2011 McGraw-Hill/Irwin
4-1
Copyright by the McGraw-Hill Companies, Inc. All rights reserved.
Key Concepts and Skills
Be
able to compute the future value of an
investment made today
Be able to compute the present value of cash to
be received at some future date
Be able to compute the return on an investment
The Time Value of Money
Which
would you rather have ?
$100
today - or
$100 one year from today
Sooner
is better !
The Time Value of Money
How
about $100 today or $115 one year from
today?
It depends
on the time value of money
Basic Definitions
Present
Value earlier money on a time line
Future Value later money on a time line
Interest rate exchange rate between earlier
money and later money
Discount
rate
Cost of capital
Opportunity cost of capital
**** Required return ****
Future Values
You
invest $1000 for one year at 5% per year.
What is the future value in one year?
Interest
= 1000(.05) = 50
Value in one year = principal + interest = 1000 +
50 = 1050
Future Value (FV) = 1000(1 + .05) = 1050
Suppose
you leave the money in for another
year. How much will you have two years from
now?
FV
= 1000(1.05)(1.05) = 1000(1.05)2 = 1102.50
Future Values: General Formula
FV
= PV(1 + r)t
FV
= future value
PV = present value
r = period interest rate, expressed as a decimal
t = number of periods
Effects of Compounding
Compound
interest
Previous example:
FV
= 1102.50 = 1000 + 50 interest + 52.50 interest
The extra 2.50 comes from the interest of .05(50) =
2.50 earned on the first interest payment
Simple
interest assumes you withdraw
your interest each period
How
much interest in 1st year? 2nd year?
Calculator Keys
HP
10B II
FV
= future value
PV = present value
I/YR = period interest rate
Interest is entered as a percent, not a decimal
N = number of periods
Remember to clear the registers (GOLD, C ALL)
after each problem
Preemptive Calculator Checks
1. Switch decimals:
<GOLD> = 6 to change to 6 decimal places
2. One PMT per year
1 <GOLD> PMT
3. Clear everything
<GOLD> C (ALL)
Future Values
Suppose
you invest the $1000 from the previous
example for 5 years. How much would you have?
1
P/YR
5 N
5 I/YR
-1000 PV (negative deposit = cash out of pocket)
FV ?
The
effect of compounding is small for a small
number of periods, but increases as the number of
periods increases.
Graphic View of the Compounding Process
Relationship among FV, I/YR, and N
5
Future Value of $1
4
i= 15%
3
i= 10%
2
i= 5%
i= 0%
1
0
0
1
2
4
6
8
Periods
10
Future Values
Suppose
you had a relative deposit $10 at 5.5%
interest 200 years ago. How much would the
investment be worth today?
200
N
5.5 I/YR
-10 PV
FV? 447,189.84
What
is the effect of compounding?
Simple
interest = 10 + 200(10)(.055) = $120
Compounding added $447,069.84 to the value of the
investment
Compounding
Quick Quiz Part 1
What
is the difference between simple interest
and compound interest?
Suppose you have $500 to invest and you
believe that you can earn 8% per year over the
next 15 years.
How
much would you have at the end of 15 years
using compound interest?
How much would you have using simple interest?
Present Values
How
much do I have to invest today to have
some amount in the future?
FV
= PV(1 + r)t
Rearrange solve to for PV = FV / (1 + r)t
Discounting
amount.
= finding the PV of a future
Present Values
You
want to begin saving for you daughters
college education and you estimate that she
will need $150,000 in 17 years. If you feel
confident that you can earn 8% per year, how
much do you need to invest today?
17
N
8 I/YR
150,000 FV
PV? -40,540.34
Present Values
Your
parents set up a trust fund for you 10
years ago that is now worth $19,671.51. If the
fund earned 7% per year, how much did your
parents invest?
10
N
7 I/YR
19,671.51 FV
PV? -10,000
Present Value Important
Relationship I
For
a given interest rate the longer the time
period, the lower the present value
What
is the present value of $500 to be received in
5 years? 10 years? The discount rate is 10%
5 years: 5 N; 10 I/YR; 500 FV
PV? -310.46
10 years: 10 N; 10 I/YR; 500 FV
PV? -192.77
Present Value Important
Relationship II
For
a given time period the higher the
interest rate, the smaller the present value
What
is the present value of $500 received in 5
years if the interest rate is 10%? 15%?
Rate = 10%: 5 N; 10 I/YR; 500 FV
PV? -310.46
Rate = 15%; 5 N; 15 I/YR; 500 FV
PV? -248.58
Graphic View of the Discounting Process
Relationship among PV, I/YR, and N
Present Value of $1
1
i= 0%
0.8
0.6
i= 5%
0.4
i= 10%
0.2
0
2
4
6
8
i= 15%
10
12
14
16
18
Periods
20
Quick Quiz Part 2
What
is the relationship between present value
and future value?
Suppose you need $15,000 in 3 years. If you
can earn 6% annually, how much do you need
to invest today?
If you could invest the money at 8%, would
you have to invest more or less than at 6%?
How much?
The Basic PV Equation - Refresher
PV
= FV / (1 + r)t
There are four parts to this equation
PV,
FV, r and t
If we know any three, we can solve for the fourth
Discount Rate
Want
to know what the implied interest rate is
in an investment?
Rearrange FV equation and solve for r
FV
= PV(1 + r)t
r = (FV / PV)1/t 1
Discount Rate
You
are looking at an investment that will pay
$1200 in 5 years if you invest $1000 today.
What is the implied rate of interest?
r
= (1200 / 1000)1/5 1 = .03714 = 3.714%
Calculator
5 N
-1000 PV (you pay 1000 today)
1200 FV (you receive 1200 in 5 years)
I/YR? 3.714%
Discount Rate
Suppose
you have a 1-year old son and you
want to provide $75,000 in 17 years towards
his college education. You currently have
$5000 to invest. What interest rate must you
earn to have the $75,000 when you need it?
Discount Rate
Suppose
you are offered an investment that
will allow you to double your money in 6
years. What is the implied rate of interest?
Finding the Number of Periods
Start
with basic equation and solve for t
FV
= PV(1 + r)t
t = ln(FV / PV) / ln(1 + r)
Or,
you can make your life easier and use the
financial keys on the calculator
Number of Periods
If
you can earn 10% per year, how long will it
take for an investment to double in value?
Comprehensive Problem
You
have $10,000 to invest for five years.
How much additional interest will you earn if
the investment provides a 5% annual return,
when compared to a 4.5% annual return?
How long will it take your $10,000 to double
in value if it earns 5% annually?
What annual rate has been earned if $1,000
grows into $4,000 in 20 years?
30