WEEK 5: LECTURE 5
TIME VALUE OF
MONEY
PRINCIPLES OF FINANCIAL
ANALYSIS
Lecturer: Chara Charalambous
1
Lecturer: Chara Charalambous
2
LEARNING OUTCOMES
1. Understand what is meant by "the time value of
money."
2. Understand the relationship between present
and future value.
3. Calculate both the future and present value of:
(a) an amount invested today; (b) a stream of
equal cash flows (an annuity)
4. Distinguish between an “ordinary annuity” and
an “annuity due.”
5. Use interest factor tables and understand how
they provide a shortcut to calculations
Lecturer: Chara Charalambous
3
Definition of 'Time Value of Money - TVM'
• The idea that money available at the present time is
worth more than the same amount in the future due to
its possible earning capacity. This core principle of
finance means that money can earn interest when
invested and so any amount of money is worth more the
sooner it is received.
• A dollar received today worth more than a dollar
expected to receive in future because the sooner a dollar
received the quicker it can be invested to earn a positive
return.
The time value of money is the central concept in finance
theory.
Lecturer: Chara Charalambous
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• Money received sooner rather than later
allows one person to use the funds for
investment or consumption purposes. This
concept is refer to as the TIME VALUE OF
MONEY.
Lecturer: Chara Charalambous
5
Time Lines
One of the most important tools in time value of money
analysis is the time line, which is used to help us picture
what is happening in a particular situation.
TIME:
Time 0 is today, Time 1 is one period from today
(e.g. one year), Time 2 is two periods from today
Lecturer: Chara Charalambous
6
Time Lines
TIME:
5%
Cash
outflow: -100
Cash
inflow: + ?
Here the interest rate if I invest my money for each of the four
periods is 5%. A cash outflow is a payment of cash for
investments and is made at Time 0: because is money given
out of my pocket it has a minus sign. At Time 4 I have a cash
inflow : a receipt of cash from an investment. The inflow is
unknown to me and I have to find it thus I symbolize it with ?
And is a positive amount (+) because I will receive money in my
pocket. Note that no cash flows occur at Time 1,2 and 3.
Lecturer: Chara Charalambous
7
Example
• Congratulations!!! You have won a cash prize!
You have two payment options: A - Receive
$10,000 now OR B - Receive $10,000 in three
years. Which option would you choose?
Lecturer: Chara Charalambous
8
If you're like most people, you would choose to receive the $10,000
now. After all, three years is a long time to wait. Why would any
rational person postpone payment into the future when he or she
could have the same amount of money now? For most of us, taking
the money in the present is just basic natural. So at the most basic
level, the time value of money demonstrates that, all things being
equal, it is better to have money now rather than later.
Back to our example: by receiving $10,000 today, you are able to
increase the future value of your money by investing and gaining
interest over a period of time. For Option B, you don't have time on
your side, and the payment received in three years would be your
future value. To illustrate, we have provided a timeline:
Lecturer: Chara Charalambous
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If you are choosing Option A, your future value will be $10,000
plus any interest acquired over the three years. The future value
for Option B, on the other hand, would only be $10,000. So how
can you calculate exactly how much more Option A is worth,
compared
to
Option
B?
Let's
take
a
look.
Lecturer: Chara Charalambous
10
Future Value Basics
• If you choose Option A and invest the total amount at a simple
annual rate of 4.5%, the future value of your investment at the
end of the first year is $10,450:
• Future value of investment at end of first year: = ($10,000 x 0.045)
+ $10,000 = $10,450
• Can be written as: $10,000 x [(1 x 0.045) + 1] =>
=> 10,000*(0.045+1)= 10,450
• So the equation for the future value can be written as:
FV= PV(i+1)
FV = Future Value
PV = Present Value – the money invested-the
capital
i = the interest rate or r = the rate of return
on the money invested
Lecturer: Chara Charalambous
11
• If the $10,450 left in your investment account at the end of the
first year is left untouched and you invested it at 4.5% for
another year, how much would you have? To calculate this, you
would take the $10,450 and multiply it again by 1.045 (0.045
+1). At the end of two years, you would have $10,920:
• Future value of investment at end of second year:
= $10,450 x (1+0.045) = $10,920.25
The above calculation, then, is equivalent to the following
equation:
Future Value = $10,000 x (1+0.045) x (1+0.045)
• The equation can be represented as the following:
Lecturer: Chara Charalambous
12
• If we were investing our money for 3 years the equation would
be: 10,000*(1+0.045) 3 = $11,411.66
• So the equation for the future value can be written as
n
FV= PV(i+1) FV = Future Value
PV = Present Value – the money
invested-the capital
i = the interest rate
n = the number of years or
periods I have invested my money
The process of going from today’s values, or present
values (PV), to future values (FV) is called compounding.
