## MEASURING MARKET LIQUIDITY RISK OF A PORTFOLIO USING LIQUIDITY ADJUSTED VALUE AT RISK MODEL

MEASURING MARKET LIQUIDITY RISK OF A PORTFOLIO USING LIQUIDITY ADJUSTED VALUE AT RISK MODEL.

ABSTRACT: The paper introduces an enhanced liquidity adjusted value at risk measure named the LAVAR applied to a sample of listed securities in a financial market. Variance covariance method is used to compute the value at risk and then the liquidity adjusted value at risk is used to incorporate the market liquidity risk into the value at risk. Cost of Liquidity of each security held in the investor’s portfolio is determined by use of bid ask spread. This COL will be used to incorporate the market liquidity risk with VaR to get the LAVAR of the security. Weighted Sum Approach(WSA) is used to obtain the COL, VaR and LAVAR of the portfolio. The paper involves intraday trading therefore determining intraday VaR.

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Keywords: market liquidity risk, bid ask spread, cost of liquidity, variance covariance method, intraday VaR, LAVAR, weighted average approach.

CHAPTER ONE.

INTRODUCTION.

Background
Value at risk (VaR) is a statistical technique used to measure and quantify the level of financial risk within a firm or investment portfolio over a specific time frame. This metric is most commonly used by investment and commercial banks to determine the extent and occurrence ratio of potential losses in their institutional portfolios. AR calculations can be applied to specific positions or portfolios as a whole or to measure firm-wide risk exposure. Asset managers and ordinary investors care about liquidity, insofar as it a?ects the return on their investments, because illiquid securities cost more to buy and sell. Illiquidity, which is opposite to liquidity, eats into an investor’s return. Another important aspect of liquidity is its e?ect on a portfolio evaluation: portfolio liquidation, including illiquid assets, may reduce the value of a portfolio signi?cantly. From another point of view, a positive relationship between expected stock returns and illiquidity levels has been found (Amihud and Mendelson,1986, Brennan and Subrahmanyam,1996, Datar et al., 1998) which opens up new investment opportunities.

According to Sarr and Lybek (2002), liquid markets should exhibit five characteristics:
Tightness. Which refers to low transaction costs.

Immediacy. Which represents the high speed order execution.

Depth. Which refers to existence of limit orders.

Breadth. Meaning small market impact of large market orders.

Resiliency. Which means a flow of new orders to correct market imbalances.

In order to understand liquidity risk, we need to understand liquidity first. Liquidity describes the trade-off between how quickly an asset or security can be bought or sold in the market and how large the degree of affecting the asset or asset or security’s price. In a liquid market, an asset or security can be sold quickly without reducing the price much. However, in a relatively illiquid market, sell an asset or security will need to reduce its price to some degree. There are two types of liquidity:
Market liquidity, which is the ability that a market allows assets to be bought and sold without affecting the asset’s price to a significant degree. Cash is the most liquid asset.

Funding liquidity, which is the ability an institution ensures its payments with immediacy.

Therefore, liquidity risk is the risk that an asset or security could not be sold quickly enough without causing great change in the price that can lead to great losses. There are two types of liquidity risk respectively:
Market(asset) liquidity risk, which is the risk that a position cannot be closed (or an asset cannot be sold) quickly enough without influencing the market price to a significant degree.

Fund (cash flow) liquidity risk, which is the risk that an institution is unable to repay its liabilities or meet its obligations when they come due, which leads to default.

Here in this paper, we will focus on market liquidity risk. Traditionally market liquidity risk is measured by the following four aspects.

Width, which is also called bid-ask spread. It is the difference between the ask price and the bid price. It can also be calculated in percentage form, which is proportional spread. It is the difference divided by the average of the ask and bid price. It is measured in a price dimension.

Depth, it is the market ability to absorb the exit of a position without changing the price dramatically, which is that the number of assets can be bought without having price appreciation. It is measured in a quantity dimension.

Resiliency. It is the time the market need to go back from incorrect price. It is measured in time dimension.

Volume. Another measure becoming popular recently. It is the amount of a certain security traded during a certain time period.

