# Simulated VaR with differently distributed processes

I am attempting to calculate the one-month 95th and 99th percentile profits for a two-year portfolio of energy-generating assets over the next three months. This means that the calculation has two components: 1) monthly electricity generation during the aforementioned two-year span and 2) energy futures prices (for each month in the two aforementioned year span).

My analysis to date has shown that: a) the generation of each asset follows its own distribution family (based on minimizing the sum of square errors between empirical percentiles and theoretical distribution percentiles) and b) electricity forward prices (not their returns!) for the energy markets follow ARIMA models with significant variables, significant residual normality, insignificant residual autocorrelation, and insignificant heteroskedasticity. I have ~40 observations per month per electricity-generating asset and about 40 observations per electricity contract (liquidity increases substantially during the final 2 months, or 40 trading days, of an electricity contract -- which substantially changes its behavior)

I propose:

1. Running N energy generation simulations per month per asset
2. Simulating N electricity forward prices per month
3. Taking the product of (1) and (2)
4. Summing (3) across all assets
5. Taking the desired quantile (either q=0.05 or q=0.01) of (4)

Is this approach sound? Generation data are decidedly non-normal (even with the help of Box-Cox transformations and/or standardization and/or differencing) and seasonal (mean reverting at the monthly level) and different sites have starkly different distributions (e.g., Normal, Laplace, Levy, Johnson's Bounded, Cauchy) across months, so I've ruled out simpler VaR approaches like variance-covariance and historical VaR.

I am hesitant because it does not explicitly account for correlation among: a) assets, b) between an asset and its respective forward price, or c) prices of successively expiring forward contracts, or d) generation of successive months.

Your help and time are immensely appreciated!

• Why do you say it doesn't account for correlations? You could measure those correlations and generate normal variates based on them, no? – Oscar Jun 17 '20 at 16:26
• Thank you for your reply, Oscar. Could you please elaborate? I'm not aware of a closed-form solution for combining non-i.i.d. random variables. And, even so, do I care about it if I'm just adding the simulation outputs? Or is any relationship reflected in the shape of the values? I suppose the question is: do I need to model a new, correlation-adjusted process and then generate random values, or can I just generate random values from independent processes and combine as needed. Please be patient with me; this is my first time embarking on such a project. – CasusBelli Jun 17 '20 at 16:31
• Sure, I'll use the example of a portfolio based on equities, I'm not sure to what degree it can be extended to your situation. First you find the covariance matrix for your underlying risk factors and then take cholezky decomposition of said matrix. Now you simulate N normal variates (independent) and you multiply the cholezky matrix with your vector of independent normal variates to get a vector of correlated normal variates. You then use these correlated normal variates to drive your stochastic processes that describe the evolution of the underlying in your simulation. – Oscar Jun 17 '20 at 16:43
• Thank you Oscar. I think I understand. You're saying that I can still use the linear correlation between the two variables, irrespective of their individual distributions, by dumping them into a vector V and taking the dot product of V with the Cholesky matrix? Also, as this is my first time encountering the Cholesky matrix, how is it different from the usual variance-covariance matrix? I'm afraid some of the technical materials out there have gone a bit over my head (it's been over 10 years since I've used linear algebra). – CasusBelli Jun 17 '20 at 16:58
• I'm not sure about the case of different distributions, my example was assuming 2 normal random variables (for example equity returns). The cholezky decomposition is an LU decomposition of a matrix (the L part). I couldn't do it on paper either without looking it up either but it's not anything fancy – Oscar Jun 17 '20 at 17:04