4-1
Chapter 4
The Time Value of Money
Chapter Outline






Future Values and Compound Interest
Present Values
Multiple Cash Flows
Level Cash Flows: Perpetuities and Annuities
Inflation and the Time Value of Money
Effective Annual Interest Rates
Semih Yildirim
ADMS 3530
4-2
Introduction
• Money
Problems …
As a person or a financial manager you will often have
to compare cash payments which occur at different
dates: Cash flows now, versus cash flows later.
 To make optimal decisions, you must understand the
relationship between a dollar received (paid) today and
a dollar received (paid) in the future.


Two basic types of cash flow problems:
 Present
Value (PV) problems.
 Future Value (FV) problems.
Semih Yildirim
ADMS 3530
4-3
Introduction
• Present Value (PV) Problems
 PV problems involve calculating the value
today of future cash flow(s).
 For example:
 Interest
rates are 7%. If I need to have $100,000
saved in 10 years, how much money must I put
aside today to create that cash flow?
 Interest rates are 12%. If I need to create an
income of $5,000 per year for 10 years, how
much money must I put aside today to create that
cash flow?
Semih Yildirim
ADMS 3530
4-4
Introduction
• Future Value (FV) Problems
 FV problems involve calculating the value an
investment will grow to after earning interest.
 For example:
 Interest
rates are 5%. If I invest $1,000 today,
how much will it be worth in 8 years?
 Interest rates are 10%. If I open an account
and invest $2,500 per year, how much will it be
worth in 12 years?
Semih Yildirim
ADMS 3530
4-5
Future Values
•
Compound Interest vs Simple Interest
 There are two types of interest you may receive:


Compound interest.
Simple interest.
1. Simple Interest

Simple interest means that interest is earned only on
your original investment:


No interest is earned on the interest.
Example:
Assume interest rates are 6%.
 You invest $100 in an account paying simple interest.
How much will the account be worth in 5 years?
Semih Yildirim
ADMS 3530
4-6
Future Values
• Simple


Interest
You earn interest only on the amount invested.
Therefore you would earn:
 $100
x 6% = $6.00 per year for 5 years.
Answer – you would have $130 after 5 years:
Balance in your account:
$100
0
$106
1
$112
2
$118
3
$124
4
Semih Yildirim
$130
5
Period (t)
ADMS 3530
4-7
Future Values
2. Compound Interest




Compound interest means that interest is earned on
interest.
The result: the income you earn would be higher than it
would be with simple interest. Why?
Because you earn interest on both the original investment
and the interest earned in previous years.
Try the example again using compound interest:
 Interest rates are 6%. You invest $100 in an account
paying compound interest. How much will the account
be worth in 5 years?
Semih Yildirim
ADMS 3530
4-8
Future Values
• Compound

Interest
You earn interest on your interest:
 $100
x 6% = $6.00 the first year.
 $106 x 6% = $6.36 the second year.
 $112.36 x 6% = $6.74 the third year … etc.
After 5 years you would have $133.82 :
Balance in your account:
$100
0
$106
1
$112.36
2
$119.10
3
$126.25 $133.82
4
Semih Yildirim
5
Period (t)
ADMS 3530
4-9
Future Values
• Formula
for Calculating FV
FV = Investment x (1 + r)t

Where r is the interest rate and t is the number of periods.

Try the example again using the formula above:
 Interest
rates are 6%. You invest $100 in an account
paying compound interest. How much will the account
be worth in 5 years?
FV = $100 x (1 + 0.06)5
= $100 x 1.3382
= $133.82
Semih Yildirim
ADMS 3530
4-10
Why Compound Interest?
Future Value of a Single $1,000 Deposit
Future Value (Dollars)
20000
10% Simple
Interest
7% Compound
Interest
10% Compound
Interest
15000
10000
5000
0
1st Year 10th
Year
20th
Year
30th
Year
Semih Yildirim
ADMS 3530
4-11
Manhattan Island Sale
Peter Minuit bought Manhattan Island for $24 in 1626.
Was this a good deal?
To answer, determine $24 is worth in the year 2000,
compounded at 8%.
FV  $24  (1  .08)
 $75.979 trillion
374
FYI - The value of Manhattan Island land is
well below this figure.
Semih Yildirim
ADMS 3530
4-12
Present Value

Assume interest rates are 10%.

