Copyright © 2008 Pearson Education Canada10-1 Contemporary Business Mathematics With Canadian...

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


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.


Finding n

FV = PV(1+i)n

(1 )n

FVi

PV

ln ln(1 )FV

n iPV

ln

ln(1 )

FVPV

ni


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.


Rule of 70

How long will it take money to double at 6%compounded annually?

n = 70 = 11.67 years 6


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.


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)


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


Finding i FV = PV(1+i)n

(1+i)n = FV PV (1+i) = FV (1/n) PV i = FV (1/n) - 1

PV


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.


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)


Electronic Calculator Solution


Determining Effective Rate f

f = (1+i)m - 1

where m is the number of compoundingperiods per year


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%


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.


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


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)


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)


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%.


Calculator Solution


An Alternate Calculator Solution


Summary

• Effective and equivalent interest rates can be used to compare different nominal rates with varying compounding intervals.

Copyright © 2008 Pearson Education Canada10-1 Contemporary Business Mathematics With Canadian...

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