Value at Risk – Methods – Variance Covariance

This method assumes that the daily returns follow a normal distribution. From the distribution of daily returns we estimate the standard deviation (σ). The daily VaR is simply a function of the standard deviation and the desired confidence level. For example, at the 99% confidence level the VaR is equal to 2.33 × σ. To determine the VaR for a
J-day holding period the square root rule will be applied, that is, the J-day VaR= √J × (daily VaR). For example, for a standard deviation, σ, of 100 basis points and an equity portfolio of Rs.10 million, the daily VaR at the 99% confidence level will be Rs.233,000. In other words the maximum daily loss of Rs.233,000 on the equity portfolio will be exceeded only 1% of the time or once in 100 days.

In the Variance-Covariance 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, (σ).

(a)    Simple Moving Average (SMA)

Under this approach the returns are given equal weight when calculating the underlying volatility.

Where Rt is the rate of return at time t and E(R) is the mean of the return distribution, i.e.

‘n’ represents the window length/ look back period or the number of observations used in the calculations.

(b)    Exponentially Weighted Moving Average (EWMA)

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, λ, and placing exponentially declining weights on historical data.

The EWMA variance formula is:

Where λ is a number greater than 0 and less than 1.

The smaller the value of λ 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). If we expect volatility to be constant we would apply a high decay factor (giving a more equal weight to older observations).

One special property of the weights used in the EWMA formula is that their sum to infinity will always equal to 1. We may not have an infinite set of historical data. We just have to make sure that our data set is large enough so that this sum is close to 1 or alternatively we can rescale the weights so that the sum is 1. In our analysis we have used λ = 0.94. In addition, the data sets are large enough so that the sum of weights is very close to one. Hence, no rescaling of weights has been applied.