; Loan amortisation &gt; LL4 &gt; Conclusion
Ordinary Annuity, Solving for PV:
3-year Ordinary Annuity of \$100 at 10%
• \$100 payments still occur at the end of each
period
17
What is time value of money? &gt; FV of lump sum &gt; PV of lump sum &gt; LL1 &gt; FV of ordinary annuity &gt; PV of ordinary annuity &gt; FV of
annuity due &gt; PV of annuity due &gt; PV of perpetuity &gt; LL2 &gt; PV of uneven CF stream &gt; Three types of interest rates &gt; Why EAR? &gt;
When is each rate used? &gt; LL3 &gt; Loan amortisation &gt; LL4 &gt; Conclusion
Ordinary Annuity, Solving for PV:
3-year Ordinary Annuity of \$100 at 10%
• \$100 payments still occur at the end of each
period, but now there is no FV.
PMT
PV 
I
INPUTS
OUTPUT

1 
1 
N 
 1  I  
3
10
N
I/YR

100 
1
PV 
1 

0.10  1  0.10 3 
PV
100
0
PMT
FV
-248.69
18
What is time value of money? &gt; FV of lump sum &gt; PV of lump sum &gt; LL1 &gt; FV of ordinary annuity &gt; PV of ordinary annuity &gt; FV of
annuity due &gt; PV of annuity due &gt; PV of perpetuity &gt; LL2 &gt; PV of uneven CF stream &gt; Three types of interest rates &gt; Why EAR? &gt;
When is each rate used? &gt; LL3 &gt; Loan amortisation &gt; LL4 &gt; Conclusion
How will things change if we want to
solve the FV and PV of a 3-year annuity
due of \$100 at 10%?
Annuity Due, Solving for FV:
3-Year Annuity Due of \$100 at 10%
Ordinary
Annuity
0
\$100
10%
\$100
1
\$100
2
\$100
\$100
\$100
3
Annuity Due
19
What is time value of money? &gt; FV of lump sum &gt; PV of lump sum &gt; LL1 &gt; FV of ordinary annuity &gt; PV of ordinary annuity &gt; FV of
annuity due &gt; PV of annuity due &gt; PV of perpetuity &gt; LL2 &gt; PV of uneven CF stream &gt; Three types of interest rates &gt; Why EAR? &gt;
When is each rate used? &gt; LL3 &gt; Loan amortisation &gt; LL4 &gt; Conclusion
Annuity Due, Solving for FV:
3-Year Annuity Due of \$100 at 10%
• Now, \$100 payments occur at the beginning
of each period.
• FVAdue= FVAord(1 + I) = \$331(1.10) = \$364.10
• Alternatively, set calculator to “BEGIN”
mode and solve for the FV of the annuity:
BEGIN
INPUTS
OUTPUT
3
10
0
-100
N
I/YR
PV
PMT
FV
364.10
20
What is time value of money? &gt; FV of lump sum &gt; PV of lump sum &gt; LL1 &gt; FV of ordinary annuity &gt; PV of ordinary annuity &gt; FV of
annuity due &gt; PV of annuity due &gt; PV of perpetuity &gt; LL2 &gt; PV of uneven CF stream &gt; Three types of interest rates &gt; Why EAR? &gt;
When is each rate used? &gt; LL3 &gt; Loan amortisation &gt; LL4 &gt; Conclusion
How will things change if we want to solve
the FV and PV of a 3-year annuity due of
\$100 at 10%?
Annuity Due, Solving for PV:
3-Year Annuity Due of \$100 at 10%
Ordinary
Annuity
0
\$100
10%
\$100
1
\$100
2
\$100
\$100
\$100
3
Annuity Due
21
What is time value of money? &gt; FV of lump sum &gt; PV of lump sum &gt; LL1 &gt; FV of ordinary annuity &gt; PV of ordinary annuity &gt; FV of
annuity due &gt; PV of annuity due &gt; PV of perpetuity &gt; LL2 &gt; PV of uneven CF stream &gt; Three types of interest rates &gt; Why EAR? &gt;
When is each rate used? &gt; LL3 &gt; Loan amortisation &gt; LL4 &gt; Conclusion
Annuity Due, Solving for PV:
3-Year Annuity Due of \$100 at 10%
• Again, \$100 payments occur at the
beginning of each period.
• PVAdue = PVAord(1 + I) = 248.69(1.10)=\$273.55
• Alternatively, set calculator to “BEGIN”
mode and solve for the PV of the annuity:
BEGIN
INPUTS
OUTPUT
3
10
N
I/YR
PV
100
0
PMT
FV
-273.55
22
What is time value of money? &gt; FV of lump sum &gt; PV of lump sum &gt; LL1 &gt; FV of ordinary annuity &gt; PV of ordinary annuity &gt; FV of
annuity due &gt; PV of annuity due &gt; PV of perpetuity &gt; LL2 &gt; PV of uneven CF stream &gt; Three types of interest rates &gt; Why EAR? &gt;
