Analysis of the 260-day Value at Risk of a Portfolio of Four Shares - Assignment Example

?Introduction This report aims to present the findings of an analysis of the 260-day Value at Risk (VAR) of a portfolio of four shares. The purpose of this analysis is to measure financial risk as well as to quantify and manage the risk of a portfolio. A short discussion of Value at Risk in general will be provided, followed by a review of the key questions motivating this analysis. A summary of the data used to conduct the analysis and the methodology employed also will be followed by presentation of the main findings of the analysis, along with a discussion of any limitations and possible recommendations. Background: Value at Risk (VAR) According to Choudhry (2006), the VAR of a portfolio is the maximum loss expected to occur with a certain probability over a given time period. It is the level of return comprising of a given probability (usually, 5, 2.3, or 1 percent) of experiencing a return of less than that level. Value-at-Risk was first used in the late 1980’s by major financial firms to measure the risk of their trading portfolios. Since then, Value-at-Risk is widely used quantitative tool to measure market risk. According to Hull (2005), “VaR answers the question: how much can one lose with X% probability over a pre-set horizon”. More precisely VaR is an amount (say V dollars), where the probability of losing more than V dollars is over some future time interval, T days. Value-at-Risk has become widely used by corporate treasurers, fund managers, financial institutions, brokerage firms and investment funds to gauge their financial risk. In addition, bank regulators use Value-at-Risk in determining how much capital a bank should possess to reflect the market risks it is bearing (ibid). The aim of this project was to implement various VAR methods that consist of Analytic VAR, historical (Bootstrap) VAR and Monte Carlo (MC) VAR simulation as alternative approaches to calculating VAR, by using data from four portfolios namely; Johnson Matthey PLC, Kazakhmys PLC, Rolls-Royce Holdings PLC and Xstrata PLC. These portfolios are listed in the FTSE index, which are among the largest 100 UK companies by full market value. The FTSE index1 is the most widely used of the FTSE Group's indices and is frequently reported on UK news bulletins as a measure of business prosperity, because it represents about 80% companies of the market capitalization of the whole London Stock Exchange. The companies listed in the FTSE index are determined quarterly according to their market capitalization. These companies must meet a number of requirements set out by the FTSE Group, including having a full listing on the London Stock Exchange and meeting certain tests on nationality, free float, and liquidity. In the FTSE, share prices are weighted by market capitalization, so that the larger companies make more of a difference to the index than smaller companies do. The first company is Johnson Matthey PLC. The company is world renown in refining and distribution of gold, silver, and platinum group metals in 30 countries on six continents. The company is organised in different divisions that includes Precious Metal Products division (the sole marketing arm for Anglo Platinum), Johnson Matthey's Environmental Technologies Catalysts division that produces emission control products, fuel cells, and process catalysts. The company also has Fine Chemicals and Catalysts division that make base and precious metals catalysts and chemicals. Johnson Matthey PLC has an average market capitalization of ? 43.90 billion. The second company under focus is Kazakhmys PLC. Kazakhmys PLC is a company that specializes in copper. It undertakes copper mining, processing, smelting, and refining as well as making of copper cathode and rod products. It is among the top ten copper producers in the world, with an annually production of about 350,000 tons of copper cathode that are used in computers, electric motors, automobiles, and other products. Additionally, Kazakhmys processes and sells by-products such as gold, silver, and zinc. Kazakhmys PLC has an average market capitalization of ? 49.36 billion. The third company under consideration is Rolls-Royce Holdings PLC. Rolls-Royce Holdings PLC is a British manufacturer of aircraft engines and propulsion as well as manufacturer of power systems and services for use on land, at sea and in the air. The Company operates in civil aerospace, defense aerospace, marine and energy. Rolls-Royce Holdings PLC has an average market capitalization of ?13.58 billion. The fourth company to be investigated is Xstrata PLC. Xstrata PLC is a global mining company that is organized into five global commodity businesses. Xstrata Alloys, Xstrata Copper, Xstrata Coal, Xstrata Nickel and Xstrata Zinc. These commodity businesses are engaged in coal, ferrochrome, vanadium, zinc, copper, platinum, gold, cobalt, lead, and silver respectively in various regions in the world. Xstrata PLC has an average market capitalization of ? 