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I suppose there are roughly two approaches to predict portfolio returns.

Either predict the returns of all underlying stocks and aggregate all individual stock predictions, or predict the portfolio returns directly.

What would be better? Or e.g. what would the advantages/disadvantages be of both approaches?

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Most models of return are based on "characteristics" or "exposures" of the stocks. These are different things but they share the property that they are linear across a portfolio. You find the characteristic (or exposure) of the portfolio by forming the weighted average of the characteristics (exposures) of the stocks in the portfolio.

As a result, the 2 approaches you suggest give the same result: For example in CAPM there is only one exposure, known as Beta. You can either compute the expected return of each stock and then form their weighted average, or you can first form the beta of the portfolio and apply the CAPM to the portfolio beta. The results are identical. The same is true for FF3, FF5 and other linear models of return.

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A portfolio is often a collection of securities. A portfolio represents the weighted sum of each security's return. I believe when you say "aggregate all the prediction" is a synonym of "predict the portfolio return." To my knowledge, calculating portfolio returns are always driven by the underlying security exposure to risk.

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It rather depends on how you mean separate estimation versus joint estimation. For example, if by separate you mean that you would run a separate regression for each security, one at a time, versus a vector regression, then you will always be less accurate with separate estimations unless the variables are intrinsically independent.

However, nothing prevents you from creating a vector regression. It would give you both the predictions for the portfolio and the parts. If you are using a coherent method then the sum of the parts will equal the value of the whole. If you are not, then that is not automatically true.

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