The **Drawdown Analysis** section within the **Advanced Benchmarks** module provides a comprehensive analysis of the drawdowns in a portfolio. These metrics help traders understand the severity and frequency of losses and their impact on the overall performance.

Key Metrics

Drawdown Deviation

A standard deviation-type statistic using individual drawdowns.
\[
\text{Drawdown Deviation} = \sqrt{\sum_{j=1}^{d} \frac{D_{j}^2}{N}}
\]
Where:
- \(D_{j}\) is the \(j^{th}\) drawdown over the entire period.
- \(d\) is the total number of drawdowns in the entire period.

Ulcer Index

Similar to drawdown deviation, but combines the impact of time "underwater" with the depth of drawdown by selecting the negative return for each period below the previous peak or high-water mark.
\[
\text{Ulcer Index} \; UI = \sqrt{\sum_{i=1}^{N} \frac{D_{i}^{'2}}{N}}
\]
Where:
- \(D_{i}^{'}\) is the drawdown since the previous peak in period \(i\).

Pain Index

Measures the average depth of drawdowns.
\[
\text{Pain Index} \; PI = \sum_{i=1}^{N} \frac{\left| D_{i}^{'} \right|}{n}
\]

CALMAR Ratio

A Sharpe-type measure that uses maximum drawdown rather than standard deviation to reflect the investor’s risk.
\[
\text{CALMAR Ratio} \; CR = \frac{\tilde{r} - \tilde{r}_{F}}{D_{\text{Max}}}
\]
Where:
- \(\tilde{r}\) is the annualized portfolio return.
- \(\tilde{r}_{F}\) is the annualized risk-free rate.
- \(D_{\text{Max}}\) is the maximum drawdown.

Sterling Ratio

Replaces maximum drawdown in the CALMAR ratio with the average drawdown over the period of analysis.
\[
\text{Sterling Ratio} \; SR_{d} = \frac{\tilde{r} - \tilde{r}_{F}}{\left| \sum_{j=1}^{d} \frac{D_{j}}{d} \right|}
\]

Burke Ratio

A Sharpe-type measure that uses the square root of the sum of the squares of each drawdown to penalize major drawdowns as opposed to many mild ones.
\[
\text{Burke Ratio} \; BR_{d} = \frac{\tilde{r} - \tilde{r}_{F}}{\sqrt{\sum_{i=1}^{d} D_{j}^2}}
\]

Modified Burke Ratio

The Burke Ratio with the denominator replaced by drawdown deviation.
\[
\text{Modified Burke Ratio} \; MBR_{d} = \frac{\tilde{r} - \tilde{r}_{F}}{\sqrt{\sum_{i=1}^{d} \frac{D_{j}^2}{N}}}
\]

Martin Ratio

Similar to the Modified Burke Ratio but using the Ulcer Index in the denominator.
\[
\text{Martin Ratio} \; MR = \frac{\tilde{r} - \tilde{r}_{F}}{\sqrt{\sum_{i=1}^{N} \frac{D_{i}^{'2}}{N}}}
\]

Pain Ratio

The equivalent to the Martin Ratio but using the Pain Index.
\[
\text{Pain Ratio} \; PR = \frac{\tilde{r} - \tilde{r}_{F}}{\sum_{i=1}^{N} \frac{D_{i}^{'}}{N}}
\]

Gauges

Graphical information about strategy main statistics performance.

Returns Distribution

A histogram of portfolio return distribution with indications of VaR (Value at Risk) and GaR (Gain at Risk) values.

Cumulative Histogram

Graphics of cumulative frequency return distribution.

P/L

Graphics of portfolio return distribution depending on sample number.

Variable Explanations

- \(D_{j}\): \(j^{th}\) drawdown over the entire period.

- \(d\): Total number of drawdowns in the entire period.

- \(D_{i}^{'}\): Drawdown since the previous peak in period \(i\).

- \(N\): Number of periods.

- \(\tilde{r}\): Annualized portfolio return.

- \(\tilde{r}_{F}\): Annualized risk-free rate.

- \(D_{\text{Max}}\): Maximum drawdown.

These drawdown metrics provide a detailed view of the severity and frequency of losses in a portfolio, helping traders to better understand and manage the risks associated with drawdowns in their trading strategies.