Market Risk Measurement

An in-depth guide on various market risk measurement metrics including VaR, CVaR, Beta, and Drawdowns.

TL; DR

• Value-at-Risk (VaR) and Conditional VaR (CVaR) are metrics used to estimate potential losses in financial portfolios.

• VaR can be calculated using parametric (assuming normal distribution) or historical methods, while CVaR uses historical data to assess the average loss beyond the VaR threshold.

• Beta measures a portfolio's sensitivity to market movements and is calculated using covariance and variance over a rolling period.

• Drawdown metrics such as LDD, LTuW, and MDD provide insights into the depth and duration of losses, which can inform risk management and model evaluation strategies.

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Tail Risk

Value-at-Risk (VaR)

QUANTIT VaR (1-day 95%)

• Interpretation: There is an expectation that a daily loss of this magnitude could occur 2-3 times per year.

• This approach combines 50% Parametric VaR and 50% Historical VaR to correct for the Fat Tail issue inherent in Normal Distribution, using a mixture of Norm idea.

QUANTIT CVaR (1-day 95%)

• Interpretation: There is an expectation that a daily loss greater than this value could occur 2-3 times per year.

• The Conditional VaR is calculated using 100% Historical CVaR.

Parametric VaR

• Assumption: Normal distribution.

• Return Length: Rolling 252 Trading Days.

• Calculation: Parametric VaR = 1-day stddev * Z-score.

Python Code for Parametric VaR

from scipy.stats import norm
Value_at_Risk = pd.Series(index=Returns.index, name='ParVaR')
for i in range(0, len(Returns) - Period_Interval):
    if i == 0:
        Data = Returns[-(Period_Interval):]
    else:
        Data = Returns[-(Period_Interval + i):-i]
    stdev = np.std(Data)
    Value_at_Risk[-i - 1] = stdev * norm.ppf(Confidence_Interval)

Historical VaR

• Assumption: Estimation from historical data. Non-Parametric.

• Return Length: Rolling 252 Trading Days.

• Calculation: Historical VaR = 95% Percentile of PnL Series.

Python Code for Historical VaR

Value_at_Risk = pd.Series(index=Returns.index, name='HSVaR')
for i in range(0, len(Returns) - Period_Interval):
    if i == 0:
        Data = Returns[-(Period_Interval):]
    else:
        Data = Returns[-(Period_Interval + i):-i]
    Value_at_Risk[-i - 1] = -np.percentile(Data, 1 - Confidence_Interval)

Conditional VaR (CVaR)

• Assumption: Estimation from Historical Data.

• Return Length: Rolling 252 Trading Days.

• Calculation: CVaR = Average of values that are above 95% of left tail of PnL series.

Other VaR Measures

• Volatility-adjusted VaR (Morgan Stanley Market Risk Management): Adjusts VaR based on current market volatility, using a rolling 60-day EWMA with a factor of 0.95.

• Stressed VaR (Morgan Stanley Market Risk Management): Evaluates how severe VaR can get over time, using a rolling 252 trading days starting from a stressed period.

Sensitivity

Beta

• Beta (ACWI): Calculated as the Covariance between 6 Month Returns of Account and Returns of ACWI divided by the Variance of 6 Month Return of ACWI.

• Beta (K200): Similar to Beta (ACWI) but with Returns of K200.

• These values are rolled over a 120 Trading Day period.

Loss

Local Drawdown (LDD)

• Calculation: LDD = min(0, current day's Drawdown).

• Interpretation: By looking at the average data and distribution of Time under Water (TuW), it is possible to estimate how long one might have to wait to return to profitability after a drawdown occurs.

Local Time under Water (LTuW)

• Calculation: LTuW = min(0, count of days since the most recent drawdown began).

Maximum Drawdown (MDD)

• Calculation: MDD = min(drawdown path).

• Interpretation: If the MDD increases over time, it may indicate a regime change or that the model is fitting to noise, suggesting a need to consider upgrading the model or halting operations.