# Asset allocation and GARCH models

I am trying to solve an asset allocation problem and I am having some troubles grasping the concept. I am working with excess returns on 4 stock indices and I am obtaining the excess returns forecasts for period $t+1$ as the conditional mean forecasts from ARMA(1,0)-GARCH(1,1). For each excess returns series I am using a different distribution, one that suits best.

Ideally I would like to use an asset allocation strategy that is taking the expected mean and the second, third and fourth co-moment matrices into consideration. And here is my problem, since I am building the strategy at time $t$ how can I obtain the co-moment matrices of innovations necessary for my asset allocation strategy.

For the covariance matrix my idea was to bind my univariate GARCH models in to a CCC-GARCH model and obtain the covariance matrix for time $t+1$ as $H_{t+1}=D_{t+1}RD_{t+1}$ where $H$ is the covariance matrix, $R$ is the unconditional correlation matrix of standardized residuals up to time $t+1$ and $D$ is a matrix with conditional standard deviation forecasts at the diagonal.

However, all the times I am encountering the same problem: how to obtain the standardized residuals for period $t+1$, which are necessary for calculating the 2nd, 3rd and 4th co-moment matrices? I need to choose the weights for each index at time $t$ and therefore cannot obtain the residuals as observed returns minus the forecast, as I do not yet know the observed return. Could you give me some advice on how to develop such a strategy? Some papers are obtaining the co-moment matrices directly from the parameters of various multivariate distributions, however, I would like to use different ones for each series. Any help on this matter would be greatly appreciated.

• It is not clear to me...Why don't you know the realized returns ? – Malick Jan 5 '16 at 16:44
• @Malick Because I am obtaining the weights in period $t$, based on the forecasts for period $t+1$. So in period $t$ i know my one-step ahead forecasts, but I do not know what the realized return in period $t+1$ will be. – Masher Jan 5 '16 at 16:48
• humm.. still not clear to me because at time $t$ you never need to know the $t+1$ realized returns to forecast. Also did you notice that the correlation matrix is time-invarriant $R_{t}=R$ in the CCC model so you can keep your first estimate of $R$ (based on past std residuals) to forecast. – Malick Jan 5 '16 at 16:56
• What you say about $R$ is true, for the $t+1$ forecast of the covariance matrix I use the first estimate of $R$ as it is time-invariant in this specification. Here is a link to a paper, based on which I want to construct the asset allocation strategy: hec.unil.ch/ejondeau/publications/… What I want to do is described in chapter 1.2. Maybe this will be helpful in understanding the nature of my problem. As you can see in the formulas for the covariance matrix and for the co-moment matrices the authors refer to realized returns in period $t+1$. – Masher Jan 5 '16 at 17:03
• I'm not an expert about these models so I may be wrong, but in the appendix (see first equation p117) components are written without the realized return terms but only as the sum of weighted returns forecast. – Malick Jan 5 '16 at 18:52