Calculating Variance-Covariance (VCV) Value at Risk (VaR)
This method assumes that the daily returns follow a normal distribution. From the distribution of daily returns we estimate the standard deviation (Sigma). The daily Value at Risk (VaR) is a function of the standard deviation and the desired confidence level. In the Variance-Covariance (VCV) method the underlying volatility may be calculated either using a simple moving average (SMA) or an exponentially weighted moving average (EWMA). Mathematically, the difference lies in the method used to calculate the standard deviation (Sigma). This methodology is specified in more detail below.
Determining SMA volatility
Under the VCV-SMA Value at Risk (VaR) approach the returns calculated in steps P4 &P5 above are given equal weight when calculating the underlying volatility as given by the following formula:
‘n’ represents the number of return observations used in the calculations.
In our look back period there were 5 observed rates. This resulted in 4 return observation, i.e. n = 4 in the formulas above. Detailed steps for SMA volatility are given below:
Step A1: Calculate the mean of the distribution
Sum the returns over the series and divide by the number of returns in the series. For the portfolio return series this is calculated as follows:
Alternatively this may be arrived at by applying the Excel’s “AVERAGE” to the return series
Step A2: Calculate the variance of the distribution
At each point in the return series calculate the difference of the return from the mean calculated in step A1 above. Square the result and then sum over all squared differences. Divide the resulting sum by the number of returns in the series less one. For the portfolio return series this is as follows:
Date | Returns ( R) | (R-E( R))^{2} |
02/03/2010 | 0.8175% | 0.0013% |
03/03/2010 | 0.6062% | 0.0002% |
04/03/2010 | -0.5002% | 0.0092% |
05/03/2010 | 0.9058% | 0.0020% |
Alternatively, this may be arrived at by applying the excel function “VAR” to the return series
Step A3: Calculate the SMA volatility
The daily SMA volatility is equal to the square root of the variance calculated in step A2 above, i.e. it is the standard deviation or Sigma. For the portfolio return series this is as follows:
Alternatively this may be arrived at by applying the excel function “STDEV” to the return series
Determining EWMA Volatility
The SMA approach gives equal importance to all observations used in the look back period and does not account for the fact that information tends to decay or become less relevant over time. The EWMA method, on the other hand, gives more importance to recent information and hence places greater weight on more recent returns. This is achieved by specifying a parameter, Lambda, (0< Lambda <1)
and placing exponentially declining weights on historical data.
The EWMA variance formula is:
Step B1: Specifying Lambda
In general, the EWMA methodology places more emphasis on recent data as higher weights are assigned through the formula to more recent data. However, the Lambda value determines the weight-age of the data in the formula and the sample size actually considered. The smaller the value of Lambda the quicker the weight decays. If we expect volatility to be very unstable then we will apply a low decay factor (giving a lot of weight to recent observations and effectively considering a smaller sample as weights taper to zero more quickly). If we expect volatility to be constant we would apply a high decay factor (giving more equal weights to older observations).
Because we are using a small sample size in our illustration we have used a Lambda of 0.5. However, an industry standard is to set Lambda to 0.94.
Step B2: Determining Weights
As given in the formula above the weights are calculated at each data point as follows:
λ=0.5 | t-1 | 1-λ | λ^{t-1} | Weights (%)=(1-λ)× λ^{t-1} |
02/03/2010 | 3 | 0.5 | 0.125 | 6.25% |
03/03/2010 | 2 | 0.5 | 0.25 | 12.50% |
04/03/2010 | 1 | 0.5 | 0.5 | 25.00% |
05/03/2010 | 0 | 0.5 | 1 | 50.00% |
Step B3: Scaling Weights
One special property of the weights used in the EWMA formula is that their sum to infinity will always equal to 1. However, it is not possible to have an infinite set of historical data. So if the sum of weights is not close to one then adjustments need to be made. These adjustments include either expanding the data set or the look back period to ensure that it is large enough so that this sum of weights is close to 1 or alternatively weights have to be rescaled so that their sum equals 1. This rescaling is achieved by dividing the weights calculated in Step B2 by 1-λ^{n}, where n is the number of return observations. This is illustrated in our example as follows:
Weights | Scaled Weights =Weights÷(1-λ^{n}) | |
02/03/2010 | 6.25% | 6.67% |
03/03/2010 | 12.50% | 13.33% |
04/03/2010 | 25.00% | 26.67% |
05/03/2010 | 50.00% | 53.33% |
93.75% | 100.00% |
Step B4: Calculating the EWMA variance
The first step in calculating the variance is to calculate the squares of the returns at each data point. Next multiply the squared series with the weights applicable to that data point and then sum the resulting weighted squared series. This is illustrated for the portfolio return series below:
dd/mm/yyyy | Scaled Weights | Returns ( R) | R^{2} | Scaled Weights×R^{2} |
02/03/2010 | 6.67% | 0.8175% | 0.0067% | 0.0004% |
03/03/2010 | 13.33% | 0.6062% | 0.0037% | 0.0005% |
04/03/2010 | 26.67% | -0.5002% | 0.0025% | 0.0007% |
05/03/2010 | 53.33% | 0.9058% | 0.0082% | 0.0044% |
Sum(Variance)= | 0.0060% |
Step B5: Calculating the EWMA volatility
The daily EWMA volatility is obtained by taking the square root of the result in Step B4 above.
