1. Introduction
Tail risks, which can be quantified by risk measurements such as quantile, expectile, and entropic value at risk [
1], highlight the potential for serious losses that could affect investors, financial institutions, and the overall stability of financial markets, making their measurement critical in the financial fields. Understanding and managing tail risk helps mitigate adverse consequences and maintain financial resilience. A voluminous literature provides econometric tools to measure tail risk ([
2,
3,
4,
5] and references therein). However, the presence of price-limit trading policies in certain markets complicates the accurate depiction of tail risk, as these constraints may distort the manifestation of tail risk. Price limits, widely endorsed across global stock and futures exchanges, serve as a safeguard procedure for investors and a deterrent against market manipulation [
6,
7,
8]. By imposing restrictions on daily price fluctuations, these policies challenge the applicability of existing tail risk measures, typically designed for unrestricted markets. Consequently, this scenario urgently calls for tailored approaches in modeling tail risk within markets subject to price limits.
The challenges posed by the price-limit policy are twofold. From an econometric point of view, the dispute on the pros and cons of the price-limit trading policy lasts for ages. Some argue that the price-limit policy may lead to ineffectiveness or even destructive market behavior [
9,
10,
11], while others believe that a price-limited policy can reduce market manipulation risk and improve market efficiency [
12,
13,
14]. The disorderly effectiveness of this policy makes it challenging to quantify its impact on tail risk. From a statistical point of view, this policy results in the censoring of observations, and ignoring such censoring could cause substantial bias and size distortion in measuring tail risk, even if the censored probability is tiny. The potential biases also occur in modeling the volatility of returns with price limits. To handle this case, Wei [
15] develops a censored-GARCH model to recognize the unobservable feature of price-limited data and Hsieh and Yang [
16] subsequently propose a censored stochastic volatility approach based on the censored-GARCH model to further improve the computational efficiency. However, both censored approaches rely on the algebraic relationship between the observed and latent returns (e.g., Equations (2)–(4) in [
15]), which breaks down when dealing with the market’s aggregated information, such as tail risk, since it is jointly determined by multiple stocks. In addition, we present a simple analysis of the SSE50 (the SSE50 is a value-weighted price index that represents the performance of the top 50 firms listed on the Shanghai Stock Exchange, selected based on their market capitalization, liquidity, and other criteria) in
Figure 1 to further illustrate the damage in measuring the tail risk caused by ignoring the censoring nature. The histogram in the left panel reveals a notable probabilistic stacking of observations triggered by the price limit, whereas the right panel demonstrates that the uncensored fitting method leads to an underestimation of tail risk (quantile).
In the domain of tail risk assessment, Extreme Value Theory (EVT; [
17]) stands out as a potent instrument. This theory encompasses two principal methods: one approach is fitting the maximum observations using the generalized extreme value distribution (GEV) and is commonly referred to as the Maxima-GEV or Block Maxima (BM) method (e.g., [
18]); another approach involves the Peak-over-Threshold (POT) method [
19], which employs the Generalized Pareto Distribution (GPD) to approximate the conditional behavior of random variables that exceed specific high thresholds. Several studies have extensively explored the implications of EVT in assessing tail risk within the context of price limits. Oh et al. [
20] assume that the conditional tail distribution of extreme returns obeys a power law and obtain an inferred estimation of tail risk under price limits. Subsequently, Ji et al. [
21] introduce a general framework of the self-exciting point process with the truncated generalized Pareto distribution to measure the extreme risks in the price-limited stock markets. Nevertheless, the former ignore the censored nature of extreme returns when estimating the tail index, a critical aspect of accurate tail risk estimation. The latter use a truncated distribution rather than a censored structure to self-adapt to the price limits, which prevents this approach from modeling the latent return and results in their risk measurements continuing to be constrained by price limits, lacking sensitivity to extreme risk events. In addition, both methods ignore a dynamic treatment for the tail index, which has been demonstrated to be necessary by Massacci [
22], Zhao et al. [
23], and Shen et al. [
24], etc. These studies have revealed substantial evidence that the tail risk in financial markets without price limits exhibits significant dynamics over time. Intuitively, these dynamic features of tail risk would also be present in price-limited markets.
For a deeper understanding of tail risk dynamics in price-limited markets, this paper focuses on modeling the time-varying tail features when observations beyond some threshold are censored. We propose a novel censored autoregressive conditional Fréchet model, which accommodates the censoring, heavy-tailed, volatility clustering, and extreme event clustering nature of financial data. The CAcF model incorporates a flexible observation-driven time evolution scheme of the parameters
(volatility index) and
(tail index) of a Fréchet (Type-II GEV) distribution, and the censoring feature into the modeling, allowing for a more explicit exploration of the time-varying tail behavior in price-limited equity markets. Moreover, we employ three typical observation-driven functions to decompose the tail risk from varying risk preference perspectives. (Risk preference is a pivotal factor in economic behavior, directly influencing the choice and behavior of investors in risk investment decisions [
25,
26,
27]).
To empirically illustrate our findings, we utilize stock data from companies included in the SSE50, CSI300 (the CSI300 is a broader index that encompasses the top 300 firms listed on the Shanghai Stock Exchange and Shenzhen Stock Exchange), and TW50 (the TW50 is a market capitalization-weighted stock index developed by the Taiwan Stock Exchange in cooperation with the Financial Times and Stock Exchange (FTSE), which comprises the 50 companies with the largest market capitalization listed on the Taiwan Stock Exchange). In terms of our proposed model, we offer a maximum likelihood estimation (MLE) procedure for model estimation. To quantify tail risk, we adopt the entropic value at risk, which incorporates self-information via entropy and allows for a more flexible and robust representation of risk. We have also derived closed-form expressions for entropic value at risk and censored probability within this framework, providing a convenient approach for out-of-sample prediction. The empirical estimation results demonstrate that the CAcF model can effectively monitor time-varying behaviors of tail risk and provide satisfactory forecasting performance. This suggests its potential value in warning against financial tail risks. In addition, the tail risk of price-limited stock markets is significantly underestimated when censoring is not taken into account. Moreover, our analysis results show that the CAcF-type models with different risk preferences yield varied interpretations of risk. Specifically, risk-preferred investors perceive that hitting limit-down will reduce the potential risk at the next moment, whereas risk-averse investors interpret it conversely. These findings align with the principles of investment psychology and market dynamics. Finally, we study the impact of widening price limits by comparing the performance of the tail risks of TW50 over periods with different price limits. The evidence shows that widening price limits would lead to a decrease in the incidence of extreme events (hitting limit-down), but a significant increase in tail risk.
This paper provides a twofold contribution to the growing literature on tail risk measurement in financial markets. From a statistical modeling point of view, we propose a dynamic tail risk model for price-limited financial markets. The CAcF model incorporates a flexible observation-driven time evolution scheme for the key parameters and accommodates many important empirical characteristics of financial data. We demonstrate that the CAcF model can be derived from a general factor model, which ensures that the dynamic model is theoretically feasible. From an econometric point of view, the CAcF model offers a new perspective to illustrate tail risk dynamics when price limits exist in financial markets. Real applications show that tail risk is seriously underestimated when the price-limited constraint is ignored. Moreover, this study provides valuable insights for policymakers to develop more effective price-limited policies from a risk management perspective.
The rest of this paper is structured as follows.
Section 2 introduces the framework of the CAcF model and derives a maximum likelihood estimation procedure.
Section 3 presents the empirical results and analysis. Finally, we conclude the paper in
Section 4.