Transcript Foundations of Finance Chapter 5 The Time Value of Money Arthur J. Keown

Foundations of Finance
Arthur J. Keown
J. William Petty
John D. Martin
David F. Scott, Jr.
Chapter 5
The Time Value of Money
Chapter 5
The Time Value of Money
Learning Objectives
 Explain the mechanics of
compounding, which is how money
grows over a time when it is
invested.
 Be able to move money through time
using time value of money tables,
financial calculators, and
spreadsheets.
 Discuss the relationship between
compounding and bringing money
back to present.
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Foundations of Finance
Pearson Prentice Hall
Chapter 5
The Time Value of Money
Learning Objectives
 Define an ordinary annuity and
calculate its compound or future
value.
 Differentiate between an ordinary
annuity and an annuity due and
determine the future and present
value of an annuity due.
 Determine the future or present
value of a sum when there are
nonannual compounding periods.
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Foundations of Finance
Pearson Prentice Hall
Chapter 5
The Time Value of Money
Learning Objectives
• Determine the present value of an
uneven stream of payments
• Determine the present value of a
perpetuity.
• Explain how the international setting
complicates the time value of money.
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Foundations of Finance
Pearson Prentice Hall
Chapter 5
The Time Value of Money
Principles Used in this
Chapter
• Principle 2: The Time Value of
Money – A Dollar Received Today
Is Worth More Than a Dollar
Received in The Future.
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Foundations of Finance
Pearson Prentice Hall
Chapter 5
The Time Value of Money
Simple Interest
Interest is earned on principal
$100 invested at 6% per year
1st year
interest is $6.00
2nd year
interest is $6.00
3rd year
interest is $6.00
Total interest earned: $18.00
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Foundations of Finance
Pearson Prentice Hall
Chapter 5
The Time Value of Money
Compound Interest
• When interest paid on an
investment during the first
period is added to the
principal; then, during the
second period, interest is
earned on the new sum.
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Foundations of Finance
Pearson Prentice Hall
Chapter 5
The Time Value of Money
Compound Interest
Interest is earned on previously earned
interest
$100 invested at 6% with annual
compounding
1st year interest is $6.00 Principal is $106.00
2nd year interest is $6.36 Principal is $112.36
3rd year interest is $6.74 Principal is $119.11
Total interest earned:
$19.11
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Foundations of Finance
Pearson Prentice Hall
Chapter 5
The Time Value of Money
Future Value
- The amount a sum will grow in
a certain number of years when
compounded at a specific rate.
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Foundations of Finance
Pearson Prentice Hall
Chapter 5
The Time Value of Money
Future Value
FV1 = PV (1 + i)
Where FV1 = the future of the investment at
the end of one year
i= the annual interest (or discount)
rate
PV = the present value, or original
amount invested at the beginning
of the first year
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Foundations of Finance
Pearson Prentice Hall
Chapter 5
The Time Value of Money
Future Value
What will an investment be worth
in 2 years?
$100 invested at 6%
FV2= PV(1+i)2 = $100 (1+.06)2
$100 (1.06)2 = $112.36
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Foundations of Finance
Pearson Prentice Hall
Chapter 5
The Time Value of Money
Future Value
• Future Value can be increased
by:
• Increasing number of years of
compounding
• Increasing the interest or
discount rate
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Foundations of Finance
Pearson Prentice Hall
Chapter 5
The Time Value of Money
Future Value Using Tables
FVn = PV (FVIFi,n)
Where FVn = the future of the investment at
the end of n year
PV = the present value, or original
amount invested at the beginning
of the first year
FVIF = Future value interest factor or
the compound sum of $1
i= the interest rate
n= number of compounding periods
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Foundations of Finance
Pearson Prentice Hall
Chapter 5
The Time Value of Money
Future Value
What is the future value of $500
invested at 8% for 7 years? (Assume
annual compounding)
Using the tables, look at 8% column, 7
time periods. What is the factor?
FVn= PV (FVIF8%,7yr)
= $500 (1.714)
= $857
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Foundations of Finance
Pearson Prentice Hall
Chapter 5
The Time Value of Money
Future Value Using Calculators
Using any four inputs you will find the 5th.