In this paper we are going to incorporate market liquidity risk into Value at Risk using the Liquidity Adjusted Value at Risk. (LAVAR). LAVAR is a common method of incorporating market liquidity risk. Market liquidity risk manifests itself as market risk: inability to sell an asset drives the market price down, or worse, renders the market price indecipherable. Market liquidity is a problem created by the interaction between buyers and sellers in the financial markets. If the seller’s position is larger relative to the market, this is called endogenous liquidity risk (a feature of the seller). if the market has withdrawn the buyers, this is called exogenous liquidity risk (a characteristic of the market which is a collection of buyers); a typical indicator here is an abnormally wide bid-ask spread. This common way of to include market liquidity financial risk model (not necessarily a valuation model) is to adjust or “penalize” the measure by adding/subtracting one-half the bid-ask spread.

COST OF LIQUIDITY (COL).

Cost of liquidity represents the liquidity cost of a round trip which means the cost of buying and selling a stock today. In order to calculate COL, we will use bid-ask spread. It is the difference between the highest ask price today and the lowest bid price today.

St=PH,t-PL,t , where St is the bid ask spread, PH,t is the highest ask price today and PL,t is the lowest bid price today.
Spread can also be expressed in percentage term, which is proportional spread:
proportional St=PH,t-PL,tPM,t ,
where PM,t is the average of ask price and bid price. The full spread represents the liquidity cost of a round trip, which means the cost of buying and selling the stock today. Here, we will only need to consider the liquidity cost of selling the stock, hence the cost of liquidity (COL) is only half of the spread.

COLt=12 x St=12PH,t-PL,t;COLt=12 x PM,t x proportional St;Percentage COLt=12 x proportional St=12PH,t-PL,tPM,t. 1.2. Statement of the problem.

Nowadays, hedge funds still use Vale-at-Risk(VaR) to measure the market risk. However, this measure can cause problems because when the volume of the position is large enough to cause price effect on the spread, the trading price is not at the mid-price. This has a result that the real price will depend on both the value of the spread and the price effect of the trading volume. And thus, the market liquidity plays an important role. To sum up, we see that the normal VaR concept lacks a rigorous treatment of market liquidity risk
This paper is devoted to incorporation of liquidity risk into VaR model, which could better reflect poor liquidity in the VaR framework. In order to incorporate the liquidity, risk this paper uses LAVAR model. With this measure, we could measure different stocks liquidity risk. When we will need to quantify the liquidity risk, we will use the concept of Cost of Liquidity (COL) which is half of the spread.

1.3. General Objective.

To measure market liquidity risk of a portfolio using Liquidity Adjusted Value at Risk (LAVAR) model.

Specific Objectives.

To determine cost of liquidity (COL) of each asset held in the portfolio and the cost of liquidity (COL) of the portfolio using the weighted sum approach(WSA).

To determine the liquidity adjusted value at risk (LAVAR) of each asset held in the portfolio and the liquidity adjusted value at risk (LAVAR) of the portfolio using the weighted sum approach(WSA).

To compare COL and LAVAR of each asset held in the portfolio.

1.6. Justification of the study.

Understanding the market liquidity risk of an asset before taking a position is essential to an investor or an investment firm. LAVAR is an important technique of incorporating market risk with VaR. Determining COL and LAVAR of positions held or to be held in a portfolio will help investors and fund managers to care about liquidity, in so far as it effects the return on their investments, because illiquid assets cost more to buy and sell. Illiquidity, which is opposite to liquidity, cuts into an investor’s returns. This leads to the significance of this study as it will help financial investors to incorporate liquidity costs in their expected returns.

CHAPTER TWO.

LITERATURE REVIEW
The risk that a given security or asset cannot be traded quickly enough in the market to prevent or minimize a loss is termed as market liquidity risk. The last decade has seen considerable amount of research work directed towards managing market liquidity risk while pricing an option. According to Acharya and Pederson (2005), liquidity is risky and has commonality. Which means that it varies with time for an individual asset and for the whole financial market. Their liquidity adjusted Capital Asset Pricing Model provides a unified theoretical framework that explains the empirical findings that returns sensitivity to market liquidity is priced (Pastor and Stambaugh,2003), that average return is priced (Amihud and Mendelson,1986) and that liquidity commoves with returns and predicts future returns (Amihud,2002;Chordia et al.,2001;Bekaert et al.,2007). In other words, this model implies that investors should consider the asset’s performance and tradability both in market downturns and when liquidity “dries up”. Brunnermeier and Pedersen (2009) provide a model that links an asset’s market liquidity and trader’s funding liquidity. The model explains empirically documented features that market liquidity;
Can suddenly “dry up”. Means that market liquidity can suddenly seize to exist.