Lecturer: Chara Charalambous
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Present Value Basics
• If you received $10,000 today, the present value would of course be $10,000
because present value is what your investment gives you now if you were to
spend it today. If $10,000 were to be received in a year, the present value of the
amount would not be $10,000 because you do not have it in your hand now, in
the present. To find the present value of the $10,000 you will receive in the
future, you need to pretend that the $10,000 is the total future value of an
amount that you invested today. In other words, to find the present value of the
future $10,000, we need to find out how much we would have to invest today in
order to receive that $10,000 in the future.
• In order to calculate present value, or the amount that we would
have to invest today, we are going to use the FV equation but this
time the unknown number it will be the PV. So we will follow the
steps below :
n
• Original equation : FV= PV(i+1) => PV =
FV
(i+1) n
The factor
1
n
(i+1)
is called the discounting factor
Lecturer: Chara Charalambous
14
• Let's walk backwards from the $10,000 offered in
Option B. Remember, the $10,000 to be received in
three years is really the same as the future value of an
investment
• So, here is how you can calculate today's present value
of the $10,000 expected from a three-year investment
earning 4.5%: PV = FV
= 10,000 = $8762.97
n
3
(i+1)
(0.045+1)
Lecturer: Chara Charalambous
15
• So the present value of a future payment of
$10,000 is worth $8,762.97 today if interest rates
are 4.5% per year. In other words, choosing Option
B is like taking $8,762.97 now and then investing it
for three years. The equations above illustrate that
Option A is better not only because it offers you
money right now but because it offers you
$1,237.03 ($10,000 - $8,762.97) more in cash!
Furthermore, if you invest the $10,000 that you
receive from Option A, your choice gives you a
future value that is $1,411.66 ($11,411.66 $10,000) greater than the future value of Option B.
Lecturer: Chara Charalambous
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Interest Tables
 The Future
Value Interest Factor for i and n is defined
n
as (1 + i), and these factors can be found using a table:
•
•
•
•
•
•
Period
1%
2%
3%
4%
5% 6%
1
1.0100 1.0200 1.0300 1.0400 1.0500 1.0600
2
1.0201 1.0404 1.0609 1.0816 1.1025 1.1236
3
1.0303 1.0612 1.0927 1.1249 1.1576 1.1910
4
1.0406 1.0824 1.1255 1.1699 1.2155 1.2625
5
1.0510 1.1041 1.1593 1.2167 1.2763 1.3382
Lecturer: Chara Charalambous
17
Interest Tables
• The Present Valuen Interest Factor for i and n is
defined as 1/(1 + i), and these factors can be found
using a table:
• Period
1% 2%
3%
4% 5% 6%
• 1
0.9901 0.9804 0.9709 0.9615 0.9524 0.9434
• 2
0.9803 0.9612 0.9426 0.9246 0.9070 0.8900
• 3
0.9706 0.9423 0.9151 0.8890 0.8638 0.8396
• 4
0.9610 0.9238 0.8885 0.8548 0.8227 0.7921
• 5
0.9515 0.9057 0.8626 0.8219 0.7835 0.7473
Lecturer: Chara Charalambous
18
Annuity
• An annuity represents series of equal
payments (or receipts) occurring for a specified
number of equity distant periods.
• Ordinary Annuity: Payments or receipts occur
at the end of each period.
