Mathematics Advanced: Financial Mathematics

Module 2 for Mathematics Advanced (HSC)

Table of Contents
NOTE: This guide assumes that you fully understand the principles of the preliminary Advanced mathematics course.

Simple and Compound Interest

  • Simple interest $(I)$ is calculated with $\color{orange}I=PRn$, an Arithmetic Progression
    • $P$ is the principal (initial) sum,
    • $R$ is the rate of interest per unit of time
    • $n$ is the number of time intervals which have passed
    • If the question asks for the total amount, add $P$ to $I$ at the end
  • Compound interest is found by $\color{orange}A_{n}=P(1+R)^{n}$, a Geometric Progression
    • $A_n$ is the amount of interest after $n$ units of time
    • To find the interest (without the initial amount), subtract $P$ from $A_n$
  • Depreciation is a form of compound in terest, where the value decreases over time
  • Depreciation is expressed as $\color{orange}A_{n}=P(1-R)^{n}$ (also a Geometric Progression)
    • $R$ is the rate of depreciation per unit time
    • To find the interest (without the initial amount), subtract $P$ from $A_n$

Annuities

  • Annuities are compound interest investments, from which equal payments are recieved on a regular basis, for a fixed period of time

Practice Question

Minho deposits 200$ per month at the start of each month into an annuity which pays 6% p.a. for 20 years. How much will the account hold after the full 20 years?

Toggle Answer

  • After 1 month, the account has $200(1+0.005)$ dollars

  • After 2 months, $200(1.005)^{2}+200(1.005)$

  • After $n$ months, we have $200(1.005^{n}+1.005^{n-1}+…+1.005)$

  • The geometric progression in the brackets is: $$S_{(20\times 12)}=\frac{1.005(1.005^{240})-1}{1.005-1}=464.3511$$

  • Therefore, $464.3511\times 200=92870.22$ $ after 20 years

Present and Future Values

  • The Future value $(FV)$ is the total value of an investment at the end of its term, including all interest
  • The Present value $(PV)$ is the single lump of money that could be initially invested to yield a given future value over a given period
  • Present values are calculated using the compound interest formula
  • Future value is calculated using a variant of the compound interest formula: $$\color{orange}FV=PV(1+r)^{n}$$

Loan Repayments

  • Loans are usually repaid through regular installments, with compound interest charged on the balance owed
  • $\color{orange}A_n = \text{principle + interest - installments + interest}$
    • The loan is paid off when $A_{n}=0$

Practice Question

Michael takes out $10000 to buy a car. He will repay the loan in 5 years, paying 60 equal monthly instalments, beginning 1 month after he takes out the loan. Interest is 6% p.a. compounded monthly. How much is the monthly installment?

Toggle Answer

Method 1:

Let $M$ be the monthly installment:

  • $A_{1}=10000(1.005)-M$
  • $A_{2}=(10000(1.005)-M)(1.005)-M$
  • $\therefore A_{2}=10000(1.005)^{2}-1.005M-M$
  • $A_{60}=0=10000(1.005)^{60}-M(1+1.005+…+1.005^{59})$

GP inside the brackets is $\frac{10000(1.005^{60})}{\frac{1.005^{60}-1}{0.005}}=\$193.33$

Method 2 (Speed Hack):

  • $A_{n}=10000(1.005)^{n}-M(1+1.005+…+1.005^{n-1})$
  • $10000(1.005)^{60}=M(1+1.005+…+1.005^{59})$

GP inside brackets is $S_{60}=\frac{1.005^{60}-1}{0.005}=69.77$

  • $\therefore M=\frac{10000(1.005)^{60}}{69.77}$
  • $=\$193.33 $

Jackson Taylor
Jackson Taylor
Post Writer

I’m a Year 12 student at Sydney Tech.

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Pranav Sharma
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Year 12 Student, site owner and developer.

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