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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

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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

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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

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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

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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

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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

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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 investors point of view, compound interest is preferable –From a borrowers point of view simple interest is preferable

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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

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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

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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

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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…)

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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

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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)

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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)

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6-15 Copyright 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 6.3 Present value We would like to know how much we must invest today to accumulate a specified amount at some future time This is called the present value (or discounted value) at compound interest Where P = present value S = compound interest

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6-16 Copyright 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 6.3 Present value (cont…) The present value factor (factor or discount factor) is the amount by which you multiply the specified amount in order to obtain the original principal. Example A plumber wishes to have an amount of $30 000 at the end of 10 years. The bank pays an interest rate of 8% per annum compounded annually. How much money will the plumber have to invest with the bank now?

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6-17 Copyright 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 6.3 Present value (cont…) Solution So Hence, the plumber must invest an amount of $13 895.80 now to accumulate the specified amount of $30 000 at the end of 10 years.

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6-18 Copyright 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 6.3 Present value (cont…) Calculation of the interest rate The compound interest formula can also be used to find the interest rate charged when a principal of P has accumulated to an amount S after n periods Note that the value of i obtained is the interest rate per period. To obtain the nominal rate of interest (per annum), multiply this value of i by the number of periods in a year. E.g., if interest is compounded quarterly, i should be multiplied by 4

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6-19 Copyright 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 6.3 Present value (cont…) Example Suppose that $1000 has accumulated to $1460 in 20 years with interest compounded quarterly. What annual rate of compound interest was used? Solution Since the interest was compounded quarterly, multiply this value of i by 4 to obtain 0.004 74 × 4 = 0.018 96. This corresponds to an interest rate of 1.896% per annum.

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6-20 Copyright 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 6.3 Present value (cont…) Calculation of the number of periods –The compound interest formula can also be used to find the number of periods (n) required for an amount P to accumulate to an amount S when interest is at a rate of i per period –That is, the value of i must be written as the interest rate per period. Note also that the value of n obtained will not always be a whole number and so we can only approximate the actual number of periods required

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6-21 Copyright 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 6.3 Present value (cont…) Example Find how long it will take for $800 to accumulate to $1500 if interest is at 5% per annum, compounded quarterly. Solution Hence, the number of periods required is 50.56. Since in this case a period is a quarter, the amount of time is approximately equal to 50 4 = 12.64 years.

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6-22 Copyright 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e Summary We have discussed the difference between simple and compound interest We calculated compound interest We have also compared calculations of simple and compound interest We calculated the present and accumulated values of a principal of money We solved problems that involve transposing the compound interest formula

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