Abstract
Risk preference has constantly been one of the vital issues in economics and finance. In this study, the time series and term structure of Chinese investors’ implicit risk aversion are investigated using the implicit risk aversion extraction method, and the dynamic term structure of Chinese investors’ implicit risk aversion is modeled using the Vasicek model. In the empirical process, the SSE 50 ETF option data from March 2015 to July 2018 are adopted to extract model-free risk-neutral skewness. The standard deviation, skewness, and kurtosis of reality measure are extracted using the Shanghai 50 ETF data, and then the monthly implied risk aversion time series and term structure of Chinese investors are obtained. As indicated by the results of this study, the risk preference of Chinese investors exhibits significant time series characteristics, and it will show risk-loving and risk-averse phenomena at certain times. Moreover, for different periods in the future, differences are generated in investors’ risk preferences, suggesting an aversion to short-term risk and a certain tolerance for long-term risk, i.e., there are significant characteristics of the term structure. Besides, the term structure of implied risk aversion of Chinese investors is dynamically modeled using the Vasicek model. To be specific, its first principal component can account for 90% of the change in the term structure. The “level factor” refers to the critical factor load, and the level of long-term implied risk aversion reaches 0.658. Furthermore, the term structure of implied risk aversion exhibits the characteristics of mean regression. Next, more effective research results on investors’ risk preference are achieved, and the time-varying characteristics and term structure characteristics of investors’ risk preference are investigated. The result suggests that the long-term risk preference of Chinese investors approaches 1, and there is a significant feature of “mean regression,” i.e., a vital finding of this study.
1. Introduction
In finance theory and practice, risk aversion has been confirmed as one of the core issues in economics. Risk preference refers to the core of the utility function, determining market risk price and random discount factor, and it can apply to asset pricing, investor decision, and risk management. However, investors’ risk aversion is slight and difficult to estimate. Furthermore, it can be considered a psychological state of investors, such that it turns out to be a difficulty in financial research.
When risk aversion is unlikely to be measured accurately, some scholars have attempted to avoid accurate measurement, use alternative variable approximate description, and set risk aversion parameters using experience and other methods for relevant analysis. However, specific and accurate risk aversion parameters are commonly required in extensive studies, such that the above three methods cannot be employed. Scholars have adopted the survey/experiment method and the historical estimation method for analysis to estimate risk aversion parameters more accurately. In theory, the survey/experiment method can most directly indicate investors’ preference for risk. In practice, however, this method is difficult to serve as a mainstream research method for its heavy workload and complex operation. The historical method has been confirmed as a common practice in existing research on risk aversion, whereas its estimated risk aversion parameters are risk aversion levels in historical samples, such that it fails to capture investors’ future risk aversion at a specific moment. Besides, it has a limited role in predicting the future development of financial markets, asset pricing, investment decision making, and risk management. Given the above-described reasons, the implied risk aversion method is employed for analysis in this study. Compared with other methods to extract risk aversion, implied risk aversion shows the advantages as follows. First, it takes on critical economic significance. Implied risk aversion conforms to the basic principle of asset pricing, and risk aversion can be calculated from the difference between the real world and the risk-neutral world. The abovementioned idea is highly consistent with the nature of risk aversion taking on strong economic significance. Second, compared with the historical method, implied risk aversion contains sufficient information regarding investors’ expectations of future market trends, such that implied risk aversion becomes more forward-looking. Third, implied risk aversion exhibits better timeliness than the survey method.
In this study, the implicit risk aversion extraction method is adopted to study the risk appetite of Chinese investors. For China’s stock market, individual investors account for a large proportion, and they are easily affected by the macro-environment, external information, and subjective emotions, such that a significant “herding effect” can be generated. Accordingly, this study attempts to investigate the time-varying characteristics of Chinese investors’ risk preference and observe the changes in Chinese investors’ risk preference in different periods using the implied risk aversion method. Second, since Chinese investors are “short-sighted,” investors are more focused on short-term risks. Thus, the advantages of the implied risk aversion method are further exploited, the term structure of investors’ implied risk aversion is extracted, the “short-sighted” phenomenon of Chinese investors is further verified, and the long-term risk preference level of Chinese investors is obtained through modeling. This level of long-term risk aversion can serve as a vital parameter for future research on asset pricing, risk management, and investment portfolio in China. Moreover, the SSE 50 ETF option data and model-free risk-neutral skewness are extracted from March 2015 to July 2018. The standard deviation, skewness, and kurtosis of reality measurement are extracted from the Shanghai 50 ETF data. Subsequently, the implied risk aversion time series and term structure of Chinese investors are obtained. The first principal component of the implied risk aversion term structure of Chinese investors is obtained through the principal component analysis. The Vasicek model is employed to model the implied risk aversion term structure, and the dynamic characteristics of Chinese investors’ risk aversion term structure are obtained.
