In FIS Front Arena – Scenario Value at Risk (VaR) we discussed computation of scenario/stressed VaR. Today will discuss more sophisticated parametric VaR. It is also called variance-covariance or delta-normal method. This blog is going to be mathematical in nature due to the computation involved. But again, I will try to keep it as simple as possible. First, we shall understand calculation involved and expected result. We shall then get into action in Prime!

Parametric VaR formula

VaR = µ – σ * z * P

Where:
µ = mean
σ = volatility, also called standard deviation
z = z score
P = position value

In standard normal distribution, mean returns are assumed to be zero.
This implies VaR = – σ * z * P

Example

Lets compute 95% VaR for following equity positions:

P1 = 2,500 INR and quantity 80 = 200,000
P2 = 2,000 INR and quantity 150 = 300,000

Also, lets assume below standard deviation and correlation for these positions.
σ1 = 0.028
σ2 = 0.040
ρ12 = 0.8

Total position = P1 + P2 = 500,000

Weight of P1 = W1 = 200000/500000 = 0.4
Weight of P2 = W2 = (1 – W1) = (1 – 0.4) = 0.6

Portfolio volatility = sqrt(W1² * σ1² + W2² * σ2² + 2 * W1 * σ1 * W2 * σ2 * ρ12)
= sqrt(0.4² * 0.028² + 0.6² * 0.040² + 2 * 0.4 * 0.028 * 0.6 * 0.040 * 0.8)
= 0.033638

z score for 95% = 1.645

95% VaR = -0.033638 * 1.645 * 500000 = -27,667.255 INR

Now, let’s get into Prime and cross check the VaR.

Equity Portfolio

Risk Factor Setup

Volatility and Correlation Input files

Trading Manager

Observations

Our calculated PVaR value of 27,667 is quite close to PVaR calculated by Prime. That could be due to rounding issues in our calculation.
Portfolio PVaR of 27,665 is less than combined VaR of each stock which is 9,211 + 19,738 = 28,949.
This is because the correlation between Lax_Stock1 and Lax_Stock2 is less than 1.
Portfolio PVaR would have been even smaller if correlation was negative. That is why it is recommended to diversify portfolio with assets that are negatively correlated.

Conclusion

This time too we analysed a simple portfolio so that we don’t get diverted from the goal of this blog, which is to understand how to configure and interpret PVaR results in Front Arena Prime. I am sure you can apply this knowledge to more complex portfolios.