Risk Statistics
The **Risk Statistics** section within the **Advanced Benchmarks** module provides a comprehensive analysis of various risk measures and ratios. These metrics help traders understand the risk-adjusted performance of their strategies and compare them to benchmarks.
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Key Metrics
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Sharpe Ratio
The Sharpe Ratio helps investors understand the return of an investment compared to its risk.
\[
\text{Sharpe Ratio} \; SR = \frac{\tilde{r} - \tilde{r}_{F}}{\tilde{\sigma}}
\]
Where:
- \(\tilde{r}\) is the annualized portfolio return.
- \(\tilde{r}_{F}\) is the annualized risk-free rate.
- \(\tilde{\sigma}\) is the annualized portfolio risk (standard deviation of return).
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Roy Ratio
Similar to the Sharpe Ratio, but it uses the annualized minimum target return instead of the risk-free rate.
\[
\text{Roy Ratio} \; RR = \frac{\tilde{r} - \tilde{r}_{T}}{\tilde{\sigma}}
\]
Where:
- \(\tilde{r}_{T}\) is the annualized minimum target return or "Disaster Level".
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Downside Risk
Measures the variability of underperformance below a minimum target rate.
\[
\text{Downside Risk} \; \sigma_{D} = \sqrt{\sum_{i=1}^{N} \frac{\min[(r_{i} - r_{T}, 0)]^2}{N}}
\]
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Upside Risk
Refers to the variability of returns that exceed a given target.
\[
\text{Upside Risk} \; \sigma_{U} = \sqrt{\sum_{i=1}^{N} \frac{\max[(r_{i} - r_{T}, 0)]^2}{N}}
\]
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Sortino Ratio
An extension of the Sharpe Ratio, where annualized downside risk is used instead of annualized portfolio risk.
\[
\text{Sortino Ratio} = \frac{\tilde{r} - \tilde{r}_{T}}{\tilde{\sigma}_{D}}
\]
Where:
- \(\tilde{\sigma}_{D}\) is the annualized portfolio downside risk.
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Omega Ratio
The gain-loss ratio that captures the information in the higher moments of a return distribution.
\[
\text{Omega Ratio} \; \Omega = \frac{\text{Upside Potential}}{\text{Downside Potential}} = \frac{\sum_{i=1}^{N} \max(r_{i} - r_{T}, 0)}{\sum_{i=1}^{N} \max(r_{T} - r_{i}, 0)}
\]
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MAD Ratio
Similar to the Sharpe Ratio but uses mean absolute deviation in the denominator.
\[
\text{MAD Ratio} = \frac{\tilde{r} - \tilde{r}_{F}}{\text{MAD}}
\]
Where:
- \(\text{MAD}\) is the mean absolute deviation.
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Skewness-Kurtosis Ratio
The ratio of portfolio return skewness to kurtosis.
\[
\text{Skewness-Kurtosis Ratio} = \frac{\varsigma}{k}
\]
Where:
- \(\varsigma\) is the portfolio return skewness.
- \(k\) is the kurtosis.
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M Squared
The risk-adjusted return for comparing portfolios with different levels of risk.
\[
\text{M Squared} \; M^{2} = \tilde{r} + SR * (\tilde{\sigma}_{b} - \tilde{\sigma})
\]
Where:
- \(\tilde{\sigma}_{b}\) is the annualized market risk (standard deviation of benchmark return).
- \(SR\) is the Sharpe Ratio.
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Arithmetic Information Ratio
Similar to the Sharpe Ratio, but uses arithmetic excess return and tracking error.
\[
\text{Arithmetic Information Ratio} \; IR_{A} = \frac{\tilde{a}}{\tilde{\sigma}_{A}} = \frac{\text{Annualized Excess Return}}{\text{Annualized Tracking Error}}
\]
Where:
- \(\tilde{a}\) is the annualized arithmetic excess return.
- \(\tilde{\sigma}_{A}\) is the annualized tracking error of arithmetic excess return.
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Geometric Information Ratio
Similar to the Arithmetic Information Ratio, but uses geometric excess return.
\[
\text{Geometric Information Ratio} \; IR_{G} = \frac{\tilde{g}}{\tilde{\sigma}_{G}}
\]
Where:
- \(\tilde{g}\) is the annualized geometric excess return.
- \(\tilde{\sigma}_{G}\) is the annualized tracking error of geometric excess return.
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Alternative Sharpe Ratio
A variation of the Sharpe Ratio that accounts for the variability of the risk-free rate.
\[
\text{Alternative Sharpe Ratio} = \frac{\tilde{r} - \tilde{r}_{F}}{\tilde{\sigma} - \tilde{\sigma}_{F}}
\]
Where:
- \(\tilde{\sigma}_{F}\) is the variability of the risk-free rate.
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Adjusted Sharpe Ratio
An adjusted version of the Sharpe Ratio that incorporates penalties for negative skewness and excess kurtosis.
\[
\text{Adjusted Sharpe Ratio} \; ASR = SR \left[ 1 + \left( \frac{\varsigma}{6} \right) * SR - \left( \frac{K - 3}{24} \right) * SR^2 \right]
\]
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Revised Sharpe Ratio
A revised version of the Sharpe Ratio that acknowledges the variability of the risk-free rate.
\[
\text{Revised Sharpe Ratio} = \frac{\tilde{r} - \tilde{r}_{F}}{\tilde{\sigma} * (r_{i} - r_{F_{i}})}
\]
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Arithmetic Tracking Error
The arithmetic divergence between the price behavior of a portfolio and a benchmark.
\[
\text{Arithmetic Tracking Error} \; \sigma_{A} = \sqrt{\frac{\sum_{i=1}^{N} (a_{i} - \bar{a})^2}{N}}
\]
Where:
- \(a_{i}\) is the arithmetic excess return in period \(i\) (\(r_{i} - b_{i}\)).
