Beta Coefficient (BETA)

The Beta Coefficient measures the volatility and systematic risk of a security or portfolio relative to a benchmark (typically a market index). Beta quantifies how much a security's price moves in relation to overall market movements, making it a crucial metric for risk assessment and portfolio construction.

Usage

require 'sqa/tai'

# Stock and benchmark prices (e.g., AAPL vs S&P 500)
stock_prices = [150.0, 152.5, 151.0, 153.5, 155.0, 154.0,
                156.5, 158.0, 157.0, 159.5, 161.0, 160.0,
                162.5, 164.0, 163.0, 165.5, 167.0, 166.0]

benchmark_prices = [4200, 4220, 4215, 4235, 4250, 4245,
                    4260, 4275, 4270, 4285, 4300, 4295,
                    4310, 4325, 4320, 4335, 4350, 4345]

# Calculate 14-period Beta
beta = SQA::TAI.beta(stock_prices, benchmark_prices, period: 14)

puts "Current Beta: #{beta.last.round(2)}"

Parameters

Parameter	Type	Required	Default	Description
prices1	Array	Yes	-	Array of stock/security prices
prices2	Array	Yes	-	Array of benchmark prices (market index)
period	Integer	No	5	Lookback period for calculation

Returns

Returns an array of beta coefficient values. The first period values will be nil. Beta can be any real number: - Positive values: Moves with the market - Negative values: Moves opposite to the market - Zero: Independent of market movements

Interpretation

Beta Range	Interpretation	Volatility	Risk Profile
β > 1.5	Highly aggressive	Very high volatility	High risk/high reward
β = 1.0 - 1.5	Aggressive	Above market volatility	Above-average risk
β = 1.0	Market neutral	Same as market	Average risk
β = 0.5 - 1.0	Defensive	Below market volatility	Below-average risk
β = 0.0 - 0.5	Highly defensive	Low volatility	Low risk/low reward
β ≈ 0	Market independent	No correlation	Diversification benefit
β < 0	Inverse	Negative correlation	Hedge position

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

Detailed Beta Values

Beta	Meaning	Example
2.0	Moves 2x the market	If market rises 1%, stock rises 2%
1.5	50% more volatile	If market rises 1%, stock rises 1.5%
1.0	Moves with market	If market rises 1%, stock rises 1%
0.5	50% less volatile	If market rises 1%, stock rises 0.5%
0.0	No correlation	Stock moves independently of market
-0.5	Moves opposite	If market rises 1%, stock falls 0.5%
-1.0	Perfect inverse	If market rises 1%, stock falls 1%

Example: Basic Beta Analysis

stock_prices = load_historical_prices('TSLA')
spy_prices = load_historical_prices('SPY')  # S&P 500 ETF

beta = SQA::TAI.beta(stock_prices, spy_prices, period: 60)
current_beta = beta.last

puts <<~ANALYSIS
  Stock Beta Analysis:
  Beta: #{current_beta.round(2)}

  Risk Profile: #{
    case current_beta
    when 1.5..Float::INFINITY then "Highly Aggressive - High Risk"
    when 1.0...1.5 then "Aggressive - Above Average Risk"
    when 0.5...1.0 then "Defensive - Below Average Risk"
    when 0.0...0.5 then "Highly Defensive - Low Risk"
    when -Float::INFINITY...0.0 then "Inverse - Hedge Position"
    end
  }

  Interpretation:
  #{
    if current_beta > 1
      "Stock is #{((current_beta - 1) * 100).round(0)}% more volatile than the market"
    elsif current_beta < 1 && current_beta > 0
      "Stock is #{((1 - current_beta) * 100).round(0)}% less volatile than the market"
    elsif current_beta < 0
      "Stock moves inversely to the market (hedge)"
    else
      "Stock moves independently of the market"
    end
  }
ANALYSIS

