Copyright © 2008 Pearson Education Canada 10-1 Contemporary Business Mathematics With Canadian Applications Eighth Edition S. A. Hummelbrunner/K. Suzanne Coombs PowerPoint: D. Johnston Chapter 10 Compound Interest— Further Topics
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Contemporary Business Mathematics With Canadian Applications
Eighth Edition S. A. Hummelbrunner/K. Suzanne Coombs
PowerPoint: D. Johnston
Chapter 10
Compound Interest—Further Topics
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ObjectivesAfter completing chapter ten, the student will
be able to :
• Determine the number of conversion periods.
• Find equated dates.
• Compute periodic and nominal rates of interest.
• Compute the effective rate of interest.
• Compute equivalent rates of interest.
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Finding n
FV = PV(1+i)n
(1 )n
FVi
PV
ln ln(1 )FV
n iPV
ln
ln(1 )
FVPV
ni
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Rule of 70Should be used for Approximate Answers Only
The number of conversion periods (n) required for money to double in value is approximately equal to : 70 i where i is the periodic interest rate.
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Rule of 70
How long will it take money to double at 6%compounded annually?
n = 70 = 11.67 years 6
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Equated Date
• The equated date is the date on which a single sum of money is equal to the sum of two or more dated values.
• Use t = 0 as the focal date when setting up the equation of value for finding an equated date.
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Finding the Equated DateA payment of $1000 is due today (t=0) and another payment of $3000 is due in 5 years. Find the date at which a payment of $3800 will settle the debt. t=0 ? 5 years 1000 3800 3000 (Continued)
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Finding the Equated Date (continued)
t=0 ? 5 years
1000 3800 3000
The equation of value using t=0 as the focal date and a rate of 4% compounded annually is: 1000 + 3000(1.04) –5 = 3800(1.04) –n
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Finding i FV = PV(1+i)n
(1+i)n = FV PV (1+i) = FV (1/n) PV i = FV (1/n) - 1
PV
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Effective Rate of Interest
• The effective rate is the equivalent rate of interest compounded annually.
• Effective rates can be used to compare nominal rates of interest.
• An effective rate can be found by computing the future value (FV) of $1 after one year and then subtracting the $1 of principal.
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Finding an Effective Rate
Find the effective rate that is equivalent to a nominal rate of 7.2% compounded monthly. Find the FV of $1 at 7.2% compounded monthly and subtract the $1 of principal. $1 (1 + .072/12) 12 - $1 = 7.4424% (continued)
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Electronic Calculator Solution
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Determining Effective Rate f
f = (1+i)m - 1
where m is the number of compoundingperiods per year
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Calculating Effective RatesNominal rate is 6%
Compounding Effective Rate
Semi-annual (1.03)2 – 1 = 6.09%
Quarterly (1.015)4-1 = 6.14%
Monthly (1.005)12-1 = 6.17%
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Equivalent Rates
• Interest rates that increase a given principal to the same future value over the same time period are equivalent.
• Find equivalent rates by equating the accumulated or FV of $1 for rates under consideration based on a selected time period, usually a year.
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Equivalent RatesAssume one year time period
Principal $1 $1 $1
NominalRate
12.55% 12.18% 12%
n 1 2 4
i 0.1255 0.0609 0.03
FV 1(1.1255) 1(1.0609)2 1(1.03)4
FV 1.1255 1.1255 1.1255
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Finding Equivalent Rates
Find the nominal rate compounded semi-annually that is equivalent to an annual rate of 6% compounded annually. Let the PV = 1 and the time period be one year. Find the FV at each rate. (continued)
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Finding Equivalent Rates (continued)
Consider 6% compounded annually. FV = 1.06 Consider the semi-annual rate i. FV = (1+i)2 The equation of value is:
(1+i)2 = 1.06 i = 1.06 0.5 - 1 = 0.029563 (continued)
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Finding the nominal rate(continued)
Recall that i is a semi-annual rate.
Multiply the semi-annual rate 0.029563 by 2to get the required nominal rate.
Nominal rate = 0.029563 x 2 = 5.91%.
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Calculator Solution
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An Alternate Calculator Solution
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Summary
• Effective and equivalent interest rates can be used to compare different nominal rates with varying compounding intervals.