Choosing a Covariance Structure for Return Forecasts
Summary
The document asks whether portfolio optimizers should reshape the covariance of expected-return forecasts, given that forecast covariance can be transformed independently of the market-return risk matrix. The response cautions that raw forecast covariance is generally a poor proxy for future market covariance: forecasts represent conditional means, while the risk matrix represents conditional variance and correlation. It cites a portfolio-management text for the claim that forecasts have little effect on volatility and correlation estimates, and that any such effect relates more to forecaster skill than to the particular forecast.
For refining forecasts, the answers suggest blending them with a prior, such as market-implied returns, or using a multivariate skill-and-signal approach or a factor-model correlation structure. A second answer argues that over a sufficiently long history, forecast covariance should approach its sample covariance absent systematic covariance error, and proposes matching the transformed history to that sample or to a Bayesian prior. These are competing practical perspectives, not a demonstrated universal optimum; the appropriate structure depends on the reliability of the data and prior assumptions.
Key ideas
- Forecast covariance describes the variation of signals and should not automatically replace the market-return risk matrix.
- The response distinguishes conditional expected returns from conditional volatility and correlation.
- Forecasts can be blended with a prior, including market-implied returns or a factor-based structure.
- A historical forecast covariance can be compared with its sample covariance, though systematic error may undermine that approach.
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# Is there an optimal covariance one would want forecasts to have?
# 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.
## Answer by Ram Ahluwalia (score 4)
https://quant.stackexchange.com/a/3392
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.
## Answer by Konsta (score 0)
https://quant.stackexchange.com/a/3394
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.)Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.