- Basic Finance

1. The Time Value of Money
TOPIC 3

2. Learning Objectives
1. Define the time value of money
2. The significance of time value of money in financial
management
3. Define and understand the conceptual and
calculation of future and present value in cash
flows
4. Define the meaning of compounding and
discounting
5. Work with annuities and perpetuities
WRMAS 2

3. TIME VALUE OF MONEY
 Basic Principle : A dollar received today is worth more than a
dollar received in the future.
• This is due to opportunity costs. The opportunity cost of
receiving $1 in the future is the interest we could have earned if
we had received the $1 sooner.
• Example
Invest RM1 today at a 6% annual interest rate. At the end of the
year you will get $1.06.
SO You can say:
1. The future value of RM1 today is $1.06 given a 6% interest
rate a year. OR WE CAN SAY
2. The present value of the $1.06 you expect to receive in
one year is only $1 today.
WRMAS 3

4.  Translate $1 today into its equivalent in the future
(compounding) – Future Value
Today Future
?
 Translate $1 in the future into its equivalent today
(discounting)- Present Value
Today Future
?
WRMAS 4

5. SIGNIFICANCE OF TIME VALUE OF
MONEY
• This concept is so important in understanding financial
management.
• We must take this time value of money into consideration
when we are making financial decisions.
• It can be used to compare investment alternatives and to
solve problems involving loans, mortgages, leases, savings,
and annuities.
WRMAS 5

6. COMPOUND INTEREST AND FUTURE
VALUE
• Future value is the value at a given future date of an amount
placed on deposit today and earning interest at a specified rate.
• Compound interest is interest paid on an investment during the
first period is added to the principal; then, during the second
period, interest is earned on the new sum (that includes the
principal and interest earned so far). (Process of determining FV)
• Principal is the amount of money on which interest is paid.
WRMAS 5-6

7. Simple Interest
• Interest is earned only on principal.
• Example: Compute simple interest on $100 invested
at 6% per year for 3 years.
– 1st year interest is $6.00
– 2nd year interest is $6.00
– 3rd year interest is $6.00
– Total interest earned: $18.00
5-7

8. Compound Interest and Future
Value
Example:
Compute compound interest on $100 invested at 6% for 3
years with annual compounding.
1st year interest is $6.00 Principal is $106.00
2nd year interest is $6.36 Principal is $112.36
3rd year interest is $6.74 Principal is $119.11
Total interest earned: $19.10
WRMAS 8

9. The Equation for Future Value
• We use the following notation for the various inputs:
– FVn = future value at the end of period n
– PV = initial principal, or present value
– r = annual rate of interest paid. (Note: On financial
calculators, I is typically used to represent this rate.)
– n = number of periods (typically years) that the money is
left on deposit
OR
FVn = PV (1+r)n FVn = PV (FVIFr,n)
WRMAS 9

10. Future Value Example
Example: What will be the FV of $100 in 2 years at interest rate of
6%?
Manually Table
FV2= $100 (1+.06)2 FV2= PV(FVIF6%,2)
= $100 (1.06)2 = $100 (1.1236)
= $112.36 SAME ANSWER! = $112.36
Exercise
Jane Farber places $800 in a savings account paying 6% interest
compounded annually. She wants to know how much money will be
in the account at the end of 5 years.
WRMAS 10

11. Future Value
Changing I, N and PV
• Future Value can be increased or
decreased by changing:
Increasing number of years of compounding
(n)
Increasing the interest or discount rate (i)
Increasing the original investment (PV)
WRMAS 11

12. FV- Changing i, n, and PV
Exercise
(a) You deposit $500 in a bank for 2 years. What is the FV at 2%?
What is the FV if you change interest rate to 6%?
FV at 2% = 500 (1.0404) = ?
i ( OR ) FV ?
FV at 6% = 500 (1.1236) = ?
(b) Continue same example but change time to 10 years. What is
the FV now?
n ( OR ) FV ?
FV at 6% = 500 (1.7908) = ?
(c) Continue same example but change contribution to $1500.
What is the FV now?
PV ( OR ) FV ?
FV at 6%, year 10 = 1500 (1.7908) = ?
WRMAS 12

13. Future Value Relationship
We can increase the FV by:
1. Increasing the number of years for which money is invested;
and/or
2. Investing at a higher interest rate.
WRMAS 13

14. FV- Finding I and n
Example 1
At what annual rate would the following have to be invested ; $500
to grow to $1183.70 in 10 years.
1183.70 = 500 (FVIFi,,10)
1183.70/500 = FVIFi,10
2.3674 = FVIFi,10 (refer to FVIF table)
2.3674 = 9%
Example 2
How many years will the following take? $100 to grow to $672.75 if
invested at 10% compounded annually.
$672.75 = $100 (FVIF10%,n)
672.75/100 = FVIF10%,n
6.7275 = FVIF10%,n (refer to FVIF table)
6.7275 = 20 years
WRMAS 14