29.98 billion. Data analysis As mentioned above, the analysis provided are intended to measure financial risk, using Analytic VAR, historical (Bootstrap) VAR and Monte Carlo (MC) VAR simulation using data from four portfolios provided. Value at Risk records the actual loss that would occur if the returns were below a certain probability threshold of the distribution. A VAR statistic has three components: a time (a day, a month or a year), a confidence level (typically either 95% or 99%), and a loss amount (expressed either in percentage or loss in dollar). In analyzing the performance of the portfolio investment, the total return index is used due to the fact that total return index is an index that calculates the performance of a group of stocks assuming that all dividends and cash distributions are reinvested. Total return index is also used because it is a more accurate measure of actual performance that includes dividends, interest, rights offerings and other distributions realized over a given period (Alexander 2009). To be able to invest in a portfolio of investment by using the solver, the figure below shows that there are some shares that will bring a higher return than other and hence they end up dominating those bring a lower return, the solver shows that the total weight is 22.1% with respective portfolio weights being shown below:= Johnson Matthey PLC =0.12% Kazakhmys PLC=18.20% Rolls-Royce Holdings PLC=3.78% Xstrata PLC 0.00% As a result, the portfolio investment will have an expected return of 0.0099% and a standard deviation of 0.7856% with a minimum expected return of 0.0372% and a maximum return of 0.0741%. As a result, it is important to use the efficient frontier curve to be able to pick a portfolio of investment. The efficient frontier curve is shown below: The efficient frontier curve has a slope of 0.31512 with the optimal portfolio weight being: Johnson Matthey PLC=17% Kazakhmys PLC=-40% Rolls-Royce Holdings PLC=95% Xstrata PLC=28% Table1: An analysis of the four-share portfolio shows the relationship between their mean and their standard deviation: Johnson Matthey PLC Kazakhmys PLC Rolls-Royce Holdings PLC Xstrata PLC Mean Return 0.037% 0.039% 0.074% 0.048% Stdev of Returns 1.976% 4.049% 2.217% 3.289% Table 2: Correlations between Returns Johnson Matthey PLC Kazakhmys PLC Rolls-Royce Holdings PLC Xstrata PLC Johnson Matthey PLC 100% 63% 49% 55% Kazakhmys PLC 63% 100% 52% 82% Rolls-Royce Holdings PLC 49% 52% 100% 44% Xstrata PLC 55% 82% 44% 100% As shown above, Johnson Mathew PLC has mean of 0.037% and a standard deviation of 1.967% bins frequency -30.88% 0 -21.88% 0 -12.88% 0 -3.88% 57 5.12% 1487 14.12% 27 23.12% 0 32.12% 0 41.12% 0 Its return distribution is of a normal curve with its share price being around 5.12% more than 1487 days in the ten years. This relationship can be shown in the histogram below: The graph shows that the share prices have been normally distributed. Kazakhmys PLC has a mean of 0.039% and a standard deviation of 4.049%. The values of its returns are shown in the table below: bins frequency -30.88% 1 -21.88% 1 -12.88% 10 -3.88% 154 5.12% 1303 14.12% 92 23.12% 8 32.12% 2 41.12% 0 This value in the above table shows that the shares of Kazakhmys PLC are normally distributed with it rate of return at 5.12% at most of the time; this is also evident with the histogram below that shows the normality of the curve and the normal distribution. Rolls-Royce Holdings PLC has a mean of 0.074% and a standard deviation of 2.217% with a normal distribution that is shown in the table and diagram below: bins frequency -30.88% 0 -21.88% 0 -12.88% 0 -3.88% 54 5.12% 1496 14.12% 21 23.12% 0 32.12% 0 41.12% 0 The figure above shows that Rolls-Royce Holdings PLC is normally distributed with the share prices bringing a rate of return of 5.12% almost 1496 times. Xstrata PLC has a mean return of 0.048% and a standard deviation of 3.289%. The value of its return and frequency are shown below. bins frequency -30.88% 0 -21.88% 0 -12.88% 9 -3.88% 151 5.12% 1317 14.12% 90 23.12% 4 32.12% 0 41.12% 0 As shown above, the shares had a constant return of around 5.12% for 1317 days. As a result, it has a normal distribution as shown in the histogram below. 