Ledoit–Wolf Covariance Shrinkage on SSE Large Caps

A minimum-variance portfolio is only as good as the covariance that goes into it. On a panel the size of a few dozen Shanghai large caps, a one-year daily sample covariance is invertible and still noisy: the smallest eigenvalues wander, and the optimizer treats that noise as a reason to take extreme weights.

This study asks a narrow question. On the same SSE names, the same month-end calendar, and the same long-only cap, does shrinking the covariance toward its diagonal change the out-of-sample wealth path relative to the raw sample covariance and relative to equal weight?

The sleeve is the repository's Shanghai large-cap list (banks, energy, consumer, and industrials with .SS tickers). The benchmark path is the Shanghai Composite. Prices are Yahoo Finance adjusted closes from 2012, stored as wide CSVs under data/exchange-prices/china/.

Let $X$ be the $T\times N$ matrix of daily simple returns over the trailing year, demeaned by column, and let $S$ be the sample covariance. The shrinkage target $F$ is the diagonal of $S$: variances are kept, every covariance is pulled toward zero. The intensity is the Ledoit–Wolf plug-in that minimises expected Frobenius loss,

$$\hat{\kappa }=\min \left(1,\max \left(0,\frac{\hat{\varphi }}{T\Vert S-F{\Vert }_F^2}\right)\right),\hat{\Sigma }=\hat{\kappa }F+(1-\hat{\kappa })S$$

where $\hat{\varphi }$ is the average squared error of the rank-one outer products $x_t{x}_t^{\top }$ around $S$. A ridge of $10^{-8}$ is added before the solve.

The long-only book is the analytical minimum-variance solution ${\hat{\Sigma }}^{-1}1$, negative weights clipped to zero and renormalised, then capped at 12% with the excess spread over the uncapped names in proportion to their current weight. That is a projection, not a full quadratic program, and it is applied identically to the shrunk matrix and to the sample matrix.

Rebalance is monthly. The gross return is $w_{t-1}^{\top }{r}_t$. Net return subtracts 10 bps times the L1 distance between successive weight vectors. Equal weight is the third book, with the same cost rule, so the comparison is not a cost-free optimiser against a cost-free benchmark.

The chart compounds the three net books and the Shanghai Composite. The test underneath is a Newey–West t-test on the monthly gap between the shrunk book and the sample-covariance book. A positive estimate means shrinkage added return after costs; it does not mean the book beat the index.

The target is the diagonal, which is the right target when residual correlations are the noisy part. A constant-correlation target, or a one-factor target built from the Shanghai Composite, would be a different estimator.

Yahoo Finance adjusted closes are not a point-in-time index membership file. The constituent lists are a liquid large-cap panel, fixed in the repository, not the official index as it stood on each rebalance date.

The risk-free rate is set to zero in every Sharpe ratio. A local bill rate would lower the ratios and would not, by itself, change a long-short spread.

Costs are a flat one-way haircut on weight turnover. They do not include borrow fees, lot-size rules, auction imbalance, stamp duty, or the Shanghai and Hong Kong short-sale bans that bind a live short book.

This is a research record. It is not a recommendation to trade the sleeve.