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VaR methods for the dynamic impawn rate of steel in inventory financing under autocorrelative return
European Journal of Operational Research 223 (2012) 106–115
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European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor
Stochastics and Statistics
VaR methods for the dynamic impawn rate of steel in inventory financing under autocorrelative return He Juan a,⇑, Jiang Xianglin b, Wang Jian a, Zhu Daoli c, Zhen Lei d a
College of Traffic, Transportation and Logistics, Southwest Jiaotong University, Chengdu 610031, China Institute for Financial Studies, Fudan University, Shanghai 200433, China c School of Economics and Management, Tongji University, Shanghai 200092, China d Department of Credit Risk Management in Chengdu, Huaxia Bank 61000, China b
a r t i c l e
i n f o
Article history: Received 20 June 2011 Accepted 1 June 2012 Available online 13 June 2012 Keywords: Finance AR (1)-GARCH (1,1)-GED Long-term VaR forecasting Dynamic impawn rate Inventory financing
a b s t r a c t This paper proposes the way of setting the dynamic impawn rate by dividing the impawn periods into different risk windows. In an efficient financial market, the return is hypothetically independent, while in a pledged inventory market where spot transactions predominate, the return is auto-correlative. Therefore, the key to setting the impawn rate is to predict the long-term risk. In this experiment, using the database of spot steel, we established a model with the formula AR (1)-GARCH (1,1)-GED, forecasting the VaR of steel during the different risk windows in the impawn period through a method of out-of-sample, and got the impawn rate according with the risk exposure of banks. The results of our experiment indicated that the introduction of coefficient K into the model can significantly improve bank risk coverage and reduce its efficiency loss. Besides, the impawn rate obtained by the model correlates positively with the lowest price in the future risk windows. Ó 2012 Elsevier B.V. All rights reserved.
1. Introduction In recent years, supply chain finance has appeared as the competition between the supply chains predominate over that between the enterprises. Euromoney magazine defines the supply chain finance as the most popular topic in banking transaction service over the past few years, and predicts that the needs of this business will continue to grow in the next few years. However, different from that of western countries, the supply chain finance in China is a kind of pledge business performed by commercial banks, key enterprises, SMEs (small and medium enterprises), warehouse supervision enterprises, and so on (Feng, 2007). Besides, the pledges in China are mainly the inventories which are traded on the spot market (such as materials, products, and half-finished products) rather than the rights pledges (accounts receivable, accounts payable, and derivatives) which are widely used in western countries. In China, supply chain finance is not only an extended business of modern logistics but more importantly, it can help to solve the financing problems that many domestic enterprises are facing. Besides, upon the international background of credit deflation, the supply chain finance is replacing the traditional liquidity loans (Shen Zhen Development Bank, 2009). According to McKinsey prediction, the foreign trade will amount to USD1500 billion and the
⇑ Corresponding author. Tel.: +86 13708179996. E-mail addresses: [email protected], [email protected] (H. Juan). 0377-2217/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.ejor.2012.06.005
international trade financing may reach USD5–10 billion by 2011 in China. On the one hand, the increasing demand for supply chain finance will provide a great potential for the development of related services and products. On the other hand, the risk bore by supply chain finance has been restricting the development itself. According to ‘‘National Income Accounts and Statistical Yearbook’’ (People’s Bank of China, 2006), the current inventory of all enterprises amount to 5.1294 trillion Yuan in China, 3.0326 trillion Yuan of which are from small and medium enterprises, and 102.4 billion Yuan are from farmers. If the discount rate for loans is 50%, these financial assets can generate secured loans of about 2.6 trillion Yuan, which equals to new loans of financial institutions in 1 year. However, due to the lack of risk management techniques for inventory financing, most banks did not fully make use of these inventory resources. In inventory financing, inventory are used as the pledges for risk averse and in this sense, the pledged inventory are evaluated to find whether its value can maintain its capability of guarantee, which is referred to as impawn rate (usually called loan-to-value ratio in other literatures). In setting the impawn rate of inventory pledge loan, the impawn rate of inventory and the interest rate of loans should be considered since these two indexes help to reflect and manage guarantee capability of inventory. However, in current domestic financial market, the risk averse is mainly ensured through adjusting the impawn rate of inventory rate rather than interest rate. What is more, in practice, ‘‘management approach from supply chain financing activities’’ (The following simply defined as ‘‘management approach’’) stipulates that the
H. Juan et al. / European Journal of Operational Research 223 (2012) 106–115
impawn rate shall not exceed the highest level of 70% and the impawn period shall not exceed 1 year. Obviously, current banking still sets impawn rate based on their experiences and the impawn rate obtained in this way would not accord with the risk exposure of banks. Therefore, it is important to set impawn rate not only for the risk averse of supply chain finance but also for the development of this business. Recently, the domestic and foreign scholars have made many beneficial explorations on volatility and risk management of pledges. Cossin and Huang (2003) set enterprises’ probability of default through the simplified method and then obtained the impawn rate according with the risk exposure of banks. Jokivuolle and Peura (2003), in their works, obtain enterprises’ probability of default through the structured method and established the relationship between the loan loss and the impawn rate. Buzacott and Zhang (2004) analyze how the setting of the interest rate and the impawn rate influence the enterprises profitability. Caldentey and Haugh (2009) investigate the supply contracts with financial hedging, and they find that the producer always prefers the flexible contract with hedging to the flexible contract without hedging, while the retailer might or might not prefer the flexible contract with hedging depending on model parameters. Lee and Rhee (2011) explicitly assume positive costs of inventory financing in a supply chain,and they show that a supplier can make its retailer work for the entire supply chain with trade-credit and derive the optimal risk premium in the trade-credit for supply chain coordination. Chen and Cai (2011) study the value of 3PL firms in budget-constrained supply chains, and found that all players can be better off under 3PL financing than under bank financing. As far as the domestic researches concerned, given that the price of the pledged inventory is fluctuant randomly, Li et al. (2007) implement risk estimation strategies of ‘‘main body + debt’’ and analyzed loan-to-value ratio decision of banks with downside risk constraint when price distribution of the stock at the end of the loan follows general distribution and several special distributions. On the whole, the results of these researches have played positive and practical role in helping us understand and capture the actual volatility and risk level of pledged inventory. However, the quantitative models above are all based on the mathematical optimization method and take the expected revenue of banks as the objective function. In other words, theoretical modeling and the cases based on individual samples are many while the empirical researches based on large number of samples are scarce. With the rapid development of modern financial risk management technology, the domestic and overseas researchers made considerable progress in using risk management tools to manage inventory financing. Among these tools, VaR (Value at Risk) has been widely used in academia and in practice to measure and manage risk exposure since it is proposed by Morgan in the 90th of last century (Jorion, 2001). Although VaR is able to show the expected value loss of the financial assets over a period of time and at a given confidence level, the results usually are not precise because the most commonly used parametric method of calculating VaR depends on both the probability distribution and the volatility of the returns. Many empirical researches show that the returns of asset usually have leptokurtosis, fat-tails and volatility clustering. However, the traditional way to obtain VaR is based on the assumption that the returns are independent in an efficient financial market, missing the fat tails and the volatility clustering of returns. Whereas, GARCH model, extended by Bollerslev (1986) can effectively describe the clustering and the heteroskedasticity of returns. For this reason, GARCH model is introduced in the financial risk management field, and widely used to predict volatility in financial practice. For instance, Ricardo (2003) uses GARCH model to predict volatility with fat tails to deal with possible maximum loss. Chinese scholars like Gong and Chen (2005) describe the
107
fat-tails, the volatility clustering of financial time series, constructing the GARCH models to calculate time-varying VaR based on the volatility and distributions of returns of stock index and copper futures in China. However, VaR-GARCH model is mainly used to study stock index, bonds and futures of commodities, which usually have effective risk control measures, such as margin system and price limits. In addition, since they have good liquidity and short settlement time, VaR-GARCH is used to forecast their short-term risk of two weeks, especially daily risk. On the contrary, the pledges which are usually traded in the spot market lack the risk control measures and have low liquidity and long settlement time; therefore, the key to inventory financing is to predict the long-term risk, in other words, to predict VaR of N months later based on past samples (Juan, 2009). Most of current researches on VaR are based on the assumption that in an efficient financial market, the return is independent and therefore they mainly use square-root rule of Risk Metrics to forecast VaR. Anderson argued that the long-term conditional variance equals to the sum of daily conditional variances; he also based his argument on the same assumption. Whereas, a large amount of empirical researches show that in an efficient financial security market, the return could be independent while it is auto-correlative in a pledged inventory market. That means, in the market where the spot transaction predominate, the long-term risk is usually higher that the sum of daily risk. Therefore, in this paper, we proposes the way of setting the dynamic impawn rate by dividing the impawn periods into different risk windows according to the macroeconomic environment, the credit level of counterparty, the liquidity of pledged inventory and the risk preference of banks. Instead of using Risk Metrics, we establishes a model with the formula AR (1)-GARCH (1,1)-GED so as to better describe the auto-correlation, the heteroscedastic and the leptokurtosis of returns, providing long-term VaR to deal with the problem between data frequency and forecast frequency. Since the out-of-sample method is more accurate in predicting volatility, we forecast the VaR of steel during the different risk windows in the impawn period through an out-of-sample method. Besides, in this paper, we set the parameter K to improve risk coverage and we establish the Hit sequences based on the failure rate of back testing to ensure the reliability of the research. The remainder of this paper is organized as follows: Section 2 is model set-up, including the setting of AR (1)-GARCH model under auto-correlative return and the model for setting the dynamic impawn rate; Section 3 presents empirical analysis, including the sample selection, the data characteristic description and the detailed result analysis; in Section 4, the models are evaluated and tested. In conclusion, discussion is given with further research areas.
2. Model assumption and set-up 2.1. Model assumption Since the price of pledged inventory, like financial assets, fluctuates, we follow the common international practice: banks must have tools and methods to timely evaluate the value of pledged inventory. Thus, we set up models based on the following assumptions. (1) Logistics enterprises closely cooperate with banks. (2) The impawn rate varies under different risk windows; considering the macroeconomic environment, the credit level of counterparty, the liquidity of pledged inventory and the risk preference of banks, banks choose different risk window.
