1.
Chapter Five
– Time Value of Money

2.
Learning Goals
LG1 Understand the concepts of future value and present value, their calculation for
single amounts, and the relationship between them.
LG2 Find the future value and the present value of both an ordinary annuity and an
annuity due, and find the present value of a perpetuity.
LG3 Calculate both the future value and the present value of a mixed stream of
cash flows.
LG4 Understand the effect that compounding interest more frequently than annually
has on future value and the effective annual rate of interest.
LG5 Describe the procedures involved in (1) determining deposits needed to
accumulate a future sum, (2) loan amortization, (3) finding interest or growth rates, and
(4) finding an unknown number of periods.
© 2012 PEARSON EDUCATION 5-2

3.
3
Based on positive time preference
~ a ringgit today is worth more than a ringgit expected in
the future
TVM tools are used to;
 Calculate deposits required to accumulate a future sum
 Amortize loans by calculating loan payments schedules
 Determine interest or growth rates of money streams
 Evaluate perpetuities
 Find the required rate of return

4.
The Role of Time Value in Finance
•Most financial decisions involve costs & benefits that are
spread out over time.
•Time value of money allows comparison of cash flows from
different periods.
•Question: Your father has offered to give you some money and
asks that you choose one of the following two alternatives:
◦ RM1,000 today, or
◦ RM1,100 one year from now.
•What do you do?

5.
The Role of Time Value in Finance
(cont.)
•The answer depends on what rate of interest you could earn
on any money you receive today.
•For example, if you could deposit the RM1,000 today at 12%
per year, you would prefer to be paid today.
•Alternatively, if you could only earn 5% on deposited funds,
you would be better off if you chose the RM1,100 in one year.
5-5

6.
Future Value versus Present Value
•Suppose a firm has an opportunity to spend RM15,000
today on some investment that will produce RM17,000
spread out over the next five years as follows:
•Is this a wise investment?
•To make the right investment decision, managers need
to compare the cash flows at a single point in time.
5-6
Year Cash flow
1 $3,000
2 $5,000
3 $4,000
4 $3,000
5 $2,000

7.
7
Basic Concepts

8.
8
TMV Solution Methods
1. Numerical – using regular calculator w/o
financial functions
2. Interest Tables - given with the text book
a. Present Value Interest Factor
b. Present Value Interest Factor for Annuity
c. Future Value Interest Factor
d. Future Value Interest Factor for Annuity
3. Financial Calculator
4. Worksheet

9.
9
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.
CF0 CF1 CF3
CF2
Time 0 1 2 3
i%
End of period
2 & beginning
of period 3

10.
10
Time line illustrations
100 100
100
0 1 2 3
i%
100
0 1 2
5%
RM100 lump sum due in 2 years
End of
period 2
PV
FV
3 year RM100 ordinary annuity (fixed periodic payment)
10%

11.
11
100 50
75
0 1 2 3
i%
-50
Uneven cash flow stream
CF0 CF1
CF2 CF3

12.
12
12
Key Description
Clear all data
No of payment per year
No of year
Annual interest rate
Present value
Future value
(No keyed in) x (P/YR)
Begin End
Calculates amortization table
C ALL
P/YR
PV
FV
I/YR
N
xP/YR
BEG/END
AMORT

13.
13
Future and Present Values

14.
14
Future Value
Future Value – the value at a given future date of an amount placed
on deposit today and earning interest at a specified
rate. Found by applying compound interest over a
specified period of time.
Ending amount of your account at the end of n periods
In determining the final value of a cash flow or series of cash flows,
compound interest will be applied.
What is compound interest? Is it the same thing as simple interest?
The process of going from today’s values, or PV to future value is
called compounding.

