A Research Based On POT-CAViaR Model of Extreme Risk Measure

This paper assumes value of a financial market at the time of t is V_t, the loss on the subsequent period shall be: (1)Lt(l)=Vt+1−VtL_{t}(l)=V_{t+1}-V_{t}

Assume the cumulative distribution function of random variable L_t(l) is F_L, under the probability level P, this paper defines VaR_1−p as a possible maximum loss: (2)VaR1−p=inf{x|FL(x)≥1−p}VaR_{1-p}=\inf \left\{x|F_{L}(x)\geq 1-p\right\}

Engle and Manganelli proposed the CAViaR method in 2004, the predication effect of VaR was improved somewhat for this method considering no distribution shape but modeled the data directly with the quantile theory [12]. Financial assets income series are represented by Y, the corresponding distribution function shall be F_Y (y) = P_r(Y ≤ y); according to the relationship between quantile and VaR, the opposite number of P quantile of the random variable Y is numerically equal to the VaR under the reliability level (1-p), which means, Qy¯(P)=VaRy¯(1−P)Q_{\bar{y}}(P)=VaR_{\bar{y}}(1-P) , so that the CAViaR model in general pattern shall be: (3)Qt(P)=β1+∑t=1mβ2iQs(P)+∑j=1nβ3i(t−j)Q_{t}(P)=\beta_{1}+\sum\limits_{t=1}^{m} \beta_{2 i} Q_{s}(P)+\sum\limits_{j=1}^{n} \beta_{3 i}(t-j)

In which, θ = (β₁,β₂₁ ··· β_2i,β₃₁ ··· β_2i) is the parameter vector to be estimated, L(t − j) is the function form of the lag information set; according to the form in (3), Engle and Manganelli [12] proposed the following common CAViaR models: (4)Q1(P){β1+β2Q1−1(P)+β3|yt−1|SAVβ1+β2Q1−1(P)+β3max(y1−1,0)+β4min(y1−1,0)ASβ1+β2Qt−12(P)+β2yt−12 IGARCHQ_{1}(P)\left\{\begin{array}{ll}{\beta_{1}+\beta_{2} Q_{1-1}(P)+\beta_{3}\left|y_{t-1}\right|} & {S A V} \\{\beta_{1}+\beta_{2} Q_{1-1}(P)+\beta_{3} \max \left(y_{1-1}, 0\right)+\beta_{4} \min \left(y_{1-1}, 0\right) AS} \\{\sqrt{\beta_{1}+\beta_{2} Q_{t-1}^{2}(P)+\beta_{2} y_{t-1}^{2}} \quad \text { IGARCH }}\end{array}\right.

The SAV model is also called the absolutely symmetrical model indicating the same impact of good news and bad news to VaR; AS model is also called the non-symmetrical model indicating the difference of bad news to VaR.

For the CAViaR model estimation, the minimum absolute deviation model proposed by Koenkerand Basset [20] and the optimization algorithm proposed by Engle and Manganelli [12] are combined to estimate parameter θ. Assume there exists a quantile equation: (5)yt=ft(β)+εtθy_{t}=f_{t}(\beta)+\varepsilon_{t\theta}

In which Q_θ (ɛ_tθ) = 0,ɛ ∼ N(0,1), Q_ɛ is the quantile equation, so that the estimation function of the quantile regression equation can be defined as follows: (6)θ^=argminθ1T∑i=1TP−I(yt<Qt(P;θ))[yt−Qt(P;θ)]\hat{\theta}=\arg \mathop {\min }\limits_\theta \frac{1}{T} \sum\limits_{i=1}^{T} P-I\left(y_{t}<Q_{t}(P ; \theta)\right)\left[y_{t}-Q_{t}(P ; \theta)\right]

In this formula, T is the total sample, and the corresponding conditional quantile Q_t(P;θ)can be estimated after estimating the parameter θ to estimate VaR.

When p is not at the extreme quantile level, the quantile level can be obtained by minimizing the objective function; however, when the quantile level is approaching to 1 or 0, the quantile regression will not be robust enough because the tail data, especially the thick-tail distribution is rather sparse. To solve this problem, the POT model of extreme value theory is introduced in this paper to combine the CAViaR and POT model organically through a certain method to construct the POT-CAViaR-based model. The specific process is as follows:

Firstly, the estimated quantile value Q_t (p₁) under the P₁ quantile point shall be estimated through the CAViaR model and standardized to obtain the residual quantile series of extreme value: (7)Zt=εt(p1)Qt(p1)=yt−Qt(p1)Qt(p1)=ytQt(p1)−1Z_{t}=\frac{\varepsilon_{t}\left(p_{1}\right)}{Q_{t}\left(p_{1}\right)}=\frac{y_{t}-Q_{t}\left(p_{1}\right)}{Q_{t}\left(p_{1}\right)}=\frac{y_{t}}{Q_{t}\left(p_{1}\right)}-1

In which, ɛ_t (p₁) refers to the residual of quantile point p₁ at t time point, and y_t represents the value of sample series at t time point.