In this study, various extraction methods and information content of implied risk aversion are systematically explored, which has a good theoretical contribution. First, the extraction of investors’ risk preference is complex, whereas it has been extensively employed. The research on the extraction of implied risk aversion can completely overcome the difficulty in the estimation of the implied risk aversion and provide a novel perspective for future research on asset pricing and explaining the mystery of equity premium. Second, through the comparison of different extraction methods of implied risk aversion, insights into the advantages and disadvantages of different methods can be gained, guidance is provided to further expand the theoretical modeling of implied risk aversion in the future, and the differences and similarities of implied risk aversion obtained using a wide variety of methods can be understood, and a reference is provided for model selection of empirical extraction of implied risk aversion in the future. Third, after the factors and information content of the time series of implied risk aversion are investigated, whether there is impurity information in the implied risk aversion can be verified, and necessary empirical evidence can be presented for the further extraction of more pure implied risk aversion.
From the perspective of actual contribution, the extraction of implied risk aversion also profoundly affects practical work. The research results of this study serve as a bridge between theoretical research and practical work, and a novel method can be developed to evaluate investors’ risk preference. First, one of the focuses of this study is to extract the implied risk aversion of market investors, and the change in investors’ risk attitude and psychology generally serves as a vital early warning indicator of market fluctuations and even crises. The implied risk aversion coefficient extracted in this study can provide a valuable reference for regulators to gain insights into the market and grasp investors’ psychology. Second, this study proposes that the implied risk aversion extracted from market data covers information regarding investor sentiment. In general, risk aversion is relatively stable, and the changes we observe in risk aversion often arise from changes in sentiment. In fact, risk aversion extracted from other methods is subjected to the same problem. As revealed by the conclusions of this study, future research on risk aversion should be conducted based on an irrational framework, and investor sentiment should be separated from real risk aversion. Third, from the perspective of implied risk aversion, this study further confirms that the posterior risk premium calculated using post-event data cannot truly indicate investors’ prior risk aversion. As indicated by the actual information content of the implied risk aversion, the implied risk aversion exhibits a significant prediction ability for the trading volume of bonds and the trend of option prices.
Several limitations remain in this study. This study only empirically demonstrates that the implied risk aversion extracted from market data contains investor sentiment, and it does not optimize the existing methods. The utility function is set under the irrational framework of investors to eliminate the interference of investors’ situation on the implied risk aversion.
This paper is organized as follows. Section 1 is the introduction, in which the research background of this study is primarily introduced. A literature review is presented in Section 2. In Section 3, theoretical research is presented, placing a focus on the extraction method of implied risk aversion, the calculation method of higher-order moments, and the dynamic modeling method of risk aversion term structure. The empirical analysis is illustrated in Section 4, in which the Shanghai 50 ETF option data and the Shanghai 50 ETF index are primarily adopted to estimate the implied risk aversion coefficient and term structure of Chinese investors. The conclusion of this study is drawn in Section 5.
2. Literature Review
For the issue of investor risk preference measurement, there are a considerable number of research results. Based on the development of risk aversion theory, risk aversion can fall into two stages (i.e., inaccurate measurement and accurate measurement).
