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Incorporating Liquidity Risk in Value-at-Risk ∗ Based on Liquidity Adjusted Returns Lei Wu † Research Institute of Economics and Management, Southwestern University of Economics and Finance Abstract In this paper, based on Acharya and Pedersen’s [Journal of Financial Eco- nomics (2006)] overlapping generation model, we show that liquidity risk could inﬂuence the market risk forecasting through at least two ways. Then we argue that traditional liquidity adjusted VaR measure, the simply adding of the two risk measure, would underestimate the risk. Hence another approach, by mod- eling the liquidity adjusted returns (LAr) directly, was employed to incorporate liquidity risk in VaR measure in this study. Under such an approach, China’s stock market is speciﬁcally studied. We estimate the one-day-ahead “standard” VaR and liquidity adjusted VaR by forming a skewed Student’s t AR-GJR model to capture the asymmetric eﬀect, non-normality and excess skewness of return, illiquidity and LAr. The empirical results support our theoretical ar- guments very well. We ﬁnd that for the most illiquidity portfolio, liquidity risk represents more than 22% of total risk. We also ﬁnd that simply adding of the two risk measure would underestimate the risk. The accuracy testing show that our approach is more accurate than the method of simply adding. JEL Classiﬁcation Codes: G11; G12; G18 Keywords: Value-at-Risk(VaR); Liquidity risk; Liquidity adjusted returns; Skewed Student’s t; GJR model ∗This research was supported by a grant from the “Project 211 (Phase III )” of the Southwestern University of Finance and Economics. †Research Institute of Economics and Management, Southwestern University of Economics and Fi- nance, 55 Guanghuacunjie, Chengdu, Sichuan, 610074, China. Email address: leiwufr@gmail.com 3VaR.¥i\6Ä5ºxµÄu6Ä5 NÂÃÇ{ ∗ É [† ÜHã²ŒÆ²L†+nïÄ Á ‡ ©3Acharya and Pedersen (2006)ªVU“.Ä:þ§y² 6Ä5 ºxŒ±ÏL–ü«ªK½|ºx"âd§·‚?˜ÚØy§ÏLò½ |ºxÚ6Ä5ºx{üƒ\ ¼6Ä5NVaR{¬$ºx"Ï d§·‚3ùŸ©Ù¥†é6Ä5NÂÃÇï§l ýÿ6Ä5N VaR§¿^ù«{ïÄ ¥I¦½|"››ÂÃÇS!š6Ä5¤ SÚ6Ä5NÂÃÇS¥šé¡5!š5Ú 5A:§·‚ï á ˜‡ t©ÙAR-GJR.ýÿ/IO0VaRÚ6Ä5NVaR"¢y (JéÐ|± ·‚nØ.ýÿ"·‚uyé6Ä5 ]|Ü ó§6Ä5ºxŒ±Óoºx22%"·‚„uy{üƒ\{(¢$ ºx"ÚOu(Jy¢·‚{'{üƒ\{°(" JEL©aaÒ: G11; G12; G18 ''…c: 3xdŠ(VaR)¶6Ä5ºx¶6Ä5NÂÃÇ¶ t©Ù¶GJR. ∗ ©ÜHã²ŒÆ/211ó§0nÏï‘8]Ï" oAŽ¤Ñ½1u~55ÒÜHã²ŒÆ²L†+nïÄ§610074" >fe‡µleiwufr@gmail.com 1 Introduction Value-at-Risk (VaR) has been widely used in risk measurement by many ﬁnancial institutions. But a fundamental assumption underlying the traditional VaR models is that assets can be traded in a liquid market. In reality, however, the capital market is not as liquid as we expect. Investors face not only the market risk but also the liquidity risk. Moreover, the bankruptcy of Long-Term Capital Management 1 (LTCM) tell us that the illiquidity “is a big source of risk to an investor” . Hence, it is necessary to incorporate liquidity risk in the VaR measure. 2 Few studies has focused on this issue . For instance, Bangia et al. (1999) esti- mate the worst increase the bid-ask spread may suﬀer. They add it to the “standard” VaR and then obtain a liquidity adjusted VaR measure. More recently, Angelidis and Benos (2006) estimate a trade volume dependent model based on the com- ponents of the bid-ask spread and then incorporate a parametric liquidity risk in “standard” VaR measure. However, the most of these researches modeling the mar- ket risk and liquidity risk separately, and then add the two risk measure—value at market risk and value at liquidity risk. But recent development in asset pricing and market microstructure theories point out that liquidity risk should be priced (see, for example, Amihud and Mendelson, 1986; Holmstr¨ om and Tirole, 2001; Acharya and Pedersen, 2006). Most empirical evidences also show that illiquidity securities should have higher expected returns (see, for instance, Brennan et al., 1998; Ami- hud, 2002; P´astor and Stambaugh, 2003). Hence, it maybe inaccurate in calculating liquidity adjusted VaR if we omit the relation between liquidity risk and market risk. In this paper, based on Acharya and Pedersen’s (2006) overlapping generation (OLG) model, we show that liquidity risk can inﬂuence the market risk forecasting through two ways. Then the accuracy of simply adding of the two risk measure would also be inﬂuenced through two ways: 1) First, as been shown in existed literatures, returns are low when illiquidity increases. Therefore, the value at market risk would increase and the simply adding would underestimate the risk. 2) Second, we show that the volatility of returns would be ampliﬁed if the volatility of illiquidity risk is large. Then the value at market risk would also increase and the simply adding would underestimate the risk, too. At a word, just add the two risk measure would underestimate the liquidity adjusted VaR. 1 2The Economist, September 23, 1999. See the section 1 of Angelidis and Benos(2006) for a nice survey. 