Value at Risk (VAR) of a portfolio of 4 shares - Assignment Example

An analysis of the Value at Risk (VAR) of a portfolio of 4 shares Introduction This is a report prepared to show the analysis of the Value at Risk (VAR) of a portfolio of 4 shares. The shares are based on four entities namely kingfisher PLC, GKN PLC, Admiral PLC, and Burberry Group PLC all from different companies. To begin with we look at a quick view of importance for valuing risk. Financial institutions face a wide number of risks. These risks can be defined as the extent of uncertainty towards future net returns of the institution. These risks can be classified into four main categories namely; Credit risk, operational risk, liquidity risk and market risk. Credit risk deals with the potential loss resulting from inability of a counterpart to adhere to its obligations. It is characterized by three basic components this being; credit exposure, loss in the event of default and probability of default. Liquidity risk is mainly caused by unforeseen outsized and stressful off-putting cash flow over a short span of time. A firm may be obliged to put up for sale some of its assets at a markdown, if it has vastly illiquid assets and suddenly requires liquidity. Market risk looks at the variations in market conditions and stipulates the uncertainties likely to occur to future earnings. Finally, operational risk includes the risk of regulatory and fraud. It mainly takes into account the errors made in settling transactions or instructing expenditure. Market risk is the most prominent; it highlights the potential economic loss as a result of a decrease in the portfolio’s market value. Value at Risk (VaR) is the best measure that financial analyst can use to compute this risk. VaR is defined as a portfolio’s maximum potential loss value of financial instruments over a certain horizon with given probability. In this report, we are using the data obtained of four different companies. These data are composed of the total return indices of the four companies for the last ten years. VaR is a challenging statistical problem, though its existing models for calculations employ different methodologies they still follow a general structure. This structure involves three steps: a) Mark to market the portfolio, b) Approximate the distribution of returns, c) Calculate the VaR of the portfolio. The difference in the methods that are used to find VaR lie in step 2, because of the way they address the hitch of how to approximate the possible variations in the significance of the portfolio. For example, CAViaR models do not take account of the distribution matter; the quartile of the distribution is calculated directly in this case. There are a number of methods used in calculating the VaR value; in this report the main methods to be used are the Monte Carlo, Analytic and Bootstrap VAR. The report gives detailed results of all the three methodologies in a systematic manner, with data sample of a 260-day from the provided data of the portfolio shares for the four companies. The data are based on the total return index which takes into account the dividend level which is essential in valuing shares, unlike the price data sample. Background to the data sample The data are sampled from a ten year record of four individual companies, Kingfisher PLC, GKN PLC, Admiral PLC and Burberry PLC. Kingfisher PLC; is a company operating in the retail industry, founded in 1982 by Paternoster Stores Ltd. It expanded through successive acquisitions like Superdrug and B&Q. The company is a multinational now headquartered in London, UK. The company provides products such as home appliances, garden supplies & plants, tools and hardware mostly home improvement products. It deals with brands such as B&Q, Brico Depot, Screwfix and Castorama. Its chain of stores is nearly 900 spread across eight countries in Asia and Europe. GKN PLC; found in the automotive and aerospace industry, its origin dates back to 1759 in the early stages of the industrial revolution. The company is a multinational headquartered in Redditch, UK. The company earlier was known as Guest, Keen and Nettlefolds deals with aerospace and automotive components. GKN has undergone substantial expansion over the years, for example, in 1966; it acquired Birfield Ltd in a diversification programme. Birfield incorporated Laycock engineering based in Sheffield and a company that manufactured constant velocity joints called Hardy Spicer Ltd based in Birmingham, England. These joints did not have very many applications then, but with the exploitation of the front wheel drive in the 1970’s and 80’s saw it increase the share of the parts in the world market by 2002. The company continues to expand even up to date like in 2008 it acquired a portion of the Airbus plant in Filton near Bristol. Admiral PLC is another company data was obtained from, operating in the insurance industry. Founded in 1993, with its headquarters in Cardiff; at that time, it was the largest financial service provider in the city. With time, the company has been joined by a number of different, successful brands. The company mainly deals with motor insurance. Burberry PLC is found in the fashion industry; founded by Thomas Burberry in 1958 the company deals with luxury fashion. It manufactures fashion accessories, clothing and fragrance. The company