At the stage of inaccurate measurement, the risk aversion for investors turns out to be a difficult problem. Several researchers have suggested that risk aversion is unobservable and difficult to estimate, and it is a psychological state of investors. The main research results are presented as follows. (1) Try to avoid this problem when you need to use the risk aversion coefficient. As early as 1964, Sprenkle proposed the option pricing model, whereas the major drawback of this model lies in the need to use the risk aversion of investors as an input parameter in the pricing formula. Moreover, Boness [1], Samuelson, Thorp and Kassouf [2] proposed a similar option pricing formula. The abovementioned models are difficult to promote probably because they cannot accurately obtain the risk aversion coefficient of investors. In 1973, Black and Scholes proposed the Black–Scholes–Merton option pricing model. The BSM formula converts option pricing from a realistic measure to a risk-neutral measure, successfully bypassing risk preferences and risk premiums, and it has been extensively accepted by academia. Thus far, a considerable number of option pricing models are still being expanded under the BSM framework, whereas not all research can circumvent the core issue of risk aversion, such that numerous scholars are beginning to think about other risk aversion metrics. (2) Approximate indicator substitution method: alternative indicators are adopted to approximate the investor’s risk aversion (e.g., VIX index, implied volatility, corporate bond credit spread, and swap rate spread), whereas volatility and other indicators primarily express risk changes instead of risk preferences. (3) Empirical setting method: the empirical setting method is limited by the research methods and the effect of the research samples, and the results are generally quite different. For instance, since Mehra and Prescott’s [3] study of the mystery of equity premiums, the extensively used relative risk aversion coefficient is mostly 2–4, while Arrow [4] considered the synthesis and correlation-based economics of predecessors since the value of risk aversion should be 1. A few years later, Friend and Blume [5] suggested that the risk aversion value should be 2 by investigating investors’ portfolios. Mehra and Prescott [3] have suggested that risk aversion should be 55. Furthermore, Cochrane and Hansen [6] reported that the value of risk aversion ranges from 40 to 50.
However, accurate risk aversion coefficients are required from numerous studies, and the above three treatment schemes cannot be employed. However, if the precise value of risk aversion cannot be determined, a considerable number of studies often cannot reach clear conclusions and must stay at the level of theoretical models. Accordingly, researchers have made considerable efforts to accurately estimate the risk aversion parameters.
At the stage of accurate measurement of risk preference, the main measurement methods can fall into three categories as follows. (1) Investigative experiment method: survey/experimental method is to set the correlation between risk aversion parameters and survey questions and experimental methods in advance and subsequently infer the risk aversion parameters from the questionnaire and experimental results (e.g., Abdellaoui et al. [7]). (2) Historical estimation method: the historical estimation method represents the specific form of setting utility function and risk aversion parameter, which adopts historical sample data of stock price to estimate risk aversion parameters (e.g., Chue [8]; Ryan [9]; and Bommier et al. [10]). (3) Implied risk aversion: after 2000, extracting the implied risk aversion coefficient from the option data has become a new research hotspot. The core of the implied risk aversion method is elucidated as follows. In accordance with the principle of asset pricing, if the risk-neutral probability distribution and the realistic measure probability distribution of the asset price are estimated from the option price and the underlying asset price, respectively, it can be estimated from the difference between them. It falls into four categories as follows. A class of researchers favors the “direct method,” thus gauging objective and risk-neutral probability density function from the price information of the underlying assets as well as its options, respectively. Subsequently, the implied risk aversion is directly estimated (see Aït-Sahalia and Lo [11]; Jackwerth [12]; Rosenberg and Engle [13]; Grith et al. [14]; Gagnon and Power [15]; Liao and Sung [16]; Hughston [17]; Didisheim et al. [18]; and so on). The advantage of this method is that it is simple, direct, and easy to understand. Utility functions or other models are not required to be set, and the model risk is small. However, the disadvantage is that the implied risk aversion obtained is not a certain value, but a curve, termed the “implied risk aversion smile,” creating certain obstacles to the subsequent research.
Parameter estimation method of the second kind: some researchers have employed the “maximum likelihood estimation (MLE)” method. To be specific, this method estimates the risk-neutral probability density from option prices. Subsequently, the MLE method has been adopted to obtain risk preference parameters. On that basis, the parameters can be converted to an objective probability distribution approximate to the real situation, such that the implied risk aversion is derived (e.g., Bliss and Panigirtzoglou [19]; Panigirtzoglou [20]; Liu et al. [21]; Wang [22]; Sinha and Kamaiah [23]; and Kyriacou et al. [24]). When using the maximum likelihood method, it is necessary to set the specific form of the utility function, the common ones are the power utility function and the exponential utility function, and the corresponding form of the absolute risk aversion function is also set.
Some researchers prefer to artificially define a dynamic random process or a probability distribution of underlying assets under risk-neutral and objective measures. This method is known as “modeling method” (see Bollerslev et al. [25]; Bedoui and Hamdi [26]; and Zhu et al. [27]). Compared with other methods, the model-setting method needs to consider that the movement path of the set target asset is subject to certain model-setting risks.
