Chapter 5
Calculators
Introduction to
Valuation: The
Time Value of
Money
McGraw-Hill/Irwin
Copyright © 2010 by The McGraw-Hill Companies, Inc. All rights reserved.
Real Life Example:
Car Financing
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
• Be able to compute the number of periods that
equates a present value and a future value given
an interest rate
• Be able to use a financial calculator and a
spreadsheet to solve time value of money
problems
Time Value of Money
You are given 2 options:
1.Option A: You receive $10,000 now.
2.Option B: You will receive $10,000
in three years.
Which option will you choose? Why?
Why is there a time value of money?
5C-2
Time Value of Money
Time Value Purposes
It can be used to determine:
a) Damages in Court Cases
b) Contract Values
c) Investment Values
In this chapter, our primary focus is to
identify the value of an investment,
either now or in the future.
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
Basic Definitions (cont.)
Present Value
T = 0 (Y2015)
r = 10 %
Future Value
T = 1 (Y2016)
t
Ex: What is the future value of your
investment in a year if you put $ 100 in a
saving account that earns r = 10%
interest annually?
FV = PV + r*PV = PV (1+ r) = 100(1.1) = $ 110
Basic Definitions (cont.)
Types of interest rate:
1. Simple Interest Rate: Interest
earned only on the original principal
amount invested
2. Compound Interest Rate:
– Compound means you reinvest your
interest earned over time.
– In this case, you earn interest on both
the principal amount and the reinvested
interest amount accumulated from the
previous periods.
Basic Definitions (cont.)
Ex: What is the future value of your
investment in 2 years if you put $ 100 in
a saving account that earns r = 10%
interest annually and r is a simple
interest rate?
FV2 = Principal + Interest earned in Y1 +
Interest earned in Y2
FV2 = 100 + 10%*100 + 10%*100 = $120
Or, FV2 = 100 + 2*10%*100 = $120
Simple interest formula
FV = PV(1 + r*t)
In which,
FV is the future value of the investment
PV is the present value of the investment
r
is the simple interest rate
t
is the number of periods
Basic Definitions (cont.)
Ex: What is the future value of your
investment in 2 years if you put $ 100 in
a saving account that earns r = 10%
interest annually and r is a compound
interest rate?
Y1: you earn 100 + 10%*100 = $ 110
A compound interest rate means that
in Y2 you earn 10% interest on both the
$100 principal and the $10 interest from
Y1.
Y2: you get 110 + 10%*(100+10) = $ 121
Compound interest formula
FV = PV(1 +
t
r)
In which,
FV is the future value of the investment
PV is the present value of the investment
r
is the compound interest rate
t
is the number of periods
Previous Example
Ex: What is the future value of your
investment in 2 years if you put $ 100
in a saving account that earns r = 10%
interest annually and r is a compound
interest rate?
Answer:
FV2 = PV(1+r)2 = 100(1+0.1)2 = $ 121
Calculator Keys
Texas Instruments BA-II Plus (APPENDIX D)
§ FV = future value; PV = present value (+/-)
§ I/Y = period interest rate
• P/Y must equal 1 for the I/Y to be the period rate
– [2nd] [I/Y] 1 [ENTER]
• Interest is entered as a percent, not a decimal (Ex:
15%, how would you enter the number?
§ N = number of periods
§ Clear the registers (CLR TVM) after each
problem
§ Should show 9 decimal places on your
calculator when you are performing
calculations.
• [2nd] [Format] 9 [ENTER]
Periods vs. Years
If you earn interest annually for the
next 3 years. How many times will you
get the interest out of your investment?
Now, if you earn interest every six
months for the next 3 years. How
many times will you get the interest out
of your investment?
FUTURE VALUE
Simple vs. Compound
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/Y; -10 PV
§ CPT FV = 447,189.84
What is the effect of compounding?
