Analysis of the Value at Risk (VaR) of a Portfolio of 4 Shares - Essay Example

?Analysis of the Value at Risk (VaR) of a Portfolio of 4 Shares Value at Risk (VaR) has grown to be an accepted measure by the analysts to identify the risk associated with the markets. VaR is understood as the greatest possible transformation in portfolio value within the given probable period. The VaR has many approaches; it is used in analyzing the risk management, to assess the performance of risk takers and to make accurate approximation in the probable market. The main aim of the study is to review and assess the performance of the important methodologies of the univariate VaR, by giving special importance to the underlying statements and to consider the logical flaws. In this study, we highlight the various methodologies like the value at risk, Mont Carlo VaR analysis, bootstrap method of analysis and the portfolio analysis. The study in this context analyzes the performance of the shares of the 4 companies namely Amec PLC, Lloyds Banking Group PLC, Lonmin PLC and Tesco PLC. Introduction: Value-at-Risk: The introduction of Value-at-Risk (VaR) as an established method for measuring market risk is an element of the advancement of risk management. The relevance of VaR has been extensive from its early use in security houses to profit-making banks and business and from marketplace risk to credit risk. Subsequent to the foreword in October 1994 by the Risk metrics by JP Morgan, the VaR is an assessment of the worst estimated failure that a firm may bear over a stage of time that has been particular by user, under standard market circumstances and a specific level of assurance. This evaluation may be attained in various ways, by means of a numerical model or by Computer calculated models. VaR is a calculation of market risk. It is the highest loss which can happen by incurring N % confidence above the property period of n days. VaR is the predictable loss of a portfolio over a particular time stage for a lay down level of probability. For instance, if every day VaR is declared as ?100,000 to a 95% level of confidence and throughout the day there is simply a 5% probability, then the next day loss is better than ?100,000. VaR dealings the potential failure in market value of a portfolio by means of expected instability and correlation. The “correlation” is considered as the correlation that is present between the market value of diverse appliance in a bank’s portfolio. VaR is considered inside a given confidence gap, typically 95% or 99%; it seeks to compute the probable losses from a place or portfolio under various normal situations. The description of regularity is vital and is fundamentally a statistical conception that varies by the organization and by risk management system. Considering merely, the most frequently used VaR models suppose that the price of resources in the financial markets go behind a standard distribution. To execute VaR, all of a firm’s situations data must be meet into one centralized database. Once this is absolute, the general risk has to be designed by combined risks from specified instruments within the whole portfolio. The possible shift in each gadget (that is the single risk factor) has to be incidental from past every day price movements above a given examination period. For dictatorial purpose, this stage is at least one year. Hence, the data where the VaR estimates are supported must confine all appropriate daily market shifts over the preceding year.  VaR is simply a measure of a bank’s risk experience; it an instrument for computing market risk experience. There is no one VaR integer for a single portfolio, as diverse methodologies used for scheming VaR produced dissimilar results. The VaR number confines only those risks that can be calculated in quantitative terms; it does not confine risk exposure such as prepared risk, liquidity risk, regulatory risk or autonomous risk. Assumption of Normality: An allocation is explained as usual, if there is greatest probability that any examination of the populace sample will have an importance that is close to the mean, and a small probability of including a price that is distant from the mean. The normal distribution curve is used by numerous VaR models, which suppose that asset returns go behind a standard pattern. A VaR model uses the standard curve to approximation the losses that an organization may experience over a specified time period. The normal distributions show the possibility of a specific observation moving an assured distance from the mean calculation ways. There are diverse methods for calculating VaR. They are: The variance/covariance /parametric method Historical simulation; Monte Carlo simulation. Variance-Covariance Method: This technique assumes that the returns on risk factors are usually dispersed, the correlations among risk factors are stable and the delta (the price sensitivity in transformation of risk factor) of every portfolio component is stable. By means of the correlation method, the instability of every risk factor is taken out from the historical examination period. Historical data on speculation returns is consequently necessary. Historical Simulation