15. Exercise-Finding i and n
a) How many years will the following take :
i. $100 to grow to $298.60 if invested at 20% compounded
annually
ii. $550 to grow to $1,044.05 if invested at 6% compounded
annually
b) At what annual rate would the following have to be invested :
i. $200 to grow to $497.65 in 5 years
ii. $180 to grow to $485.93 in 6 years
WRMAS 15

16. DISCOUNT INTEREST AND PRESENT
VALUE
 Present value reflects the current value of a future payment or
receipt.
 How much do I have to invest today to have some amount in
the future?
 Finding Present Values (PVs) = discounting
Example
You need RM400 to buy textbook next year. Earn 7% on your
money. How much do you have to put today?
WRMAS 16

17. PRESENT VALUE
 Formula of Present Value (PV):
FVn
PV = or PV = FVn (PVIFi,n)
(1+i )n
Where;
FVn = the future value of the investment at the end of n years
n = number of years until payment is received
i = the interest rate
PV = the present value of the future sum of money
FVIF = Future value interest factor or the compound sum $1
[ 1/(1+i)n ] is also known as discounting factor
WRMAS 17

18. Present Value
Example :
What is the PV of $800 to be received 10 years from
today if our discount rate is 10%.
Manually
PV = 800/(1.10)10
= $308.43
Table
SAME ANSWER!
PV = $800 (PVIF 10%,10yrs)
= $800 (0.3855)
= $308.40
WRMAS 18

19. Present Value Exercise
Exercise 1 (finding PV)
Pam Valenti wishes to find the present value of $1,700 that will be
received 8 years from now. Pam’s opportunity cost is 8%.
Exercise 2 (changing i)
Find the PV of $10,000 to be received 10 years from today if our
discount rate is:
a) 5% b) 10% c) 20% i ( OR ) PV ?
Exercise 3 (finding n)
How many years will it take for your initial investment of RM7,752
to grow to RM20,000 with a 9% interest ?
n ( OR ) PV ?
WRMAS 19

20. Present Value Relationship
PV is lower if:
1. Time period is longer; and/or
2. Interest rate is higher.
WRMAS 20

21. ANNUITY
An annuity is a series of equal payments for a specified numbers
of years. These cash flows can be inflows of returns earned on
investments or outflows of funds invested to earn future returns.
There are 2 types of annuities*:
- An ordinary annuity is an annuity for which the cash flow
occurs at the end of each period (much more frequently in
finance)
- An annuity due is an annuity for which the cash flow occurs
at the beginning of each period.
Note: An annuity due will always be greater than an ordinary
annuity because interest will compound for an additional period.
WRMAS 21

22. Ordinary Annuity-PV
a) Present Value of Annuity (PVA)
• Pensions, insurance obligations, and interest owed on bonds are
all annuities. To compare these three types of investments we
need to know the present value (PV) of each.
Formula:
PVAn = PMT [1-(1+i)-n] PVAn = PMT (PVIFAi,n)
i or
WRMAS 22

23. Ordinary Annuity-FV
b) Future Value of Annuity (FVA)
• Depositing or investing an equal sum of money at the end of
each year for a certain number of years and allowing it to
grow.
Formula
FVAn = PMT (1+ i)n -1 or FVAn = PMT (FVIFAi,n)
i
WRMAS 23

24. FV of Annuity: Changing PMT, N & r
1. What will $5,000 deposited annually for 50 years be worth
at 7%?
– FV= $2,032,644
– Contribution = 250,000 (= 5000*50)
2. Change PMT = $6,000 for 50 years at 7%
– FV = 2,439,173
– Contribution= $300,000 (= 6000*50)
3. Change time = 60 years, $6,000 at 7%
– FV = $4,881,122
– Contribution = 360,000 (= 6000*60)
4. Change r = 9%, 60 years, $6,000
– FV = $11,668,753
– Contribution = $360,000 (= 6000*60)
24

25. ANNUITY DUE
• Remember-Annuity due is ordinary annuities in which all
payments have been shifted forward by one time period.
a) Future Value of Annuity Due (FVAD):
FVADn = PMT (FVIFAi,n) (1+i)
b) Present Value of Annuity Due (PVAD) formula:
PVADn = PMT (PVIFAi,n) (1+i)
WRMAS 25