The moment condition method is a method that only uses the first four moment conditions (i.e., mean, volatility, skewness, and kurtosis, mainly volatility, skewness, and kurtosis) in the probability distribution to calculate the implied risk aversion. The moment condition contains considerable information regarding the distribution (in financial research, it is generally believed that the first four moments contain most of the information needed), such that it is reasonable to consider using moments to replace the complete probability distribution, construct the correlation between the risk-neutral moment and the reality moment, and then determine the risk aversion coefficient, which is the moment condition method (see Bakshi et al. [28]; Bakshi and Madan [29]; Faccini et al. [30]; Rong et al. [31]; and Faccini et al. [32]).
Comparing the three measurement methods of risk preference, we can find that although the survey experimental method is simple, it has at least two problems. First, the workload is large and the operation is complex. For instance, the questionnaire survey needs to take at least three months. If we consider the change in risk aversion before and after the financial crisis, it will take at least three years, and the cost is low. Second, we need to be careful when designing questionnaires and experiments because the possible problems and experimental methods reflect people’s expectations rather than their risk attitudes. Accordingly, risk aversion parameters are difficult to be estimated through investigation/experiment as the mainstream research method. Although the operation of the historical method is not complicated, its most significant disadvantage is that the estimated risk aversion parameters include the risk aversion level in the historical sample, i.e., an estimate of the facts that have occurred. It cannot capture the risk aversion of investors in the future at a particular moment, and it exerts limited effect on predicting the future development of the financial market, asset pricing, investment decision making, and risk management. On the other hand, the implied risk aversion method has its inherent advantages though it is more difficult to implement than the other two methods. First, it takes on obvious economic significance. The implied risk aversion conforms to the basic principle of asset pricing, and the risk aversion is derived from the difference between the real world and the risk-neutral world, highly consistent with the nature of risk aversion and taking on significant economic significance. Second, compared with the historical method, the implied risk aversion covers sufficient investors’ expected information regarding the future market trend, since a considerable number of option data should be adopted to extract the implied risk aversion. Besides, the data contain extremely rich expected information, which can give timely and high-frequency feedback to market participants for making judgments and developing opinions on the asset price trend. Accordingly, the implied risk aversion extracted from the option price is more forward-looking, indicating people’s views on the future. Third, compared with the survey method, the implied risk aversion is more time-sensitive. The implied risk aversion method can obtain the risk preference of investment at different times, saving a lot of experimental investigation time. Fourth, using the option data with different maturities, the investors’ risk aversion can be obtained under different maturities, and the term structure of implied risk aversion can be formed, which is impossible for any other method. In brief, this study will use the implied risk aversion method to extract the risk aversion time series and term structure while dynamically modeling the term structure.
3. Methodology
3.1. Implied Risk Aversion Based on Higher Moment Method
The utility function of investors be power utility function is set as . The price of the stock at time T is expressed as represents risk-neutral probability density, and represents the realistic measure probability density. The logarithm return of the stock in period is . In this way, this study can establish the correlation between the risk-neutral probability density of return rate and the probability density of realistic measure .
For simplicity, it is assumed that the mean of the probability density of the realistic measure is 0 (assume ). Three additional higher moment conditions in the definition of are expressed as
The moment-generating function of probability density of realistic measure is defined as . Taylor expansion of the moment-generating function is conducted:
Using the same method, the moment-generating function of risk-neutral probability density is:
The above formula suggests that the moment-generating function of risk-neutral probability density can be determined from the probability density of realistic measure . Using the property of moment-generating function, the correlation between each order moment of risk-neutral probability density and each order moment of realistic measure probability density can be obtained:
, Using the most basic definition of skewness of reality measure, the skewness of reality measure is expressed as
Using the relation among standard deviation, skewness, kurtosis, and moments of each order, it yieldswhere denotes implied risk aversion, represents risk-neutral skewness, and , , and represent the standard deviation, skewness, and kurtosis of realistic measure, respectively. Thus, the core problem of extracting implied risk aversion using high-order moments is to estimate the neutral skewness of risk and the standard deviation, skewness, and kurtosis of reality measure based on appropriate methods, and the implied risk aversion coefficient can be inversely determined after substituting into the formula.