1 Hence, we employ another method to incorporate liquidity risk in VaR measure. We modeling the liquidity adjusted returns (LAr) directly, where the LAr is equal to returns minus illiquidity cost. Since LAr itself considering the market risk and the liquidity risk simultaneously, this approach can avoid the underestimate problem in the simply adding of value at market risk and value at liquidity risk. Moreover, LAr satisﬁes Artzner et al’s (1999) “future value” viewpoint because it is the actually value of one unit of assets when investors need to liquidate the assets. Under such an approach, China’s stock market is speciﬁcally studied, a market considered as a very important emerging market. We form a skewed Student’s t AR- GJR model to estimate the one-day-ahead “standard” VaR and liquidity adjusted VaR. We ﬁnd that for the most illiquidity portfolio, liquidity risk represents more than 22% of total risk. We also ﬁnd that simply adding of the two risk measure would underestimate the risk. The accuracy testing based on Kupiec’s (1995) statistic show that our approach is more accurate than the method of simply adding. Three main contributions are belong to this paper. Firstly, we propose a more accurate approach to modeling liquidity adjusted VaR. Secondly, this study adds to the evidence on the importance of liquidity risk in VaR measure. Lastly, there are rarely studies considering China’s stock market about such issue in international journals. But China has became the most important emerging market and its stock market has been opened to international investors. Hence we need more researches, such as this paper, to study the characteristic of the risk in China’s stock market. The remainder of this paper is organized as follows. In Section 2, we will show the inaccuracy of the method of simply adding. Section 3 presents the data and econometric model. In Section 4, we will report the empirical results. Section 5 concludes the paper. 2 Theoretic framework 2.1 market risk and liquidity risk In this subsection, we would ﬁnd the relation between liquidity risk and market risk by simplifying Acharya and Pedersen’s (2006) OLG model. The model assumes generation t, which is born at time t ∈ {...,−2,−1,0,−1,2,...} and lives in period t and t + 1, consists of N agents. Agent n of generation t only has an endowment n etin period t. We assume he trades in period t and t + 1 and maximizes his expect utility function −Etexp(−γx t+1), where γ is his constant absolute risk aversion and 2 xt+1 is his consumption at time t + 1. Since there is no other source of income, the utility function is equal to −E etp(−γW t+1 ), where W t+1 is the derived wealth by trading. Suppose there are two kinds of asset, the risky asset with total of S shares and the riskfree asset. At time t, the risky asset has a share price of P , atd has a per- share illiquidity cost of Ct. Hence, agents can buy the risky asset at P but must sell it at Pt− C .tP atd C are toth random variables. Uncertainty about P is what t generates the market risk in this model. Similarly, uncertainty about C generatts the liquidity risk. Moreover, the illiquidity cost C istassumed to be autoregressive process of order one, that is ¯ C ¯ C t C + ρ (C t−1 − C) + η t (1) ¯ C where C ∈ R , + ∈ [0,1], and η ts an independent identically distributed process C with zero mean and variance V ar(η t) = Σ . We assume the gross return of the riskfree asset is R (R > 1). For the the risky asset’s (net) return P t rt= P − 1 (2) t−1 and its relative illiquidity cost C t c t , (3) P t−1 3 we can obtain two implications : 4 Proposition 1. Returns are low when illiquidity increases, Cov tc t+1 ,rt+1) < 0. (4) There are lots of empirical evidences consistent with this proposition both in developed markets and emerging markets. For examples, Amihud (2002) ﬁnds a negative relation between the return on size portfolios traded in NYSE and their corresponding unexpected illiquidity; Bekaert et al.(2007) ﬁnds a negative relation between the returns and illiquidity for emerging markets. Proposition 2. The conditional variance of returns increases with the conditional variance of illiquidity, ∂V ar t(rt+1) > 0. (5) ∂V ar tC t+1) 3See appendix for proves. 4 This proposition is similar to Proposition 3 in Acharya and Pedersen(2006). 3 The relation between the second moments of returns and illiquidity always be omitted in literatures. However, it is important to consider variance in risk mea- surement. Practically, we would ﬁnd that both kinds of relation could inﬂuence the accuracy of the simply adding of the two risk measure. 2.2 Incorporating liquidity risk in VaR Value-at-Risk (VaR) is deﬁned as the worst outcome that is expected to occur over a predetermined period and at a given conﬁdence level (say 1 − α). The tra- ditional VaR measure, VaR(r), focuses on the market risk but doesn’t consider the liquidity risk. In fact, we should compute VaR(r + (−c)) (6) if we want to incorporate liquidity risk in VaR. Using −c here because c is deﬁned as illiquidity in the assumption. In most of literatures, the market risk and the liquidity risk were modeled separately. The liquidity adjusted VaR was calculated by simply adding of the two risk measure, that is VaR(r + (−c)) = VaR(r) + VaR(−c). (7) But as shown above, there exit at least two kinds of relation between the liquidity risk and the market risk. We now analyze the impact of the two kinds of relationship on the