15.
15
Present Value
•It is based on the idea that a dollar today is worth more
than a dollar tomorrow.
•Discounting cash flows is the process of finding present
values; the inverse of compounding interest.
Present Value is the current dollar value of a future amount—the
amount of money that would have to be invested today at a given
interest rate over a specified period to equal the future amount.
Beginning amount in your account

16.
16
Future Value & Present Value
BASIS FOR
COMPARISON
COMPOUNDING DISCOUNTING
Meaning The method used to
determine the future
value of present
investment
The method used to
determine the present value
of future cash flows
Concept If we invest some money
today, what will be the
amount we get at a
future date
What should be the amount
we need to invest today, t get
a specific amount in the future
Use of Compound Interest Rate Discount Rate
Known Present Value Future Value
Factor Future Value Factor or
Compounding Factor
Present Value Factor or
Discounting Factor
Formula FV=PV (1 + r)^n PV = FV / (1 + r)^n

17.
17
Simple
interest; 105 110
2nd period; (Principal) [(2 x 0.05) + 1 ] = RM110
or,
[100 (1.05) - 100] + 100(1.05) = RM110
0 1 2
5%
100
0 1 2
5%
Compounding
interest;
100 105 110.25
2nd period; (Principal + interest)(1 + i)
(RM100 + RM5) (1 + 0.05) = RM110.25
Future Value

18.
18
Mimi has decided to invest RM100 in a
savings account that earns 12% interest. She
invested for two years. How much would she
earn at the end of year 2?
Bill Smith deposited RM80 in a savings
account for 4 years at annual interest rate of
8%. What is Bill’s interest and compounded
amount?
Bill Smith deposited RM80 in a savings
account for 4 years at annual interest rate of
8%. What is Bill’s simple interest and
maturity value?
Activity 1

19.
19
Future Value : One period case
Case 1: Given that a car dealer offers a car for RM20,000 in cash or
RM25,000 on credit for one year. Given an annual i.r. of 10%,
which payment is better for you & which for the car dealer?
0 1
i% = ?
20,000 25,000
Given RM20,000 now, in 1 year at ir of 10%, the money
deposited will be ?
20,000
0 1
10%
FV = ?
Future Value

20.
20
FV1 = PV + INT
= PV + PV (i)
= PV ( 1 + i )
Numerical Solution (N/S) ;
FV1 = PV ( 1 + i )
= 20,000 ( 1 + 0.1)
= RM22,000
Future Value Interest Factor for i & n (FVIFi,n)
~ the future value of RM1 left on deposit for n
periods at a rate of i percent per period
~ where ( 1 + I )n = FVIFi,n
Tabular Solution (T/B) ;
FV1 = PV (FVIFi,n)
= 20,000 ( 1.1000)
= RM22,000
Future Value FV1 = PV + (1 + i) n

21.
21
21
Find the FV of RM20,000 given an interest rate of 10% in one year.
Data Key
Clear all data
1
No of payment per
year
1 No of year
10
Annual interest
rate
20,000 Present value
22,000.00
C All
P/YR
PV
FV
I/YR
N
+ / -

22.
22
Present Value : One period case
FV1 = PV ( 1 + i )n , so PV = FV1
(1 + i) n
C2 : Given the annual ir of 10%, at what amount of cash
would the car dealer be indifferent to receiving RM25,000
at time 1?
0 1
PV = ?
10%
25,000

23.
23
n
n
i
i
PVIF
)
1
(
1
)
( ,


Numerical solution ;
PV = FV1 / (1 + i)
= 25,000 = RM22,727.27
1 + 0.1
Tabular Solution ;
Present Value Interest Factor for i & n (PVIFi,n)
~ the present value of RM1 due n periods in the future
discounted at i percent per period
PV = FV (PVIFi,n)
= 25,000 (0.9091)
= RM 22,727.50
~ where;

24.
24
Find the PV of RM25,000 given an interest rate of 10% in one year.
Data Key
Clear all data
1
Payment per year
1
10
Annual interest
rate
25,000
-22727.27273
C ALL
P/YR
FV
PV
I/YR
N

25.
25
1. What is the future value of RM10,000 twenty years from now given interest
rate of 6%?
2. What is the present value of RM100,000 ten years from now given the
same annual rate of 6%
3. What will the present value be RM1,000 to be received eight years from
today if the discount rate is 5%?
4. Mandy plans to invest RM50,000 in a fixed deposit that pays 4% interest
annually for five years. Later, she plans to invest in another marketable
securities that can bring 6% interest per annum for another two years.
Advise Mandy how much her money would grown in seven years
Activity 2

26.
26

27.
27

28.
28
Future Value & Present Value : Multi – period case
Important terms;
Compound interest – interest earned on the principal & on the
accumulated interest
Discount interest rate – the rate that will make the future value
equivalent to the present value
Fair (Equilibrium) Value – the price at which investors are indifferent
btw buying or selling a security
0 1 2
discounting
5%
compounding

29.
29
The discount rate is often also referred to as the opportunity
cost, the required return, and the cost of capital.

