6-1 copyright 2010 mcgraw-hill australia pty ltd powerpoint slides to accompany croucher,...

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

Chapter 6

Compound Interest

Introductory Mathematics & Statistics

6-2Copyright 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

6-3Copyright 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

6-4Copyright 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 beginningi = rate of interest per period (expressed as a fraction or

decimal)n = number of periods for which interest is accumulatedS = accumulated value at the end of n periods

ni1PS

6-5Copyright 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

ni 1factoronAccumulati

6-6Copyright 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

1factoronaccumulatiP

1i1P

Pi1P

principaloriginalvaluedaccumulateinterestcompoundofAmount

n

n

6-7Copyright 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

6-8Copyright 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

1ki1Pi

6-9Copyright 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

6-10Copyright 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e

6.2 Calculation of compound interest (cont…)

ExampleSuppose $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

3n,05.0i,8000$P

157625.1

05.01

i1rfactoonAccumulati3

n

6-11Copyright 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…)

9261$

05.18000$

i1PSvaluedAccumulate3

n

1261$

1157625.18000$

1factoronaccumulatiinterest compoundofAmount

P

6-12Copyright 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

6-13Copyright 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)

3284n,01.04

04.0i,6000$P

37494068.1

01.01

i1rfactoonAccumulati32

n

6-14Copyright 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)

64.8249$

01.16000$

i1PSvaluedAccumulate32

n

64.2249$

137494068.16000$

1factoronaccumulatiinterestcompoundofAmount

P

6-15Copyright 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

WhereP = present value

S = compound interest

ni1SP

6-16Copyright 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?

n)i1(rfactovaluePresent

6-17Copyright 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.

10n,08.0i,30000$S

46319349.0

08.01

i1factorvaluePresent10

n

80.13895$

46319349.030000$

i1SP n

6-18Copyright 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

1P

Si

n1

6-19Copyright 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.

80420,1460$,1000$ nSP

00474.0

11000

1460

1P

Si

801

n1

6-20Copyright 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

i1logPS

logn

6-21Copyright 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.

0125.04

05.0i,1500$S,800$P

56.50

0125.01log800

1500log

n

6-22Copyright 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