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(3) Considering that the inventory financing is short-term financing practice (less than 1 year), we assume that the loan interest rate does not change during the impawn periods.
tion (GED) for better approximating the fat-tails of the innovations. For this reason, we assume zt follows GED. The density function of GED is,
f ðx; v Þ ¼ 2.2. Model set up Considering that the returns of the pledged inventory have autocorrelation, heteroscedastic and leptokurtosis and fat-tails, we use the formula AR (1)-GARCH (1,1)-GED to estimate the volatility. Furthermore, based on VaR, we calculate the long-term price risk, which indicates the maximum possible loss over a predefined time horizon at certain level of confidence established previously. The risk-free value of pledged inventory, which can be obtained by subtracting the VaR, is also the amount of loan. Thus, we organize the steps as follows: 2.2.1. Yield rate of pledged inventory
Rt ¼ ln Pt ln Pt1
ð1Þ
where Rt is defined as the logarithmic return, Pt is the price at time t. 2.2.2. Volatility of yield rate of pledged inventory In financial studies, the volatility of financial assets is the standard deviation of the return rate. Likewise, the volatility of pledged inventory is the standard deviation of its return. Recent empirical studies indicated that financial assets have volatility clustering, which contribute to the heteroscedasticity of the return rate. For this reason, GARCH model was used so as to better describe this trait above. Besides, current researches show that GARCH (1,1) model might describe most of the time-varying variance of financial series. Therefore, we use the GARCH (1,1) to forecast the volatility of inventory, establishing the conditional mean equation and conditional variance equation as follows:
Rt ¼ lt þ et ¼ lt þ rt zt ; zt i:i:d:; Eðzt Þ ¼ 0; Varðzt Þ ¼ 1
r2t ¼ a0 þ a1 e2t1 þ b1 r2t1
ð2Þ ð3Þ
where lt is the conditional mean. AR (p) or ARMA (p,q) process is usually introduced to capture the autocorrelation of returns.Moreover,AR (1) process is proved to be a simple and valid model to describe the autocorrelation of return (Mcneil and Frey, 2000). Thus, we assume lt = qRt1. et rtzt is the stochastic disturbing term, also called residual term, which is a white noise process.In this formula,zt is the Innovations;r2t is the conditional variance of et at time t; a0 is constant; a1 is the ARCH parameter, b1 is the GARCH parameter,and a0 > 0,a1 > 0,b1 > 0. The model is stationary on condition that a1 + b1 < 1,jqj < 1.The unconditional variance of et can be calcua0 lated as V L ¼ r21 ¼ 1aa10b1 ¼ 1k wherek = a1 + b1.In most current financial literatures, the conditional variance and unconditional variance of et are used to describe the volatility of returns since log-returns are assumed as independent to each other,where q = 0, Rt = et = rtzt,Rt is a standard GARCH (1,1) process (Bollerslev, 1986).However,it is not valid in inventory financing,since the logreturns of pledges are autocorrelative (q – 0). As mentioned above,Rt follows AR (1)-GARCH (1,1) process, the unconditional vara0 eL ¼ e L equals VL.1 iance of Rt is that V , if q = 0, the V ð1q2 Þð1kÞ In the current practice, the Innovation zt is usually assumed to follow normal distribution. It usually has tails in practice. The student distribution was firstly introduced by Bollerslev (1987), while Nelson (1991) suggested the so-called Generalized Error DistribuThe proof of unconditional Variance of et andRt is presented in Appendix A.
c2
ð4Þ
where c = [2(2/v)C(1/v)/C(3/v)]1/2, when v = 2, zt follows normal distribution, for v < 2, the distribution of zt has fatter tails than normal distribution. 2.2.3. VaR and impawn rate Considering the volatility of pledged inventory price and the time period from risk identification to risk treatment, to calculate VaR in inventory financing is to forecast the long-term risk rather than the daily ones, which not only meets the demand of emerging business (supply chain finance, etc.) but also the regulations of Basel Accord II or even the latest version Basel Accord III that banks must report longer term VaR to supervision institutions. Unfortunately, as mentioned in the introduction, current researches are mainly concerned with the short-term risks, while missing the long-term risk. Besides, the best-known method to forecast long-termpffiffiffirisk is square-root rule with the formula VaRðTÞ ¼ VaRð1Þ T . This method is valid only when the return follows the independent normal distribution with a zero trend. Whereas, the return usually has leptokurtic and fat tails, so the scaling is no longer valid when applied to the evaluation of longterm risk in inventory financing. For this reason, in order to obtain more precise VaR, Dowd et al. (2004) revised square-root rule as follows:
h pffiffiffii VaRðTÞ ¼ Pt 1 exp lt T þ F 1 a rt T
ð5Þ
where Pt is the initial value of unit pledged inventory (for convenience, this paper denotes the unit price as the initial value). F 1 a denotes the left quantile at the certain level of confidence. Compared with the square root rule, the revised model has made much improvement and is able to avoid day-to-day prediction of volatility. However, it still relies on the square root rule more or less. In order to improve the two approaches above, Jorion (2001); Tsay (2005); and Andersen et al. (2006) argue that the conditional variance of the long time horizon equals to the sum of daily conditional variances under the hypothesis that in an efficient financial market, the return is independent (q = 0).2
l½t þ 1 : t þ T ¼
T X Rtþi
ð6Þ
i¼1
r2 ½t þ 1 : t þ T ¼ TV L þ
1 kT 2 rtþ1 V L 1k
ð7Þ
As mentioned in the introduction, the pledged inventory that is usually traded in spot market has low liquidity, and therefore the return is auto-correlative. In this sense, long-term risk is usually higher than the sum of daily risk. Considering both the autocorrelation and the time-varying volatility, the AR (1)-GARCH (1,1) model was proposed to forecast the long-term conditional variance (Brummelhuis and Kaufmann, 2004; Kaufmann, 2004).3
r^ 2 ½t þ 1 : t þ T ¼
2T 1 qT 2 1q V T 2 q þ q L 1q 1 q2 ð1 qÞ2 1
1 kT qT kT 2 q2T kT þ r2tþ1 V L 2q þq 1k qk q2 k
!#
Hence, the formula (5) can be revised in the following way, 2
1
1 exp jx=cjv 2 Cð1=v Þ
v ð1þ1=v Þ
3
The proof of formula (7) is presented in Appendix B. The proof of formula (8) is presented in Appendix C.
ð8Þ
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H. Juan et al. / European Journal of Operational Research 223 (2012) 106–115 5,600
Pt
5,200 4,800
SK
Margin call, Replenishment
SV
4,400 4,000 3,600
T
3,200
Fig. 1. The mechanism design of setting impawn rate.
h
VaRðTÞ ¼ P t 1 exp
1
l½t þ 1 : t þ T þ F a r^ ½t þ 1 : t þ T
i
2,800 2006
In practice, it is necessary for a commercial bank to set appropriate risk windows (risk holding period) to measure risk and to control risk (i.e. price risk) of inventory financing by setting impawn rate dynamically. It is well known that the risk window and the confidence level are the two key parameters to calculate VaR. Risk window T is defined as settlement time, which ideally is equal to the maximum since in banking practice, most of assets are monetary assets of good liquidity and therefore these areas are mainly concerned with daily VaR. As for the risk window T of inventory pledges, theoretically they are defined according to the liquidity of supply chain financial market, the sample size and the position adjustment of pledged assets; while, in practice, risk window T of inventory pledges can be defined according to its liquidity, the level of counterparty and some financial indicators such as the level of solvency and profitability. In addition, the level of confidence is set as 99% according to the recommendation of internal model of market risk measurement of commercial bank which was issued by China Banking Regulatory Commission. After setting the risk window and the confidence level based on theoretical analysis as well as practical study, we calculate the VaR during the risk window, and obtain the risk-free value which is the amount of loan as mentioned above. The ratio between the riskfree value and current price of pledged inventory is the impawn rate.
x¼
Pt VaR Sk 100% ¼ 100% Pt Pt
2007
2008
2009
ð9Þ
ð10Þ
where x is defined as impawn rate. Obviously, SK denotes the riskfree value. According to the Basel Accord (both Basels I and III), banks are suggested to predict market risk through Internal Model Approach (IMA). In practice, it is proved that through IMA, the VaR at the confidence level of 99%, especially in extreme markets, cannot cover the risk perfectly, and therefore multiplicative factor K is introduced to improve its risk coverage. This paper is written based on the principle. Although the AR (1)-GARCH (1,1)-GED model can describe the autocorrelation, the leptokurtic and the fat tailed and the volatility clustering of log-returns to a certain extent in this paper, the risk could be underestimated. Based on these factors, considering the liquidity of pledged inventory, the credit level of counterparty and the cost of replenishment and the closed position, we introduce the warning level and parameter K to further mitigate the risk of banks. As can be seen from Fig. 1, during the risk window T, when the price of steel falls below the warning level, margin call or replenishment is done till the value of steel regresses above the warning line. For convenience, we assume that closed position would be done if companies refuse to do that in time. In order to improve the risk coverage, we set the correction factor K = 1.1–1.2 (according to Basel accord in principle, which references the guidelines of 4 We only provide the necessity of the introduction of parameter K in this paper, the optimization problem of K is the separate subject in our future research.
Fig. 2. Time series of the steel rebar price in Shanghai. Source: http://www.96369.org/ Quotation/HistoryQuotationN.aspx?AreaID=1.
stock collateral issued by China Securities Regulatory Commission, 2004).4 In banking practice, for reduction of adverse selection and avoiding moral hazard to some extent, banks should set K for different pledges according to the macroeconomic environment, the credit level of counterparty, the liquidity of pledged inventory. In the empirical analysis followed, we assume previously K = 1.1, the modified model will be:
x¼
Pt VaR 1 SK 1=K SV 100% ¼ 100% ¼ 100% Pt K Pt Pt
ð11Þ
where SV is defined as the amount of pledged loan. 3. Empirical analysis 3.1. Sample selecting and descriptive statistics 3.1.1. Sample selecting In this section, in order to evaluate the models, we use the steel rebar (u HRB335) which are widely used in the industries of real estate and infrastructure industries as samples.5 The data set is obtained from XiBen new line stock and ShangHai futures exchange from the period of September 5th, 2005 to December 31st of 2009 (shown in Fig. 2). The data from September 5th of 2005 to December 31st of 2008 will be used be to evaluate the parameters; the rest will be used as test samples.6 Then we carry on a series of simulated pledge with the impawn contract’s starting date as January 1st of 2009 and set the impawn period to be the maximum of 12 months. Obviously, it is important to trade off the impawn period and the risk holding period. Generally speaking, the longer the risk holding period is, the more radical the banks are; the shorter the holding period is, the more conservative the banks are. According to normal banking practice, we set the risk holding period respectively as 1 week, 2 weeks, 1 month, 2 months, 3 months, 4 months, 5 months, 6 months, 7 months, 8 months, 9 months, 10 months, 11 months, 12 months.7 5 Steel Rebar has been transacted officially at the Shanghai futures exchange on the March 27, 2009. 6 In 2009, in response to the international financial crisis, China government launched packages of plans and policy measures ‘‘guaranteed growth, expand domestic demand, readjusting structure and benefit people’s livelihood’’. Under the action of above policy, China economy gets rid of financial crisis’s puzzle gradually, GDP restores by the season, promote the steel products to expend, stimulates the steel and iron good immovable property to be able to release, the steel enterprises turn losses into profits gradually, steel industry recovers slowly. However, the high yield, low exports, high inventory lead to a situation of excess domestic supply. The interaction of steel futures, electronic and the spot market, the price of domestic steel market performs ups and downs of the trend. Therefore this article analyzes time series in 2009, which contains two different stages, relatively stable and the volatility of the financial markets around the world, so our model plays the role of good testing. 7 Regarding to the risk windows, banks may definitely establish them according to own need, this paper only provides a model of dynamic management risk.
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H. Juan et al. / European Journal of Operational Research 223 (2012) 106–115 .12
R .08
.04
.00
-.04
-.08 06M01
06M07
07M01
07M07
08M01
08M07
Fig. 3. The linear graph of log-return of steel rebar in Shanghai.
Table 2 The results of ARCH effect test.
Table 1 The results of ADF test of log-returns.
Augmented Dickey–Fuller test statistic Test critical values: 1% Level 5% Level 10% Level
t-Statistic
Prob.⁄
Heteroskedasticity test: ARCH
13.82975 3.437695 2.864672 2.568491
0.0000
F-statistic Obs⁄R-squared
3.1.2. Descriptive statistics of log-returns of steel rebar price As shown in Fig. 3, the log-return series show significant volatility clustering which indicates that ARCH effect may exist in log-returns. Since GARCH model can be established only when the time series is stationary, we carry on Augmented Dickey–Fuller test (i.e. ADF unit root test) the time series. As shown in Table 1, in all cases, the null hypothesis is rejected at any level of significance (1%, 5%, 10%), so the time series of daily log-return is stationary. Fig. 4 indicates that the autocorrelation and the partial correlation are significant in some cases, especially in first order. It is reasonable to denote Rt = qRt1 + et. As can be shown in Table 2, the value is significant, which indicates that the ARCH effect exists in the residuals of daily logreturn. Fig. 5 shows that in almost all cases, the null hypothesis of normality is rejected at any level of significance, as there is evidence of significant excess kurtosis and positive skewness. 3.2. Empirical results As presented in Table 3, the constant term of conditional mean is 1.87E07, which is close to zero while not significant in all cases.