You have just won a lottery and must choose between
the following two options:


Receive $1,000,000 today.
Receive $1,000,000 five years from now.
Which option gives you the biggest “winnings”
The above example demonstrates a basic financial principle:
A dollar received today is worth more than a dollar received
tomorrow.
The key question is:
How much less valuable is a dollar received tomorrow as
versus a dollar received today?
That question is answered by using the interest rate (also known
as the discount rate) to calculate the PV of the second option.
Semih Yildirim
ADMS 3530
4-13
Present Value
• Formula
for Calculating PV
PV = Future Value x 1/(1 + r)t


You have been offered $1 million five years
from now. Interest rates are 10%.
What is that worth to you in today’s dollars?
PV = $1.0 million x 1/ (1 + 0.10)5
= $1.0 million x 0.620921
= $620,921
Semih Yildirim
ADMS 3530
4-14
Present Value
• More


Money Problems …
Thus, you could have $1 million today.
Or you could have the second option, which
equates to $620,921 in today’s dollars.
$1 million now
vs
The equivalent
of $620,921 now
You knew before that the first option was better,
but now you can calculate exactly how much better
off you are:

$379,079
better off!
Semih Yildirim
ADMS 3530
4-15
Present Value vs Future Value
• PV and FV are related!
 Have you noticed that $620,921 becomes $1
million (and that $1 million requires $620,921) if
you have a time period of 5 years and a discount
rate of 10%?
PV at 10%
$1,000,000
$620,921
FV at 10%
Semih Yildirim
ADMS 3530
4-16
Present Value vs Future Value
• PV and FV are related!
 $620,921 invested for 5 years at 10% grows to $1
million.
 Or, working it in reverse, if rates are 10%, and you
need $1 million in 5 years, you must put aside
$620,921 right now.
FV = PV x (1 + r)t
PV = FV x 1/(1 + r)t
= $620,921 x (1 + 0.10)5
= $1 million x 1/ (1 + 0.10)5
= $620,921 x 1.61051
= $1 million x 0.620921
= $1 million
= $620,921
Semih Yildirim
ADMS 3530
4-17
Present Value vs Future Value
• PV

and FV are related!
To calculate the FV of money which is available now (PV)
to be invested for t years at an interest rate r, multiply the
PV by (1+r)t.
To calculate the PV of a future payment, run the process
in reverse and divide the FV by (1+r)t.
 (1+r)t
is called the future value factor.
 r is called the discount rate
 Finding the PV is often called discounting
Semih Yildirim
ADMS 3530
4-18
Present Value vs Future Value
• Two


Key Principles for Financial Calculations
Think of the example in which we compared
receiving $1 million today against $1 million
received 5 years from now.
You should see from that example that:
 A dollar
received today is worth more than a dollar
received tomorrow.
Lesson: The value of cash flows received at
different times can never be directly compared.
You must first discount all cash flows to a
common date and then compare them.
Semih Yildirim
ADMS 3530
4-19
Present Value vs Future Value
• Finding
the Unknown …
FV = PV x (1 + r)t
PV = FV x 1/(1 + r)t
The FV and PV formulas
have many applications.
Note that the variables used
in these two equations are:
FV
PV
r
t
Given any three variables
in the equation, you can
always solve for the
remaining variable!
Semih Yildirim
ADMS 3530
4-20
Present Value vs Future Value
Example
You just bought a new computer for $3,000. The
payment terms are 2 years same as cash. If you can
earn 8% on your money, how much money should
you set aside today in order to make the payment
when due in two years?
PV 
3000
(1.08)2
 $2,572
Semih Yildirim
ADMS 3530
4-21
Present Value vs Future Value
Semih Yildirim
ADMS 3530
4-22
Present Value vs Future Value
Semih Yildirim
ADMS 3530
4-23
Multiple Cash Flows
• Future



Value Calculations
So far, we have looked at problems involving only a single
cash flow.
This is unrealistic – most business investments will
involve multiple cash flows over time.
We need a method for coping with such streams of cash
flows!


