Why EMA Isn't What You Think It Is

Many new traders adopt the Exponential Moving Average (EMA) believing it's simply a "better Simple Moving Average (SMA)". This common misconception leads to fundamental misunderstandings about how EMA works and when to use it.

EMA and SMA differ at their core. SMA use a window of finite number of data points, giving equal weight to each data point in the calculation period. This makes SMA a Finite Impulse Response (FIR) filter in signal processing terms. Remember that FIR means that "all that we need is the 'period' number of data points" to calculate the filter value. Anything beyond the given period is not relevant to FIR filters – much like how a security camera with 14-day storage automatically overwrites older footage, making last month's activity completely invisible regardless of how important it might have been.

EMA, however, is an Infinite Impulse Response (IIR) filter. It uses ALL historical data, with each past price having a diminishing - but never zero - influence on the calculated value. This creates an EMA response that extends infinitely into the past—not just for the last N periods. IIR filters cannot be precise if we give them only a 'period' number of data to work on - they will be off-target significantly due to lack of context, like trying to understand Game of Thrones by watching only the final season and wondering why everyone's so upset about that dragon lady going full pyromaniac.

If we only consider a number of data points equal to the EMA's period, we are capturing no more than 86.5% of the total weight of the EMA calculation. Relying on he period window alone (the warm-up period) will provide only 1 - (1 / e^2) weights, which is approximately 1−0.1353 = 0.8647 = 86.5%. That's like claiming you've read a book when you've skipped the first few chapters – technically, you got most of it, but you probably miss some crucial early context.

▶️ What is period in EMA used for?

What does a period parameter really mean for EMA? When we select a 15-period EMA, we're not selecting a window of 15 data points as with an SMA. Instead, we are using that number to calculate a decay factor (α) that determines how quickly older data loses influence in EMA result. Every trader knows EMA calculation: α = 1 / (1+period) – or at least every trader claims to know this while secretly checking the formula when they need it.

Thinking in terms of "period" seriously restricts EMA. The α parameter can be - should be! - any value between 0.0 and 1.0, offering infinite tuning possibilities of the indicator. When we limit ourselves to whole-number periods that we use in FIR indicators, we can only access a small subset of possible IIR calculations – it's like having access to the entire RGB color spectrum with 16.7 million possible colors but stubbornly sticking to the 8 basic crayons in a child's first art set because the coloring book only mentioned those by name.

For example:

• Period 10 → alpha = 0.1818
  
• Period 11 → alpha = 0.1667
  

What about wanting an alpha of 0.17, which might yield superior returns in your strategy that uses EMA? No whole-number period can provide this! Direct α parameterization offers more precision, much like how an analog tuner lets you find the perfect radio frequency while digital presets force you to choose only from predetermined stations, potentially missing the clearest signal sitting right between channels.

Sidenote: the choice of α = 1 / (1+period) is just a convention from 1970s, probably started by J. Welles Wilder, who popularized the use of the 14-day EMA. It was designed to create an approximate equivalence between EMA and SMA over the same number of periods, even thought SMA needs a period window (as it is FIR filter) and EMA doesn't. In reality, the decay factor α in EMA should be allowed any valye between 0.0 and 1.0, not just some discrete values derived from an integer-based period! Algorithmic systems should find the best α decay for EMA directly, allowing the system to fine-tune at will and not through conversion of integer period to float α decay – though this might put a few traditionalist traders into early retirement. Well, to prevent that, most traditionalist implementations of EMA only use period and no alpha at all. Heaven forbid we disturb people who print their charts on paper, draw trendlines with rulers, and insist the market "feels different" since computers do algotrading!

▶️ Calculating EMAs Efficiently

The standard textbook formula for EMA is:

EMA = CurrentPrice × alpha + PreviousEMA × (1 - alpha)

But did you know that a more efficient version exists, once you apply a tiny bit of high school algebra:

EMA = alpha × (CurrentPrice - PreviousEMA) + PreviousEMA

The first one requires three operations: 2 multiplications + 1 addition. The second one also requires three ops: 1 multiplication + 1 addition + 1 subtraction.

That's pathetic, you say? Not worth implementing? In most computational models, multiplications cost much more than additions/subtractions – much like how ordering dessert costs more than asking for a water refill at restaurants.

Relative CPU cost of float operations:

• Addition/Subtraction: ~1 cycle
  
• Multiplication: ~5 cycles (depending on precision and architecture)
  

Now you see the difference? 2 * 5 + 1 = 11 against 5 + 1 + 1 = 7. That is ≈ 36.36% efficiency gain just by swapping formulas around! And making your high school math teacher proud enough to finally put your test on the refrigerator.

▶️ The Warmup Problem: how to start the EMA sequence right

How do we calculate the first EMA value when there's no previous EMA available? Let's see some possible options used throughout the history:

1. Start with zero: EMA(0) = 0. This creates stupidly large distortion until enough bars pass for the horrible effect to diminish – like starting a trading account with zero balance but backdating a year of missed trades, then watching your balance struggle to climb out of a phantom debt for months.
   
2. Start with first price: EMA(0) = first price. This is better than starting with zero, but still causes initial distortion that will be extra-bad if the first price is an outlier – like forming your entire opinion of a stock based solely on its IPO day price, then wondering why your model is tanking for weeks afterward.
   