31.55798 134.1414
Prob. F (5, 855) Prob. chi-square (5)
0.0000 0.0000
The parameter q which considers the autocorrelation is significant. In addition, the value of AIC and SC is reasonable. Therefore, the model of AR (1)-GARCH (1,1)-GED is acceptable. Thus, the conditional mean and conditional variance can be rewritten as follows:
Rt ¼ 0:1440Rt1 þ et 2 t
ð12Þ 2 t1
r ¼ 3:40E 06 þ 0:1115e
2 t1
þ 0:7452r
ð13Þ
From Eq. (13), we can obtain a1 + b1 = 0.86 < 1, which denotes that the time series of daily log-return is stationary. Thus, the long-term variance of et can be calculated through the following formula: V L ¼ 1aa10b1 ¼ 2:3724E 05. The parameter v is 0.853 obtained through Eviews. The corresponding quantile on left is 2.87, when the confidence level is 99%. Considering all these above, we get the empirical results as in Table 4. 4. Model evaluation 4.1. Back testing of long-term risk In order to test the accuracy of the AR (1)-GARCH (1,1)-GED model under autocorrelation with heavy tails, we have to test the risk coverage level of VaR. The most widely used models for back testing of VaR are the failure rate method (see Kupiec, 1995) and the internal back-testing model of Basel accord. Both
Fig. 4. The results of autocorrelation test.
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700
Series: R Sample 9/05/2005 12/31/2008 Observations 867
600 500
Mean 0.000129 Median 0.000000 Maximum 0.113759 Minimum -0.070068 Std. Dev. 0.009075 1.038078 Skewness Kurtosis 42.97649
400 300 200
Jarque-Bera Probability
100
57887.78 0.000000
0 -0.05
0.00
0.05
0.10
Fig. 5. The results of normal assumption test.
Table 3 Parameters estimates of AR (1)-GARCH (1,1)-GED model. c
q
a0
a1
b1
v
AIC/SC
1.87E07 (0.9979)
0.1440 (0.0000)
3.40E06 (0.0000)
0.1115 (0.0000)
0.7452 (0.0000)
0.853 (0.0000)
8.4295 8.3965
Therefore, the Hit function is established to test the accuracy of long term price risk forecasting. Put another way, we have to observe the exceptions in which the expected price of inventory (risk-free value) is higher than the actual price.
Hit ¼
of the two models are applied to examine the exceptions in which the actual loss is greater than the daily VaR. For instance, when the time horizon is 1 year and the confidence level is 99%, the number of exceptions of less than seven times is accepted by the failure rate method while Basel II internal green light area only accepts the number of less than four times. However, both of these two methods are not suitable for back testing of long-term risk in inventory financing.
1; P tþi < Pt VaR
ð14Þ
0; or else
where Sk = Pt VaR, which denotes the expected price (risk-free value, the warning level in this paper). f is the observed number of exceptions in the sample. f/N should be close to 1 a statistically, if the bias is too large or too small, it means that the model cannot forecast the price risk correctly. In the following, comparisons are made between the Risk Metrics and the GARCH (1,1)-GED model and the results are presented as follows: The main results (see Tables 5–7) indicate that the AR (1)GARCH (1,1)-GED model outperforms both the Risk Metrics
Table 4 The empirical results of different risk windows of 12 months impawn period. Risk window 1 week 2 weeks 1 month 2 months 3 months 4 months 5 months 6 months 7 months 8 months 9 months 10 months 11 months 12 months Size N Pt VaR SK SV
x
5 3580 120 3460 3145 0.88
12 3580 191 3389 3081 0.86
23 3580 264 3316 3015 0.84
43 3580 358 3222 2929 0.82
65 3580 437 3143 2858 0.80
87 3580 501 3079 2799 0.78
108 3580 554 3026 2751 0.77
130 3580 604 2976 2706 0.76
153 3580 650 2930 2663 0.74
174 3580 689 2891 2628 0.73
196 3580 727 2853 2593 0.72
218 3580 763 2817 2561 0.72
239 3580 794 2786 2532 0.71
261 3580 827 2753 2502 0.70
Table 5 The results of the Risk Metrics (square root rule) during 12-month impawn period. Risk window 1 week 2 weeks 1 month 2 months 3 months 4 months 5 months 6 months 7 months 8 months 9 months 10 months 11 months 12 months Size N Pt VaR SK f f/N
5 3580 97 3483 0 0
12 3580 150 3430 0 0
23 3580 208 3371 0 0
43 3580 285 3295 0 0
65 3580 350 3230 10 15.4%
87 3580 405 3175 15 17.4%
108 3580 451 3128 0 0
130 3580 495 3085 0 0
153 3580 537 3043 0 0
174 3580 573 3007 0 0
196 3580 608 2972 0 0
218 3580 641 2939 0 0
239 3580 671 2909 0 0
261 3580 701 2879 0 0
Table 6 Results of the GARCH (1,1)-GED model during 12-month impawn period. Risk window 1 week 2 weeks 1 month 2 months 3 months 4 months 5 months 6 months 7 months 8 months 9 months 10 months 11 months 12 months Size N Pt VaR SK f f/N
5 3580 106 3474 0 0
12 3580 165 3415 0 0
23 3580 228 3352 0 0
43 3580 309 3271 0 0
65 3580 377 3203 8 12.3%
87 3580 433 3147 3 3.5%
108 3580 480 3100 0 0
130 3580 523 3057 0 0
153 3580 564 3016 0 0
Note: Due to the page limit, the details of calculating VaR in Tables 5 and 6 is not shown in this paper.
174 3580 599 2981 0 0
196 3580 632 2948 0 0
218 3580 664 2916 0 0
239 3580 692 2888 0 0
261 3580 721 2859 0 0
112
H. Juan et al. / European Journal of Operational Research 223 (2012) 106–115
Table 7 Results of the AR (1)-GARCH (1,1)-GED model during 12-month impawn period. Risk window 1 week 2 weeks 1 month 2 months 3 months 4 months 5 months 6 months 7 months 8 months 9 months 10 months 11 months 12 months Size N Pt VaR SK f f/N SV f1 f1/N
5 3580 120 3460 0 0 3145 0 0
12 3580 191 3389 0 0 3081 0 0
23 3580 264 3316 0 0 3015 0 0
43 3580 358 3222 0 0 2929 0 0
65 3580 437 3143 3 4.6% 2858 0 0
87 3580 501 3079 0 0 2799 0 0
108 3580 554 3026 0 0 2751 0 0
130 3580 604 2976 0 0 2706 0 0
153 3580 650 2930 0 0 2663 0 0
174 3580 689 2891 0 0 2628 0 0
196 3580 727 2853 0 0 2593 0 0
218 3580 763 2817 0 0 2561 0 0
239 3580 794 2786 0 0 2532 0 0
261 3580 827 2753 0 0 2502 0 0
Table 8 The impawn rate and the corresponding lowest value under different risk windows during 12-month impawn period. Risk window 1 week 2 weeks 1 month 2 months 3 months 4 months 5 months 6 months 7 months 8 months 9 months 10 months 11 months 12 months PL,T
3580 0.88
x
3580 0.86
3580 0.84
3330 0.82
3140 0.80
3140 0.78
3140 0.77
x
PL,T
x
1.000000 0.869665
0.869665 1.000000
PL,T
3140 0.74
3140 0.73
3140 0.72
3140 0.72
3140 0.71
3140 0.70
words, the closer the coefficient of correlation is to 1, the more effective the model is. Table 8 presents the lowest price of steel rebar in different risk windows and corresponding impawn rate during both 6 months and 12 months time horizon, the coefficient of correlation is 0.87, which is close to 1(see Table 9). This shows that the corrected model is efficient.
Table 9 The correlation metrics of x and PL,T. 12 months
3140 0.76
and the GARCH (1,1) models in the accuracy of long-term price risk forecasting. Unfortunately, for AR (1)-GARCH (1,1)-GED model, the failure rates in the 3 months is 4.6%. In both two impawn periods, the failure rates are beyond 1% (corresponding to 99% confidence level). This indicates that the price risk is not covered completely by AR (1)-GARCH-GED model. Therefore, we have to test whether the price series Pt puncture the loan value SV which is corrected by the parameter K. Table 7 presents that the exceptions did not happen. In this sense, the revised model improved the risk coverage level significantly.
4.3. The analysis of efficiency of model As mentioned in the introduction, according to the experience method, the impawn rate is almost lower than 70%. This method may control risk while cause higher efficiency loss. In the following, we will compare the model of setting the impawn rate in this paper with the experience method, and then we introduce two indicators: efficiency loss h1, risk rate h2. To facilitate the process, we select the upper limit (70%) of impawn rate based on banks’ experience.
PtþT SV 100% Pt SV 100% h2 ¼ PtþT
h1 ¼ 4.2. Testing between the impawn rate and lowest value of the steel rebar The impawn rate reflects the banks’ expectation of lowest price of pledges in the future risk windows. Thus, statistically speaking, the impawn rate obtained by an effective model should positively correlate with the lowest value, although it is almost impossible for a certain model to forecast it due to various factors. In other
ð14Þ ð15Þ
where Pt+rT is defined as the price at period-end of risk window; h2 < 1 shows that the risk is under control. As can be seen from Tables 10 and 11 and Fig. 6, the two indicators negatively correlate with each other; it means that the risk is significantly reduced while the efficiency loss is minimized.
Table 10 Analysis of the efficiency and the risk based on the experience of 12-month impawn period. Risk window
1 week (%)
2 weeks 1 month 2 months 3 months 4 months 5 months 6 months 7 months 8 months 9 months 10 months 11 months 12 months (%) (%) (%) (%) (%) (%) (%) (%) (%) (%) (%) (%) (%)
h1 h2
33 68
33 68
36 66
23 75
21 77
24 75
27 72
36 66
49 59
31 69
27 72
27 72
30 70
34 67
Table 11 Analysis of the efficiency and the risk in AR (1)-GARCH (1,1)-GED model. Risk window
1 week (%)
2 weeks 1 month 2 months 3 months 4 months 5 months 6 months 7 months 8 months 9 months 10 months 11 months 12 months (%) (%) (%) (%) (%) (%) (%) (%) (%) (%) (%) (%) (%)
h1 h2
15 85
17 83
22 79
11 88
11 88
15 84
21 79
31 71
44 63
28 72
25 75
26 74
30 71
35 67
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H. Juan et al. / European Journal of Operational Research 223 (2012) 106–115
100%
efficiency loss of experience risk rate of experience efficiency loss of model risk rate of model
80% 60% 40% 20% 0%
Fig. 6. Comparison between AR (1)-GARCH (1,1)-GED model and experience method during 12 months impawn period.
Additionally, the risk rates according to both two methods are less than 1, so the risk is under control. As for the efficiency loss, it is still great according to the experience method, even if we set the impawn rate of upper limit (70%). Although a certain degree of efficiency loss still exists in the model used in this paper, it has been greatly improved compared with the experience method.
