RRG Algorithm Explained

Overview

The Relative Rotation Graph (RRG) algorithm, developed by Julius de Kempenaer, visualizes the relative strength and momentum of securities against a benchmark. It plots securities on a 2D graph where:

X-axis (RS-Ratio): Relative strength indicator
Y-axis (RS-Momentum): Rate of change of relative strength

The Core Algorithm

Step 1: Calculate Relative Strength (RS)

# Simple ratio of security price to benchmark price
RS = (security_price / benchmark_price) * 100

Step 2: Calculate RS-Ratio (Double Smoothing)

The RS-Ratio uses double smoothing to normalize the relative strength around 100:

# First smoothing: Apply WMA to RS values
RS_smooth = WMA(RS, window=10)

# Second smoothing: Create a benchmark by smoothing again
RS_benchmark = WMA(RS_smooth, window=10)

# Normalize by dividing
RS_Ratio = (RS_smooth / RS_benchmark) * 100

Step 3: Calculate RS-Momentum

Two methods are commonly used:

Method 1: Percentage Change (Default)

# Compare current RS-Ratio to N periods ago
RS_Momentum = (RS_Ratio / RS_Ratio_N_periods_ago) * 100

# Example: If RS-Ratio was 98 ten periods ago and is 102 now
# RS_Momentum = (102/98) * 100 = 104.08

Method 2: Z-Score Normalization

# Calculate rate of change
ROC = (RS_Ratio - RS_Ratio_previous) / RS_Ratio_previous

# Normalize using z-score
RS_Momentum = 100 + (ROC - mean(ROC)) / std(ROC)

The Four Quadrants

         RS-Momentum ↑
                    |
    Improving       |  Leading
    RS-Ratio < 100  |  RS-Ratio > 100
    Momentum > 100  |  Momentum > 100
  ←-----------------|------------------→ RS-Ratio
         100        |        100
    Lagging        |  Weakening
    RS-Ratio < 100  |  RS-Ratio > 100
    Momentum < 100  |  Momentum < 100
                    |

Leading: Strong and accelerating (best performers)
Weakening: Strong but decelerating (losing momentum)
Lagging: Weak and decelerating (worst performers)
Improving: Weak but accelerating (potential opportunities)

Weighted Moving Average (WMA)

Unlike simple moving averages, WMA gives more weight to recent data:

def calculate_wma(data, window):
    """
    WMA with linearly increasing weights
    Weights: [1, 2, 3, ..., window]
    """
    weights = np.arange(1, window + 1)
    wma = []
    
    for i in range(window - 1, len(data)):
        window_data = data[i - window + 1:i + 1]
        weighted_sum = np.sum(window_data * weights)
        wma.append(weighted_sum / weights.sum())
    
    return np.array(wma)

# Example: data=[10, 20, 30], window=3
# Weights=[1, 2, 3]
# WMA = (10*1 + 20*2 + 30*3) / (1+2+3) = 140/6 = 23.33

Complete Python Implementation

import numpy as np
import pandas as pd

def calculate_rrg(security_prices, benchmark_prices, window=10, momentum_period=10):
    """
    Calculate RRG indicators for a security relative to benchmark
    
    Parameters:
    - security_prices: array of security closing prices
    - benchmark_prices: array of benchmark closing prices
    - window: smoothing window (default 10)
    - momentum_period: lookback for momentum calculation (default 10)
    
    Returns:
    - DataFrame with RS, RS-Ratio, and RS-Momentum
    """
    
    # Step 1: Calculate Relative Strength
    rs = (security_prices / benchmark_prices) * 100
    
    # Step 2: Calculate RS-Ratio (double smoothing)
    rs_smooth = calculate_wma(rs, window)
    rs_benchmark = calculate_wma(rs_smooth, window)
    rs_ratio = (rs_smooth / rs_benchmark) * 100
    
    # Step 3: Calculate RS-Momentum (percentage method)
    rs_momentum = []
    for i in range(momentum_period, len(rs_ratio)):
        current = rs_ratio[i]
        previous = rs_ratio[i - momentum_period]
        momentum = (current / previous) * 100
        rs_momentum.append(momentum)
    
    # Align arrays (due to window reductions)
    min_length = len(rs_momentum)
    
    return pd.DataFrame({
        'RS': rs[-min_length:],
        'RS_Ratio': rs_ratio[-min_length:],
        'RS_Momentum': rs_momentum
    })

