Incorporating Liquidity Risk In Var Models

Incorporating Liquidity Risk in VaR Models First Version: September 2000 This Version: June 2002

LE SAOUT ERWAN♣

Abstract Conventional Value at Risk models lack a treatment of liquidity risk. Neglecting liquidity risk leads to an underestimation of overall risk and misapplication of capital for the safety of financial institutions. Standard Value at Risk model assumes that any quantity of securities can be traded without influencing market prices. In reality, most markets are less than perfectly liquid. Recent financial turbulences, as the collapse of LCTM, prove that liquidity is a significant risk for market players. The main aim of this article is to demonstrate the importance of the liquidity risk component in financial markets. First, we make a survey of literature explaining how the liquidity risk can be incorporated into one single measure of market risk. Then, we apply the Liquidity Adjusted Value at Risk model provided by Bangia, Diebold, Schuermann et Stroughair (1999) on the French stock market: our results indicate that the exogenous liquidity risk defined by Bangia, Diebold, Schuermann et Stroughair (1999) can represent more than an half of market risk for illiquid stocks. We extend this model to show that endogenous liquidity risk, which refers to liquidity fluctuations driven by individual action such as the size of the investors’ position, is also a very important component of overall risk. Key-words : Liquidity Risk, Value at Risk, Depth, Bid-Ask Spread, Weight Average Spread.

♣

Paris 1 University - CREFIB & CREREG. Correspondance :[email protected] This document is available through internet: http://www.market-microstructure.org I want to thank Professors Patrick Navatte, Patrice Fontaine, Jean-Pierre Gourlaouen, Jacques Hamon, Michel Levasseur, and Charles-André Vailhen. I also received many helpful comments from participants in presentations at the 2000 IAE Meeting (Biarritz), the 2001 AFFI Meeting (Namur) and IRG Workshop. All remaining errors are mine.

Incorporating Liquidity Risk in Value at Risk Models

Introduction The Russian financial crisis, which took place in July 1998 and the distribution of the effects that followed it on all the world financial places, was at the origin of numerous debates. Many wondered about the components of this systematic crisis. If the underestimate of credit risks is at the origin of the Russian monetary crisis, one of the factors prevailing in certainly due to the correlation of the risks. According to the IMF, the factor, which started this process of interdependance of the risks, is exchange market illiquidity. In times of crisis, liquidity tends to dry up suddenly; we can observe a decline of the liquidity’s offer and the increase of assets ’correlation, what has the consequence of making ineffective the diversification. This liquidity risk is with difficulty predictable in spite of numerous models including the value at risk models because it’s very difficult to measure liquidity. This paper is organised as follows: in a first part, we examine the literature dealing with the consideration of the liquidity risk in the evaluation of market risk. In a second part, we adapt the VaR model adjusted by the liquidity proposed by Bangia, Diebold, Shuermann and Stroughair (1999) on the French stocks market.

1.

Liquidity Risk Management

Liquidity becomes a major stake for stock exchange authorities as shown by the numerous current reorganisation projects. Liquidity is the ability to transact quickly at low cost a large size position. However, no agreement exits on the proper measurement of liquidity. Terms such as “large” and “quickly” tend to be subjective. Hence, following BIS Report (1999), we may define asset liquidity according to at least one of three dimensions: depth, tightness and resiliency. Tightness, measured by bid-ask spread, indicates how far transaction price diverges from the mid-price. Depth defines the maximum number of shares that can be traded without affecting prevailing quoted market prices. Finally1, resiliency denotes the speed with which price fluctuations resulting from trades are dissipated or how quickly markets clear order imbalances.

1 Another uses concept is immediacy, which refers to the time between the order placement and its execution. This dimension incorporates element of tightness, depth and resiliency.

