While most investors focus on volatility, the covariance matrix can significantly affect performance and is hard to measure. A large covariance matrix with N assets will have N(N+1)/2 free parameters and T degrees of freedom based on the number of observations. A large portfolio will be hard to calculate and difficult to use out of sample.
Work has also been done on shrinkage estimates, which suggests that the raw sample covariance should contribute only about 20% to the new covariance matrix. Linear shrinkage does better than nonlinear out-of-sample estimates. Overall, shrinking toward zero correlation will help long-short portfolios because they rely less on extreme values.
The problem is that, in optimization, weights depend on risk aversion, the inverse of the covariance matrix (the precision matrix), and expected returns. Estimation errors are amplified during inversion. A 10% mistake in the covariance matrix will cause the precision matrix to take much larger values, which can be catastrophic.
Additionally, illiquidity can distort covariance in hidden ways. Illiquidity creates positive autocorrelation in the return series, so measured volatility is lower than true volatility and measured covariance is lower than true covariance.
These issues are one reason funds should focus on covariance across the environment and try to adjust for relative volatility and risk contribution. It is not just the overall volatility that is an issue but the link across markets which defines diversification and the risk hidden within portfolios.
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