# Mathematics Advanced: Financial Mathematics

Module 2 for Mathematics Advanced (HSC)

## Table of Contents

## 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 $