Formulas · Published May 8, 2026 · 5-min read
The Exponential Moving Average is defined by a simple recursive formula: alpha times today's close plus (1 minus alpha) times yesterday's EMA. Here is the math, the worked example on NIFTY 50, and how to compute it in Excel and Python.
The Exponential Moving Average (EMA) is one of the most-used technical indicators in trading, and its formula is deceptively simple: a single recursive equation that takes today’s close, yesterday’s EMA, and a smoothing factor called alpha. This guide walks through the formula step by step, derives where alpha comes from, computes a worked example on NIFTY 50 daily data, and shows how to implement EMA in Excel, Python, and pandas.
The Exponential Moving Average for a closing-price series P at time t is defined recursively:
α = 2 / (N + 1)
EMA_t = α × P_t + (1 − α) × EMA_t-1
Two values matter:
The recursion needs a starting value, EMA_0. The most common conventions are:
EMA_0 = P_0. Simple but the first few EMA values are biased toward that single starting bar.EMA_N = (P_0 + P_1 + … + P_N−1) / N, then run the recursion from there. This is what TradingView, MetaTrader, and most charting platforms use.ewm(adjust=False) with no seed: pandas effectively starts with EMA_0 = P_0. Same as option 1.After about 3 × N to 4 × N bars, the choice of seed has almost no impact on the current EMA — the exponential decay washes it out.
The choice of α = 2 / (N + 1) is not arbitrary. It is engineered so that the center of mass of the EMA’s weighted-average lookback equals the same lookback you would expect from an N-period simple moving average. In other words, an N-period EMA “feels” like it is averaging over N bars, even though it technically incorporates the entire history with declining weights.
If you unroll the recursion, the implicit weight schedule for an N-period EMA is:
The sum of all those weights is exactly 1, which means the EMA is a true weighted average. The choice α = 2/(N+1) makes the effective lookback (the weighted-average bar age) equal to (N−1)/2, the same as an SMA of length N.
Suppose the last 10 NIFTY 50 daily closes are:
| Day | Close |
|---|---|
| t−9 | 24,100 |
| t−8 | 24,200 |
| t−7 | 24,150 |
| t−6 | 24,250 |
| t−5 | 24,300 |
| t−4 | 24,180 |
| t−3 | 24,360 |
| t−2 | 24,420 |
| t−1 | 24,500 |
| t | 24,580 |
For a 10-period EMA, α = 2/11 ≈ 0.1818.
Step 1 — Seed. Use the first close as EMA_0 = 24,100 (the simplest convention). Or use the SMA of all 10 = 24,304 for a more centered seed. Let’s go with the SMA seed: EMA_t-10 = 24,304.
Step 2 — Run the recursion forward. This requires bars after the seed. Suppose the next bar is P = 24,650.
EMA = 0.1818 × 24,650 + (1 − 0.1818) × 24,304
= 4,481 + 19,886
= 24,367
Step 3 — Continue. For each subsequent bar, take 18.18% of today’s close and add 81.82% of yesterday’s EMA. After enough bars, the EMA settles into its long-run behaviour — a smooth line that hugs price during trends and curves through ranges.
In Excel, with closes in column A starting at A2, drop this formula in B2 to seed and B3 onwards to compute:
B2: =A2 (seed — first close)
B3: =(2/11) * A3 + (1 - 2/11) * B2 (10-period EMA recursion)
Drag B3 down to fill the column. The constant 2/11 ≈ 0.1818 is the alpha for a 10-period EMA — change to 2/(N+1) for any other length.
In pandas, ewm does the work:
import pandas as pd
closes = pd.Series([24100, 24200, 24150, 24250, 24300, 24180, 24360, 24420, 24500, 24580])
ema = closes.ewm(span=10, adjust=False).mean()
print(ema.tolist())
The adjust=False flag uses the recursive form above. With adjust=True (the default), pandas uses an alternative finite-window adjustment that produces slightly different values for the first few bars and converges to the same long-run series. For technical-indicator work, adjust=False is standard and matches what trading platforms produce.
//@version=5
indicator("Custom EMA", overlay=true)
length = input.int(10, title="Length")
alpha = 2 / (length + 1)
ema_val = ta.ema(close, length)
plot(ema_val, color=color.orange, linewidth=2)
TradingView’s built-in ta.ema uses the same recursion with the SMA-seed convention — virtually identical to pandas ewm(adjust=False) after a few dozen bars.
These are the conventions you’ll see on every NIFTY 50 / BANK NIFTY chart:
The 9/21 5-min crossover is what intraday momentum traders watch most; the 50/200 daily is what financial-news desks reference in trend reporting.
EMA reacts faster than SMA at trend changes because the most recent bar weighs more. It also catches reversals slightly earlier. The trade-off: EMA is slightly noisier in ranges, producing more whipsaws on choppy days.
For a side-by-side comparison on NIFTY 50 — including average lag in trading days, total cross counts over 5 years, and forward-return profile — see our EMA vs SMA comparison page. The data shows that 50/200 EMA on NIFTY daily flips, on average, several trading days before the 50/200 SMA at trend changes.
The EMA formula is small but powerful: a single recursion blends today’s close with yesterday’s average, weighted by a smoothing factor that controls how much memory the line has. Once you understand alpha and the recursion, every “EMA” on every charting platform is just this same calculation under the hood.
For traders, the formula matters because it tells you exactly how reactive an EMA is to a fresh bar — about 18% of a 10-EMA’s value comes from the latest close, about 4% of a 50-EMA’s value, and less than 1% of a 200-EMA’s value. That’s why short-period EMAs are useful for intraday entries and long-period EMAs for trend definition.
For the live application of EMA on Indian indices, see the NIFTY EMA regime and methodology pages — both show the formulas above applied in real time to NIFTY 50 and BANK NIFTY data.