Based on constructing the quantile residuals of standard extreme values, it is assumed that there exists the quantile point p₂, p₁ > p₂; the POT model is adopted to estimate it for the corresponding quantile Q_t (p₂). According to the related concepts of quantile, it can be obtained that: (8)Pr[yt≤Qt(p2)]=Pr[yt≤Qt(p1)−Qt(p1)+Qt(p2)]=Pr(ytQt(p1)−1≥Qt(p2)Qt(p1)−1)=p2P_{r}\left[y_{t} \leq Q_{t}\left(p_{2}\right)\right]=P_{r}\left[y_{t} \leq Q_{t}\left(p_{1}\right)-Q_{t}\left(p_{1}\right)+Q_{t}\left(p_{2}\right)\right]=P_{r}\left(\frac{y_{t}}{Q_{t}\left(p_{1}\right)}-1 \geq \frac{Q_{t}\left(p_{2}\right)}{Q_{t}\left(p_{1}\right)}-1\right)=p_{2}

It can be obtained through the combination of formula (7) and (8) that: (9)Pr[Zt≥Qt(p2)Qt(p1)−1]=p2P_{r}\left[Z_{t} \geq \frac{Q_{t}\left(p_{2}\right)}{Q_{t}\left(p_{1}\right)}-1\right]=p_{2}

According to the quantile conversion, there will be: (10)Qzt(1−p2)=Qt(p2)Qt(p1)−1Q_{zt}\left(1-p_{2}\right)=\frac{Q_{t}\left(p_{2}\right)}{Q_{t}\left(p_{1}\right)}-1

Formula (10) indicates the quantile level Q_zt (1 − p₂) of the standardized extreme value residual series Z_t under 1 − p₂ ; the solution process is also the modelling process of Z_t through POT model. The key point of POT modelling is to determine the threshold, generally through the ways of Hill plot, excess expected function graph and Du Mouchel 10% principle [21]. Considering Du Mouchel 10 is more frequently applied now, in this paper, the Du Mouchel 10% principle is uniformly selected to determine the threshold μ; the functional distribution of the standardized extreme value residual series of the super-threshold shall be: (11)Fu(y)=P(x−u≤y|x>u)=F(u+y)−F(u)1−F(u)F_{u}(y)=P(x-u \leq y | x>u)=\frac{F(u+y)-F(u)}{1-F(u)}

When the threshold is big enough, according to the extreme value theory [22], there exists F_u(y) ≈ G_ξ,σ (y): (12)Fu(y)≈Gξ,σ(y)=1−(1−yξσ)−1xiξ≠0F_{u}(y) \approx G_{\xi, \sigma}(y)=1-\left(1-\frac{y \xi}{\sigma}\right)^{-\frac{1}{xi}}\xi\neq 0

The G_ξ,σ (y) function is in GPD distribution, in which σ and ξ respectively refer to the scale parameter and shape parameter; the larger ξ, the fatter the tail; the value of σ and ξ can be obtained through the maximum likelihood estimation. Under the condition of F_u(y) ≈ G_ξ,σ (y), it can be obtained as follows through the combination of formula (11) and (12): (13)F(y)=F(u)+Gξ,σ(y−u)[1−F(u)]F(y)=F(u)+G_{\xi, \sigma}(y-u)[1-F(u)]

For the convenient study, N is used to represent the total sample, n refers to the number of super-threshold, the threshold distribution function F(u) is represented as N−nN\frac{N-n}{N} , substituting it into formula (13) in combination with the estimated values of σ and ξ, there will be: (14)F^(y)=1−nN(1+ξ^y−uσ^)−1/ξ\widehat{F}(y)=1-\frac{n}{N}\left(1+\hat{\xi} \frac{y-u}{\widehat{\sigma}}\right)^{-1 / \xi}