Determining SMA and EWMA daily VaR
The daily Value at Risk (VaR) is simply a function of the standard deviation or volatility and the desired confidence level. Specifically:
Value at Risk (VAR) = λ × z-value of standard normal cumulative distribution corresponding with a specified confidence level
For example for a confidence level of 99% the z-value is 2.326 (Excel’s function ‘NORMSINV(.99) may be used to determine the z-value) and the
and the daily Value at Risk (VaR)=2.326 λ. For our sample portfolio, the VCV Value at Risk (VaR)s at the 99% confidence level work out to:
SMA | EWMA | |||
σ | Daily VaR = 2.326σ | σ | Daily VaR = 2.326σ | |
OGDC | 0.7476% | 1.7391% | 0.7305% | 1.6995% |
WTI | 1.1683% | 2.7178% | 1.3120% | 3.0523% |
FX | 0.0907% | 0.2111% | 0.1376% | 0.3201% |
PIB Prices | 0.0137% | 0.0318% | 0.0135% | 0.0315% |
Portfolio | 0.6506% | 1.5136% | 0.7732% | 1.7988% |
Determining Historical Simulation daily Value at Risk (VaR)
Historical simulation is a non-parametric approach of estimating Value at Risk (VaR), i.e. the returns are not subjected to any functional distribution. Value at Risk (VaR) is estimated directly from the data without deriving parameters or making assumptions about the entire distribution of the data. This methodology is based on the premise that the pattern of historical returns is indicative of future returns.
Step H1: Ordered return series derived in Steps P4 & P5
The first step is to order these daily returns in ascending order. Each ordered return corresponds to an index number. In our example this is illustrated as follows for the portfolio return series:
Index Number | R (sorted in ascending order) |
1 | -0.5002% |
2 | 0.6062% |
3 | 0.8175% |
4 | 0.9058% |
Step H2: Determine the index value corresponding to 1- confidence level%
This is given by the number of return observations × (1-confidence level%). The resulting number is truncated, or rounded down to an integer, i.e. if the resulting number is 1.6 the index value will be equal to 1. In our example, however because of the small data size the resulting number works out to 4 × (1-0.99) = 0.04. Following the methodology this results in an index number of 0. However, as this is not a valid number the next highest number, i.e.1 will be used as the index value in our example.
Step H3: Identify the daily historical Value at Risk (VaR)
The daily historical Value at Risk (VaR) is the absolute value of the return in the ordered series in Step H1 that corresponds to the index value derived in Step H2. For the portfolio return series this is the absolute value of the return at index number 1, i.e. 0.5002%
Scaling of the daily VaR
Step S1: Determine the holding period
The holding period is the time it would take to liquidate the asset/ portfolio in the market. In Basel 2 for most instances a ten-day holding period is a standard requirement.
Step S2: Scaling the daily Value at Risk (VaR)
To determine the Value at Risk (VaR) for a J-day holding period the square root rule will be applied, that is, the J-day VaR= √J × (daily VaR).
For the portfolio the Holding VaR for each approach is as follows:
Approach | Daily VaR | 10-day Holding VaR % = Daily VaR × √ 10 | 10-day Holding VaR Amount =VaR% × Portfolio Value |
SMA | 1.5136% | 4.7864% | 3,092.62 |
EWMA | 1.7988% | 5.6883% | 3,675.36 |
Historical | 0.5002% | 1.5819% | 1,022.12 |
The maximum loss that we could experience in our portfolio over a 10-day holding period with 99% probability is PKR 3,675.36 using a EWMA Value at Risk (VaR) approach. In other words there is a 1% chance the losses will exceed this amount in a 10-day holding period.
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