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The arithmetic mean is the simplest and most widely used measure of a mean, or average. It simply involves taking the sum of a group of numbers, then dividing that sum by the count of the numbers used in the series. For example, take the numbers 34, 44, 56, and 78. The sum is 212. The arithmetic mean is 212 divided by four, or 53.
People also use several other types of means, such as the geometric mean and harmonic mean, which comes into play in certain situations in finance and investing. Another example is the trimmed mean, used when calculating economic data such as the consumer price index (CPI) and personal consumption expenditures (PCE).
The arithmetic mean maintains its place in finance, as well. For example, mean earnings estimates typically are an arithmetic mean. Say you want to know the average earnings expectation of the 16 analysts covering a particular stock. Simply add up all the estimates and divide by 16 to get the arithmetic mean.
The same is true if you want to calculate a stock’s average closing price during a particular month. Say there are 23 trading days in the month. Simply take all the prices, add them up, and divide by 23 to get the arithmetic mean.
The arithmetic mean is simple, and most people with even a little bit of finance and math skill can calculate it. It’s also a useful measure of central tendency, as it tends to provide useful results, even with large groupings of numbers.
The arithmetic mean isn’t always ideal, especially when a single outlier can skew the mean by a large amount. Let’s say you want to estimate the allowance of a group of 10 kids. Nine of them get an allowance between $10 and $12 a week. The tenth kid gets an allowance of $60. That one outlier is going to result in an arithmetic mean of $16. This is not very representative of the group.
In this particular case, the median allowance of 10 might be a better measure.
The arithmetic mean also isn’t great when calculating the performance of investment portfolios, especially when it involves compounding, or the reinvestment of dividends and earnings. It is also generally not used to calculate present and future cash flows, which analysts use in making their estimates. Doing so is almost sure to lead to misleading numbers.
The arithmetic mean can be misleading when there are outliers or when looking at historical returns. The geometric mean is most appropriate for series that exhibit serial correlation. This is especially true for investment portfolios.
For these applications, analysts tend to use the geometric mean, which is calculated differently. The geometric mean is most appropriate for series that exhibit serial correlation. This is especially true for investment portfolios.
Most returns in finance are correlated, including yields on bonds, stock returns, and market risk premiums. The longer the time horizon, the more critical compounding and the use of the geometric mean becomes. For volatile numbers, the geometric average provides a far more accurate measurement of the true return by taking into account year-over-year compounding.
The geometric mean takes the product of all numbers in the series and raises it to the inverse of the length of the series. It’s more laborious by hand, but easy to calculate in Microsoft Excel using the GEOMEAN function.
The geometric mean differs from the arithmetic average, or arithmetic mean, in how it’s calculated because it takes into account the compounding that occurs from period to period. Because of this, investors usually consider the geometric mean a more accurate measure of returns than the arithmetic mean.
Let’s say that a stock’s returns over the last five years are 20%, 6%, -10%, -1%, and 6%. The arithmetic mean would simply add those up and divide by five, giving a 4.2% per year average return.
The geometric mean would instead be calculated as (1.2 x 1.06 x 0.9 x 0.99 x 1.06)1/5 -1 = 3.74% per year average return. Note that the geometric mean, a more accurate calculation in this case, will always be smaller than the arithmetic mean.
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Washington Wizards vs. Portland Trail Blazers Full Game Highlights | Feb 14 | 2022-2023 NBA Season
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The exponential moving average (EMA) is a technical chart indicator that tracks the price of an investment (like a stock or commodity) over time. The EMA is a type of weighted moving average (WMA) that gives more weighting or importance to recent price data. Like the simple moving average (SMA), the EMA is used to see price trends over time, and watching several EMAs at the same time is easy to do with moving average ribbons.
The EMA is designed to improve on the idea of an SMA by giving more weight to the most recent price data, which is considered to be more relevant than older data. Since new data carries greater weight, the EMA responds more quickly to price changes than the SMA does.
The formula for calculating the EMA is a matter of using a multiplier and starting with the SMA. There are three steps in the calculation (although chart applications do the math for you):
The calculation for the SMA is the same as computing an average or mean. That is, the SMA for any given number of time periods is simply the sum of closing prices for that number of time periods, divided by that same number. So, for example, a 10-day SMA is just the sum of the closing prices for the past 10 days, divided by 10.
The mathematical formula looks like this:
SMA=nA1+A2+…+Anwhere:An=Price of an asset at period nn=Number of total periods
The formula for calculating the weighting multiplier looks like this:
Weighted multiplier=2÷(selected time period+1)=2÷(10+1)=0.1818=18.18%
In both cases, we’re assuming a 10-day SMA.
So when it comes to calculating the EMA of a stock:
EMA=Price(t)×k+EMA(y)×(1−k)where:t=todayy=yesterdayN=number of days in EMAk=2÷(N+1)
The weighting given to the most recent price is greater for a shorter-period EMA than for a longer-period EMA. For example, an 18.18% multiplier is applied to the most recent price data for a 10-day EMA, as we did above, whereas for a 20-day EMA, only a 9.52% multiplier weighting is used.
There are also slight variations of the EMA arrived at by using the open, high, low, or median price instead of using the closing price.
Traders sometimes watch moving average ribbons, which plot a large number of moving averages onto a price chart, rather than just one moving average. Though seemingly complex based on the sheer volume of concurrent lines, ribbons are easy to see on charting applications and offer a simple way of visualizing the dynamic relationship between trends in the short, intermediate, and long term.
Traders and analysts rely on moving averages and ribbons to identify turning points, continuations, and overbought/oversold conditions, to define areas of support and resistance, and to measure price trend strengths.
Defined by their characteristic three-dimensional shape that seems to flow and twist across a price chart, moving average ribbons are easy to interpret. The indicators trigger buy and sell signals whenever the moving average lines all converge at one point. Traders look to buy on occasions when shorter-term moving averages cross above the longer-term moving averages from below and look to sell when shorter moving averages cross below from above.
To construct a moving average ribbon, simply plot a large number of moving averages of varying time period lengths on a price chart at the same time. Common parameters include eight or more moving averages and intervals that range from a two-day moving average to a 200- or 400-day moving average.
For ease of analysis, keep the type of moving average consistent across the ribbon—for example, use only exponential moving averages or simple moving averages.
When the ribbon folds—when all of the moving averages converge into one close point on the chart—trend strength is likely weakening and possibly pointing to a reversal. The opposite is true if the moving averages are fanning and moving apart from each other, suggesting that prices are ranging and that a trend is strong or strengthening.
Downtrends are often characterized by shorter moving averages crossing below longer moving averages. Uptrends, conversely, show shorter moving averages crossing above longer moving averages. In these circumstances, the short-term moving averages act as leading indicators that are confirmed as longer-term averages trend toward them.
The preferred number and type of moving averages can vary considerably between traders, based on investment strategies and the underlying security or index. But EMAs are especially popular because they give more weight to recent prices, lagging less than other averages. Some common moving average ribbon examples involve eight separate EMA lines, ranging in length from a few days to multiple months.
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