26. Earlier, we examined this “ordinary”
annuity:
500 500 500
0 1 2 3 …….……5
Using an interest rate of 5%, we find that:
• The FVA (at 3) is $2,818.50
• The PVA (at 0) is $2,106.00
HOW ABOUT ANNUITY DUE?
• FVAD5 (annuity due) = PMT{[(1 + r)n – 1]/r}* (1 + r)
= 500(5.637)(1.06)
= $2,987.61
• PVAD0 = $2,818.80
WRMAS 26

27. Annuity Exercise
Exercise 1
Fran Abrams wishes to determine how much money she will have at
the end of 5 years if he chooses annuity A that earns 7% annually
and deposit $1,000 per year.
Exercise 1
Branden Co., a small producer of plastic toys, wants to determine the
most it should pay to purchase a particular annuity. The annuity
consists a cash flows of $700 at the end of each year for 5 years. The
required return is 8%.
Exercise 3
Determine the answers for exercise 1 and 2 on annuity due.
WRMAS 27

28. FV and PV With Non-annual Periods
Non-annual periods : not annual compounding but occur
semiannually, quarterly, monthly…
– r = stated rate/# of compounding periods
– N = # of years * # of compounding periods in a year
• If semiannually compounding :
FV = PV (1 + i/2)m x 2 or FVn = PV (FVIFi/2,nx2)
• If quarterly compounding :
FV = PV (1 + i/4)m x 4 or FVn = PV (FVIFi/4,nx4)
• If monthly compounding :
FV = PV (1 + i/12)m x 12 or FVn = PV (FVIFi/12,nx12)
How about PV?
WRMAS 28

29. Compound Interest With Non-annual Periods
Example 1: If you deposit $100 in an account earning 6% with
semiannually compounding, how much would you have in the
account after 5 years?
Manually Table
FV5 = PV (1 + i/2)m x 2 FV5 = PV (FVIFi/2, nx2 )
= 100 (1 + 0.03 )10 = 100 (FVIF 3%,10)
= 100 (1.3439) = $134.39 = 100 (1.3439) = $134.39
Example 2: If you deposit $1,000 in an account earning 12% with
quarterly compounding, how much would you have in the
account after 5 years?
Manually Table
FV5 = PV (1 + i/4)m x 4 FV5 = PV (FVIFi/4, nx4 )
= 1000 (1 + 0.03)20 = 1000 (FVIF 3%,20)
= 1000 (1.8061) = $1806.11 = 1000 (1.8061) =$1806.11
WRMAS 29

30. Exercise-Non Annual
Exercise 1
How much would you have today, if RM1,000 is being discounted at
18% semiannually for 10 years.
Exercise 2
Calculate the PV of a sum of money, if RM40,000 is discounted back
quarterly at 24% per annum for 10 years.
Exercise 3
Paul makes a single deposit today of $400. The deposit will be invested
at an interest rate of 12% per year compounded monthly. What will be
the value of Paul’s account at the end of 2 years?
Exercise 4
Consider a 10-year mutual fund in which payments of $100 are made at
the beginning of each month. What is the amount today if the annual
rate of interest is 5%?
WRMAS 30

31. Quoted Vs. Effective Rate
• We cannot compare rates with different compounding periods.
5% compounded annually is not the same as 5%
compounded quarterly.
• To make the rates comparable, we must calculate their
equivalent rate at some common compounding period by using
effective annual rate (EAR).
• In general, the effective rate > quoted rate whenever
compounding occurs more than once per year.
WRMAS 31

32. Quoted Vs. Effective Rate
Example 1
RM1 invested at 1% per month will grow to RM1.126825
(=RM1.00(1.01)12) in 1 year. Thus even though the interest rate may
be quoted as 12% compounded monthly, the EAR is:
EAR = (1 + .12/12)12 – 1 = 12.6825%
Example 2
Fred Moreno wishes to find the effective annual rate associated
with an 8% quoted rate (r = 0.08) when interest is compounded (1)
annually (m = 1); (2) semiannually (m = 2); and (3) quarterly (m = 4).
WRMAS 32

33. PERPETUITY
• A perpetuity is an annuity that continues forever.
• The present value of a perpetuity is
PV = PP
i
PV = present value of the perpetuity
PP = constant dollar amount provided by the perpetuity
i = annuity interest (or discount rate)
Example
What is the present value of $2,000 perpetuity discounted back to
the present at 10% interest rate?
= 2000/.10 = $20,000
WRMAS 33

34. Perpetuity Exercise
Exercise
What is the Present Value of the following :
- A $100 perpetuity discounted back to the present at
12%
- A $95 perpetuity discounted back to the present at
5%
i ( OR ) P? $ ( OR ) PV ?
WRMAS 34

35. WRMAS 35