When is each rate used? &gt; LL3 &gt; Loan amortisation &gt; LL4 &gt; Conclusion
What is the PV of a perpetuity that pays
\$100 per year at 10%?
• Perpetuity: An annuity that lasts forever
PV = PMT/I
= \$100/0.1 = \$1,000
23
What is time value of money? &gt; FV of lump sum &gt; PV of lump sum &gt; LL1 &gt; FV of ordinary annuity &gt; PV of ordinary annuity &gt; FV of
annuity due &gt; PV of annuity due &gt; PV of perpetuity &gt; LL2 &gt; PV of uneven CF stream &gt; Three types of interest rates &gt; Why EAR? &gt;
When is each rate used? &gt; LL3 &gt; Loan amortisation &gt; LL4 &gt; Conclusion
Lessons Learnt 2
• Ordinary annuity is the series of equal cash flows
that occur at the end of the period
• Annuity due is the series of equal cash flows that
occur at the beginning of the period
• FVdue = FVord(1+I)
• PVdue = PVord(1+I)
• Perpetuity is the annuity that lasts forever
24
What is time value of money? &gt; FV of lump sum &gt; PV of lump sum &gt; LL1 &gt; FV of ordinary annuity &gt; PV of ordinary annuity &gt; FV of
annuity due &gt; PV of annuity due &gt; PV of perpetuity &gt; LL2 &gt; PV of uneven CF stream &gt; Three types of interest rates &gt; Why EAR? &gt;
When is each rate used? &gt; LL3 &gt; Loan amortisation &gt; LL4 &gt; Conclusion
What is the PV of this uneven CF stream?
0
1.
2.
3.
4.
10%
1
2
3
4
100
300
300
-50
530
590
598
Impossible to solve!
25
What is time value of money? &gt; FV of lump sum &gt; PV of lump sum &gt; LL1 &gt; FV of ordinary annuity &gt; PV of ordinary annuity &gt; FV of
annuity due &gt; PV of annuity due &gt; PV of perpetuity &gt; LL2 &gt; PV of uneven CF stream &gt; Three types of interest rates &gt; Why EAR? &gt;
When is each rate used? &gt; LL3 &gt; Loan amortisation &gt; LL4 &gt; Conclusion
What is the PV of this uneven cash flow
stream? 1st Method
0
1
10%
100
2
3
4
300
300
-50
90.91
247.93
225.39
-34.15
530.08 = PV
26
What is time value of money? &gt; FV of lump sum &gt; PV of lump sum &gt; LL1 &gt; FV of ordinary annuity &gt; PV of ordinary annuity &gt; FV of
annuity due &gt; PV of annuity due &gt; PV of perpetuity &gt; LL2 &gt; PV of uneven CF stream &gt; Three types of interest rates &gt; Why EAR? &gt;
When is each rate used? &gt; LL3 &gt; Loan amortisation &gt; LL4 &gt; Conclusion
What is the PV of this uneven cash flow
stream? 2nd Method
• 2nd Method: Input cash flows in the calculator’s
“CFLO” register:
– CF0 = 0
– CF1 = 100
– CF2 = 300
– CF3 = 300
– CF4 = -50
• Press NPV button, enter I/YR = 10, press CPT button
to get NPV = \$530.087. (Here NPV = PV.)
27
What is time value of money? &gt; FV of lump sum &gt; PV of lump sum &gt; LL1 &gt; FV of ordinary annuity &gt; PV of ordinary annuity &gt; FV of
annuity due &gt; PV of annuity due &gt; PV of perpetuity &gt; LL2 &gt; PV of uneven CF stream &gt; Three types of interest rates &gt; Why EAR? &gt;
When is each rate used? &gt; LL3 &gt; Loan amortisation &gt; LL4 &gt; Conclusion
Will the FV of a lump sum be larger or smaller if
compounded more often, holding the stated I%
constant?
0
10%
1
2
3
100
133.10
Annually: FV3 = \$100(1.10)3 = \$133.10
0
100
5%
1
1 year
2
3
2 years
4
5
3 years
6
134.01
Semiannually: FV6 = \$100(1.05)6 = \$134.01
• LARGER, as the more frequently compounding
occurs, interest is earned on interest more often.
28
What is time value of money? &gt; FV of lump sum &gt; PV of lump sum &gt; LL1 &gt; FV of ordinary annuity &gt; PV of ordinary annuity &gt; FV of
annuity due &gt; PV of annuity due &gt; PV of perpetuity &gt; LL2 &gt; PV of uneven CF stream &gt; Three types of interest rates &gt; Why EAR? &gt;
When is each rate used? &gt; LL3 &gt; Loan amortisation &gt; LL4 &gt; Conclusion