1

Incorporating Liquidity Risk in Value at Risk Models

1.1

Liquidity risk: a definition

Liquidity risk is the risk of loss arising from the cost of liquidating a position. Liquidity risk increases when markets are not liquid. Typically, market illiquidity manifests itself in the form of important costs of trade, a weak number of trades and wide bid-ask spread2. These factors mean that investors who wish to settle a position are going to have to pay significant costs to do it: they have to bear important cost of trade, relatively long period of wait because of the absence of counterpart or still sell quickly to an unfavourable price. It is obvious that most of the market know liquidity troubles. There are only few markets that can offer an adequate level of liquidity to investors. However, liquidity of these markets cannot be guaranteed all the trading day or during crisis where liquidity dries out. So, liquidity risk is an important factor, at least potentially, which is often ignored by investors. According to Dowd (1998), it’s possible to distinguish two types of liquidity risk. The first one is the “normal” liquidity risk which increase according to the exchanges on markets considered as little liquid. The second type is more insidious. It’s about the liquidity risk arising during stock market crises where the market loses its current liquidity level: investors who settles its positions registers so a loss more important than during normal circumstances. Consequently, we should modify our conception of the value at risk and take into account of liquidity. The relation between liquidity costs and the value at risk is indicated by the figure 1 below. It indicates situation of liquid and illiquid positions. We can settle our liquid position quickly and to obtain the market price without significant liquidity cost. On the other hand, we can settle our illiquid position only by paying conversely proportional costs of liquidation to the period when it was necessary to close its position: the more the period the investor will have agreed to liquidate his positions is long, the less the costs will be high. However, it is necessary to be careful in the fact which during the wait, the price can vary in a very unfavourable way. So, ceteris paribus, an illiquid asset a most important value at risk if we take into account liquidity cost.

2

As defined by Dowd (1998).

2

Incorporating Liquidity Risk in Value at Risk Models

Now we have demonstrated clearly the necessity of taking into account the liquidity costs in the estimation of value at risk models, we have to know how to do it. [INSERT FIGURE 1] 1.2

Modelling liquidity risk: a survey

The last few years have witnessed increasing interest in the measurement and management of liquidity risk. Liquidity risk has been investigated in several ways. Chordia, Roll and Subrahmanyam (2001), Hasbrouck and Seppi (2001) and Huberman and Halka (2001) investigate commonalities in liquidity. Authors consider the existence of a systematic and a specific liquidity risk. Amihud and Mendelson (1986), Brennan and Subrahmanyam (1996), and Jacoby, Fowler and Gottesman (2000) develop a CAPM model and examine the relationship between expected return and the liquidity level. Finally, the last but not the least way, different answers have been proposed to consider liquidity costs in Value at Risk models. Prior research investigated the optimal execution strategy for liquidating portfolios. Jarrow and Subramanian (1997) consider optimal liquidation of an investment portfolio over a fixed horizon. They derive the optimal execution strategy by determining the sales schedule that will maximise the expected total sales values. In the same way, Bertsimas and Lo (1998) derive dynamic optimal trading strategies that minimize the expected cost of execution over an exogenous time horizon. Then, they obtain an optimal sequence of trades as a function of market conditions. Almgren and Chriss (1998) consider the problem of portfolio liquidation with the aim of minimizing a combination of volatility risk and transaction costs arising from permanent and temporary market impact. From a simple linear cost model, they build an efficient frontier in the space of time-dependant probability. They consider the risk-reward tradeoff both from the point of view of classic mean-variance optimization, and the standpoint of Value at Risk. This analysis leads to general insights into optimal portfolio trading, and to several applications including a definition of liquidity-adjusted Value at Risk. Hisata and Yamai (2000) propose a Liquidity Adjusted Value at Risk model based on the framework presented by Almgren and Chriss (1998). Unlike Almgren and Chriss (1998), Hisata and Yamai (2000) turn the sales period into an endogenous variable. This model incorporates the mechanism of the market impact

3

Incorporating Liquidity Risk in Value at Risk Models

caused by the investor’s own dealings through adjusting Value at Risk according to the level of market liquidity and the scale of the investor’s position. According to Lawrence and Robinson (1995), the best way to capture liquidity issues within the VaR would be to match the VaR time horizon with the time investor believes it could take to exit the portfolio. For example, if investors believe liquidity is a problem for the given portfolio, he could estimate the time needed to exit the positions and use this as his VaR time horizon. As this time horizon is increased (due to the illiquidity of the portfolio), his reported VaR would also increase to reflect higher risk. Lawrence and Robinson (1995) appear among the first ones to wonder about the fact that conventional VaR models often exclude asset liquidity risk. From an example of estimation of Value at Risk, the authors find that the largest amount of money a position could lose, with a given degree of confidence, over a one day time horizon is underestimated. Indeed, according to Lawrence and Robinson (1995), the liquidation of a portfolio during a trading day generate an additional liquidity cost. The more the time horizon is weak, the more we underestimate liquidity risk and this, especially the investor’s position is important. The act to settle a position, in itself, will have a consequence, unfavourable for the investor, on the price range. So, on the illiquid markets, consequent volume of assets should bear a high liquidity risk that current Value at Risk models don’t take into account. From this report, to Lawrence and Robinson (1995) provide a generic model of Value at Risk by deriving the optimal execution strategy incorporating the market risk using a mean-standard deviation approach. In the same way, Haberle and Persson (2000) provide a modelling with easier calculation. The authors propose a method based on the notion of orderly liquidation3. This method assumes that single investor can liquidate a fraction of the daily trading volume without significant impact on the market price. Le Saout (2000) distinguishes interday and intraday Value at Risk. The author proposes a new intraday measure of liquidity risk, which is constructed from the return during a market event defined by a volume movement. His results indicate that we can distinguish an exogenous liquidity risk, which refers to liquidity fluctuation driven by factors beyond individual investors’ control, from an endogenous liquidity risk, which