Set to P/YR = 1 and END mode.
INPUTS
N
10
PV
-100
PMT
0
FV
179.10
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OUTPUT
I/YR
6
Foundations of Finance
Pearson Prentice Hall
Chapter 5
The Time Value of Money
Future Value Using Spreadsheets
Spreadsheets and the Time Value of Money
If we invest $500 in a bank where it will earn 8 percent compounded
annually, how much will it be worth at the end of 7 years?
rate (I) =
number of periods (n) =
payment (PMT) =
present value (PV) =
type (0=at end of period) =
Future value =
8%
7
0
$500
0
$856.91
Excel formula: FV = (rate, number of periods, payment, present value, type)
Entered in cell d13: = FV(d7,d8,d9,-d10,d11)
Notice that present value ($500) took a negative value
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Foundations of Finance
Pearson Prentice Hall
Chapter 5
The Time Value of Money
Present Value
The current value of a future payment
PV = FVn {1/(1+i)n}
Where FVn = the future of the investment at
the end of n years
n= number of years until payment is
received
i= the interest rate
PV = the present value of the future sum
of money
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Foundations of Finance
Pearson Prentice Hall
Chapter 5
The Time Value of Money
Present Value
What will be the present value of $500
to be received 10 years from today if
the discount rate is 6%?
PV = $500 {1/(1+.06)10}
= $500 (1/1.791)
= $500 (.558)
= $279
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Foundations of Finance
Pearson Prentice Hall
Chapter 5
The Time Value of Money
Present Value Using Tables
PVn = FV (PVIFi,n)
Where PVn = the present value of a future sum of
money
FV = the future value of an investment at
the end of an investment period
PVIF = Present Value interest factor of $1
i= the interest rate
n= number of compounding periods
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Foundations of Finance
Pearson Prentice Hall
Chapter 5
The Time Value of Money
Present Value
What is the present value of $100
to be received in 10 years if the
discount rate is 6%?
PVn = FV (PVIF6%,10yrs.)
= $100 (.558)
= $55.80
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Foundations of Finance
Pearson Prentice Hall
Chapter 5
The Time Value of Money
Present Value Using Calculators
Using any four inputs you will find the 5th.
Set to P/YR = 1 and END mode.
INPUTS
N
10
I/YR
6
PMT
0
FV
100.00
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OUTPUT
PV
-55.84
Foundations of Finance
Pearson Prentice Hall
Chapter 5
The Time Value of Money
Annuity
• Series of equal dollar payments
for a specified number of years.
• Ordinary annuity payments
occur at the end of each period
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Foundations of Finance
Pearson Prentice Hall
Chapter 5
The Time Value of Money
Compound Annuity
• Depositing or investing an equal
sum of money at the end of
each year for a certain number
of years and allowing it to grow.
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Foundations of Finance
Pearson Prentice Hall
Chapter 5
The Time Value of Money
Compound Annuity
FV5 = $500 (1+.06)4 + $500 (1+.06)3
+$500(1+.06)2 + $500 (1+.06) + $500
= $500 (1.262) + $500 (1.191) +
$500 (1.124) + $500 (1.090) +
$500
= $631.00 + $595.50 + $562.00 +
$530.00 + $500
= $2,818.50
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Foundations of Finance
Pearson Prentice Hall
Chapter 5
The Time Value of Money
Illustration of a 5yr $500 Annuity
Compounded at 6%
0
1
2
3
4
5
500
500
500
500
500
6%
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Foundations of Finance
Pearson Prentice Hall
Chapter 5
The Time Value of Money
Future Value of an Annuity
FV = PMT {(FVIFi,n-1)/ i }
Where FV n= the future of an annuity at
the end of the nth years
FVIFi,n= future-value interest factor or sum of
annuity of $1 for n years
PMT= the annuity payment deposited or
received at the end of each year
i= the annual interest (or discount) rate
n = the number of years for which the
annuity will last
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Foundations of Finance
Pearson Prentice Hall
Chapter 5
The Time Value of Money
Compounding Annuity
What will $500 deposited in the bank
every year for 5 years at 10% be
worth?