Classifications of Interest Rates
• Nominal rate (INOM) – also called the quoted or
stated rate. An annual rate that ignores
compounding effects.
– Stated in contracts. Periods must also be given,
e.g. 8% quarterly compounding or 8% daily interest
compounding
• Periodic rate (IPER) – amount of interest charged
each period, e.g. monthly or quarterly.
- IPER = INOM/M, where M is the number of
compounding periods per year. M = 4 for quarterly
and M = 12 for monthly compounding
29
What is time value of money? &gt; FV of lump sum &gt; PV of lump sum &gt; LL1 &gt; FV of ordinary annuity &gt; PV of ordinary annuity &gt; FV of
annuity due &gt; PV of annuity due &gt; PV of perpetuity &gt; LL2 &gt; PV of uneven CF stream &gt; Three types of interest rates &gt; Why EAR? &gt;
When is each rate used? &gt; LL3 &gt; Loan amortisation &gt; LL4 &gt; Conclusion
Classifications of Interest Rates
• Effective (or equivalent) annual rate (EAR =
EFF%) – the annual rate of interest actually
being earned, accounting for
compounding.
– EFF% for 10% semiannual investment
EFF% = ( 1 + INOM/M )M – 1
= ( 1 + 0.10/2 )2 – 1 = 10.25%
– Should be indifferent between receiving 10.25%
annual interest and receiving 10% interest,
compounded semiannually.
30
What is time value of money? &gt; FV of lump sum &gt; PV of lump sum &gt; LL1 &gt; FV of ordinary annuity &gt; PV of ordinary annuity &gt; FV of
annuity due &gt; PV of annuity due &gt; PV of perpetuity &gt; LL2 &gt; PV of uneven CF stream &gt; Three types of interest rates &gt; Why EAR? &gt;
When is each rate used? &gt; LL3 &gt; Loan amortisation &gt; LL4 &gt; Conclusion
Why is it important to consider effective
rates of return?
• Investments with different compounding
intervals provide different effective returns.
• To compare investments with different
compounding intervals, you must look at
their effective returns (EFF% or EAR).
31
What is time value of money? &gt; FV of lump sum &gt; PV of lump sum &gt; LL1 &gt; FV of ordinary annuity &gt; PV of ordinary annuity &gt; FV of
annuity due &gt; PV of annuity due &gt; PV of perpetuity &gt; LL2 &gt; PV of uneven CF stream &gt; Three types of interest rates &gt; Why EAR? &gt;
When is each rate used? &gt; LL3 &gt; Loan amortisation &gt; LL4 &gt; Conclusion
Why is it important to consider
the effective rates of return?
• See how the effective return varies between
investments with the same nominal rate
(10%), but different compounding intervals.
–
–
–
–
EARANNUAL
EARQUARTERLY
EARMONTHLY
EARDAILY (365)
10.00%
10.38%
10.47%
10.52%
??
32
What is time value of money? &gt; FV of lump sum &gt; PV of lump sum &gt; LL1 &gt; FV of ordinary annuity &gt; PV of ordinary annuity &gt; FV of
annuity due &gt; PV of annuity due &gt; PV of perpetuity &gt; LL2 &gt; PV of uneven CF stream &gt; Three types of interest rates &gt; Why EAR? &gt;
When is each rate used? &gt; LL3 &gt; Loan amortisation &gt; LL4 &gt; Conclusion
When is each rate used?
• INOM - Written into contracts, quoted by
banks and brokers. Not used in calculations
or shown on time lines.
• IPER - Used in calculations and shown on
time lines. If M = 1, INOM = IPER = EAR.
• EAR - Used to compare returns on
investments with different payments per
year.
33
What is time value of money? &gt; FV of lump sum &gt; PV of lump sum &gt; LL1 &gt; FV of ordinary annuity &gt; PV of ordinary annuity &gt; FV of
annuity due &gt; PV of annuity due &gt; PV of perpetuity &gt; LL2 &gt; PV of uneven CF stream &gt; Three types of interest rates &gt; Why EAR? &gt;
When is each rate used? &gt; LL3 &gt; Loan amortisation &gt; LL4 &gt; Conclusion
Example: What is the FV of \$100 after three
years under 10% semiannual compounding?
 I NOM 
FVN  PV1 