5. Conclusions Considering the long-term risk window of inventory pledges due to its insufficient liquidity, we established the formula of AR (1)-GARCH (1,1)-GED to better describe the autocorrelation, the fat-tails and the volatility clustering of return, forecasting the long-term price risk; furthermore, considering the macroeconomic environment, the credit level of counterparty, the liquidity of pledged inventory and the risk preference of banks, we set dynamic impawn rate by dividing the impawn period into different risk windows. In the end, the conclusions are arrived as follows: First, the time series of return of rebar price in Shanghai have significant autocorrelation, fat-tails, and volatility clustering. Compared with other models such as the Risk Metrics and those revised models, the AR (1)-GARCH (1,1)-GED model have more accuracy in forecasting long-term VaR. However, it is important to note that the failure rate under 3 month risk window is higher than what can be accepted at the confidence level, which indicates that none of the models is able to forecast the long-term risk perfectly, even when considering the autocorrelation, the fat-tails and the volatility clustering. Therefore, when banks and supervision institutions use mature and sound VaR models to forecast risk, they should also consider the macroeconomic environment and the liquidity of pledged inventory. For the reason, parameter K was introduced into this paper. The results (in Table 7) show that, the risk coverage level has been improved remarkably via K. In addition, banks can get the upper limit of K through stress tests. For instance, stress test can be carried onto get upper limit of K regarding the price plunge of steel in 2008 in order to deal with extreme cases. Second, the impawn rate obtained from the model significantly positively correlate with the low price in the future risk window. Hence, this model could reasonably reflect the banks’ expectation about the risk of pledged steel. Moreover, compared with the empirical method widely used in banking, the model used in this paper can effectively control the risk, while have the higher financing efficiency. In this sense, it may better enhance the attraction of inventory financing by reducing the adverse selection and moral hazard to some extent. Finally, it is also important to keep in mind that, different from financial assets, the pledged inventories have autocorrelation, fattails and so on. Substantial empirical works are left for future re-
search to capture the unique characteristics of pledged inventories and determine the K and the T of the inventory financing to improve the performance of risk forecasting. Furthermore, with the Copula theory better describing the dependence structure between two or more random variables, the pledged inventory can be expanded from single commodity to portfolio which includes no less than two kinds of commodities. Undoubtedly, this will cater to the trend of supply chain financing in China. The last but not least, we will introduce the real option theory and stochastic differential equation (SDE) to study the mechanism for setting impawn rate in a novel viewpoint rather than VaR. The specific work is still left for future research.
Acknowledgments This paper is supported by the NSFC (Natural Science Foundation of China) (71003082), the Fund Projects for Doctoral Degree of The National Ministry of Education of China (200806131007), the Scientific Fund Projects for National Post Doctors of China (20080430602) and Project 985 Platform School of Economics of Fudan University and is the result of her Postdoctoral research in Institute of Financial Studies of Fudan University. Acknowledgement goes to Professor Chen Xuebin of Institute of Financial Study of Fudan University, Professor Zhang Jinqing for his suggestion on annual statistics and model revision, and Luo Cheng in Shanghai Futures Exchange for his statistical support. The authors would like to thank the anonymous referees for their helpful comments and suggestions, which improved the contents and composition substantially.
Appendix A. Proof of the unconditional variance of et and Rt
Rt ¼ qRt1 þ et
ðA:1Þ
et ¼ rt zt ; zt i:i:d:; Eðzt Þ ¼ 0; Varðzt Þ ¼ 1
ðA:2Þ
r2t ¼ a0 þ a1 e2t1 þ b1 r2t1
et follows a GARCH (1,1) process, if t ? 1, the Eðe2t1 Þ ¼ r2t ¼
r2t1 ¼ r21 . Thus, the unconditional a0 . V L ¼ r21 ¼ 1aa10b1 ¼ 1k
variance
of
et
equals
Rt follows AR (1)-GARCH (1,1) process, using (A.1) recursively leads to,
Rt ¼ qRt1 þ et ¼ qðqRt2 þ et1 Þ þ et ¼ qnþ1 Rtn1 þ et þ qet1 þ þ qn etn If jqj < 1,limn?1qn = 0, which yields
ðA:3Þ
114
H. Juan et al. / European Journal of Operational Research 223 (2012) 106–115
Rt ¼ et þ qet1 þ q2 et2 þ ;
ðA:4Þ
Thus, the unconditional variance of Rt equals
2
e L ¼ E½Rt EðRt Þ2 V 2 ¼ E et þ qet1 þ q2 e2t2 þ 2
2
^2
r
½t þ 1 : t þ T ¼ E4
T X Rtþi
!2
3
" #!2 T X 5 jltþ1 ; rtþ1 E Rtþi jltþ1 ; rtþ1
i¼1 2
i¼1
ðA:12Þ
ðA:5Þ
¼ E½et þ q2 E½et1 þ q4 E½et2 þ 2
In the AR (1)-GARCH (1,1) model, the daily log-returns exist first-order autocorrelation (q – q), the T-days conditional variance of Rt is:
4
¼ ð1 þ q þ q þ ÞV L
Using recursively the conditional mean equation, we get
" # " # T T T Ti X X X X i1 s E Rtþi jltþ1 ; rtþ1 ¼ ltþ1 q þ E ri zi q jltþ1 ; rtþ1
0 ¼ 11q2 V L ¼ ð1q2aÞð1kÞ
e L reduces again to VL. If q = 0, the V
i¼1
i¼1
s¼0
i¼1
T
q ¼ ltþ1 1 1q
Appendix B
ðA:13Þ
B.1. Proof of the formula (7)
Moreover 2 3 2 !2 T X E4 Rtþi jltþ1 ; rtþ1 5 ¼ E4
T
r
2
1k 2 ½t þ 1 : t þ T ¼ TV L þ rtþ1 V L 1k
2
r
T X Rtþi
½t þ 1 : t þ T ¼ E4
3
!2
jrtþ1 5 E
¼
r^ 2 ½t þ 1 : t þ T ¼ ¼
ðA:7Þ
h
r2t jr1 ¼ E a0 þ a1 ðrt1 zt1 Þ2 þ b1 r2t1 jr1
ðA:8Þ
T 2 X 1qTiþ1 E r2tþi jrtþ1 1q
T 2 X 1qTiþ1 E r2tþi jrtþ1 1q
i¼1
ðA:9Þ
Hence
r2tþ1 V L
r2tþ1 V L
i
ðA:15Þ
T T X r2tþ1 V L X ð1 qTiþ1 Þ2 þ ð1 ð1 qTiþ1 Þ2 ki1 qÞ2 i¼1
i¼1
ðA:16Þ
i¼1 1qT 1q
2T 2 1q 1q2
þq
ðA:17Þ
i¼1 T
ðA:10Þ
T T h i X X E R2tþi jrtþ1 ¼ TV L þ ki1 r2tþ1 V L
T
T
2T
T
¼ 1k 2q qqk þ q2 qq2 k 1k k k
which yields
T
V L þ ki1
T T X X ð1 qTiþ1 Þ2 ki1 ¼ ð1 2qTiþ1 þ q2T2iþ2 Þki1 i¼1
h i E R2tþi jrtþ1 ¼ E r2tþi jrtþ1 ¼ V L þ ða1 þ b1 Þi1 r2tþ1 V L ¼ V L þ ki1 r2tþ1 V L
i¼1
2 h
¼ T 2q
¼ V L þ ða1 þ b1 Þt1 ðr21 V L Þ
¼ TV L þ 1k 1k
1qTiþ1 1q
T T X X ð1 qTiþ1 Þ2 ¼ ð1 2qTiþ1 þ q2T2iþ2 Þ
1 ða1 þ b1 Þt1 ¼ a0 þ þ ða1 þ b1 Þt1 r21 1 ða1 þ b1 Þ
r2 ½t þ 1 : t þ T ¼
jltþ1 ; rtþ1 5
If jqj < 1, we get
Using this formula recursively leads to,
r2t jr1
T X
¼ ð1V qL Þ2
i
¼ a0 þ ða1 þ b1 ÞE r2t1 jr1
E
q
i¼1
h i E R2tþi jrtþ1 ¼ E e2tþi jrtþ1 ¼ E r2tþi jrtþ1
þ
s¼0
r i zi
3
i¼1
and
2
i¼1
!2 s
Let (A.10), (A.13), (A.14) Substitute (A.12),
i¼1
ðA:6Þ
þ
Ti X
ðA:14Þ
i¼1
q
T X
i¼1
T h i X ¼ E R2tþi jrtþ1
E
T
q ltþ1 1 1q
" #!2 T X Rtþi jrtþ1
i¼1
i1
i¼1
In the standard GARCH (1,1) model, the log-returns is independent (q = 0, Rt = et), the T-days conditional variance of Rt is: 2
ltþ1
i¼1
T X
i¼1
ðA:11Þ
and while if Considering equations (A.16), (A.17), (A.15) can be written as 1 1 qT 1 q2T r^ 2 ½t þ 1 : t þ T ¼ V L T 2q þ q2 2 2 1q 1q ð1 qÞ !# 1 kT qT kT 2 q2T kT þ r2tþ1jt V L 2q þq ðA:18Þ 1k qk q2 k If q = 0, the formula (A.18) reduces again to (A.11).
Appendix C
References
C.1. Proof of the formula (8)
Andersen, T.G., Bollerslev, T., Christoffersen, P.F., Diebold, F.X., 2006. Volatility and correlation forecasting. In: Handbook of Economic Forecasting. North Holland Press, Amsterdam, pp. 798–814. Bollerslev, T., 1986. Generalized autoregressive conditional heteroscedasticity. Journal of Econometric 31 (3), 307–327. Bollerslev, T., 1987. A conditionally heteroscedastic time series model for speculative prices and rates of return. Review of Economics and Statistics 69 (3), 542–547. Brummelhuis, R., Kaufmann, R., 2004. Time Scaling for GARCH(1,1) and AR(1)GARCH(1,1) Process. Working paper, 1–48. Buzacott, J.A., Zhang, R.Q., 2004. Inventory management with asset-based financing. Management Science 50(24), 1274–1292.
r^ 2 ½t þ 1 : t þ T ¼
1 ð1 qÞ
2
1 qT 1 q2T V L T 2q þ q2 2 1q 1q
1 kT qT kT q2T kT þ r2tþ1 V L 2q þ q2 2 1k qk q k
!#
H. Juan et al. / European Journal of Operational Research 223 (2012) 106–115 Caldentey, R., Haugh, M.B., 2009. Supply contracts with financial hedging. Operations Research 57 (1), 47–65. Chen, X.F., Cai, G.S., 2011. Joint logistics and financial services by a 3PL firm. European Journal of Operational Research 214 (3), 579–587. Dowd, K., Blake, D., Cairns, A., 2004. Long-term value at risk. In: The Journal of Risk Finance. Winter/Spring, pp. 52–57. Feng, G.Z., 2007. Analysis of logistics financing business innovation in China. Forecasting 26 (1), 49–54. Gong, R., Chen, Z.C., 2005. To evaluate VaR of China stock marketing comparatively by using GARCH family model and comment. The Journal of Quantitative & Technical Economics 2005(7), 67–81. Jokivuolle, E., Peura, S., 2003. Incorporating collateral value uncertainty in loss given default estimates and loan-to-value ratios. European Financial Management 9 (3), 299–314. Jorion, P., 2001. Value at Risk. McGraw-Hill, New York. Juan, H., 2009. Comprehensive Evaluation and Measurement of Logistic Financing Risk. Fudan University, Shang Hai. Kaufmann, R., 2004. Long-Term Risk Management. ETH Zurich. Kupiec, P., 1995. Techniques for verifying the accuracy of risk measurement models. Journal of Derivative 32 (2), 173–184. Lee, C.H., Rhee, B.D., 2011. Trade credit for supply chain coordination. European Journal of Operational Research 214 (1), 136–146.