3

The liquidation doesn’t impact the market price.

4

Incorporating Liquidity Risk in Value at Risk Models

refers to liquidity fluctuations driven by individual actions such as the investors’ position. Bangia, Diebold, Schuermann and Stroughair (1999) approach the liquidity risk from another point of view. The summer of 1998 was exceptionally turbulent times for financial markets. During this period, The authors noticed that losses have been amplified by the increase of liquidity risk. Bangia, Diebold, Schuermann and Stroughair (1999) develop a Value at Risk model, which take into account bid-ask spreads. Thus model allows an increase of the Value at Risk when investors decide to liquidate in the same time their portfolio. According to Bangia, Diebold, Schuermann and Stroughair (1999) and Shamroukh (2000), Value at Risk models mustn’t ignore the volatility of the bid-ask spread even if it is likely to be of at least one order of magnitude than the volatility of prices. Value at Risk models, which adjust the holding period upward in line with the inherent liquidity of the position, imply that the liquidation takes place throughout the holding. Shamroukh (2000) argues that scaling the holding period to account for orderly liquidation can only be justified if we allow the portfolio to be liquidated throughout the holding period. Before any modelling, Bangia, Diebold, Schuermann and Stroughair (1999) study the nature of the concept that is liquidity. They make the distinction between an endogenous liquidity and an exogenous liquidity. Exogenous liquidity is the result of market characteristics; it affects all market players without attribute the responsibility of the degradation of the liquidity level to one investor rather than another one. This degradation is the result of a collective action. In contrast, endogenous liquidity is specific to one’s position in the market. Concretely, the exhibition at the liquidity risk of an investor is determined by its position: the more the size is important, and the more the endogenous liquidity risk increases. Figure 2 describes this relationship. [INSERT FIGURE 2] Market risk is characterised by the uncertainty of the prices or returns due to market movements. In a market without friction, risk management deals exclusively with the distribution of returns. A more rigorous management implies so the consideration of the “frictions” such as the assets liquidity especially when important positions have to be liquidated quickly. In this case, investors won’t get the mid-price. Bangia, Diebold,

5

Incorporating Liquidity Risk in Value at Risk Models

Schuermann and Stroughair (1999) argue that the deviation of this liquidation price from the mid-price is important components to model in order to capture the overall risk. So, the authors estimate the Value at Risk in two stages: uncertainty that increases from asset returns and uncertainty due to liquidity risk. Figure 3 summarizes their market taxonomy. [INSERT FIGURE 3]

2.

An empirical analyse on the French Stock Market

In this section, we apply the Liquidity adjusted Value at Risk model provided by Bangia, Diebold, Schuermann and Stroughair (1999) on the French stock market. First, we describe briefly the organisation of the French stock market and the database that were used. Then, we proceed to the estimation of the BDSS’ model. In a third time, we comment our results. Finally; we extend the modelling of the bid-ask spread to weight average spread which allows us to show the importance of the endogenous liquidity component in the Liquidity adjusted Value at Risk model. 2.1.

Data

The Paris Bourse is a computerized limit order market in which trading occurs continuously from 9 a.m. to 5.35 p.m. The trading day takes place in five stages: The pre-opening phase, the market opening and the continuous market From 7.45 a.m. to 9.00 a.m., the market is in its pre-opening phase and orders are accumulated in the centralized order book without any transactions taking place. Each time an order enters the system, a theoretical equilibrium price is computed. At 9.00 a.m., the market opens with a batch auction. Depending on the limit orders received, the central computer automatically calculates the opening price at which the largest number of bids and asks can be matched. From 9.00 a.m. to 5.25 p.m., trading takes place on a continuous basis, and the arrival of a new order immediately triggers one (or several) transaction(s) if a matching order (or orders) exists on the centralized book. The execution price is the price limit placed on

6

Incorporating Liquidity Risk in Value at Risk Models

the matching order in the book. Assuming identical price limits, orders are executed as they arrive: first entered, first matched. From 5.25 p.m. to 5.30 p.m., the market is in its pre-closing phase and orders are accumulated in the order book without any transactions taking place. At 5.30 p.m., the market closes with a batch auction. Depending on the limit orders received, the central computer automatically calculates the closing price at which the largest number of bids and asks can be matched.

