Variance (VAR)

The Variance indicator measures the dispersion of price returns around their mean, representing volatility squared. It quantifies how much prices fluctuate from the average, providing a statistical measure of risk and uncertainty in asset prices.

Usage

require 'sqa/tai'

prices = [44.34, 44.09, 44.15, 43.61, 44.33, 44.83,
          45.10, 45.42, 45.84, 46.08, 46.03, 46.41,
          46.22, 45.64, 46.21, 46.25, 46.08, 46.46]

# Calculate 5-period variance
variance = SQA::TAI.var(prices, period: 5)

debug_me "Current variance: #{variance.last.round(4)}"

Parameters

Parameter	Type	Required	Default	Description
prices	Array	Yes	-	Array of price values
period	Integer	No	5	Number of periods for calculation
nbdev	Float/Array	No	1.0	Number of standard deviations (for advanced calculations)

Returns

Returns an array of variance values. The first period-1 values will be nil. Variance is always non-negative, with higher values indicating greater dispersion/volatility.

Formula

Variance = Σ(Price - Mean)² / N

Where:
  - N = number of periods
  - Mean = average price over period
  - Variance = VAR
  - Standard Deviation = √Variance

Interpretation

Variance Level	Interpretation	Risk Assessment
High Variance	Large price swings, high volatility	Higher risk, wider stops needed
Low Variance	Stable prices, low volatility	Lower risk, tighter stops possible
Rising Variance	Volatility expanding	Risk increasing, position size down
Falling Variance	Volatility contracting	Risk decreasing, may increase size

Note: Array elements should be ordered from oldest to newest (chronological order)

Example: Variance-Based Risk Assessment

prices = load_historical_prices('AAPL')

# Calculate variance with different periods
var_short = SQA::TAI.var(prices, period: 5)
var_medium = SQA::TAI.var(prices, period: 20)
var_long = SQA::TAI.var(prices, period: 60)

current_var = var_short.last
medium_var = var_medium.last
long_var = var_long.last

# Compare current variance to longer-term averages
if current_var > medium_var * 2
  debug_me "HIGH RISK: Variance spike detected!"
  debug_me "Current variance: #{current_var.round(4)}"
  debug_me "20-period variance: #{medium_var.round(4)}"
  debug_me "Risk level: ELEVATED - Consider reducing position size"
elsif current_var < medium_var * 0.5
  debug_me "LOW RISK: Variance compression detected"
  debug_me "Current variance: #{current_var.round(4)}"
  debug_me "20-period variance: #{medium_var.round(4)}"
  debug_me "Risk level: LOW - Market is quiet"
else
  debug_me "NORMAL RISK: Variance in typical range"
  debug_me "Current variance: #{current_var.round(4)}"
end

Example: Position Sizing Based on Variance

prices = load_historical_prices('TSLA')

variance = SQA::TAI.var(prices, period: 20)
stddev = SQA::TAI.stddev(prices, period: 20)

current_price = prices.last
current_variance = variance.last
current_stddev = stddev.last

# Risk parameters
account_value = 100_000
risk_per_trade = account_value * 0.02  # 2% risk

# Average variance over last 60 days for comparison
avg_variance = variance.compact[-60..-1].sum / 60.0

# Adjust position size based on current vs average variance
variance_ratio = current_variance / avg_variance

if variance_ratio > 2.0
  # High volatility - reduce position size
  risk_adjustment = 0.5  # Cut risk in half
  adjusted_risk = risk_per_trade * risk_adjustment

  debug_me <<~HEREDOC
    HIGH VOLATILITY REGIME
    Current variance: #{current_variance.round(4)}
    Average variance: #{avg_variance.round(4)}
    Variance ratio: #{variance_ratio.round(2)}x
    Reducing position size by 50%
    Risk allocation: $#{adjusted_risk.round(2)} (was $#{risk_per_trade.round(2)})
  HEREDOC
elsif variance_ratio < 0.5
  # Low volatility - can increase position size slightly
  risk_adjustment = 1.25  # Increase risk by 25%
  adjusted_risk = risk_per_trade * risk_adjustment

  debug_me <<~HEREDOC
    LOW VOLATILITY REGIME
    Current variance: #{current_variance.round(4)}
    Average variance: #{avg_variance.round(4)}
    Variance ratio: #{variance_ratio.round(2)}x
    Increasing position size by 25%
    Risk allocation: $#{adjusted_risk.round(2)} (was $#{risk_per_trade.round(2)})
  HEREDOC
else
  # Normal volatility
  adjusted_risk = risk_per_trade

  debug_me <<~HEREDOC
    NORMAL VOLATILITY REGIME
    Current variance: #{current_variance.round(4)}
    Average variance: #{avg_variance.round(4)}
    Variance ratio: #{variance_ratio.round(2)}x
    Using standard position size
    Risk allocation: $#{adjusted_risk.round(2)}
  HEREDOC
end

