In the formula:

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  Represents the idiosyncratic return of the stock $ε_{i,d}$, which is estimated by the linear regression model $R_{i,d} = α_i + β_iR_{m,d} + γ_iR_{n,d} + ε_{i,d}$. Among them, $R_{i,d}$ is the return of stock i on day d, $R_{m,d}$ is the market return, and $R_{n,d}$ is the industry return. $\alpha_i$ is the intercept term of stock i, $\beta_i$ is the exposure of stock i to market risk, and $\gamma_i$ is the exposure of stock i to industry risk. $\epsilon_{i,d}$ represents the idiosyncratic return of stock i on day d, that is, the return after deducting the market and industry effects, which reflects the company's own unique risks and returns.

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  It is the threshold used to define extreme tail events, which represents the degree to which the return rate deviates from the mean. Positive and negative k define the boundaries of positive and negative extreme return events respectively. Usually, k takes a value between 1 and 2. For example, 1.5 times the standard deviation can be used as the threshold, that is, k = 1.5. The size of this value will affect the calculation of the tail probability. It is recommended to select an appropriate threshold based on the data distribution and specific application scenarios.

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  Represents the probability density function of the characteristic return x, which describes the probability distribution of the characteristic return within each value range.

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  It represents the probability that the characteristic return x is greater than or equal to k, that is, the probability of a positive extreme event occurring.

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  It represents the probability that the characteristic return rate x is less than or equal to -k, that is, the probability of a negative extreme event occurring.