3. Use SMA for warmup: This is the tradition from the pencil-and-paper era of technical analysis – when calculators were luxury items and "algorithmic trading" meant your broker had neat handwriting. We first calculate an SMA over the initial period, then kickstart the EMA with this average value. It's widely used due to tradition, not merit, creating a mathematical Frankenstein that uses an FIR filter (SMA) during the initial period before abruptly switching to an IIR filter (EMA). This methodology is so aesthetically offensive (abrupt kink on the transition from SMA to EMA) that charting platforms hide these early values entirely, pretending EMA simply doesn't exist until the warmup period passes – the technical analysis equivalent of sweeping dust under the rug.
   
4. Use WMA for warmup: This one was never popular because it is harder to calculate with a pencil - compared to using simple SMA for warmup. Weighted Moving Average provides a much better approximation of a starting value as its linear descending profile is much closer to the EMA's decay profile.
   

These methods all share one problem: they produce inaccurate initial values that traders often hide or discard, much like how hedge funds conveniently report awesome performance "since strategy inception" only after their disastrous first quarter has been surgically removed from the track record.

▶️ A Better Way to start EMA: Decaying compensation

Think of it this way: An ideal EMA uses an infinite history of prices, but we only have data starting from a specific point. This creates a problem - our EMA starts with an incorrect assumption that all previous prices were all zero, all close, or all average – like trying to write someone's biography but only having information about their life since last Tuesday.
But there is a better way. It requires more than high school math comprehension and is more computationally intensive, but is mathematically correct and numerically stable. This approach involves compensating calculated EMA values for the "phantom data" that would have existed before our first price point.

Here's how phantom data compensation works:

1. We start our normal EMA calculation:
   

   EMA_today = EMA_yesterday + α × (Price_today - EMA_yesterday)

2. But we add a correction factor that adjusts for the missing history:
   

   Correction = 1 at the start
   Correction = Correction × (1-α) after each calculation

3. We then apply this correction:
   

   True_EMA = Raw_EMA / (1-Correction)

This correction factor starts at 1 (full compensation effect) and gets exponentially smaller with each new price bar. After enough data points, the correction becomes so small (i.e., below 0.0000000001) that we can stop applying it as it is no longer relevant.

Let's see how this works in practice:

• For the first price bar:
  Raw_EMA = 0
  Correction = 1
  True_EMA = Price (since 0 ÷ (1-1) is undefined, we use the first price)
  
• For the second price bar:
  Raw_EMA = α × (Price_2 - 0) + 0 = α × Price_2
  Correction = 1 × (1-α) = (1-α)
  True_EMA = α × Price_2 ÷ (1-(1-α)) = Price_2
  
• For the third price bar:
  Raw_EMA updates using the standard formula
  Correction = (1-α) × (1-α) = (1-α)²
  True_EMA = Raw_EMA ÷ (1-(1-α)²)
  

With each new price, the correction factor shrinks exponentially. After about -log₁₀(1e-10)/log₁₀(1-α) bars, the correction becomes negligible, and our EMA calculation matches what we would get if we had infinite historical data.

This approach provides accurate EMA values from the very first calculation. There's no need to use SMA for warmup or discard early values before output converges - EMA is mathematically correct from first value, ready to party without the awkward warmup phase.

Here is Pine Script 6 implementation of EMA that can take alpha parameter directly (or period if desired), returns valid values from the start, is resilient to dirty input values, uses decaying compensator instead of SMA, and uses the least amount of computational cycles possible.

// Enhanced EMA function with proper initialization and efficient calculation
ema(series float source, simple int period=0, simple float alpha=0)=>
// Input validation - one of alpha or period must be provided
if alpha<=0 and period<=0
runtime.error("Alpha or period must be provided")
// Calculate alpha from period if alpha not directly specified
float a = alpha > 0 ? alpha : 2.0 / math.max(period, 1)

// Initialize variables for EMA calculation
var float ema = na      // Stores raw EMA value
var float result = na   // Stores final corrected EMA
var float e = 1.0       // Decay compensation factor
var bool warmup = true  // Flag for warmup phase

if not na(source)
    if na(ema)
        // First value case - initialize EMA to zero
        // (we'll correct this immediately with the compensation)
        ema := 0
        result := source
    else
        // Standard EMA calculation (optimized formula)
        ema := a * (source - ema) + ema
        
        if warmup
            // During warmup phase, apply decay compensation
            e *= (1-a)                  // Update decay factor
            float c = 1.0 / (1.0 - e)   // Calculate correction multiplier
            result := c * ema           // Apply correction
            
            // Stop warmup phase when correction becomes negligible
            if e <= 1e-10
                warmup := false
        else
            // After warmup, EMA operates without correction
            result := ema

result  // Return the properly compensated EMA value

▶️ CONCLUSION

EMA isn't just a "better SMA"—it is a fundamentally different tool, like how a submarine differs from a sailboat – both float, but the similarities end there. EMA responds to inputs differently, weighs historical data differently, and requires different initialization techniques.
By understanding these differences, traders can make more informed decisions about when and how to use EMA in trading strategies. And as EMA is embedded in so many other complex and compound indicators and strategies, if system uses tainted and inferior EMA calculatiomn, it is doing a disservice to all derivative indicators too – like building a skyscraper on a foundation of Jell-O.

The next time you add an EMA to your chart, remember: you're not just looking at a "faster moving average." You're using an INFINITE IMPULSE RESPONSE filter that carries the echo of all previous price actions, properly weighted to help make better trading decisions.
EMA done right might significantly improve the quality of all signals, strategies, and trades that rely on EMA somewhere deep in its algorithmic bowels – proving once again that math skills are indeed useful after high school, no matter what your guidance counselor told you.