# Multivariate GARCH with respect to Value at Risk

For the last few weeks I studied GARCH with respect to individual assets. Now I want to combine assets and execute a multivariate (DCC) GARCH test. Let us say that I have a portfolio containing four assets. By using a software called G@RCH I obtained the conditional variance of the portfolio. Now I want to use this conditional variance of the portfolio to calculate the Value at Risk (abbreviated as VaR).

With univariate GARCH one could multiply the variance with the probability density function to obtain the Value at Risk. However, I’m not sure how I should perform this procedure for a multivariate GARCH!

I want to explain my thoughts and I hope someone can confirm or improve my way of thinking! I have a multivariate DCC student t-distribution GARCH which gives me the conditional variance for day t+1. Now I make an equally weighted portfolio times series from the individual returns (so 25%*asset1+25%*asset2+25%*asset3+25%*asset) to perform an univariate student t-distribution GARCH procedure to obtain the density function. Then I multiply this density function with the conditional variance to obtain the Value at Risk for the portfolio.

Is it correct to assume that the univariate GARCH distribution function will give me the correct density function as input for the Value at Risk formula?

• Did you read any literature on VaR? For instance Jorion's VaR book? There are many ways of doing VaR with GARCH in it. You have to pick one that fits your problem. – Aksakal Dec 24 '14 at 15:20
• Yes, actually I had to study the whole book last year and I passed the course. As mentioned in the question, I have already performed the univariate GARCH models (EGARCH, GJRGARCH, APARCH etc.) on individual assets with various VaR models (gaus, student, skewed student, GED). However, it still remains unclear for me how to perform the study when using multivariate GARCH in combination with VaR. Should I just aggregate the 4 individual returns and get a density function from the aggregation? – DavidjeK Dec 24 '14 at 16:04
• it's not clear what you're trying to aggregate, returns or market values? You can't aggregate returns if they're t distributed without some approximation – Aksakal Dec 24 '14 at 16:07

Not sure if this will be of any help but here is how I would think about it.

The joint distribution of the individual returns on the assets in the portfolio is sufficient to obtain VaR of the portfolio. (Is the joint distribution necessary? I am not sure.)

Step 1: Do you have the joint distribution? Since you are using a DCC model, you have assumed some unconditional joint distribution for the error terms. Given a fitted DCC model, you have a conditional joint distribution of the returns of the elements in the portfolio, for each time point in the sample $t=1,...,T$. Also, from a fitted DCC model it is straightforward to obtain the conditional joint distribution for the next period ($t+1$).

Step 2: Given the joint distribution, how to obtain the distribution of a (weighted) sum of the individual returns? A multivariate normal case is the simplest, see the answer by @emcor. Other multivariate distributions could be more tricky, but hopefully the link in the answer by @emcor gives an accessible solution to the problem.

I hope breaking down the problem as above will help identify what step (or sub-step) is the tricky one so that you can work on it further.

• Thanks a lot! For the research I will use both the multivariate normal distribution and the multivariate student t-distribution (to account for fat tails) for my research. Using the software G@RCH, I can obtain one-day ahead forecast of the conditional portfolio variance and the used degrees of freedom. By performing this process h times I get h degrees of freedom and h conditional variance. This found degrees of freedom (I believe this is based on the used observations within DCC) will then be implemented in the following VaR formula VaR_α(L)=-μ_p-√((v-2)/v) σ_p t_(1-α,v) – DavidjeK Dec 26 '14 at 12:02
• The formula is very unclear, I'm sorry for that. For that reason I uploaded a picture of the formula.. static.afbeeldinguploaden.nl/1412/89767/58g2nl.PNG And I also made a screenshot of the output of the software, so that in the future people with similar problems can fall back on this site :) static.afbeeldinguploaden.nl/1412/89768/SxtWu.PNG (It's output of a cDCC model !) – DavidjeK Dec 26 '14 at 12:11
• I'm glad you found it helpful! I have to admit I have never worked much on VaR in particular (just a bit on GARCH and DCC models). But if you need confirmation or comments regarding your solution, you may want to update you original question or post a new question to get broader attention. (Comments on a particular answer like mine might get less noticed.) – Richard Hardy Dec 26 '14 at 21:44
• Thanks for the advice! First I will try to find the answers myself, and when I’m stuck again I will update the question! Sorry to bother you, but I was wondering what kind of misspecification tests or forecasting tests you performed as you didn’t work much on Value at Risk. Maybe I could include your knowledge and experience to look at the results from a different angle! Thanks for the advice! I compared my theory with some friends, and they were convinced – DavidjeK Dec 27 '14 at 13:04
• Well, I was trying to fit univariate GARCH models to a number of commodity returns and then a DCC model on top of that. I checked the individual GARCH models by running Li-Mak test (for remaining conditional heteroskedasticity) and Ljung-Box test on standardized (by inverse fitted standard deviations from GARCH models) returns for remaining autocorrelations. After DCC you may try applying system-wide tests for the whole covariance matrix (a la multivariate Li-Mak) and the mean equations for returns (a la multivariate Ljung-Box); but I did not do these at the time (although I should have done). – Richard Hardy Dec 27 '14 at 22:16

A portfolio $$X_p=\sum_{i=1}^nw_iX_i$$ is a univariate random variable.

The distribution of the sum must be calculated by convolution. This paper says to have found an expression: http://www.tandfonline.com/doi/abs/10.1080/02331888.2012.694447?journalCode=gsta20

E.g. for the normal distribution: $$X_p=\sum_{i=1}^nw_iX_i\sim N\left(\sum \mu_i,w'\Sigma w\right)$$ Then you get the corresponding portfolio VaR by the known formula for normal VaR with the above parameters.

I'll suggest you an easy way to start with. It's called sometimes historical portfolio VaR. Get the series of portfolio values and the returns: $$V_t=\sum_{i=1}^4n_{it}p_{it}$$ $$\Delta V_t=\sum_{i=1}^4n_{it}(p_{i,t-1}-p_{i,t-1})$$ $$r_t=\frac{\Delta V_t}{V_{t-1}}$$ Here, $n_{it},p_{it}$ are number of shares and the price of a stock $i$ on day $t$.

Next, Apply GARCH to the portfolio returns. Compute VaR using the obtained conditional variance for tomorrow.

• Well, correct me if I'm wrong, but the main advantage of using multivariate GARCH is to account for time-varying correlation between the individual assets. My study field is based on the difference of Emerging-Developed markets, and for that reason it is very important to take this time-varying correlation into account. By using multivariate GARCH I obtained the daily conditional variance of the portfolio. It is unclear to me with which density function it should be multiplied to obtain the Value at Risk. Historical would indeed solve this problem, but research showed it's inaccruacy. – DavidjeK Dec 24 '14 at 16:06
• If you this method you are accounting for all the correlation implicitly. – Aksakal Dec 24 '14 at 16:08

When applying multivariate GARCH models, you are supposed to use the Variance/Covariance approach.

Let's assume that you have obtained the conditional variance forecast, say $$H_t$$.

Then, assuming a vector of portfolio weights $$w_t$$ at time $$t$$, the portfolio variance will be $$\sigma_p^2=w_t H_t w_t$$.

Depending on whether you have defined a level equation (say a conditional mean, or VAR etc.) or not, you should have a value for $$\mu_{p,t}=w_t r_t$$.

Then, the Value-at-Risk of the portfolio should be:

$$VaR(\alpha)=-\mu_{p,t}- \sigma_p F^{-1}(\alpha)$$,

with $$F^{-1}(\alpha)$$ denoting the quantile of the distribution you assumed to obtain the $$H_t$$ matrix.