FV = PMT {(FVIFi,n-1)/ i }
Simplified this equation is:
FV5 = PMT(FVIFAi,n)
= $500(5.637)
= $2,818.50
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Foundations of Finance
Pearson Prentice Hall
Chapter 5
The Time Value of Money
Future Value of an Annuity
Using Calculators
Using any four inputs you will find the 5th.
Set to P/YR = 1 and END mode.
INPUTS
N
5
PV
0
I/YR
6
PMT
500
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OUTPUT
FV
-2,818.55
Foundations of Finance
Pearson Prentice Hall
Chapter 5
The Time Value of Money
Present Value of an Annuity
• Pensions, insurance obligations,
and interest received from
bonds are all annuities. These
items all have a present value.
• Calculate the present value of
an annuity using the present
value of annuity table.
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Foundations of Finance
Pearson Prentice Hall
Chapter 5
The Time Value of Money
Present Value of an Annuity
Calculate the present value of a $500
annuity received at the end of the
year annually for five years when the
discount rate is 6%.
PV = PMT(PVIFAi,n)
= $500(4.212)
= $2,106
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Foundations of Finance
Pearson Prentice Hall
Chapter 5
The Time Value of Money
Annuities Due
• Ordinary annuities in which all
payments have been shifted
forward by one time period.
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Foundations of Finance
Pearson Prentice Hall
Chapter 5
The Time Value of Money
Amortized Loans
• Loans paid off in equal
installments over time
– Typically Home Mortgages
– Auto Loans
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Foundations of Finance
Pearson Prentice Hall
Chapter 5
The Time Value of Money
Payments and Annuities
If you want to finance a new
machinery with a purchase
price of $6,000 at an interest
rate of 15% over 4 years, what
will your payments be?
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Foundations of Finance
Pearson Prentice Hall
Chapter 5
The Time Value of Money
Future Value Using Calculators
Using any four inputs you will find the 5th.
Set to P/YR = 1 and END mode.
INPUTS
N
4
PV
6,000
I/YR
15
FV
0
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OUTPUT
PMT
-2,101.59
Foundations of Finance
Pearson Prentice Hall
Chapter 5
The Time Value of Money
Amortization of a Loan
• Reducing the balance of a loan via
annuity payments is called
amortizing.
• A typical amortization schedule
looks at payment, interest, principal
payment and balance.
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Foundations of Finance
Pearson Prentice Hall
Chapter 5
The Time Value of Money
Amortization Schedule
Yr.
Annuity
Interest Principal
1
$2,101.58
$900.00 $1,201.58 $4,798.42
2
$2,101.58
719.76
1,381.82
3,416.60
3
$2,101.58
512.49
1,589.09
1,827.51
4
$2,101.58
274.07
1,827.51
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Foundations of Finance
Balance
Pearson Prentice Hall
Chapter 5
The Time Value of Money
Compounding Interest with
Non-annual periods
If using the tables, divide the percentage by
the number of compounding periods in a
year, and multiply the time periods by the
number of compounding periods in a year.
Example:
8% a year, with semiannual compounding for
5 years.
8% / 2 = 4% column on the tables
N = 5 years, with semiannual compounding
or 10
Use 10 for number of periods, 4% each
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Foundations of Finance
Pearson Prentice Hall
Chapter 5
The Time Value of Money
Perpetuity
• An annuity that continues forever is
called perpetuity
• The present value of a perpetuity is
PV = PP/i
PV = present value of the perpetuity
PP = constant dollar amount
provided by the of perpetuity
i = annuity interest (or discount
rate)
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Foundations of Finance
Pearson Prentice Hall
Chapter 5
The Time Value of Money
The Multinational Firm
• Principle 1- The Risk Return Tradeoff – We
Won’t Take on Additional Risk Unless We
Expect to Be Compensated with Additional
Return
• The discount rate is reflected in the rate of
inflation.
• Inflation rate outside US difficult to predict
• Inflation rate in Argentina in 1989 was
4,924%, in 1990 dropped to 1,344%, and in
1991 it was only 84%.
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Foundations of Finance
Pearson Prentice Hall