# Monte Carlo & VaR - Application Issues

#### MattRobinson

Hi Guys,

First real forum post! I have been playing around with Monte Carlo (MC) with respect to Value at Risk and have hit a wall so to speak. A problem stemming from my lack of experience with this process.

I have come across the Benson_Zangari methodology (attached the paper) which simulates a vector of returns Y via the following:
$Y_j = \sum_{i=1}^n R_{ij}Z_i = \sqrt{\frac{1 - \lambda}{1 - \lambda^n}}\sum_{i=1}^n \lambda^{(i-1)/2}f_j(n-i+1)Z_i, \qquad i = 1, \dots,n \quad j = 1, \dots, k.$
Where k is the number of risk factors (i.e assets in a portfolio); n is the number of historical observations for each risk factor and lambda is the exponential smoothing factor. Z is a vector of iid standard normal random variables

This is quite simple and easy to replicate in C# or Python, i obtain a covariance matrix (R'R) and the simulated returns vector.

The Monte Carlo is applied to the simulated returns with the addition of Ito's drift term. Just to keep things simple (ignoring sampling & analysis horizons), the returns are generated via MC by:
$Y_a = R'Z - \frac{\sigma^2_j}{2}.$
Where the variance is obtained from the covariance matrix.

My confusion comes from the actual calculation of VaR from the set of simulations above. Do i go about the standard way of simulated profit/loss, ranking then for each asset and then calculating VaR (This wouldnt calculate VaR for the portfolio as VaR is not additive, additionally i cannot include the quantity of each asset in the calculation).

So how do i go about it?

The first thing that comes to my head is using the MPT formulas for k asset portfolio return and variance.

I can calculate the weights for each asset using the NAV and the quantity of each asset in the portfolio and use that in the formula, however how would i go about the MC process?

Any guidance would be extremely appreciated, I think I am missing out some key theory concerning Monte Carlo VaR for an N asset portfolio.

Matt

#### Attachments

• benson_zangari.pdf
290.1 KB · Views: 27

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