Example: Portfolio Beta Construction

# Portfolio allocation
portfolio = {
  'TSLA' => { weight: 0.20, prices: load_prices('TSLA') },
  'AAPL' => { weight: 0.30, prices: load_prices('AAPL') },
  'JNJ'  => { weight: 0.25, prices: load_prices('JNJ') },
  'GLD'  => { weight: 0.15, prices: load_prices('GLD') },
  'TLT'  => { weight: 0.10, prices: load_prices('TLT') }
}

spy_prices = load_prices('SPY')

# Calculate individual betas
portfolio_beta = 0.0

portfolio.each do |ticker, data|
  beta = SQA::TAI.beta(data[:prices], spy_prices, period: 252)
  stock_beta = beta.last
  weighted_beta = stock_beta * data[:weight]

  portfolio_beta += weighted_beta

  puts "#{ticker}: Beta=#{stock_beta.round(2)}, Weight=#{(data[:weight]*100).round(0)}%, Contribution=#{weighted_beta.round(2)}"
end

puts <<~PORTFOLIO

  Portfolio Summary:
  Overall Beta: #{portfolio_beta.round(2)}

  Risk Profile: #{
    if portfolio_beta > 1.2
      "Aggressive - Outperforms in bull markets, underperforms in bear markets"
    elsif portfolio_beta > 0.8
      "Balanced - Moves roughly with the market"
    else
      "Defensive - Better downside protection, lower upside potential"
    end
  }

  Expected Movement:
  If market rises 10%, portfolio expected to rise #{(portfolio_beta * 10).round(1)}%
  If market falls 10%, portfolio expected to fall #{(portfolio_beta * 10).round(1)}%
PORTFOLIO

Example: Beta-Based Market Strategy

stock_prices = load_prices('NVDA')
spy_prices = load_prices('SPY')

beta = SQA::TAI.beta(stock_prices, spy_prices, period: 60)
current_beta = beta.last

# Market trend detection
sma_200 = SQA::TAI.sma(spy_prices, period: 200)
market_trend = spy_prices.last > sma_200.last ? :bullish : :bearish

puts <<~STRATEGY
  Market-Based Beta Strategy:

  Market Trend: #{market_trend.to_s.upcase}
  Stock Beta: #{current_beta.round(2)}

  Trading Strategy:
STRATEGY

case market_trend
when :bullish
  if current_beta > 1.5
    puts "  Action: OVERWEIGHT - High beta stock in bull market"
    puts "  Rationale: Maximize upside capture"
    puts "  Target allocation: 150% of benchmark weight"
  elsif current_beta > 1.0
    puts "  Action: MARKET WEIGHT - Moderate beta"
    puts "  Rationale: Participate in upside with reasonable risk"
    puts "  Target allocation: 100% of benchmark weight"
  else
    puts "  Action: UNDERWEIGHT - Low beta in bull market"
    puts "  Rationale: Look for higher beta opportunities"
    puts "  Target allocation: 50% of benchmark weight"
  end

when :bearish
  if current_beta < 0.5
    puts "  Action: OVERWEIGHT - Low beta stock in bear market"
    puts "  Rationale: Defensive positioning for downside protection"
    puts "  Target allocation: 150% of benchmark weight"
  elsif current_beta < 1.0
    puts "  Action: MARKET WEIGHT - Moderate defensive position"
    puts "  Rationale: Balance protection with opportunity"
    puts "  Target allocation: 100% of benchmark weight"
  else
    puts "  Action: UNDERWEIGHT - High beta in bear market"
    puts "  Rationale: Reduce exposure to volatility"
    puts "  Target allocation: 25-50% of benchmark weight"
  end
end

Example: Beta for Hedging

# Portfolio to hedge
portfolio_value = 1_000_000
portfolio_prices = load_portfolio_prices('MY_PORTFOLIO')
spy_prices = load_prices('SPY')

# Calculate portfolio beta
portfolio_beta = SQA::TAI.beta(portfolio_prices, spy_prices, period: 60).last

puts <<~HEDGE
  Portfolio Hedging Analysis:

