copyright © 2015, 2011, and 2007 pearson education, inc. 1 chapter 10 compound interest and...

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Chapter 10Compound Interest and

Inflation

Section 1

Compound Interest

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Objectives

1. Use the simple interest formula I = PRT to calculate compound interest

2. Identify interest rate per compounding period and number of compounding periods.

3. Use the formula M = P(1 + i)n to find compound amount.

4. Use the table to find compound amount.

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Present Value & FutureValue

Present Value – value of an investment right now

Future Value, Future Amount, Compound Amount – amount in an investment at a specific future date

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Future Value

Depends on:

1.Compound interest—Compound interest results in a greater future value than simple interest.

2.Interest rate—A higher rate results in a greater future value.

3.Length of investment—An investment held longer usually results in a greater future value.

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Use Simple Interest Formula I = PRTto Calculate Compound Interest

Compound Interest – calculated on previously credited interest in addition to the original principal

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Finding Future Value

1. Use I = PRT to find simple interest for the period.

2. Add principal at the end of the previous period to the interest for the current period to find the principal at the end of the current period.

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Example

George Morton wants to compare simple interest to compound interest on a $3000 investment.(a)Find the interest if funds earn 8% simple interest for 1 year.(b) Find the interest if funds earn 8% interest compounded every 6 months for 1 year.(c) Find the difference between the two.(d) Find the effective rate for both.

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Example (cont)

(a) Simple interest on $3000 at 8% for 1 year is found as follows.

I = PRT = $3000 × .08 × 1 = $240

(b) Interest for first 6 months= PRT = $3000 × .08 × 1/2 = $120

Principal at end of first 6 months

= Original principal + Interest

= $3000 + $120 = $3120

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Example (cont)

(b) Interest for second 6 months= PRT = $3120 × .08 × 1/2 = $124.80Principal at end of 1 year= $3120 + $124.80 = $3244.80

Interest earned in the second 6 months ($124.80) is greater than that earned in the first 6 months ($120) because the interest earned becomes part of the principal, and therefore earns interest.

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Example (cont)

(b) Total Compound Interest

= $120 + $124.80 = $244.80

(c) Difference in interest

= 244.80 – 240 = $4.80

The difference of $4.80 over a year does not seem like much, but compound interest leads to huge differences when applied to larger sums of money over long time periods.

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Example (cont)

(d) The effective interest rate is the interest for the year divided by the original investment.

8% simple interest

6% compounded

Although they have the same nominal rate (8,), the compound interest investment has a larger effective interest rate due to compounding.

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Example

The Peters hope to have $5000 in 4 years for a down payment on a new car. They invest $3800 in an account that pays 6% interest at the end of each year, on previous interest in addition to principal. (a) Find the excess of compound interest over simple interest after 4 years. (b) Will they have enough money at the end of 4 years to meet their goal of a down payment?

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Example (cont)

First calculate interest using I = PRT. Find the new principal by adding the interest earned to the preceding principal.

Year P × R × T = Interest CompoundAmount

1 $3800.00 × .06 × 1 = $228.00 $4028.00

2 $4028.00 × .06 × 1 = $241.68 $4269.68

3 $4269.68 × .06 × 1 = $256.18 $4252.86

4 $5252.86 × .06 × 1 = $271.55 $4797.41

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Example (cont)

Compound Interest= $4797.41 – $3800 = $997.41

Simple Interest= $3800 × .06 × 4 = $912

Difference = $997.41 – $912 = $85.41

(b) No, but almost! They will be short of their goal by $5000 – $4797.41 = $202.59.

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Compounding Period

Time period over which the interest is calculated and added to principal

For example, 8% compounded quarterly means that interest will be calculated and added to principal at the end of each quarter. This requires four interest-rate calculations in one year.

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Identify Interest Rate Per Compounding Period

Interest rate applied at the end of each compounding period

Divide the annual interest rate by the number of compounding periods in one year

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Identify Number of Compounding Periods

Total number of compounding periods in the investment is the product of the number of years in the term of the investment and the number of compounding periods per year

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Example

Find the interest rate per compounding period and the number of compounding periods over the life of each loan.

(a) 6% compounded semiannually, 2 years

(b) 9% per year, compounded monthly,4 years

(c) 7% per year, compounded quarterly, 4 years

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Example (cont)

(a) 6% compounded semiannually is6% ÷ 2 = 3% credited at the end of each 6 months2 years × 2 periods per year = 4 compounding periods in 4 years

(b) 9% compounded monthly results in 9% ÷ 12 = 0.75% credited at the end of each month

4 years × 12 periods per year = 48 compounding periods in 4 years

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Example (cont)

(c) 7% compounded quarterly results in 7% ÷ 4 = 1.75% credited at the end of each quarter

4 years × 4 periods per year = 16 compounding periods in 4 years

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Use M = P(1 + i)n to Find Compound Amount

The formula for compound interest uses exponents, which is a short way of writing repeated products.

For example,

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Use M = P(1 + i)n to Find Compound Amount

Maturity Value = M = P(1 + i)n

Interest = I = M – P

where

P = initial investment

n = total number of compounding periods

i = interest rate per compounding period

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Example

An investment at Wells Fargo pays 6% interest per year compounded semiannually. Given an initial deposit of $3200, (a) use the formula to find the compound amount after 4 years, and (b) find the compound interest.

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Use the Table to Find a Compound Amount

The value of (1 + i)n can be found using a calculator or in the compound interest table

Interest rate i at the top of the table is the interest rate per compounding period

n far left or far right column of the table is the total number of compounding periods

In the body of the table is the compound amount for each $1 in principal

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Finding Compound Amount

Compound amount

= Principal × Number from compoundinterest table

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Example

In each case, find the interest earned on a $2000 deposit.

(a)For 3 years, compounded annually at 4%

(b) For 5 years, compounded semiannually at 6%

(c) For 6 years, compounded quarterly at 8%

(d) For 2 years, compounded monthly at 12%

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Example (cont)

(a) in 3 years, there are 3 × 1 = 3 compounding periods

interest rate per compounding period is 4% ÷ 1 = 4%

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Example (cont)

(a) Compound amount = M

= $2000 × 1.12486 = $2249.72

Interest earned = I

= $2249.72 – $2000 = $249.72

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Example (cont)

(b) in 5 years, there are 5 × 2 = 10 semiannual compounding periodsinterest rate per compounding period is 6% ÷ 2 = 3%

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Example (cont)

(b) Compound amount = M

= $2000 × 1.34392 = $2687.84

Interest earned = I

= $2687.84 – $2000 = $687.84

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Example (cont)

(c) in 6 years, there are 6 × 4 = 24 quarterly compounding periods

interest rate per compounding period is 8% ÷ 4 = 2%

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Example (cont)

(c) Compound amount = M

= $2000 × 1.60844 = $3216.88

Interest earned = I

= $3216.88 – $2000 = $1216.88

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Example (cont)

(d) in 2 years, there are 2 × 12 = 24 monthly compounding periods

interest rate per compounding period is 12% ÷ 12 = 1%

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Example (cont)

(d) Compound amount = M

= $2000 × 1.26973 = $2539.46

Interest earned = I

= $2539.46 – $2000 = $539.46