To calculate the FV of multiple cash flow stream, you need to
calculate the FV of each CF and then add up these FVs.
To calculate the PV of multiple cash flow stream, you need to
calculate the PV of each CF and then add up these PVs
Semih Yildirim
ADMS 3530
4-24
FV of Multiple Cash Flows
Doing Future Value Calculations
•
•
•
Assume interest rates are 8%. You make 3 deposits to your bank
account: $1,200 today, $1,400 one year later, $1,000 two years later.
How much money will you have in your account 3 years from now?
Calculate what each cash flow will be worth at the specified future
date and add up these future values.
$1,200
0
$1,400
1
$1,000
2
3
FV in Year 3:
$1,080.00 = $1,000 x 1.08
$1,632.96 = $1,400 x (1.08)2
$1,511.65 = $1,200 x (1.08)3
$4,224.61
Semih Yildirim
ADMS 3530
4-25
PV of Multiple Cash Flows
EXAMPLE

Assume interest rates are 8%.

You wish to buy a car making three installments:





$8,000 today
$4,000 one year later.
$4,000 two years later.
How much money would you have to place in an
account today to generate this stream of cash flows?
If Your auto dealer gives you the choice to pay
$15,500 cash now, or make three payments, which do
you prefer?
Semih Yildirim
ADMS 3530
4-26
Multiple Cash Flows
• Present Value Calculations
 You would need to place $15,133.06 in an account
today to generate the desired cash flows:
$8,000
PV today:
0
$4,000
1
$8,000.00
$4,000 / (1.08) = $3,703.30
$4,000 / (1.08)2 = $3,429.36
$15,133.06
Semih Yildirim
ADMS 3530
$4,000
2
4-27
Example: FV & PV of Mixed Stream
(4% Interest Rate)
Compounding
- $12,166.5
$3,509.6
$5,624.3
$4,326.4
FV
$6,413.8
$3,120.0
-$10,000
0
$2,884.6
$3,000
1
$5,000
2
$4,000
$3,000
3
$2,000.0
4
5
End of Year
$4,622.8
PV
$5,271.7
$3,556.0
$2,564.4
$1,643.9
Discounting
Semih Yildirim
ADMS 3530
4-28
Multiple Cash Flows
• Special




Situations
In the previous examples, we worked with
multiple cash flows of different sizes.
Sometimes we have a situation in which a
series of equal cash flows is involved:
Any sequence of equally spaced, level cash
flows is called an Annuity.
If the payment Stream lasts forever, it is called
a perpetuity.
Semih Yildirim
ADMS 3530
4-29
PV of an Annuity: the Long Method
In previous examples, we have worked with multiple cash flows
of different sizes.
 Suppose we now need to calculate the PV of a stream of
level future cash flows.
 We could use the same procedure as before:


•
Calculate what each cash flow would be worth today, i.e. get
its PV.
Add up these present values.
EXAMPLE


Assume interest rates are 10%. You wish to buy a car
making three installments of $4000, first installment will be
a year from today
How much money would you have to place in an account
today to generate this stream of cash flows?
Semih Yildirim
ADMS 3530
4-30
PV of an Annuity: the Long Method

You would need to place $9,947.41 in an account
today to generate the desired cash flows:
$4,000
PV today:
0
1
$4,000
2
$4,000 / (1.10) = $3,636.36
$4,000 / (1.10)2 = $3,305.79
$4,000 / (1.10)3 = $3,005.26
$9,947.41
Semih Yildirim
ADMS 3530
4,000
3
4-31
PV of Annuity Formula
t
1
1 


1

(
1

r
)
PVA  C   
t  or PVA  C  

r
r
(
1

r
)
r




C = cash payment
r = interest rate
t = Number of periods cash payment is received
•
PVannuity = $4,000 x [1/0.10 – 1/(0.10 (1 + 0.10)3)]
= $4,000 x 2.48685
= $9,947.41
Using the PV of an annuity calculation, we get the same
answer as before:

Put aside $9,947.41 to fund the cash flows.
Semih Yildirim
ADMS 3530
4-32
Calculating the FV of an Annuity


Suppose interest rates are 10% and you decide to
save $4,000 per year for 20 years. How much will
you have saved for your retirement?
This is a FV problem. We could use the same
procedure as we used for multiple cash flows of
different sizes:
 Calculate
what each cash flow would be worth in, 20
years, i.e. get its FV and Add up these future values.
Can you see the problem with using this method?
 Calculating the FV this way would mean working out
the FV for 20 separate cash flows ... 