§ Simple interest = 10 + 200(10)(.055) =
120.00
§ Compounding added $447,069.84 to
the value of the investment
Simple vs. Compound
Interests on Interests
$447,069.84
$110.00
$10.00
Future Value as a General
Growth Formula
Suppose your company expects to increase
unit sales of widgets by 15% per year for the
next 5 years. If you sell 3 million widgets in
the current year, how many widgets do you
expect to sell in the fifth year?
• 5 N;15 I/Y; 3,000,000 PV
• CPT FV = -6,034,072 units
(remember the sign convention)
APPENDIX D
Example: Compute the future value
of $ 2,250 at a 17% annual rate for 30
years.
Answer: $ 249,895.46
Example: Compute the future value
of $ 2,250 at a 17% semi-annual rate
for 30 years.
Answer: $ 300,584.66
PRESENT VALUE
The one and only formula
If the Future value:
FV = PV(1 + r)t
How do you compute the PV?
PV =
t
FV/[(1+r) ]
Present Value
• When we talk about discounting,
we mean finding the present value of
some future amount.
• When we talk about the “value” of
something, we are talking about the
present value unless we specifically
indicate that we want the future
value.
APPENDIX D (Cont.)
Example: What is the present value of
your investment now if you expect to
have $ 75,000 in 18 years and the
annual interest rate is 14.08%?
Answer: $ 7,003.10
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: N = 5; I/Y = 10; FV = 500
CPT PV = -310.46
§ 10 years: N = 10; I/Y = 10; FV = 500
CPT 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%: N = 5; I/Y = 10; FV = 500
CPT PV = -310.46
• Rate = 15%; N = 5; I/Y = 15; FV = 500
CPT PV = -248.59
DISCOUNT RATE
OR
INTEREST RATE
The one and only formula
If the Future value:
FV = PV(1 + r)t
How do you compute the interest rate
r?
r = (FV/PV)1/t - 1
Example 1
You are looking at an investment that will
pay $1,200 in 5 years if you invest $1,000
today. What is the implied rate of interest?
• r = (1,200 / 1,000)1/5 – 1 = .03714 =
3.714%
• Calculator – the sign convention matters!!!
§
§
§
§
N=5
PV = -1,000 (you pay 1,000 today)
FV = 1,200 (you receive 1,200 in 5 years)
CPT I/Y = 3.714%
APPENDIX D (Cont.)
Example: Assume that the total cost of
a college education will be $ 75,000
when your child enters college in 18
years. Now you have $ 7,000 to invest.
What rate of interest you must earn
on your investment to cover the cost of
your child’s college education in 18
years?
Answer: 14.08%
NUMBER OF PERIODS
The one and only formula
If the Future value:
FV = PV(1 + r)t
How do you compute the # of periods?
t = ln(FV / PV) / ln(1 + r)
Example 1
You want to purchase a new car, and you
are willing to pay $20,000. If you can
invest at 10% per year and you currently
have $15,000, how long will it be before
you have enough money to pay cash for
the car?
§ I/Y = 10; PV = -15,000; FV = 20,000
§ CPT N = 3.02 years
APPENDIX D (Cont.)
Example: How many year does it take
for you to have $ 250,000 out of a $
5,000 investment that earns 10%
interest per year?
Answer: 41.05 years
Spreadsheet Example
• Use the following formulas for TVM
calculations
–
–
–
–
FV(rate,nper,pmt,pv)
PV(rate,nper,pmt,fv)
RATE(nper,pmt,pv,fv)
NPER(rate,pmt,pv,fv)
• The formula icon is very useful when you
can’t remember the exact formula
• Click on the Excel icon to open a
spreadsheet containing four different
examples.
Table 5.4
5C-36
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?
Comprehensive Problem
N=5
PV = -10,000
At I/Y = 5, the FV = 12,762.82
At I/Y = 4.5, the FV = 12,461.82
The difference is attributable to interest. That difference is 12,762.82 –
12,461.82 = 301
To double the 10,000:
I/Y = 5
PV = -10,000
FV = 20,000
CPT N = 14.2 years
Note, the rule of 72 indicates 72/5 = 14 years, approximately.
N = 20
PV = -1,000
FV = 4,000
CPT I/Y = 7.18%
End of Chapter