Has commonality across securities
Is related to volatility
Is subject to “flight to quality”
Co-moves with the market
Importantly, the model link’s a security market illiquidity and risk premium to its margin requirement (i.e. finding use) and the general shadow cost of funding.

There are many alternative measures of measures of liquidity in literature. Measures that have appeared in the literature include quoted bid-ask spreads, effective bid-ask spreads, turnover, the ratio of absolute returns-to-volume, and adverse- selection and market-making cost of components of price impact. Korajczyk and Sadka (2008) estimate latent factor models of liquidity and a measure of global, across-measure systematic liquidity by estimating a latent factor model pooled across all measures. The results show that there is commonality, across assets, for each individual measure of liquidity and that these common factors are correlated across measures of liquidity. Return shocks are contemporaneously correlated with liquidity shocks and lead changes in liquidity. Additionally, shocks to liquidity tend to die out slowly over time.

Liquidity risk is neglected by widely used risk management measures such as VaR. Derivative users generally calculate a VaR measure for their derivatives portfolio and by not taking into account the liquidity risk component; they underestimate the portfolio risk exposures. VaR I an estimate of the maximum potential loss that may be incurred on a position for a given period of time horizon and a specified level of confidence. Since the publication of the market-risk-management system Risk Metrics of JP Morgan of 1994, VaR has gained increasing acceptance and is now considered as industry’s standard tool to measure market risk. In calculating VaR, it is assumed that the positions concerned can be liquidated or hedged within a fixed and fairly short timeframe (in general one day to ten days), that the liquidation of positions will have no impact on the market and the that the bid-ask spread will remain stable irrespective of the size of the position, in essence a perfect market is assumed. The price referred to is often the mid-price or the last known market price. However, the quoted market price cannot be used as a basis for valuating a portfolio that is to be sold on a less that perfectly liquid market: in practice, account must be taken of its orderly liquidation value or even its distress liquidation value.

Jarrow and Subramanian (1997) were among the first to estimate liquidity-adjusted VaR (LAVAR), taking account of the expected execution lag in closing a position and the market impact of prices being adversely affected by a quantity discount that varies with the size of the trade. The model requires three quantities which increase the loss level-namely a liquidity discount, the volatility of the liquidity discount and the volatility of the time horizon of liquidation. Whilst this model is robust and fairly easy to implement, estimating these quantities is by no means trivial. Indeed, some may only be determined empirically with the accompanying introduction of significant bias. Bangia et al. (2002) propose similar measures of LAVAR, they classify the liquidity risk into two different categories:
The exogenous illiquidity that depends on the general conditions of the market
By focusing on the exogenous risk, they construct an LAVAR measure for both the underlying instrument and the bid-ask spread. Specifically, they adjust the VaR number for “fat” tails and for the variation of the bid-ask spread.

Hisata and Yamai (2000) propose a practical framework for the quantification of LAVAR which incorporates the market liquidity of financial products. The framework incorporates the mechanism of the market impact caused by the investor’s own dealings through adjusting VaR according to the level of market liquidity and the scale of the investor’s position. In addition, they propose a closed-form solution for calculating LAVAR as well as a method of estimating portfolio LAVAR. Angelidis and Benos (2006) relax the traditional, yet unrealistic, assumption of a perfect, frictionless financial market where investors can either buy or sell any amount of stock without causing significant price changes. They extend the work of Madhavan et al. (1997) (who argue that the traded volume can explain price movements) and develop a liquidity VaR measure based on spread components. Under this framework, the liquidity risk is decomposed into endogenous components, thereby permitting an assessment of the liquid risk of a specific position.

Stange and Kaserer (2011) analyze the importance of liquidity risk using a comprehensive liquidity measure, weighted spread, in a Value-at-Risk (VaR) framework. The weighted spread measure extract liquidity costs by order size from the limit order book. Using a unique, representative data set of 160 German stocks over 5.5 years, they show that liquidity risk is an important risk component. Liquidity risk increases the total price risk by over 25%, even at 10-day horizons and for liquid blue chip stocks and especially in larger, yet realistic order sizes beyond € 1 million. When correcting for liquidity risk, it is commonly assumed that liquidity risk can simply be added to price risk. The empirical results show that this is not correct, as the correlation between liquidity and price is non-perfect and total risk is thus overestimated. According to Ernst et al. (2012) liquidity costs, which measure market liquidity, are non-normally distributed, displaying fat tails and skewness. Most liquidity risk models either ignore this fact or use the historical distribution to empirically estimate worst losses. They suggest a parametric approach based on the Cornish–Fisher approximation to account for non-normality in liquidity risk. They show how to implement this methodology in a large sample of stocks and provide evidence that it produces much more accurate results than alternative empirical risk estimation.

