principles of
corporate
finance
Chapter 2a
Lecturer
Sihem Smida
The Time Value of
Money
1- 2
Chapter Objectives
 Distinguish between simple and compound interest.
 Calculate the present value and future value of a single amount for both
one period and multiple periods.
 Calculate the present value and future value of multiple cash flows.
 Calculate the present value and future value of annuities.
 Compare nominal interest rates (NIR) and effective annual interest rates
(EAR).
 Distinguish between the different types of loans and calculate the present
value of each type of loan.
1-2
1- 3
What is cash flow?
Cash flow is the money that is moving
(flowing) in and out of your business in a
month.
 Cash is coming in: from customers who are buying
your products or services. If customers don't pay at
time of purchase, some of your cash flow is coming
from collections of accounts receivable. Revenus,
profit, rent
 Cash is going out: in the form of payments for
expenses, like rent or a mortgage, in monthly loan
1-3
payments, and in payments for taxes and other
accounts payable.
1- 4
Time Value Terminology
0
PV
1
2
3
4
FV
 Future value (FV) is the amount an investment is worth after one or
more periods.
 Present value (PV) is the current value of one or more future cash flows
from an investment.
1-4
If we can measure the value of cash
flow, we can:
 Translate Rs.1 today into its equivalent in the
future (compounding).
Today
Future
?
 Translate Rs.1 in the future into its equivalent
today (discounting).
Today
?
Future
1- 5
1- 6
Simple interest
Simple interest refers to interest earned only
on the original capital investment amount.
– FV = original capital (PV) + interest
– Interest = PV × r×t
FV = PV(1 + r×t )
Example :
PV = 100 ; r = 10% ; t = 3 ; FV?
FV = 100(1 + 0.1×3 )
FV = 130
1- 7
Exercises
Exercise 1:
What will 6000$ which are deposit in an
account worth after 6 months earning 12%
(simple interest )per year ?
Exercise 2:
You deposit 5000$ in bank, after 7months
you distinguish that the amount needed is
5408.3.
Calculate the simple interest rate?
1- 8
Interest Rate Terminology
Compound interest refers to interest earned
on both the initial capital investment and on
the interest reinvested from prior periods.
1-8
1- 9
Future Value of a Lump Sum
 Example:
You invest $100 in a savings account that earns 10 per cent
interest per year (compounded) for three years.
After one year:
After two years:
After three years:
$100  (1 + 0.10) = $110
$110  (1 + 0.10) = $121
$121  (1 + 0.10) = $133.10
1-9
1- 10
Future Value of a Lump Sum
The accumulated value of this investment at the
end of three years can be split into two
components:
– original principal
$100
– interest earned $33.10
Using simple interest, the total interest earned
would only have been $30. The other $3.10 is
from compounding.
1-10
1- 11
Future Value of a Lump Sum
In general, the future value, FVt , of $1 invested
today at r per cent for t periods is:
FVt  PV  1  r 
t
The expression (1 + r)t is the future value
interest factor (FVIF). Refer to Table A.1.
1-11
1- 12
Time Value Terminology
The number of time periods between the
present value and the future value is
represented by ‘t’.
The rate of interest for discounting or
compounding is called ‘r’.
All time value questions involve four values:
PV, FV, r and t. Given three of them, it is
always possible to calculate the fourth.
1-12
1- 13
Future Value of a Lump Sum
Example:
 What will $1000 amount to in five years time if interest is 12 per cent per
year, compounded annually?
 From the example, now assume interest is 12 per cent per year,
compounded monthly.
 Always remember that t is the number of compounding periods, not the
number of years.
1-13
1- 14
Interpretation
The difference in values is due to the larger
number of periods in which interest can
compound.
Future values also depend critically on the
assumed interest rate—the higher the interest
rate, the greater the future value.
1-14
1- 15
Future Values at Different Interest Rates
Number of
periods
Future value of $100 at various interest rates
5%
10%
15%
20%
1
$105.00
$110.00
$115.00
$120.00
2
$110.25
$121.00
$132.25
$144.00
3
$115.76
$133.10
$152.09
$172.80
4
$121.55
$146.41
$174.90
$207.36
5
$127.63
$161.05
$201.14
$248.83
1-15
1- 16
Present Value of a Lump Sum
You need $1000 in three years time. If you can earn 10 per
cent per year, how much do you need to invest now?
Discount one year: $1000 (1 + 0.10) –1 = $909.09
Discount two years:
$909.09 (1 + 0.10) –1
Discount three years:
$826.45 (1 + 0.10) –1
= $826.45
= $751.32
1-16
1- 17
Interpretation
 In general, the present value of $1 received in t periods of time,
earning r per cent interest is:
PV  FV  1  r 
FV

1  r t
t
 The expression (1 + r)–t is the present value interest factor (PVIF).
Refer to Table A.2.
1-17
1- 18
Present Value of a Lump Sum
Example:
Your father promise to give you 10 000 Ryal in 10 years
time. If interest rates are 12 per cent per year, how much is
that gift worth today?
PV  10 000  1  0.12 
 10 000  0.3220
 3220 ryal
10
1-18
1- 19
Present Values at Different Interest Rates
Number of
periods
Present value of 100 Ryal at various interest
rates
5%
10%
15%
20%
1
95.24
90.91
86.96
83.33
2
90.70
82.64
75.61
69.44
3
86.38
75.13
65.75
57.87
4
82.27
68.30
57.18
48.23
5
78.35
62.09
49.72
40.19
1-19
1- 20
Future Value of Multiple Cash Flows
Example:
 You deposit $1000 now, $1500 in one year, $2000 in two years and $2500
in three years in an account paying 10 per cent interest per year. How
much do you have in the account at the end of the third year?
1-20
1- 21
Solutions
 Solution 1
– End of year 1:
– End of year 2:
– End of year 3:
($1000  1.10) + $1500 =
($2600  1.10) + $2000 =
($4860  1.10) + $2500 =
$2600
$4860
$ 846
$ 8306
 Solution 2
$1000  (1.10)3
$1500  (1.10)2
$2000  (1.10)1
$2500  1.00
Total
=
=
=
=
=
$1331
$1815
$2200
$2500
$7846
1-21
1- 22
Present Value of Multiple Cash Flows
Example:
 You will deposit $1500 in one year’s time, $2000 in two years time
and $2500 in three years time in an account paying 10 per cent interest
per year. What is the present value of these cash flows?
1-22
1- 23
 Solution
$2500  (1.10) –3
$2000  (1.10) –2
$1500  (1.10) –1
Total
=
=
=
=
$1878
$1653
$1364
$4895
1-23