• Annuity Due: Payments or receipts occur at
the beginning of each period
Lecturer: Chara Charalambous
19
Examples of Annuities
•
•
•
•
•
Student Loan Payments
Car Loan Payments
Insurance Premiums
Mortgage Payments
Retirement Savings
Lecturer: Chara Charalambous
20
Parts of an Annuity
(Ordinary Annuity)
End of
Period 1
0
Today
End of
Period 2
End of
Period 3
1
2
3
$100
$100
$100
Equal Cash Flows
Each 1 Period Apart
Lecturer: Chara Charalambous
21
Parts of an Annuity
(Annuity Due)
Beginning of
Period 1
Beginning of
Period 2
0
1
2
$100
$100
$100
Today
Beginning of
Period 3
3
Equal Cash Flows
Each 1 Period Apart
Lecturer: Chara Charalambous
22
What is the difference between an ordinary
annuity and an annuity due?
Ordinary Annuity
0
i%
1
2
3
PMT
PMT
PMT
1
2
3
PMT
PMT
Annuity Due
0
i%
PMT
Lecturer: Chara Charalambous
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Example 1 :If I deposit $1000 at the end of each
year for 3 years in a saving account that pays
7% interest per year, how much will I have at
the end of the 3 years?
Lecturer: Chara Charalambous
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Example of an
Ordinary Annuity -- FVA
Cash flows occur at the end of the period
0
1
2
3
4
7%
$1,000
$1,000
$1,000
$1,000
$1,070
$1,145
$3,215 = FVA3
FVA3 = $1,000(1.07)2 + $1,000(1.07)1 +$1,000(1.07)0=
$1,145 +$1070 +$1000= $3,215
Lecturer: Chara Charalambous
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n-1
n-2
1
0
FVA n = PMT(1+i) + PMT(1+i) +…+ R(1+i) +PMT(1+i)
=>
n
FVA n = PV (1+i) – 1
i
Lecturer: Chara Charalambous
26
Example 2 :If I deposit $1000 at the beginning
of each year for 3 years in a saving account
that pays 7% interest per year, how much will I
have at the end of the 3 years?
Lecturer: Chara Charalambous
27
Example of an
Annuity Due -- FVAD
Cash flows occur at the beginning of the period
0
1
2
3
4
7%
$1,000
$1,000
$1,000
$1,070
$1,145
$1,225
3
2
$3,440 = FVAD3
FVAD3 = $1,000(1.07) + $1,000(1.07) + $1,000(1.07)
= $1,225 + $1,145 + $1,070 = $3,440
1
• FVADn = R(1+i)n + R(1+i)n-1 + ... + R(1+i)2 + R(1+i)1
=>
n
FVA = PV (1+i) – 1 * (1+i)
i
Lecturer: Chara Charalambous
29
• Example 3: you are offered a 3-year annuity
with payments of $ 1000 at the end of each
year. So you have to deposit the payments in a
saving account that pays 7% interest per year.
How much you have to deposit today?
Lecturer: Chara Charalambous
30
Example of the Present Value of an
Ordinary Annuity
Cash flows occur at the end of the period
0
1
2
3
$1,000
$1,000
4
7%
$934.58
$873.44
$816.30
$1,000
$2,624.32 = PVA3
$1,000 / (1.07)1 +
$1,000 / (1.07)2 +
$1,000 / (1.07)3
= $934.58 + $873.44 + $816.30
= $2,624.32
PVA3 =
PVAn = R/(1+i)1 + R/(1+i)2 + ... + R/(1+i)n =>
PVAn =FV 1-1/(1+i)
i
n
Lecturer: Chara Charalambous
32
• Example 3: you are offered a 3-year annuity
with payments of $ 1000 at the beginning of
each year. So you have to deposit the
payments in a saving account that pays 7%
interest per year. How much you have to
deposit today?
Lecturer: Chara Charalambous
33
Example of an
Annuity Due -- PVAD
Cash flows occur at the beginning of the period
0
1
2
3
4
7%
$1,000.00
$ 934.58
$ 873.44
$1,000
$1,000
$2,808.02 = PVADn
PVADn = $1,000/(1.07)0 + $1,000/(1.07)1 +
$1,000/(1.07)2 = $2,808.02
PVAn = R/(1+i)0 + R/(1+i)1 + ... + R/(1+i)n-1 =>
PVAn =FV 1-1/(1+i)
i
n
* (1+i)
Lecturer: Chara Charalambous
35
Exercises
Lecturer: Chara Charalambous
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