# James-Stein estimator for superior estimates of returns in m.v. portfolio optimization

I am currently learning about statistical techniques to enhance the estimation of input parameters in a m.v. optimization. Specifically I have some doubts about the James-Stein estimator applied as an estimation-error correction while modeling the asset returns.

Considering both [Quantitative Portfolio Optimization, Asset allocation and Risk management - Mikkel Rassmussen - 2003] and [Efficient Asset Management a practical guide to Stock Portfolio optimization and asset allocation - O. Michaud - 2008] apparently there are two different formulae for computing the "shrinkage factor" $$\xi$$:

1. $$\xi = Min\bigg[1, \ \frac{N-2}{k \cdot (\vec{r}_{H} - \vec{r}_{G} \cdot \vec{e})^T \cdot \sum^{-1} \cdot \ (\vec{r}_{H} - \vec{r}_{G} \cdot \vec{e})}\bigg]$$ where $$k$$ is the number of observations in the estimation period of all asset classes, $$e$$ is a vector 1s, $$\vec{r}_{H}$$ is the vector of $$N$$ historical means of each asset class, $$\vec{r}_{G}$$ is the "grand mean" (mean of all historical means), $$\sum^{-1}$$ is the inverted covariance matrix.

2. $$\xi = Max\bigg[0, \ 1- \ \frac{(N - 3) \ \cdot \ \vec{\sigma^2}}{(\vec{r}_{H} - \vec{r}_{G})^2}\bigg]$$ where $$\vec{\sigma^2}$$ is the vector of variances of each asset class.

Could someone be so kind to explain me the difference between $$N-2$$ and $$N-3$$ (why is it so and what are the effects on the estimation process)? Moreover am I missing something or the first approach is more effective because it takes into account the covariance matrix rather than simply the variance of each asset class? Finally why multiplying by $$\vec{e}$$ vector of 1s?

• This often happens when you try to do research, you can end up more confused than when you started ;) For what it is worth the first formula agrees with the article by Ph. Jorion: Improved Estimation for Markowitz Portfolios using James-Stein Estimators (1986) JFQA, which is often cited (and is online). I feel comfortable with it. Does Michaud say where he got his formula? (maybe Jobson and Korkie or somewhere else?) Jul 9, 2020 at 17:27
• Thank you for the reply. If I am not mistaken it seems that Michaud refereed to the "positive rule empirical James-Stein estimator" by Efron & Morris (1977, p. 123). Moreover is $\sum^{-1}$ the entire inverted covariance matrix or just its diagonal? (I ask this because $\xi$ should be a vector of shrinkage coefficients rather than a matrix) Jul 9, 2020 at 19:03
• James-stein is an outdated precursor to portfolio shrinkage inferior to Ledoit-wolf so dont see why you'd want to spend any time on it. Shrinkage in turn has been outperformed by denoised or detoned covariance portfolio construction which is very recent, and clustering methods Jul 12, 2020 at 1:26
• @develarist Because that is what I am currently learning. I know about Ledoit-Wolf applied when estimating covariance matrix but this is not topic. I specified Stein estimator to estimates expected returns. Jul 17, 2020 at 18:16