sells through its franchises and branded stores across the world, also through third party stores. The company famously known as the luxury British house of fashion and its distinctive tartan pattern; has seen the company being granted Royal Warrants by HRH The Prince of Wales and HM Queen Elizabeth II. The company has been putting up different strategies and programs in promoting its market share across the world. In 2006, it began selling online first in the US, then UK and in 2007 it offered the service to the rest of Europe. The portfolio comprises of return indices, which are linearly dependent on all the alterations in the values of the shares, such that the portfolio return and the index returns are linearly dependant. The index returns of the shares are jointly distributed. The implication of these statements is that the portfolio return is distributed. The portfolio indices are a sample of 260 days ranging from 22nd Oct. 2009 to 22nd Oct. 2010, from the ten year records of four companies being used in the report Kingfisher, GKN, Admiral and Burberry. The statistical parameters for the data are as indicated in the below table: MEANS AND STANDARD DEVIATIONS kingfisher GKN Burberry grp Admiral grp Mean Return 0.042% 0.047% 0.042% -0.078% Stdev of Returns 1.845% 2.629% 2.432% 1.753% CORRELATIONS BETWEEN RETURNS Kingfisher GKN Burberry grp Admiral grp Kingfisher 100% 58% 46% 40% GKN 58% 100% 62% 33% Burberry grp 46% 62% 100% 34% Admiral grp 40% 33% 34% 100% The sample represents the ten year records for the companies giving a standard deviation between the shares for each company. The results of the data can be plotted into a histogram to the clear picture of the correlation among the data sample. Empirical returns to each share and portfolio, overlaid with the normal distribution with the same mean and standard deviation of returns The mean is represented by 1 while the standard deviation by 2. Analytic VAR Also, known as the variance covariance, this method derives a probability distribution this approach is quite simple but limited by the problems encountered while deriving distribution probabilities. The method requires estimation of two factors the expected return which is average and the standard deviation. This values help in plotting a normal distribution curve. The data sample for the 260-day portfolio for the four shares at a 95% and 99% confidence level will be; 95% confidence level = -1.65 ? std. Deviation ? v260 99% confidence level = -2.33 ? std. deviation ? v260 A portfolio cannot be efficient enough if it does not give the possibility of obtaining an upper return without escalating the standard deviation. The graph below displays the returns against the portfolio expected returns. Graph of the expected returns against the standard deviation for the portfolio The standard deviation obtained for the 260 day total index data of the four shares: Kingfisher PLC 1.845%, GKN PLC 2.629%, Admiral PLC 2.432% and Burberry PLC 1.753%. For kingfisher PLC: 95% = -1.65 ? 1.845 = -3.044% ? v260 = -49.08% 99% = -2.33 ? 1.845 = -4.2989% ? v260 = -69.32% The results for Kingfisher show that with a 95% confidence level the company is unlikely to make a 49.08% loss. With a more precise confidence level of 99% it shows that the risk is high but does not exceed 69.32% in a 260 day period. For GKN PLC: 95% = -1.65 ? 2.629 = -4.3379% ? v260 = -69.95% 99% = -2.33 ? 2.629 = -6.1256% ? v260 = -98.77% GKN has a remarkably high risk according to the results generated by the variance covariance method. The results indicate that at a 99% confidence level the company shares can incur a loss of up to 98% in a 260 period of time. This can be due to its daily high risk of around 6%. But with a 95% confidence level the shares are not as risky as compared to the 99% confidence level. Burberry PLC: 95% = -1.65 ? 2.432 = - 4.0128% ? v260 = -64.704% 99% = -2.33 ? 2.432 = -5.667% ? v260 = -91.38% The shares for Admiral also show signs of being risky but not to the high extent of the GKN. For these shares a 95% confidence level shows that the shares have a risk of losing 64.7% of its value. Although looking at a shorter period the risk is very minimal. Admiral PLC: 95% = -1.65 ? 1.753 = -2.8925% ? v260 = -46.64% 99% = -2.33 ? 1.753 = -4.0845% ? v260 = -65.86% For a 260-day period the results show that the risk is not very high. For example at 95% confidence level the risk is that the shares are can only lose up to 46% of its value. This means that given the shares are rewarded dividends in short periods like one month shows that the shares have a low risk exposure. This method has a number of advantages when used in calculating the VAR value. These advantages include; the method is easy and fast to implement. Portfolio entry entities are the main requirements needed to carry out the calculation. The data sets are easily accessible. And finally the measure for convexities can be achieved with a few modifications. On the other hand what has an advantage does not lack its limitations. Some of the disadvantages of this methodology are that; it fails to revalue positions, when it comes to complex or discontinuous payoffs it lacks the capacity to account for them. When it comes to multiple time horizons the method cannot be applied. The method also assumes a normal distribution pattern or a similar distribution; this is not