Comparison with Other Implementations

1. This Implementation (Production-Grade)

Uses WMA for better responsiveness
Double smoothing for RS-Ratio
Two momentum calculation methods
Handles missing data and edge cases

2. Simple Open-Source Versions

# Common simplified approach
rs = prices / benchmark
rs_ratio = sma(rs, 10) / sma(sma(rs, 10), 10) * 100
rs_momentum = (rs_ratio - rs_ratio.shift(10)) / rs_ratio.shift(10) * 100 + 100

Uses SMA instead of WMA (less responsive)
Basic momentum calculation
Minimal error handling

3. Commercial Platforms (Bloomberg, StockCharts)

Proprietary smoothing algorithms (often enhanced WMA or custom)
May use different default periods (some use 14 or 20)
Additional filters to reduce whipsaws
Optimized for their specific user base (institutional vs retail)

Key Differences Across Implementations

Why Smoothing Methods Matter

The choice of smoothing algorithm significantly impacts RRG signals:

# Example: Same data, different smoothing methods
import numpy as np

# Simulated price movement: sudden 5% jump
prices = np.array([100, 100, 100, 100, 105, 105, 105, 105, 105, 105])

# SMA (Simple Moving Average) - Equal weights
sma_weights = [0.2, 0.2, 0.2, 0.2, 0.2]  # All equal
sma_result = 103  # Slow to react

# WMA (Weighted Moving Average) - Linear weights
wma_weights = [0.067, 0.133, 0.200, 0.267, 0.333]  # Recent data weighted more
wma_result = 104  # Faster reaction

# EMA (Exponential Moving Average) - Exponential weights
ema_alpha = 0.3  # Recent data gets 30% weight
ema_result = 103.5  # Moderate reaction

Impact on Trading Signals

Method	Signal Timing	False Signals	Best For
WMA	Early (1-2 days faster)	More frequent	Active traders
SMA	Late (1-2 days slower)	Less frequent	Long-term investors
EMA	Moderate	Balanced	General use

Real-World Example

# Sector rotation signal detection
# Scenario: Tech sector starting to outperform

# Day 10: Tech enters "Improving" quadrant
wma_rs_ratio = 99.5  # Just below 100
wma_momentum = 101   # Above 100 - WMA detects improvement

# Same day with SMA
sma_rs_ratio = 98    # Further below 100
sma_momentum = 99.5  # Still below 100 - SMA misses early signal

# Result: WMA users enter position 2-3 days earlier

Momentum Calculations

Method	Formula	Interpretation
Percentage	(Current/Previous) × 100	105 = 5% increase
Rate of Change	((Current-Previous)/Previous) × 100	5 = 5% increase
Z-Score	Normalized ROC	Statistical measure

Window Parameters

Shorter windows (5-7): More responsive, more noise
Standard (10): Balanced for most use cases
Longer windows (15-20): Smoother, better for trends

Practical Usage

# Example: Calculate RRG for a sector ETF vs SPY
sector_data = pd.read_csv('XLK_prices.csv')
spy_data = pd.read_csv('SPY_prices.csv')

rrg_results = calculate_rrg(
    sector_data['close'].values,
    spy_data['close'].values,
    window=10,
    momentum_period=10
)

# Current position
current_rs_ratio = rrg_results['RS_Ratio'].iloc[-1]
current_rs_momentum = rrg_results['RS_Momentum'].iloc[-1]

# Determine quadrant
if current_rs_ratio > 100 and current_rs_momentum > 100:
    quadrant = "Leading"
elif current_rs_ratio > 100 and current_rs_momentum < 100:
    quadrant = "Weakening"
elif current_rs_ratio < 100 and current_rs_momentum < 100:
    quadrant = "Lagging"
else:
    quadrant = "Improving"

Key Takeaways

RRG measures relative performance in two dimensions: strength (RS-Ratio) and momentum
Double smoothing in RS-Ratio calculation reduces noise while maintaining responsiveness
WMA vs SMA makes a significant difference in catching trend changes
Window size is the main tuning parameter - adjust based on your time horizon
Quadrant transitions are important signals but should be confirmed over multiple periods

The algorithm's elegance lies in its simplicity: it transforms absolute prices into relative measures that oscillate predictably around 100, making cross-sectional comparison intuitive and actionable.

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