Lower Tail Risk Beta
factor.formula
Conditional CAPM model based on lower tail events:
Lower tail risk beta estimation based on extreme value theory:
$\tau_j(k/n)$ - Joint Exceedance Probability:
In the formula:
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The return on stock j at time t.
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The market return at time t.
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The lower tail risk beta value of stock j represents the sensitivity of the stock return to the market return when the market is extremely down.
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Model error term
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The significance level is usually 5%, which means that the probability that the market return is less than -VaR is $\alpha$, that is, $P(R_m^t < -VaR_m(\alpha)) = \alpha$.
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The VaR value of stock j is estimated using the lower tail extreme value of historical returns. $VaR_j(k/n)$ represents the negative value of the kth minimum return of stock j in the past n trading days, that is, the maximum loss among the first k losses. k represents the number of samples used to estimate VaR, which is usually a small proportion of n, such as 5%, corresponding to a 5% confidence level.
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The VaR value of the market is estimated using the lower tail extreme value of the historical return. $VaR_m(k/n)$ represents the negative value of the kth minimum return of the market in the past n trading days, that is, the maximum loss among the first k losses. k is the same as the k value used in $VaR_j(k/n)$.
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The number of tail samples used to calculate VaR and lower-tail risk beta is usually equal to a small fraction of n, such as $k \approx \alpha * n$ (for example, when $\alpha=0.05$, it means taking the largest 5% loss in n trading days), where n is the number of trading days in the calculation period.
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Indicator function, which is 1 when the condition is met, otherwise it is 0.
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The joint exceedance probability of stock j and market return being less than their corresponding VaR values at the same time is the proportion of trading days in the past n trading days when stock j and market return simultaneously fall below their respective VaR values.
factor.explanation
Tail Beta measures the sensitivity of individual stock returns to market returns when the market falls to an extreme, that is, how the stock's returns will change when the market experiences extreme negative returns. Stocks with high tail betas mean that their returns may fall more when the market experiences extreme downside risks. This factor can help investors identify stocks with greater risk exposure during extreme market fluctuations, thereby conducting risk management and asset allocation. Compared with traditional beta, tail beta pays more attention to risk exposure in extreme market situations and can serve as an effective supplement to traditional risk indicators.