M 

M N
23
 0.10 
FV3S  \$1001 

2 

6
FV3S  \$100(1.05)  \$134.01
Quarterly compounding?
FV3Q  \$100(1.025)  \$134.49
12
34
What is time value of money? &gt; FV of lump sum &gt; PV of lump sum &gt; LL1 &gt; FV of ordinary annuity &gt; PV of ordinary annuity &gt; FV of
annuity due &gt; PV of annuity due &gt; PV of perpetuity &gt; LL2 &gt; PV of uneven CF stream &gt; Three types of interest rates &gt; Why EAR? &gt;
When is each rate used? &gt; LL3 &gt; Loan amortisation &gt; LL4 &gt; Conclusion
Example: What’s the FV of a 3-year \$100
ordinary annuity, if the quoted interest rate
is 10%, compounded semiannually?
• Payments occur annually, but
compounding occurs every six months.
• Cannot use normal annuity valuation
techniques.
0
5%
1
2
100
3
4
100
5
6
100
35
What is time value of money? &gt; FV of lump sum &gt; PV of lump sum &gt; LL1 &gt; FV of ordinary annuity &gt; PV of ordinary annuity &gt; FV of
annuity due &gt; PV of annuity due &gt; PV of perpetuity &gt; LL2 &gt; PV of uneven CF stream &gt; Three types of interest rates &gt; Why EAR? &gt;
When is each rate used? &gt; LL3 &gt; Loan amortisation &gt; LL4 &gt; Conclusion
Compound Each Cash Flow
0
5%
1
2
3
100
\$100(1.05)4
4
5
100
\$100(1.05)2
6
100
110.25
121.55
331.80
36
What is time value of money? &gt; FV of lump sum &gt; PV of lump sum &gt; LL1 &gt; FV of ordinary annuity &gt; PV of ordinary annuity &gt; FV of
annuity due &gt; PV of annuity due &gt; PV of perpetuity &gt; LL2 &gt; PV of uneven CF stream &gt; Three types of interest rates &gt; Why EAR? &gt;
When is each rate used? &gt; LL3 &gt; Loan amortisation &gt; LL4 &gt; Conclusion
Lessons Learnt 3
• FV of a lump sum will be larger if compounded
more often, holding the stated I% constant because
as the more frequently compounding occurs,
interest is earned on interest more often.
• Three types of interest rates—nominal rate, periodic
rate and effective annual rate (EAR)
• EAR is used to compare returns on investments with
different payments per year.
37
What is time value of money? &gt; FV of lump sum &gt; PV of lump sum &gt; LL1 &gt; FV of ordinary annuity &gt; PV of ordinary annuity &gt; FV of
annuity due &gt; PV of annuity due &gt; PV of perpetuity &gt; LL2 &gt; PV of uneven CF stream &gt; Three types of interest rates &gt; Why EAR? &gt;
When is each rate used? &gt; LL3 &gt; Loan amortisation &gt; LL4 &gt; Conclusion
Loan Amortisation
• Amortised loan: A loan that is repaid in
equal payments over its life
• Amortised loans are widely used for home
retirement plans, etc.
38
What is time value of money? &gt; FV of lump sum &gt; PV of lump sum &gt; LL1 &gt; FV of ordinary annuity &gt; PV of ordinary annuity &gt; FV of
annuity due &gt; PV of annuity due &gt; PV of perpetuity &gt; LL2 &gt; PV of uneven CF stream &gt; Three types of interest rates &gt; Why EAR? &gt;
When is each rate used? &gt; LL3 &gt; Loan amortisation &gt; LL4 &gt; Conclusion
Example
• You take out a \$1,000 loan to buy a used
car. The loan is to be repaid in three equal
payments at the end of each of the next
three years. Construct an amortisation
schedule, with annual rate of 10%.
39
What is time value of money? &gt; FV of lump sum &gt; PV of lump sum &gt; LL1 &gt; FV of ordinary annuity &gt; PV of ordinary annuity &gt; FV of
annuity due &gt; PV of annuity due &gt; PV of perpetuity &gt; LL2 &gt; PV of uneven CF stream &gt; Three types of interest rates &gt; Why EAR? &gt;
When is each rate used? &gt; LL3 &gt; Loan amortisation &gt; LL4 &gt; Conclusion
How much do you need to pay at the
end of each of the next three years?
0
1
2
3
10%
-PMT
-PMT
-PMT
• PMT = 1000/3 = \$333?
• PMT = (1000*1.13)/3 = \$443.67?
• Somewhere in between: Ordinary annuity?
\$1000
– Three equal payments at the end of each of the
next three years
40
What is time value of money? &gt; FV of lump sum &gt; PV of lump sum &gt; LL1 &gt; FV of ordinary annuity &gt; PV of ordinary annuity &gt; FV of
annuity due &gt; PV of annuity due &gt; PV of perpetuity &gt; LL2 &gt; PV of uneven CF stream &gt; Three types of interest rates &gt; Why EAR? &gt;
When is each rate used? &gt; LL3 &gt; Loan amortisation &gt; LL4 &gt; Conclusion
Step 1:
Find the Required Annual Payment
PMT
PV 
I

