Core Concepts of Financial Mathematics

The core concepts of financial mathematics include understanding percentages and their role in calculating changes in value or price, and exploring two types of interest – simple and compound. These concepts are explained through and applied to practical scenarios like mark-ups, discounts, and GST, demonstrating their relevance in everyday financial decisions. These form the foundation for understanding more complex financial topics and applications.

Use this page to revise the following concepts of financial mathematics:

• Percentage calculations
• Interest
• Compound interest

Percentage calculations

Percentages measure a portion or part of an amount relative to its total value, with \(100\%\) representing the whole. This concept is widely used in financial mathematics because changes in value depend on the initial amount—a measure of relative value. This provides a standardised way of comparing changes, assessing efficiencies and understanding growth or decline across a range of scenarios.

Without percentages, relying on absolute values alone can be misleading. For example, consider the case of spending \(\$100\) to earn back \(\$1\) compared to spending \(\$10\) to earn back \(\$1\). In this case, the absolute gain of \(\$1\) is identical. However, the use of percentages can reveal the effectiveness of spending: in the first scenario there is a return of \(1\%\), whilst in the second there is a return of \(10\%\). This demonstrates how percentages allow for more meaningful comparisons, making them a fundamental tool in financial analysis.

Percentage Increase and Decrease

Percentage change can be understood in two key ways:

• As a value relative to the whole: For example, \(120\%\) of the original represents the total amount as a whole, including the original value and any change
• As a change relative to the original amount: For example, \(120\%\) of the original can also be described as a \(20\%\) increase from the original value.

It is important to understand it from both perspectives as this is fundamental in calculating percentage change.

To calculate percentage change relative to the original amount, we can use the following equation:

This equation describes the change in value \(\left(V_f - V_i\right)\) as a proportion of the initial value, by dividing it by \(V_i\). Multiplying by \(100\) converts this proportion to a percentage.

Mark-ups and discounts

There are situations where the goal is not to determine the percentage change but to apply it. This is commonly seen in cases of mark-ups and discounts:

• Mark-ups involve increasing the original price by a specified percentage, often used in retail or pricing strategies.
• Discounts involve reducing the original price by a specified percentage, typically seen in sales or promotions.

To apply a percentage change, there are two methods:

1. Determine the percentage change and adjust the original amount
   

   \[V_f = V_i + (\Delta\% \times V_i)\]

2. Calculate the new value as a proportion of the original amount
   

   \[V_f = (1 + \Delta\%) \times V_i\]

Both methods require converting the percentage into its decimal form to simplify calculations. This can be achieved by dividing \(\Delta\%\) by \(100\).

Goods and Services Tax (GST)

Goods and Services Tax (GST) is a tax applied to the provision of goods and services, calculated as a \(10\%\) mark-up on the original price. This means the final price includes the base cost of the good or service plus an additional \(10\%\) of that amount as tax.

Depending on the situation, prices may be displayed inclusive or exclusive of GST.

• GST-inclusive: The price already includes the \(10\%\) GST, meaning the amount shown is the total the customer will pay.
• GST-exclusive: The price does not include the \(10\%\) GST, and the tax is added to the displayed price at the point of sale to calculate the final total.

This distinction is important for understanding the true cost of goods or services, particularly in contexts like business invoicing or retail pricing.

When a price is exclusive of GST, calculating the final price is equivalent to calculating a mark-up of \(10\%\). However, when calculating the original price from a GST-inclusive price, the calculation is different. The GST-inclusive price represents \(110\%\) of the original. Simply subtracting \(10\%\) off this price would result in \(10\%\) of the \(110\%\) (original), which is subtracting \(11\%\) off the price.

To calculate the original price, the final price must be divided by \(1.1\) (the decimal form of \(110\%\)):

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Interest

Interest is the cost of borrowing money, typically calculated as a percentage of the principal or initial amount borrowed. This percentage, known as the interest rate , is applied over different time periods during which the cost is accrued , and is usually calculated at the end of that time period. For example, interest can be calculated on a yearly basis, referred to as “per annum” (p.a.) or annually, which means interest accrues at the end of the year. Interest may also be calculated twice a year (semiannually), quarterly (every \(3\) months), monthly or even weekly or daily.