  Portfolio Value: $#{portfolio_value.to_s.reverse.gsub(/(\d{3})(?=\d)/, '\\1,').reverse}
  Portfolio Beta: #{portfolio_beta.round(2)}

  Hedge Requirements:
HEDGE

if portfolio_beta > 1.0
  # Need to hedge more than portfolio value
  hedge_ratio = portfolio_beta
  hedge_value = portfolio_value * hedge_ratio

  puts "  High Beta Portfolio - Requires enhanced hedge"
  puts "  Hedge Ratio: #{hedge_ratio.round(2)}"
  puts "  SPY Short Position Needed: $#{hedge_value.to_s.reverse.gsub(/(\d{3})(?=\d)/, '\\1,').reverse}"
  puts "  SPY Shares to Short: #{(hedge_value / spy_prices.last).round(0)}"

elsif portfolio_beta < 1.0 && portfolio_beta > 0
  # Need to hedge less than portfolio value
  hedge_ratio = portfolio_beta
  hedge_value = portfolio_value * hedge_ratio

  puts "  Low Beta Portfolio - Requires reduced hedge"
  puts "  Hedge Ratio: #{hedge_ratio.round(2)}"
  puts "  SPY Short Position Needed: $#{hedge_value.to_s.reverse.gsub(/(\d{3})(?=\d)/, '\\1,').reverse}"
  puts "  SPY Shares to Short: #{(hedge_value / spy_prices.last).round(0)}"

else
  puts "  Portfolio beta near zero or negative - Review hedge strategy"
end

Example: Multi-Period Beta Analysis

stock_prices = load_prices('AMZN')
spy_prices = load_prices('SPY')

# Calculate different period betas
beta_30d = SQA::TAI.beta(stock_prices, spy_prices, period: 30).last
beta_90d = SQA::TAI.beta(stock_prices, spy_prices, period: 90).last
beta_252d = SQA::TAI.beta(stock_prices, spy_prices, period: 252).last

puts <<~MULTIPERIOD
  Multi-Period Beta Analysis:

  30-Day Beta:  #{beta_30d.round(2)} (Short-term volatility)
  90-Day Beta:  #{beta_90d.round(2)} (Medium-term trend)
  252-Day Beta: #{beta_252d.round(2)} (Long-term characteristic)

  Beta Stability: #{
    if (beta_30d - beta_252d).abs < 0.2
      "STABLE - Consistent risk profile across timeframes"
    elsif beta_30d > beta_252d + 0.3
      "INCREASING - Short-term volatility rising"
    else
      "DECREASING - Short-term volatility falling"
    end
  }

  Risk Assessment:
  - Current positioning reflects #{beta_30d > 1 ? 'aggressive' : 'defensive'} short-term behavior
  - Historical baseline shows #{beta_252d > 1 ? 'aggressive' : 'defensive'} long-term character
  #{
    if (beta_30d - beta_252d).abs > 0.5
      "- WARNING: Significant beta shift detected - review position sizing"
    else
      "- Risk profile stable - maintain current strategy"
    end
  }
MULTIPERIOD

Advanced Techniques

1. Rolling Beta Analysis

Track how beta changes over time to identify regime changes:

stock_prices = load_prices('NFLX')
spy_prices = load_prices('SPY')

# Calculate rolling 60-day beta
rolling_betas = []
60.upto(stock_prices.length - 1) do |i|
  beta = SQA::TAI.beta(
    stock_prices[i-60..i],
    spy_prices[i-60..i],
    period: 60
  ).last
  rolling_betas << beta
end

avg_beta = rolling_betas.sum / rolling_betas.length
current_beta = rolling_betas.last

if current_beta > avg_beta * 1.3
  puts "Beta spike detected - increased market sensitivity"
elsif current_beta < avg_beta * 0.7
  puts "Beta drop detected - reduced market correlation"
end

