VaR

Application of VaR methodology to risk management

in the stock market in China

Ying Fan *,Yi-Ming Wei,Wei-Xuan Xu

Institute of Policy and Management,Chinese Academy of Sciences,P.O.Box 8712,Beijing 100080,China Abstract

This paper applies the new risk management tool,Value at Risk (VaR)methodology,to the stock market in China.From the comparison between the predicted VaR and real return,the calculated results are mostly satis?ed with the con?dence level at 95%.

Keywords:Value at risk methodology;Risk management;Exponential weighted moving average

1.Introduction

Financial risk management is always one of the important topics either in theory or in practice.In the last 25years,international ?nancial market has developed greatly,and ?nancial storms have much in?uence on human’s entire economic behavior with the mode of over imagination.In early 1990s,a kind of new risk management methodology was developed,which is Value at Risk (VaR)methodology.VaR methodology is becoming to be the international standard of risk measures (Wang,Tang,&Shi,1999).VaR is the maximum amount we expect to lose over some target period.A formal de?nition is described as following:

Let R be a random variable denoting the return of a portfolio,f eR Tthe probability density function of R ;and c a con?dence level.The probability that the return is less than R p is:

prob ?R ,R p ?

eR p

f eR Td R ?12c e1TThe de?nition of VaR has two types:absolute VaR and relative VaR.Absolute VaR represents the maximum likely loss relative to the present position:

VaR eabsolute T?2R p W

e2T

doi:10.1016/j.cie.2003.12.018*Corresponding author.

E-mail address:yfan@https://www.360docs.net/doc/a012625171.html, (Y.Fan).

And the relative VaR represents the maximum likely loss relative to the expected return,which is often more convenient to deal with:

VaR erelative T?2R p W tm W e3Twhere m is the expected return and W is the position.

There are many methods to calculate VaR,which ?t different market conditions,data set and precision requirements.Generally,we can classify them into three types (Dowd,1998):

?variance–covariance method

?historical simulation method

?Monte Carlo simulation method

The detailed discussion about historical and Monte Carlo simulation methods was given by Dowd (1998).In view of the reality of the Chinese stock market,this paper selects the variance–covariance method to estimate VaR of stock market in China.Especially,the exponential weighted moving average (EWMA)method was used and the related problem was solved.The following will give the optimal decay factors in Shenzhen and Shanghai stock market.Furthermore,the daily VaR of the two markets was forecasted on the real data.

2.EWMA method to estimate VaR

Let {r t }denote a time series of return of a certain ?nancial instrument.On the hypothesis of random walk,r t is normally distributed with mean m and variance s 2t

r t ,N em ;s 2t Te4TFor the return of every day,often assume m ?0(Fan,2000).For a certain con?dence level c ;its correspondent quartile in standard normal distribution is a ;we can derive

VaR erelative T?2as t W e5TWhen a portfolio holds two or more types of assets,their correlation coef?cient must be calculated.In formula (5),W is the position of the portfolio as it is known at a certain time t :a is determined by the con?dence level c :So,while we estimate VaR with the variance–covariance method,the main question is how to estimate the standard derivation of the return distribution,which is a measurement of volatility.

From formula (4),we can use historical observations of r t to estimate s t ;formula (6)is the estimation using historical data length T :

^s t ?????????????????1T X t i ?t 2T t1r 2i

v u u t e6TThis method to estimate variance is called the simple moving average model (SMA).Its characteristics are putting ?xed equal weights for every data e1=T T:So the result would be depend heavily on the selection of T :

Y.Fan et al./Computers &Industrial Engineering 46(2004)383–388

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One improved model for SMA is EWMA,whose weights are different for every observation,the latest observation carry the highest weight in the volatility estimation ^s

t ??????????????????????e12l TX 1i ?0

l i r 2t 2i v u u t e7TIn formula (7),the parameter l e0,l ,1Tis often referred to as the decay factor.This parameter determines the relative weights that are applied to the observations (returns)and the effective amount of data used in estimating volatility.

When we estimate volatility,we pay attention to three important issues that arise.

2.1.Accuracy of estimation:calculation of the tolerance level

Since the weight of r 2t 2i ;e12l Tl i ,approaches the limit 0ei !t1T;the estimation of volatility can be calculated proximately in limited sample length K :In that case,de?ne tolerance level L K

L K ?e12l TX

1i ?K l i e8T

In tolerance level L K ;proximate estimation of standard derivation is ^s t ??????????????????????e12l TX K 21i ?0l i r 2t 2i

v u u t e9T

2.2.The effective data length

L K ?l K e12l TX

1i ?0l i ?l K e10T

We can transform formula (8)for Eq.(10).

