Conditional Value at Risk (CVaR)

This Pine Script implements the Conditional Value at Risk (CVaR), a risk metric that evaluates the potential losses in a financial portfolio beyond a certain confidence level, incorporating both the Value at Risk (VaR) and the expected loss given that the VaR threshold has been breached.

Key Features:

Input Parameters:

length: Defines the observation period in days (default is 252, typically used to represent the number of trading days in a year).

confidence: Specifies the confidence interval for calculating VaR and CVaR, with values between 0.5 and 0.99 (default is 0.95, indicating a 95% confidence level).

Logarithmic Returns Calculation: The script computes the logarithmic returns based on the daily closing prices, a common method to measure financial asset returns, given by:

Log Return=ln⁡(PtPt−1)
Log Return=ln(Pt−1​Pt​​)

where PtPt​ is the price at time tt, and Pt−1Pt−1​ is the price at the previous time point.

VaR Calculation: Value at Risk (VaR) is estimated as the percentile of the returns array corresponding to the given confidence interval. This represents the maximum loss expected over a given time horizon under normal market conditions at the specified confidence level.

CVaR Calculation: The Conditional VaR (CVaR) is calculated as the average of the returns that fall below the VaR threshold. This represents the expected loss given that the loss has exceeded the VaR threshold.

Visualization: The script plots two key risk measures:
VaR: The maximum potential loss at the specified confidence level.
CVaR: The average of the losses beyond the VaR threshold.

The script also includes a neutral line at zero to help visualize the losses and their magnitude.
Source and Scientific Background:

The concept of Value at Risk (VaR) was popularized by J.P. Morgan in the 1990s, and it has since become a widely-used tool for risk management (Jorion, 2007). Conditional Value at Risk (CVaR), also known as Expected Shortfall, addresses the limitation of VaR by considering the severity of losses beyond the VaR threshold (Rockafellar & Uryasev, 2002). CVaR provides a more comprehensive risk measure, especially in extreme tail risk scenarios.

References:

Jorion, P. (2007). Value at Risk: The New Benchmark for Managing Financial Risk. McGraw-Hill Education.

Rockafellar, R.T., & Uryasev, S. (2002). Conditional Value-at-Risk for General Loss Distributions. Journal of Banking & Finance, 26(7), 1443–1471.