Estimating Asset-Class Returns from Portfolio Returns and Weights

Summary

The document considers whether unknown asset-class returns can be recovered by regressing observed portfolio returns on known allocation weights. It notes a potential identification problem because the weights in each portfolio sum to one, creating a relationship among the regressors. The answer clarifies that a single portfolio return cannot identify multiple underlying asset-class returns: it supplies one equation for several unknowns.

With returns and allocations from multiple managers or portfolios, the asset-class returns can be estimated if there are at least as many independent observations as asset classes. When every portfolio’s weights sum to one, the model can be specified without an intercept or one asset class can be omitted to avoid the redundant regressor. The response gives a simple two-asset, one-portfolio example and describes the two-manager, two-asset case as a solvable linear system. It does not discuss noisy observations, changing weights, or statistical uncertainty, so practical estimation still depends on sufficiently varied allocations and suitable data.

Key ideas

• A single portfolio return cannot determine multiple asset-class returns because it provides fewer equations than unknowns.
• Multiple portfolios with known weights and returns can provide the observations needed for estimation.
• When portfolio weights sum to one, omit the intercept or drop one asset class to remove redundancy.
• The number of observations must be at least the number of asset classes, and allocation variation matters for identification.

Tags

# Linear regression on portfolio return (to estimate asset class/factor returns)


# Linear regression on portfolio return (to estimate asset class/factor returns)












I know the portfolio return and the share in each asset class, but don't have the return on those asset classes. My idea is to estimate the return on the different asset classes by a linear regression of the form

$$ r_{\text{ptf}} = \beta_1 s_1 + \dots + \beta_n s_n + \epsilon $$

where $s_i$ are the known shares of asset classes $\{1,\dots,n\}$ and $\beta_i$ are the coefficients to be estimated which should represent in the end the asset class return.

I feel like that I will have problems with multicollinearity as my shares of asset classes will sum up to $1$. Any way to solve this problem? Note that some portfolios might be only invested in some of the asset classes and zero in others.

## Answer by Enrico Schumann (score 0)

https://quant.stackexchange.com/a/81869

Suppose someone told you that they earned 20% on their portfolio, and had 60% in equities and 40% in bonds. For a single portfolio, there is simply no way to extract the actual returns of the asset classes from the sum (i.e. portfolio return). Mathematically, you have one equation and two unknowns. So there is an infinity of solutions.

If you knew the portfolio returns and allocations of more than one manager -- at least as many managers as asset classes --, you could estimate them via regression. If your weights sum to 1, then you can either leave out the constant or drop one asset class.

With two managers, you would solve a linear system (two equations, two unknowns).

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.