Method: The historic simulation method for calculating VaR is the easiest and evades some of the consequences of the correlation method. Particularly, the three main assumptions after correlation (usually circulated returns, regular correlations, and stable deltas) are not required in this case. For historical simulation, the model analyze potential loss incurred via real historical returns in the risk factors and so confine the non-normal allocation of risk factor returns. Monte Carlo Simulation Method: The third method, Monte Carlo simulation is extra flexible than the prior ones. As with historical simulation, Monte Carlo simulation permits the risk manager to use real historical distributions for risk factor returns moderately than including to suppose normal returns. A large number of indiscriminately produced simulations are run further in time by means of instability and correlation estimate preferred by the risk manager. Each simulation will be dissimilar, but in totality the simulations will combine to the selected statistical parameters (that is, historical distributions and instability and association estimates). This technique is much practical than the preceding two models and consequently much likely to approximate VaR more precisely. Conversely its execution necessitates influential computers and at hand is also a necessary to achieve calculation that is longer. Background to the Data Sample: Portfolio analysis consists of focusing each of the association's services and products through a succession of finer screens. During the time of scarce resources, it is essential to screen out services and programs that are not significant to most members. Those who appeal to an additional limited part can be funded by those desiring the service or product rather than by dues. A portfolio analysis instrument permits a user to evaluate and contrast investment choices according to a quantity of qualitative and quantitative criteria, containing different types of cost and risk. Portfolio analysis is a useful method to assess the services and products that make up an association with industry portfolio for strategic planning; such an instrument can make holistic, top-down depictions of options and their implications, possibly over a lot of years into the future. For investment planning, such an instrument can help in balancing plans and in marginal analysis that is measuring where to subtract or add the marginal dollar. Such procedures are important in facilities based planning. “Portfolio management involves decisions about investment (both increase and reductions). These investment decisions are designed to optimize risk and return in the portfolio; meaning that, as well as the level of profits generated by the portfolio, the level of risk (volatility) in those profits would also be considered and managed” (Ryals 2008, p. 177). Portfolio analysis assists to decide which of these services and products should be highlighted and which should be phased out, founded on objective criteria. All associations are involved in excess of one business. Some of these contain meetings and conventions, publishing, education and training, research, government representation, public relations, standards setting, etc. “The investment process includes investment policy and investment management. Investment policy requires the identification and statement of investor risk preference, portfolio investment objective, and passive asset allocation. The latter includes determining the amount in the out of portfolio cash reserves and portfolio stock/bond allocation along with annual portfolio rebalancing” (Haslem 2003, p. 146). The main aim of the project is to conduct an analysis of the Value at Risk (VaR) of a portfolio of 4 shares. This portfolio analysis, which includes the four shares are Amec PLC, Lloyds Banking Group PLC, Lonmin PLC and Tesco PLC. The study generally aims in analyzing the shares by the different financial models utilized and by the variety of analyses of the statistical models like the analytical Variation method, the boot strap method and Monte Carol Var. Table of Portfolio Analysis: The 4 shares of the four companies are used for portfolio analysis are the Amec PLC, Lloyds Banking Group PLC, Lonmin PLC and Tesco PLC. Entity Name: Amec PLC Lloyds Banking Group PLC Lonmin PLC Tesco PLC Key: C000087214 C000015577 C000015578 C000015623 Currency: GBP GBP GBP GBP 2002 318.050 174.690 117,990.700 47,745.050 2003 604.000 193.700 158,292.100 65,648.060 2004 718.520 223.010 138,194.300 84,159.000 2005 857.220 247.190 252,784.600 88,877.130 2006 1,081.990 308.530 482,429.600 111,328.900 2007 2,197.230 271.200 503,722.900 134,221.200 2008 1,319.290 79.350 151,344.600 104,111.300 2009 2,170.480 64.350 341,068.000 128,123.300 The above table shows portfolio analysis of each company. Following are the mean and standard deviation of each company: Histogram: A histogram is a sign of a frequency distribution by rectangles whose width represents class intervals and whose regions are comparative to the matching frequencies. Histograms are in general used in statistics to reveal how many of a particular type of variable happens within a particular range. Following are the histogram of the empirical returns to each share and the portfolio: Above the graph represents the histogram of the empirical returns to each share and the portfolio. In 2007 Lonmin plc shows high empirical returns. 