always the case which makes it in appropriate to apply to certain data. Monte Carlo VAR Monte Carlo is another useful method/tool for assessing VAR; it focuses on loss probability exceeding a marked value other than the whole distribution. In Monte Carlo method its first two steps are similar to the ones for Variance-covariance method. The differences start to occur at the third stage. The difference comes in where instead of calculating the variance and covariance, this method adopts the simulation route. The probability distributions are specified for each market risk factors with explanation specifically indicating how these factors move together. The method generates indiscriminate market scenarios using preset parameters for correlation and price instability and computes the profit and loss for each. Simulations generated in large numbers are run forward in time through correlation and volatility estimates. The simulations though different, in total they will cumulate to the selected statistical parameters. This method has a higher chance of estimating the VaR accurately as compared to the Variance Covariance and Historical analysis method. Its key disadvantage is that, it requires use of ultra powerful computers and it is also time consuming as compared to the other methodologies such as Variance Covariance. Assuming a normal distribution for the data provide for the four entities. The sample comprising of a 260 day portfolio we reduce the sample to 200 days so as to obtain a definite figure, when checked against 95% and 99% confidence level we find that: 99% = (1-0.99) ? 260 = 2.6 = 3rd position 95% = (1-0.95) ? 260 = 13th position Comparing with distribution of each company for the 260-day analysis, after running the trials for the 260-day analysis returns for each share the results used were used to construct a histogram and the 3rd and 13th positions for each share taken. The results showed that: For kingfisher the results indicated that below the 3rd position the outcomes range between -43.45% and -48.79%. Meaning at a 99% confidence level the risk is less than 43%. At the 13th position the value generated was -32.41% which shows that the worst outcome for this share is below 32%. For GKN share, the generated results show that at the third position of the 260-day sample the value was approximately 59.72%. This shows the 99% confidence level assumption for the portfolio returns. At the 13th position which gives the 95% confidence level the results indicate -45.10%. Admiral results showed -53.56% and -32.67% at the 3rd and 13th positions respectively. From the results its assumed that the share is less likely to encounter a risk higher than 32% at a 95% confidence level. Finally the Burberry share showed -30.37% at the 3rd position and -22% at the 13th position showing that the Burberry share is secure as compared to the other three. It low risk rate if checked under the 95% confidence level with chances being below 22%. The method has a number of advantages and disadvantages of its own. The advantages include; the method accommodates a large number of statistical assumptions and models. The method also captures risks that are non-linear and it can be used on data from multiple periods of time. Its limitations on the other hand it does not capture or provide results for a distribution with ‘fat tails’. It is a very intensive process requiring a lot of time and attention. It is also exceedingly difficult to understand with limited transparency. Historical analysis The method re-arranges the historical returns from the data in order from the lowest to the highest value. The method is built around the notion that from a risk perception history will repeat itself. The method is the simplest in the calculation of VaR, of all the three methods. This is because it does not look into many of the pitfalls from the correlation method. These pitfalls are the three key assumptions in correlation; constant deltas, distributed returns and the constant correlations. Historical simulation computes the potential losses by use of definite historical returns from risk factors ending up with the unusual distribution. This results to the assumption that crashes and sometimes rare events can be included in the results. Both the risk factor returns and correlations are the actual past movements and correlations. To begin with, the time series data is taken from each market risk factor the same way it is done in the variance covariance method. The difference is in this approach the data is not used to estimate the covariance together with the variance; since all the changes over time in the portfolio yield necessary information required for computing the Value at Risk. At 99% and 95% confidence levels the returns would be: For the 260-day returns; 99% = 0.99 ? 260 = 257.4 95% = 0.95 ? 