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Li, Y.X., Feng, G.Z., et al., 2007. Research on loan to value ratio of inventory financing under randomly fluctuant price. System Engineering Theory & Practice 12, 4248. Mcneil, A.J., Frey, R., 2000. Estimation of tail-related risk measures for heteroscedastic financial time series: an extreme value approach. Journal of Empirical Finance 7 (10), 271–300. Nelson, D.B., 1991. Conditional heteroscedasticity in asset returns: a new approach. Econometrica 59 (2), 347–370. Ricardo, A., 2003. The estimation of market VaR using GARCH models and a heavy tail distributions. Working Paper Series, 1–28. Shen Zhen Development Bank, C.E.I.B.S., 2009. Supply Chain Finance. Shang Hai Far East Press, Shang Hai. Tsay, R.S., 2005. Analysis of Financial Time Series, second ed. John Wiley & Sons, Inc.
Further reading Cossin, D., Huang, Z.J., Aunon, N.D., 2003. A frame work for collateral risk control determination. European central bank working paper series, 1–41. Liu, Q.F., Zhong, W.J., 2006. Market risk measurement of copper futures in China based on VaR GARCH models. Journal of Systems Engineering 8, 429–433. White, H., 2000. A reality check for data snooping. Econometrica 68 (2), 1097–1126.
Contents lists available at SciVerse ScienceDirect
European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor
Stochastics and Statistics
VaR methods for the dynamic impawn rate of steel in inventory financing under autocorrelative return He Juan a,⇑, Jiang Xianglin b, Wang Jian a, Zhu Daoli c, Zhen Lei d a
College of Traffic, Transportation and Logistics, Southwest Jiaotong University, Chengdu 610031, China Institute for Financial Studies, Fudan University, Shanghai 200433, China c School of Economics and Management, Tongji University, Shanghai 200092, China d Department of Credit Risk Management in Chengdu, Huaxia Bank 61000, China b
a r t i c l e
i n f o
Article history: Received 20 June 2011 Accepted 1 June 2012 Available online 13 June 2012 Keywords: Finance AR (1)-GARCH (1,1)-GED Long-term VaR forecasting Dynamic impawn rate Inventory financing
a b s t r a c t This paper proposes the way of setting the dynamic impawn rate by dividing the impawn periods into different risk windows. In an efficient financial market, the return is hypothetically independent, while in a pledged inventory market where spot transactions predominate, the return is auto-correlative. Therefore, the key to setting the impawn rate is to predict the long-term risk. In this experiment, using the database of spot steel, we established a model with the formula AR (1)-GARCH (1,1)-GED, forecasting the VaR of steel during the different risk windows in the impawn period through a method of out-of-sample, and got the impawn rate according with the risk exposure of banks. The results of our experiment indicated that the introduction of coefficient K into the model can significantly improve bank risk coverage and reduce its efficiency loss. Besides, the impawn rate obtained by the model correlates positively with the lowest price in the future risk windows. Ó 2012 Elsevier B.V. All rights reserved.
1. Introduction In recent years, supply chain finance has appeared as the competition between the supply chains predominate over that between the enterprises. Euromoney magazine defines the supply chain finance as the most popular topic in banking transaction service over the past few years, and predicts that the needs of this business will continue to grow in the next few years. However, different from that of western countries, the supply chain finance in China is a kind of pledge business performed by commercial banks, key enterprises, SMEs (small and medium enterprises), warehouse supervision enterprises, and so on (Feng, 2007). Besides, the pledges in China are mainly the inventories which are traded on the spot market (such as materials, products, and half-finished products) rather than the rights pledges (accounts receivable, accounts payable, and derivatives) which are widely used in western countries. In China, supply chain finance is not only an extended business of modern logistics but more importantly, it can help to solve the financing problems that many domestic enterprises are facing. Besides, upon the international background of credit deflation, the supply chain finance is replacing the traditional liquidity loans (Shen Zhen Development Bank, 2009). According to McKinsey prediction, the foreign trade will amount to USD1500 billion and the
⇑ Corresponding author. Tel.: +86 13708179996. E-mail addresses: [email protected], [email protected] (H. Juan). 0377-2217/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.ejor.2012.06.005
international trade financing may reach USD5–10 billion by 2011 in China. On the one hand, the increasing demand for supply chain finance will provide a great potential for the development of related services and products. On the other hand, the risk bore by supply chain finance has been restricting the development itself. According to ‘‘National Income Accounts and Statistical Yearbook’’ (People’s Bank of China, 2006), the current inventory of all enterprises amount to 5.1294 trillion Yuan in China, 3.0326 trillion Yuan of which are from small and medium enterprises, and 102.4 billion Yuan are from farmers. If the discount rate for loans is 50%, these financial assets can generate secured loans of about 2.6 trillion Yuan, which equals to new loans of financial institutions in 1 year. However, due to the lack of risk management techniques for inventory financing, most banks did not fully make use of these inventory resources. In inventory financing, inventory are used as the pledges for risk averse and in this sense, the pledged inventory are evaluated to find whether its value can maintain its capability of guarantee, which is referred to as impawn rate (usually called loan-to-value ratio in other literatures). In setting the impawn rate of inventory pledge loan, the impawn rate of inventory and the interest rate of loans should be considered since these two indexes help to reflect and manage guarantee capability of inventory. However, in current domestic financial market, the risk averse is mainly ensured through adjusting the impawn rate of inventory rate rather than interest rate. What is more, in practice, ‘‘management approach from supply chain financing activities’’ (The following simply defined as ‘‘management approach’’) stipulates that the
H. Juan et al. / European Journal of Operational Research 223 (2012) 106–115
impawn rate shall not exceed the highest level of 70% and the impawn period shall not exceed 1 year. Obviously, current banking still sets impawn rate based on their experiences and the impawn rate obtained in this way would not accord with the risk exposure of banks. Therefore, it is important to set impawn rate not only for the risk averse of supply chain finance but also for the development of this business. Recently, the domestic and foreign scholars have made many beneficial explorations on volatility and risk management of pledges. Cossin and Huang (2003) set enterprises’ probability of default through the simplified method and then obtained the impawn rate according with the risk exposure of banks. Jokivuolle and Peura (2003), in their works, obtain enterprises’ probability of default through the structured method and established the relationship between the loan loss and the impawn rate. Buzacott and Zhang (2004) analyze how the setting of the interest rate and the impawn rate influence the enterprises profitability. Caldentey and Haugh (2009) investigate the supply contracts with financial hedging, and they find that the producer always prefers the flexible contract with hedging to the flexible contract without hedging, while the retailer might or might not prefer the flexible contract with hedging depending on model parameters. Lee and Rhee (2011) explicitly assume positive costs of inventory financing in a supply chain,and they show that a supplier can make its retailer work for the entire supply chain with trade-credit and derive the optimal risk premium in the trade-credit for supply chain coordination. Chen and Cai (2011) study the value of 3PL firms in budget-constrained supply chains, and found that all players can be better off under 3PL financing than under bank financing. As far as the domestic researches concerned, given that the price of the pledged inventory is fluctuant randomly, Li et al. (2007) implement risk estimation strategies of ‘‘main body + debt’’ and analyzed loan-to-value ratio decision of banks with downside risk constraint when price distribution of the stock at the end of the loan follows general distribution and several special distributions. On the whole, the results of these researches have played positive and practical role in helping us understand and capture the actual volatility and risk level of pledged inventory. However, the quantitative models above are all based on the mathematical optimization method and take the expected revenue of banks as the objective function. In other words, theoretical modeling and the cases based on individual samples are many while the empirical researches based on large number of samples are scarce. With the rapid development of modern financial risk management technology, the domestic and overseas researchers made considerable progress in using risk management tools to manage inventory financing. Among these tools, VaR (Value at Risk) has been widely used in academia and in practice to measure and manage risk exposure since it is proposed by Morgan in the 90th of last century (Jorion, 2001). Although VaR is able to show the expected value loss of the financial assets over a period of time and at a given confidence level, the results usually are not precise because the most commonly used parametric method of calculating VaR depends on both the probability distribution and the volatility of the returns. Many empirical researches show that the returns of asset usually have leptokurtosis, fat-tails and volatility clustering. However, the traditional way to obtain VaR is based on the assumption that the returns are independent in an efficient financial market, missing the fat tails and the volatility clustering of returns. Whereas, GARCH model, extended by Bollerslev (1986) can effectively describe the clustering and the heteroskedasticity of returns. For this reason, GARCH model is introduced in the financial risk management field, and widely used to predict volatility in financial practice. For instance, Ricardo (2003) uses GARCH model to predict volatility with fat tails to deal with possible maximum loss. Chinese scholars like Gong and Chen (2005) describe the
107
fat-tails, the volatility clustering of financial time series, constructing the GARCH models to calculate time-varying VaR based on the volatility and distributions of returns of stock index and copper futures in China. However, VaR-GARCH model is mainly used to study stock index, bonds and futures of commodities, which usually have effective risk control measures, such as margin system and price limits. In addition, since they have good liquidity and short settlement time, VaR-GARCH is used to forecast their short-term risk of two weeks, especially daily risk. On the contrary, the pledges which are usually traded in the spot market lack the risk control measures and have low liquidity and long settlement time; therefore, the key to inventory financing is to predict the long-term risk, in other words, to predict VaR of N months later based on past samples (Juan, 2009). Most of current researches on VaR are based on the assumption that in an efficient financial market, the return is independent and therefore they mainly use square-root rule of Risk Metrics to forecast VaR. Anderson argued that the long-term conditional variance equals to the sum of daily conditional variances; he also based his argument on the same assumption. Whereas, a large amount of empirical researches show that in an efficient financial security market, the return could be independent while it is auto-correlative in a pledged inventory market. That means, in the market where the spot transaction predominate, the long-term risk is usually higher that the sum of daily risk. Therefore, in this paper, we proposes the way of setting the dynamic impawn rate by dividing the impawn periods into different risk windows according to the macroeconomic environment, the credit level of counterparty, the liquidity of pledged inventory and the risk preference of banks. Instead of using Risk Metrics, we establishes a model with the formula AR (1)-GARCH (1,1)-GED so as to better describe the auto-correlation, the heteroscedastic and the leptokurtosis of returns, providing long-term VaR to deal with the problem between data frequency and forecast frequency. Since the out-of-sample method is more accurate in predicting volatility, we forecast the VaR of steel during the different risk windows in the impawn period through an out-of-sample method. Besides, in this paper, we set the parameter K to improve risk coverage and we establish the Hit sequences based on the failure rate of back testing to ensure the reliability of the research. The remainder of this paper is organized as follows: Section 2 is model set-up, including the setting of AR (1)-GARCH model under auto-correlative return and the model for setting the dynamic impawn rate; Section 3 presents empirical analysis, including the sample selection, the data characteristic description and the detailed result analysis; in Section 4, the models are evaluated and tested. In conclusion, discussion is given with further research areas.
2. Model assumption and set-up 2.1. Model assumption Since the price of pledged inventory, like financial assets, fluctuates, we follow the common international practice: banks must have tools and methods to timely evaluate the value of pledged inventory. Thus, we set up models based on the following assumptions. (1) Logistics enterprises closely cooperate with banks. (2) The impawn rate varies under different risk windows; considering the macroeconomic environment, the credit level of counterparty, the liquidity of pledged inventory and the risk preference of banks, banks choose different risk window.