The database, BDM, we use contains millions data points on transactions: time stamped stock prices volume, bid/ask spread…We have extracted from the database BDM bidask spread of 41 shares – on the period from October 1st, 1997 to January 3rd, 2000 (that it represents a little more than 500 consecutive trading days). We opt for a daily data to estimate Value at Risk in a one day time horizon. This means that we estimate the maximum loss for a one-day period. 2.2.

Modelling Liquidity Adjusted Value at Risk

Following Bangia, Diebold, Schuermann and Stroughair (1999), we estimate the LAVaR model by proceeding to a decomposition in two types of risk: the price risk which correspond to the potential of loss connected to the depreciation of the asset, and the liquidity risk which corresponds to the liquidity cost supported by the investor who want to sell his position. This model is described below. P _ VaR = Pt (1 − e −2 ,33θσ t ) ECL =

[ (

1 α P S + aσ~t 2

)]

Loss* = Pt (1 − e −2 ,33θσ t ) +

Where:

(1) (2)

[ (

1 α P S + aσ~t 2

)]

Loss* is the Liquidity Adjusted Value at Risk

ECL is the Exogenous Cost of Liquidation, Pt is the mid-price at date t ,

7

(3)

Incorporating Liquidity Risk in Value at Risk Models

θ designs a correction factor for current VaR to take account of leptokurtic

return distribution, σ t is the volatility of the asset at date t , Pα

is the price Value at Risk at date t ,

S is the average bid-ask spread, a corresponds to a scaling factor that corrects the bid-ask spread distribution. σ~t represents the volatility of bid-ask spread.

Estimation of the current VaR The first stage consists in modelling the conventional Value at Risk from the mid-price. Generally, the assumption of normality is violated, that’s why Bangia, Diebold, Schuermann and Stroughair (1999) propose to correct the fat-tails of returns distributions by incorporating a correction factor θ . This factor is such that θ = 1 if the security return distribution is normal, and θ > 1 is an increasing function of the tail causing the deviation from normality. To estimate this correction factor, the author consider a relationship between the kurtosis K and the factor θ : K θ = 1 + φ ⋅ Ln  3

(4)

Where φ is a constant value depends on the tail probability. Following the authors, we estimated the value of this constant by regressing the right hand side of equation (1) with historical Value at Risk for the stocks, which composed our sample. So, we have got a value of 0.039. Then, we are able to estimate the correction factor for our sample. Figure 4 displays the return distribution and the cumulative frequency of one stock, i.e. Saint-Gobain chosen for its liquidity and capitalization. We can notice that return distribution isn’t very far from normality. However, in our sample, as shown in table 1, some securities such as Clarins and Montupet have a return distribution that doesn’t look like a Gaussian distribution. Their Kurtosis are over 10. But, generally, return distributions of the stocks are near the normality distribution. [INSERT FIGURE 4] [INSERT TABLE 1]

8

Incorporating Liquidity Risk in Value at Risk Models

Estimation of the Exogenous Cost of Liquidity The second unknown value that we have to estimate is the scaling factor a that we find in the equation of Exogenous Cost of Liquidity (Eq. 2). Following Bangia, Diebold, Schuermann and Stroughair (1999), we have considered two sub-samples: liquid stocks compose the first one whereas illiquid stocks compose the second sub-sample. Then, we estimate the scaling factor by regressing the right hand side of the Exogenous Cost of Liquidity equation with the historical Value at Risk of the bid-ask spread for each subsample. Figure 5 displays the bid-ask spread distribution and the cumulative frequency of Saint-Gobain. Skewness and kurtosis are relatively high. [INSERT FIGURE 5] Hence, we get scaling factors equal to 6.724 for high liquid stock and 7.809 for little liquidity stock. Effectively, the greater is the deviation from normality, the larger is the scaling factor. Estimation of the Liquidity Adjusted Value at Risk Tables 2 reports the decomposition of market risks (uncertainty in asset returns and due to liquidity risk) for Saint-Gobain stock. [INSERT TABLE 2] Our results can be interpreted as follows: the worst return, given a 99% confidence level, is estimated like this: 99% rSaint − Gobain = −1,129 * 0,022 * 2,33 = −5,84%

