How Is the Exponential Moving Average (EMA) Formula Calculated?

What Is the Exponential Moving Average (EMA)?

The exponential moving average (EMA) is a technical chart indicator that tracks the price of an investment such as a stock or a commodity over time.

Like the simple moving average (SMA), the EMA tracks price trends over time. But the EMA is a weighted moving average that gives more importance to recent price data within the timeline being tracked.

Watching several EMAs at the same time is easy to do with moving average ribbons.

Traders and analysts rely on moving averages and ribbons to identify turning points, continuations, and overbought or oversold conditions. They also can be used to define areas of support and resistance and to measure the strength of price trends.

Calculating EMA

The EMA is designed to improve on a simple moving average by giving more weight to the most recent price data, which is considered more relevant to investors than older data. Since new data carries greater weight, the EMA responds more quickly than the SMA to price changes.

The formula for calculating the EMA starts with the SMA and uses a multiplier. There are three steps in the calculation (although chart applications do the math for you):

1. Compute the SMA
2. Calculate the multiplier for weighting the EMA
3. Calculate the current EMA

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 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:

$\begin{aligned}&\text{SMA} = \frac { A_1 + A_2 + … + A_n }{ n } \\&\textbf{where:} \\&A_n = \text{Price of an asset at period } n \\&n = \text{Number of total periods} \\\end{aligned}$​SMA=nA1​+A2​+…+An​​where:An​=Price of an asset at period nn=Number of total periods​

The formula for calculating the weighting multiplier looks like this:

$\begin{aligned} \text{Weighted multiplier} &= 2 \div (\text{selected time period} + 1) \\ &= 2 \div (10 + 1) \\ &= 0.1818 \\ &= 18.18\% \\ \end{aligned}$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:

$\begin{aligned} &EMA = \text{Price}(t) \times k + EMA(y) \times (1-k) \\ &\textbf{where:}\\ &t=\text{today}\\ &y=\text{yesterday}\\ &N=\text{number of days in EMA}\\ &k=2 \div (N + 1)\\ \end{aligned}$​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.

Using the EMA: Moving Average Ribbons

Traders sometimes watch moving average ribbons, which plot a number of moving averages onto a price chart rather than just one moving average.

Though seemingly complex based on the sheer number 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.

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.

How to Create a Moving Average Ribbon

To construct a moving average ribbon, plot a large number of moving averages of varying time lengths on a price chart.

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.

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 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 Bottom Line

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, responding earlier than other averages.

Some common moving average ribbon examples involve eight separate EMA lines, ranging in length from a few days to many months.