Transcript Chapter 9 - slc

Contemporary Business Mathematics
With Canadian Applications
Eighth Edition
S. A. Hummelbrunner/K. Suzanne Coombs
PowerPoint: D. Johnston
Chapter 9
Compound Interest—
Future Value and Present Value
Copyright © 2008 Pearson Education Canada
9-1
Objectives
After completing chapter nine, the student will
be able to:
• Calculate interest rates and number of
compounding periods
• Compute future (maturity) value.
• Compute the present value of future sums of
money.
• Discount long-term promissory notes.
• Solve equivalent value problems.
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Compound Interest
• Interest for a specified time period is added
to the original principal.
• The sum of the principal and interest
becomes the new principal for the next time
period.
• The amount of compound interest for the
first period is the same as for simple interest
but is greater for the following periods.
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9-3
Compounding of Interest
Principal = 10000, Rate = 10% p.a.
Term = 4 years
Year
Principal
Interest
Amount
1
10000
1000
11000
2
11000
1100
12100
3
12100
1210
13310
4
13310
1331
14641
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Formula for Future Value
FV = PV(1 + i)
n
or
S = P(1 + i)
n
FV = S = Future or Maturity Value
PV = P = Original Principal
i
Periodic Interest Rate
n
Number of compounding periods
over term of loan
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Compounding Frequencies
(determining periodic rate of interest)
Compounding Length of
Frequency
period
Number of
annual periods
Annual
12 months
1
Semi-annual
6 months
2
Quarterly
3 months
4
Monthly
1 month
12
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Determining Periodic Rate of Interest (i)
The nominal rate of interest is the stated
annual rate of interest.
In the equations we will use we need to
periodic rate of interest i.e. i. To calculate i
we use this formula.
Nominal (Annual) Rate
i
Number of Compounding Periods Per Year
i = j/m
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Finding Periodic Rate of Interest
i= j/m
j = nominal or annual rate of interest
m = number of yearly compounding periods
Compound interval
Annual
Periodic rate i
8%
Semi-annual
8% =4%
2
8% = 2%
4
Quarterly
Monthly
8% = 0.66%
12
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9-8
Determining Compounding
Factor (1+i)n
12% compounded
monthly for 3 years
(1.01)36
n=12x3=36
7% compounded
semi-annually for 8
years
(1.035)16
10% compounded
quarterly for 5 years
(1.025)20
n=2X8=16
n=4X5=20
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Calculation of Future Value
A principal of $10,000 is invested at an annual
rate of 10% for four years. Find the FV.
n
FV = P(1+i)
4
FV =10000(1.10) = $14641
Interest earned = 14641 – 10000 = $4641
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9-10
Simple Interest vs. Compound
Interest
9-11
Comparison of Simple and
Compound Interest
Principal $10000
Term 5 years
Simple Interest
Rate 6% compounded
annually
Compound Interest
FV = 10000(1+.06x5) FV=10000(1+.06)
= $13000
=$13382.26
5
Note the difference of $382.26.
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Future Value of an Investment
FV = PV(1+i)n
Find the future value (accumulated or maturity
value) of a savings certificate with a principal
of $5000 earning interest at 4% compounded
quarterly for five years at the end of the fiveyear term.
20
FV = $5000(1.01) = 5000(1.22019004) =
$6100.95
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9-13
Finding FV When n Is a
Fraction
Find the future or accumulated value of $6000
invested for 4 years, 5 months at 4%
compounded quarterly.
17.666667
FV = 6000(1.01)
= 7153.12
Note : There are 4x4 or 16 quarters in 4 years.
There are 1 and 2/3 quarters in 5 months (5/3).
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9-14
Computing Present Value
(Discounting)
Find the principal that will amount to $10,000
in 6 years at 4% compounded semi-annually.
FV = PV(1+i)n
10000 = PV(1.02)12
PV = 10000 = 10000(1.02) –12 = $7884.93
(1.02)12
(continued)
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Compound Discount
(continued)
Compound Discount = FV – PV
Compound Discount = 10000 – 7884.93
= $2115.07
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Formula for Present Value
FV = PV(1+i)n
n
Divide both sides by (1+i)
PV = FV = FV(1+i)
(1+i)n
.
-n
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Calculating Present Value
Find the present value of an amount of $10000
due four years from today if the interest rate is
8% compounded semi-annually.
PV = 10000 =
8
(1.04)
10000(1.04) –8
PV = $7306.90
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9-18
Long-term Promissory Notes
• Term of note longer than one year.
• Can be bought and sold at any time before
maturity.
• Subject to compound interest.
• No requirement to add the 3 days of grace
in determining legal due date.
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9-19
Proceeds of Long-term
Promissory Note
The discounted value (or proceeds) is the
PRESENT VALUE of the MATURITY
VALUE at the date of discount.
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9-20
D
Promissory Note Diagram
Discount Period
issue
discount
maturity
date
date
date
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Calculating Proceeds of a NonInterest-bearing Note
Find the proceeds of a non-interest bearing
note for $3000 discounted 2 years before
maturity. The interest rate is 9% compounded
monthly.
Since this is a non-interest bearing note, the
maturity value is equal to the face value.
PV = Proceeds =3000(1.0075)
–36
Copyright © 2008 Pearson Education Canada Inc.
= $2292.45
9-22
Discounting an Interestbearing Note
• Step 1 -- Find the maturity value of the
note.
• Step 2 -- Find the present value at the
discount date of the maturity value.
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9-23
Finding the Proceeds for an
Interest-bearing Note
On April 1, 2004, a three-year promissory note for $5000 is
issued with an interest rate of 8% compounded semi-annually.
The note is discounted on April 1, 2006 at 9% compounded
quarterly. Find the proceeds of the note.
Step 1 – Calculate the Maturity value = 5000(1.04)6 = $6326.60
Step 2- Calculate the Proceeds = 6326.60(1.0225) –4 = $5787.85
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Equivalent Values
• Equivalent values are the dated values of an
original sum of money.
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Finding Equivalent Values
• Select a focal date. The focal date is a
specific date chosen to compare the time
values of one or more dated sums of money.
• If the due date of the payment falls before
the focal date, use the FV formula.
• If the due date of the payment falls after the
focal date, use the PV formula.
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Calculating Equivalent Values
A payment of $7000 is due 5 years from now.
Money is worth 4% compounded annually.
The focal date is 5 years.
t=0
5 years
8 years
7000
Find the equivalent value today (t=0).
–5
7000(1.04) = 5753.49
Find the equivalent value 8 years from today.
3
7000(1.04) = 7874.05
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Summary
• With compound interest, earned interest is
added to the principal and thus “interest is
earned on interest” resulting in exponential
growth.
• The future value of an investment at
compound interest can be expressed by the
formula FV = P(1+i)n .
(continued)
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9-28
Summary
(continued)
• The present value of a future amount at
compound interest can be expressed by the
formula PV = FV(1+i) -n .
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