30.
30
C3 : Find the FVo RM100 left for 3 years in an account paying
10 percent, annual compounding;
FV = ?
0 1 2 3
10%
100
FV1 = PV + INT
= PV + PV (i)
= PV ( 1 + i )
FV2 = FV1 (1 + i)
= PV ( 1 + i ) (1 + i)
= PV (1 + i)2
FV3 = PV (1 + i)3
N/S;
FV3 = 100 (1.10)3 = 133.10
FVn = PV (1 + i)n = PV (FVIFi,n)
= 100 (1.10)3 = 100 (1.3310)
= RM133.10
T/S;

31.
31
31
Data Key
1
3
10
-100
133.10
C ALL
P/YR
PV
FV
I/YR
N
+/-

32.
32
C4 : Find the PV of RM100 to be received in 3 years if the
appropriate ir is 10 percent, compounded annually:
100
0 1 2 3
10%
PV = ?
PVn = FV
(1 + i)n
PV3 = FV
(1 + i)3
N/S;
PVn = FVn/ (1 + i)n = FVn 1 n = FVn(PVIFi,n)
1 + i
= 100 (1/1.10)3 = 100 (0.7513)
= RM75.13
T/S;

33.
33
Data Key
1
3
10
100
-75.13
C ALL
P/YR
FV
PV
I/YR
N

34.
34
n (periods)
and
i (interest rate)

35.
35
Solving for n in TVM problems
C5: How long will it take a firm’s sales to double, if sales are
growing at a 15% rate?
0 15%
n = ?
n - 1
RM1 RM2
FVn = PV (1 + i)n
2 = 1 (1.15)n
2 = (1.15)n
FVn = PV (FVIFi,n)
(FVIFi,n) = FV / PV
= 2 / 1 = 2
T/S;
Look in FVIF Table for
(FVIF15%,n) = 2
n  5 periods
N/S;

36.
36
0 15%
n = ?
n - 1
RM1 RM2
Financial Calculator Solution ;
Data Key
1
1
15
2
4.96
C ALL
P/YR
FV
PV
I/YR
N
+/-

37.
37
Solving for interest rate
C6: What annual ir will cause RM100 to grow to RM125.97 in 3 years?
125.97
0 1 2 3
i = ?
100
T/S;
100 (1 + i) 100 (1 + i)2 100 (1 + i)3
100 (1 + i)3 = 125.97
100 (FVIFi,3) = 125.97
FVIFi,3 = 1.2597
Look at Row 3 of FVIF Table.
1.2597 is in the 8% column

38.
38
Tutorial Activity
• Differentiate between compounding and discounting
• What is future value and present value

39.
39
Activity 1
• You are offered RM2,500 today, RM10,000 in 12 years or RM25,000 in 25 years. Assuming
you can earn 10% on your money, which should you choose?
• Azrul has decided to place RM500, which he received as a birthday gift, in a saving account
paying 4% interest. How much will accrue to Azrul’s account in six years time.
• Faizal intends to buy a new car, the MyVi for RM57,650, but only has RM20,000 in cash. How
many years will it take for RM20,000 to grow to RM 57,650 if it is invested at 10% interest
compounded annually?
FVn = PV (FVIFi,, n)
• Let us say Faizal intends to buy the Myvi in five years’ time. At what rate must his RM20,000
be compounded annually for it to grow to RM57,650 in five years?