3.2. Estimation of the Risk-Neutral Skewness
Notably, the risk-neutral skewness at time t is one of the critical variables in equation (4). Since the options are priced in the risk-neutral world, we can estimate risk-neutral skewness with different maturities from the option price. This study uses the method in the white paper published by CBOE in 2011 to calculate the risk-neutral skewness. The formula is as follows:
Among them,
is the forward price; is the closest and less than the exercise price of ; is the exercise price of the out-of-money contract, where the call option satisfies , and the put option satisfies ; is the interval of the exercise price; is the risk-free rate; and is the market price of the option contract at time t with the exercise price .
3.3. Standard Deviation, Skewness, and Kurtosis under Realistic Measurement
As indicated by equation (4), the standard deviation, skewness, and kurtosis should be known under the realistic measure, i.e., the predicted value of the real moment from t to T at time t. The estimation methods are elucidated as follows.
From the perspective of forecasting methods, there are three different approaches. (1) Historical method: assume that history will repeat itself, and the high-order moments that have occurred in history are taken as the predicted future high-order moments. (2) Extrapolation: build a model and establish a time series correlation between historical moments and future moments (e.g., GARCH model). (3) Post-test method: assuming rational expectation, i.e., the predicted value is equal to the true occurrence value, the higher-order moment obtained by the post-test method is used as the predicted value. This study studies the term structure of implied risk aversion and uses data of different periods. GARCH and other models are relatively suitable for daily data. The ARCH effect is not obvious in long-term data, so this study does not use extrapolation. The post-test method needs to assume that the investor is a rational expectation. From the existing research, there is a large consolidation of whether the investor is a rational expectation, and at a certain point in time, the future market situation cannot be known, such that no post-test method is used. In brief, the historical method is employed in this study, i.e., the historical moment as the predicted value of the future moment, which is also one of the most used practices in practice.
3.4. Implied Risk Aversion Term Structure Based on the Vasicek Model
Using the method of (5) in this study, we can get the implied risk aversion under different expiration times, i.e., the risk aversion term structure. This study uses the Vasicek model in the interest rate model to model the implied risk aversion term structure. The Vasicek model is a well-known short-term interest rate model proposed in 1977. It assumes that the instantaneous interest rate conforms to an O-U process. In this study, it is adopted to investigate the term structure of the implied risk aversion. The model is expressed aswhere , , and are constants and represents a standard Brownian motion. To be more economical to market parameters, this model is often written aswhere expresses the equilibrium implied risk aversion level and is the mean recovery speed. If the current implied risk aversion level is lower than the average implied risk aversion level , the drift term will cause the risk aversion level to increase. If the current implied risk aversion level is higher than the average implied risk aversion level , the drift term will cause the risk aversion level to decrease. Regardless of the risk aversion level above or below equilibrium, immediate risk aversion always exhibits a pattern that fluctuates around its long-term average, and the greater the deviation ( and difference) between the two, the faster the recovery rate. The Vasicek model often has a negative interest rate when portraying interest rate changes. This is inconsistent with reality, so scholars have proposed a CIR model to circumvent this problem. But in the change in the implied risk aversion, there is no such problem because when the investor prefers the risk, the risk aversion is negative. From this perspective, the Vasicek model can characterize the implied risk aversion mean regression, which can be negative.
This study writes it in a discrete form:where , , long-term risk aversion level is , and mean recovery speed is . We can use the maximum likelihood method to estimate the parameters of the model (5).
4. Empirical Analysis
4.1. Data
This study takes China SSE 50 ETF index and 50 ETF options as empirical research objects, and the data come from the Wind database. The basic information of SSE 50 ETF options is shown in Table 1.
During the estimation of the risk-neutral skewness, the sample period of the SSE 50 ETF option employed in this study ranges from March 26, 2015, to August 23, 2018, which has experienced market ups and downs. In this study, the options for the next working day of the option expiration date (4th Wednesday) are selected as the research object since the remaining period of the current option can be considered 1 month, 2 months, and the last two quarterly months. The time series and term structure of implied risk aversion can be obtained using the above-described options with different maturities. We process the data as follows: (1) use the option settlement price as the date price; (2) remove the missing value; and (3) only the out-of-money option price is retained.
The data from September 22, 2014, to August 23, 2018, are adopted during the estimation of the realistic measure of higher moments. The data sample is 6 months ahead of schedule since the data from the past 6 months should be adopted to calculate the standard deviation, skewness, and kurtosis with a maturity of 6 months as the predicted value of the actual higher moment. The statistical nature of the return of SSE 50 ETF (different maturities) is listed in Table 2. The one-year fixed deposit rate serves as a risk-free rate.
