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The term VaR was first introduced by JP Morgan in 1994 (Leong, 1997, cited in Grayling p.23). The VaR has grown to be the industry standard for measuring portfolio risks. Even the Bank for International Settlements (BIS) and the central bank around the world set the capital adequacy requirements for market risk in terms of VaR estimate. Linsmeier and Pearson (2000) described VaR as summary of statistical measure of possible portfolio losses with a given confidence level over specified period.
The objective of this report is to estimate and analyse the Value at Risk (VaR) for the given four (4) portfolio assets using Variance - Covariance method and Historical method. The distinction between the two methods as well as the sensitivity of the VaR value to the sample sizes will also be discussed. The outline of the sections will be as follows - section 2 and 3 briefly outlined the assumptions behind VaR and the VaR parameters. Section 4 describes the VaR methodology used in this report. Data, results and Discussion are summarized in Section 5 and 6. Finally, Section 7 concludes the report.
2. The assumptions behind VaR Calculation
According to Allen et.al. (2004), there are several statistical assumptions for VaR calculation. First assumption is stationarity requirement. Stationarity refers to assumption that certain percentage of fluctuation in returns has the same probability of occurrence at any point of time. A related assumption is the random walk assumption of inter-temporal unpredictability. That is, day-to-day fluctuations in returns are independent and there is no relevant information available at time t to forecast prices at time t + 1. Another assumption is the non-negativity requirement, which stipulates that financial assets with limited liability cannot attain negative values. However, derivatives (e.g., forwards, futures, and swaps) fail to meet this assumption. The time consistency requirement states that the portfolio does not change over the holding period. For multiple asset portfolios, the assumption is the change in the portfolio return is a liner combination of all the changes in the portfolio assets. And the most questionable assumption is the asset returns are normally distributed. Despite many evidence shows that most assets are not normally distributed, this assumption is still a standard assumption in finance.
3. VaR Parameters
Three aspects need to be kept in mind when judging the VaR of a portfolio. In the first place, we need to know the initial value of the portfolio and two arbitrarily parameters - the duration and the confidence level. Dowd (2002) emphasized that the choice of the two parameters often depends on the use of VaR number and also consideration for back-testing.
Allen et.al (2004) mentioned that VaR calculations require assumptions about the possible future values of the portfolio in the future. There are at least three ways to calculate a rate of return from period t to t + 1 as shown below:
The absolute change method violates the stationarity requirement. The simple return as a measure of the change in the underlying factor meets the stationarity requirement, but it does not comply with the time consistency requirement. And continuously compounded return satisfies both stationarity and time consistency requirement. However, for short duration asset and small R, the r is â‰ˆ R.
4. VaR Methodology used
4.1 Historical Method
This is probably the simplest non-parametric method with no assumption of shape of the distribution of asset returns. The historical distribution of returns to the assets in the portfolio is used to simulate the portfolio's VaR on the assumption that the asset is held over the time period cover in the historical data set.
The assumptions of the historical simulation approach are firstly there are no underlying assumptions of normality distribution. Secondly, the approach is based on the assumption of history repeating itself which may not be true in reality. Lastly, each day in the time series used in measuring the VaR has equally weightage. This could be a potential problem if there is a trend in the variability - lower in the earlier periods and higher in the later periods (Benninga and Wiener, 1998).
In historical method, we first identify sample of historic returns of different assets in the portfolio over observation period. The weight of assets in the portfolio is used to stimulate the hypothetical distribution of portfolio changes. By ordering all the resulting data, the relevant percentile of historical return leads us to the VaR of our portfolio, with 95% confidence level; the VaR equals the value at the 5% quantile of worst outcomes (Dowd, 1998).
4.2 Variance-Covariance Method
Variance-covariance method relates the VaR of a portfolio to the standard deviation of the portfolio returns. Generally, the larger the variance of a portfolio return, the more likely the occurrence of large swings in the portfolio value and the larger the VaR (Van den Goorbegh and Vlaar, 1999).