So, the Price Value at Risk, at 1% is the following: 99% −5,84% PSaint = 182,06 euros, so P _ VaR = 10,94 euros. − Gobain = 193e

The integration of liquidity risk reduces the price expectation: ECL =

1 [182.06(0.208% + 6.724 ⋅ 0.041% )] = 0.43 2

Hence, PSa* int −Gobain = 182,06 − 0.43 = 181,63 euros. Hence, the overall value at risk, given a 99 percent confidence level for a one day time horizon is:

9

Incorporating Liquidity Risk in Value at Risk Models

* LossSaint − Gobain = 193 − 181,63 = 11,37 euros (5,89%).

The liquidity risk component represent only 3.8% of market risk for Saint-Gobain, nevertheless table 3 indicates that the percentage can be higher for other stocks. Hence, concerning the illiquid stock Fromagerie Bel, the liquidity risk represents more than a half of the market risk. [INSERT TABLE 3] To complete this analysis, we have studied the relation between capitalisation and the Exogenous Cost of Liquidation. Table 4 reports results of the following regression: ECL = a + b ⋅ Capi

(5)

Our results indicate that Exogenous Cost of Liquidation is negatively correlated with the capitalisation. Lower is the capitalisation of the society, more investors are exposed to liquidity risk. This result isn’t a surprise but can be considered like an another proof that small caps funds are aware of liquidity risk. [INSERT TABLE 4] 2.3.

Discussion

The major interest of the model proposed by Bangia, Diebold, Schuermann and Stroughair (1999) isn’t the estimation of Value at Risk because it is based on historical simulation, but the decomposition of the market risk in two types: return risk and liquidity risk. Nevertheless, the application of this model reveals some limits. The first concerns the assumption that in adverse market environments extreme event in returns and extreme events in spreads happen concurrently. The examination of the data showed us that the widest spreads don’t appear during period of extreme price movements. So, the Liquidity Adjusted Value at Risk overestimates the overall risk market. The second limit concern trading volume and position size. The spread indicates the tightness but not the resiliency. The model deals only exogenous liquidity that implies that overall risk is underestimated.

10

Incorporating Liquidity Risk in Value at Risk Models

That’s why, to conclude, we have built the same model with the Weight Average Spread4 for Saint-Gobain stock instead of the bid-ask spread. Table 4 reports our results. The component liquidity risk has strongly increased: 20.94% against 3.81%. In the case of Saint-Gobain, the Weight Average Spread is calculated from 5000 stocks. This means that the endogenous liquidity risk supported by an investor who hold 5000 stocks is 17.13%. [INSERT TABLE 5] Everybody knows the principle of diversification: volatilities of asset risks don’t add directly. Volatility of portfolio of risks is less than the sum of the individual of the individual risks’volatilities. One of the consequences of our analyse is that diversification reduce also endogenous liquidity risk. Financial institutions, which want to reduce its value at risk, have to diversify its portfolio and so reduce the size of its positions.

Conclusion Liquidity risk is an aspect of market risk that has been largely neglected by standard value at risk model. This negligence is certainly due to the fact that no single measure captures the various aspect of liquidity in financial markets. In this paper, we apply the Liquidity Adjusted Value at Risk model provides by Bangia, Diebold, Shuermann and Stroughair (1999) on the French stock market and extend it incorporating the modelling of the Weight Average Spread. We demonstrated that liquidity risk, endogenous and exogenous, can be a very important component of risk market. This means that nowadays, the risk is underestimated. These results are not without consequences on the safety of financial institutions.

4

See Appendix for a definition of Weight Average Spread at Paris Bourse.

11

Incorporating Liquidity Risk in Value at Risk Models

Figures and Tables

FIGURE 1 Value at Risk and holding period VaR

VaR ( Illiquid Position )

VaR ( Liquid Position ) Holding Period Source : Dowd (1998)

FIGURE 2 Effect of position size on liquidation value Price

Point of endogenous illiquidity Ask

Bid Position size

Quote depth

Source : Bangia, Diebold, Schuermann and Stroughair (1999).

12

Incorporating Liquidity Risk in Value at Risk Models

FIGURE 3 Taxonomy of market risk Uncertainty in Market Value

Uncertainty in Asset Returns

Uncertainty due to Liquidity Risk

Exogenous Illiquidity

Endogenous Illiquidity

Source : Bangia, Diebold, Schuermann and Stroughair (1999).