above method. Without loss generality, we focus on the one-step-ahead VaR computed in time t VaR t(rt+1 + (−c t+1)) = VaR (t t+1 ) + VaR (tc t+1 ). (8) Firstly, according to proposition 1, we have Cov (ct t+1 ,rt+1) < 0. Since VaR (−t t+1) refers to the worst increase the illiquidity may suﬀer, the probability that r t+1 would be low will increase when we consider the liquidity risk. Therefore, if we isolate the calculating of VaR t(rt+1) from the calculating of VaR (−c t t+1 ), we would underes- timate the risk. Contrarily, since VaR (r ) is deﬁned as the worst outcome the t t+1 return may occur, the probability that c t+1 would be high will increase when we con- sider the VaR measure of the market risk. Hence, we would also underestimate the risk if we isolate the calculating of VaR (tc t+1 ) from the calculating of VaR (rt t+1). Secondly, according to proposition 2, we have ∂V art(t+1) > 0. We know that ∂V art(Ct+1 the one-step-ahead VaR measure of the market risk is computed as r r r VaR (t t+1) = ▯ t+1 + z α t+1 , (9) 4 where ▯ r is the conditional mean of the asset’s return and σ r the conditional t+1 t+1 r standard variance of the return. z α is the left quantile at α for the empirical dis- r tribution of the return. Since σ t+1 increases with the conditional standard variance of the illiquidity risk, VaRt(rt+1) would be high (in absolute value) when we con- 5 sider the liquidity risk . Hence, we would underestimate the risk if we isolate the calculating of VaR tr t+1). To sum up, the simply adding of the two risk measure would underestimate the liquidity adjusted VaR. We suggest that it is more accurate to model the liquidity adjusted returns (LAr) directly, where LAr is equal to returns minus illiquidity cost. Since LAr itself considering the market risk and the liquidity risk simultaneously, this approach can avoid the underestimate problem in the simply adding of value at market risk and value at liquidity risk. Moreover, LAr satisﬁes Artzner’s (1999) “future value” viewpoint because it is the actually value of one unit of assets when investors need to liquidate the assets. Then we should calculate VaR(LAr) in this paper. To highlight the importance of the liquidity risk, we deﬁne and compute the relative liquidity risk proportion ℓ = VaR(LAr) − VaR(r) . (10) VaR(LAr) 3 Data and econometric models China’s stock market is speciﬁcally studied in this paper, transaction data cover the period from 2 January 2001 to 31 December 2008. Before describing our data set in detail, we ﬁrst introduce the illiquidity measure used in this study. 3.1 The illiquidity measure The concept of (il)liquidity is elusive. Literatures about liquidity focus on one kind or several kinds of liquidity proxy because it is not observed directly. For examples, Amihud and Mendelson (1986) use the bid-ask spread relating to trading cost; Pastor and Stambaugh (2003) form a monthly liquidity measure by regressing individual stock’s return minus the market return on the lagged individual stock’s return and the lagged signed dollar trading volume using daily data; Amihud (2002) deﬁned illiquidity as the average ratio of the daily absolute return to the dollar trading volume on that day; Bekaert et al.(2007) construct the proportion of zero 5If we don’t consider the liquidity risk, it is same to assume the mean and variance of illiquidity are both equal to zero. 5 daily returns observed over the relevant month for emerging market as liquidity measure. In the literatures of liquidity adjusted VaR, bid-ask spread is a widely used illiquidity measure. For instance, Bangia et al. (1999) classify illiquidity into the exogenous illiquidity and the endogeneous illiquidity and employ the bid-ask spread to represent the former. Based on the components of the bid-ask spread, Angelidis and Benos (2006) use order-based proxies of liquidity. In emerging markets, however, detailed transaction data of the bid-ask spreads are not widely available, especially for long time series. Hence, we employ Amihud (2002)’s illiquidity measure using only daily data. Particularly, the illiquidity of stock i in day t is i |rt| ILLIQ = t i , (11) V t where r and V iare the return and yuan volume (in ten millions) for stock i on t t day t, respectively. This illiquidity measure has been widely used in empirical studies, and has been shown to be a valid instrument for the illiquidity. ILLIQ is positively related to price impact to capture the price reaction to trading volume (Liu, 2006). Hasbrouck (2002) ﬁnds that the Spearman (Pearson) correlation between ILLIQ and a measure of Kyle’s (1985) lambda is 0.737 (0.473) in the USA. Yuan volume in ten millions means that we assume the investor’s position is ten millions yuan, and the illiquidity cost is positively related to trade demands. Therefore, ILLIQ captures both the exogenous illiquidity and the endogeneous illiquidity in Bangia et al. (1999). ILLIQ, in terms of return impact, can be viewed as the cost of 10 millions yuan trade. But China’s stock market experiences a rapidly growth in the sample period— the market capitalizations of the market portfolio increase by almost 832 7 percent from January 2001 to December 2008 . Obviously, 10 millions yuan trade was more substantial in January 2001 than December 2008, so ILLIQ tend to be smaller in magnitude later in the period. Hence follow P´ astor and Stambaugh (2003), we construct the scaled series (m h/m )1LLIQ , where m is thehtotal value of the t market at the end of month h corresponding to day t, and m 1 is the total value of the market at the end of January 2001. Finally, the illiquidity measure for stock i at day t we used is m c = min{ h ILLIQ ,10.00}. (12) t m 1 t 6Yuan is the units of Renminbi (RMB, the Chinese currency). 