Recent work has begun to incorporate vanishing liquidity in times of crisis. Le Saout (2002) provides a good review of liquidity risk in VaR models and gives a comprehensive overview of recent research in the field. Lawrence and Robinson (1995: 64) were among the first to identify and establish that conventional VaR models often exclude asset liquidity risk. They argued that the best way to capture liquidity issues within the VaR framework would be to match the VaR time horizon with the time investors believed it could take to exit or liquidate the portfolio. They established that the liquidation of a portfolio over several trading days generated additional liquidity costs. Diebold et al. (1996) pointed out that the scaling of volatilities by the square root of time is only applicable if log changes of price returns are i.i.d. (independently and identically distributed) and, in addition, normally distributed. They noted that high frequency financial asset returns are not i.i.d. and that, even if they are conditional mean independent they are definitely not mean variance independent (see also Bollersev, Chou ; Kroner 1992: 20 and Diebold ; Lopez 1995: 433 for evidence of strong volatility persistence in financial asset returns.) Diebold et al. (1996) showed that scaling by the square root of time magnifies the volatility fluctuations i.e. scaling results in large conditional variance fluctuations of long horizon returns, when in fact the opposite is true. Jarrow and Subramanian (1997: 171, 2001: 450) considered optimal liquidation of an investment over a fixed horizon. They characterized the costs and benefits of block sales versus slow liquidation and they proposed an endogenous liquidity adjustment to the standard VaR measure. The model requires three quantities which increase the loss level – namely a liquidity discount, the volatility of the liquidity discount and the volatility of the time horizon to liquidation. The authors themselves acknowledge that traders or firms must collect time series data on the shares traded, prices received and time to execution in order to estimate these quantities. Whilst this model is robust and fairly easy to implement, estimating these quantities is by no means trivial. Indeed, some may only be determined empirically with the accompanying introduction of significant bias. Fernandez (1999: 2) examined liquidity risk in the aftermath of the 1998 LTCM liquidity crisis. He argued that: “…financial markets are undergoing rapid structural change, which may be contributing to liquidity risk. These changes along with rising homogeneity of market participants’ behavior are increasing concentration and ‘herding behavior’ and eliminating ‘friction’ which may prove disadvantageous in a market correction.” (Fernandez, 1999: 3)

This section provided a literature review of recent research in the field of liquidity risk as well as insight into some of the methods which have been developed to mitigate and manage it. Hisata and Yamai (2000: 90) provide – to our knowledge – the only coherent portfolio approach to liquidity risk. The next section will explore the possibility of combining Jarrow and Subramanian’s (1997, 2001) – henceforth JS-model (for evaluating individual instrument liquidity-adjusted VaR – henceforth LAVAR) and standard portfolio theory to produce a robust portfolio LAVAR approach under normal trading conditions, i.e. endogenous liquidity risk. This technique represents a variation on Hisata and Yamai’s (2000: 90) portfolio approach, but also incorporates several elements discussed by them. The aim is thus to construct a liquidity adjusted VaR (LAVAR) at a portfolio level. Whilst many LAVAR models exist, the JS model is increasing in importance as the endogenous liquidity model of choice (for example, see Umut (2004: 322). Although Çetin’s (2004) work is currently enjoying some popularity – see Jarrow and Protter (2005: 12) – more work is required before the adjustments recommended can be effectively and robustly implemented into existing VaR models). The JS model’s results will be used later to combine individual LAVARs into a portfolio LAVAR. No attempt will be made here to reproduce in full the underlying theoretical framework of the JS model. Nevertheless, it is instructive to provide a brief summary of the structure and constituents of the JS model equations. Having established this JS model overview, the individual instrument LAVARs will be combined using standard portfolio theory to produce a portfolio LAVAR equation. This formula will then be tested on actual profit and loss and accompanying non-liquidity-adjusted VaR data from several South African equity portfolios and the results compared.