13

Incorporating Liquidity Risk in Value at Risk Models

FIGURE 4 Return Distribution and Cumulative Frequency 12%

6%

10%

5%

8%

4%

6%

3%

4%

2% 1%

2%

0% 0%

-7,0% -6,5% -6,0% -5,5% -5,0% -4,5% -4,0% -3,5% -7% -6% -4% -3% -1% 1% normal

2%

4%

5% 7% normal

Saint-Gobain

Saint-Gobain

FIGURE 5 Bid-Ask spread Distribution and Cumulative Frequency 35%

60%

30%

50%

25%

40%

20% 30% 15% 20%

10%

10%

5%

0%

0% 0,10%

0,15%

0,20%

normal

0,25%

0,30%

0,35%

0,35%

0,33%

0,30%

normal

Saint-Gobain

14

0,28%

0,25%

0,23%

Saint-Gobain

0,20%

Incorporating Liquidity Risk in Value at Risk Models

TABLE 1 Estimation of Price Value at Risk Name

Price (euros)

Kurtosis

Volatility

Correction factor

ACCOR

48

6,088

0,024

1,241

44,77

AIR LIQUIDE

169

3,427

0,021

1,045

160,51

ALCATEL

227,7

4,728

0,033

1,155

208,43

ATOS

172,8

7,804

0,029

1,325

157,83

BIC

44,01

4,036

0,025

1,101

41,30

BNP

92,5

4,928

0,028

1,169

85,60

BONGRAIN

331,4

5,864

0,020

1,228

313,33

BOUYGUES

636

7,667

0,026

1,319

587,15

CANAL +

135

7,571

0,026

1,315

124,80

CAP GEMINI

255,9

6,764

0,032

1,276

232,82

CARREFOUR

183,5

4,848

0,022

1,163

173,09

CASINO

115,1

6,428

0,019

1,259

108,85

CLARINS

118

11,628

0,023

1,461

109,10

114,1

10,769

0,022

1,435

105,98

28

5,334

0,021

1,196

26,44

149,1

7,176

0,026

1,297

137,89

FROM. BEL

700

5,305

0,016

1,194

669,05

INFOGRAMES.

35

4,685

0,025

1,152

32,76

LABINAL

110

4,106

0,026

1,107

102,80

LAFARGE

115,5

3,282

0,024

1,031

109,11

L'OREAL

789

4,448

0,024

1,134

740,04

LVMH

444

3,896

0,024

1,089

418,01

39,79

4,688

0,025

1,152

37,16

MONTUPET

33

13,168

0,029

1,503

29,77

MOULINEX

10,1

5,090

0,026

1,180

9,40

PARIBAS

110,4

8,890

0,024

1,369

102,32 376,04

CLUB MED DYNACTION ELF

MICHELIN

PENAUILLE

VaR 99% (euros)

400

5,730

0,022

1,220

88,45

4,399

0,024

1,130

83,11

SAINT-GOBAIN

193

4,388

0,022

1,129

182,06

SEB

74,8

6,177

0,026

1,246

69,27

SEITA

42,2

4,965

0,023

1,171

39,66

SKIS ROSS.

16

6,682

0,020

1,272

15,07

SODEXHO

168

3,777

0,023

1,078

158,58

REXEL

SPIR COM.

77,5

10,968

0,024

1,441

71,46

SUEZ

159,9

4,446

0,018

1,134

152,56

TECHNIP

104,6

4,377

0,026

1,128

97,78

TOTAL

132

4,624

0,023

1,147

124,05

USINOR

18,92

6,094

0,026

1,241

17,55

VALEO

76,2

3,453

0,025

1,048

71,69

VIVENDI

87,2

3,793

0,018

1,080

83,32

ZODIAC

208

5,726

0,023

1,220

194,94

15

Incorporating Liquidity Risk in Value at Risk Models

TABLE 2 Liquidity Risk Summarised of Saint-Gobain stock Saint-Gobain Price on January, 3rd (euros)

193

Return volatility

0,022

Correction factor

1,129

Price component

10.94

Average bid-ask spread

0,208%

Bid-ask spread volatility

0,041%

Liquidity component

0.43

Liquidity Adjusted Value at Risk (euros)

11.37

% Liquidity Component

3.81%

16

Incorporating Liquidity Risk in Value at Risk Models

TABLE 3 Estimation of Liquidity Adjusted Value at Risk Summary Name

Price (euros)

