Percentages, Ratios, and Conversions in Finance

Income Yield vs. Capital Gain Yield

1. Income Yield: This measures the "cash flow" generated by the investment, such as dividends paid out to shareholders. It is the ratio of the annual income to the initial purchase price. $\text{Income Yield} = \frac{\text{Annual Income}}{\text{Purchase Price}}$
2. Capital Gain Yield: This measures the change in the asset's market value. It is the percentage change from the purchase price to the selling price. $\text{Capital Gain Yield} = \frac{\text{Selling Price} - \text{Purchase Price}}{\text{Purchase Price}}$

Total Rate of Return

The Total Rate of Return is the combined effect of both the income earned and the change in capital value. To find this, you add the income yield to the capital gain yield. If the capital gain is negative (a loss), it will reduce the total return.

Ratios and Proportions

Ratios allow us to compare quantities or divide a total amount into specific parts.

Simplifying and Solving Ratios

A ratio should always be expressed in its lowest terms. This often requires converting decimals or fractions into whole numbers before simplifying. For example, a ratio like $0.11 : 0.044 : 3.3$ must be scaled up to whole numbers to be useful.

When dealing with proportions, we use the relationship between two ratios to find an unknown variable ($x$ or $y$). This can involve simple two-part ratios or extended ratios involving three or more terms (e.g., $a:b:c = d:e:f$).

Proportional Allocation

Ratios are frequently used for proportional allocation, such as dividing expenses or ingredients. If three roommates share an apartment, they might divide the internet bill based on the ratio of their room sizes. To solve these, you first find the value of "one part" of the ratio by dividing the total amount by the sum of the ratio parts, then multiply that unit value by the specific ratio required for each person.

Currency and Unit Conversion

In a globalized economy, being able to convert between different currencies and measurement systems is vital.

Currency Exchange and Commission

To convert from one currency to another, you multiply the amount by the given exchange rate. However, financial institutions often charge a commission—a percentage-based fee for the service. When calculating the total cost of purchasing a foreign currency, you must account for this fee, which increases the amount of your home currency required for the transaction.

Unit Comparison and True Value

When comparing prices across different regions (for example, gas prices in the US vs. Canada), you must perform a two-step conversion to find the "true value".

1. Currency Conversion: Convert the foreign price into your local currency.
2. Unit Conversion: Convert the units (e.g., gallons to litres) so that you are comparing the same standard of measurement.

Summary Checklist for Unit 2:

• Did I compound percentage increases instead of adding them?
• Did I include both dividends and price changes when calculating total return?
• Are my ratios simplified to the lowest whole numbers?
• Did I include the bank's commission when calculating currency costs?