# Could we estimate a portfolio's volatility using a GARCH on the portfolio returns?

Estimating the volatility of a portfolio is typically done by first estimating the covariance matrix. This, however, can be difficult to do accurately and predictivly. This paper gives a nice summary of the various methods.

But why make it so complicated?

Let's say there are $n$ securities $s_1, s_2 \dots s_n$, which at time $t$ has a price of $p_{i,t}$.

You're interested in the portfolio with weights $w_i$ in security $s_i$.

Why not take the time series of the portfolio value $\sum w_i p_{i,t}$ and do a normal GARCH estimate on that?

This technique seems more straight forward and probably just as accurate.

Am I missing something?

Update 10/7: To be clear, I would like to estimate the current volatility of the portfolio.

• The choice of the model you want to use to estimate volatility depends probably on the use you want to make of this measure... You should add this info in your question. Are you trying to estimate past volatility? are you trying to predict future volatility? for what purpose? – SRKX Oct 7 '15 at 5:29
• It seems you cross-posted, please delete the other question on Cross Validated. – SRKX Oct 8 '15 at 1:00

Yes, you can use Multivariate GARCH model to estimate the volatility of a portfolio. For example, the Constant Conditional Correlation(CCC) GARCH model. In the CCC GARCH model, it says there is a constant correlation between portfolio and the model is defined as:

Once you have estimated the correlation matrix, the the composed volatility can be computed by the product $w'H_tw$.

• I'm assuming for $R$, you use the entire time series to estimate the correlation. That's why it's called constant correlation? – JPN Oct 10 '15 at 22:18
• Yes,@JPN. But if you want a dynamic one, then check the model: Dynamic Conditional Correlation GARCH model or Asymmetric Dynamic Correlation GARCH model. – Fly_back Oct 11 '15 at 17:18

If you want to use GARCH to estimate past local volatility of the portfolio you can do but, but you'd use GARCH to model the portfolio returns, not prices.

Then you will be able to build a range of possible volatilities in the futures given a certain confidence level and you would have a local volatility $\sigma_t$ for each historical point.

It depends on what you are trying to do. First of all, you would estimate GARCH on the portfolio returns, not the portfolio value, as @SRKX points out.

If you are trying to forecast what the volatility of the portfolio will be in the future, then the danger is the portfolio weights you have today are not the same as what you held in the past. For instance, suppose I was fully invested in Treasuries over the GARCH estimation period, then I sold all the bonds and bought out of the money call options on the S&P 500. The GARCH forecast would be too low because it does not reflect the new portfolio characteristics. How important this issue is depends on how much trading the portfolio does. If there is no portfolio turnover during the GARCH estimation period, then it's a reasonable enough approach.

If instead of trying to forecast GARCH volatility of the current portfolio, you are just trying to evaluate what the historic volatility of the portfolio was, then this is a good approach.