- \(\bar{a}\) is the mean arithmetic excess return.
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Arithmetic Relative Skewness
The skewness of arithmetic excess return.
\[
\text{Arithmetic Relative Skewness} \; \varsigma_{A} = \sum_{i=1}^{N} \left( \frac{a_{i} - \bar{a}}{\sigma_{A}} \right)^3 \frac{1}{N}
\]
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Arithmetic Relative Kurtosis
The kurtosis of arithmetic excess return.
\[
\text{Arithmetic Relative Kurtosis} \; K_{A} = \sum_{i=1}^{N} \left( \frac{a_{i} - \bar{a}}{\sigma_{A}} \right)^4 \frac{1}{N}
\]
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Arithmetic Adjusted Information Ratio
An arithmetic information ratio that penalizes negative skewness and positive excess kurtosis in the tracking error.
\[
\text{Arithmetic Adjusted Information Ratio} = IR_{A} \left[ 1 + \left( \frac{\varsigma_{A}}{6} \right) * IR_{A} - \left( \frac{K_{A} - 3}{24} \right) * IR_{A}^2 \right]
\]
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Geometric Tracking Error
The geometric divergence between the price behavior of a portfolio and a benchmark.
\[
\text{Geometric Tracking Error} \; \sigma_{G} = \sqrt{\frac{\sum_{i=1}^{N} (g_{i} - \bar{g})^2}{N}}
\]
Where:
- \(g_{i}\) is the geometric excess return in period \(i\) \(\left( \frac{1 + r_{i}}{1 + b_{i}} \right) - 1\).
- \(\bar{g}\) is the mean geometric excess return.
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Geometric Relative Skewness
The skewness of geometric excess return.
\[
\text{Geometric Relative Skewness} \; \varsigma_{G} = \sum_{i=1}^{N} \left( \frac{g_{i} - \bar{g}}{\sigma_{G}} \right)^3 \frac{1}{N}
\]
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Geometric Relative Kurtosis
The kurtosis of geometric excess return.
\[
\text{Geometric Relative Kurtosis} \; K_{G} = \sum_{i=1}^{N} \left( \frac{g_{i} - \bar{g}}{\sigma_{G}} \right)^4 \frac{1}{N}
\]
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Geometric Adjusted Information Ratio
A geometric information ratio that penalizes negative skewness and positive excess kurtosis in the tracking error.
\[
\text{Geometric Adjusted Information Ratio} = IR_{G} \left[ 1 + \left( \frac{\varsigma_{G}}{6} \right) * IR_{G} - \left( \frac{K_{G} - 3}{24} \right) * IR_{G}^2 \right]
\]
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Annualized Downside Risk
The annualized measure of downside risk.
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Annualized Upside Risk
The annualized measure of upside risk.
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Upside Potential Ratio
The average sum of returns above the target.
\[
\text{Upside Potential Ratio} \; \mu_{U} = \sum_{i=1}^{N} \frac{\max \left[ (r_{i} - r_{T}), 0 \right]}{N}
\]
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Adjusted M Squared
M Squared in which the Sharpe Ratio is replaced by the Adjusted Sharpe Ratio.
\[
\text{Adjusted M Squared} = \tilde{r} + ASR * (\tilde{\sigma}_{b} - \tilde{\sigma})
\]
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Variable Explanations
- \(\tilde{r}\): Annualized portfolio return.
- \(\tilde{r}_{F}\): Annualized risk-free rate.
- \(\tilde{\sigma}\): Annualized portfolio risk (standard deviation of return).
- \(\tilde{r}_{T}\): Annualized minimum target return or "Disaster Level".
- \(r_{i}\): Portfolio return in period \(i\).
- \(r_{T}\): Minimum target return.
- \(\sigma_{D}\): Downside risk.
- \(\sigma_{U}\): Upside risk.
- \(\text{MAD}\): Mean absolute deviation.
- \(\varsigma\): Portfolio return skewness.
- \(k\): Kurtosis.
- \(\tilde{\sigma}_{b}\): Annualized market risk (standard deviation of benchmark return).
- \(SR\): Sharpe Ratio.
- \(\tilde{a}\): Annualized arithmetic excess return.
- \(\tilde{\sigma}_{A}\): Annualized tracking error of arithmetic excess return.
- \(\tilde{g}\): Annualized geometric excess return.
- \(\tilde{\sigma}_{G}\): Annualized tracking error of geometric excess return.
- \(\tilde{\sigma}_{F}\): Variability of the risk-free rate.
- \(ASR\): Adjusted Sharpe Ratio.
- \(a_{i}\): Arithmetic excess return in period \(i\) (\(r_{i} - b_{i}\)).
- \(\bar{a}\): Mean arithmetic excess return.
- \(g_{i}\): Geometric excess return in period \(i\) \(\left( \frac{1 + r_{i}}{1 + b_{i}} \right) - 1\).
- \(\bar{g}\): Mean geometric excess return.
- \(K\): Kurtosis.
- \(CVaR\): Conditional Value at Risk.
- \(GaR\): Gain at Risk.
- \(VaR\): Value at Risk.
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These risk statistics provide a detailed view of the risk-adjusted performance of a portfolio, helping traders to better understand and manage the risks associated with their trading strategies.