So,we know the relationship between decay factor l ;effective amount of data K and tolerance level L K :

2.3.Determining the decay factor

The important problem is:how to determine the decay factor.Now,if we de?ne the variance forecast error as 1t t1l t ?r 2t t12s 2t t1l t ;it then follows that the expected value of the forecast error is zero,i.e.E e1t t1l t T?E er 2t t1T2s 2t t1l t ?0:Based on this relation a natural requirement for choosing l is to minimize average squared errors which is given by

RMSE ???????????????????????????????1T X T t ?1

er 2t t12^s 2t t1l t el TT2v u u t e11T

where T is the days of prediction period.Y.Fan et al./Computers &Industrial Engineering 46(2004)383–388385

Based on above RMSE criteria,Morgan have given some optimal decay factors of daily VaR predicting for some ?nancial instruments of some country in a technique document named RiskMetrics (Table 1).

From Table 1,we observe that the decay factors of different instruments in different countries vary signi?cantly.This implies that in a certain country with a certain background of economy and culture,the market memory length for every instrument is different from others.In RiskMetrics,Shenzhen and Shanghai stock markets in China were not included.

3.VaR estimation of stock market in China based on EWMA method

For the stock market in China,we estimate the Value at Risk for every day.Denote P t is the value of index at time t ;the time interval is 1day.Suppose r t ?ln eP t T2ln eP t 21T;we use r t replace R t :We estimate VaR of Shanghai and Shenzhen index with EWMA.The same period of sample data are selected for proper comparison from January 3,1994to February 23,1998.The length of data used for prediction is more than 1000days,which implies that the tolerance level is smaller than 0.001%when l is greater than 0.85according to formula (10).We obtained the following results.Figs.1and 2show the comparison between predicted VaR and real return.

Table 1

Optimal decay factors based on volatility forecasts (Morgan,1996)

Country Foreign exchange Five-year swaps 10-year zero prices Equity indices One-year money

market rates

Austria 0.945––––

Australia 0.9800.9550.9750.9750.970

Belgium 0.9450.9350.9350.9650.850

Canada 0.9600.9650.960–0.990

Switzerland 0.9550.835–0.9700.980

Germany 0.9550.9400.9600.9800.970

Denmark 0.9500.9050.9200.9850.850

Spain 0.9200.9250.9350.9800.945

France 0.9550.9450.9450.985–

Finland 0.995–––0.960

Great Britain 0.9600.9500.9600.9750.990

Hong Kong 0.980––––

Ireland 0.990–0.925––

Italy 0.9400.9600.9350.9700.990

Japan 0.9650.9650.9500.9550.985

Netherlands 0.9600.9450.9500.9750.970

Norway 0.975––––

New Zealand 0.9750.980–––

Portugal 0.940–––0.895

Sweden 0.985–0.980–0.885

Singapore 0.9500.935–––

United States –0.9700.9800.9800.965

ECU –0.950––

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?The optimal l for Shenzhen stock market reaches to 0.86.

?The optimal l for Shanghai stock market reaches to 0.88.

?From the data of recent years,we know that the ?uctuation of the stock market of China is large and that the ?uctuation of Shenzhen market is larger than that of Shanghai market,which is correspondence to the reality.The value of l which was calculated with EWMA method could better re?ect the ?uctuation of stock market of China and the memory length of the markets.?Applying the optimum decay factor l ;we use the EWMA method to predict the VaR of Shanghai and Shenzhen market and make their comparisons with real daily return in the con?dence level c ?95%:

?We had forecasted VaR of Shenzhen index in the 475days continually,and the results show that the days in which the negative return over VaR is 28,the ratio is (28/475)?5.89%.

?At the same time,the result of Shanghai market show that the days in which the negative return over VaR is 24days,the ratio is (24/475)?5.05%.

?From the comparison between the predicted VaR and real return,the calculated results are mostly satis?ed with the con?dence level at 95%.

4.Remark

The study of this paper shows that VaR methodology can be applied to risk management in stock market in China,which is an ef?cient tool to measure market risk.Promisin opportunities exist for the application of this methodology in the area of risk management of China besides stock market.This research was supported by the grant (No.70371064)from NSFC.

References

Dowd,K.(1998).Beyond value at risk.New York:Wiley.

Fan,Y.(2000).VaR methodology and its application to stock market risk analysis in China.Chinese Journal of Management Science ,8(3),26–32.(in Chinese).

Morgan,J.P.(1996).RiskMetrics technology document (4th ed.).

Wang,Z.,Tang,G.,&Shi,S.(1999).VaR methodology for ?nancial risk analysis.Chinese Journal of Science ,51(6),15–18.(in Chinese).Y.Fan et al./Computers &Industrial Engineering 46(2004)383–388

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