6-1 Copyright  2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e Chapter 6 Compound Interest Introductory Mathematics & Statistics

6-2 Copyright  2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e Learning Objectives Distinguish between simple and compound interest Calculate compound interest Compare calculations of simple and compound interest Calculate the present and accumulated values of a principal of money Solve problems that involve transposing the compound interest formula

6-3 Copyright  2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 6.1 Introduction We are now considering the case in which the interest due is added to the principal at the end of each interest period and this interest itself also earns interest from that point onwards In this case, the interest is said to be compounded, and the sum of the original principal plus total interest earned is called the accumulated value or maturity value The difference between the accumulated value and original principal is called compound interest

6-4 Copyright  2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 6.1 Introduction (cont…) Compound interest formula Where: P = principal at the beginning i = rate of interest per period (expressed as a fraction or decimal) n = number of periods for which interest is accumulated S = accumulated value at the end of n periods

6-5 Copyright  2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 6.1 Introduction (cont…) The accumulation factor is the factor by which you multiply the original principal in order to obtain the accumulated value The value of the accumulation factor is independent of the value of the beginning principal, P

6-6 Copyright  2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 6.1 Introduction (cont…) The actual amount of compound interest earned after n years is the difference between the accumulated value and the original principal

6-7 Copyright  2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 6.1 Introduction (cont…) Comparison of the calculation of simple interest and compound interest from first principles –For any given principal P, given the same interest rate i and the same period of an investment or loan, compound interest will always have a value greater than simple interest –From an investor’s point of view, compound interest is preferable –From a borrower’s point of view simple interest is preferable

6-8 Copyright  2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 6.1 Introduction (cont…) –The amount of simple interest earned each year is a constant: P × i = Pi –The amount of compound interest earned in the first year is also Pi –However, the amount of compound interest earned in the second year is Pi(1 + i), which is greater than Pi –The amount of compound interest earned in the third year is Pi(1 + i) 2, which is also greater than Pi –Amount of compound interest earned in the kth year –Amount of simple interest earned each year = Pi

6-9 Copyright  2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 6.2 Calculation of compound interest In many instances the interest may be compounded using other time periods, such as semi-annually, monthly, weekly or even daily This rate, when expressed as a rate per annum, is known as a nominal rate of interest The interest rate is divided by the number of periods per year for which the interest is compounded The number of time periods (n) is now the total number of time periods involved

6-10 Copyright  2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 6.2 Calculation of compound interest (cont…) Example Suppose $8000 is invested at a compound interest rate of 5% per annum. Find the accumulation factor, accumulated value and amount of compound interest earned after 3 years. Solution

6-11 Copyright  2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 6.2 Calculation of compound interest (cont…) Solution (cont…)

6-12 Copyright  2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 6.2 Calculation of compound interest (cont…) Example A company secretary has an investment opportunity in which a lending institution offers her an interest rate of 4.0% compounded quarterly. She decides to invest an amount of $6000 under the scheme for 8 years. Calculate: (a) the accumulation factor (b) the accumulated value after 5 years (c) the total compound interest earned

6-13 Copyright  2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 6.2 Calculation of compound interest (cont…) Solution (a)

6-14 Copyright  2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 6.2 Calculation of compound interest (cont…) Solution (cont…) (b) (c)