# Calculate position size using standard deviation (√variance)
stop_distance = current_stddev * 2  # 2 standard deviations
position_size = adjusted_risk / stop_distance

debug_me <<~HEREDOC
  POSITION DETAILS
  Entry price: $#{current_price.round(2)}
  Stop distance: $#{stop_distance.round(2)} (2σ)
  Position size: #{position_size.round(0)} shares
  Total value: $#{(position_size * current_price).round(2)}
  Stop loss: $#{(current_price - stop_distance).round(2)}
HEREDOC

Example: Volatility Regime Detection

prices = load_historical_prices('SPY')

variance = SQA::TAI.var(prices, period: 20)

# Calculate rolling statistics on variance
recent_variance = variance.compact[-60..-1]
mean_variance = recent_variance.sum / recent_variance.length
stddev_variance = Math.sqrt(
  recent_variance.map { |v| (v - mean_variance) ** 2 }.sum / recent_variance.length
)

current_var = variance.last

# Classify volatility regime
if current_var > mean_variance + (2 * stddev_variance)
  regime = "EXTREME HIGH"
  color = :red
  action = "Defensive - reduce exposure, tighten stops"
elsif current_var > mean_variance + stddev_variance
  regime = "ELEVATED"
  color = :yellow
  action = "Cautious - reduce position sizes"
elsif current_var < mean_variance - stddev_variance
  regime = "LOW"
  color = :green
  action = "Opportunistic - can increase exposure"
else
  regime = "NORMAL"
  color = :blue
  action = "Standard risk management"
end

debug_me <<~HEREDOC
  VOLATILITY REGIME ANALYSIS
  Current variance: #{current_var.round(4)}
  Mean variance (60d): #{mean_variance.round(4)}
  StdDev of variance: #{stddev_variance.round(4)}

  Regime: #{regime}
  Recommended action: #{action}

  Z-Score: #{((current_var - mean_variance) / stddev_variance).round(2)}
HEREDOC

Example: Comparing Risk Across Assets

# Compare variance across different assets for portfolio allocation
symbols = ['AAPL', 'MSFT', 'TSLA', 'SPY', 'TLT']

risk_analysis = symbols.map do |symbol|
  prices = load_historical_prices(symbol)
  variance = SQA::TAI.var(prices, period: 20)
  stddev = SQA::TAI.stddev(prices, period: 20)

  current_price = prices.last
  current_variance = variance.last
  current_stddev = stddev.last

  # Normalize by price (coefficient of variation)
  cv = (current_stddev / current_price) * 100  # As percentage

  {
    symbol: symbol,
    price: current_price.round(2),
    variance: current_variance.round(4),
    stddev: current_stddev.round(2),
    cv_percent: cv.round(2)
  }
end

# Sort by coefficient of variation (relative risk)
debug_me "COMPARATIVE RISK ANALYSIS (20-day period)\n"
debug_me "=" * 70

risk_analysis.sort_by { |d| d[:cv_percent] }.each do |data|
  debug_me <<~HEREDOC
    #{data[:symbol]}:
      Price: $#{data[:price]}
      Variance: #{data[:variance]}
      Std Dev: $#{data[:stddev]}
      Coefficient of Variation: #{data[:cv_percent]}% (relative risk)

  HEREDOC
end

# Determine portfolio allocation based on variance
total_cv = risk_analysis.sum { |d| d[:cv_percent] }

debug_me "\nRISK-ADJUSTED ALLOCATION (Inverse Variance Weighting):\n"
risk_analysis.each do |data|
  # Inverse of CV for allocation (lower risk = higher allocation)
  inverse_cv = 1.0 / data[:cv_percent]
  total_inverse = risk_analysis.sum { |d| 1.0 / d[:cv_percent] }
  allocation = (inverse_cv / total_inverse) * 100

  debug_me "#{data[:symbol]}: #{allocation.round(1)}% (lower variance = higher allocation)"
end

Example: Variance Breakout Trading

prices = load_historical_prices('NVDA')

variance = SQA::TAI.var(prices, period: 20)
stddev = SQA::TAI.stddev(prices, period: 20)

# Calculate variance of variance (second moment)
var_history = variance.compact[-60..-1]
avg_variance = var_history.sum / var_history.length

current_var = variance.last
current_price = prices.last

# Detect variance breakouts (volatility expansion)
if current_var > avg_variance * 2.0
  debug_me <<~HEREDOC
    VARIANCE BREAKOUT DETECTED!
    Current variance: #{current_var.round(4)}
    60-day avg variance: #{avg_variance.round(4)}
    Ratio: #{(current_var / avg_variance).round(2)}x

    INTERPRETATION:
    - Volatility is expanding rapidly
    - Expect larger price swings
    - Potential trend change or acceleration
    - Use wider stops
    - Consider reducing position size
    - Look for directional breakout
  HEREDOC