The underlying key VaR assumption of this method is similar to RiskMetrics approach introduced by JP Morgan. First, it is assumed that asset volatilities and correlations are relatively stable and can be estimated using historical data. In reality, however, volatilities and correlations change dramatically and but sensitivity check of the data may reduce this error. Secondly, the portfolio change is assumed to follow normality assumption. And lastly, to facilitate the analysis of VaR for large portfolio asset, RiskMetrics use the Risk-Mapping methodology to map the risk in the individual asset (Leong, 1997, cited in Grayling p.23). In this assignment, the risk mapping process is not explored as we only analyse four portfolio assets.
According to Leong (1997), to calculate VaR for the portfolio assets, we need to daily return of the portfolio and translate it into volatility return of the asset. Next, as each asset movement will also affect the portfolio return, we need to calculate the covariance / correlation coefficients between the assets' returns. The volatility of portfolio is calculated using modern portfolio theory and summarized by following equation:
The data used in this assignment consists of four assets namely British Petroleum (BP), Mark & Spencer (M&S), GKN and Legal & Gen. Throughout the analysis, the simple return is used to calculate the daily return of asset and one day holding period is used. The VaR of the $100m portfolio is calculated based on allocation of $25m in BP, $30m in M&S, $20m in GKN and $25m in Legal & Gen and each stock is assumed to be synchronous and at same currency.
There are a total time series of 1289 daily data running from 15 March 2005 to 12 March 2010. Table 1 shows daily return statistics of the 4 assets.
6. Results & Discussion
The result of the VaR calculation appears in Table 2. Different method produces different results and several observations could be made on:
Effect of confidence level on VaR
The higher the confidence level, the higher the VaR value, this means that at 99% confidence level, there is 1% chance that the portfolio will suffer a loss greater or equal to statistic shown.
As shown in Table 1, all assets have fat-tailed and M&S and GKN are left skewed - longer left tail. Pafka and Kondor (2001) concluded that the effect of fat tails is insignificant at 95% confidence level, however, for higher fat tail level, the simple variance-covariance method will result in underestimating of risk. That's probably explained why at 99% confidence level, variance-covariance method yields lower VaR than historical method.
Sensitivity of VaR on sample sizes
Market factors / risks have varying degrees of impact on the asset's price sensitivity. Vastly different risk views may be produced by different data sets used. The length of observation period influences the overall validity of the VaR as longer period tend to provide a larger sampling of possible factor volatility changes but can hide sharp changes in the market's trend (Payant, 1997). Furthermore, Beder (1996) stated that it is importance to determine which data points are to be excluded. Inclusion of outliers events caused by onetime events will produce different VaR results. In this assignment, the data sets used are from March 2005 till March 2010, and each year has different market risks. For instance, per global financial risk map of year 2007 till 2010 as shown in Figure 1, market players in 2007 has the highest risk appetite but the risk appetite reduced significantly in 2008 due to global financial crisis. Thus, due to different market risks, the VaR calculated using full data sets given (1289 days) is different from VaR of 1000 days or VaR of 250 days.
Period of year 2008/2009 was period with higher volatility due to global financial crisis. IMF Global Financial Stability Report October 2010 reported that the global financial system is gradually progressing towards financial stability. Thus, in future year with stability period, if we would like to have 'true' estimation of VaR, we may need to exclude the data sets collected in 2008 as the volatility may no longer valid.
Effect of volatility clustering
As shown in Figure 2, when daily return of each asset is plotted against the time, we can observe that the volatility is dynamic and there is volatility clustering effect. Volatility tends cluster for certain period of time till the market condition adjust / stabilize, i.e. year 2009 was the year with high volatility period however, volatility was stabilized from year 2010 onwards after the market adjustment due to various factor, i.e. government intervention, changing of risk appetite of market players etc.
