Multivariate GARCH with respect to Value at Risk

Question

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?

Answer 1

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.

Answer 2

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.

Answer 3

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.

Answer 4

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.