Calculation of 10-Day Portfolio VaR: Advantages and Disadvantages of Historical Simulation (Essay Sample)

Calculation of 10-day portfolio VaR
Value at Risk is a measure of financial risk that is widely used. It gives a method of determining the quantity of risk of a portfolio and managing the risk (Marrison, 2002). In calculating the 10-day VaR, three ASX-listed companies were selected. The companies were Azure Healthcare Ltd (AZV.AX), Azonto Petroleum Ltd (APY.AX), and National Australia Bank Limited (NAB.AX). The respective returns for the respective companies (in AUD) were obtained for the period between. The calculations were done using two different methods, the historical simulation and the Variance Covariance parameter approach (VCV approach).
Justification for chosen calculation methods
Historical simulation
Historical simulation was chosen because it offers an implementation of the full valuation in a straightforward manner. Market states are simulated and produced through the addition of the period-to-period changes in market variables in a specific time series to the base case. The historical simulation assumes primarily that there is a full representation of the set of possible future scenarios by the events of a particular historical window. In this method, a set of changes in the risk factor are collected over a given historical window. The scenarios obtained are then assumed to represent all the possibilities that might happen in the immediate future (Alexander, 2008).
Advantages and disadvantages of historical simulation
A major advantage of historical simulation is that no assumptions are made about the changes in risk factor being from a specific distribution. As a result, this methodology is consistent with changes in risk factor originating from any distribution. Historical simulation also does not involve any estimation of the statistical parameters like variances or covariance. It is also continuously exempt from the unavoidable estimation errors. It is easy to explain and defend even to an audience that is not technical yet important, like the corporate board of directors. No assumptions are needed on the distributions of the risk factors, there is ease revaluating the full portfolio based on the scenery data, and it is intuitively simplistic and obvious (Hull, 2008).
Historical simulation also has certain disadvantages. Accomplishing the purest form of historical simulation is difficult since the method poses the requirement of data about risk factors spanning over a long historical period. This requirement is usually necessary so that what might happen in the future can be adequately represented. The method also does not involve distributional assumptions. The scenarios used in calculating VaR are restricted to the ones that took place in the historical sample (Bohdalova, 2007). The past is not always a good way of modeling for the future. There is also a high possibility of gaining results that are erroneous if the sample size is not sufficient (Boyarshinov, 2016).
Variance-covariance method
The variance covariance (VCV) method makes use of a historical sample of the risk factors of a portfolio value. It functions almost like the historical simulation. However, unlike the historical simulation, the variance covariance method assumes that there are normally distributed logarithmic yields for the risk factors. The estimate of the VaR is then equated to a sample quartile of the portfolio yield volatility. The volatility of the portfolio yield is calculated depending on the covariance between portfolio risk factor yields.
Advantages and disadvantages of Variance-covariance method
There are many advantages of variance covariance method. They are easy to implement as compared to the historical simulation method and the Monte Carlo Simulation. The method requires less historical data in comparison to the historical simulation method. In most cases, it has an acceptable precision and accuracy.
The method also has its disadvantages. VCV method has a low quality of estimates for securities whose prices have nonlinear dependence on the risk factors. The assumption of the logarithmic distribution of the yields of the risk factors is not always correct. It also ignores the risk of extreme events that can cause significant losses in the value of the portfolio (Boyarshinov, 2016).
The Monte Carlo Simulation method was not chosen because of its high dependence on the resource which makes it very time-consuming. The sample sizes of observations that are used in deriving the distribution kind and parameters are also often insufficient. The insufficiency can lead to incorrect estimates of VaR.
Calculation of VaR using Variance-covariance method
In this method, the simple moving average (SMA) method was used. The calculations involved include the SMA daily volatility, SMA daily VaR, and Portfolio holding SMA VaR.
The portfolio comprises equal exposures of 1000 shares for all the stocks. The market price for AZV.AX, APY.AX and NAB.AX were AUD 0.068, 0.015 and 28.94 respectively. The historical price data for all the three companies were obtained for the period of 14th July 2015 to 13th May 2016. The daily time series are presented in the extract in Figure 1.this period is called the look-back period, which is the period over which the risk is evaluated.
