expanding Method¶

The expanding() method computes statistics over all data from the start up to each point, with a window that grows over time.

Basic Expanding¶

Create expanding window calculations.

1. Cumulative Mean¶

import pandas as pd
import yfinance as yf

aapl = yf.download('AAPL', start='2023-01-01', end='2024-01-01')
aapl['Cumulative_Mean'] = aapl['Close'].expanding().mean()
print(aapl[['Close', 'Cumulative_Mean']].head(10))

2. Cumulative Sum¶

aapl['Cumulative_Sum'] = aapl['Volume'].expanding().sum()

3. Cumulative Standard Deviation¶

aapl['Cumulative_STD'] = aapl['Close'].expanding().std()

How Expanding Works¶

At time t, includes observations 1 to t.

1. Window Growth¶

# Day 1: window = [1]
# Day 2: window = [1, 2]
# Day 3: window = [1, 2, 3]
# Day N: window = [1, 2, ..., N]

2. Formula for Mean¶

At time \(t\):

\[\text{Expanding Mean}_t = \frac{1}{t} \sum_{i=1}^{t} x_i\]

3. vs Rolling¶

# Rolling: fixed window size
# Expanding: window grows from 1 to N

min_periods Parameter¶

Minimum observations before calculating.

1. Default (min_periods=1)¶

s.expanding().mean()  # Starts from first value

2. With Minimum¶

s.expanding(min_periods=5).mean()  # First 4 values are NaN

3. Use Case¶

# Wait for enough data before computing volatility
s.expanding(min_periods=20).std()

Common Applications¶

Typical expanding calculations.

1. Cumulative Returns¶

returns = aapl['Close'].pct_change()
cumulative = (1 + returns).expanding().prod() - 1

2. Running Statistics¶

aapl['Running_Max'] = aapl['Close'].expanding().max()
aapl['Running_Min'] = aapl['Close'].expanding().min()

3. Progressive Estimates¶

# Track how estimates converge
aapl['Mean_Estimate'] = aapl['Close'].expanding().mean()

Financial Example¶

Track cumulative performance metrics.

1. Cumulative Sharpe¶

import numpy as np

returns = aapl['Close'].pct_change()
aapl['Cum_Mean'] = returns.expanding().mean()
aapl['Cum_Std'] = returns.expanding().std()
aapl['Cum_Sharpe'] = aapl['Cum_Mean'] / aapl['Cum_Std'] * np.sqrt(252)

2. All-Time High¶

aapl['ATH'] = aapl['Close'].expanding().max()
aapl['Drawdown'] = aapl['Close'] / aapl['ATH'] - 1

3. Since Inception Returns¶

first_price = aapl['Close'].iloc[0]
aapl['Since_Inception'] = aapl['Close'] / first_price - 1

Expanding vs cumsum/cumprod¶

Comparison with built-in cumulative functions.

1. cumsum¶

# Equivalent for sum
s.expanding().sum()
s.cumsum()  # Same result, faster

2. cumprod¶

# Equivalent for product
s.expanding().prod()
s.cumprod()  # Same result, faster

3. When to Use Each¶

# Use cumsum/cumprod for simple cumulative operations
# Use expanding() for mean, std, custom functions