4.2. Estimation Results of Risk-Neutral and Realistic Measure Higher Moments
As indicated by equation (5), the key to estimating the implied risk aversion coefficient is to estimate the risk-neutral skewness and the realistic measure standard deviation, skewness, and kurtosis of the different maturities.
4.2.1. Risk-Neutral Skewness
The risk-neutral skewness of the expiration period of 1 month, 2 months, and the last two quarterly months can be determined using the SSE 50 ETF option data. The cubic spline method is adopted in this study to interpolate the risk-neutral skewness for determining the risk-neutral skewness with the maturity of 1, 2, 3, 4, 5, and 6 months (Figure 1), so as to further clarify the characteristics of the risk aversion term structure in depth.
The skewness index, i.e., the “Black Swan Index,” is set to measure the market’s concern regarding unexpected events. In general, the skewness index is positively correlated with the stock market, suggesting that investors are increasingly worried about unexpected events in the context of market prosperity. From the trend of the SSE 50 ETF skewness index, it is also in line with this feature: in May 2017, the implied skewness index is negative for most of the time, corresponding to the fact that the Chinese stock market is in a period of steady fluctuation. Since May 2017, the implied skewness index has been positive for most of the time, and there has been a significant fluctuation. In the above period, Chinese stocks underwent a period of steady rise and large fluctuations, as reflected in the skewness index for positive bias and large fluctuations.
4.2.2. Realistic Measure Standard Deviation, Skewness, and Kurtosis
Using the SSE 50 ETF yield data for the period corresponding to the history, the standard deviation, skewness, and kurtosis of the measured degree are calculated (Figures 2–4).
4.3. Characteristics of Implied Risk Aversion Time Series
Since the one-month option trading is the most active and the information reflected is the most adequate, this study uses the 1-month option price as a sample to examine the variation characteristics of the implied risk aversion in time series.
To be specific, the one-month option settlement price on the second day of each month’s expiration date (usually around the 20th) is adopted to calculate the investor’s expected value for the risky neutrality of the next 1-month period. Using historical data to estimate the expected value of the investor’s actual volatility, skewness, and kurtosis of the asset price in the next month substituting (5), and using the method of rolling regression, we can get the implied risk aversion of next month.. The corresponding time series can be obtained by linking the estimated results for the respective month. Figure 5 and Table 3 present the time series of 1-month implied risk aversion and their statistical characteristics, respectively.
As depicted in Figure 5 and Table 3, first, the risk preference of Chinese investors has significant time-varying characteristics from 2015 to 2018. To be specific, from December 2015 to February 2017, the implied risk aversion coefficient of Chinese investors was relatively stable, fluctuating around 1. The above result suggests that in this period, the risk preference of Chinese investors has not changed significantly. The performance of stocks indicates that China’s stock market was relatively stable in this period, such that the risk preference of investors varied slightly in this period. However, since February 2017, the volatility of China’s stock market has intensified, especially after May 2017, China’s stock market has been subjected to a sharp rise and fall. In the above-described process, the risk preference of Chinese investors has changed notably, with an implied risk aversion coefficient of over 4. This result suggests that in this period, investors have faced greater market volatility and increased uncertainty in the future market and have become more cautious about future investment, as directly manifested by the implied risk aversion coefficient of investors. Second, Chinese investors occasionally show risk-loving behavior, i.e., the implied risk aversion coefficient is less than 0. For instance, the implied risk aversion coefficient was less than 0 in April 2018. At this time, China’s stock market experienced a staged rebound. Investors considered that the stock market might have a rebound opportunity after a sharp decline. On that basis, investors turned out to be risk-loving.
4.4. Basic Characteristics of the Implied Risk Aversion Term Structure
In this study, four risk skewness indexes with different durations can be determined using the data of four different expiration dates of China SSE 50 ETF option. The cubic spline method is adopted to interpolate the risk skewness of different durations. The realistic measure standard deviation, skewness, and kurtosis of 1 month to 6 months are calculated using the historical rate of return. Thus, a static implied risk aversion term structure can be obtained (Figure 6 and Table 4).
