VaR calculations are becoming increasing important for all the investors, bankers, and regulators. There are several ways to calculate VaR; the historical method, the Variance – Covariance Method, and Monte Carlo method. In this analysis, we will use the returns of stocks from Dow Jones Industrial Average for the last 40 years in order to make a comparison between different VaR calculation methods. We will try to determine how that observation time, normality of the data, time frame, and variation of the data influence different ways of calculating VaR.
Implementation of Historical Methods
Data
Historical simulation is one of the popular methods used in the calculation of VaR. The data used in this case consists of 481 monthly prices of 4 stocks: American Express, Boeing, Du Pont, and the Walt Disney. In this VaR analysis, the assumptions used include the following:
- We have an investment portfolio of 4 assets with an investment of $1,000 each.
- The historical time period based on which the calculation will be done is as follows:
- The calculation will be done for one month, 95% VaR.
- All the historical periods are equally weighted.
The historical calculation will be done using the data from the last one year (12 observations) to the last 40 years (480 observations). Then it will be seen how the VaR values change as the observation periods change from 12 to 480.
Result
As can be seen from the above graphs that the VaR calculations continue to increase when the data is used for up to 10 years. After 10 years, all the four companies show relatively less variations in VaR. There may be a few possible reasons for the same.
- As the above graphs show that the returns of Boeing are not at all normally distributed for the one year or five years histogram. The one year returns have only 12 observations in it. Therefore, due to the lack of data, there is a loss of tail. This is the primary reason for Boeing (also other companies) to have got minimum VaR value in the first and second year.
- As the number of observations increases, there is a high chance for the market returns to be more equally distributed around the mean. Therefore, when the number of observations goes fairly up (more than 10 years or 120 observations), the normality of the distribution still holds good.
However, it is to be also analyzed why all the four stocks have stabilized after 10 years and not afterwards.
As the above graphs show that all the four stocks have exhibited a decline around the 400th observation. This may have happened owing to some financial distress, market corrections, depression or recession. Probably, in this case, all these four companies’ returns were majorly corrected during and after the 2007 financial downturn. Before that, companies witnessed some volatility, but they were not as major as what happened during 2007. That is why as more and more data are used for the VaR calculation beyond the year 2007, the VaR values do not display high levels of volatility.
One of the major problems with historical simulation method is that it is highly dependent on the outliers. Also, if the historical data based VaR calculation is done using equal weights, then for a large data set, the data of older volatility and returns may not reflect the current market situation correctly. In the data set used for this VaR calculation, it can be observed that the market returns and variations are relatively low during the first 20 years, whereas the market volatility has increased highly in the last 10 years. Therefore, making the prediction of VaR for September 2014 at 95% confidence level on the basis of data, which includes the period of the 1970s and the 1980s, may produce conservative outcome. This problem can be partially overcome using the following parameters wisely:
- Observation Period: It is important to define observation period that is not too big or not too small. Too big an observation period will create old data affecting the present result. Two small a data set may not represent the asset fully and may only capture the present volatility.
- Weighted Average: It is also important to give more weightage to the present historical observation than the older historical observation.
- Time Frame: Time frame is also important. For the long term investors, a time frame of a year or month based observation is important, whereas for the daily investors, it is important to consider historical returns on a daily basis.
Implementation of Variance - Covariance Method
Data
In order to understand the strengths and weaknesses of the Variance - Covariance Method, two portfolios of data have been chosen. Portfolio A contains the stocks of Du Pont, Johnson & Johnson, and Merck & Co. Portfolio B consists of American Express, 3M, and United Technologies.
Portfolio A: Jarque - Bera Normality Test: Asymptotic p Value: 0.05582
Portfolio B: Jarque - Bera Normalality Test: Asymptotic p Value: < 2.2e-16
First, a Jarque - Bera Normality Test was conducted on the two portfolios to see that the expected returns of the portfolio follow the normal distribution or not. The null hypothesis was that the returns of those portfolios follow normal distribution. Looking at the p values, it can be said that portfolio A’s returns are normally distributed with 95% confidence level, whereas the null hypothesis for portfolio B is rejected. Therefore, it can be concluded that the returns of portfolio B are not normally distributed.
For the Variance – Covariance Method, the three basic assumptions are as follows:
- Returns on risk factors are normally distributed.
- The correlations between the risk factors are constant.
- Price sensitivity of each portfolio constituent is also constant.
As normality is a big assumption in the Variance – Covariance Method and it has been seen in the previous sections that the number of observations below 10 years may not satisfy the normality assumption. Therefore, a minimum observation period of 10 years will be used for calculating the VaR in this method. First, the return of the portfolio for the 121st month will be predicted by using the historical data from June 1974 to May 1984 (10 years data). Once the calculation will be complete, the next time interval will be considered by using the data of the returns for the next 120 months (from July 1974 to June 1984). The same procedure will be repeated for calculating the VaR for the 122nd month. This procedure will be repeated for 359 more time intervals, and the corresponding VaR will be calculated for each time interval.