VaR

Average spread

99%

LAVaR 99%

% Liquidity

ACCOR

48

44,77

0,259%

44,59

5,05%

AIR LIQUIDE

169

160,51

0,215%

160,14

4,16%

ALCATEL

227,7

208,43

0,172%

208,02

2,11%

ATOS

172,8

157,83

0,502%

156,49

8,19%

BIC

44,01

41,30

0,417%

40,97

10,82%

BNP

92,5

85,60

0,189%

85,32

3,93%

BONGRAIN

331,4

313,33

0,811%

306,17

28,39%

BOUYGUES

636

587,15

0,369%

582,88

8,04%

CANAL +

135

124,80

0,315%

124,22

5,38%

CAP GEMINI

255,9

232,82

0,307%

231,36

5,94%

CARREFOUR

183,5

173,09

0,151%

172,80

2,71%

CASINO

115,1

108,85

0,264%

108,42

6,47%

CLARINS

118

109,10

0,514%

108,06

10,51%

114,1

105,98

0,633%

104,51

15,28%

28

26,44

0,886%

25,97

23,08%

149,1

137,89

0,231%

137,34

4,65%

FROM. BEL

700

669,05

1,376%

635,26

52,20%

INFOGRAMES

35

32,76

0,640%

32,32

16,54%

LABINAL

110

102,80

0,739%

101,17

18,51%

CLUB MED DYNACTION ELF

LAFARGE

115,5

109,11

0,241%

108,74

5,60%

L'OREAL

789

740,04

0,197%

738,34

3,35%

LVMH

444

418,01

0,197%

416,98

3,80%

MICHELIN

39,79

37,16

0,230%

37,03

4,75%

MONTUPET

33

29,77

0,971%

28,77

23,60%

MOULINEX

10,1

9,40

0,661%

9,26

17,21%

PARIBAS

110,4

102,32

0,243%

101,82

5,76%

PENAUILLE

400

376,04

1,434%

363,16

34,96%

REXEL

88,45

83,11

0,777%

81,76

20,26%

SEB

74,8

69,27

0,693%

68,37

13,91%

SEITA

42,2

39,66

0,646%

39,06

19,20%

SKIS ROSS.

16

15,07

0,904%

14,76

25,44%

SODEXHO

168

158,58

0,436%

157,08

13,71% 22,70%

SPIR COM.

77,5

71,46

0,862%

69,69

SUEZ

159,9

152,56

0,159%

152,18

4,81%

TECHNIP

104,6

97,78

0,653%

96,57

15,12%

TOTAL

132

124,05

0,187%

123,76

3,61%

USINOR

18,92

17,55

0,320%

17,47

5,52%

VALEO

76,2

71,69

0,387%

71,03

12,84%

VIVENDI

87,2

83,32

0,162%

83,10

5,48%

ZODIAC

208

194,94

0,573%

192,51

15,65%

17

Incorporating Liquidity Risk in Value at Risk Models

TABLE 4 Regression between the Exogenous Cost of Liquidation and the capitalisation

ECL = a + b ⋅ Capi Coefficients 0,014 -4,93E-10

Constant Capitalisation

R²=0,319

Standard Error 0,002 1,15E-10

t-student 8,826 -4,279

F=18,312

TABLE 5 Liquidity Risk Summarised of Saint-Gobain stock Weight Average Spread Saint-Gobain Price on January, 3rd (euros)

193

Return volatility

0.022

Correction factor

1.129

Price component

10.94

Average WAS

1,15%

Bid-ask spread volatility

0,31%

Liquidity component

2.90

Liquidity Adjusted Value at Risk (euros) % Liquidity Component

13.84 20.94%

18

Incorporating Liquidity Risk in Value at Risk Models

Appendix: The Weighted Average Spread In 1994, a new block market was created to best meet domestic and international investors' demand. Member firms are now allowed to buy and sell large blocks in a single transaction and with a set price. This new rule is consistent with the recent changes in French tax law: the FF 4,000 cap on stamp duty for transactions effective after July 1993, and the complete exemption of non-residents from stamp duty since January 1994. The Stock Exchange Council selects eligible stocks. They include all component stocks of the CAC 40 index and other stocks with market capitalization on a similar scale. The number of shares in each transaction must exceed what is referred to as "Normal Market Size", a figure based on average trade volumes in that stock. Total amount per trade may not be less than FF 500,000. Finally block trades must take place at a price which falls within the Weighted Average Spread (WAS) for a standard-size block, as this results from visible buy and sell orders placed through the CAC central trading system. This spread is calculated by adding up orders within the system to reach standard block size and computing the average per share. TABLE A1 Weighted Average Spread Estimation (example) CAC Central Market XYZ TNB 15000 Buy-orders Quantity 3200 2330 2270 2530 4760