From the perspective of the mean value of the term structure, the implied risk aversion of investors tends to decline as the term is extended. This feature shows that Chinese investors pay more attention to recent risks, are more averse to recent risks, and have a higher tolerance for investment risks in long-term periods, and the risk aversion coefficient tends to zero. The above result is mutually confirmed by the short-sighted behavior of Chinese investors. The principal component analysis is conducted to extract the principal component analysis of implied risk aversion. The variance contribution rate of the first principal component reaches 90.14%, suggesting that the first principal component of the implied risk aversion term structure can account for over 90% of the change in the overall term structure. Further, factor analysis is conducted to analyze the factor loading of the implied risk aversion term structure. The results are presented in Figure 7.
The following conclusions can be drawn according to Figure 7. First, the ratio of the first factor reaches 89.22%, suggesting that the factor accounts for nearly 90% of the change in the structure of the implied risk aversion. It shows that in the static term structure of implied risk aversion, we must pay attention to the change in the first principal component. Second, as revealed by the principal component analysis, the change in the term structure of static implied risk aversion can be explained by using two principal components. The first principal component presents a horizontal state, which is termed the “horizontal factor.” The second principal component exhibits the characteristics of rising first and then falling. It has the characteristics of curvature, which is termed the “curvature factor.” The level factor approaches 1, which means that in the long run, Chinese investors are risk-averse, and their risk aversion level approaches 1. This conclusion can be further verified by the dynamic implied risk aversion modeling below. The situation that Chinese investors are risk-averse in the short term and do not care much about risk in the long term is mainly caused by the “curvature factor,” but this is not the mainstream phenomenon of Chinese investors’ risk preference.
4.5. Dynamic Modeling of the Implied Risk Aversion Term Structure
4.5.1. Parameter Estimation of the Vasicek Model
Through the research in Sections 4.2 and 4.3, this study obtains the implied risk aversion term structure at different time points and different expiration dates, which can be considered as a static implied risk aversion term structure, to further explain the dynamic characteristics of the investor preference changes. This section uses the Vasicek model to dynamically model the implied risk aversion term structure. The sample data applied are the implied risk aversion coefficients of different time points with different expiration calculated in the previous paragraph. The parameter estimation results can be obtained (Table 5).
From the parameter estimation results, and are significantly different from 0 at 5% and 1%. The long-term risk aversion level reaches 0.6850. The above-described result suggests that in the long run, the Chinese investors’ implied risk aversion level is 0.685, such that the Chinese investors are risk-averse. This conclusion verifies the results of the level factor in Figure 7, suggesting that the implied risk aversion level of Chinese investors approaches 1 in the short and long term. The above conclusion is consistent with the empirical value of the implied risk aversion coefficient. The mean recovery speed reaches −0.4998. As revealed by the estimation result of this parameter, the Chinese investors’ implied risk aversion term structure exhibits significant characteristics of mean recovery. This finding suggests that the investors’ risk preference is a relatively stable value, which may fluctuate in the short term, whereas investors’ risk aversion tends to be stable in the long run. As indicated by the result, it is inappropriate to assume an infinitely high-risk aversion coefficient (e.g., 150) in the existing research. The risk aversion level of Chinese investors fluctuates around 1, and it is unlikely to have an extreme value of the implied risk aversion coefficient.
4.5.2. Prediction Effect of Implied Risk Aversion Term Structure Based on the Vasicek Model
Using the parameter estimation results in Section 4.5.1, the mean prediction error (RMSE) and mean absolute error (MAE) are adopted to evaluate the dynamic prediction effect of the implied risk aversion term structure. The evaluation results are listed in Table 6.
The prediction effect of Table 6 suggests the Vasicek model can be adopted to predict the implied risk aversion term structure of Chinese investors. The predicted RMSE and MAE values reach 0.4685 and 0.3922, respectively, and the prediction error is 2 times the standard deviation. As revealed by the above result, the Vasicek model can be adopted to dynamically model the implied risk aversion of Chinese investors, which exhibits certain rationality.
5. Conclusion
In this study, the main existing methods of studying risk preference are systematically summarized. By comparing different methods, the extraction method of implicit risk aversion is selected to study the time series and term structure of Chinese investors’ implicit risk aversion. As indicated by the result, the existing research has primarily focused on the improvement of methods, whereas no detailed papers have been published on some practical issues of investors’ risk preference (e.g., whether risk preference has time-varying characteristics and term structure characteristics, whether investors’ attitude towards risk has “short-sighted” phenomenon, and how much investors’ long-term risk preference is). To address the shortcomings of existing research, the implicit risk preference of Chinese investors is studied using a relatively mature extraction method of high-order moment implicit risk aversion.