Is there an easier way?
Semih Yildirim
ADMS 3530
4-33
FV of Annuity Formula
 (1  r )  1
FVA  C  

r


t
C = cash payment
r = interest rate
t = Number of periods cash payment is received
FVannuity = $4,000 x [ ((1 + 0.10)20 – 1) / 0.10 ]
= $4,000 x 57.27499949
= $229,100

Using the FV of an annuity calculation, we see that you
will have $229,100 in your account when you retire in 20
years.
Semih Yildirim
ADMS 3530
4-34
Examples of Special Annuities
Semih Yildirim
ADMS 3530
4-35
Annuities Due
Semih Yildirim
ADMS 3530
4-36
Annuities Due
Semih Yildirim
ADMS 3530
4-37
Annuities Due
Semih Yildirim
ADMS 3530
4-38
Examples of Special Annuities
Semih Yildirim
ADMS 3530
4-39
Annuities with missing CFs
Semih Yildirim
ADMS 3530
4-40
Annuities with missing CFs
Semih Yildirim
ADMS 3530
4-41
PV of Perpetuity
• Perpetuities
 The PV of a perpetuity is calculated by dividing the
cash payment by the interest rate:
PV of a perpetuity =

C
r
=
Cash Payment
Interest rate
The interest rate on a perpetuity is calculated by
dividing the cash payment by the PV:
Interest rate on a perpetuity =
C
PV
=
Semih Yildirim
Cash Payment
Present Value
ADMS 3530
4-42
PV of Perpetuity
• Example
 What
would you pay to own a guaranteed
income of $1,000 per year to be received
forever, if interest rates are 4%?
PV of a perpetuity =
C
r
=
Cash Payment
Interest rate
=
$1,000
4%
=
$25,000
Semih Yildirim
ADMS 3530
4-43
Growing Cash Flows
• Cash


Flows Growing at a Constant Rate
What if the cash flows in a financial problem
are not equal, but are instead growing at a
constant rate?
For example:
 Assume
the discount rate is 8%. You are thinking of
buying a condo which generates $12,000 per year
in net cash flow in perpetuity. These cash flows
grow at 3% per year.
What is the maximum price you
should pay for this condo?
Semih Yildirim
ADMS 3530
4-44
Growing Cash Flows
• Valuing

Growing Perpetuities
The PV of a growing perpetuity is calculated by
dividing the cash payment by the discount rate
less the growth rate:
PV of a perpetuity =
=
=
C
r-g
=
Cash Payment
Discount Rate – Growth Rate
$12,000
0.08 – 0.03
$240,000
Semih Yildirim
ADMS 3530
4-45
Growing Cash Flows
• Cash



Flows Growing at a Constant Rate
In the previous problem we assumed that the
cash flows grew at a constant rate forever.
It may be more reasonable to assume a
constant growth rate for a limited time period.
For example:
 Assume
the condo in the previous problem
generates $12,000 per year in net cash flow for 20
years. These cash flows grow at 3% per year.
Now, what is the maximum price you
should pay for this condo?
Semih Yildirim
ADMS 3530
4-46
Growing Cash Flows
• Valuing

Finite Growing Cash Flows
The PV of cash flows which grow at a constant
rate for a limited time period (T) is calculated by:
PV of cash flow =
t

C
1 g  
 1  

r  g   1  r  
  1  .03 20 
12,000
=
 1  
 
0.08  0.03   1  .08  
=
$240,000 * 0.6125
=
$147,000
Semih Yildirim
ADMS 3530
4-47
Inflation, Real and Nominal Rates
•
•
•
Inflation - Rate at which prices as a whole are increasing.
Nominal Interest Rate - Rate at which money invested
grows.
Real Interest Rate - Rate at which the purchasing power of
an investment increases.
The relation between nominal and real rates and the
(expected) future inflation rate is
1+ nominal interest rate
1  real interest rate =
1+inflation rate
approximation formula
Real int. rate  nominal int. rate - inflation rate
Semih Yildirim
ADMS 3530
4-48
Inflation, Real and Nominal Rates
Example
If the interest rate on one year govt. bonds is 5.0%
and the inflation rate is 2.2%, what is the real
interest rate?
1 + real interest rate = 1+.050
1+.022
1 + real interest rate = 1.027
Real interest rate = .027 or 2.7%
Approximation = .050 - .022 or .028 or 2.8%
Semih Yildirim
ADMS 3530
4-49
Frequency of Compounding
General Formula: FVt= PV0(1 + [r/m])mt
t:
Number of Years
m:
Compounding Periods per Year
r:
Annual Interest Rate
FVt,m: FV at the end of Year t
PV0:
PV of the Cash Flow today
Impact of Frequency
Julie Miller has $1,000 to invest for 2 years at an annual interest rate of 12%.
Annual
Semi
Qrtly
Monthly
Daily
FV2
FV2
FV2
FV2
FV2
= 1,000(1+ [.12/1])(1)(2) = 1,254.40
= 1,000(1+ [.12/2])(2)(2) = 1,262.48
= 1,000(1+ [.12/4])(4)(2) = 1,266.77
= 1,000(1+ [.12/12])(12)(2) = 1,269.73
= 1,000(1+[.12/365])(365)(2)= 1,271.20
Semih Yildirim
ADMS 3530
4-50
Effective Interest Rates
• EAR




