1. Introductory Mathematics & Statistics
Chapter 6
Compound Interest
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2. 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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3. 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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4. 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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5. 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. 6.1 Introduction (cont…)
The actual amount of compound interest earned after n years is the difference between the accumulated value and the original principal
Copyright  2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e

7. 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
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8. 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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9. 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
Copyright  2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e

10. 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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11. 6.2 Calculation of compound interest (cont…)
Solution (cont…)
Copyright  2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e

12. 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
Copyright  2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e

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

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

15. 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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16. 6.3 Present value (cont…) Example
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 $ 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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17. 6.3 Present value (cont…) Solution So
Hence, the plumber must invest an amount of $ now to accumulate the specified amount of $ at the end of 10 years.
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18. 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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19. 6.3 Present value (cont…) Example Solution
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 × 4 = This corresponds to an interest rate of 1.896% per annum.
Copyright  2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e

20. 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
Copyright  2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e

21. 6.3 Present value (cont…) Example Solution
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 Since in this case a period is a quarter, the amount of time is approximately equal to = years.
Copyright  2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e

22. 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
Copyright  2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e