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Please consider the following estimator for the expected returns specified in the paper "Improving on risk parity: Hedging forecast uncertainty" by Peter Rappoport, J.P. Morgan, October 2012.

$$m_{FH}=(1-\omega)\cdot m +\omega \cdot(b_{RP}s)$$

Where $m$ is the $N\times1$ vector of average returns $s$ the $N\times1$ vector of standard deviations, $\omega$ the shrinkage coefficient computed as for the Bayes-Stein estimator.

Specifically, $b_{RP}s$ is the prior towards the average returns are shrunk in relation to $\omega$.

The article specifies that $b_{RP}$ is computed as follow.

$$b_{RP}=\frac{s^\mathsf{T}\bullet\Sigma^{-1}\bullet m}{s^\mathsf{T}\bullet\Sigma^{-1}\bullet s}$$

Can anyone explain me what $b_{RP}$ represents and why is multiplied element-wise by $s$? The author specified that $b_{RP}$ is the generalised linear regression coefficient of $m$ on $s$ but he keeps highlighting in the paper that such estimator shrunk the "the plugin portfolio towards the risk parity portfolio). I am confused about it, is $b_{RP}$ the expected return of the risk-parity portfolio (to be thorough the author assumes that correlation of securities is constant therefore securities weights of the risk-parity portfolio are proportional to $1/s$, whatever proportional means)?

Moreover by considering the original Bayes-Stein estimator developed by Jorion (1986) the prior is the expected return of the global minimum variance portfolio which is a single value not a vector of values. Am I right saying that $m_{FH}$ uses multiple priors since $b_{RP}s$ is a scaled vector of volatilities?

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    $\begingroup$ One simple idea to forecast returns (or excess returns) is that they should be proportional to the asset's standard deviation. And $b_{rp}$ can be seen as the constant of proportionality in this relationship. The reward for taking an additional unit of risk. $\endgroup$
    – nbbo2
    May 27, 2021 at 18:40
  • $\begingroup$ So within such framework, the idea behind multiplying $b_{rp}$ (that is the linear regression coefficient) and the standard deviations $s$ (that are one of the OLS model variable) is obtaining the OLS estimates of the expected returns and then use them as priors, am I right? Am I missing something about the risk-parity portfolio under the assumption of constant correlation? $\endgroup$
    – Nipper
    May 27, 2021 at 18:58
  • $\begingroup$ Yes, that is what I think. We don't know $b_{rp}$ but we estimate it with an OLS like procedure. $\endgroup$
    – nbbo2
    May 27, 2021 at 19:30

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