Copyright © 2008 Pearson Education Canada

Chapter 9

Compound Interest—

Future Value and Present Value

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Contemporary Business Mathematics With Canadian Applications

Eighth Edition S. A. Hummelbrunner/K. Suzanne Coombs

PowerPoint: D. Johnston

9-2Copyright © 2008 Pearson Education Canada Inc.

ObjectivesAfter 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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Compounding of InterestPrincipal = 10000, Rate = 10% p.a.

Term = 4 yearsYear 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)

CompoundingFrequency

Length ofperiod

Number ofannual 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

Number of Compounding Periods Per Yeari

i = j/m

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Finding Periodic Rate of Interest i= j/m

j = nominal or annual m = number of yearly

rate of interest compounding periods

Compound interval Annual

Periodic rate i 8%

Semi-annual 8% =4% 2

Quarterly 8% = 2% 4

Monthly 8% = 0.66% 12

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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

n=2X8=16

10% compounded quarterly for 5 years

(1.025)20

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. FV = P(1+i)n FV =10000(1.10)4 = $14641 Interest earned = 14641 – 10000 = $4641

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Simple Interest vs. Compound Interest

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Comparison of Simple and Compound Interest

Principal $10000 Term 5 years

Rate 6% compounded annually

Simple Interest Compound Interest

FV = 10000(1+.06x5) = $13000

FV=10000(1+.06)5 =$13382.26

Note the difference of

$382.26.

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Future Value of an InvestmentFV = 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 five-year term. FV = $5000(1.01)20 = 5000(1.22019004) = $6100.95

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Finding FV When n Is a Fraction

Find the future or accumulated value of $6000invested for 4 years, 5 months at 4%compounded quarterly.

FV = 6000(1.01)17.666667 = 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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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 ValueFV = PV(1+i)n

Divide both sides by (1+i)n .

PV = FV = FV(1+i) -n (1+i)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 = 10000(1.04) –8 (1.04)8

PV = $7306.90

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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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Proceeds of Long-term Promissory Note

The discounted value (or proceeds) is thePRESENT VALUE of the MATURITYVALUE at the date of discount.

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Promissory Note DiagramD

issue discount maturity

date date date

Discount Period

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Calculating Proceeds of a Non-Interest-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 = $2292.45

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Discounting an Interest-bearing 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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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). 7000(1.04) –5 = 5753.49 Find the equivalent value 8 years from today. 7000(1.04)3= 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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Summary (continued)

• The present value of a future amount at compound interest can be expressed by the formula PV = FV(1+i) -n .