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[h] IB Mathematics AI HL Flashcards- Amortization and annuities

[q] **Amortisation and Annuities**

ANNUITIES — These are when fixed sums of money are paid to someone or into some account at fixed periods for a length of time, whilst gaining interest. Some concepts involving geometric series will be relevant here.

E.G. 1: You invest $2000 at the end of each year, for 8 years, at a fixed interest rate of 5%. What will be the value at the end of the 8 years?

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Firstly, we must understand that this is not just $16,000 increased by 5%, eight times. The first $2000 increases by 5%, seven times, but as the second $2000 is only in the account for 6 years, it increases by 5%, six times. The third $2000 increases five times, and so on. Bear in mind that to increase by 5%, we multiply by (1 + 0.05), or 1.05.

This is all shown in the table to the right:

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The total F.V. is the sum of all the F.V.s:

i.e.: F.V. = 2000 + 2000(1.05) + 2000(1.05)² + … + 2000(1.05)⁷

which is a geometric series, with U₁ = 2000 and r = 1.05, n = 8.

Using \( S_n = \frac{U_1(1 – r^n)}{1 – r} \), then F.V. = \( \frac{2000(1.05^8)}{1.05} = \frac{9554.91}{-0.05} = 19098.22 \)

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AMORTISATION — In IB, we are interpreting this term as almost an opposite of annuity. This is when you are paying back some kind of loan, with a periodic payment, but interest is charged after each payment. So, the amount of interest charged changes each year, making it complicated to calculate a fixed periodic repayment that would work for the desired no. of periods.

You could use a spreadsheet for this too, or use the following formula:

\[

R = \frac{i \times P}{1 – (1 + i)^{-n}}

\]

where \( i = \) interest rate, \( P = \) amount owed, \( R = \) periodic payment, \( n = \) no. of periods

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