Figure SEQ Figure \* ARABIC 1: Time series data for AZV, APY and NAB
It is first important to determine the return series. To obtain this, the natural logarithms of the ratio of successive prices are determined. This is shown in Figure 2.
The formula shown in the figure is LN(Cell B19/Cell B18). In cell B19 there is 0.15 while in cell B18 there is 0.15. calculating the LN(0.15/0.15) gives 0 as shown in cell H19.
Figure SEQ Figure \* ARABIC 2: Return Series
The next step is to calculate the daily volatility is calculated using the formula:
Rt = rate of return at time, t.
E(R) = mean of return distribution
The squared differences of Rt are summed over E(R) across all data points then the results are divided by the number of returns in the series minus one so that the variance can be obtained (Alexander, 2008).
After this, the square rood to the result is calculated, which is the standard deviation (SMA volatility) of the return series. The method used in this calculation, however is the use of the STDEV function in EXCEL, which is applied to the return series as illustrated in Figure 3. The daily volatility is determined as 0.494429 for the portfolio.
Figure SEQ Figure \* ARABIC 3: Calculating the daily volatility
Getting the SMA VaR is determining how much will be lost over a particular holding period with a particular probability. The daily volatility is multiplied by the z-value of the inverse of the standard normol cumulative distribution function (CDF) which corresponds with a specific confidence level, which is the NORMSINV (confidence level). For example, in our case, the confidence level is 99%, so it is expressed in excel as NORMSINV (99%). Figure 4 shows how the daily VaR is calculated
Figure SEQ Figure \* ARABIC 4: Daily Volatility
Figure 4 show that the daily VaR is 1.150214. The VaR is the Daily VaR multiplied by the square root of the holding period in days. The square root function in excel is SQRT. Figure 5 shows how the calculation is done in excel.
Figure SEQ Figure \* ARABIC 5: VaR
Calculating VaR using the Historical Simulation method
In this method, there is no assumption that is made about the return distribution that is underlying. The first step is to obtain the return series and reorder them into ascending order from the smallest to the largest returns. This is done by use of the Sort function in excel. After reordering the values, the number of returns in the series is counted using the COUNTA formula as shown in Figure 6 (Bohdalova, 2007).
Figure SEQ Figure \* ARABIC 6: Determining the number of returns
From our series, there are 218 returns. The daily VaR is then calculated as the return that is corresponding to the index number. It is calculated by subtracting the confidence level then multiplying it with the number of returns at the point where the result is rounded down to the nearest integer. The integer is the index number for a particular return. This is shown in Figure 7.
Figure SEQ Figure \* ARABIC 7: Historical Simulation
The 10-day VaR is then calculated by multiplying the daily VaR by the square root of the holding period- 10 days. Figure 8 shows how this is done.
Figure SEQ Figure \* ARABIC 8: 10-day VaR.
From our calculations, the 10-day VaR for AZV.AX, APY.AX and NAB.AX are 0.705641867, 0.909730591 and 0.150688793 respectively. The respective amounts of worst case loss are AUD 47.98364698, 13.64595886 and 4360.933676.
In order to calculate the portfolio VaR through the historical simulation, the returns are combined to get the portfolio change as shown in Figure 9.
Figure SEQ Figure \* ARABIC 9: Calculating portfolio change
The obtained values are then sorted in an ascending order from the least to the largest value. The number of observation is equivalent to the index, which is manually assigned. After this, the VaR is calculated for each portfolio change as shown in Figure 10.
Figure SEQ Figure \* ARABIC 10: Calculating VaR historic
Cell D10 contains the sum of all portfolio values, it is shown in Figure 11.
Figure SEQ Figure \* ARABIC 11: Sum of portfolio values
To get the VaR that corresponds to our rounded down Number of Observation (Index), the IF function is used in column R as shown in Figure 12.
Figure SEQ Figure \* ARABIC 12: Obtaining the VaR
All possible values of VaR obtained by the IF function are then summed up to get the overall VaR. This step is not very necessary since only one value is obtained by the IF function. However, it is required to set the result at an obvious position. It is shown in figure 13.
Figure SEQ Figure \* ARABIC 13: VaR
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