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(3) Considering that the inventory financing is short-term financing practice (less than 1 year), we assume that the loan interest rate does not change during the impawn periods.
tion (GED) for better approximating the fat-tails of the innovations. For this reason, we assume zt follows GED. The density function of GED is,
f ðx; v Þ ¼ 2.2. Model set up Considering that the returns of the pledged inventory have autocorrelation, heteroscedastic and leptokurtosis and fat-tails, we use the formula AR (1)-GARCH (1,1)-GED to estimate the volatility. Furthermore, based on VaR, we calculate the long-term price risk, which indicates the maximum possible loss over a predefined time horizon at certain level of confidence established previously. The risk-free value of pledged inventory, which can be obtained by subtracting the VaR, is also the amount of loan. Thus, we organize the steps as follows: 2.2.1. Yield rate of pledged inventory
Rt ¼ ln Pt ln Pt1
ð1Þ
where Rt is defined as the logarithmic return, Pt is the price at time t. 2.2.2. Volatility of yield rate of pledged inventory In financial studies, the volatility of financial assets is the standard deviation of the return rate. Likewise, the volatility of pledged inventory is the standard deviation of its return. Recent empirical studies indicated that financial assets have volatility clustering, which contribute to the heteroscedasticity of the return rate. For this reason, GARCH model was used so as to better describe this trait above. Besides, current researches show that GARCH (1,1) model might describe most of the time-varying variance of financial series. Therefore, we use the GARCH (1,1) to forecast the volatility of inventory, establishing the conditional mean equation and conditional variance equation as follows:
Rt ¼ lt þ et ¼ lt þ rt zt ; zt i:i:d:; Eðzt Þ ¼ 0; Varðzt Þ ¼ 1
r2t ¼ a0 þ a1 e2t1 þ b1 r2t1
ð2Þ ð3Þ
where lt is the conditional mean. AR (p) or ARMA (p,q) process is usually introduced to capture the autocorrelation of returns.Moreover,AR (1) process is proved to be a simple and valid model to describe the autocorrelation of return (Mcneil and Frey, 2000). Thus, we assume lt = qRt1. et rtzt is the stochastic disturbing term, also called residual term, which is a white noise process.In this formula,zt is the Innovations;r2t is the conditional variance of et at time t; a0 is constant; a1 is the ARCH parameter, b1 is the GARCH parameter,and a0 > 0,a1 > 0,b1 > 0. The model is stationary on condition that a1 + b1 < 1,jqj < 1.The unconditional variance of et can be calcua0 lated as V L ¼ r21 ¼ 1aa10b1 ¼ 1k wherek = a1 + b1.In most current financial literatures, the conditional variance and unconditional variance of et are used to describe the volatility of returns since log-returns are assumed as independent to each other,where q = 0, Rt = et = rtzt,Rt is a standard GARCH (1,1) process (Bollerslev, 1986).However,it is not valid in inventory financing,since the logreturns of pledges are autocorrelative (q – 0). As mentioned above,Rt follows AR (1)-GARCH (1,1) process, the unconditional vara0 eL ¼ e L equals VL.1 iance of Rt is that V , if q = 0, the V ð1q2 Þð1kÞ In the current practice, the Innovation zt is usually assumed to follow normal distribution. It usually has tails in practice. The student distribution was firstly introduced by Bollerslev (1987), while Nelson (1991) suggested the so-called Generalized Error DistribuThe proof of unconditional Variance of et andRt is presented in Appendix A.
c2
ð4Þ
where c = [2(2/v)C(1/v)/C(3/v)]1/2, when v = 2, zt follows normal distribution, for v < 2, the distribution of zt has fatter tails than normal distribution. 2.2.3. VaR and impawn rate Considering the volatility of pledged inventory price and the time period from risk identification to risk treatment, to calculate VaR in inventory financing is to forecast the long-term risk rather than the daily ones, which not only meets the demand of emerging business (supply chain finance, etc.) but also the regulations of Basel Accord II or even the latest version Basel Accord III that banks must report longer term VaR to supervision institutions. Unfortunately, as mentioned in the introduction, current researches are mainly concerned with the short-term risks, while missing the long-term risk. Besides, the best-known method to forecast long-termpffiffiffirisk is square-root rule with the formula VaRðTÞ ¼ VaRð1Þ T . This method is valid only when the return follows the independent normal distribution with a zero trend. Whereas, the return usually has leptokurtic and fat tails, so the scaling is no longer valid when applied to the evaluation of longterm risk in inventory financing. For this reason, in order to obtain more precise VaR, Dowd et al. (2004) revised square-root rule as follows:
h pffiffiffii VaRðTÞ ¼ Pt 1 exp lt T þ F 1 a rt T
ð5Þ
where Pt is the initial value of unit pledged inventory (for convenience, this paper denotes the unit price as the initial value). F 1 a denotes the left quantile at the certain level of confidence. Compared with the square root rule, the revised model has made much improvement and is able to avoid day-to-day prediction of volatility. However, it still relies on the square root rule more or less. In order to improve the two approaches above, Jorion (2001); Tsay (2005); and Andersen et al. (2006) argue that the conditional variance of the long time horizon equals to the sum of daily conditional variances under the hypothesis that in an efficient financial market, the return is independent (q = 0).2
l½t þ 1 : t þ T ¼
T X Rtþi
ð6Þ
i¼1
r2 ½t þ 1 : t þ T ¼ TV L þ
1 kT 2 rtþ1 V L 1k
ð7Þ
As mentioned in the introduction, the pledged inventory that is usually traded in spot market has low liquidity, and therefore the return is auto-correlative. In this sense, long-term risk is usually higher than the sum of daily risk. Considering both the autocorrelation and the time-varying volatility, the AR (1)-GARCH (1,1) model was proposed to forecast the long-term conditional variance (Brummelhuis and Kaufmann, 2004; Kaufmann, 2004).3
r^ 2 ½t þ 1 : t þ T ¼
2T 1 qT 2 1q V T 2 q þ q L 1q 1 q2 ð1 qÞ2 1
1 kT qT kT 2 q2T kT þ r2tþ1 V L 2q þq 1k qk q2 k
!#
Hence, the formula (5) can be revised in the following way, 2
1
1 exp jx=cjv 2 Cð1=v Þ
v ð1þ1=v Þ
3
The proof of formula (7) is presented in Appendix B. The proof of formula (8) is presented in Appendix C.
ð8Þ
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H. Juan et al. / European Journal of Operational Research 223 (2012) 106–115 5,600
Pt
5,200 4,800
SK
Margin call, Replenishment
SV
4,400 4,000 3,600
T
3,200
Fig. 1. The mechanism design of setting impawn rate.
h
VaRðTÞ ¼ P t 1 exp
1
l½t þ 1 : t þ T þ F a r^ ½t þ 1 : t þ T
i
2,800 2006
In practice, it is necessary for a commercial bank to set appropriate risk windows (risk holding period) to measure risk and to control risk (i.e. price risk) of inventory financing by setting impawn rate dynamically. It is well known that the risk window and the confidence level are the two key parameters to calculate VaR. Risk window T is defined as settlement time, which ideally is equal to the maximum since in banking practice, most of assets are monetary assets of good liquidity and therefore these areas are mainly concerned with daily VaR. As for the risk window T of inventory pledges, theoretically they are defined according to the liquidity of supply chain financial market, the sample size and the position adjustment of pledged assets; while, in practice, risk window T of inventory pledges can be defined according to its liquidity, the level of counterparty and some financial indicators such as the level of solvency and profitability. In addition, the level of confidence is set as 99% according to the recommendation of internal model of market risk measurement of commercial bank which was issued by China Banking Regulatory Commission. After setting the risk window and the confidence level based on theoretical analysis as well as practical study, we calculate the VaR during the risk window, and obtain the risk-free value which is the amount of loan as mentioned above. The ratio between the riskfree value and current price of pledged inventory is the impawn rate.
x¼
Pt VaR Sk 100% ¼ 100% Pt Pt
2007
2008
2009
ð9Þ
ð10Þ
where x is defined as impawn rate. Obviously, SK denotes the riskfree value. According to the Basel Accord (both Basels I and III), banks are suggested to predict market risk through Internal Model Approach (IMA). In practice, it is proved that through IMA, the VaR at the confidence level of 99%, especially in extreme markets, cannot cover the risk perfectly, and therefore multiplicative factor K is introduced to improve its risk coverage. This paper is written based on the principle. Although the AR (1)-GARCH (1,1)-GED model can describe the autocorrelation, the leptokurtic and the fat tailed and the volatility clustering of log-returns to a certain extent in this paper, the risk could be underestimated. Based on these factors, considering the liquidity of pledged inventory, the credit level of counterparty and the cost of replenishment and the closed position, we introduce the warning level and parameter K to further mitigate the risk of banks. As can be seen from Fig. 1, during the risk window T, when the price of steel falls below the warning level, margin call or replenishment is done till the value of steel regresses above the warning line. For convenience, we assume that closed position would be done if companies refuse to do that in time. In order to improve the risk coverage, we set the correction factor K = 1.1–1.2 (according to Basel accord in principle, which references the guidelines of 4 We only provide the necessity of the introduction of parameter K in this paper, the optimization problem of K is the separate subject in our future research.
Fig. 2. Time series of the steel rebar price in Shanghai. Source: http://www.96369.org/ Quotation/HistoryQuotationN.aspx?AreaID=1.
stock collateral issued by China Securities Regulatory Commission, 2004).4 In banking practice, for reduction of adverse selection and avoiding moral hazard to some extent, banks should set K for different pledges according to the macroeconomic environment, the credit level of counterparty, the liquidity of pledged inventory. In the empirical analysis followed, we assume previously K = 1.1, the modified model will be:
x¼
Pt VaR 1 SK 1=K SV 100% ¼ 100% ¼ 100% Pt K Pt Pt
ð11Þ
where SV is defined as the amount of pledged loan. 3. Empirical analysis 3.1. Sample selecting and descriptive statistics 3.1.1. Sample selecting In this section, in order to evaluate the models, we use the steel rebar (u HRB335) which are widely used in the industries of real estate and infrastructure industries as samples.5 The data set is obtained from XiBen new line stock and ShangHai futures exchange from the period of September 5th, 2005 to December 31st of 2009 (shown in Fig. 2). The data from September 5th of 2005 to December 31st of 2008 will be used be to evaluate the parameters; the rest will be used as test samples.6 Then we carry on a series of simulated pledge with the impawn contract’s starting date as January 1st of 2009 and set the impawn period to be the maximum of 12 months. Obviously, it is important to trade off the impawn period and the risk holding period. Generally speaking, the longer the risk holding period is, the more radical the banks are; the shorter the holding period is, the more conservative the banks are. According to normal banking practice, we set the risk holding period respectively as 1 week, 2 weeks, 1 month, 2 months, 3 months, 4 months, 5 months, 6 months, 7 months, 8 months, 9 months, 10 months, 11 months, 12 months.7 5 Steel Rebar has been transacted officially at the Shanghai futures exchange on the March 27, 2009. 6 In 2009, in response to the international financial crisis, China government launched packages of plans and policy measures ‘‘guaranteed growth, expand domestic demand, readjusting structure and benefit people’s livelihood’’. Under the action of above policy, China economy gets rid of financial crisis’s puzzle gradually, GDP restores by the season, promote the steel products to expend, stimulates the steel and iron good immovable property to be able to release, the steel enterprises turn losses into profits gradually, steel industry recovers slowly. However, the high yield, low exports, high inventory lead to a situation of excess domestic supply. The interaction of steel futures, electronic and the spot market, the price of domestic steel market performs ups and downs of the trend. Therefore this article analyzes time series in 2009, which contains two different stages, relatively stable and the volatility of the financial markets around the world, so our model plays the role of good testing. 7 Regarding to the risk windows, banks may definitely establish them according to own need, this paper only provides a model of dynamic management risk.
