Sample Covariance Matrix
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A Sample Covariance Matrix is a covariance matrix based on a population sample.
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- Counter-Example(s):
- See: Constant-Correlation Covariance Matrix, Random Matrix, Covariance Matrix Estimation, Sample Covariance, Sample Mean.
References
2015
- (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/Sample_mean_and_sample_covariance Retrieved:2015-2-16.
- The sample mean or empirical mean and the sample covariance are statistics computed from a collection of data on one or more random variables. The sample mean is a vector each of whose elements is the sample mean of one of the random variablesthat is, each of whose elements is the arithmetic average of the observed values of one of the variables. The sample covariance matrix is a square matrix whose i, j element is the sample covariance (an estimate of the population covariance) between the sets of observed values of two of the variables and whose i, i element is the sample variance of the observed values of one of the variables. If only one variable has had values observed, then the sample mean is a single number (the arithmetic average of the observed values of that variable) and the sample covariance matrix is also simply a single value (the sample variance of the observed values of that variable).
2004
- (Ledoit & Wolf, 2004) ⇒ Olivier Ledoit, and Michael Wolf. (2004). “Honey, I Shrunk the Sample Covariance Matrix.” In: Journal of Portfolio Management, 31(1).
- NOTE: This is a paper for practitioners showing how to shrink optimally the sample covariance matrix towards the constant-correlation covariance matrix. The resulting estimator improves the out-of-sample performance of portfolio managers
- ABSTRACT: The central message of this paper is that nobody should be using the sample covariance matrix for the purpose of portfolio optimization. It contains estimation error of the kind most likely to perturb a mean-variance optimizer. In its place, we suggest using the matrix obtained from the sample covariance matrix through a transformation called shrinkage. This tends to pull the most extreme coefficients towards more central values, thereby systematically reducing estimation error where it matters most. Statistically, the challenge is to know the optimal shrinkage intensity, and we give the formula for that. Without changing any other step in the portfolio optimization process, we show on actual stock market data that shrinkage reduces tracking error relative to a benchmark index, and substantially increases the realized information ratio of the active portfolio manager.
- QUOTE: ... The crux of the method is that those estimated coefficients in the sample covariance matrix that are extremely high tend to contain a lot of positive error and therefore need to be pulled downwards to compensate for that. Similarly, we compensate for the negative error that tends to be embedded inside extremely low estimated coefficients by pulling them upwards."