40.
40
1. What is the future value of RM10,000 twenty years from now given interest
rate of 6%?
2. What is the present value of RM100,000 ten years from now given the
same annual rate of 6%
3. What will the present value be RM1,000 to be received eight years from
today if the discount rate is 5%?
4. Mandy plans to invest RM50,000 in a fixed deposit that pays 4% interest
annually for five years. Later, she plans to invest in another marketable
securities that can bring 6% interest per annum for another two years.
Advise Mandy how much her money would grown in seven years
Activity 2

41.
41
Annuities & Perpetuities

42.
42

43.
43

44.
44

45.
45

46.
46
Annuities
An annuity is a series of equal payments made at fixed intervals
for a specific number of periods
Ordinary annuity - payments occur at the end of each
period
- eg. Students loan
Annuity due – payments are made at the beginning of
each period
- eg. Monthly rentals, insurance premiums

47.
47
What is the difference between an ordinary annuity and
an annuity due? Both are 3-yr annuities ( 3 pmts)
Ordinary Annuity
PMT PMT
PMT
0 1 2 3
i%
PMT PMT
0 1 2 3
i%
PMT
Annuity Due

48.
48
Future Value of an Ordinary Annuity
C7: What is the future value of an ordinary annuity of RM100 per period
for 3 yrs if the ir is 10 percent, compounded annually?
Time line approach;
0 1 2 3
10%
100 100 100
110
121
331
100(1 + i)2
100(1 + i)
+
Twice
compounding

49.
49
N/S; FVA3 = PMT (1 + i) + PMT (1 + i)1 + PMT (1 + i)2
= 100 (1) + 100 (1.10) + 100 (1.21)
= RM331
T/S;
FVAn = PMT (FVIFAi,n)
FVA3 = 100 (FVIFA10%,3)
= 100 (3.3100)
= RM331
FC;
= 331.00
10 3
-100
Make sure no BGN sign
PMT I/YR N FV
P/YR
1

50.
50
Future Value of an Annuity Due
C8: What is the future value of RM100 payments made at beginning of
each year for 3 yrs in a saving account that pays 10 percent,
compounded annually?
Time line approach;
0 1 2 3
10%
100 100 0
110
121
133.10
100(1 + i)2
100(1 + i)
+
Triple
compounding
100
100 (1 + i)3
364.10

51.
51
N/S; FVAD3 = PMT (1 + i)3 + PMT (1 + i)2 + PMT (1 + i)1
= 100 (1.331) + 100 (1.21) + 100 (1.10)
= RM364.10
T/S;
FVADn = FVA3 (1 + i) or = PMT (FVIFA10%,3) (1 + i)
= 331 (1.10) = 100 (3.3100) (1.10)
= RM364.10 = 364.10
FC;
= 364.10
10 3
-100
Make sure BGN sign
P/YR PMT I/YR N
1
FV
BEG/END

52.
52
Present Value of an Ordinary Annuity
C9: What is the PV of an annuity of RM100 per period for 3 years if the
ir is 10 percent annually?
Time line approach;
0 1 2 3
10%
100 100
100 /(1 + i)2
100 / (1 + i)
Triple
discounting
0
100 / (1 + i)3
90.91
82.64
75.13
248.68
+
100

53.
Activity Three
1. What is the present value of a 5 year RM 1,000 ordinary annuity discounted back to the
present at 10%
2. How much must Sharifah deposit at the end of each year in a savings account earning 10%
annual interest to accumulate RM10,000 at the end of six years?
3. What is the present value of a 4year RM1,000 annuity discounted back to the present of 9%?
4. What is the present value of a RM10,000 perpetuity discounted at 8%?
53

54.
Activity Four
1. Danny deposits RM 1200 at the end of each year into an ordinary annuity that pays 6%
annual interest at the beginning of each year. What is the future value at 5 years?
2. Kamal deposit RM100 each month into an annuity pays of 6% annual interest at the end of
each year. How much money she have at this account at the end of 5 years?
3. Rose wants to have RM800,000 in an annuity by the time he retires 30 years from now. If the
annuity pays a fixed interest of 5% at the end of each year, how much money should she deposit
into his account for the next 30 years?
54

55.
Activity Five
1. Mazni has decided to invest RM100 in a savings account paying 12% interest for two year.
How much will she earn?
2. What is the present value of a RM600 payment you expect to receive in three years if the
rate of interest is 12% compounded annually?
3. Liza is certain of receiving RM50,000 10 years from now. She wishes to sell this future
amount to Amrul. What maximum price can Amrul afford to pay if he intends to earn 7% per
year over the next 10 years? RM
4. In order for Lai Ming to enjoy a holiday in Japan, she will need RM 7,480. She is able to
save RM 729.04 at the end of each year for this purpose. She has discovered that a mutual
fund pays 7% interest compounded annually. How long will it be before Lai Ming can embark
on her holiday?
5. What is the present value of a 5year ordinary annuity with annual payment of RM200 and
discounted at 15% interest?
6. What is the present value of 16 year annuity if the payments are RM2,000 per year and the
rate of return is 7%?
55

