# Financial Mathematics

## Table of Contents

## Interest

- Interest is essentially the cost of using someone else’s money
- When you borrow money from a bank or lender, you are charged a percentage of that intial amount extra when it comes time to pay it back
- In most cases, interest is paid back
*periodically*(over time) rather than as a*lump sum*(all at once) - There are two main kinds of interest: simple and compound

### Simple Interest

Simple interest is where interest is calculated based on the principle (initial) amount, and the time since the loan started

Simple interest $(I)$ is calculated with $\color{orange}I=(P\times R\times n)$

- $P$ 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

For example, if you take a $\$ $100 loan at 6% simple interest per year, every year you wait adds $\$ $6 to the amount you have to pay back $\color{orange}(\$100\times 6\%\times 1=\$6,\$100\times 6\%\times 2=\$12\text{, and so on})$

If the question asks for the total amount, add $P$ to $I$ at the end

- Basically, use the formula $\color{orange}I=(P\times R\times n)+P$ instead

### Compound Interest

- Compound interest is where the interest in each period is calculated on the principle,
**PLUS any interest earned until that point**Most loans, debts, and repayments are compound interest. If the type of interest isn’t specified, it’s almost definitely compound. - The formula for compound interest is $\color{orange}FV=PV(1+r^{n})$
- $FV$ stands for “future value” or “final value” (same as $I$ in simple interest)
- $PV$ stands for “present value” or “principal value”
- $r$ is the rate per period (for example 6% per year)
- $n$ is the number of periods passed

#### Increasing Future Value

- There are 3 main ways to increase the future value of an investment:
- Increase the principal value (basically, more money = more money)
- Increase the frequency of the compounding periods (e.g. make $r$ per month rather than per year)
- Increase the interest rate (7% is more than 6%)

Remember, if you’re the one paying for it (e.g. a loan or debt), you want to do the OPPOSITE of those.

### Comparing Investment Strategies (Question Guide)

Usually, questions involving interest tend to involve comparing investment strategies. An example question would be:

Heidi goes to the bank to take out a loan of 1000 dollars over 3 years. The bank offers her two options: the first with a compounding interest rate of 5% per annum (compounding annually), and the second with a compounding rate of 4% per annum (compounding monthly). Which is the better deal for Heidi?

## Toggle Solutions

- Figure out whether you need to find the smaller or larger value.
- In the case of a “which is the better deal” question involving a loan, smaller is better.

- Calculate the future value of the first option
- In this case, $FV=1000\times (1+0.05)^{3}=\$1157.63$

- Repeat step 2 for each option
- Option 2: $FV=1000\times(1+\frac{0.04}{12})^{12\times 3}=\$1012.07$

- Answer the question.
- Since Heidi will get a better deal from a lower final sum, option 2 is the better deal for her.

## Investment Graphs

### Simple Interest

- Simple interest demonstrates a linear relationship, with the x-axis as $n$ (number of time periods), and the y-axis as $I$ (the interest earned).
- To draw a simple interest graph:
- Construct a table of values for $I$ and $n$ using the simple interest formula.
- Draw a number plane with the $n$ horizontal axis and $I$ vertical axis, then plot the points.
- Join the points to make a straight line.

##### Example

Draw a graph of the simple interest earned over a period of 10 years, where the initial amount is $10, and the rate of interest is 8% p.a.

### Compound Interest

- Compound interest demonstrates an exponential curve, with the x-axis as $n$ (number of time periods), and the y-axis as $FV$ (the future value).
- To draw a compound interest graph:
- Construct a table of values for $FV$ and $n$ using the compound interest formula.
- Draw a number plane with the $n$ horizontal axis and $FV$ vertical axis, then plot the points.
- Join the points to make an exponential curve.

## Appreciation and Inflation

- Appreciation is when an item increases in value.
- The rate of financial appreciation can often be expressed using the compound interest formula.

$$\color{orange}{FV=PV(1+r)^{n}}$$

- Inflation is when the value of money goes down. When inflation occurs, the price of goods and services increases.
- Usually, inflation is between 2% and 3%.

- This increase in the price of things can also be expressed using the compound interest formula.

### Declining-Balance Depreciation

- Declining-balance depreciation occurs when the value of the item decreases by a fixed percentage each time period.
- Declining-balance depreciation has a slightly modified version of the compound interest formula:
- $\color{orange}{S=V_{0}(1-r)^{n}}$
- $S$ is the final or “salvage” value, $V_0$ is the initial value, $r$ is the rate of depreciation per time period, and $n$ is the number of time periods.

### Reducing-Balance Loans

- Reducing-balance loans are loans where the interest is calculated on the outstanding amount, rather than the total amount.
- These use more complicated formulae, so you’ll typically be given a two-way table, which you can then use to determine the amount outstanding.