Financial MathsChapter 16

16 D and E – compound interest and depreciation

Compound interest

• When interest is added to the principal (the initial balance that needs to be paid) and this new balance is used to calculate the next interest payment, this is called compound interest.

• We can calculate compound interest by calculating simple interest one period at a time or all at once using a formula.

16D – worked example (one year at a time)• Kyna invests $8000 at 8% p.a. for 3

years with interest paid at the end of each year. Find the compounded value of the investment by calculating the simple interest on each year separately.

• Principle (P) = 8000

• Interest rate (R) = 8%

• Calculate the interest (first year) = 8% of 8000

= 8000 x .08

= 640

• Principle (second year) = initial principle + interest from first year

= 8000 + 640

= 8640

• Now, calculate the interest for the second year based on the principle from the second year

• Interest (second year) = of 8640

= 8640 x .08

= 691.20

• Principle for the third year = 8640 + 691.20

= 9331.20

• Interest (third year) = 8% of 9331.20

=746.50

• The total compounded value is the initial principle plus the compounded interest from each of the three years

= 8000 + 640 + 691.20 + 746.50

= 10077.70

The value of Kyna’s investment after three years is $10,077.70

Same example

Year Principle Interest Total

1 8000 8000 x 0.08 = 640 8640

2 86408640 x 0.08 = 691.20

9331.20

3 9331.209331.20 x 0.08 = 746.50

10077.70

This type of calculation can be set up in a table such as below:

Calculating compound interest• A formula for calculating annual compound interest is:

• Where,

• A = final amount

• P = principal amount (initial investment)

• r = annual interest rate (as a decimal e.g. 7% = 0.07)

• n = number of times the interest is compounded per year (e.g. yearly, quarterly, monthly, fortnightly)

• t = number of years

16D – worked example (using the formula)

Calculate the value of an investment of $4000 at 6% p.a. for 2 years with interest compounded quarterly.

• Principle (P) = 4000

• r = 6% = 0.06 (needs to be expressed as a decimal to use in the formula)

• n = 4 (interest is compounded/calculated four times per year/quarterly)

• t = 2 (number of years)

• Apply the formula:

The value of this investment after two

years is $4505.97

16D – worked example (using the formula)

William has $14 000 to invest. He invests the money at 9% p.a. for 5 years with interest compounded annually.

a. Calculate the amount to which this investment will grow.

• Principle (P) = 14000

• r = 9% = 0.09 (needs to be expressed as a decimal to use in the formula)

• n = 1 (interest is compounded/calculated once per year)

• t = 5 (number of years)

• Apply the formula:

The amount to which this investment will grow is $21,540.74

b. Calculate the compound interest earned on the investment.

Interest earned = total investment – initial principle

= 21540.74 – 14000

=$7540.74

The interest earned over 5 years is $7540.74

Class work

• Ex 16D page 551 questions 1ace, 2ace, 6, 7, 8, 10, 12, 15 (extension)

Depreciation

• Depreciation is the reduction in the value of an item as it ages over a period of time.

• For example, cars are known for going down in value based on how old they are (unless it is a collectible vintage car)

• We use the formula:

where

• A = amount

• P = principle (original value)

• r = rate of depreciation (as a decimal)

• t = time (in years)

16E worked example

• A farmer purchases a tractor for $115 000. The value of the tractor depreciates by 12% p.a. Find the value of the tractor after 5 years.

• A = amount (what we want to figure out)

• P = principle/initial amount ($115000)

• r = rate (12% = 0.12)

• t = time (5 years)

• Apply the formula:

Value after 5 years is $60,689.17

Written off items

• When the value of an item falls below a certain value it is said to be written off.

• Use trial-and-error methods to calculate the length of time that the item will take to reduce to this value.

• This means you need to guess a number of years (t) and trial them to see which t value fits for the value to fall below the written-off level.

Class work

• Ex 16E page 554 questions 1, 3, 4, 6, 7, 9, 11, 12