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R .08
.04
.00
-.04
-.08 06M01
06M07
07M01
07M07
08M01
08M07
Fig. 3. The linear graph of log-return of steel rebar in Shanghai.
Table 2 The results of ARCH effect test.
Table 1 The results of ADF test of log-returns.
Augmented Dickey–Fuller test statistic Test critical values: 1% Level 5% Level 10% Level
t-Statistic
Prob.⁄
Heteroskedasticity test: ARCH
13.82975 3.437695 2.864672 2.568491
0.0000
F-statistic Obs⁄R-squared
3.1.2. Descriptive statistics of log-returns of steel rebar price As shown in Fig. 3, the log-return series show significant volatility clustering which indicates that ARCH effect may exist in log-returns. Since GARCH model can be established only when the time series is stationary, we carry on Augmented Dickey–Fuller test (i.e. ADF unit root test) the time series. As shown in Table 1, in all cases, the null hypothesis is rejected at any level of significance (1%, 5%, 10%), so the time series of daily log-return is stationary. Fig. 4 indicates that the autocorrelation and the partial correlation are significant in some cases, especially in first order. It is reasonable to denote Rt = qRt1 + et. As can be shown in Table 2, the value is significant, which indicates that the ARCH effect exists in the residuals of daily logreturn. Fig. 5 shows that in almost all cases, the null hypothesis of normality is rejected at any level of significance, as there is evidence of significant excess kurtosis and positive skewness. 3.2. Empirical results As presented in Table 3, the constant term of conditional mean is 1.87E07, which is close to zero while not significant in all cases.
31.55798 134.1414
Prob. F (5, 855) Prob. chi-square (5)
0.0000 0.0000
The parameter q which considers the autocorrelation is significant. In addition, the value of AIC and SC is reasonable. Therefore, the model of AR (1)-GARCH (1,1)-GED is acceptable. Thus, the conditional mean and conditional variance can be rewritten as follows:
Rt ¼ 0:1440Rt1 þ et 2 t
ð12Þ 2 t1
r ¼ 3:40E 06 þ 0:1115e
2 t1
þ 0:7452r
ð13Þ
From Eq. (13), we can obtain a1 + b1 = 0.86 < 1, which denotes that the time series of daily log-return is stationary. Thus, the long-term variance of et can be calculated through the following formula: V L ¼ 1aa10b1 ¼ 2:3724E 05. The parameter v is 0.853 obtained through Eviews. The corresponding quantile on left is 2.87, when the confidence level is 99%. Considering all these above, we get the empirical results as in Table 4. 4. Model evaluation 4.1. Back testing of long-term risk In order to test the accuracy of the AR (1)-GARCH (1,1)-GED model under autocorrelation with heavy tails, we have to test the risk coverage level of VaR. The most widely used models for back testing of VaR are the failure rate method (see Kupiec, 1995) and the internal back-testing model of Basel accord. Both
Fig. 4. The results of autocorrelation test.
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700
Series: R Sample 9/05/2005 12/31/2008 Observations 867
600 500
Mean 0.000129 Median 0.000000 Maximum 0.113759 Minimum -0.070068 Std. Dev. 0.009075 1.038078 Skewness Kurtosis 42.97649
400 300 200
Jarque-Bera Probability
100
57887.78 0.000000
0 -0.05
0.00
0.05
0.10
Fig. 5. The results of normal assumption test.
Table 3 Parameters estimates of AR (1)-GARCH (1,1)-GED model. c
q
a0
a1
b1
v
AIC/SC
1.87E07 (0.9979)
0.1440 (0.0000)
3.40E06 (0.0000)
0.1115 (0.0000)
0.7452 (0.0000)
0.853 (0.0000)
8.4295 8.3965
Therefore, the Hit function is established to test the accuracy of long term price risk forecasting. Put another way, we have to observe the exceptions in which the expected price of inventory (risk-free value) is higher than the actual price.
Hit ¼
of the two models are applied to examine the exceptions in which the actual loss is greater than the daily VaR. For instance, when the time horizon is 1 year and the confidence level is 99%, the number of exceptions of less than seven times is accepted by the failure rate method while Basel II internal green light area only accepts the number of less than four times. However, both of these two methods are not suitable for back testing of long-term risk in inventory financing.
1; P tþi < Pt VaR
ð14Þ
0; or else
where Sk = Pt VaR, which denotes the expected price (risk-free value, the warning level in this paper). f is the observed number of exceptions in the sample. f/N should be close to 1 a statistically, if the bias is too large or too small, it means that the model cannot forecast the price risk correctly. In the following, comparisons are made between the Risk Metrics and the GARCH (1,1)-GED model and the results are presented as follows: The main results (see Tables 5–7) indicate that the AR (1)GARCH (1,1)-GED model outperforms both the Risk Metrics
Table 4 The empirical results of different risk windows of 12 months impawn period. Risk window 1 week 2 weeks 1 month 2 months 3 months 4 months 5 months 6 months 7 months 8 months 9 months 10 months 11 months 12 months Size N Pt VaR SK SV
x
5 3580 120 3460 3145 0.88
12 3580 191 3389 3081 0.86
23 3580 264 3316 3015 0.84
43 3580 358 3222 2929 0.82
65 3580 437 3143 2858 0.80
87 3580 501 3079 2799 0.78
108 3580 554 3026 2751 0.77
130 3580 604 2976 2706 0.76
153 3580 650 2930 2663 0.74
174 3580 689 2891 2628 0.73
196 3580 727 2853 2593 0.72
218 3580 763 2817 2561 0.72
239 3580 794 2786 2532 0.71
261 3580 827 2753 2502 0.70
Table 5 The results of the Risk Metrics (square root rule) during 12-month impawn period. Risk window 1 week 2 weeks 1 month 2 months 3 months 4 months 5 months 6 months 7 months 8 months 9 months 10 months 11 months 12 months Size N Pt VaR SK f f/N
5 3580 97 3483 0 0
12 3580 150 3430 0 0
23 3580 208 3371 0 0
43 3580 285 3295 0 0
65 3580 350 3230 10 15.4%
87 3580 405 3175 15 17.4%
108 3580 451 3128 0 0
130 3580 495 3085 0 0
153 3580 537 3043 0 0
174 3580 573 3007 0 0
196 3580 608 2972 0 0
218 3580 641 2939 0 0
239 3580 671 2909 0 0
261 3580 701 2879 0 0
Table 6 Results of the GARCH (1,1)-GED model during 12-month impawn period. Risk window 1 week 2 weeks 1 month 2 months 3 months 4 months 5 months 6 months 7 months 8 months 9 months 10 months 11 months 12 months Size N Pt VaR SK f f/N
5 3580 106 3474 0 0
12 3580 165 3415 0 0
23 3580 228 3352 0 0
43 3580 309 3271 0 0
65 3580 377 3203 8 12.3%
87 3580 433 3147 3 3.5%
108 3580 480 3100 0 0
130 3580 523 3057 0 0
153 3580 564 3016 0 0
Note: Due to the page limit, the details of calculating VaR in Tables 5 and 6 is not shown in this paper.
174 3580 599 2981 0 0
196 3580 632 2948 0 0
218 3580 664 2916 0 0
239 3580 692 2888 0 0
261 3580 721 2859 0 0
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Table 7 Results of the AR (1)-GARCH (1,1)-GED model during 12-month impawn period. Risk window 1 week 2 weeks 1 month 2 months 3 months 4 months 5 months 6 months 7 months 8 months 9 months 10 months 11 months 12 months Size N Pt VaR SK f f/N SV f1 f1/N
5 3580 120 3460 0 0 3145 0 0
12 3580 191 3389 0 0 3081 0 0
23 3580 264 3316 0 0 3015 0 0
43 3580 358 3222 0 0 2929 0 0
65 3580 437 3143 3 4.6% 2858 0 0
87 3580 501 3079 0 0 2799 0 0
108 3580 554 3026 0 0 2751 0 0
130 3580 604 2976 0 0 2706 0 0
153 3580 650 2930 0 0 2663 0 0
174 3580 689 2891 0 0 2628 0 0
196 3580 727 2853 0 0 2593 0 0
218 3580 763 2817 0 0 2561 0 0
239 3580 794 2786 0 0 2532 0 0
261 3580 827 2753 0 0 2502 0 0
Table 8 The impawn rate and the corresponding lowest value under different risk windows during 12-month impawn period. Risk window 1 week 2 weeks 1 month 2 months 3 months 4 months 5 months 6 months 7 months 8 months 9 months 10 months 11 months 12 months PL,T
3580 0.88
x
3580 0.86
3580 0.84
3330 0.82
3140 0.80
3140 0.78
3140 0.77
x
PL,T
x
1.000000 0.869665
0.869665 1.000000
PL,T
3140 0.74
3140 0.73
3140 0.72
3140 0.72
3140 0.71
3140 0.70
words, the closer the coefficient of correlation is to 1, the more effective the model is. Table 8 presents the lowest price of steel rebar in different risk windows and corresponding impawn rate during both 6 months and 12 months time horizon, the coefficient of correlation is 0.87, which is close to 1(see Table 9). This shows that the corrected model is efficient.
Table 9 The correlation metrics of x and PL,T. 12 months
3140 0.76
and the GARCH (1,1) models in the accuracy of long-term price risk forecasting. Unfortunately, for AR (1)-GARCH (1,1)-GED model, the failure rates in the 3 months is 4.6%. In both two impawn periods, the failure rates are beyond 1% (corresponding to 99% confidence level). This indicates that the price risk is not covered completely by AR (1)-GARCH-GED model. Therefore, we have to test whether the price series Pt puncture the loan value SV which is corrected by the parameter K. Table 7 presents that the exceptions did not happen. In this sense, the revised model improved the risk coverage level significantly.
4.3. The analysis of efficiency of model As mentioned in the introduction, according to the experience method, the impawn rate is almost lower than 70%. This method may control risk while cause higher efficiency loss. In the following, we will compare the model of setting the impawn rate in this paper with the experience method, and then we introduce two indicators: efficiency loss h1, risk rate h2. To facilitate the process, we select the upper limit (70%) of impawn rate based on banks’ experience.
PtþT SV 100% Pt SV 100% h2 ¼ PtþT
h1 ¼ 4.2. Testing between the impawn rate and lowest value of the steel rebar The impawn rate reflects the banks’ expectation of lowest price of pledges in the future risk windows. Thus, statistically speaking, the impawn rate obtained by an effective model should positively correlate with the lowest value, although it is almost impossible for a certain model to forecast it due to various factors. In other
ð14Þ ð15Þ
where Pt+rT is defined as the price at period-end of risk window; h2 < 1 shows that the risk is under control. As can be seen from Tables 10 and 11 and Fig. 6, the two indicators negatively correlate with each other; it means that the risk is significantly reduced while the efficiency loss is minimized.
Table 10 Analysis of the efficiency and the risk based on the experience of 12-month impawn period. Risk window
1 week (%)
2 weeks 1 month 2 months 3 months 4 months 5 months 6 months 7 months 8 months 9 months 10 months 11 months 12 months (%) (%) (%) (%) (%) (%) (%) (%) (%) (%) (%) (%) (%)
h1 h2
33 68
33 68
36 66
23 75
21 77
24 75
27 72
36 66
49 59
31 69
27 72
27 72
30 70
34 67
Table 11 Analysis of the efficiency and the risk in AR (1)-GARCH (1,1)-GED model. Risk window
1 week (%)
2 weeks 1 month 2 months 3 months 4 months 5 months 6 months 7 months 8 months 9 months 10 months 11 months 12 months (%) (%) (%) (%) (%) (%) (%) (%) (%) (%) (%) (%) (%)
h1 h2
15 85
17 83
22 79
11 88
11 88
15 84
21 79
31 71
44 63
28 72
25 75
26 74
30 71
35 67
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H. Juan et al. / European Journal of Operational Research 223 (2012) 106–115
100%
efficiency loss of experience risk rate of experience efficiency loss of model risk rate of model
80% 60% 40% 20% 0%
Fig. 6. Comparison between AR (1)-GARCH (1,1)-GED model and experience method during 12 months impawn period.
