Risk-based Portfolios
Shrinking the sample covariance matrix
ctrucios@unicamp.br
Instituto de Matemática, Estatística e Computação Científica (IMECC),
Universidade Estadual de Campinas (UNICAMP).
Motivation
Motivation
Motivation
Which is the best estimator of the populational mean in a multivariate setting (\(N > 3\))?
- Stein (1956) and James and Stein (1961) answer this question with an unecpected result: It is not the sample mean!
- In dimensions \(N > 3\), a better estimator than the sample mean can be obtained by shrinking the sample mean to a target vector
- This results was so unexpected and revolucionary that it took a while to be digested and embraced by the academic community.
Motivation
- Years later, a similar idea was applied (mainly by Olivier Ledoit and Michael Wolf) to the covariance matrix in large dimensions, giving rise to the shrinkage estimators for the covariance matrix.
- Ledoit and Wolf have devoted almost 20 years of their careers to the development of shrinkage estimators, which have been successfully applied in several fields:
- Chemistry
- Electromagnetics
- Genetics
- Neuroscience
- Psychology
- Image and Speech recognition
- Economics and Finance
Shrinkage Estimators
Shrinkage Estimators
Any Shrinkage estimator for the large-dimensional covariance matrix has three ingredients:
- An estimator with no structure (\(S\))
- An estimator with a lot of structure (\(F\))
- A Shrinkage constant (\(\delta\)) (also known as Shrinkage intensity)
S, F and \(\delta\)
- The estimatior with no structure: 🏄♀️ the sample covariance matrix.
- The Shrinkage constant: 💻 data-driven.
- The estimator with a lot of structure: 🤷
- Should involve only a small number of free parameters (i.e, a lot of structure)
- Should reflect important characteristics of the unknown quantity to be estimated.
- Many alternatives
Shrinkage Estimators
Linear Shrinkage Estimators
- Easy to understand
- Easy to proof
- Easy to implement
Non-Linear Shrinkage Estimators
- Hard to understand
- Hard to proof
- Hard to implement
- More flexivel
Notation
- \(\Sigma\): The true (an unknown) covariance matrix
- \(S\): The sample covariance matrix
- \(\mathbb{I}\): The identify matrix
- \(F\): A target matrix (with a lot os structure)
- \(\hat{\Sigma}\): The shrinkage estimator of the covariance matrix
- \(T\): Sample size
- \(N\): Number os variables (assets)
Linear Shrinkage Estimators
Linear Shrinkage Estimators
A linear shrinkage estimator is given by:
\[\hat{\Sigma} = (1 - \delta)S + \delta F\]
| \(F\) | Reference |
|---|---|
| \(\sigma^2 \times \mathbb{I}\) | A well-conditioned estimator for large-dimensional covariance matrices (Ledoit and Wolf 2004a) |
| Single factor of Sharpe (1963) (CAPM) | Improved estimation of the covariance matrix of stock returns with an application to portfolio selection (Ledoit and Wolf 2003) |
| \(f_{ij} = \sqrt{\sigma_{ii}\sigma_{jj}}\rho\) | Honey, I shrunk the sample covariance matrix (Ledoit and Wolf 2004b) |
| \(f_{ij} = \eta\) and \(f_{ii} = \sigma^2\) | Essays on risk and return in the stock market (Ledoit 1995) |
- \(\sigma^2\): common variance across assets
- \(\rho\): common correlation across assets
- \(\eta\): common covariance across assets
Linear Shrinkage Estimators
Proof
Queremos minimizar \[\mathbb{E} || \hat{\Sigma} - \Sigma ||^2,\] em que \(\hat{\Sigma} = \delta \nu I + (1 - \delta)S\).