vs APR
So far, we have used annual interest rates
applied to annual cash flows.
But interest can be applied daily, weekly,
monthly, semi-annually – or for any other
convenient time period.
The simplest way to deal with this situation is
to convert to annual rates.
There are two ways to convert a periodical
interest rate to an annual basis:
 Calculate
the Effective Annual Rate (EAR).
 Calculate the Annual Percentage Rate (APR).
Semih Yildirim
ADMS 3530
4-51
Effective Interest Rates
• Annual


Percentage Rate (APR)
The annual percentage rate (APR) is an
interest rate that is annualized using simple
interest.
For example:
 Your
credit card charges 1.5% per month. What is
the annual charge?
APR = Quoted rate x Number of Periods Per Year
= 1.5% x 12
= 18%
Semih Yildirim
ADMS 3530
4-52
Effective Interest Rates
• Effective Annual


Interest Rate (EAR)
The effective annual interest rate (EAR) is an
interest rate that is annualized using
compound interest.
For example:
 Your
credit card charges 1.5% per month. What is
the annual charge?
EAR = (1 + period rate) # of Periods Per Year-1
= (1 + 0.015)12-1
= 19.56%
Semih Yildirim
ADMS 3530
4-53
Effective Interest Rates
• Calculating

the EAR
Convert the APR to a period rate and then apply the
equation:
(1 + Period Rate) m-1
Quoted APR: 12% (a)
Compounding
Period
1 year
Semiannually
Quarterly
Monthly
Daily
(m = Number of periods per year)
(b)
(c)
Periods
per Year
(m)
1
2
4
12
365
Period Rate
(= a/b)
12.0000%
6.0000%
3.0000%
1.0000%
0.0329%
Semih Yildirim
EAR
m
= (1+ c) - 1
12.0000%
12.3600%
12.5509%
12.6825%
12.7475%
ADMS 3530
4-54
Home Mortgage
•
Home Mortgage Monthly Payment

Assume we are on October 1st and you negotiate a
home mortgage of $250,000 to pay during 25 years
with equal monthly installments starting at the end
of October with a monthly interest rate of 0.5%.
What would be your monthly payment ?
C
C
C
C
C
1
2
298
299
300
PV = $250,000
Semih Yildirim
ADMS 3530
4-55
Home Mortgage
C
C
C
C
C
1
2
298
299
300
 1

1
$250,000  C  

300 
0
.
5
%
0
.
5
%

(
1

0
.
5
%)


$250,000  C 
$155.21
$250,000
C
 $1,610.75
$155.21
Semih Yildirim
ADMS 3530
4-56
Home Mortgage in Canada
(See Example 4.19 pages 113)
•
Mortgage rates in Canada are APRs for
6-month interest rates





They must be converted for use in calculations
Assuming interest rate is 6% (compounded semiannually) , we first calculate the 6-month interest
rate: 6% /2 = 3% per 6-month
The EAR is then given by (1+3%)2 – 1 = 6.09%
The monthly rate is then given by
(1+6.09%)1/12 – 1 = 0.4939%
This monthly rate will be used in calculations
Semih Yildirim
ADMS 3530
4-57
Home Mortgage in Canada
(See Example 4.19 pages 113)
C
C
C
C
C
1
2
298
299
300


1
1
$250,000  C  

300 
0
.
4939
%
0
.
4939
%

(
1

0
.
4939
%)


$250,000  C 
$156.30
$250,000
C
 $1,599.52
$156.30
Semih Yildirim
ADMS 3530