In this study, the implied risk aversion time series and implied risk aversion term structure of Chinese investors are extracted using the higher-order moment implied risk aversion extraction method from 2015 to 2018. First, the risk-neutral bias of different time points and different maturities is extracted using the Shanghai Stock Exchange 50 ETF option data. Second, using the common calculation methods of using volatility, skewness, and kurtosis, the realistic measures of volatility, skewness, and kurtosis are extracted at different time points and maturities. Third, the time series of implied risk aversion is determined using the risk-neutral and high-order moments of realistic measures of one month, and the term structure of implied risk aversion is obtained using the risk-neutral and high-order moments of realistic measures of different maturities. The structure suggests that the implied risk aversion coefficient of Chinese investors exhibits significant time-varying characteristics. In most cases, the implied risk aversion coefficient of Chinese investors ranges from 0 to 4.5, i.e., risk aversion. Occasionally, there will be less than 0, i.e., love risk. Fourth, the principal component analysis method is adopted to conduct the static analysis of the term structure of implied risk aversion, the “level factor” and “curvature factor” of the term structure of implied risk aversion are obtained, and the “level factor” approaches 1, thus accounting for 90% change in the term structure of implied risk aversion. Fifth, the term structure of implied risk aversion is dynamically modeled using the Vasicek model. As indicated by the result, the risk aversion coefficient of Chinese investors is 0.658 in the long run, such that the conclusion that the “level factor” approaches 1 can be verified. The term structure of dynamic implied risk aversion exhibits the characteristics of significant mean reversion.
This study has a certain contribution to the existing research. First, the implied risk aversion of Chinese investors is examined based on the Chinese market. China has been recognized as the largest developing country, and the risk preference of Chinese investors is a problem worthy of in-depth study. Second, the term structure of implied risk aversion is investigated, which is an effective supplement to the existing research; the research scope of implied risk aversion is further expanded, and a new direction for follow-up research is provided. Third, this study attempts to use Vasicek model to dynamically model the term structure of implicit aversion, i.e., a meaningful attempt in the field of risk preference research. Fourth, as indicated by the result of the principal component analysis and dynamic modeling, the implied risk aversion coefficient of Chinese investors approaches 1. The above conclusion can continue to be applied to subsequent scientific research.
This study takes on certain significance in China’s capital market:(i)First, this study has investigated the time series and term structure of implied risk aversion of Chinese investors for the first time. This study intuitively shows the level of risk preference of Chinese investors and effectively makes up for the deficiency of quantitative analysis of Chinese investors’ risk preference, which takes on great significance in the policy making, structural development, and investor education of China’s capital market.(ii)Second, the time series of implied risk aversion of Chinese investors are built. In subsequent research, an in-depth study can be conducted on the changes in implied risk aversion in different periods, and the predictive ability of implied risk aversion can be explored, which takes on critical significance in managing risks in the capital market and investigating the correlation between risk preference and financial crisis.(iii)Third, this study studies the implied risk-averse term structure of Chinese investors and obtains the attitudes of Chinese investors towards risks of different periods in the future. As indicated by the results, Chinese investors prefer short-term risks over long-term risks. This conclusion can be adopted to manage investors’ expectations, and it is vital to stabilizing China’s stock market in depth.
Of course, this study also has some limitations:(i)First, Chinese investors’ implied risk aversion is extracted using the method of high-order moment implied risk aversion. In the future, the innovation of extraction methods of implied risk aversion can be investigated, and an extraction model of implied risk aversion can be built under the framework of investor irrationality to make the extraction method more consistent with the actual situation of irrational behavior in the capital market.(ii)Second, the application value of implied risk aversion time series is explored in depth. For instance, the predictive effect of implied risk aversion on asset prices is studied using the regression model, machine learning, and other methods, and how to integrate implied risk aversion information into risk management is investigated, so as to make the application value of implied risk aversion more prominent.(iii)Third, the application value of implied risk aversion term structure is explored (e.g., the correlation between implied risk aversion term structure, implied volatility term structure, and implied skewness term structure). Furthermore, the correlation between implied risk aversion term structure and stock market crash risk is further analyzed.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This study was supported by the Fundamental Research Funds for the Central Universities (2022ZDPYSK02), National Nature Science of China (71871120), Suqian Sci & Tech Program—“Research on Science and Technology Finance Boosting the Development of Science and Technology SMEs in Suqian under the Background of the Epidemic” (S202005), and Jiangsu Blue Project.