56.
56
Present Value of an Annuity Due
C10: How much lump sum today to make it equivalent with a 3 year
annuity paying RM100 at beginning of each year?
Time line approach;
0 1 2 3
10%
100 100
100 /(1 + i)2
100 / (1 + i)
Double
discounting
100
90.91
82.64
273.55
+
Make sure BGN sign

57.
57
An annuity due will always be greater than an
otherwise equivalent ordinary annuity because
interest will compound for an additional period.

58.
58
Perpetuities
- is a stream of equal payments expected to continue
forever
- a type of annuity
PV (Perpetuity) = payment = PMT
interest rate i
the current price

59.
59
C11: A perpetual bond promised to pay RM100 per year in perpetuity. What would the bond’s
worth today if the opportunity cost, or discount rate was 5 percent
PV (Perpetuity) = RM100 = RM2000
0.05
As the interest rate increases, the perpetuity’s value will drop.
When ir = 10%;
PVp = 100 = RM1000
0.1

60.
60
Uneven Cash Flow Stream
Payment (PMT) - equal cash flows at regular intervals
Cash flow (CF) - uneven cash flows
Examples of uneven cash flows;
- common stock’s dividend
- returns from fixed asset investments
~ production income
~ rentals

61.
61
Present Value of an Uneven Cash Flow Stream
C12: Find PV of the following cash flows stream, discounted at 10%
0 1 2 3 4
10%
0 100 300 300 -50
PV = CF0 1 0 + CF1 1 1 + CF2 1 2
1 + i 1 + i 1 + i
+ CF3 1 3 + CF4 1 4
1 + i 1 + i
CF0 CF1 CF2 CF3 CF4

62.
62
For cash flow calculation ;
Key
Clear all
No of periods per year
Cash flow j
No of consecutive times CFj occurs
Internal rate of return per year
Net present value
C ALL
P/YR
Nj
CFj
IRR/YR
NPV

63.
63
Key
0
100
300
300
50
10
530.09
C ALL
I/YR
CFj
NPV
CFj
CFj
CFj
+/-
Key
0
100
300
2
50
10
530.09
C ALL
I/YR
CFj
NPV
CFj
CFj
Nj
+/-
CFj CFj

64.
64
0 1 2 3 4 5
10%
0 100 50 200 200 200
C13: Find the PV of the following c/f discounted at 10%
497.38
200 (PVIFA10%,3)
= 200 (2.4869)
100(1/1 + i)1
50(1/1 + i)2
497.38(1/1 + i)2 or
200 (PVIFA10%,3) or 200 (1/1 + i)3
200 (PVIFA10%,4) or 200 (1/1 + i)4
200 (PVIFA10%,5) or 200 (1/1 + i)5
90.91
41.32
411.03
543.26
150.26
136.60
124.18

65.
65
Future Value of an Uneven Cash Flow Stream
C14: Find FV of the following cash flows stream, compounded at 10%
0 1 2 3 4 5
10%
0 100 50 200 200 200
FV = CF5 (1 + i)0 + CF4 (1 + i)1 + CF3 (1 + i)2
+ CF2 (1 + i)3 + CF1 (1 + i) 4 + CF0 (1 + i)5
CF0 CF1 CF2 CF3 CF4 CF5

66.
66
0 1 2 3 4 5
10%
0 100 50 200 200 200
420 ( 1 + i)
200 (FVIFA10%,2) = 200 (2.1)
200 (1 + i)2
200 (1 + i)
50 (1 + i)3 = 50 (1.331)
100 (1 + i)4 = 100 (1.4641)
220
242
66.55
146.41
NFV = 874.96
462
462

67.
67
0 1 2 3 4 5
10%
0 100 50 200 200 200
Data Key
1
0
100
50
200
3
C ALL
P/YR
I/YR
CFj
CF0 CF1 CF2 CF3 CF4 CF5
NFV=?
Data Key
10
543.28
5
0
847.93
CFj
Nj
CFj
NPV
N
PMT
FV
CFj
Using formula;
NFV = NPV ( 1 + i)n
= 543.26 (1 + 0.1)5
= 874.93

68.
Different Compounding Periods
- annually / semiannually / quarterly / monthly / daily
compounding
- the quoted interest rate is normally the annual one.
- If bank promised 10% annual interest rate semiannually
what does it means ?
~ interest will be added each 6 months but
will the interest be 10% as quoted or more/less?

