Historical average

What Is Historical Average?

The historical average, also known as the arithmetic mean, is a fundamental concept in statistical analysis used to represent the central tendency of a dataset over a past period. It is calculated by summing all values within a given series and then dividing by the total number of data points. In the realm of quantitative finance, the historical average provides a concise summary of past performance, whether it pertains to asset returns, economic indicators, or other financial metrics. This simple yet powerful measure helps analysts and investors understand the typical value or behavior of a variable over a specified historical period, serving as a baseline for comparison and initial insights.

History and Origin

The concept of the arithmetic mean, which forms the basis of the historical average, has roots stretching back to ancient civilizations. Early forms of averaging were used by Babylonian astronomers around 2000 BCE for calculating planetary positions and by Egyptians for trade calculations. However, the first systematic approach to calculation matching modern usage is often attributed to the Persian mathematician Al-Khwarizmi (780–850 CE), who introduced decimal-based calculation methods. T⁷he arithmetic mean gained further prominence and theoretical grounding with the development of probability theory by mathematicians like Abraham de Moivre (1677–1754) and Carl Friedrich Gauss (1777–1855), establishing its centrality in modern statistics and error theory. Over centuries, as data collection became more formalized, the historical average became an indispensable tool for summarizing numerical observations across various fields, including economics and, eventually, finance.

Key Takeaways

The historical average is the sum of a set of values divided by the number of values in the set, providing a simple measure of central tendency.
It is widely used in finance to summarize past data, such as asset returns, interest rates, or economic growth.
While easy to calculate and understand, the historical average has limitations, particularly when used to forecast future performance, as past results do not guarantee future outcomes.
It is sensitive to outliers and does not account for compounding effects over time, which can lead to misleading interpretations for investment performance.
Despite its limitations, it serves as a valuable starting point for investment analysis and benchmarking.

Formula and Calculation

The formula for the historical average (arithmetic mean) is straightforward:

\text{Historical Average} = \frac{\sum_{i=1}^{n} x_i}{n}

Where:

(\sum_{i=1}^{n} x_i) represents the sum of all individual data points ((x)) in the series.
(n) represents the total number of data points in the series.

For example, to calculate the historical average daily closing price of a stock over a month, one would sum the closing prices for each trading day and divide by the number of trading days. This calculation forms a foundational performance metric.

Interpreting the Historical Average

Interpreting the historical average involves understanding what it represents within its specific context. A historical average provides a snapshot of "what was typical" over a past period. For instance, if the historical average annual return for a stock market index over the last 50 years is 10%, it suggests that, on average, an investment in that index grew by 10% per year over that span. However, this figure does not imply that the market will return 10% every year, nor does it guarantee future returns.

When evaluating a historical average, it is important to consider the period length, the presence of volatility, and any significant events that occurred within that period. A long historical period might smooth out short-term fluctuations, revealing broader market trends, while a shorter period might highlight more recent behavior. Investors often use the historical average as a baseline for risk assessment and to understand long-term patterns, but they must be cautious about projecting it directly into the future.

Hypothetical Example

Consider an investor who owns a mutual fund and wants to calculate its historical average annual return over the past five years. The annual returns are as follows:

Year 1: +12%
Year 2: +8%
Year 3: -5%
Year 4: +15%
Year 5: +10%

To calculate the historical average annual return:

Sum the annual returns: (12 + 8 + (-5) + 15 + 10 = 40)
Count the number of years: (5)
Divide the sum by the number of years: (40 / 5 = 8)

The historical average annual return for this mutual fund over the past five years is 8%. This figure provides a quick understanding of the fund's general performance over this period, aiding in preliminary investment decisions.

Practical Applications

The historical average finds numerous practical applications across various facets of finance and economics. In financial forecasting, analysts often use historical averages of sales, earnings, or expenses as a starting point to project future financial statements. For example, a company might use its average revenue growth from the past five years to estimate next year's revenue.

In portfolio management, historical averages of asset class returns are frequently used to set expectations for future asset allocation models. For instance, the long-term average return of equities versus bonds helps inform strategic portfolio construction. Furthermore, economists use historical averages of economic indicators like GDP growth, inflation rates, or unemployment figures to analyze economic cycles and set monetary policy. For instance, the Federal Reserve's historical average of the federal funds rate provides context for current policy decisions. This ⁶data, available from sources like the St. Louis Federal Reserve's FRED database, helps illustrate the typical level of interest rates over long periods.

L⁵imitations and Criticisms

While useful for summarizing past data, the historical average has significant limitations, particularly when applied to future predictions in dynamic financial markets. A primary criticism is that "past performance is not indicative of future results." Relyi⁴ng solely on historical averages can lead to an unwarranted optimism, especially when the underlying conditions that generated those averages have changed. For e³xample, a period of exceptionally high returns or low interest rates in the past may not be reproducible in the future due to shifts in economic fundamentals or market structure.

Academic research highlights that for forecasting future cumulative returns, the arithmetic average can lead to excessively high expectations, especially over longer horizons, if not properly adjusted. Outli²ers, or unusually high or low values, can also heavily skew the historical average, making it less representative of the typical experience. For instance, a single year of extreme market gains or losses can significantly distort a multi-year historical average, reducing its effectiveness as a reliable measure. Financial planning firms, acknowledging these issues, have increasingly shifted towards using projected, forward-looking returns rather than solely relying on historical averages.

H¹istorical Average vs. Moving Average

The historical average and the moving average are both measures of central tendency based on past data, but they differ fundamentally in their calculation and application.

Feature	Historical Average (Arithmetic Mean)	Moving Average
Calculation	Sum of all data points in a fixed, specified period, divided by the count. The period is static.	Average of data points over a specified period that shifts or "moves" forward with each new data point.
Focus	Represents the average over a definitive, unchanging past period.	Represents the average over a recent, continuously updated period, smoothing out short-term fluctuations.
Purpose	Provides a static historical benchmark; useful for long-term aggregate understanding.	Identifies recent market trends and reduces noise in data; used for technical analysis and short-term signals.
Responsiveness	Not responsive to recent changes outside its fixed historical window.	Highly responsive to recent changes, as older data drops out and new data enters the calculation.

While the historical average provides a fixed reference point for a defined past, the moving average offers a dynamic, evolving view of recent trends, making it particularly valuable for technical analysis and identifying current momentum in asset valuation.

FAQs

What is the difference between a historical average and an average?

The terms "historical average" and "average" are often used interchangeably in finance, with "historical average" simply emphasizing that the average is derived from past data. "Average" is a broader statistical term for a measure of central tendency, which can also include current or forecasted data. In most financial contexts, when someone refers to an "average," they are usually implying a historical average.

Is historical average a good predictor of future performance?

No, the historical average is generally not considered a reliable standalone predictor of future performance. While it provides insight into past trends, financial markets are influenced by numerous unpredictable factors such as economic cycles, geopolitical events, and technological advancements that may not be reflected in historical data. It is widely cautioned that past performance does not guarantee future results.

When should the historical average be used?

The historical average is best used for understanding past behavior, establishing a baseline for comparison, or for long-term strategic analysis. It is useful for aggregating data over significant periods to observe underlying patterns or for benchmarking. However, for precise financial forecasting or short-term decision-making, it should be used in conjunction with other more sophisticated analytical tools and current market conditions.

Can the historical average be skewed?

Yes, the historical average can be significantly skewed by outliers, which are extremely high or low values within the dataset. For instance, a single year with an abnormally high or low return in an investment portfolio can disproportionately influence the overall historical average, making it less representative of the typical performance. This sensitivity to extreme values is a key limitation.