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IBMathsDPAnalysis & Approaches (AA)HLRevision Notes1. Number & AlgebraSequences & SeriesCompound Interest & Depreciation

Compound Interest & Depreciation (DP IB Analysis & Approaches (AA)): Revision Note

Amber

Written by: Amber

Reviewed by: Mark Curtis

Updated on

DP Analysis & Approaches (AA): HLDP Applications & Interpretation (AI): HLDP Applications & Interpretation (AI): SLDP Analysis & Approaches (AA): SL

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Compound Interest

What is compound interest?

  • Compound interest is where interest is added to both:

    • the initial investment

      • how much you originally put in

    • and to any interest that has already been added

      • over the previous years

  • This means compound interest grows faster than simple interest

    • Simple interest is when the same amount of interest is paid in each year

  • For compound interest you need

    • a timeframe (e.g. every year for ten years)

    • a nominal annual rate of interest, r% (e.g. increasing by 3% each year)

What does compounding monthly mean?

  • If interest of 6% per annum is divided by 12 and paid in every month ( 0.5% each month), then it is compounding monthly

  • In general, if r percent sign per annum is paid compounding monthly

    • then this means r over 12 percent sign interest is paid in each month

Examiner Tips and Tricks

Compounding monthly and compounding annually over the same time frame do not give the same amount of interest - compounding monthly gives more!

What are compounding periods?

  • Compounding periods are the time intervals after which interest is paid into your account

    • e.g. if r percent sign per annum is compounded monthly then

      • the compounding period is 'a month'

      • the number of compounding periods per year is 12

  • The letter k is used for the number of compounding periods per year

    • Compounding annually means k equals 1

    • Compounding half-yearly means k equals 2

    • Compounding quarterly means k equals 4

    • Compounding monthly means k equals 12

  • In general, if r percent sign p.a. (per annum) is paid across k compounding periods per year

    • then this means r over k percent sign interest is paid in after each compounding period

What is the formula for compound interest?

  • The formula for calculating compound interest is

    F V equals P V cross times open parentheses 1 plus fraction numerator r over denominator 100 k end fraction close parentheses to the power of k n end exponent

    • Where

      • F V is the future value

      • P V is the present value

      • n is the number of years

      • k is the number of compounding periods per year

      • r % is the nominal annual rate of interest

Examiner Tips and Tricks

The formula for compound interest is given in the formula booklet.

Examiner Tips and Tricks

Whilst your GDC may have a finance package, a lot of exam questions are designed around the formula, so it is often easier to use that.

Worked Example

Kim invests MYR 2000 (Malaysian Ringgit) in an account that pays a nominal annual interest rate of 2.5% compounded monthly.

Calculate the amount that Kim will have in her account after 5 years, to the nearest 10 MYR.

ai-sl-1-3-1-ci-we

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Depreciation

What is depreciation?

  • Depreciation is when the value of something falls over time

    • e.g. the price of a car

  • If the depreciation is occurring at a constant rate (e.g. 5% per year) then it is compound depreciation

What is the formula for compound depreciation?

  • The formula for calculating compound depreciation is

    F V equals P V cross times open parentheses 1 minus r over 100 close parentheses to the power of n

    • Where

      • F V is the future value

      • P V is the present value

      • n is the number of years

      • r% is the rate of depreciation

Examiner Tips and Tricks

The formula for compound depreciation is not given in the formula booklet, but it can be found from the compound interest formula by

  • setting k equals 1

  • changingplus to minus

Examiner Tips and Tricks

If you are using the finance package on your GDC for a depreciation question, make the interest rate negative.

Worked Example

Kyle buys a new car for AUD $14 999.  The value of the car depreciates by 15% each year.

(a) Find the value of the car after 5 years.

ai-sl-1-3-1-deprciationa

(b) Find the number of years and months it will take for the value of the car to be approximately AUD $9999.

ai-sl-1-3-1-deprciationb

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    Author
    Reviewer
    Amber

    Author: Amber

    Expertise: Maths Content Creator

    Amber gained a first class degree in Mathematics & Meteorology from the University of Reading before training to become a teacher. She is passionate about teaching, having spent 8 years teaching GCSE and A Level Mathematics both in the UK and internationally. Amber loves creating bright and informative resources to help students reach their potential.

    Mark Curtis

    Reviewer: Mark Curtis

    Expertise: Maths Content Creator

    Mark graduated twice from the University of Oxford: once in 2009 with a First in Mathematics, then again in 2013 with a PhD (DPhil) in Mathematics. He has had nine successful years as a secondary school teacher, specialising in A-Level Further Maths and running extension classes for Oxbridge Maths applicants. Alongside his teaching, he has written five internal textbooks, introduced new spiralling school curriculums and trained other Maths teachers through outreach programmes.