According to formula (2), the risk value under the reliability level p can be obtained as: (15)VaR1−pθ=F^−1(y)=u−σξ^{1−[Nn(1−p)]−ξ^}VaR_{1-p}^{\theta}=\widehat{F}^{-1}(y)=u-\frac{\sigma}{\hat{\xi}}\left\{1-\left[\frac{N}{n}(1-p)\right]^{-\hat{\xi}}\right\}

Therefore, according to formula (1)–(15) mentioned above, the extreme value risk measurement model based on the POT-CAViaR model can be obtained as follows: (16){Qt(P)=β1+∑i=1mβ2iQii(P)+∑j=1nβ3iL(t−j)θ^=argminθ1T∑i=1TP−I(yi<Qi(P;θ))[yt−Qi(P;θ]Fu(y)≈Gt,σ(y)=1−(1−yξσ)−1ζξ≠0VaR1−pθ=F^−1(y)=u−σ^ξ^{1−[Nn(1−p)]t^}\left\{\begin{array}{l}{Q_{t}(P)=\beta_{1}+\sum_{i=1}^{m} \beta_{2 i} Q_{i i}(P)+\sum_{j=1}^{n} \beta_{3 i} L(t-j)} \\{\hat{\theta}=\arg \min _{\theta} \frac{1}{T} \sum_{i=1}^{T} P-I\left(y_{i}<Q_{i}(P ; \theta)\right)\left[y_{t}-Q_{i}(P ; \theta]\right.} \\{F_{u}(y) \approx G_{t, \sigma}(y)=1-\left(1-\frac{y \xi}{\sigma}\right)^{-\frac{1}{\zeta}} \xi \neq 0} \\{V a R_{1-p}^{\theta}=\widehat{F}^{-1}(y)=u-\frac{\hat{\sigma}}{\hat{\xi}}\left\{1-\left[\frac{N}{n}(1-p)\right]^{\hat{t}}\right\}}\end{array}\right.

To test the effect of POT-CAViaR model, this paper adopts the posteriori test method proposed by Kupiec, the basic concept of which is: to compare the estimated value and actual value measured by the model; it fails if it is larger than the actual loss, the failure rate can be obtained by dividing the total days of observation by the total days of failure and then comparing it with the preset VaR; the closer it is to the actual VaR, the better the effect is. Refer to document [23] for the specific details.

To test the model constructed earlier in this paper, the Shanghai-Shenzhen 300 Index (HS300), Hong Kong Hang Seng Index as well as the Nikkei225 Index (N225) are taken as the research samples; in which HS300 Index represents the emerging capital market while the HSI Index and N225 Index refer to the mature capital market.

As the Shanghai-Shenzhen 300 Index was issued since April 8^th, 2005, the selective interval of samples in this paper shall be: April 8^th, 2005–March 19^th, 2017, approximately 12 years with almost 29000 sample points for each index. During the research, the samples are divided into estimation samples and test samples in this paper, in which the interval of estimation samples is April 8^th, 2005–December 17^th, 2014 (2176 samples in total); the interval of test samples is December 18^th, 2014–March 19^th, 2017 (500 samples in total). The data are from Dazhihui Software; the daily income yield is defined as follows: (17)Xt=100(lnpt−lnpt−1)X_{t}=100\left(\ln p_{t}-\ln p_{t-1}\right)

It can be seen from Table 1 that all sample series are of the left-skewed form (skewness <0) with leptokurtosis and fat-tail characteristics (kurtosis>3); what's more, with the J-B test, all sample series under the 1% significance level are significantly different from the normal distribution. To further test the distribution characteristics of the sample series, QQ graph of various sample series (Fig.2). Taking the leftmost HS300 as an example, the upper and lower tails of series HS300 are obviously derived from the normal distribution and presenting the fat-tail characteristics; it can be concluded that the sample series presenting the typical “leptokurtosis and fat-tail characteristics” characteristics. According to Fig.1, compared with HSI series and N225 series, the HS300 series have more extreme values than the latter; generally, the sample series are of non-symmetric distribution and typical “leptokurtosis and fat-tail characteristics of financial yield in addition to certain explosion and agglomeration characteristics.”

Fig. 1

Fig. 2

Considering the satisfying characteristics of CAViaR model, the SAV model, AS model and IGARCH model mentioned earlier are adopted in this paper to fit the HS 300, HSI and N225 income yield series; the specific results are shown in Table 2.

According to Table 2, β₂ is completely significant, it means that tail quantile of the sample series has obvious volatility clustering characteristics while the meaning stays above 0.85 with obvious short-term memory characteristics. In the meantime, β₃ is obvious especially for the AS model and IGARCH model; the quantile predication of model has certain explanatory ability at the external information impact. To further test the predication effect of SAV model, AS model and IGARCH model, the Kupiec test introduced earlier is adopted for analysis. See Table 3 for details.

