The **Extreme Risk Analysis** section within the **Advanced Benchmarks** module provides a comprehensive analysis of the potential extreme risks associated with a portfolio. These metrics help traders understand the worst-case scenarios and the potential impacts on their portfolios.
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Key Metrics
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Value at Risk (VaR)
VaR measures the worst expected loss over a given time interval under normal market conditions at a given confidence level.
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Historical VaR
This method calculates VaR by reorganizing actual historical returns based on current portfolio holdings, ordering them from best to worst. The value at risk is determined at the \( K \)th percentile for a \( K\% \) confidence level.
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Parametric VaR
This method assumes that portfolio returns are normally distributed. Only the mean return and standard deviation of returns are required to calculate VaR.
\[
\text{Parametric VaR} = \bar{r} + z_{c} * \sigma
\]
Where:
- \(\bar{r}\) is the mean portfolio return.
- \(\sigma\) is the portfolio standard deviation.
- \(z_{c}\) is the z-score for the desired confidence level (-1.65 for 95% confidence and -2.33 for 99% confidence).
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Gain at Risk (GaR)
GaR measures the best expected win over a given time interval under normal market conditions at a given confidence level.
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Modified VaR
VaR adjusted for the kurtosis and skewness of the return distribution.
\[
\text{Modified VaR} = \bar{r} + \left[ z_{c} + \frac{z_{c}^2 - 1}{6} * \varsigma + \frac{z_{c}^3 - 3z_{c} * (K - 3)}{24} - \frac{2z_{c}^3 - 5z_{c}}{36} * \varsigma \right]
\]
Where:
- \(\varsigma\) is the skewness of the return distribution.
- \(K\) is the kurtosis of the return distribution.
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Reward to VaR
A Sharpe-type measure that uses VaR as the risk measure in the denominator.
\[
\text{Reward to VaR} = \frac{\tilde{r} - \tilde{r}_{F}}{VaR_{1 - \alpha}}
\]
Where:
- \(\tilde{r}\) is the annualized portfolio return.
- \(\tilde{r}_{F}\) is the annualized risk-free rate.
- \(VaR_{1 - \alpha}\) is the value at risk at the \(1 - \alpha\) confidence level.
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Double VaR
A gain-to-loss ratio with potential upside in the numerator and value at risk in the denominator.
\[
\text{Double VaR} = DVaR_{\beta, \alpha} = \frac{GaR_{1 - \beta}}{VaR_{1 - \alpha}}
\]
Where:
- \(GaR_{1 - \beta}\) is the best-ranked return with \(1 - \beta\) confidence (gain at risk or potential upside).
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Conditional VaR (CVaR)
The average return conditional on the return being more negative than the value at risk.
\[
CVaR_{1 - \alpha} = \frac{\left| \sum_{i=1}^{N} \left\langle r_{i} | r_{i} < VaR_{1 - \alpha} \right\rangle \right|}{n_{VaR}}
\]
Where:
- \(n_{VaR}\) is the number of returns that are more negative than the value at risk.
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Conditional Gain at Risk (CGaR)
The average return conditional on the return exceeding the gain at risk.
\[
CGaR_{1 - \beta} = \frac{\left| \sum_{i=1}^{N} \left\langle r_{i} | r_{i} > GaR_{1 - \beta} \right\rangle \right|}{n_{GaR}}
\]
Where:
- \(n_{GaR}\) is the number of returns that are greater in value than the potential upside (gain at risk).
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Conditional Sharpe Ratio
A reward-to-VaR ratio that uses CVaR in the denominator.
\[
\text{Conditional Sharpe Ratio} = \frac{\tilde{r} - \tilde{r}_{F}}{CVaR}
\]
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Modified Sharpe Ratio
A reward-to-modified VaR ratio.
\[
\text{Modified Sharpe Ratio} = \frac{\tilde{r} - \tilde{r}_{F}}{MVaR}
\]
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Tail Risk
A variance-type calculation of the returns in the tail of the distribution, giving greater weight to the more extreme returns in the tail.
\[
\text{Tail Risk} = \sqrt{\frac{\sum_{i=1}^{N} \left( \left\langle r_{i} | r_{i} < VaR_{1 - \alpha} \right\rangle - CVaR_{1 - \alpha} \right)^2}{n_{VaR}}}
\]
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Tail Ratio
A Sharpe-type measure using tail risk in the denominator.
\[
\text{Tail Ratio} = \frac{\tilde{r} - \tilde{r}_{F}}{TVaR}
\]
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Rachev Ratio
A gain-loss ratio with tail gain in the numerator and tail loss (conditional VaR) in the denominator.
\[
\text{Rachev Ratio} \; R_{\beta, \alpha} = \frac{CGaR_{1 - \beta}}{CVaR_{1 - \alpha}}
\]
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Drawdown at Risk (DaR)
Similar to VaR, but ranks continuous negative returns (drawdowns) in the return series from best to worst.
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Conditional Drawdown at Risk (CDaR)
The average of drawdowns that exceed the drawdown at risk.
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Reward to Conditional DaR
A Sharpe-type ratio with conditional drawdown in the denominator.
\[
\text{Reward to Conditional DaR} = \frac{\tilde{r} - \tilde{r}_{F}}{CDaR}
\]
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Variable Explanations
- \(\bar{r}\): Mean portfolio return.
- \(\tilde{r}\): Annualized portfolio return.
- \(\tilde{r}_{F}\): Annualized risk-free rate.
- \(r_{i}\): Portfolio return in period \(i\).
- \(r_{T}\): Minimum target return or "Disaster Level".
- \(\sigma\): Portfolio standard deviation.
- \(\sigma_{D}\): Downside risk.
- \(z_{c}\): Z-score for confidence level.
- \(\varsigma\): Skewness of the return distribution.
- \(K\): Kurtosis of the return distribution.
- \(VaR_{1 - \alpha}\): Value at Risk at the \(1 - \alpha\) confidence level.
- \(GaR_{1 - \beta}\): Gain at Risk at the \(1 - \beta\) confidence level.
- \(n_{VaR}\): Number of returns more negative than VaR.
- \(n_{GaR}\): Number of returns greater than GaR.
- \(CVaR_{1 - \alpha}\): Conditional Value at Risk at the \(1 - \alpha\) confidence level.
- \(CGaR_{1 - \beta}\): Conditional Gain at Risk at the \(1 - \beta\) confidence level.
- \(TVaR\): Tail Value at Risk.
- \(CDaR\): Conditional Drawdown at Risk.
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These extreme risk metrics provide a detailed view of the potential worst-case scenarios for a portfolio, helping traders to better understand and manage extreme risks in their trading strategies.