Conditional VaR Beta
factor.formula
Conceptual model of conditional VaR Beta:
Conditional VaR Beta estimation formula based on extreme value theory:
Auxiliary parameters $alpha_{n,k}$ of sample quantile estimation (the formula here is wrong, has been corrected, and has nothing to do with tau_j):
Estimation of the probability of conditional excess event occurrence $tau_j(k/n)$:
The meaning of each parameter in the formula is as follows:
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The excess return of asset j at time t is equal to the asset return minus the risk-free rate of return.
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The excess return of the market portfolio (e.g., a benchmark index) at time t is equal to the market portfolio return minus the risk-free rate.
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The residual term of the return on asset j represents the part that the model cannot explain and includes non-systematic risk.
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The conditional VaR Beta of asset j measures the sensitivity of asset j's return to the market return when the market experiences an extreme decline.
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The significance level is usually 5% (i.e. 0.05), which means that considering historical data, the probability that the market return is lower than its VaR value is $bar{p}$, that is, $P(R_m^t < -VaR_m(bar{p})) = bar{p}$. For example, if $bar{p}$ is 0.05, it means that we are concerned about the performance of the market return in the extreme case of the worst 5% decline in historical data.
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The value at risk (VaR) of asset j at a confidence level of $(1-k/n)$ represents the kth loss value of asset j after the losses (negative value of returns) are sorted from small to large in the historical data. It can also be understood as the k+1th largest loss of asset j in the historical data. Here, it is assumed that there are k days with losses exceeding the VaR value in n trading days.
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The value at risk (VaR) of the market portfolio at a confidence level of $(1-k/n)$ represents the kth loss value of the market portfolio after the losses (negative values of returns) are sorted from small to large in historical data. It can also be understood as the k+1th largest loss of the market in historical data. Here, it is assumed that there are k days with losses exceeding the VaR value in n trading days.
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The number of days in n trading days when the return loss of an asset or market portfolio exceeds its VaR value. Usually, k ≈ p*n, where p is the significance level, such as 5%. For example, when n=250 (a year of trading days) and the significance level is 5%, k is approximately equal to 12 or 13.
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The negative return of the market portfolio at time t is $X_t^{(m)} = -R_m^t$. The negative returns (losses) of the market portfolio over n trading days are arranged in ascending order as $X_1^{(m)} leq X_2^{(m)} leq ... leq X_n^{(m)}$.
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The negative return of asset j at time t is $X_t^{(j)} = -R_j^t$. The negative returns (losses) of asset j over n trading days are arranged in ascending order as $X_1^{(j)} leq X_2^{(j)} leq ... leq X_n^{(j)}$.
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The conditional probability of asset j and the market portfolio experiencing extreme losses at the same time is estimated. Specifically, it is the frequency of the losses of asset j and the market portfolio exceeding their respective VaR values at the same time in n trading days, where $I{cdot}$ is an indicative function, which takes the value of 1 when the conditions in the brackets are met, and 0 otherwise. The larger $tau_j(k/n)$ is, the more likely asset j is to suffer a large loss when the market falls extremely.
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The auxiliary parameter of the sample quantile estimation is the average loss size of the market return in the case of extreme decline (i.e., the k-day with the largest loss value), which reduces the impact of extreme values by logarithmic transformation of negative market returns. Specifically, it represents the mean of the logarithmic difference between the first k values and the kth value after sorting the logarithm of the negative market return value.
factor.explanation
The conditional VaR Beta indicator captures the sensitivity of individual stock returns to market returns when the market experiences extreme negative returns. Unlike traditional CAPM beta, conditional VaR Beta focuses on the systematic risk exposure when market tail risk events occur. A higher conditional VaR Beta indicates that when the market falls sharply, the returns of individual stocks are likely to fall sharply as well, so the stock is more exposed to extreme market risks and has higher risks. Conversely, a lower conditional VaR Beta means that individual stocks are more resistant to declines when the market falls extremely. In quantitative investment, conditional VaR Beta is often used as a risk management and asset allocation tool to measure and control the tail risk of a portfolio.