1 
1 
N 
 1  I  
PMT
1000 
0.10


1
1 
3


1

0
.
10


• All input information is already given, just remember that
the FV = 0 because the reason for amortising the loan
and making payments is to retire the loan.
INPUTS
OUTPUT
3
10
-1000
N
I/YR
PV
0
PMT
FV
402.11
41
What is time value of money? &gt; FV of lump sum &gt; PV of lump sum &gt; LL1 &gt; FV of ordinary annuity &gt; PV of ordinary annuity &gt; FV of
annuity due &gt; PV of annuity due &gt; PV of perpetuity &gt; LL2 &gt; PV of uneven CF stream &gt; Three types of interest rates &gt; Why EAR? &gt;
When is each rate used? &gt; LL3 &gt; Loan amortisation &gt; LL4 &gt; Conclusion
Step 2:
Find the Interest Paid in Year 1
• The borrower will owe interest upon the initial
balance at the end of the first year. Interest
to be paid in the first year can be found by
multiplying the beginning balance by the
interest rate.
INTt = Beg balt(I)
INT1 = \$1,000(0.10) = \$100
42
What is time value of money? &gt; FV of lump sum &gt; PV of lump sum &gt; LL1 &gt; FV of ordinary annuity &gt; PV of ordinary annuity &gt; FV of
annuity due &gt; PV of annuity due &gt; PV of perpetuity &gt; LL2 &gt; PV of uneven CF stream &gt; Three types of interest rates &gt; Why EAR? &gt;
When is each rate used? &gt; LL3 &gt; Loan amortisation &gt; LL4 &gt; Conclusion
Step 3:
Find the Principal Repaid in Year 1
• If a payment of \$402.11 was made at the
end of the first year and \$100 was paid
toward interest, the remaining value must
represent the amount of principal repaid.
PRIN = PMT – INT
= \$402.11 – \$100
= \$302.11
43
What is time value of money? &gt; FV of lump sum &gt; PV of lump sum &gt; LL1 &gt; FV of ordinary annuity &gt; PV of ordinary annuity &gt; FV of
annuity due &gt; PV of annuity due &gt; PV of perpetuity &gt; LL2 &gt; PV of uneven CF stream &gt; Three types of interest rates &gt; Why EAR? &gt;
When is each rate used? &gt; LL3 &gt; Loan amortisation &gt; LL4 &gt; Conclusion
Step 4:
Find the Ending Balance after Year 1
• To find the balance at the end of the
period, subtract the amount paid toward
principal from the beginning balance.
END BAL
= BEG BAL – PRIN
= \$1,000 – \$302.11
= \$697.89
44
What is time value of money? &gt; FV of lump sum &gt; PV of lump sum &gt; LL1 &gt; FV of ordinary annuity &gt; PV of ordinary annuity &gt; FV of
annuity due &gt; PV of annuity due &gt; PV of perpetuity &gt; LL2 &gt; PV of uneven CF stream &gt; Three types of interest rates &gt; Why EAR? &gt;
When is each rate used? &gt; LL3 &gt; Loan amortisation &gt; LL4 &gt; Conclusion
Constructing an Amortisation Table:
Repeat Steps 1–4 Until End of Loan
• Interest paid declines with each payment as
the balance declines.
YEAR
1
2
3
TOTAL
BEG BAL
PMT
\$1,000
\$ 402
698
402
366
402
–
\$1,206
INT
\$100
70
36
\$206
PRIN
\$ 302
332
366
\$1,000
END BAL
\$698
366
0
–
45
What is time value of money? &gt; FV of lump sum &gt; PV of lump sum &gt; LL1 &gt; FV of ordinary annuity &gt; PV of ordinary annuity &gt; FV of
annuity due &gt; PV of annuity due &gt; PV of perpetuity &gt; LL2 &gt; PV of uneven CF stream &gt; Three types of interest rates &gt; Why EAR? &gt;
When is each rate used? &gt; LL3 &gt; Loan amortisation &gt; LL4 &gt; Conclusion
Lessons Learnt 4
• Amortised loan: A loan that is repaid in
equal payment over its life.
– Amortisation schedule
• Outstanding principal/loan = PV of all future
payments
46
What is time value of money? &gt; FV of lump sum &gt; PV of lump sum &gt; LL1 &gt; FV of ordinary annuity &gt; PV of ordinary annuity &gt; FV of
annuity due &gt; PV of annuity due &gt; PV of perpetuity &gt; LL2 &gt; PV of uneven CF stream &gt; Three types of interest rates &gt; Why EAR? &gt;
When is each rate used? &gt; LL3 &gt; Loan amortisation &gt; LL4 &gt; Conclusion
Some Food for Thought
• What determines the discount rate?
• What if the cash flows are riskier?
• Consider two banks:
– Bank A = Very safe
– Bank B = Not so safe, i.e. some chance that you
will not get back your deposit
– If both banks offered you the same interest rates,
who would you bank with?
– What would make you feel indifferent between
the two banks?
47
What is time value of money? &gt; FV of lump sum &gt; PV of lump sum &gt; LL1 &gt; FV of ordinary annuity &gt; PV of ordinary annuity &gt; FV of
annuity due &gt; PV of annuity due &gt; PV of perpetuity &gt; LL2 &gt; PV of uneven CF stream &gt; Three types of interest rates &gt; Why EAR? &gt;
When is each rate used? &gt; LL3 &gt; Loan amortisation &gt; LL4 &gt; Conclusion
Where do we stand?
• Future value of lump sum after N years:
– FVN
= PV(1 + I)N
• Present value of lump sum due in N years:
– PV
= FVN /(1 + I)N
• Future value of ordinary annuity:
•
•
•
•
•
FV 