CHAPTER THREE RESEARCH METHODOLOGY.

One approach for a full portfolio level treatment for liquidity risk is suggested in Bangia et al. (2002). They suggest computing the portfolio-level bid and ask series by taking the weighted sum of the bids and asks of the instruments. However, Bangia et al. (2002) assume that the returns are normally distributed while computing the portfolio LAVAR estimates using this approach. Many studies (Stange and Kaserere 2011, Ernst et al. 2012) show that the assumption of normally distributed returns is rejected for most financial time series, including those for individual stocks, exchange rates, precious metals etc. Using the above approach, we will determine the COL of the portfolio and the LAVAR of the portfolio using the Weighted Sum Approach.

To achieve or specific objectives we will require he intraday bid and ask prices of twenty assets that we will include in our portfolio. Since VaR will use one-day horizon for our investment we will require price data for only one day. We will acquire our data online from yahoo finance on 29 FEB 2018. We will require the highest ask prices and lowest bid prices then tabulate the data and proceed to compute the cost of liquidity of each asset in the portfolio.

THE VARIANCE-COVARIANCE METHOD OF CALCULATING VALUE AT RISK.

This method assumes that stock returns are normally distributed. In other words, it requires we estimate only two factors-an expected (average) return and a standard deviation. The method uses a certain level of confidence set by the investor.

VaR?=value of the investment x volatility x Z?VaR?=W x Vx Z?Where, W= value of the position on an asset, V= one-day volatility and Z? is he normal deviation. The VaR of the entire portfolio will be given by;
LIQUIDITY ADJUSTED VALUE AT RISK MODEL.

the assumptions in LAVAR model.

The expected returns are normally distributed.

For intraday trading, the drift (positive expected) is assumed to be zero.

Liquidity adjustment increases the VaR by one-half the spread. VaR at a certain confidence level is given by;
VaR?=value of the investment x volatility x Z?Liquidity CostLC=0.5 x Stliquidity adjusted VaR=value of an investment x -drift%+volatility x Z?+LCIn this case, we will use intraday trading and therefore our volatility will be daily volatility of each asset included in the portfolio. Drift is the position positive expected return but for the investment horizon being daily the daily expected return will return down to zero(a common practice). therefore;
liquidity adjusted VaR=value of an investment x volatility x Z?+LCIf we let the position value to be W, daily volatility to be V then;
LAVAR=W x V x Z?+LC
where ? is the level of significance. It is also assumed that returns are normally distributed that’s why we use the left tail of normal distribution to signify losses.

Cost of liquidity of an asset will be given
COLt=12 x St=12PH,t-PL,t………………………………………………………………………… 1

The cost of liquidity for the entire portfolio will be;
COLtp=i=1NWiCOLti ………………………………..2
where Wi the weight of the ith asset in the portfolio, N is the number of assets in the portfolio and COLti is the cost of liquidity of ith asset in the portfolio. Our N will be 20 in this case.

VaR is a number that represents the potential change in a portfolio/ass’s future value. This value change is measured in terms of;
The horizon over which the portfolio’s/asset’s value is measured.

‘The degree of confidence” chosen by the investor or he risk manager.

In our case our investment horizon will be one day because we will use the intraday trading for traders who wish to close their positions within a day. To compute the VaR of an asset over a 1- day horizon wih C% chance (confidence level) that the actual loss in the asset’s value does not exceed VaR estimate will be;
VaR?=W x Vx Z? ……………………(i).

Where, W= value of the position on an asset, V=one-day volatility and Z? is he normal deviation. The VaR of the entire portfolio will be given by;
VaRP?=i=1NWiVaR(?)i ……………………(ii).

where Wi is the weight of the ith asset in the portfolio and VaR(?)i is the VaR of the ith asset and N is he number of assets in the portfolio. We will also need to incorporate the market liquidity risk to VaR. To achieve this, we will use the liquidity adjusted Value at Risk (LAVAR). LAVAR is will be computed by adding a half of spread to the VaR. We will add one half spread because we will already be holding the assets and only selling cost will be incurred. LAVAR will be gotten by;
LAVAR?=W x V x Z?+0.5LC……………………………………………………………3.

Where W is the value of the position, V is the intraday volatility of the asset and LC is the liquidity cost given by;
LC=PH,t-PL,t; where PH,t is the highest ask price and PL,t is the lowest bid price.