260 = 247 Kingfisher PLC; 99% gives -3.48% = -3.48 ? v260 = -56.11% 95% gives -2.10% = -2.10 ? v260 = -33.86% GKN PLC; 99% gives -4.76% = -4.76 ? v260 = -76.75% 95% gives -3.84% = -3.84 ? v260 = -61.91% Burberry PLC; 99% gives -4.50% = -4.50 ? v260 = -72.56% 95% gives -2.76% = -2.76 ? v260 = -44.50% Admiral PLC; 99% gives -3.27% = -3.27 ? v260 = -52.73% 95% gives -2.30% = -2.30 ? v260 = -37.09% For the 130-day returns; 99% = 0.99 ? v130 = 128.7 95% = 0.95 ? v130 = 123.55 Kingfisher PLC; 99% gives 0% = 0 ? v130 = 0% 95% gives 0% = 0 ? v130 = 0% GKN PLC; 99% gives 0.05% = 0.05 ? v130 = 0.57% 95% gives 0.14% = 0.14 ? v130 = 1.60% Burberry PLC; 99% gives 0.10% = 0.10? v130 = 1.14% 95% gives 0.18% = 0.18 ? v130 = 2.05% Admiral PLC; 99% gives 0% = 0 ? v130 = 0% 95% gives 0.12% = -0.12 ? v130 = 1.37% Histogram for the Historical data ranging from the lower quartile to the upper quartile The advantages of using this approach are that it is exceptionally easy to understand and easily captures non-linear risks. It has an actual distribution; also unlike the Monte Carlo method it incorporates ‘fat tails’. Some of its disadvantages are that; it can be data intensive and its assumption that the past events are a representation of the future which is not always the case. Discussion The results from the three approaches show varying result; this is because of the nature of the different approaches and the factors that are considered in generating the results. The Analytic approach shows results with extremely high risk levels for example 98% for GKN at a 99% confidence level. This is not likely the case because of the assumptions undertaken that tend to ignore pertinent factors such as non-linear risks. The Monte Carlo results are generated from a random data sample that is generated with many considerations at hand. The results generated are also convincing and can provide a reliable Value at Risk estimate. The historical approach is not a very suitable approach for this data sample. This is seen when the two data samples are used the 130-day and the 260-day. The results show extreme variations indicating that the data cannot be exceptionally reliable. The three approaches all have advantages and baggage when used in estimating the Value at risk of the four shares. The Variance –covariance approach due to its delta gamma and normal requires one to draw strong assumptions on the return distribution. This usually makes the method easier once the assumptions are drawn. The historical simulation does not rely on any assumptions to be made, but assumes data being used is a representational sample of future risks. The Monte Carlo approach is very flexible when it comes to selecting distributions for returns and incorporating external data and skewed judgments. This approach is also the most demanding in computational point of view. The difference brought about the three approaches is the function of the input data used. For example, the Variance Covariance and the Bootstrap methods would generate the same value for Value at Risk; if the same data used in Bootstrap is normally distributed and then the same data applied in estimating the Variance Covariance matrix. Same way the Monte Carlo approach and Variance-covariance will yield similar results, if the input data in the Monte Carlo method is normally distributed with steady variances and means. According to the difficulties encountered and accuracy required in calculating the Value at Risk it is always crucial to be keen on the method to select for the estimation. The data provided also provides a basis that should be considered in choosing the approach to adopt. From the data and results in this report the best recommended method for calculating the Value at Risk would be the Analytic approach. The approach is a bit tasking but it gives a more reliable estimate considering the various factors that it puts into account. The Historical approach would have been also appropriate had all the companies’ ten year records matched. The Monte Carlo approach assumes a lot of factors that ought to be considered. This makes its results not very reliable as far as the types of portfolio in this report are concerned. Conclusion Finding Value at Risk might look simple but the effort and keenness required is a lot. After analysing the data provided and generating results through the three methods, certain conclusions can be drawn from these results. First it is evident that the three methods due to their specific assumptions, steps and form in which the data is used, they end up generating dissimilar values. This shows that any of the three methods cannot be used in the place of the other. This is because the three handle differently the data provided in the selected portfolio. Therefore before choosing a method first one has to look at the objectives and the sample distribution. In this portfolio after the analysis it is clear that the best approach to go with is the Analytic approach also known as the Variance-covariance. References: Choudhry, M., The Bond and Money Markets: Strategy, Trading, Analysis, Butterworth-Heinemann 2001 Harper, D. 2010. An Introduction To Value at Risk (VAR). Ivestopedia. May 27, 2010. Retrieved from; http://www.investopedia.com/articles/04/092904.asp#axzz1iTq4XbWe Engle, R., 1982, Autogregressive Conditional Heteroskedasticity with Estimates of the Variance of UK Inflation, Econometrica, 50, 987{1008. Nocera, J. (January 4, 2009), Risk Mismanagement, The New York Times Magazine Jorion, P., 1997, Value at Risk. Irvine, Chicago. Vlaar, P.J.G., 1998, Value at Risk models for Dutch bond portfolios, Journal of Banking and Finance, forthcoming (available at http://www.dnb.nl). Hendricks, D., 1996, Evaluation of Value-at-Risk Models Using Historical Data, Federal Reserve Bank of New York Economic Policy Review April: 39{69. 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