PMT
1  I N  1
I
Present value of ordinary annuity:
PMT 
1 
PV 
1



Annuity due
I  1  I N 
Uneven cash flow streams
Nominal rates, periodic rates, effective annual rates
Loans amortisation
- Amortised schedule
48
Lecture 2 Revisiting Question of the Week
You are saving for retirement. You plan to retire when you
reach 65 years old. To live comfortably during retirement, you
plan to withdraw money from your savings account \$48,000
each year. Suppose today is your 25th birthday, and you
decide, starting today and continuing on every year up to
your 64th birthday, that you will put the same amount into a
savings account. If the interest rate is 2.4% per annum,
compounded annually, how much must you set aside each
year to make sure that you will have enough money in the
account on your 65th birthday in order to be able to withdraw
\$48,000 per year when you retire and you expect to live until
your are 90 years old (i.e., the first withdrawal is on your 65th
birthday and the last withdrawal occurs on your 89th
birthday)?
49
Lecture 2 Revisiting Questions of the Week
25th
0
| | |
PMT ….
64th 65th
39 40
| |
PMT
|
PMT …
|
|
89th 90th
64 65
|
|
|
PMT
-48k -48k
………
-48k
Step 1: Finding the amount of money you need to have at 65
years old in order to withdraw 48k every year when you retire.
Present value at t = 40
50
Lecture 2 Revisiting Questions of the Week
25th
64th 65th
89th 90th
0
39 40
64 65
| | |
PMT ….
| |
PMT
|
PMT …
|
|
|
|
|
PMT
-48k -48k
………
-48k
916,040
Step 2: Finding the amount of money you need to save each year,
starting today (t=0), until you are 64 years old (t=39).
PMT
51
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