Using this data set, the following will be analyzed:
- Variations in VaR with changes in the input data
- Variations in VaR calculation for normal and non-normal portfolios
Results
Portfolio A Returns
Portfolio B Returns
Using the 10 year observation period and 360 data sets, we have constructed the above VaR values. For instance, the red line at 0 level with a VaR value of 95 at 95% confidence level indicates that the use of the data of the period from June 1974 to May 1984 gave a VaR value of 95. In other words, there is a 5% chance for the portfolio A to lose $95 in the next month. This way the whole graph for portfolio A (red line) and portfolio B (black line) is constructed.
It is also evident from the above graph that there is less volatility in the prediction of VaR for portfolio A. This is primarily because 1) Portfolio A has normal distribution of returns that have already been discussed above, and 2) the three stocks in portfolio A behave in a more predictable manner without huge market corrections. Portfolio B, on the other hand, is already established as a non-normal distribution. The Variance – Covariance Method used to calculate VaR for non-normal distribution gives erratic values. Portfolio B has a thick-tailed distribution that causes huge variations in the VaR calculation. Because of the presence of outliers, these VaR values are also getting pronounced.
The black line in the above graph shows that any variation in portfolio B is magnified in the VaR calculation. For example, all the stocks in portfolio B have shown relatively stable growth with less volatility till 1999-2000. The black line shows continuous improvement in the VaR values during that time, going as low as $72. However, bank stocks get huge market corrections when the market goes through any recession. In portfolio B, American Express is the financial stock that got market correction during the bubble burst of 1999-2000 and again for heavily corrected during 2007-2008 subprime crisis. These two huge corrections of American Express stock have pushed the VaR values up. Since we have used 120 data sets (10 years), any data set, used for VaR calculations, with the data of 1999-2000 or 2007-2008 included gave high VaR values.
In the real world, it is almost impossible to find a portfolio that is normally distributed. Generally, any portfolio that contains callable bonds, options, and banking stocks is usually non-normal in nature. In such cases, the above method used provides a good approximation of VaR, but Monte Carlo simulation provides a better result.
Implementation of Monte Carlo Method
Data
In Monte Carlo method, we will be using Pfizer Inc. as the stock for analysis. The data set of the time period of June 1974 to June 2014 will be used. As Monte Carlo method is extremely computation-intensive process, we have only used a single stock for the calculation. We will use the same procedure by dividing the dataset as done in the previous section. In each of those time intervals, we will simulate 100 price paths over N times where N=10 or 100 or 1,000 or 10,000. This means that in each interval, the simulation will be running and predicting price point N times for each of those 100 price paths. As N increases, it is expected that the price path will become smoother because the variation will be random and therefore, normally distributed.
Result
Based on the past historical data, Monte Carlo simulation tries to construct the future hypothetical market conditions. For example, for N = 10, Monte Carlo simulation constructs 10 possible future scenarios and comes up with an average VaR on the basis of the outcome of those 10 scenarios. The number of possible hypothetical scenarios is infinite. Therefore, the higher the value of N, Monte Carlo simulation will be able to generate more future possible scenarios. That is why it can be seen from the above graph that as N increases, more random future scenarios are generated, and the VaR values become smoother. The major benefit of Monte Carlo simulation is that it has no underlying assumption like the Variance - Covariance Method. Non-linear portfolios are best suited for calculating VaR using MC.
Comparison: Historical Method vs Variance-covariance Method vs Monte-Carlo Method
Contrary to common belief that Monte Carlo simulation provides best VaR result, we can see from the above figure that Monte Carlo simulation has provided worst VaR figures. This is primarily because we have used only 24 historical data points based on which we are trying to predict 100 possible future scenarios. As this is a small data set, using a small number of scenario simulations will magnify the outlier effect, which has probably happened with the Monte Carlo method.
Between the Variance – Covariance and the historical method, the historical method gave better result because as the above graph shows that the data set is non-normal, there is negative skewness in the data set used for the calculation of VaR. Therefore, in this type of scenario where the data set is small and non-normal, and we do not have huge computational ability, the historical method is the most suited one.
We have also compared the three methods using the data set of 480 (40 years) observations. In that case also, the historical method outperformed the Variance – Covariance Method. Again, it can be seen from the above graph that this has happened because the data set is still not strictly normal.
What are Empirical CDF and Kernel Estimation?
The cumulative distribution function estimates can be done from non-parametric and semi-parametric method. For small dataset, linear cumulative distribution functions are used with values and weights like the one used in the above section in which there are 24 observations. It is easier to construct a linear non-parametric CDF for the dataset. From the stairstep empirical CDF, it is easier to determine statistical parameters like mean, probability, and VaR. As the number of data increases, CDF becomes more continuous, and it can approximated as a cumulative density functions.
CDF is a step-wise function, whereas the kernel estimation makes the function smooth and non-parametric. Kernel density is a continuous non-parametric density estimate, which is difficult to calculate than linear CDF, but is more accurate for large data sets. Kernel estimation can be used for VaR calculations, and it provides a huge advantage over the historical method and the Variance – Covariance Method as it is non-parametric in nature.