Additionally, the risk rates according to both two methods are less than 1, so the risk is under control. As for the efficiency loss, it is still great according to the experience method, even if we set the impawn rate of upper limit (70%). Although a certain degree of efficiency loss still exists in the model used in this paper, it has been greatly improved compared with the experience method.
5. Conclusions Considering the long-term risk window of inventory pledges due to its insufficient liquidity, we established the formula of AR (1)-GARCH (1,1)-GED to better describe the autocorrelation, the fat-tails and the volatility clustering of return, forecasting the long-term price risk; furthermore, considering the macroeconomic environment, the credit level of counterparty, the liquidity of pledged inventory and the risk preference of banks, we set dynamic impawn rate by dividing the impawn period into different risk windows. In the end, the conclusions are arrived as follows: First, the time series of return of rebar price in Shanghai have significant autocorrelation, fat-tails, and volatility clustering. Compared with other models such as the Risk Metrics and those revised models, the AR (1)-GARCH (1,1)-GED model have more accuracy in forecasting long-term VaR. However, it is important to note that the failure rate under 3 month risk window is higher than what can be accepted at the confidence level, which indicates that none of the models is able to forecast the long-term risk perfectly, even when considering the autocorrelation, the fat-tails and the volatility clustering. Therefore, when banks and supervision institutions use mature and sound VaR models to forecast risk, they should also consider the macroeconomic environment and the liquidity of pledged inventory. For the reason, parameter K was introduced into this paper. The results (in Table 7) show that, the risk coverage level has been improved remarkably via K. In addition, banks can get the upper limit of K through stress tests. For instance, stress test can be carried onto get upper limit of K regarding the price plunge of steel in 2008 in order to deal with extreme cases. Second, the impawn rate obtained from the model significantly positively correlate with the low price in the future risk window. Hence, this model could reasonably reflect the banks’ expectation about the risk of pledged steel. Moreover, compared with the empirical method widely used in banking, the model used in this paper can effectively control the risk, while have the higher financing efficiency. In this sense, it may better enhance the attraction of inventory financing by reducing the adverse selection and moral hazard to some extent. Finally, it is also important to keep in mind that, different from financial assets, the pledged inventories have autocorrelation, fattails and so on. Substantial empirical works are left for future re-
search to capture the unique characteristics of pledged inventories and determine the K and the T of the inventory financing to improve the performance of risk forecasting. Furthermore, with the Copula theory better describing the dependence structure between two or more random variables, the pledged inventory can be expanded from single commodity to portfolio which includes no less than two kinds of commodities. Undoubtedly, this will cater to the trend of supply chain financing in China. The last but not least, we will introduce the real option theory and stochastic differential equation (SDE) to study the mechanism for setting impawn rate in a novel viewpoint rather than VaR. The specific work is still left for future research.
Acknowledgments This paper is supported by the NSFC (Natural Science Foundation of China) (71003082), the Fund Projects for Doctoral Degree of The National Ministry of Education of China (200806131007), the Scientific Fund Projects for National Post Doctors of China (20080430602) and Project 985 Platform School of Economics of Fudan University and is the result of her Postdoctoral research in Institute of Financial Studies of Fudan University. Acknowledgement goes to Professor Chen Xuebin of Institute of Financial Study of Fudan University, Professor Zhang Jinqing for his suggestion on annual statistics and model revision, and Luo Cheng in Shanghai Futures Exchange for his statistical support. The authors would like to thank the anonymous referees for their helpful comments and suggestions, which improved the contents and composition substantially.
Appendix A. Proof of the unconditional variance of et and Rt
Rt ¼ qRt1 þ et
ðA:1Þ
et ¼ rt zt ; zt i:i:d:; Eðzt Þ ¼ 0; Varðzt Þ ¼ 1
ðA:2Þ
r2t ¼ a0 þ a1 e2t1 þ b1 r2t1
et follows a GARCH (1,1) process, if t ? 1, the Eðe2t1 Þ ¼ r2t ¼
r2t1 ¼ r21 . Thus, the unconditional a0 . V L ¼ r21 ¼ 1aa10b1 ¼ 1k
variance
of
et
equals
Rt follows AR (1)-GARCH (1,1) process, using (A.1) recursively leads to,
Rt ¼ qRt1 þ et ¼ qðqRt2 þ et1 Þ þ et ¼ qnþ1 Rtn1 þ et þ qet1 þ þ qn etn If jqj < 1,limn?1qn = 0, which yields
ðA:3Þ
114
H. Juan et al. / European Journal of Operational Research 223 (2012) 106–115
Rt ¼ et þ qet1 þ q2 et2 þ ;
ðA:4Þ
Thus, the unconditional variance of Rt equals
2
e L ¼ E½Rt EðRt Þ2 V 2 ¼ E et þ qet1 þ q2 e2t2 þ 2
2
^2
r
½t þ 1 : t þ T ¼ E4
T X Rtþi
!2
3
" #!2 T X 5 jltþ1 ; rtþ1 E Rtþi jltþ1 ; rtþ1
i¼1 2
i¼1
ðA:12Þ
ðA:5Þ
¼ E½et þ q2 E½et1 þ q4 E½et2 þ 2
In the AR (1)-GARCH (1,1) model, the daily log-returns exist first-order autocorrelation (q – q), the T-days conditional variance of Rt is:
4
¼ ð1 þ q þ q þ ÞV L
Using recursively the conditional mean equation, we get
" # " # T T T Ti X X X X i1 s E Rtþi jltþ1 ; rtþ1 ¼ ltþ1 q þ E ri zi q jltþ1 ; rtþ1
0 ¼ 11q2 V L ¼ ð1q2aÞð1kÞ
e L reduces again to VL. If q = 0, the V
i¼1
i¼1
s¼0
i¼1
T
q ¼ ltþ1 1 1q
Appendix B
ðA:13Þ
B.1. Proof of the formula (7)
Moreover 2 3 2 !2 T X E4 Rtþi jltþ1 ; rtþ1 5 ¼ E4
T
r
2
1k 2 ½t þ 1 : t þ T ¼ TV L þ rtþ1 V L 1k
2
r
T X Rtþi
½t þ 1 : t þ T ¼ E4
3
!2
jrtþ1 5 E
¼
r^ 2 ½t þ 1 : t þ T ¼ ¼
ðA:7Þ
h
r2t jr1 ¼ E a0 þ a1 ðrt1 zt1 Þ2 þ b1 r2t1 jr1
ðA:8Þ
T 2 X 1qTiþ1 E r2tþi jrtþ1 1q
T 2 X 1qTiþ1 E r2tþi jrtþ1 1q
i¼1
ðA:9Þ
Hence
r2tþ1 V L
r2tþ1 V L
i
ðA:15Þ
T T X r2tþ1 V L X ð1 qTiþ1 Þ2 þ ð1 ð1 qTiþ1 Þ2 ki1 qÞ2 i¼1
i¼1
ðA:16Þ
i¼1 1qT 1q
2T 2 1q 1q2
þq
ðA:17Þ
i¼1 T
ðA:10Þ
T T h i X X E R2tþi jrtþ1 ¼ TV L þ ki1 r2tþ1 V L
T
T
2T
T
¼ 1k 2q qqk þ q2 qq2 k 1k k k
which yields
T
V L þ ki1
T T X X ð1 qTiþ1 Þ2 ki1 ¼ ð1 2qTiþ1 þ q2T2iþ2 Þki1 i¼1
h i E R2tþi jrtþ1 ¼ E r2tþi jrtþ1 ¼ V L þ ða1 þ b1 Þi1 r2tþ1 V L ¼ V L þ ki1 r2tþ1 V L
i¼1
2 h
¼ T 2q
¼ V L þ ða1 þ b1 Þt1 ðr21 V L Þ
¼ TV L þ 1k 1k
1qTiþ1 1q
T T X X ð1 qTiþ1 Þ2 ¼ ð1 2qTiþ1 þ q2T2iþ2 Þ
1 ða1 þ b1 Þt1 ¼ a0 þ þ ða1 þ b1 Þt1 r21 1 ða1 þ b1 Þ
r2 ½t þ 1 : t þ T ¼
jltþ1 ; rtþ1 5
If jqj < 1, we get
Using this formula recursively leads to,
r2t jr1
T X
¼ ð1V qL Þ2
i
¼ a0 þ ða1 þ b1 ÞE r2t1 jr1
E
q
i¼1
h i E R2tþi jrtþ1 ¼ E e2tþi jrtþ1 ¼ E r2tþi jrtþ1
þ
s¼0
r i zi
3
i¼1
and
2
i¼1
!2 s
Let (A.10), (A.13), (A.14) Substitute (A.12),
i¼1
ðA:6Þ
þ
Ti X
ðA:14Þ
i¼1
q
T X
i¼1
T h i X ¼ E R2tþi jrtþ1
E
T
q ltþ1 1 1q
" #!2 T X Rtþi jrtþ1
i¼1
i1
i¼1
In the standard GARCH (1,1) model, the log-returns is independent (q = 0, Rt = et), the T-days conditional variance of Rt is: 2
ltþ1
i¼1
T X
i¼1
ðA:11Þ
and while if Considering equations (A.16), (A.17), (A.15) can be written as 1 1 qT 1 q2T r^ 2 ½t þ 1 : t þ T ¼ V L T 2q þ q2 2 2 1q 1q ð1 qÞ !# 1 kT qT kT 2 q2T kT þ r2tþ1jt V L 2q þq ðA:18Þ 1k qk q2 k If q = 0, the formula (A.18) reduces again to (A.11).
Appendix C
References
C.1. Proof of the formula (8)
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r^ 2 ½t þ 1 : t þ T ¼
1 ð1 qÞ
2
1 qT 1 q2T V L T 2q þ q2 2 1q 1q
1 kT qT kT q2T kT þ r2tþ1 V L 2q þ q2 2 1k qk q k
!#
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Li, Y.X., Feng, G.Z., et al., 2007. Research on loan to value ratio of inventory financing under randomly fluctuant price. System Engineering Theory & Practice 12, 4248. Mcneil, A.J., Frey, R., 2000. Estimation of tail-related risk measures for heteroscedastic financial time series: an extreme value approach. Journal of Empirical Finance 7 (10), 271–300. Nelson, D.B., 1991. Conditional heteroscedasticity in asset returns: a new approach. Econometrica 59 (2), 347–370. Ricardo, A., 2003. The estimation of market VaR using GARCH models and a heavy tail distributions. Working Paper Series, 1–28. Shen Zhen Development Bank, C.E.I.B.S., 2009. Supply Chain Finance. Shang Hai Far East Press, Shang Hai. Tsay, R.S., 2005. Analysis of Financial Time Series, second ed. John Wiley & Sons, Inc.
Further reading Cossin, D., Huang, Z.J., Aunon, N.D., 2003. A frame work for collateral risk control determination. European central bank working paper series, 1–41. Liu, Q.F., Zhong, W.J., 2006. Market risk measurement of copper futures in China based on VaR GARCH models. Journal of Systems Engineering 8, 429–433. White, H., 2000. A reality check for data snooping. Econometrica 68 (2), 1097–1126.