Mathematics Advanced: Financial Mathematics

Module 2 for Mathematics Advanced (HSC)

Jackson Taylor, Pranav Sharma

Last updated on August 24, 2021 3 min read Year 12, Mathematics Advanced, Financial Mathematics

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 $I = P R n$ , 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 $A_{n} = P (1 + R)^{n}$ , 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 $A_{n} = P (1 - R)^{n}$ (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 $200 ({1.005}^{n} + {1.005}^{n - 1} + \dots + 1.005)$
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 $(F V)$ is the total value of an investment at the end of its term, including all interest
The Present value $(P V)$ 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: $F V = P V (1 + r)^{n}$

Loan Repayments

Loans are usually repaid through regular installments, with compound interest charged on the balance owed
$A_{n} = principle + interest - installments + interest$
- 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?

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Method 1:

Let $M$ be the monthly installment:

$A_{1} = 10000 (1.005) - M$
$A_{2} = (10000 (1.005) - M) (1.005) - M$
$∴ A_{2} = 10000 (1.005)^{2} - 1.005 M - M$
$A_{60} = 0 = 10000 (1.005)^{60} - M (1 + 1.005 + \dots + {1.005}^{59})$

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

Method 2 (Speed Hack):

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

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

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

Year 12 Mathematics Advanced Financial Mathematics Mathematics Simple Interest Compound Interest Interest Annuities Future Value Present Value Loan Repayments Inflation

Mathematics Advanced: Financial Mathematics

Simple and Compound Interest

Annuities

Practice Question

Present and Future Values

Loan Repayments

Practice Question

Jackson Taylor

Post Writer

Pranav Sharma

Site Owner