Content: Time Value of Money
SINGLE AMOUNTS
•
Compound Interest:
Slides 2 - 10
•
Nominal Rate (APR) and Periodic Rate: Slides 11 - 14
•
Continuous
Interest:
Slides 15 – 19
ANNUITIES & PERPETUITIES
•
Annuity Problems: Slide 20
•
Future Value of an (Ordinary) Annuity: Slides 21 – 29
•
Present Value of an (Ordinary) Annuity: Slides 30 - 33
•
Amortization Schedules:
Slides 34 - 38
•
EAR:
Slides 39 – 47
•
Annuity Due:
Slides 48 – 50
SOLUTIONS TO YOU TRY PROBLEMS:
SLIDES
52 - 66
1

The Future Value Equation:
An
Amount Under Compound Interest
GENERAL EXPRESSION:
After n periods, at a periodic interest rate of
k,
a Present Value ‘PV’ grows to:
NOTATION:
(1 + k)
n
is called t
he “Future Value Factor for k and n” and
is denoted:
2
FV
n
=
PV (1 + k)
n
“
FVF
k,n
”
=
(1 + k)
n

3
The Future Value Factor –
Using a Standard Calculator
Example:
Calculate
FVF
12, 10
= (1 + .12)
10
on a standard-calculator.
Answer: Standard Calculator Method:
1.12
10
→
Keys:
FVF
12, 10
=
3.105848208
Enter:
1.12
Press:
y
x
Enter:
10
Press:
=

4
Example: A Future Value Problem
Example:
How much will $1000 be worth if it’s
deposited for ten years at a 12% annual interest
rate?
Solution:
FV
n
=
PV (1 + k)
n
FV
10
= $1000 · (1 + .12)
10
FV
10
= $1000 · 3.105848208
=
$
3,105.85
FVF
12,10
↓
PV
n
k

Example: A Future Value Problem
PROBLEM 1 (
YOU TRY
):
How much will
$1000
be
worth after
20 periods
at a
6%
periodic interest
rate?
SOLUTION:
FV
n
=
PV (1 + k)
n
5
Check answers to problems
Appear at the end of this
document.

6
The Four Generic Time-Value Problems –
Compound Interest
Problem-Solving:
Every time-value problem contains four variables.
In the case of single amounts, the four variables are:
1.
FV
n
2.
PV
3.
k
4.
n
Generally, problems will
give you three of these four variables
, and
ask you to find the fourth.

Example:
Unknown Present Value -
PV
Example:
How much do you need to deposit now in order
to have $4000 in 10 years at a 12% annual interest rate?
Solution:
FV
n
=
PV (1 + k)
n
$4000 = PV (1 + .12)
10
PV = $4000 / (1.12)
10
PV
=
$4000 / 3.105848208
=
$
1,287.89
7
FV
n
k

Example: Unknown Interest Rate, k
Problem 2 (
YOU TRY
):
What annual interest rate would you need for
a $1000 deposit to grow to $4000 in 10 years?
Solution (
YOU TRY
):
FV
n
=
PV (1 + k)
n
8
FV
PV
n
Check answers to problems
Appear at the end of this
document.

Example: Unknown Time, n
Problem 3 (
YOU TRY
):
How many years would it take for a $1000
deposit to grow to $4000 at a 12% annual interest rate?
Solution (
YOU TRY
):
FV
n
=
PV (1 + k)
n
9
Check answers to problems
Appear at the end of this
document.

Single Amounts Under Compound
Interest – Four Generic Problems
10
GENERIC PROBLEMS:
For compound interest involving a single
amount, the four generic problems have general solutions. But, it’s
probably better just to remember the basic equation and solve it as
necessary.
TO BE SOLVED
FOR:
GIVEN:
GENERAL SOLUTION:
FV
n
PV, k & n
FV
n
=
PV ( 1 + k )
n
PV
FV
n
, k & n
PV
=
FV
n
( 1 + k )
– n
k
PV, FV
n
& n
n
PV, FV
n
& k

Compounding More Frequently Than
Once Per Year:
Introducing k
nom
and m
Suppose there are m compounding periods per year
(e.g. “monthly” ↔
m = 12).

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