69.
Say that you want to deposit your money in the bank
which offer you the highest return. As you shopped
around, you come up with these rates:
Bank A : 15 percent, compounded daily
Bank B : 15.5 percent, compounded quarterly
Bank C : 16 percent, compounded annually
Which one has the best rates for
deposits?

70.
C15: A bank declares that it pays a 6% annual ir semiannually & you
want to deposit RM 100. What is FV at the end of 3rd year?
P/YR 1
PV (-)100
n 3
I% 6
FV 119.1
P/YR 1 2
PV (-)100 (-)100
N 6 6
I% 6/2 6
FV 119.41 119.41
0 1 2 3
6%
Annual compounding
Semiannual compounding
0 1 2 3 4 5 6
i%
FV3 = PV (1 + i)3
= 100 (1 + 0.06)3 = RM119.10
-100 FV = ?
-100 FV = ?
FV6 = PV (1 + i)6
= 100 (1 + 0.03)6
= RM119.41
i% = 6% /2 = 3%
n = 3 x 2 = 6

71.
Different compounding periods are used for different
types of investment
In order to compare securities with different
compounding periods, need to put them on a common basis.
Types of interest rates;
 Nominal or quoted interest rates
 Annual percentage rates (APR)
 Effective annual rates (EAR)

72.
1. Nominal or Quoted interest rates , inom
Is the contracted, or stated, or declared ir.
The rate which is given by the bank or issuer.
Annual Percentage Rate (APR)
The interest rate charged per period multiplied by the
number of periods per year.
C16: If a bank is charging 1.2% per month on car loans, what is the APR?
APR = 1.2% x 12
= 14.4%
It is the nominal rates for loan that some government
requires the bank to display to customers.

73.
Periodic rate is the nominal rate at each period;
where m is the no of compounding
periods per year
Eg: 6% compounded quarterly.
periodic rate, iper = inom / m
= 6% / 4 = 1.5%
Periodic Rate
m
i
i nom
per 

74.
So periodic rate is the rate charged by a lender or paid
by a borrower in each period.
C17: A bank charges 18% annual interest rates monthly on credit
card loans, what is the periodic rate?
iper = inom / m
= 18% / 12
= 1.5%
Bank will charge 1.5% of
interest monthly or per month
So if we delayed paying our credit card debt for a year,
will the debt be the same as we take a loan of the same
amount at 18% annual interest?

75.
Notice that iper is the rate that is shown on time lines
and used in certain calculations, not the annual rate.
C18: How much would you have at the end of the 2nd year when you
make RM100 deposit in an account that pays 12% interest
rate semiannually.
0 1/2 1 1/2 2 (x2) = N
6%
-100 FV = ?
For calculation of FV given only PV, must use this
rate not the annual rate given in this case. AND
maintain P/YR = 1.

76.
2. Effective Annual Rates (EAR) or (EFF)
The rate which would produce the same ending (future) value if
annual compounding has been used.
~ (the interest rate expressed as if it were
compounded once per year)
0 1 2 3
6%
Annual compounding
Semiannual compounding
0 1 2 3 4 5 6
3%
-100 119.41
-100 119.10
0 3
i = ?
-100 119.41
EAR ;
The annual
rate that
produces the
same FV as
if we had
compounded
at a given
periodic rate
m times per
year

77.
An investor would be indifferent between an
investment offering a 10.25% annual return and
one offering a 10% annual return, compounded
semiannually.
Why?
C19: EFF% or EAR for 10% semiannual investment.
%
EFF
10.25%
1
2
0.10
1
1
m
i
1
EAR
or
EFF%
2
m
nom




















78.
C20: What is the FV of RM100 compounded semiannually for 3 years if
inom = 10%? Would it be different if it were compounded
quarterly?
Quarterly compounding
Semiannual compounding
0 1 2 3
5%
-100 FV = ?
0 1 2 3
-100 FV = ?
2.5%
134.01
134.49