7The total value of the market at the end of January 2001 and at the end of December 2008 are 1567768 and 14602379 millions yuan, respectively. 6 3.2 Data China has two stock exchanges, the Shanghai stock exchange (SHSE) and the Shenzhen stock exchange (SZSE). The two were both inaugurated in the early 1990s, and the SZSE is relatively smaller. Chinese company can raise funds through an A or B share listing on one of the two exchanges. The A shares are held by Chinese citizens and purchased in RMB, while B shares are held by foreign parties and denominated in U.S. dollars. Since only a few ﬁrms oﬀer B shares and the B shares always experience a light trading, we focus on the A shares only. In the remainder of this paper, the A shares market of the SHSE and the A shares market of the SZSE are, respectively, abbreviated to SHSE-A and SZSE-A. The data used in this paper for the two stock exchanges are from the CSMAR China Stock Market Trading Database, which imitates CRSP and be widely used by Chinese academe and ﬁnancial companies. For each year t (2001-2008), we allocate all the ﬁrms listed in the SHSE-A and the SZSE-A into ﬁve size-portfolios (from small to big: S1, S2, S3, S4 and S5) based on their market capitalization at the end of December of year t − 1. For all stocks, the days with no trading have been eliminated from the sample. Value-weight daily returns (in percent) and illiquidity costs (in percent) on the portfolios are calculated from the ﬁrst trading day to the last trading day of year t. Particularly, for each portfolio p (p ∈ {S1, S2, S3, S4, S5}), its return at day t is p ▯ i i rt= wt t, (13) i in p and the illiquidity cost at day t is p ▯ i i ct = wt t, (14) i in p where w tre value-based weights. Suppose month h is corresponding to the day t, then we form the weights based on the market value for ﬁrm i at the end of month h − 1. Similarly, we compute the LAr (in percent) of portfolio p at day t, as p ▯ i i LAr t = w tAr t i▯in p i i i = w tr t c t. (15) i in p The number of valid observation days in the sample for each portfolio is 1932. p p p We plot r t c tnd LAr t in ﬁgure 1, ﬁgure 2 and ﬁgure 3, respectively. These 7 ﬁgures indicate that both returns time series and illiquidity costs time series exhibit volatility clustering. So, not surprisingly, the LAr time series, which is the diﬀerences between the returns and the illiquidiy costs, also exhibit volatility clustering. The high volatility in magnitude later may due to the reform of non-tradable shares and the subprime crisis. Moreover, the ﬁve portfolios experience similar price trend, that is periods of low volatility and periods of high volatility are almost the same in the ﬁve portfolios. This ﬁnding is consistent with Morck et al.’s (2000) conclusion. They ﬁnd that the systematic component of returns variation is large in emerging markets, including China. [Figure 1 about here.] [Figure 2 about here.] [Figure 3 about here.] Table 1 presents descriptive statistics for the ﬁve size-portfolios. The portfolios have almost the same average number of ﬁrms. Obviously, portfolio contain smaller ﬁrms has higher illiquidity cost, and S5 exhibits superior liquidity level. But the mean of the returns is nearly indiﬀerence among the ﬁve portfolios: E(r) for S5 is just a little higher than for S1. For each portfolio, the covariance between the returns and illiquidity costs are negative, supporting the proposition 1 we have shown. We also ﬁnd that this covariance is larger (in absolute value) for less liquid portfolios. This ﬁnding indicates that liquidity risk is more inﬂuential for less liquid assets, which is familiar in reality. The size-portfolio with smaller ﬁrms has both higher return volatility and higher illiquidity volatility than portfolio contain bigger ﬁrms except for S1. This is the evidence to prove proposition 2. [Table 1 about here.] The liquidity adjusted returns for the high capitalization stocks (S5) outper- formance the low capitalization stocks (S1) clearly. Therefore, liquidity risk is an important non-market risk we should consider. More speciﬁcally, the returns for other 4 size-portfolios exhibit negatively skewed except for S5. In contrast, illiquid- ity for all the portfolios exhibit positive skewed. After adjusting for illiquidity, LAr for all the ﬁve portfolios show positive skewed. 