In variance-covariance calculation of VaR, the volatility of portfolio is assumed to be constant. In this assignment, at 99% confidence level, for 1289 prices, the Sdev is 0.0163 and VaR is $3.8m while, for 1000 prices, the Sdev is 0.0182 with VaR of $4.23m. The result would also be different for 250 prices, simply due to different price volatility.
To conclude, to have the more 'sensible' VaR, we need to select the data set accordingly as it make more sense to use recent data to forecast future volatility. The longer period of data set used, the effect of volatility will be normalized thus could lead to under/over estimate of VaR.
Effect of skewness / kurtosis on VaR
Skewness characterizes the degree of asymmetry of a distribution around its mean with positive skew indicates a tail extending towards more positive values, '0' indicates the distribution is normal and negative skew indicates the opposite. Kurtosis characterizes the relative flatness of a distribution compared with the normal distribution. For a normal distribution, kurtosis equals to 3, value >3 refers to presence of fat tails.
Theoretically, the historical simulation and variance-covariance methods will yield the same VaR if the historical returns data is normally distributed. Table 3 below shows that the portfolio assets are asymmetric on the left with an excess of kurtosis. Thus, VaR values estimated using variance-covariance method are different from historical method, the risk are either over/under estimated.
Comparison of VaR generated by the methods
Beder (1995) mentioned that VaR may be potentially seductive but dangerous in nature. VaR calculations may differ significantly for the same portfolio as it is extremely dependent on the parameters, data, assumptions, and methodology. Each VaR method has its own strengths and weaknesses and none is superior to the others. Table 6 summarizes the comparison of VaR using variance-covariance method and historical method as extracted from Leong's article of The Right Approach (Leong, 1997, cited in Grayling p.23)
Limitation of historical method
This approach is relatively simple and can easily incorporate non-linear instruments such as options. However, historical approach has its weaknesses: Coronado (2000) agreed that the VaR estimate may not be accurate as the past condition may not replicate the current prices. Furthermore, we need enough data to have reasonable observation on tail of the distribution. However, the further we go into the past for data, the less relevant this information is to today's market. According to Dowd (1998), this approach could also generate unreliable VaR as all data points are weighted equally. In reality, there is a trend of increasing volatility within the historical time period. Take example of this assignment, the VaR of complete data set (1289 trading days) is estimated using data from 2005 till 2010, the 2005 data is treated in the same proportion as 2009 data; thus, the VaR estimated using this approach may be understated because year 2009 was the period with high volatility due to financial crisis. Furthermore, the historical approach is difficult to estimate VaR for markets where risks are volatile, structural shifts occur at regular intervals for new market/risk.
Limitation of Variance-Covariance method
Beder (1996) corroborated that in spite of simplicity, the variance-covariance approach VaR is criticised for its inadaptability for nonlinear financial instruments such as options. In addition to that, there is a large number of evidence that financial variables generally are unstable and characterized by fat tails and kurtosis excess. Furthermore, VaR estimate could become more complicated as asset returns are often negatively skewed, i.e. more observation in the left-hand-side tails than the right-hand-side one. Dowd (2002) suggested that the normality assumption is strictly applicable for dealing with a zero-skew distribution with a kurtosis of 3; otherwise this could lead to major errors in risk analysis.
Which is the better method?
There is no easy answer on which better method is to be used for VaR estimate. The methods have their own characteristics in estimating risk for options portfolio, ease of implementation, ease of communication to board directors, flexibility incorporating alternative assumptions and accuracy of the results. The best choice will be determined by which dimensions the risk manager finds most important (Linsmeier & Pearson, 2000).
Diversification of Risk
It is widely known that risks can be reduced by diversifying across assets that are imperfectly correlated (Allen et.al, 2004). Table 5 shows the undiversified VaR and diversified VaR of portfolio assets for 1289 trading days calculated using variance-covariance method. In four asset portfolio the undiversified VaR is the summation of each asset's VaR. Simple addition of exposures to risk factors implies that all factors are perfectly correlated. This summation does not represent an economic measure because risks are not additive. Generally, the likelihood of losing money on all assets is slim because the assets are not perfectly correlated. The correlation coefficient Ï always lies between -1 and +1. When equal to unity, the two variables are said to be perfectly correlated, while 0 means totally uncorrelated (Jorion, 2007).