There are two types of interest: simple and compound interest .

Simple interest

Simple interest is interest calculated solely on the principal amount. This remains fixed for the entire duration of a loan or investment.

The total amount of simple interest can be calculated using the formula:

Simple interest recurrence relation

Alternatively, the interest amount for each time period can be described using a different formula, which is particularly useful in recurrence relations and rules:

A recurrence relationship can be used to describe incremental changes to the value of a loan or investment after \(n\) number of time periods. The recurrence relationship which describes the value of a loan or investment that incurs simple interest can be described by:

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Rule for simple interest recurrence relation

A recurrence relation requires knowledge of the value from the previous time period to calculate the value for the current time period. This can become tedious as the number of time periods increases. Since the amount added is directly proportional to the number of years, the value at any time period can be simplified using the formula:

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Compound interest

Compound interest is calculated on the previous value, which includes both the principal amount, and any interest accrued up to that point. With each compounding period, the value of the loan or investment grows further, as the interest is applied to the updated amount.

Since compound interest depends on the previous value, the growth of a loan or investment can be described using recurrence relations . The relationship starts with the initial value and grows by adding interest at each step:

\[\begin{align}V_1 &= V_0 + \frac{r}{100} V_0 \\ V_2 &= V_1 + \frac{r}{100} V_1 \\ &\phantom{=}\ldots \end{align}\]

These can be simplified to:

\[\begin{align}V_1 &= \left(1+\frac{r}{100}\right)V_0 \\ V_2 &= \left(1+\frac{r}{100}\right)V_1\\ &\phantom{=}\ldots\end{align}\]

Here, \(1+ \dfrac{r}{100}\) is referred to as the growth or multiplication factor.

To simplify the recurrence relation, a single variable is introduced for the growth factor:

\[R=1+ \frac{r}{100}\]

Recurrence Relation

The recurrence relation for compound interest can now be written as:

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This example uses the same values from the simple interest worked example. It is important to highlights the difference in loan values when calculated with simple interest versus compound interest:

• At the end of Year 1: $2148 (simple interest) compared with $2148 (compound interest).
• At the end of Year 2: $2296 (simple interest) compared with $2306.95 (compound interest).
• At the end of Year 3: $2444 (simple interest) compared with $2477.67 (compound interest).

Although the difference in interest accrued using compound interest is small initially ($10.95 more in the 2nd year and $33.67 more in the 3rd year), the effect becomes increasingly significant over time. This occurs because compound interest applies not only to the principal but also to the accumulated interest at each compounding period, resulting in exponential growth.

Different compounding periods

Compounding periods refer to the intervals at which interest is calculated and added to the loan or investment. These periods are essentially time periods, but the term "compounding" is emphasised because interest is recalculated and accrued at each interval. Common compounding periods include:

• Weekly: 52 times a year
• Fortnightly: 26 times a year
• Monthly: 12 times a year
• Quarterly: 4 times a year
• Yearly: once a year

Understanding the compounding period is essential for calculations, particularly when determining the growth or multiplication factor. The rate is usually given as a rate per annum, but the compounding period may be different. For example, an investment may accrue interest at a rate per annum that compounds quarterly. The rate per quarter is determined by dividing the annual rate by the number of compounding periods per year. In this case, to determine the value after one year, four calculations must be performed, one for each quarter, applying the quarterly rate incrementally. This ensures the interest is compounded accurately for the specified time period.

This example uses the same values from the compound interest worked example above. It highlights that when compound interest is calculated in smaller increments, the total value increases, even though the annual interest rate remains the same. For instance:

• Compounding quarterly resulted in a value of $2152.16 at the end of one year.
• Compounding yearly resulted in a value of $2148 after the same period.

This difference occurs because compounding more frequently allows interest to be accrued not only on the principal but also on the smaller increases from previous compounding periods. However, this growth slows down over time and approaches a maximum value.

This happens because at a certain point, the additional interest generated from more frequent compounding becomes very small. From a practical perspective, the amount increased has an almost negligible effect on the final value.

Rule for the recurrence relation

The tedious nature of computing a recurrence relationship for a greater number of periods also exist for compound interest calculation. This can be simplified using the rule:

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