Exponential Moving Average (EMA)

Now that we've tackled Running Simple Moving Averages (SMA)...let's move on to Exponential Moving Averages (EMA). You may wonder why we're not covering Running Exponential Moving Averages? The default formula for EMA is the running method - so we're already covered.

Check out the posts below to understand the background on Exponential Moving Averages (EMA) and their calculation.

• Wapedia's Exponential Moving Average
• StockCharts.com EMA Calculation
• Exponential Versus Simple Moving Averages

Be careful with using EMAs in your backtesting. Or any of these running type of indicators. Since all of them require a starting value. If that starting value changes - your signals change. Which can happen if you switch price quote providers that have different history requirements. Should not be a big deal but something to be aware of.

Let's begin. We need to calculate our smoothing factor for the time series. Typical use in technical analysis is:
\( \alpha = 2.0 / (periods + 1.0) \)

We can use any value between 0 & 1 for the smoothing factor. Closer to one is less smooth and places greater weight on the more recent values. Use a value of 1 and you get the most recent value back. Closer to zero is more smooth and places greater weight on the older values.

Now, the formula for an EMA given our smoothing factor:
\( EMA_{today} = EMA_{yesterday} + \alpha(price_{today} - EMA_{yesterday}) \)

Coding in Python we get:

def ema(bar, series, period, prevma, smoothing=None):
    '''Returns the Exponential Moving Average of a series.

    Keyword arguments:
    bar         -- currrent index or location of the series
    series      -- series of values to be averaged
    period      -- number of values in the series to average
    prevma      -- previous exponential moving average
    smoothing   -- smoothing factor to use in the series.
        valid values: between 0 & 1.
        default: None - which then uses formula = 2.0 / (period + 1.0)
        closer to 1 to gives greater weight to recent values - less smooth
        closer to 0 gives greater weight to older values -- more smooth
    '''
    if period < 1:
        raise ValueError("period must be 1 or greater")

    if smoothing:
        if (smoothing < 0) or (smoothing > 1.0):
            raise ValueError("smoothing must be between 0 and 1")

    else:
        smoothing = 2.0 / (period + 1.0)

    if bar <= 0:
        return series[0]

    elif bar < period:
        return cumulative_sma(bar, series, prevma)

    return prevma + smoothing * (series[bar] - prevma)


def cumulative_sma(bar, series, prevma):
    """
    Returns the cumulative or unweighted simple moving average.
    Avoids averaging the entire series on each call.

    Keyword arguments:
    bar     --  current index or location of the value in the series
    series  --  list or tuple of data to average
    prevma  --  previous average (n - 1) of the series.
    """

    if bar <= 0:
        return series[0]

    else:
        return prevma + ((series[bar] - prevma) / (bar + 1.0))

Example call and results using the typical smoothing factor of 2 / (period + 1):

prices = [32.47, 32.70, 32.77, 33.11, 33.25, 33.23, 33.23, 33.0, 33.04, 33.21]
period = 5   #number of bars to average
prevsma = prevema = prices[0]   #1st day nothing to average

for bar, close in enumerate(prices):
    currentema = ema(bar, prices, period, prevema, smoothing=None)

    #running_sma defined in simple moving average blog post
    currentsma = running_sma(bar, prices, period, prevsma)

    print "Day %02d Value=%.2f %i-bar SMA=%f and EMA=%f" % (bar + 1, close, period, currentsma, currentema)
    prevema = currentema
    prevsma = currentsma

----------------------------------------------------------
Results of call:

Day 01 Value=32.47 5-day SMA=32.470000 and EMA=32.470000
Day 02 Value=32.70 5-day SMA=32.585000 and EMA=32.585000
Day 03 Value=32.77 5-day SMA=32.646667 and EMA=32.646667
Day 04 Value=33.11 5-day SMA=32.762500 and EMA=32.762500
Day 05 Value=33.25 5-day SMA=32.860000 and EMA=32.860000
Day 06 Value=33.23 5-day SMA=33.012000 and EMA=32.983333
Day 07 Value=33.23 5-day SMA=33.118000 and EMA=33.065556
Day 08 Value=33.00 5-day SMA=33.164000 and EMA=33.043704
Day 09 Value=33.04 5-day SMA=33.150000 and EMA=33.042469
Day 10 Value=33.21 5-day SMA=33.142000 and EMA=33.098313