\[\mathbb{E} || \hat{\Sigma} - \Sigma ||^2\]
\[= \mathbb{E} || \delta \nu I + (1 - \delta)S - \Sigma ||^2\]
\[= \mathbb{E} || \delta \times (\nu I - \Sigma) + (1 - \delta) \times(S - \Sigma) ||^2\]
Linear Shrinkage Estimators
Recuerde
\(|| \alpha A + \beta B ||^2 = \alpha^2 ||A||^2 + \beta^2 ||B||^2 + 2\alpha \beta <A, B>\)
\[= \mathbb{E} || \delta \times (\nu I - \Sigma) + (1 - \delta) \times(S - \Sigma) ||^2\]
\[= \mathbb{E} \Big[ \delta^2 ||\nu I - \Sigma||^2 + (1 - \delta)^2 || S - \Sigma ||^2 + 2 \delta (1 - \delta) <\nu I - \Sigma, S - \Sigma >) \Big]\]
\[= \delta^2 ||\nu I - \Sigma||^2 + (1 - \delta)^2 \mathbb{E} || S - \Sigma ||^2 + \underbrace{2 \delta (1 - \delta) <\nu I - \Sigma, \underbrace{\mathbb{E} (S - \Sigma)}_{0} >)}_{0} \Big]\]
Linear Shrinkage Estimators
\[= \delta^2 ||\nu I - \Sigma||^2 + (1 - \delta)^2 \mathbb{E} || S - \Sigma ||^2\]
Observação
Encontrar o \(\nu\) óptimo, não depende de \(\delta\). Assim, minimizando \(||\nu I - \Sigma||^2 =\nu^2 ||I||^2 + ||\Sigma||^2 - 2\nu <I, \Sigma>.\):
Derivando w.r.t \(\nu\) e igualando a zero:
\[2 \nu \underbrace{||I||^2}_{1} = 2 <I, \Sigma>\]
\[\nu = <I, \Sigma> = \mu\]
Subtituyendo \(\nu\) por su valor optimo:
Linear Shrinkage Estimators
\[= \delta^2 ||\mu I - \Sigma||^2 + (1 - \delta)^2 \mathbb{E} || S - \Sigma ||^2\]
Lema
Sejan \(\mu = <I, \Sigma>\), \(\alpha^2 = ||\Sigma - \nu I||^2\), \(\beta^2 = \mathbb{E} ||S - \Sigma||^2\) e \(\rho^2 = \mathbb{E} || S - \mu I ||^2\). Então:
\[\alpha^2 + \beta^2 = \rho^2.\]
\[= \delta^2 \alpha^2 + (1 - \delta)^2 \beta^2\]
Linear Shrinkage Estimators
\[= \delta^2 \alpha^2 + (1 - \delta)^2 \beta^2\]
Derivando w.r.t \(\delta\)
\[2 \delta \alpha^2 - 2 (1 - \delta) \beta^2.\]
Igualando a zero:
\(\delta \alpha^2 = \beta^2 - \delta \beta^2 \quad \rightarrow \quad \delta = \dfrac{\beta^2}{\alpha^2 + \beta^2} = \dfrac{\beta^2}{\rho^2}\)
Note
\[\delta = \dfrac{\beta^2}{\alpha^2 + \beta^2} = \dfrac{\mathbb{E} ||S - \Sigma||^2}{\mathbb{E} || S - \mu I ||^2} \quad e \quad 1-\delta = \dfrac{\alpha^2}{\alpha^2 + \beta^2} = \dfrac{||\Sigma - \nu I||^2}{\mathbb{E} || S - \mu I ||^2}\]
Linear Shrinkage Estimators
- Optimal \(\delta\) is an oracle!
Can be proved that
- \(m^2 - \mu^2 \rightarrow_{L2} 0\)
- \(d^2 - \rho^2 \rightarrow_{L2} 0\)
- \(b^2 - \beta^2 \rightarrow_{L2} 0\)
- \(a^2 - \alpha^2 \rightarrow_{L2} 0\)
- \(\hat{S} = \dfrac{b^2}{d^2} m I + \dfrac{a^2}{d^2}S\)
Em que \(m = <S, I>\), \(d^2 = ||S - mI||^2\), \(\bar{b}^2 = \dfrac{1}{n^2} \sum_{k = 1}^n ||x_k x_k' - S||^2\), \(b^2 = min(\bar{b}^2, d^2)\) and \(a^2 = d^2 - b^2\).
Non-Linear Shrinkage Estimators
Non-Linear Shrinkage Estimators
- The different formulations of \(F\) are essentially improvements of the generic linear shrinkage.
- Such improvements are obtained by incorporating prior information or structural assumptions about the target covariance matrix.
- Is it possible to improve the generic linear shrinkage with no previous knowledge about the target covariance matrix?
Non-Linear Shrinkage Estimators
First Idea
- The number of shrinkage intensities increases on the order of \(N^2\) 😿
- Allowing different shrinkage intensities no longer guarantees that the final estimator is positive semi-definite 😢
- So… that way doesn’t really looks like the right way, does it?
Non-Linear Shrinkage Estimators
Let \(\lambda_1, \cdots, \lambda_N\) be the eiganvalues of \(S\) and let
\[(1 - \delta) S + \delta F = \hat{\Sigma} = U \Lambda^{\ast} U\] be the spectral decomposition of \(\hat{\Sigma}\).
Can be proved that the elements \(\lambda_1^{\ast}, \cdots, \lambda_N^{\ast}\) of the diagonal matrix \(\Lambda^{\ast}\) are equal to \[\lambda_i^{\ast} = \delta \sigma^2 + (1 - \delta) \lambda_i\]
- This means that \(\hat{\Sigma}\) has the same eigenvectors than \(S\), but different eigenvalues.
- Now, a genralization of the generic linear shrinkage, seems to be more obvious: Let’s use different shrinkage intensities for the different sample eigenvalues!
Non-Linear Shrinkage Estimators
- Nowadays, there are several alternatives to the sample covariance matrix.
- A great source for codes is https://github.com/MikeWolf007/covShrinkage
- Some R packages are also available:
- nlshrink
- cvCovEst
- ShrinkCovMat
- Recent methodological contributions include the works of Ledoit and Wolf (2020), Ledoit and Wolf (2022), De Nard (2022), among others.
- In a recent paper (Trucı́os 2025), I conduct a large-scale empirical comparison using real-world data in the context of portfolio allocation.
References
Carlos Trucíos (IMECC/UNICAMP) | FCM-UNMSM 2025 | Risk-Based Portfolio Allocation | ctruciosm.github.io