2. Sector Beta Comparison

Compare individual stock beta to sector beta:

stock_beta = SQA::TAI.beta(stock_prices, spy_prices, period: 252).last
sector_beta = SQA::TAI.beta(sector_etf_prices, spy_prices, period: 252).last

relative_beta = stock_beta / sector_beta

if relative_beta > 1.2
  puts "Stock is significantly more volatile than its sector"
elsif relative_beta < 0.8
  puts "Stock is defensive relative to its sector"
end

3. Beta-Adjusted Performance

Calculate risk-adjusted returns:

stock_return = ((stock_prices.last - stock_prices.first) / stock_prices.first) * 100
market_return = ((spy_prices.last - spy_prices.first) / spy_prices.first) * 100
stock_beta = SQA::TAI.beta(stock_prices, spy_prices, period: 60).last

expected_return = market_return * stock_beta
excess_return = stock_return - expected_return

puts "Stock Return: #{stock_return.round(2)}%"
puts "Expected Return (based on beta): #{expected_return.round(2)}%"
puts "Alpha (Excess Return): #{excess_return.round(2)}%"

4. Down-Market Beta

Calculate beta specifically during market declines:

# Separate up and down market periods
down_stock_prices = []
down_spy_prices = []

spy_prices.each_with_index do |price, i|
  next if i == 0
  if price < spy_prices[i-1]  # Market down day
    down_stock_prices << stock_prices[i]
    down_spy_prices << price
  end
end

down_beta = SQA::TAI.beta(down_stock_prices, down_spy_prices, period: 30).last
full_beta = SQA::TAI.beta(stock_prices, spy_prices, period: 60).last

puts "Full-Period Beta: #{full_beta.round(2)}"
puts "Down-Market Beta: #{down_beta.round(2)}"

if down_beta > full_beta * 1.2
  puts "WARNING: Stock is significantly more volatile during declines"
elsif down_beta < full_beta * 0.8
  puts "POSITIVE: Stock shows resilience during market declines"
end

Portfolio Applications

High Beta Strategy (Bull Market)

• Target beta: 1.3 - 1.8
• Focus: Growth stocks, tech, small-caps
• Market view: Strong uptrend expected
• Risk: High volatility, larger drawdowns

Low Beta Strategy (Bear Market or Uncertainty)

• Target beta: 0.4 - 0.7
• Focus: Utilities, consumer staples, bonds
• Market view: Decline or consolidation expected
• Risk: Lower returns in bull markets

Market Neutral Strategy

• Target beta: 0.0 - 0.2
• Focus: Long/short combinations
• Market view: Stock picking, not market timing
• Risk: Position-specific risk only

Smart Beta Strategy

• Adjust beta based on market conditions
• High beta in confirmed uptrends
• Low beta in downtrends or high volatility
• Rebalance quarterly or on trend changes

Common Settings

Period	Use Case	Typical Beta Range
30	Short-term trading	More volatile, 0.5-2.0
60	Medium-term positioning	Moderate stability, 0.6-1.8
252	Annual analysis (standard)	Most stable, 0.7-1.5
504	Long-term investing	Very stable, 0.8-1.3

Limitations and Considerations

1. Historical Measure: Beta is backward-looking and may not predict future behavior
2. Market Conditions: Beta can change during market regime changes
3. Linear Assumption: Assumes linear relationship between stock and market
4. Benchmark Selection: Different benchmarks yield different betas
5. Time Period: Longer periods provide stability, shorter periods show current behavior
6. Company Changes: Mergers, business model shifts can alter beta
7. Not Absolute Risk: Only measures systematic (market) risk, not total risk

Related Indicators

• CORREL - Correlation coefficient between securities
• STDDEV - Standard deviation for volatility measurement
• VAR - Variance calculation
• LINEARREG - Linear regression analysis

See Also

• Back to Indicators
• Statistical Indicators Overview
• Basic Usage