1. Historical Method Calculating VAR In using the historical method, we simply re-organize actual historical returns, putting them in order from worst to best. It then assumes that history will repeat itself, from a risk perspective. In using this method to calculate the data provided we are able to calculate each daily return and produce a rich data set of all points. In a histogram that compares the frequency of return. At the highest point of the histogram (the highest bar). The above graph show John Mathew PLC presentation showing the "left tail" of the histogram shows the lowest 5% and 1% of daily returns. In analysing the histogram, we can say with 95% confidence that the worst daily loss will not exceed 1%. Put another way, we expect with 95% confidence that our gain will exceed -1%. At the same time, with 99% confidence, the worst daily loss will not exceed 2%. And with 95% confidence the gains realised will not exceed 2% Kazakhmys PLC graph is presented above whereby it shows the "left tail" of the histogram shows the lowest 5% of daily returns would be realised in less than 0.7%. Because these are the worst 0.7% of all daily returns, we can say with 95% confidence that the worst daily loss will not exceed 0.7% At the same time, we can say with 99% confidence that the worst daily loss will not exceed 2% and with 99% confidence, the daily gains will not exceed 2%. Rolls Royce holding PLC graph depicted above shows that with 95% confidence, the value of its shares will not get a loss of more than 1% and gain more than 1%. At the same time, we can say with 99% confidence, we can say that the value of its shares will not get a loss of more than 2% and a gain of 2%. The above graph and histogram depict Xstrata PLC showing that the left tail of the histogram that with 95% confidence, the value of its shares will not get a loss of more than 0.8% and a gain of more than 0.8%. At the same time, with 99% confidence, the value of its shares will not reduce by more than 2% and at the same time the value of the shares increasing by more than 2%. Advantages: According to Melichar (2010), Historical simulation method is relatively simple to implement if the past data is readily available for estimating VAR. Historical simulation method allows nonlinearities and non-normal distribution by relying on the actual prices. It does not rely on underlying stochastic structure of the market. Finally, historical simulation helps one get a full distribution of potential portfolios the same time, there is no need to make distributional assumptions. Limitations of historical VAR To analyze the figures, one needs a significant amount of daily rate history (at least a year) and this would create a great drawback if some assets have short or no history at all. In addition, the analyst would need significant computational power to revalue each portfolio scenario. The Historical Simulation method is cumbersome for large portfolios with complicated structures (Gregoriou and Gregoriou, 2009). Finally, the historical simulation approach has difficulty dealing with new risks and assets since there is no historical data available to compute the Value at Risk. 2. The Variance-Covariance Method This method assumes that stock returns are normally distributed. Therefore, it requires that we estimate only two factors; an expected (or average) return and a standard deviation, which allow us to plot a normal distribution curve. The normal curves have been plotted against the same actual return data to automatically know where the worst 5% and 1% lie on the curve. For a four- share portfolio, the blue curve above is based on the actual daily standard deviation of return, which is 1.976% for Johnson Matthey PLC, 4.049% for Kazakhmys PLC, 2.217% for Roll-Royce Holdings PLC and 3.289% for Xstrata PLC as shown below. The average daily return is fairly close to zero, so it assumes an average return of zero for illustrative purposes. Given that, the portfolio assets have a normal curve and assume normal distribution, the VAR metric in 260 day 95% and 99%, was obtained as follows: Confidence Johnson Matthey PLC Kazakhmys PLC Rolls-Royce Holdings PLC Xstrata PLC 95% -1.65 x 1.976% =-3.2% -1.65 x 4.049% = -6.6% -1.65 x 2.217% =-3.7% -1.65 x 3.289% =-5.4% 99% -2.33 x 1.976% =- 4.6% -2.33 x 4.049% =-9.4% -2.33 x 2.217% =- 5.2% -2.33 x 3.289%=-7.6% The answer in the data above support the normal curve showing the areas that the worst 5% and 1% lie on the curves of the four shares with 95% and 99% respectively. According to Saita (2007), the main disadvantage of Variance-Covariance Method is that it is less accurate for non-linear portfolios. 3. Monte Carlo Simulation The third method involves developing a model for future stock price returns and running multiple hypothetical trials through the model. A Monte Carlo simulation refers to any method that randomly generates trials, but by itself does not tell us anything about the underlying methodology (Hofler, 2008). When a Monte Carlo simulation of 1000 trials is performed on the four- shares based on their historical trading pattern and repeated, a different result is highly likely though the differences would be narrow. According to the result, after 1000 runs, with 99% probability, the stock price of Kazakhmys PLC would not climb to more than 115.352 and its future price will likely not fall to below 87.188. For Kazakhmys PLC, after 1000 runs to get the VAR at 95% confidence level, the 5% lowest stock price is 87.188. Hence, there is 5% likelihood that if the share rises to 115.352, a gain of 28.164will be experienced at 95% confidence level for the share. For roll Royce PLC, after 1000 runs to get the