8 3.3 Econometric models VaR is an estimation of the tails of the empirical distribution (Angelidis et al., 2004). The family of ARCH models are popularly used in modeling the daily VaR, such as RiskMetricsTM or GARCH(1, 1) under speciﬁc distribution. We would also use ARCH models in this study. Particularly, we set the conditional mean equation by ﬁtting an p orders autoregressive (AR(p)) process, that is ▯ p yt= φ 0 φiy(t − i) +tε , (16) i=1 where ytis the return series, illiquidity series or LAr series. For many stocks, the “bad” news and the “good” news have diﬀerent pronounced eﬀect on volatility— the so-called asymmetric eﬀects. This asymmetric eﬀects is important in the accuracy of VaR estimation. Brooks and Persand (2003) ﬁnd that the VaR would be underestimated if the models leave asymmetric eﬀects out of account. Hence, we employ the GJR model, which was introduced by Glosten, Jagannathan and Runkle (1993), to capture the asymmetric eﬀects. On the other hand, the Jarque-Bera tests in Tabel 1 show that all the series exhibit a non-normal distribution. And all the series show either positively skewed or negatively skewed. Therefore, to account for the non-normality and the excess skewness, we, follow Giot and Laurent (2003), assume the residuals tof the conditional mean equation (16) have a skewed Student’s t-distribution. Finally, we set the conditional variance equation to be a skewed Student’s t GJR model, that is εt = σ zt t (17) 2 2 2 2 σt = ω + α ε1 t−1 + γ1 t−1εt−1+ β 1 t−1, (18) where λtis a dummy variable that take the value 1 when εtis negative and 0 when it is positive. I1 γ > 0, negative shocks will have larger eﬀects on volatility than positive shocks— the so-called leverage eﬀects. Follow Lambert and Laurent (2001) and Giot and Laurent (2003), the innovation process ztis assumed to be (standardized) skewed Student distributed, that is 2 1sg[ξ(sz + m)|ν], if z < − s (19) ξ + ξ f(z|ξ,ν) = 2 m ′ ξ + 1sg[(sz + m)/ξ|ν], if z ≥ − s (19 ) ξ 9 where g(·|ν) is the standard Student’s t density with freedom ν and ξ is the asym- 8 metry coeﬃcient: the density is skew to the right (left), if log(ξ) > 0(< 0) . m 2 and s are respectively the mean and the variance of the non-standardized skewed Student’t distribution: ▯ ▯ ν−1 √ Γ 2 ν − 2 1 m = √ ▯ ▯ (ξ − ), (20) πΓ ν ξ 2 2 ▯ 2 1 ▯ 2 s = ξ + 2 − 1 − m . (21) ξ To estimate VaR, we need know the quantile function of the distribution. Lam- bert and Laurent (2000) show that the quantile function of the standardized skewed Student’s t distribution is skstα,ν,ξ m skstα,ν,ξ , (22) s in which, 1st [ (1 + ξ )], if α < 1 , (23) ∗ ξ α,ν 2 1+ξ2 skstα,ν,ξ= −ξst [1 − α(1 + ξ−2)], if α ≥ 1 , (23 ) α,ν 2 1+ξ2 where st is the quantile function of the standard Student’s t density. Then given α,ν conﬁdence level 1 − α, the one-day-ahead VaR estimation is given by V aR tyt+1) = E tyt+1) + skstα,ν,ξt+1. (24) 4 Empirical results Based on the estimation of the parametric model in subsection 3.3, we calculate and compare the two kinds liquidity adjusted VaR— VaR(LAr), and the simply adding of VaR(r) and VaR(−c). In addition, we will compute the liquidity compo- nent in VaR(LAr) to highlight the importance of liquidity risk. 4.1 Skewed Student’s t AR-GJR model estimation In this subsection, we ﬁrstly estimate the skewed Student’s t AR-GJR model (16)- (21) for the ﬁve size-portfolios’ returns, illiquidity costs and LAr, respectively. All the econometric models in this paper was estimated by G@RCH 5.0, an Ox package. We only report the results for the volatility speciﬁcation. Table 2 provides the 8 See Giot and Laurent (2003) for a more detailed discussion. 10 parameters’ estimation for returns and illiquidity costs. we also compute the Pearson correlation coeﬃcient between the conditional variances of return and illiquidity. [Table 2 about here.] The returns of ﬁve size-portfolios feature relatively similar volatility characteris- tics. The conditional variances exhibit strong memory eﬀects since the autoregressive coeﬃcient β i1 nearly 0.9. γ fo1 returns is positive and signiﬁcant for all portfolios, which implies that the leverage eﬀect for negative returns also exists in China’s stock market. log(ξ) is negative for all portfolios and signiﬁcant except for S5, indicating a negative excess skewness. For the illiquidity costs, memory eﬀects of the conditional variance also exist in each portfolio. But S5 show a high sensitivity to short run shock (α = 01883). log(ξ) is positive and signiﬁcant for all portfolios, indicating a positive excess skewness. γ 1 is negative and signiﬁcant for each portfolio. Notice that a negative illiquidity shock equal to a positive liquidity shock. Hence, there also exist a leverage eﬀect for negative liquidity shocks. Moreover, the absolute value of γ 1 for illiquidity costs is much larger than for returns, which implies that liquidity is much more sensitive. The bankruptcy of LTCM is a suitable example. “...LTCM’s partners, calling in from Tokyo and London, reported that their markets has dried up. There were no 9 buyers, no sellers. ...” . The Pearson correlation coeﬃcient between the conditional variances of returns and illiquidity costs is positive for all portfolios, supporting the proposition 2 we have proved. Moreover, this correlation coeﬃcient is larger for less liquid portfolios, indicating that liquidity risk is more inﬂuential for less liquid assets, similar to the conclusion in table 1. Table 3 presents the estimation results for the conditional variance models of LAr. All the portfolios exhibit relatively similar volatility characteristics for LAr. The autoregressive coeﬃcient β 1 is close to 0.9, pointing out a memory eﬀect of the conditional variance for LAr. γ f1r LAr is positive and signiﬁcant for each portfolio, indicating a leverage eﬀect for negative LAr in the conditional variance speciﬁcation. log(ξ) is negative and signiﬁcant for all portfolios, which implies a negative excess skewness. [Table 3 about here.] 9Wall Street Journal, November 16, 1998. 