Matematica Financeira Medias - Financial Business

Financial mathematics uses several types of averages (or means) to analyze and interpret data. Understanding these averages and their specific applications is crucial for making informed financial decisions. The most common averages used in financial mathematics are the arithmetic mean, the geometric mean, and the harmonic mean. The **arithmetic mean**, often referred to as the simple average, is calculated by summing a set of values and dividing by the number of values. It’s useful for understanding a typical value within a dataset, but can be easily skewed by outliers (extreme values). In financial contexts, the arithmetic mean is used to calculate the average return of an investment over a period, or the average price of a stock over a trading day. For example, if a stock had daily returns of 5%, -2%, 3%, and 1%, the arithmetic mean return would be (5 – 2 + 3 + 1) / 4 = 1.75%. While simple, this average doesn’t accurately represent the true growth of the investment due to compounding. The **geometric mean** is a more appropriate measure when dealing with rates of change or growth rates over multiple periods, especially when compounding is involved. It is calculated by multiplying all the values in a set, taking the nth root of the product, where n is the number of values. The geometric mean provides a more accurate reflection of the overall performance of an investment over time because it considers the compounding effect. Using the same example as above, the geometric mean return would be calculated as: [(1+0.05) * (1-0.02) * (1+0.03) * (1+0.01)]^(1/4) – 1 ≈ 1.70%. Notice that this is slightly lower than the arithmetic mean. This difference highlights the impact of volatility; the higher the volatility, the greater the difference between the arithmetic and geometric mean, and the more important it is to use the geometric mean to represent true investment performance. The geometric mean is often used in portfolio performance analysis and to calculate average annual growth rates (AAGR). The **harmonic mean** is used to find the average of rates or ratios, particularly when the denominator is constant. It is calculated by dividing the number of values by the sum of the reciprocals of the values. A common application in finance is calculating the average cost of shares purchased at different prices. Consider an investor who buys 100 shares each month. In month one, the share price is $10, in month two, it’s $12, and in month three, it’s $15. Using the harmonic mean, we can calculate the average price paid per share. The harmonic mean is 3 / (1/10 + 1/12 + 1/15) = $12.41. This represents the average price paid across the three months, accounting for the consistent number of shares purchased. The harmonic mean is always less than or equal to the geometric mean, which is always less than or equal to the arithmetic mean. Choosing the correct average depends on the specific financial question being addressed. The arithmetic mean offers simplicity but can be misleading with fluctuating rates. The geometric mean accurately reflects compounded growth and is ideal for investment performance analysis. The harmonic mean is vital when averaging rates or ratios, especially when dealing with fixed quantities. Recognizing the differences and appropriate uses of these averages is essential for accurate financial analysis and decision-making.