VAR at a 99% confidence level, the 1% lowest stock price, is 161.764. The original stock price was 173.78. There is a 1% likelihood that the share will rise to 189.784 and a gain of at least 28.02 will be experienced at a 99% confidence level for one share. At the same time, for roll Royce PLC, after 1000 runs to get the VaR at 95% confidence level, the 5% lowest stock price is 168.054. Hence, there is 5% likelihood that if the share rises to 189.784, a gain of 21.73 will experienced at 95% confidence level for the share. For Johnson Mathew PLC, after 1000 runs to get the VaR at a 99% confidence level, the 1% lowest stock price, is 10689.244. The original stock price was 10698.18. There is a 1% likelihood that the share will rise to 10711.488. If that happens, a gain of 13.308will experienced at a 99% confidence level for one share. At the same time, for Johnson Mathew PLC, after 1000 runs to get the VAR at 95% confidence level, the 5% lowest stock price is 10692.752. Hence, there is 5% likelihood that if the share rises to 388.345, a gain of 18.763will experienced at 95% confidence level for the share. At the same time, for Xstrata PLC, after 1000 runs to get the VaR at a 99% confidence level, the 1% lowest stock price, is 363.356. The original stock price was 371.92. There is a 1% likelihood that the share will raise to 388.345. a gain of 16.425will experienced at a 99% confidence level for one share. At the same time, for Xstrata PLC, after 1000 runs to get the VaR at 95% confidence level, the 5% lowest stock price is 365.356. Hence, there is 5% likelihood that if the share rises to 388.345, a gain of 22.989will experienced at 95% confidence level for the share. Advantages of the Monte Carlo method include incorporation of nonlinear positions, non-normal distributions, implied parameters, and user-defined scenarios. As a result, one can get a full distribution of potential portfolios (not just a specific percentile) at the same time and a manager can use various distributional assumptions like normal and T-distribution (Artzner et al, 1997). However, this method require a higher computational time since computer and data requirements are much higher than those required by other approaches. At the same time, it takes a lot of computational power and longer time to estimate results. This method is expensive to implement in terms of systems infrastructure and is subject to risk that the model might be wrong due to the underlying risk factors as well as the pricing models for different financial products. Conclusion Value at risk (VAR) measures maximum loss, which can be experienced on a given investment portfolio at a given period. VAR is used by fund managers, corporate treasurers, and financial institutions as well as by brokerage firms and investment funds to measure their financial risks. Analytical VAR, Historical VAR and Monte Carlo VAR have been used to calculate VAR for Johnson Matthey PLC, Kazakhmys PLC, Rolls-Royce Holdings PLC and Xstrata PLC. The components of VAR are time, a confidence level and a loss amount. Performance of portfolio was analyzed using total return index. Advantages of historical include possible use of nonlinearities and non-normal distribution, do not rely on underlying stochastic structure of the market and uses historical simulation. the disadvantages include use of daily rates, it is cumbersome due to need for computational capabilities and it is difficult to assess new risks and assets. The main disadvantage of Variance-Covariance Method is that it is less accurate for non-linear portfolios. Finally, Monte Carlo method incorporates nonlinear positions, non-normal distributions, implied parameters, and user-defined scenarios. However, the method requires high computational capability, high data requirements, and takes longer to obtain results. In addition, the method is expensive to implement and is subject to risk that the model might be wrong due to the underlying risk factors as well as the pricing models for different financial products. References Alexander, C 2009, Market Risk Analysis, Value at Risk Models, John Wiley & Sons, New York. Artzner, P, Delbaen, F, Eber, JM & Heath, D1997, Thinking coherently, Risk, Vol 10, No 11, pp 68-71. Choudhry, M 2006, An introduction to value-at-risk, 4th edn, John Wiley and Sons, West Sussex. Gregoriou , NG 2009, The VAR Implementation Handbook, Chapter 9 - Computational Aspects of Value at Risk, Part 9, McGraw-Hill, New York. Gregoriou, NG 2009, The VaR Modeling Handbook: Practical Applications in Alternative Investing, Banking, Insurance, and Portfolio Management, McGraw-Hill, New York. Hofler, B, 2008, Risk Measures - Value at Risk and Beyond, GRIN Verlag, Norderstedt Hull, CJ 2005, Options futures and other Derivatives, 5th edn, Pearson education, New Jersey. Melichar, A 2010, Problems of Value At Risk - A Critical View, GRIN Verlag, Norderstedt. Saita, F 2007, Value at risk and bank capital management, Academic Press, London. Read More