# Is there an optimal covariance one would want forecasts to have?

Often in a quant process, one will generate a time series of return forecasts and use them in some sort of optimization to generate a portfolio. Generally, there will be a covariance matrix of market returns used in the optimization. I haven't seen any attention ever placed on the covariance of the return forecasts themselves, however. Since we could easily adjust the covariance of the forecasts to anything we'd like via: $$\textrm{forecasts}_{\textrm{new}} = \Omega_{\textrm{new}} R_{\textrm{new}}^{1/2} R_{\textrm{old}}^{-1/2} \Omega_{\textrm{old}}^{-1} \textrm{forecasts}_{\textrm{old}}$$ where $\Omega$ are diagonal matrices of volatilities and $R$ are correlation matrices. I'm wondering if there is some optimal $\Omega_{new}$ and $R_{new}$ one would want to use, or if a transformation such as this would provably not matter?

I wouldn't necessarily expect, for example, that the covariance of the raw return forecasts would be a good forecaster of the covariance of market returns, so another way of putting this question is, should we "fix" the covariance of forecasts in some way? If one had the true future covariance of market returns, for example, wouldn't we want to impose that, and by extension, wouldn't we want to use whatever our forecasted covariance to be? I have not been able to derive a proof or find a reference that it is good or inconsequential to do any such transform.

• While "generate a time series of return forecasts" might be what could be done, in standard portfolio optimisation you take only into account the expected return for the next period. So you are suggesting to save these forcasts for a couple of periods and look at the covariance of these series? After some more reading and thinking your model pretty much looks like a VAR (vector autoregressive) model. Apr 27, 2012 at 23:49
• Yes, I am suggesting saving these historical forecasts and looking at the covariance of the series. It is not necessary to specifying the model that generates the expected returns at all, I'm simply asking whether adding any particular covariance, and specifically expected market covariance, is provably good or inconsequential. I am not suggesting a VAR model specifically at all since transforming the covariance via the stated equation does not add any dependency on lagged values.
– jdav
May 1, 2012 at 17:22

Is the covariance of the raw return forecasts a good forecaster of the covariance of market returns?

As you suggest, the covariance of the raw return forecasts is a lousy forecast of the covariance of market returns. Grinold & Kahn explain why quite eloquently in Active Portfolio Management, 2nd edition (pg. 275).

It might be tempting to augment the risk matrix with the correlations of the alpha signals, however, this "line of thought confounds the notion of conditional means (i.e. the expected return on the S&P 500 taking into consideration research) and the notion of conditional covariance (i.e. how research should influence the forecasts of variance and covariance)."

It is surprising to note that forecasts of returns have negligible effects on forecasts of volatility and correlation. It is even more surprising to note that what little effect there is has nothing to do with the forecast and everything to do with the skill of the forecaster. This welcome news makes life easier. We can concentrate on the expected return part and not worry about the risk part.

Should we "fix" the forecasts in some way?

Taking a Bayesian line of argument, if you have a strong prior then you can and should fix (i.e. blend) the forecasts with the prior. Rather than use the covariance of market returns as the prior, some use market implied returns (i.e. Black Litterman) to update their forecasts.

If these are indeed "raw" or un-refined raw forecasts, you could also use a multi-variate form of Grinold & Kahn's alpha = volatility * IC * score to update the raw forecasts.

Another choice of optimal factor structure to blend with might be correlation structure derived from a industry or style multi-factor model.

For a longer history of forecast returns their covariance should converge to the sample covariance. Else the forecast returns exhibit a systematic error with respect to their covariance. Thus, the transforming matrices should be chosen such that the history of forecasts (including the current forecast) exhibits the sample covariance. (In this process instead of the sample covariance matrix another [Bayesian] prior might be used, towards which the sample covariance matrix was shrunk in the first place, as well. This depends on your belief in the sample covariance matrix as being the true covariance matrix of your series or not.)