11 4.2 Liquidity adjusted VaR In this subsection, we calculate and compare the two kinds one-day-ahead liq- uidity adjusted VaR— VaR(LAr), and the simply adding of VaR(r) and VaR(−c). We will also compute the liquidity component ℓ in equation (10) to emphasize the importance of liquidity risk. Table 4 provides that two kinds one-day-ahead liquidity adjusted VaR and the liquidity component ℓ given α = 5% or 1% ( the conﬁdence level 1 − α is 95% or 99%). We ﬁnd that, without adjusting for liquidity risk, the VaR(r) is relatively similar for S1, S2, S3 and S4. S5 has a smaller VaR(r). But after incorporating liquidity risk in VaR, VaR(LAr) is larger for more illiquidity (lower capitalized) portfolio. Also, S5 is much outperformance S1: the diﬀerence between VaR(LAr) of S1 and S5 is 1.583 percent when α = 5% or 2.371 percent when α = 1%, but it is 0.523 or 0.675 for VaR(r). Moreover, the liquidity component ℓ is more than 22% for the low capitalization portfolios and almost 5% even for the high capitalization ones. All the above results indicate that liquidity risk is even more important and we must incorporate the liquidity risk in VaR measure. [Table 4 about here.] By comparing VaR(LAr) with VaR(r)+VaR(−c), we ﬁnd that VaR(LAr) is a little larger than the simply adding for all portfolios whenever α = 5% or 1%. It appears that the simply adding VaR(r)+VaR(−c) underestimates the risk. How- ever, we couldn’t make the ﬁnal judgement before testing the accuracy of the two approaches. We will do the test in next subsection. 4.3 Accuracy testing The accuracy testing is based on the statistic developed by Kupiec (1995). For a T-day period, suppose N is the number of empirical failure days, which is the number returns exceed (in absolute value) the forecasted VaR. If the VaR model is correctly speciﬁed, N/T should be equal to the theoretical speciﬁed VaR level α. Then the appropriate likelihood ratio statistic, under the null hypothesis that N/T = α, is: ▯ ▯ T−N N▯ ▯ N T−N N N▯▯ 2 LR uc = −2 log (1 − α) α − log (1 − ) ( ) ∼ χ 1 (25) T T Table 5 presents the number of failure days for the two kinds liquidity adjusted VaR model — VaR(LAr) and VaR(r)+VaR(−c) — for each portfolio in the sample 12 period. Based on equation (25), the 95% conﬁdence intervals of the number of failure days are (77.82, 115.38) and (10.75, 27.89) for α = 5% and α = 1%, respectively. In all portfolios, whenever α = 5% or 1%, the number of failure days of the VaR(LAr) model is closer to the theoretical number than the model of simply adding. Generally, at the 95% conﬁdence level, VaR(LAr) are more accurate than VaR(r)+VaR(−c), even though the method of simply adding also generates adequate risk predictions. [Table 5 about here.] In addition, the number of failure days predicted by VaR(r)+VaR(−c) are closer to the expected number for S5 both at α = 5% and α = 1%. This ﬁnding supports our arguments that the relationships between liquidity risk and market risk produce the inaccuracy of the method of simply adding. In fact, we ﬁnd that the negative covariance and the positive correlation between returns and illiquidity costs, which are the causations of the inaccuracy of the VaR(r)+VaR(−c) model, both are the smallest (in absolute value) for S5 in table 1 and table 2. Then it is not surprising that VaR(r)+VaR(−c) could predict more precise number of failure days for S5 than for other portfolios. 5 Conclusions In this paper, by simplifying Acharya and Pedersen’s (2006) overlapping gener- ation model, we show that liquidity risk could inﬂuence the market risk forecasting through two ways. Then the accuracy of the traditional liquidity adjusted VaR mea- sure, the simply adding of the two risk measure, would also be inﬂuenced through two ways: 1) First, returns are low when illiquidity increases. Therefore, the value at market risk would increase and the simply adding would underestimate the risk. 2) Second, we show that the volatility of returns would be ampliﬁed if the volatility of illiquidity risk is large. Then the value at market risk would also increase and the simply adding would underestimate the risk, too. At a word, just add the two risk measure would underestimate the liquidity adjusted VaR. Hence, we employ another method to incorporate liquidity risk in VaR measure. We modeling the liquidity adjusted returns (LAr) directly, where the LAr is equal to returns minus illiquidity cost. Under such an approach, China’s stock market is speciﬁcally studied. We ﬁrst construct a skewed Student’s t AR-GJR model to cap- ture the asymmetric eﬀect, non-normality and excess skewness of return, illiquidity 13 and LAr. Then we estimate the one-day-ahead “standard” VaR and liquidity ad- justed VaR. We ﬁnd