# A Bayesian-Stein based expected return estimator by J.P. Morgan

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?

• 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. May 27, 2021 at 18:40
• 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? May 27, 2021 at 18:58
• Yes, that is what I think. We don't know $b_{rp}$ but we estimate it with an OLS like procedure. May 27, 2021 at 19:30