Interpreting a Bayesian-Stein Return Prior Tied to Risk Parity
Summary
The document asks how to interpret a shrinkage estimator that blends sample average returns with a prior proportional to each asset’s volatility. Its central expression is a scalar coefficient multiplied element-wise by the vector of asset standard deviations. The coefficient is calculated using the estimated return vector and covariance matrix, and is described as a generalized regression coefficient of returns on volatilities.
The question is whether this volatility-scaled prior represents an expected return associated with a risk-parity portfolio, whose asset weights under a constant-correlation assumption are proportional to inverse volatility. It contrasts this vector prior with the single-value global minimum-variance prior in Jorion’s Bayes-Stein estimator. The document raises these interpretations but provides no answer or empirical results, so it does not establish that the scaled vector is literally the risk-parity portfolio’s expected return. Understanding it requires the estimator’s normalization and portfolio assumptions.
Key ideas
- The estimator shrinks sample mean returns toward a prior proportional to asset volatilities.
- The prior’s scalar coefficient is computed from returns, volatilities, and the inverse covariance matrix.
- The document asks whether the scaled volatility vector corresponds to a risk-parity portfolio prior.
- It contrasts the vector prior with a single global minimum-variance return prior.
- The question leaves the interpretation unresolved and supplies no empirical validation.
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# A Bayesian-Stein based expected return estimator by J.P. Morgan
# 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?Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.