that for the most illiquidity portfolio, liquidity risk represents more than 22% of total risk. We also ﬁnd that simply adding of the two risk mea- sure would underestimate the risk. The accuracy testing based on Kupiec’s (1995) statistic show that our approach is more accurate than the method of simply adding. The ﬁndings of this paper make three main contributions to literatures. Firstly, we propose a more accurate approach to modeling liquidity adjusted VaR. Secondly, this study adds to the evidence on the importance of liquidity risk in VaR measure. Lastly, there are rarely studies considering China’s stock market about such issue in international journals. But China has became the most important emerging market and its stock market has been opened to international investors. Hence we need more researches, such as this paper, to study the characteristic of the risk in China’s stock market. Appendix We ﬁrst solve the problem of investor n at time t. We assume the investor n purchases y shares of the risky asset. Then the agent’s problem is n 1 n y ∈R+ (t t+1) − 2γV ar tW t+1), (A.1) where n n f n n W t+1 = (P t+1− C t+1)y + R (e −tP y t. (A.2) From the ﬁrst order condition, we have n 1 −1 f y = γ [V art(Pt+1− C t+1)] [Et(Pt+1 − C t+1) − R P t. (A.3) ▯ Since the total supply of risky asset S = ny , then we have equilibrium conditon 1 γS Pt= R f[E tP t+1− C t+1) − N V art(Pt+1 − Ct+1)]. (A.4) We can obtain the unique stationary linear equilibrium, Pt= Υ + ΦC , t (A.5) 14 where 1 ▯R (1 − ρ ) γS R f ▯ Υ = − C + V art(− ηt) , (A.6) R − 1 R − ρ C N R − ρ C C ρ Φ = − R − ρ C. (A.7) Proof of Proposition 1 The conditional covariance between illiquidity and the gross return is 1 Covt(ct+1,Rt+1) = Cov tC t+1,Pt+1) P t ▯ C ▯ 1 ρ = 2Cov t t+1,− f C C t+1 P t R − ρ 1 ▯ ρC ▯ = − V at (t+1) P t R − ρ C < 0, (A.8) which yields the proposition. 2 Proof of Proposition 2 The conditional variance of the gross return is 1 V art(Rt+1 = 2V art(t+1) Pt ▯ C ▯ = 1 V ar − ρ C P 2 t R − ρ C t+1 t 1 ▯ ρC ▯ 2 = 2 V art(Ct+1). (A.9) Pt R − ρ C So we have ∂V art(Rt+1) 1 ▯ ρC ▯2 = 2 f C > 0, (A.10) ∂V art(Ct+1) Pt R − ρ which yields the proposition. 2 15 References Acharya, V. V. and L. H. Pedersen, 2005. Asset pricing with liquidity risk. Journal of Financial Economics, 77(2), 375–410. Amihud, Y., 2002. Illiquidity and stock returns: cross-section and time-series eﬀects. Journal of Financial Markets, 5(1), 31–56. Amihud, Y. and H. Mendelson, 1986. Asset pricing and the bid-ask spread. Journal of Financial Economics, 17(2), 223–249. Angelidis, T. and A. Benos, 2006. Liquidity adjusted value-at-risk based on the components of the bid-ask spread. Applied Financial Economics, 16(11), 835 – 851. Angelidis, T., A. Benos, and S. Degiannakis, 2004. The use of GARCH models in VaR estimation. Statistical Methodology, 1(1-2), 105–128. Artzner, P., F. Delbaen, J.-M. Eber, and D. Heath, 1999. Coherent measures of risk. Mathematical Finance, 9(3), 203–228. Bangia, A., F. X. Diebold, T. Schuermann, and J. Stroughair, 1999. Modeling liquidity risk with implications for traditional market risk measurement and man- agement. Bekaert, G., C. R. Harvey, and C. Lundblad, 2007. Liquidity and expected returns: lessons from emerging markets. Review of Financial Study, 20(6), 1783–1831. Brennan, M. J., T. Chordia, and A. Subrahmanyam, 1998. Alternative factor speci- ﬁcations, security characteristics, and the cross-section of expected stock returns. Journal of Financial Economics, 49(3), 345–373. Brooks, C. and G. Persand, 2003. The eﬀect of asymmetries on stock index return Value-at-Risk estimates. The Journal of Risk Finance, 4(2), 29–42. Giot, P. and S. Laurent, 2003. Value-at-Risk for long and short trading positions. Journal of Applied Econometrics, 18(6), 641–664. Glosten, L. R., R. Jagannathan, and D. E. Runkle, 1993. On the relation between the rxpected value and the volatility of the nominal excess return on stocks. The Journal of Finance, 48(5), 1779–1801. 16 Hasbrouck, J., 2002. Inferring trading costs from daily data: US equities from 1962 to 2001. Unpublished working paper. Holmstr¨om, B. and J. Tirole, 2001. LAPM: A liquidity-based asset pricing model. The Journal of Finance, 56(5), 1837–1867. Kupiec, P. H., 1995. Techniques for verifying the accuracy of risk measurement models. The Journal of Derivatives, (3), 73–84. Kyle, A. S., 1985. Continuous auctions and insider trading. Econometrica, 53(6), 1315–1335. Lambert, P. and S. Laurent, 2000. Modelling skewness dynamics in series of ﬁnancial data. Unpublished working paper. Lambert, P. and S. Laurent, 2001. Modelling ﬁnancial time series using GARCH- type models and a skewed Student density. Unpublished working paper. Liu, W., 2006. A liquidity-augmented capital asset pricing model. Journal of Fi- nancial Economics, 82(3), 631–671. Morck, R., B. Yeung, and W. Yu, 2000. The information content of stock markets: why do emerging markets have synchronous stock price movements? Journal of Financial Economics, 58(1-2), 215–260. P´astor, L. and R. F. Stambaugh, 2003. Liquidity risk and expected stock returns. Journal of Political Economy, 111(3), 642–685. 