Table 6 shows the correlation coefficient of the four assets respectively. The diversified VaR, which takes into account the correlation between the assets, is the square root of the variance of the portfolio. The diversified VaR is considerably lower than the sum of the four VaRs. The risk reduction is entirely due to the diversification effect of the portfolio. The four assets in the portfolio analyzed here are from different industries: BP (British Petroleum) is oil / refining company, M&S (Mark & Spencer) is mainly on commercial retailing, GKN is aircraft / automotive manufacturer and Legal & Gen is an investment / fund managing company. As the assets are from different type of industries, they are exposed to different type of risk but we can observe that the four asset are positively correlated which means when price of one asset moves down due to market shock, the same market shock effect may cause downward movement of the rest of three assets. However, as the four assets are not perfectly positive correlated, the diversification of portfolio will reduce the overall risk of the portfolio.
Risk diversification is very powerful and motivates much financial activity. However, Allen et.al (2004) emphasized that any estimated of risk diversification effect is only valid when its assumptions and parameter estimates remain stable with no major shocks in the financial market.
VaR limitation in general
VaR is a useful risk management tool, but it subject to some caveats. The accuracy of VaR depends on the parameters and assumptions used. The data set used for estimation of VaR is based on historical past data and it is generally recognized that history seldom replicate in future. This results in a dilemma while on one hand, using past data will result in inaccurate VaR but on other hand, to estimate VaR, we need sufficient data. This explains the importance of VaR validation tools such as stress testing, back testing and other independent review. In addition to that, every VaR measured makes assumptions about return distributions, which, if violated, could result in incorrect VaR value. Variance-covariance method assumes the return distribution is normal, while in historical method we assume the historical return distribution is based upon past data and used as the representative of the future distribution of returns. Thus, the VaR measured often contains some errors and the errors can be large enough to make VaR a misleading measure of risk exposure (Beder, 1995).
Apart from error of in the value, the VaR obtained does not describe the worst loss and does not provide the losses in the left tail. The estimated VaR only gives indication of the probabilities of occurrence - frequency in which a specific loss will exceed the VaR. However, from risk management perspective, managers generally care more about the size of the losses than the number of times the losses occur (Jorion, 2009).
In financial market, there are other technique evolved for calculating VaR such as MonteCarlo simulation, extreme value theory and etc. In addition to that, financial experts are constantly exploring methods to improve the VaR technique as well as to test the accuracy of the VaR obtained i.e stress testing, back testing and GARCH method. Realistically, no risk measurement model is without limitation or assumptions thus, in selecting a risk measurement approach, the risk manager has to understand the relative strength of each method and the specific situation of the firm - the composition of portfolio, availability of software / database / people skill and so on.
In this paper, we analyzed Value-at-Risk of four portfolio assets using variance-covariance and historical method. The sample price is collected from historical price from year 2005 till 2010.
Some salient points from the analysis:
Each VaR method has its advantages and limitations.
The accuracy of the VaR estimate depends on the parameters and assumptions used. For instance, when same assumption / same VaR method is used but with different set of data, it will yield different VaR results due to price volatility effect.
Generally the assets are not perfectly correlated and by diversifying the portfolio we may reduce overall portfolio risk.
To conclude, it is reckoned that the VaR is an important tool in risk measurement. It provides estimation with certain degree of confidence level on how much loss over its holding period that a firm could potentially suffer. However, VaR does not provide exact value of the worst loss. Furthermore, different VaR method will yield different results and it depends on many factors such as type of risk variables, firm's risk appetites and risk management technique. In addition to that, it is importance to supplement the VaR estimation not only with stress / back testing but also with prudent checks and control, limits, random audits and appropriate reserves (Beder, 1995).