The portfolio’s LAVAR will be given by;
LAVAR(?)P=i=1NWiLAVAR(?)i…………………………………………………4.

Where Wi will be the weight of the ith asset in the portfolio and LAVAR(?)i will be the LAVAR of ith asset in the portfolio. Z? is a standard normal (0,1) distribution.

The cost of liquidity of the assets and the portfolio will be estimated using equations 1 and 2 respectively.

Liquidity adjusted Value at Risk of the assets and the portfolio will be estimated by use of equations 3 and 4 respectively.

3.1 SAMPLING
Our sample will contain 15 stocks from all over the world and 5 major currency pairs as the assets to make the portfolio.

The following are the assets we will include in our portfolio.

STOCK ABBREVIATION
Apple Stock APPL
Tesla Stock TSL
Netflix Stock NFLX
Amazon Stock AMZN
Ge Stock GE
Disney Stock DSN
Snapchat Stock SPCT
Sunpower Stock SNPW
Total Stock TTL
Veolia Environment Stock SNPW
Microsoft Stock MCSFT
Currency pairs will include;
EUR/USD
GBP/USD
USD/JPY
EUR/GBP
AUD/JPY
3.2.DATA PRESENTATION AND ANALYSIS.

The stocks and currency pairs prices of the above assets will be downloaded from the yahoo finance website. This is because the data is readily available in Yahoo Finance. We will use intraday prices and intraday volatilities. The required data will be the ask and bid prices. We will then tabulate this data in order to compute the required values i.e. COL and LAVAR for the individual stocks and for the entire portfolio.

We will later do conclusions based on these values to determine which assets will be easier to exit without reducing the value of the portfolio. We will also compare various assets COL. We will also determine the LAVAR of the portfolio, which will give us the worst-case scenario, or the maximum value we can lose if we chose to liquidate the whole portfolio.

CHAPTER 4.

WORK PLAN AND BUDGETING
4.1. WORK PLAN AND PROJECT TIME LINE.

The following is the schedule we will follow in doing our study. We use the Gannt Chart as below.

ACTIVITIES PLACE OF ACTIVITY DECEMBER 2017 JANUARY 2018 FEBRUARY-MARCH 2018
Proposal writing JKUAT main campus JujaProposal presentation. JKUAT main campus JujaData collection
JKUAT main campus JujaReport writing and presentation JKUAT main campus JujaAll activities to be done within the timeline allocated.

4.2 BUDGETING
The proposed budget is as follows.

ITEM/ACTIVITY QUANTITY PROPOSED COST(KSH) TOTAL COSTING
Data bundles 3000
1000 4000
Printing out proposal copy. 5 copies 1500
1500
Printing out project report 5 copies 2000 2000
Binding of the study reports 5 copies 100 500
TOTAL COST 8000
References.

Bangia,A., Diebold, F.X., Schuermann, T.,& Stroughair,J.D.(2001). Modeling liquidity risk,with implications for traditional market risk measurement and management. In Risk management: The state of the art (pp. 3-13). Springer US.

Amihud and Mendelson (1986). Asset pricing and the bid-ask spread. Journal of financial Economics.

Brennan and Subrahmanyam (1996). Market Microstructure and asset pricing: On the compensation for illiquidity in stock returns. Journal of financial economics.

Datar et al (1998). Liquidity and stock returns: An alternative test. Journal of financial economics.

Jarrow and Subramanian (1997). Assessing Performance of Liquidity Adjusted Value-At-Risk. -sciedu press.

Le Saout (2002). Measuring Liquidity Risk In An Emerging Market: Liquidity Adjusted Value At Risk Approach For High Frequency Data. -EconJournals.com
Danielson J. & De Veries,C.G.(2000). Value At Risk and etreme returns. Annales d’Economie et de Statistique,239-270.

Danielson J, Jorgensen,B.N, Samorodnitsky,G,Sarma,M & de Vries,C.G(2013). Fat Tails,VaR and subadditivity. Journal of econometrics .172(2),283-291.

Erwan (2002: 8). Incorporating liquidity risk in var Models. Universitie de Rennes.www.gloriamundi.org.

Stange and Kaserere (2011), Ernst et al. (2012). Evaluation of portfolio level liquidity adjusted value at risk model formulated by accounting for non-normality. Apria 2017. Apria 2017.

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