3. Analytic VaR: The VaR risk evaluation describes risk as a market failure on a permanent portfolio in excess of an unchanging time horizon, by examining the standard markets. There are three primary methods that used for computing the Value at Risk namely analytic VaR, Monte Carlo and historical analysis. The analytical process of VaR utilizes standard portfolio theory. The portfolio in question is derived in terms of a location vector including cash flow present values symbolizing all parts of the portfolio. There are many option risk procedures in finance. Analytical VaR is also called Parametric VaR for the reason that one of its essential statement is that the return allocation belongs to a family of parametric distributions, for instance the lognormal or normal distributions. The return allocation is explained in terms of a covaiance forcastes and matrix of variance symbolizing the risk attributes of the portfolio in excess of the chosen horizon. The standard deviation of portfolio cost is represented by post pre multiplying the covariance matrix by the situation vector and receiving the squre root of the resulting scalar. The measure can be calculated rationally by building statement on the subject of return distributions in the market risks, and by using the variances and covariances diagonally in these risks. It can also be expected by operating hypothetical portfolios all through chronological information or from Monte Carlo model. Following are the Calculation of Analytical VaR: Company Name Analytical VaR of (95%) (99%) Amec PLC 304.62 432.38 Lloyds Banking Group PLC 54487.92 77063.48 Lonmin PLC 46049.88 65129.5 Tesco PLC 9770.64 13818.91 Above the table represents Analytical VaR. This analysis includes the four shares that are Amec PLC, Lloyds Banking Group PLC, Lonmin PLC and Tesco PLC. Amec PLC shows lowest value of analytical VaR that is 304.62 (95%) and 432.38 (99%). Lloyds Banking Group PLC shows 54487.92 (95%) and 77063.48 (99%). Lonmin PLC shows 46049.88 (95%) and 65129.5 (99%) and finally Tesco PLC shows 9770.64 (95%) and 13818.91 (99%). Expected Returns: Value: Amec PLC 3.73 Lloyds Banking Group PLC 0.42 Lonmin PLC 0.67 Tesco PLC 0.26 Advantages and Limitations: Analytical VaR is attractive and quick and not dreadfully demanding of computational incomes. Analytical VaR is the easiest methodology to examine VaR and is comparatively easy to apply for a fund. The input data is rather restricted, and for the reason that there are no imitations involved, the calculation time is minimum. Its simplicity is also its main trouble. Firstly, Analytical VaR assumes not only that the earlier period incomes follow a normal division, but also that the transforms in cost of the assets limited in the portfolio pursue a standard distribution. And this very rarely subsists the examination of reality. Secondly, Analytical VaR does not manage extremely well with securities that have a non-linear payoff allocation like mortgage-backed securities. Finally, if our past chains reveals heavy tails, then computing Analytical VaR utilising a normal distribution will undervalue VaR at overestimate VaR and high assurance levels at low confidence levels. And it has a number of weaknesses. In its simple structure, choices and other nonliner tools are delta probable, which says that the representative cash flow vector is a linear approximation of understanding i.e. intrinsically nonlinear. 4. Monte Carlo VaR: Monte Carlo VaR simulation involves developing the future stock price returns by running many hypothetical trials. Generating a large number of possible paths for a relevant asset price is the core of Monte Carlo simulation. We should specify probability distributions for each of the market risk factors and specify how these market risk factors move together. We need to repeat the steps until the formula yield the solution. Drawing the random numbers over a large number of times will give an idea about what output should be. While the estimation of parameters is easier, if you assume normal distributions for all variables, the power of Monte Carlo simulations comes from the freedom you have to pick alternate distributions for the variables. In addition, you can bring in subjective judgments to modify these distributions.  Advantages: Monte Carlo method is non- linear and path dependent pay off function. So it will give result as we give the input. Monte Carlo VaR simulation is not much affected by the extreme events. This will help to input all applicable data. The statistical distribution can be used to simulate the returns so that we feel comfortable with the underlying assumptions that justify the use of particular distribution. Disadvantages: The Monte Carlo calculation is a time consuming process. The increase in number of simulation may provide the accuracy but time consuming Monte Carlo method demands more calculations to complete all simulations Higher cost to develop a Monte Carlo VaR engine that can perform Monte Carlo simulations.  