Price 629 628 627 626 625

Price 630 631 632 633 634

10.49 a.m. WAS 627 - 631 Sell orders Quantity 2560 1780 5020 6370 5870

Bid : (3200*629) + (2330*628) + (2270*627) + (2530*626) + (4670*625) = 627 15000 Ask : (2560*630) + (1780*631) + (5020*632) + (5640*633) = 631 15000

19

Incorporating Liquidity Risk in Value at Risk Models

In the example above (Table A1), the weighted average spread for 15,000 shares representing the standard-size block of XYZ is 627-631. Thus, member firms can execute orders representing quantities equal to or more than 15,000 shares at a price within the current WAS of 627 to 631. They do this either by a cross-order or by acting as principal with their clients. The Weighted Average Spread for each stock eligible for block trades is calculated and disseminated on-line throughout the trading day. Block trading reporting and disclosure: - Block reporting is immediate: all block trades must be immediately reported to the Paris Bourse by brokerage firm. If two member firms are involved, both must file a declaration. - Block disclosure: the disclosure to the market is immediate when a member firm acts as a broker between two clients for a block trade. Otherwise, timing of disclosure depends on the transaction's size: trades of blocks less than five times the Normal Market Size are disclosed within two hours from the time they are reported. Trades of blocks more than five times the normal market size are disclosed when the market opens on the following business day. Using this methodology, we get the figure 9 which shows us evolution of the visible and the real Weighted Average Spreads (WAS) during the day.

20

Incorporating Liquidity Risk in Value at Risk Models

References Almgren, R. and N. Chriss, "Optimal Execution of Portfolio Transactions", The University of Chicago, Department of Mathematics, working paper, April 1999. Amihud Y. and H. Mendelson, “Asset pricing and the bid-ask spread”, The Journal of Financial Economics, 17, 1986, pp. 223-249. Bangia, A., F.X. Diebold, T. Shuermann and J.D. Stroughair, "Modeling Liquidity Risk", Risk 12, january 1999, pp. 68-73. Bank for International Settlements, “Market Liquidity: Research Findings ans Selected Policy Implications”, Report, may 1999. Bertsimas, D. and A.W. Lo, "Optimal control of execution costs", Journal of Financial Markets 1, 1998, pp. 1-50. Brennan M. et A. Subrahmanyam, “Market microstructure and asset pricing: On the compensation for market illiquidity in stock returns”, The Journal of Financial Economics, 41, 1996, pp. 341-364. Chordia T., R. Roll and A. Subrahmanyam, “Commonality in liquidity”, The Journal of Financial Economics, 56, 2000, pp. 3-28. Dowd, K., Beyond Value at Risk : the New Science of Risk Management, ed. John Wiley & Son, 1998. Häberle, R. and P.G Persson, "Incorporating Market Liquidity Constraints in Value at Risk", Banques & Marchés 44, janvier-février 2000, pp. 14-20. Hasbrouck J. and D.J. Seppi, 2000, “Common factors in prices, order flows and liquidity”, The Journal of Financial Economics, 59, 2001, pp. 383-411 Hisata Y. and Y. Yamai, “Research toward the practical application of liquidity risk evaluation methods”, Discussion Paper, IMES Bank of Japan. Huberman G. and D. Halka, “Systematic liquidity”, Journal of Financial Research, 2001, forthcoming. Jacoby G., D.J. Fowler and A.A. Gottesman, “The capital asset pricing model and the liquidity effect: A theoretical approach”, Journal of Financial Markets, 3, 2000, pp. 69-81. Jarrow, R. and A. Subramanian, "Mopping up liquidity", Risk 10, december 1997, pp. 170-173. Lawrence, C. and G. Robinson, "Liquidity, Dynamic Hedging and Value at Risk", in Risk Management for Financial Institutions, ed. Risk Publications, 1998, pp. 63-72. Le Saout, E., "Beyond the liquidity : from microstructure to liquidity risk management", Ph. D University of Rennes 1, November 2000. Shamroukh, N., “Modelling liquidity risk in VaR models”, Working Paper, Algorithmics UK, 2000.

21