According to Table 3, in terms of the number of failures, the number of failures of CAViaR model is higher than the standard 25 under the 5% reliability level. Despite the risk underestimations at different levels, the general deviation is small; the minimum P value is 0.2346 and the maximum one is 0.8384, in which the smaller P values are mainly in HS300 Index and AS Index while the larger P values are mainly in the N225 Index and IGARCH model. Generally, although the three models have all underestimated the risk to different extents, under the 5% significance level, the CAViaR model is effective for risk predication of sample, in which the IGARCH model has favorable performances with small differences. HS300 market underestimates risk more obviously, which it is mainly because that compared with the HSI market and N225 market, the HS300 market starts developing late and the market price fluctuates greatly.

Under the 1% reliability level, with different samples indexes, the number of failures of all quantile models is significantly larger than the theoretical level 5; the corresponding P value is also approaching 0; as the sample risk is seriously underestimated, the model presents obvious incompatibility and invalid predication capacity to risk. The HS 300 index performance is generally weaker than that of HSI and N225 and obviously unstable with risk predication highly deviated from the optimal level. Compared with HSI and N225 Index, as the HS 300 market starts developing later and various systems and mechanisms are incomplete, the current quantile models fail to fit the extreme value risk characteristics effectively in case of extreme risk, hence the predication capacity of the model is invalid.

As the 1% VaR of CAViaR model has been seriously underestimated, the CAViaR model and extreme value POT model are combined in this paper, so that the tail characteristics can be captured while the distribution of initial series is not considered. First, the 5% quantile level is calculated with the CAViaR model, after that, the standardized residual series of extreme value can be obtained according to the standardized quantile residual method to predicate the 1% VaR through the POT model based on standardized extreme value residual series. According to the method in the previous model construction part, the standardized residual series of extreme value shall be calculated first before the POT model is adopted for fitting. The key point for POT model fitting is to select the threshold; as mentioned above, the Du Mouchel 10% principle is adopted to select the threshold of the standardized residual series of extreme value; after which the maximum likelihood estimation is adopted to determine the scale parameter σ and shape parameter ξ of the POT model. According to Table, the shape parameter ξ is larger than 0, which means the standardized residual series of extreme value have obvious fat-tail characteristics.

To test the effectiveness of the POT model estimation, the fitting diagnosis diagrams of standardized residual series of extreme value under all models are provided in this paper; taking the HS300 standardized series of extreme value as an example, as shown in Fig. 3, most points have fell on or near the threshold distribution diagram and tail distribution diagram, and only a few points are deviated without influencing the fitting effect. POT model has a favorable fitting effect in general; the same conclusion has been made on the same fitting effect test of the other series, which won’t be listed separately due to the limited length.

Fig. 3

To test the risk estimation effect based on the CAViaR-POT, the posteriori test method mentioned earlier is adopted here for test; in the meantime, to make the comparative analysis, the VaR test result of GARCH-POT model and POT model under the 1% reliability level is calculated in this paper; see details in Table 5.

According to the test result of HS300, despite the underestimation in the risk measurement based on CAViaR-POT, the underestimation degree has been greatly reduced. Specifically, the number of failures is significantly lower than that before the model improvement and approaching the ideal level 5; compared with that of the unimproved model, the P value is also greatly improved. Therefore, the improvement of CAViaR model by the POT model improves the accuracy of risk measurement and it can characterize and predicate the extreme risk effectively.

According to the test result of HSI and N 225, the risk measurement effect based on CAViaR-POT model can be greatly improved and the predication accuracy of risk value is also enhanced; the POT-IGARCH model is comparatively superior to other models. Both the number of failures and P value fully indicate that the extreme risk predication accuracy of the improved model has been significantly improved while the model is strongly accommodative.

Generally, the extreme risk estimation accuracy based on the improved model has been improved at different levels, which means that the improved model is more accommodative with greater effectiveness; the predication accuracy of POT-IGARCH model is the highest. According to the sample market analysis, after the improvement, the extreme risk predication accuracy of the improved model for the relatively mature market HSI and N225 is higher than that for the HS300;despite the certain underestimation in the extreme value risk predication of HS 300, the difference is generally small. This might be related to the maturity of HS300 market. Considering that the HS300 market starts developing later than the HSI market and N255 market, its market price is more easily impacted by policy for system and mechanism issues, so that there will be more extreme value points.