  # Calculate recommended stop based on expanded volatility
  expanded_stop = current_price - (3 * stddev.last)  # 3σ for high vol
  debug_me "Recommended stop loss: $#{expanded_stop.round(2)} (3σ)"

elsif current_var < avg_variance * 0.3
  debug_me <<~HEREDOC
    VARIANCE COMPRESSION DETECTED!
    Current variance: #{current_var.round(4)}
    60-day avg variance: #{avg_variance.round(4)}
    Ratio: #{(current_var / avg_variance).round(2)}x

    INTERPRETATION:
    - Volatility is contracting
    - Market is consolidating
    - Potential breakout pending
    - Good time to establish positions
    - Use tighter stops
    - Wait for directional move
  HEREDOC

  # Tighter stops during compression
  compressed_stop = current_price - (1.5 * stddev.last)  # 1.5σ for low vol
  debug_me "Recommended stop loss: $#{compressed_stop.round(2)} (1.5σ)"
end

Example: Rolling Variance for Trend Analysis

prices = load_historical_prices('BTC-USD')

# Calculate variance with multiple periods
var_5 = SQA::TAI.var(prices, period: 5)
var_20 = SQA::TAI.var(prices, period: 20)
var_60 = SQA::TAI.var(prices, period: 60)

# Analyze variance trend
recent_var_5 = var_5.compact.last(10)
var_trend = if recent_var_5.last > recent_var_5.first * 1.5
  "INCREASING"
elsif recent_var_5.last < recent_var_5.first * 0.67
  "DECREASING"
else
  "STABLE"
end

debug_me <<~HEREDOC
  VARIANCE TREND ANALYSIS

  Current Variance (5-day): #{var_5.last.round(4)}
  Current Variance (20-day): #{var_20.last.round(4)}
  Current Variance (60-day): #{var_60.last.round(4)}

  Short-term trend: #{var_trend}

  MARKET REGIME:
HEREDOC

if var_5.last > var_20.last && var_20.last > var_60.last
  debug_me "  - Accelerating volatility across all timeframes"
  debug_me "  - Risk is increasing"
  debug_me "  - Consider defensive positioning"
elsif var_5.last < var_20.last && var_20.last < var_60.last
  debug_me "  - Volatility declining across all timeframes"
  debug_me "  - Risk is decreasing"
  debug_me "  - Market stabilizing"
else
  debug_me "  - Mixed volatility signals"
  debug_me "  - Monitor for regime change"
end

Common Period Settings

Period	Use Case	Risk Context
5	Short-term volatility	Day trading, quick moves
10	Swing trading	Multi-day holds
20	Standard (monthly)	Position trading
60	Quarterly trends	Long-term risk assessment
252	Annual variance	Portfolio management

Risk Management Applications

1. Dynamic Stop Losses

# Adjust stop loss based on current variance
stop_multiplier = current_var > avg_var ? 3.0 : 2.0
stop_loss = entry - (stop_multiplier * Math.sqrt(current_var))

2. Position Sizing

# Inverse variance weighting
position_size = (risk_capital / current_var) * scale_factor

3. Portfolio Allocation

# Lower variance assets get higher allocation
weight = (1.0 / variance) / total_inverse_variance

4. Volatility Filters

# Only trade when variance is in acceptable range
trade_ok = current_var.between?(min_var, max_var)

Relationship to Other Indicators

Variance is the foundation for many other statistical measures:

• Standard Deviation (STDDEV): √Variance - more intuitive scale
• Coefficient of Variation: (StdDev / Mean) × 100 - relative risk
• Bollinger Bands: Price ± (N × StdDev)
• Value at Risk (VaR): Uses variance for risk estimation
• Sharpe Ratio: Return / StdDev - risk-adjusted return

Advantages

• Mathematically rigorous measure of dispersion
• Foundation for many risk metrics
• Useful for portfolio theory and optimization
• Works across all asset classes and timeframes
• Quantifies uncertainty objectively

Limitations

• Squared units (less intuitive than standard deviation)
• Assumes normal distribution (real markets have fat tails)
• Backward-looking (uses historical data)
• Sensitive to outliers
• Doesn't distinguish between upside and downside volatility

Best Practices

1. Use with Standard Deviation: Variance and StdDev complement each other
2. Compare Relatively: Look at variance ratios, not absolute values
3. Multiple Timeframes: Check variance across different periods
4. Normalize by Price: Use coefficient of variation for cross-asset comparison
5. Combine with Trend: Variance tells magnitude, not direction
6. Regular Recalibration: Update risk models as market regimes change

Related Indicators

• STDDEV - Square root of variance (same scale as price)
• ATR - Alternative volatility measure
• BBANDS - Uses standard deviation
• BETA - Relative variance to market

See Also

• Back to Indicators
• Basic Usage
• Statistical Indicators Overview