The steps in the calculation of VAR using Monte Carlo simulation Step 1 – Determine the length of the analysis limit and divide it equally into a large number by small time increments   Step 2- Take a random number from a random number generator and update the price of the asset at the end of the first time increment.  Step 3 – Repeat Step 2 until reaching the end of the analysis limit by doing maximum round up  Step 4 – Repeat Steps 2 and 3 a large number of times to generate different paths for the stock over   Step 5 – Rank the full terminal stock prices from the smallest to the largest, read the simulated value in this series that represents the desired (1-?)% confidence level (95% or 99% generally) and deduce the relevant VAR  We can use the below equation to do the simulation  “Ri = (Si+1 - Si) / Si = ? ?t + ? ? ?t1/2 Where, Ri is the return of the stock on the ith day Si is the stock price on the ith day Si+1 is the stock price on the i+1th day ? is the sample mean of the stock price ?t is the time step ? is the sample volatility (standard deviation) of the stock price ? is a random number generated from a normal distribution” (Berry 2012). Calculation is attached herewith in excel spreadsheet.  5. Historical Analysis / Bootsrap VaR:  Historical simulations represent the simplest way of estimating the Value at Risk for many portfolios. As the name indicates, this is the method in which the past performance of the company is taken into consideration. The historical method is simply re-organizing the actual historical data. The assumptions are made by analyzing the past performances. The past data will be a good indicator for the near future. This method assumes that the history will repeat itself. Generally, to run a historical simulation, we begin with time series data on each market risk factor.  Advantages: There is no need to make any assumptions. Because all data are available from the past records. No need of bothering about volatilities and correlations at present. Only real facts and figures are taken into consideration. The distribution and all extreme events are considered. Disadvantages:  Historical analysis/bootstrap VaR completely relies on historical data and it may biased Historical VaR cannot accommodate the changes in the present market. Some times it doesn’t match the present scenario. Historical analysis lags when the portfolio contains complex securities. It cannot handle the sensitivity analysis easily. The steps in the calculation of VAR using historical simulation, Step 1 – Calculate the returns or price changes of all the assets in the portfolio between every time interval.  Step 2 – Apply the current market value and revalue the portfolio  Step 3 – Sort the redefined portfolio values in the ascending order  Step 4 – The simulated value should assign to the desired confidence level.  These steps can be formulated as follows.  “VaR1-? =µ(R)- R?   Where: VaR1-? is the estimated VaR at the confidence level 100 ? (1 - ?)% ?(R) is the mean of the series of simulated returns or P&Ls of the portfolio R? is the ?th worst return of the series of simulated P&Ls of the portfolio or, in other words, the return of the series of simulated P&Ls that corresponds to the level of significance ?” (Berry 2012). The calculation are attached herewith in excel spreadsheet. Discussion: 260 day VaR calculation is very important for analyzing the various businesses. VaR is mainly related to the calculation of market risk and is mainly used by financial institutions to assess their risks. “VaR is a single estimate of the amount by which an institution’s position in a risk category could decline due to general market movements during a given holding period” (Tsay 2010, p. 326). From a financial point of view, VaR is also viewed as a measure of loss incurred during a particular time period especially under the normal as well as extraordinary market conditions. There are three steps involved in the calculation of VaR which are mainly the historical simulation or bootstrap VaR, the analytical VaR and the Monte Carlo simulation. For the purpose of the calculation of the 260 day VaR, the four firms that have been taken into account are the Amec PLC, Lloyds Banking Group PLC, Lonmin PLC and Tesco PLC. Using the analytical VaR method 260 day, VaR calculation for Amec PLC according to the confidence interval of 95% is 304.62 while in the confidence interval of 99% is 432.38. For Lloyds Banking Group PLC 260 day VaR calculated according to the confidence interval of 95% is 54487.92 while in the confidence interval of 99% is 77063.48. Similarly, for Lonmin PLC 260 day VaR calculated according to the confidence interval of 95% is 46049.88 while in the confidence interval of 99% it is 65129.5. For Tesco PLC 260 day VaR calculated according to the confidence interval of 95% is 9770.64 while in the confidence interval of 99% is 13818.91. In the analytical method of VaR, calculation standard portfolio theory is used. Similarly, under this method the expected return calculated for the four firms Amec PLC, Lloyds Banking group PLC, Lonmin PLC and Tesco PLC is respectively 3.73, 0.42, 0.67 and 0.26 respectively. Using the Monte Carlo simulation