# Return forecasting for portfolio optimization

I have some questions related to forecasting returns and how it's used to generate the inputs for portfolio optimization.

First, I want to understand why factor models such as FF- 3-factor model are not used in practice for estimating the expected returns and covariance matrix (or different estimates given the inputs required for a particular problem) for portfolio optimization. This question originates from the answers here, if I understood the answers correctly, these factor models are used to evaluate and not forecasting. I would like to understand why this is the case. If I wanted to include, let's say, value for the forecast, would it be better to include book-to-market rather than the HML factor as an independent variable?

Second, I would like to get a better understanding of what a "real world" model for stock returns for the purpose of getting the estimates needed for portfolio optimization would look like. Could you please direct me to a book or paper (or any source, really) that shows concrete examples that you think are close to something that would be currently used? I understand that there are multiple ways to model returns, but it would be great to see at least 1 concrete example to get a better idea of the kind of inputs one would consider.

I want to understand why factor models such as FF- 3-factor model are not used in practice for estimating the expected returns and covariance matrix (or different estimates given the inputs required for a particular problem) for portfolio optimization. <...> these factor models are used to evaluate and not forecasting. I would like to understand why this is the case.

For forecasting, we need a model that has the left hand side variable leading the right hand side variables, something like $$y_{t}=f(x_{t-1})+\varepsilon_{t}$$. In the FF3f model, this is not the case; there, $$y_{t}=f(x_t)+\varepsilon_{t}$$. That is, returns on an asset are modelled as a function of contemporaneous values of some explanatory variables, namely, the three factors. This does not facilitate forecasting but only contemporaneous explanation.

• Thanks for your answers @richard-hardy and Kai. My question arises because oftentimes when papers/books show moment estimation ( see, for example De Nard, Ledoit, Wolf) schemes that use factor models, they cite FF3 as one commonly used. This leads me to think that a model such as FF3 is indeed used to estimate the covariance matrix or vector or expected returns ( which, for the purpose of portfolio optimization, is a forecast, in the sense that is guides your portfolio allocation for the next period) Dec 21, 2023 at 18:09
• I guess these authors have some unstated assumptions or they modify FF3f in some ways to make it work. Perhaps you could post a new question asking to clarify how exactly this is done in one such paper. Dec 21, 2023 at 21:31

I don't think you can use the FF3 model because the FF3 model is used to capture the returns of a portfolio of risk factors (which is why there is the negative and positive alpha mentioned in the reference post in your question). You should use a model for expected returns that forecasts the returns of YOUR portfolio.

I used to apply a historical approach (3 years) to forecast annualized returns for my portfolio. However, as you probably know, a historical approach is not the best one. I recall that there are many books on expected returns online? You can try looking up online and you will find them easily.

Maybe an industrial expert can voice out what he/she uses in their job.

• FF3 is not forecasting anything. It is about a contemporaneous relationships, not lead-lag ones. Dec 21, 2023 at 9:32
• Thanks @RichardHardy, edited my answer for clarity. Dec 21, 2023 at 9:43