79.
FVn = PV 1 + inom
mn
m
FV3 = 100 1 + 0.1 2(3)
2
= 100(1.05)6
= 134.01
EAR = ( 1 + inom / m )m - 1
= ( 1 + 0.10 / 2 )2 – 1
= 10.25%
Semiannual compounding
Quarterly compounding
FVn = PV 1 + inom
mn
m
FV3 = 100 1 + 0.1 4(3)
4
= 100(1.025)12
= 134.49
EAR = ( 1 + inom / m )m - 1
= ( 1 + 0.10 / 4 )4 – 1
= 10.38%

80.
F/C ;
Given annual i.r. of 10%, compounded semiannually;
10.25
To find APR, given the EAR of
10.25%;
10.00
The APR formula;
APR = 1 + EFF 1/m - 1 x m x 100
100
NOM%
10
2 P/YR
EFF% EAR
NOM%
P/YR
EFF%
10.25
2

81.
Compounding More Frequently
than Annually
• Compounding more frequently than once a year results in
a higher effective interest rate because you are earning
on interest on interest more frequently.
• As a result, the effective interest rate is greater than
the nominal (annual) interest rate.
• Furthermore, the effective rate of interest will
increase the more frequently interest is compounded.

82.
Nominal & Effective Rates
The nominal interest rate is the stated or contractual rate
of interest charged by a lender or promised by a borrower.
The effective interest rate is the rate actually paid or
earned.
The effective rate > nominal rate whenever compounding
occurs more than once per year
EAR > inom
EAR = inom = iper
If compounding occurs only once a year, then;

83.
Nominal & Effective Rates
C21: What is the effective rate of interest on your credit card
if the nominal rate is 18% per year, compounded monthly?
EAR = (1 + .18/12) 12 -1
EAR = 19.56%

84.
Why is it important to consider effective
rates of return?
An investment with monthly payments is different
from one with quarterly payments. Must put each
return on an EFF% basis to compare rates of
return. Must use EFF% for comparisons. See the
following values of EFF% rates at various
compounding levels.
EARANNUAL 10.00%
EARQUARTERLY 10.38%
EARMONTHLY 10.47%
EARDAILY (365) 10.52%

85.
Fractional Time Periods
Before, payments only occur at beginning or end of
periods.
What if, payments occur at some date within a period?
0 1st 2nd
month 20th day month
1%
-100 FV = ?

86.
C22: Deposits RM100 in a bank that pays 12% ir compounded
monthly. How much the amount will be then in 1 month and 20
days?
N/S;
FVn = PV (1 + i)n
= PV (1 + i)1+ 20/30
= 100(1 + 0.1)1.67
= RM101.68
F/C ;
N 20÷30+1
I/YR 1
PV (-)100
P/YR 1
FV 101.67
If use 360-day year;
iper = 0.12 /360
= 0.00033333 per day
No of days deposited;
= 50 days
FVn = PV (1 + iper)n
= 100 ( 1.0003333)50
= RM 101.68

87.
Note of Caution;
Rule 1: Money saved/deposited/invested should be in
negative sign. Money withdrawn/received should
be in positive sign.
C23: RM1000 is deposited today for a semiannual
payment of RM300 for 3 years. Given an interest
rate of 10% semiannually, how much would be left in
the account in 3 years time?
0 1 2 3
5%
-1000 300 300 300 300 300 300
FV = ?

88.
Rule 2 : If there is only PV & no PMT, either;
a. If use periodic ir, keep P/YR = 1.
b. If use nominal rate, change P/YR accordingly.
C24: RM1000 is deposited today. Given an interest rate of 10%
semiannually, how much would be in the account in 3 years time?
0 1 2 3
5%
-100 FV = ?
N 3x2 = 6
I/YR 10÷2 = 5
PV (-)100
PMT 0
P/YR 1
FV 134.01
N 3x2 = 6
I/YR 10
PV (-)100
PMT 0
P/YR 2
FV 134.01