method the mean calculated for Amec PLC is 2842.83. In the Monte Carlo VaR calculated is between various fraction levels like the 0-0.25, 0.25 – 0.5, 0.5 – 0.75 and 0.75 – 1. The VaR calculated in the fraction level 0.75 – 1 is 1. Using the Monte Carlo simulation method the mean calculated for Banking Group PLC is 73.69. In the Monte Carlo VaR calculated is between various fraction levels like the 0-0.25, 0.25 – 0.5, 0.5 – 0.75 and 0.75 – 1. The VaR calculated in the fraction level 0.75 – 1 is 1 while for the rest of the fraction levels it is 0. Using the Monte Carlo simulation method the mean calculated for Lonmin PLC is 279812. In the Monte Carlo VaR calculated is between various fraction levels like the 0-0.25, 0.25 – 0.5, 0.5 – 0.75 and 0.75 – 1. The VaR calculated in the fraction level 0.75 – 1 is 1 while for the rest of the fraction levels it is 0. Using the Monte Carlo simulation method the mean calculated for Tesco PLC is 125893.2. In the Monte Carlo VaR calculated is between various fraction levels like the 0-0.25, 0.25 – 0.5, 0.5 – 0.75 and 0.75 – 1. The VaR calculated in the fraction level 0.75 – 1 is 1 while for the rest of the fraction levels it is 0. Using the bootstrap or historical VaR calculation method the 260 day VaR calculated is 0.05 when the risk free rate is 2% and the confidence interval is 95%. The Monte Carlo simulation involves running a large number of simulations in order to predict the 260 day VaR. In the Monte Carlo simulation count if function is used to calculate the VaR and thereby at different fraction levels VaR is calculated. The three methods used in the computation of VaR are different and therefore the results according to three processes are different. Among the three processes used the Monte Carlo simulation scores over the other two methods. “In order to account for non linear exposures related to a number of structured bonds, the Monte Carlo simulation was used instead of the simpler analytical method for computing value at risk” (Advances in Risk Management of Government Debt 2005, p. 124). The VaR calculation using the analytical method involves the calculation of the value of the expected returns. This helps in better analysis of the VaR calculated. The VaR is usually calculated using the statistical model by taking the appropriate confidence intervals. “This confidence interval implies a corresponding probability that the certain loss is likely to be exceeded a certain percentage of the time” (McCool 1998, p. 77). A firm can increase or decrease it profits by altering its confidence interval. In order to increase its profits the firm can decrease its confidence interval. All the three methods used in the calculation of VaR require statistical analysis. For an accurate determination it requires that the returns of each firm should be first symmetrically distributed around the mean and the standard deviation and calculations being done. The historical method of VaR calculation is completely being rejected on the grounds that this method uses the normal distribution of asset returns. But empirical researches conducted lead to the point that asset returns are not always or most of the time normally distributed. Returns always vary especially according to the assumptions used and there are always high and low returns. In the historic mode of calculation, VaR is estimated on the historic data and this produces an inflated value in the VaR calculation. The Monte Carlo simulation method was developed in order to simultaneously calculate the several risks associated along with. Using the Monte Carlo method one can get the answers not only for the present but also the future. Using the Monte Carlo approach one can determine the returns not only about the present but also about the future, which lets the firm, take economic decisions accordingly. VaR calculation is done on the basis of the past behavior associated with change in prices and the price volatility functions associated with the firm. This helps the business in taking important financial and investment decisions associated with the firm. Reference List Advances in Risk Management of Government Debt. 2005. Organization for Economic Co-Operation and Development. Available at [Accessed on 10 January, 2012]. Berry, R 2012. An Overview of Value-at-Risk: Part III – Monte Carlo Simulations VaR. Available at < http://www.jpmorgan.com/tss/General/Risk_Management/1159380637650> [Accessed on 10 January, 2012]. Berry, R 2012. An Overview of Value-at-Risk: Part II – Historical Simulations VaR. J.P. Morgan. [Online] Available at [Accessed on 10 January, 2012]. Haslem, JA 2003. Mutual Funds: Risk and Performance Analysis for Decision Making. Blackwell Publishing. Available at [Accessed on 10 January, 2012]. McCool, TJ 1998. Risk-Based Capital: Regulatory and Industry Approaches to Capital and Risk. General Accounting Office. Available at [Accessed on 10 January, 2012]. Ryals, L 2008. Managing Customers Profitably. John Wiley & Sons Ltd. Available at [Accessed on 10 January, 2012]. Tsay, RS 2010. Wiley Series in Probability and Statistics: Analysis of Financial Time Series. 3rd Edn. John Wiley & Sons, Inc. Available at [Accessed on 10 January, 2012]. Read More