Treynor Index - Worked Example

Mathematical Programming with Julia

Julia Workshop

Treynor Index

The Treynor Index is a risk-adjusted performance metric used to evaluate the performance of an investment portfolio, especially in comparison to a benchmark. It measures the excess return of an investment portfolio relative to the risk-free rate per unit of systematic risk.

Equation:

Treynor Ratio = (Rp - Rf) / β

Where:

  • Rp = Portfolio return
  • Rf = Risk-free rate
  • β = Portfolio beta (a measure of systematic risk)

Interpretation:

  • A higher Treynor Ratio indicates better risk-adjusted performance, suggesting higher returns for a given level of systematic risk.
  • A negative Treynor Ratio indicates that the portfolio’s returns are below the risk-free rate after adjusting for systematic risk.
  • The Treynor Ratio is often used in conjunction with other risk-adjusted performance metrics like the Sharpe Ratio to provide a more comprehensive assessment of an investment portfolio’s risk-return profile.
function treynor_ratio(portfolio_return, risk_free_rate, portfolio_beta)
    """
    Calculates the Treynor Ratio for a given portfolio.

    Args:
        portfolio_return: The portfolio's return.
        risk_free_rate: The risk-free rate of return.
        portfolio_beta: The portfolio's beta (systematic risk).

    Returns:
        The Treynor Ratio.
    """

    if portfolio_beta == 0
        return Inf  # Handle the case where beta is zero (infinite Treynor Ratio)
    end

    return (portfolio_return - risk_free_rate) / portfolio_beta
end

Explanation:

  1. Function Definition:
    • treynor_ratio(portfolio_return, risk_free_rate, portfolio_beta): Defines the function with three arguments:
      • portfolio_return: The portfolio’s return.
      • risk_free_rate: The risk-free rate of return.
      • portfolio_beta: The portfolio’s beta (systematic risk).
  2. Handle Zero Beta:
    • if portfolio_beta == 0: Checks if the portfolio’s beta is zero.
      • return Inf: If beta is zero, the Treynor Ratio is mathematically undefined. This code returns Inf (infinity) to represent this case.
  3. Calculate Treynor Ratio:
    • return (portfolio_return - risk_free_rate) / portfolio_beta: Calculates the Treynor Ratio using the provided formula.

How to Use:

  1. Import the function: Copy and paste the function code into your Julia script.
  2. Prepare your data:
    • portfolio_return: The return of your portfolio.
    • risk_free_rate: The risk-free rate of return (e.g., the return on a short-term government bond).
    • portfolio_beta: The beta of your portfolio (can be obtained from financial data providers or calculated using regression analysis).
  3. Calculate the Treynor Ratio: Call the function with your data:
# Example usage
portfolio_return = 0.12  # 12% portfolio return
risk_free_rate = 0.03   # 3% risk-free rate
portfolio_beta = 1.2     # Portfolio beta

treynor_ratio_value = treynor_ratio(portfolio_return, risk_free_rate, portfolio_beta)
println("Treynor Ratio: ", treynor_ratio_value)

This will calculate and print the Treynor Ratio for the given inputs.

This script effectively implements the Treynor Ratio calculation in Julia, including a check for zero beta.