89.
Rule 3: For case with PMT or PMT and PV, N = no of
payment made, I/YR = annual interest rate, P/YR =
no of payments made per year.
0 1 2 3
8%
-100 -150 -150 -150 -150 -150 -150
FV = ?
C25: Interest is 8% compounded quarterly. Initial deposit is RM100,
and regular payments of RM150 will be made every
semiannually.
2 P/YR
3x2 = 6 N
8.08 I/YR
(-)100 PV
(-)150 PMT
1122.77 FV
8 NOM%
4 P/YR
8.24 I/YR
8.24 EFF%
2 P/YR
8.08 NOM%

90.
C26: Someone offers to sell you a note calling for the pmt of RM1000,
15 months from today for RM850. You have RM850 in the bank,
which pays a 7% nominal rate with daily compounding. Should you
buy the note or leave your money in the bank.
An Example of Everything
0 456 days
-850 1,000
iper = 7%/365 = 0.0192%
How to solve this? Have to compare both investments
on similar grounds;
Fvnote vs. FVbank PVnote vs. PVbank EARnote vs. EARbank

91.
Fvnote vs. Fvbank
Bank : FV = 850 (1.000192)456 = 927.67
Note : FV = 1,000 Buy note
(more value in future)
PVnote vs. Pvbank
Bank : PV = RM850
Note : PV = 1,000 (1.000192)-456 = 916.27
Buy note (more value now)
EARnote vs. EARbank
Bank : iper = 0.0192%
Note : 1,000 = 850 (1 + i)456, solving i = 0.0356%
Buy note (higher iper means higher EAR)

92.
C27: Cost of note = RM850
PMT = RM190 quarterly for 5 quarters
inom = 7% compounded daily
Is this a good investment?
0 91 182 274 366 456 days
-850 190 190 190 190 190
iper for bank = 7% / 365 = 0.0192%

93.
PVAnote = 190 (1.000192)-91 + 190 (1.000192)-182 + … +
190 (1.000192)-456
= 901.68
PVApocket = 850 Buy note (more value now)
EARnote ; finding iper;
inom = (iper) (m)
= (3.83) (4) = 15.3%
So for daily rate = inom / 365 = 0.0419%
Buy note coz iper,note > iper, bank = 0.0192%
-850 CFj
190 CFj
5 Nj
IRR 3.82586 Quarterly iper

94.
Loan Types
1. Pure Discount Loans
- the borrower receives money today & repays a single
sum at some time in the future
- eg. A 1-year, 10% RM100 pure discount loan, would
require the borrower to repay RM110 in one year.
2. Interest-Only Loans
- a loan that has a repayment plan that calls for the
borrower to pay interest each period & repay the
entire principal (original loan amount) at some point
in the future
- eg. With a 3-year, 10%, interest-only loan of RM1000, the
borrower would pay RM1000(0.1) = 100 in interest at the end of
1st & 2nd years. At the end of 3rd year, the borrower would
return the principal along with RM100 in interest for that year.

95.
3. Amortized Loans
- a loan that is repaid in equal payments over its life.
- eg. Car & home loans
C28: Say, borrow RM1,000 at 10% interest and have to pay equally
at the end of each of the next 3 years.
0 1 2 3
-1,000 PMT PMT PMT
10%
T/S ;
PVAn = PMT (PVIFAi,n)
1,000 = PMT(PVIFA10%,3)
PMT = 1,000 / (2.4869)
= RM402.11
F/C ; 3 N
10 I/YR
-1000 PV
0 FV
PMT 402.11

96.
Constructing an amortization table:
Repeat steps 1 – 4 until end of loan
Interest paid declines with each payment as the balance
declines.
Year Beginning
Balance (1)
PMT
(2)
Interest
(3)
Princip
Repmt (4)
End
Balance
1 RM1,000 RM402 RM100 RM302 RM698
2 698 402 70 332 366
3 366 402 37 366 0
Total 1,206.34 206.34 1,000 -
PVn(1 + i) (2) – (3) (1) – (4)

97.
97
1 P/YR
3 N
10 I/YR
-1000 PV
0 FV
PMT 402.11
1 INPUT
AMORT 1-1
= 302.11 PRIN
= 100.00 INT
= -697.89 BAL
1 INPUT 2
AMORT 1-2
= 634.43 PRIN
= 169.79 INT
= -365.57 BAL