Analytical Approach to Calculating VaR (Variance-Covariance Method)

We earlier saw how VaR can be calculated using the parametric method. We will now look at this method in detail, and also understand how VaR can be easily calculated using matrices.

VaR of a Single Asset

VaR of a single asset is the value of the asset multiplied by its volatility. Here, the volatility can be calculated at the desired confidence level.

Example:

An IBM stock is trading at $115 with a 1-year standard deviation of 20%.

In the normal distribution, 95% confidence level is 1.645 standard deviations away from the mean.

Therefore, our VaR at 95% confidence level will be:

VaR (95%) = 115* 0.20 * 1.645
= 37.835

Assumption of Normality

A distribution is described as normal if there is a high probability that any observation
form the population sample will have a value that is close to the mean, and a low
probability of having a value that is far from the mean. The normal distribution curve
is used by many VaR models, which assume that asset returns follow a normal
pattern. A VaR model uses the normal curve to estimate the losses that an institution
may suffer over a given time period. Normal distribution tables show the probability
of a particular observation moving a certain distance from the mean.

If we look along a normal distribution table we see that at -1.645 standard deviations,
the probability is 5%; this means that there is a 5% probability that an observation
will be at least 1.645 standard deviations below the mean. This level is used in many
VaR models.

VaR of a Portfolio

Generally VaR will not be calculated for a single position, but a portfolio of positions.
In such a case will require the portfolio volatility.

The portfolio volatility of a two-asset portfolio is given by:

Where:

  • w_{1} is the weighting of the first asset
  • w_{2} is the weighting of the second asset
  • \sigma_{1} is the standard deviation or volatility of the first asset
  • \sigma_{2} is the standard deviation or volatility of the second asset
  • \rho is the correlation coefficient between the two assets

The VaR will then be given by:

Where:

  • VaR (1 – \alpha ) is the estimated VaR at the confidence level 100 × (1 – \alpha )%.
  • Z_{\alpha} represents the no. of standard deviations on the left side of the mean, at the required standard deviation.

VaR of a Portfolio – Example

Let us assume that we want to calculate Parametric VaR at a 95% confidence level over a one-day horizon on a portfolio composed of two assets with the following assumptions:

  • P = $100 million
  • w_{1} = 50%
  • w_{2} = 50%
  • \sigma_{1} = 3%
  • \sigma_{2} = 5%
  • \rho = 30%

Portfolio of n Assets

If the portfolio is bigger than 2 assets, the volatility of the portfolio is expressed using matrix notation:


Where:

  • w is the vector of the weights of the n assets.
  • w' is the transpose vector of w
  • \sum is the covariance matrix of the n assets

This is the reason why this method is also known as Variance Covariance method.

Variance Covariance Method – Examples

Example 1 – Two Asset Portfolio

This spreadsheet takes the same example above and recalculates the VaR using the matrices.

Two Asset Portfolio VaR

Example 2 – Three Asset Portfolio

This spreadsheet shows the VaR calculation of a three-asset portfolio.

Three Asset Portfolio VaR

It is easy from there to expand the calculation to a portfolio of n assets. But be aware that you will soon reach the limits of Excel as we will have to calculate n(n-1)/2 terms for your covariance matrix.

Series NavigationValue at Risk (VaR)Three Methodologies for Calculating VaR

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8 Responses to “Analytical Approach to Calculating VaR (Variance-Covariance Method)”

  1. Ahmosa April 25, 2011 at 3:04 am #

    Brilliant but from where did we get the 1.645 in calculating the VAR (95%), Thank you.

    • admin April 25, 2011 at 8:47 pm #

      This value refers to the standard normal distribution curve. In the standard normal distribution curve, a 95% one-tail VaR corresponds to 1.645 standard deviations.

      • Dhiraj March 24, 2013 at 5:02 am #

        But i have this in my portion (syllabus), so i want to know how really can we calculate this. Since, i got the whole idea about variance-covariance method of VaR but I am trying calculating 1.645 using the table but still unable to get it. Please help. Thnx in advance

        • mary September 2, 2013 at 7:19 am #

          95% confidence level is same as 0.95.now u go to ur normal distribution table and inside for a number that is approximately to 0.95 .select the closest there is 0.9505 this is the closest to 0.95,now see it fall on 1.60 on the horizontal x-column and 0.05 vertical top x-row.so if u add the two u will get 1.65 which is an approximation of 1.645

  2. max May 2, 2012 at 4:54 pm #

    How to calculate daily price VAR of single asset?

  3. Mas Budhi October 9, 2012 at 9:00 pm #

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  4. JIA CHEN April 3, 2013 at 9:23 am #

    That’s very useful, but I have a question.
    I have a portfolio consists of 4 stocks, and the daily closing prices of these stocks for the past 3 years are given. I’ve calculated the portfolio values for each day and finally I have 755 portfolio values. But which portfolio value can be used to calculate the value at risk using variance and covariance method?
    Because you said VaR(1-alpha)=P*sigma*Z.
    Thank you in advance.

  5. ombo May 21, 2013 at 1:33 am #

    dear sir,

    I have portfolios consist of 3 stocks, 4 bonds, 1 time deposits and 5 mutual funds. Now i’m stuck for calculating VaR portfolio for these. Maybe you can help me out for these. It looks complicated for me. Thanks

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