A:

An arithmetic average is the sum of a series of numbers divided by the count of that series of numbers.

If you were asked to find the class (arithmetic) average of test scores, you would simply add up all the test scores of the students, and then divide that sum by the number of students. For example, if five students took an exam and their scores were 60%, 70%, 80%, 90% and 100%, the arithmetic class average would be 80%.

This would be calculated as: (0.6 + 0.7 + 0.8 + 0.9 + 1.0) / 5 = 0.8.

The reason you use an arithmetic average for test scores is that each test score is an independent event. If one student happens to perform poorly on the exam, the next student's chances of doing poor (or well) on the exam isn't affected. In other words, each student's score is independent of the all other students' scores. However, there are some instances, particularly in the world of finance, where an arithmetic mean is not an appropriate method for calculating an average.

Consider your investment returns, for example. Suppose you have invested your savings in the stock market for five years. If your returns each year were 90%, 10%, 20%, 30% and -90%, what would your average return be during this period? Well, taking the simple arithmetic average, you would get an answer of 12%. Not too shabby, you might think.

However, when it comes to annual investment returns, the numbers are not independent of each other. If you lose a ton of money one year, you have that much less capital to generate returns during the following years, and vice versa. Because of this reality, we need to calculate the geometric average of your investment returns in order to get an accurate measurement of what your actual average annual return over the five-year period is.

To do this, we simply add one to each number (to avoid any problems with negative percentages). Then, multiply all the numbers together, and raise their product to the power of one divided by the count of the numbers in the series. And you're finished - just don't forget to subtract one from the result!

That's quite a mouthful, but on paper it's actually not that complex. Returning to our example, let's calculate the geometric average: Our returns were 90%, 10%, 20%, 30% and -90%, so we plug them into the formula as [(1.9 x 1.1 x 1.2 x 1.3 x 0.1) ^ 1/5] - 1. This equals a geometric average annual return of -20.08%. That's a heck of a lot worse than the 12% arithmetic average we calculated earlier, and unfortunately it's also the number that represents reality in this case.

It may seem confusing as to why geometric average returns are more accurate than arithmetic average returns, but look at it this way: if you lose 100% of your capital in one year, you don't have any hope of making a return on it during the next year. In other words, investment returns are not independent of each other, so they require a geometric average to represent their mean.

To learn more about the mathematical nature of investment returns, check out Overcoming Compounding's Dark Side.

RELATED FAQS
  1. Do plane tickets get cheaper closer to the date of departure?

    Read about when to buy flights. See how statistics can predict optimal pricing. Read about price volatility over time. Learn ... Read Answer >>
  2. Is Colombia an emerging market economy?

    Learn the definition of an emerging market economy, and understand how Colombia, while not yet developed, meets the standards ... Read Answer >>
  3. What assumptions are made when conducting a t-test?

    Learn what a t-test is, and discover the five standard assumptions that are made regarding the validity of sampling and data ... Read Answer >>
  4. What are some of the more common types of regressions investors can use?

    Learn about the most common types of regressions investors use to model asset prices including linear regressions and multiple ... Read Answer >>
  5. What types of assets lower portfolio variance?

    Learn what type of assets reduce portfolio variance and how modern portfolio theory uses correlation coefficients. Read Answer >>
  6. When is it better to use systematic over simple random sampling?

    Learn when systematic sampling is better than simple random sampling, such as in the absence of data patterns and when there ... Read Answer >>
Related Articles
  1. Investing

    How to Prepare for the Low Return Environment Ahead

    Learn about the big takeaway from this week’s chart: Investors aiming for higher returns over the next five years should be prepared to stomach more volatility.
  2. Active Trading Fundamentals

    SandRidge's 3 Key Financial Ratios (SDOC)

    Learn more about SandRidge Energy, Inc., a history of the company's performance and financial stability through key financial ratios and its future outlook.
  3. Economics

    A Statistic About the U.S. Economy that May Surprise You

    Learn why many commentators seem to be pessimistically focused on the U.S. economy’s weak wage growth and manufacturing sector trouble.
  4. Economics

    The Current Probability of President Donald Trump

    Predict the current odds of a Donald Trump presidency, and understand the factors that have kept him on top and the looming challenges he faces.
  5. Fundamental Analysis

    Calculating the Coefficient Of Variation (CV)

    Coefficient of variation measures the dispersion of data points around the mean, a statistical average.
  6. Markets

    The Market Chart You Need to See This Week

    This week’s chart helps show why current low levels of stock market volatility look unsustainable and why now is a good time to prepare portfolios for a rockier road ahead.
  7. Economics

    Explaining Pareto Efficiency

    Pareto efficiency is an economic state where resources are allocated in the most efficient manner.
  8. Fundamental Analysis

    Intro to Stationary and Non-Stationary Processes

    Refining data points is the key to applying financial series time data to stock analysis.
  9. Economics

    What is a Bell Curve?

    The bell curve is the most common type of graphed data distribution.
  10. Economics

    What Big Data Can Tell us about the Economy

    Given recent market turbulence, it’s no wonder that investors are wondering whether the economy is heading in the right direction.
RELATED TERMS
  1. Kurtosis

    A statistical measure used to describe the distribution of observed ...
  2. Metrics

    A wide variety of tools that managers and executives can use ...
  3. IRR Rule

    A measure for evaluating whether to proceed with a project or ...
  4. Rule Of 72

    A shortcut to estimate the number of years required to double ...
  5. Black Swan

    An event or occurrence that deviates beyond what is normally ...
  6. Contagion

    The spread of market changes or disturbances from one region ...

You May Also Like

Hot Definitions
  1. Keynesian Economics

    An economic theory of total spending in the economy and its effects on output and inflation. Keynesian economics was developed ...
  2. Society for Worldwide Interbank Financial Telecommunications ...

    A member-owned cooperative that provides safe and secure financial transactions for its members. Established in 1973, the ...
  3. Generally Accepted Accounting Principles - GAAP

    The common set of accounting principles, standards and procedures that companies use to compile their financial statements. ...
  4. DuPont Analysis

    A method of performance measurement that was started by the DuPont Corporation in the 1920s. With this method, assets are ...
  5. Call Option

    An agreement that gives an investor the right (but not the obligation) to buy a stock, bond, commodity, or other instrument ...
  6. Economies Of Scale

    Economies of scale is the cost advantage that arises with increased output of a product. Economies of scale arise because ...
Trading Center