17 10 10 5 0 0 −5 0 300 600 900r_S11200 1500 1800 0 300 600 900r_S21200 1500 1800 10 10 0 0 0 300 600 900 1200 1500 1800 0 300 600 900 1200 1500 1800 r_S3 r_S4 10 0 0 300 600 900 1200 1500 1800 r_S5 Figure 1. Daily Returns (in percent) of size-portfolios (from small to big: S1, S2, S3, S4 and S5) from 2 January 2001 to 31 December 2008. 18 4 4 3 2 2 1 0 300 600 90c_S11200 1500 1800 0 300 600 900c_S21200 1500 1800 1.5 2 1.0 1 0.5 0 300 600 900 1200 1500 1800 0 300 600 900 1200 1500 1800 c_S3 c_S4 0.4 0.2 0 300 600 900 1200 1500 1800 c_S5 Figure 2. Daily illiquidity costs (in percent) of size-portfolios (from small to big: S1, S2, S3, S4 and S5) from 2 January 2001 to 31 December 2008. 19 10 10 0 0 −10 −10 0 300 600 900 1200 1500 1800 0 300 600 900 1200 1500 1800 LAr_S1 LAr_S2 10 10 0 0 −10 0 300 600 900 1200 1500 1800 0 300 600 900 1200 1500 1800 LAr_S3 LAr_S4 10 0 0 300 600 900 1200 1500 1800 LAr_S5 Figure 3. Daily Liquidity adjusted returns (LAr, in percent) of size-portfolios (from small to big: S1, S2, S3, S4 and S5) from 2 January 2001 to 31 December 2008. 20 Table 1: This table presents descriptive statistics for the ﬁve size-portfolios. We report the average number of ﬁrms for each portfolio in Panel A. E(·), Std.(·) and Skew.(·) are, respectively, the mean , the standard variance and the Skewness for the time series. Jarque-Bera is the Jarque-Bera test statistics for normality. Cov(r,c) is the covariance between return and illiquidity. The sample period is from 2 January 2001 to 31 December 2008, including 1932 observations. S1 S2 S3 S4 S5 Panel A. The average number of ﬁrms 233.61 242.27 244.50 244.32 240.49 Panel B. Descriptive statistics for returns and illiquidity costs E(r) 0.0222 0.0209 0.0224 0.0208 0.0204 Std.(r) 2.1152 2.1446 2.1242 2.0729 1.8800 Skew.(r) -0.3977 -0.3889 -0.3835 -0.3139 0.0228 Jarque-Bera 657.08 774.06 988.22 1062.3 1425.1 P-value 0.0000 0.0000 0.0000 0.0000 0.0000 E(c) 0.7345 0.4958 0.3662 0.2502 0.0931 Std.(c) 0.6056 0.4015 0.2851 0.1850 0.0679 Skew.(c) 2.4157 2.3300 2.1649 1.9721 2.0828 Jarque-Bera 7270.1 7024.0 5710.5 3671.3 4659.7 P-value 0.0000 0.0000 0.0000 0.0000 0.0000 Cov(r,c) -0.4445 -0.2995 -0.2036 -0.1245 -0.0314 Panel C. Descriptive statistics for liquidity adjusted returns E(LAr) -0.7122 -0.4749 -0.3438 -0.2293 -0.0727 Std.(LAr) 2.3937 2.3151 2.2362 2.1401 1.8979 Skew.(LAr) -1.1971 -0.9151 -0.7628 -0.5689 -0.0568 Jarque-Bera 1545.4 1104.6 1095.7 1014.7 1284.8 P-value 0.0000 0.0000 0.0000 0.0000 0.0000 21 0.000)10)1).00.654)7()62.527(0.025) 070(0.023) -0.441(0.066) s. Robust standard errors 0.032(0.029) 0.548(0.035) nal variances of returns and illiquidity. The 761 1265.413 -3773.060 2069.560 -3575.176 3787.696 42(0.017)(0)169(0.804)00).4.167(0.395)16).5.560(0.748)000).4.133(0.381)99( 5.212( .052(0.022) -0.342(0.036) 0.060(0.023) -0.335(0.032) 0. -0.142(0.029) 0.615(0.041) -0.128(0.028) 0.629(0.038) - uding 1932 observations. onditional variance models of returns and illiquidity cost 2c is the Pearson correlation coeﬃcient between the conditio 2rσ σ ( Corr S1 S2 S3 S4 S5 0.062(0.024)06.827(1.095)01)24.251(0.411)21)55.212(0.654)000)04.240(0.403) 5.7 ) 0.660 0.576 0.543 0.536 2 c 2 r ) ( -0.174(0.029) 0.602(0.054) -0.162(0.028) 0.610(0.048) ξ 1 1 1 y ω α β γ logν( CorrLog-likelihood -3883.364 37.526 -3882.678 688.234 -3831. r c Tables2 ra:mTpleisptrbildphisrreomts2thJeanesairya2tio01 rtos31ltsDfocrthbeecr 2008, incl 22 Table 3: This table presents the estimation results for the conditional variance model of LAr. Robust standard errors are reported in parentheses. The sample period is from 2 January 2001 to 31 December 2008, including 1932 observations. S1 S2 S3 S4 S5 ω 0.117(0.038) 0.079(0.027) 0.059(0.021) 0.046(0.018) 0.033(0.014) α 1 0.092(0.030) 0.091(0.027) 0.088(0.024) 0.083(0.023) 0.067(0.017) β 1 0.870(0.024) 0.880(0.021) 0.885(0.019) 0.887(0.018) 0.897(0.017) γ1 0.039(0.033) 0.043(0.029) 0.047(0.026) 0.060(0.026) 0.074(0.025) log(ξ) -0.364(0.030) -0.299(0.028) -0.252(0.029) -0.203(0.029) -0.056(0.029) ν 5.693(0.777) 5.742(0.825) 5.827(0.846) 5.678(0.794) 5.298(0.687) Log-likelihood -3995.937 -3968.288 -3901.662 -3819.144 -3594.588 23 Table 4: This table presents the one-day-ahead liquidity adjusted VaR and the liq- uidity component ℓ. VaR(r) is the one-day-ahead VaR without considering liquidity risk; VaR(LAr) is the liquidity adjusted VaR based on liquidity adjusted returns; the simply adding of VaR(r) and VaR(−c) is denoted as VaR(r)+VaR(−c). S1 S2 S3 S4 S5 α = 5% VaR(r) 4.410 4.347 4.500 4.382 3.887 VaR(LAr) 5.713 5.363 5.325 4.958 4.130 ℓ 22.8% 18.9% 15.5% 11.6% 5.88% VaR(r)+VaR(−c) 5.659 5.343 5.215 4.888 4.022 α = 1% VaR(r) 7.118 7.152 7.452 7.280 6.443 VaR(LAr) 9.151 8.665 8.652 8.123 6.780 ℓ 22.2% 17.5% 13.9% 10.4% 4.97% VaR(r)+VaR(−c) 8.863 8.557 8.466 7.997 6.667 24 Table 5: This table presents the number of failure days for the VaR(LAr) model and the VaR(r)+VaR(−c) model. The sample period is from 2 January 2001 to 31 December 2008, including 1932 observations. S1 S2 S3 S4 S5 α = 5% Theoretical Number 96.60 96.60 96.60 96.60 96.60 VaR(LAr) 97 97 97 97 97 VaR(r)+VaR(−c) 94 98 94 98 95 α = 1% Theoretical Number 19.32 19.32 19.32 19.32 19.32 VaR(LAr) 20 20 20 20 20 VaR(r)+VaR(−c) 17 16 18 18 20 25

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