Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 1 Chapter 10 Compound Interest and Inflation Section 1 Compound Interest

Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 2 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.

Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 3 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

Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 4 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.

Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 5 Use Simple Interest Formula I = PRT to Calculate Compound Interest Compound Interest – calculated on previously credited interest in addition to the original principal

Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 6 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.

Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 7 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.

Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 8 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 = $ $120 = $3120

Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 9 Example (cont) (b)Interest for second 6 months = PRT = $3120 ×.08 × 1/2 = $ Principal at end of 1 year = $ $ = $ 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.

Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 10 Example (cont) (b)Total Compound Interest = $120 + $ = $ (c)Difference in interest = – 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.

Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 11 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.

Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 12 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?

Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 13 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 Compound Amount 1 $ ×.06 × 1 = $ $ $ ×.06 × 1 = $ $ $ ×.06 × 1 = $ $ $ ×.06 × 1 = $ $

Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 14 Example (cont) Compound Interest = $ – $3800 = $ Simple Interest = $3800 ×.06 × 4 = $912 Difference = $ – $912 = $85.41 (b)No, but almost! They will be short of their goal by $5000 – $ = $

Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 15 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.

Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 16 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

Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 17 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

Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 18 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

Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 19 Example (cont) (a)6% compounded semiannually is 6% ÷ 2 = 3% credited at the end of each 6 months 2 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

Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 20 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

Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 21 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,

Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 22 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

Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 23 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.

Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 24 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

Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 25

Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 26 Finding Compound Amount Compound amount = Principal × Number from compound interest table

Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 27 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%

Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 28 Example (cont) (a) in 3 years, there are 3 × 1 = 3 compounding periods interest rate per compounding period is 4% ÷ 1 = 4%

Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 29 Example (cont) (a)Compound amount = M = $2000 × = $ Interest earned = I = $ – $2000 = $249.72

Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 30 Example (cont) (b) in 5 years, there are 5 × 2 = 10 semiannual compounding periods interest rate per compounding period is 6% ÷ 2 = 3%

Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 31 Example (cont) (b)Compound amount = M = $2000 × = $ Interest earned = I = $ – $2000 = $687.84

Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 32 Example (cont) (c) in 6 years, there are 6 × 4 = 24 quarterly compounding periods interest rate per compounding period is 8% ÷ 4 = 2%

Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 33 Example (cont) (c)Compound amount = M = $2000 × = $ Interest earned = I = $ – $2000 = $

Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 34 Example (cont) (d) in 2 years, there are 2 × 12 = 24 monthly compounding periods interest rate per compounding period is 12% ÷ 12 = 1%

Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 35 Example (cont) (d)Compound amount = M = $2000 × = $ Interest earned = I = $ – $2000 = $539.46