Estimating Covariance for Practical Markowitz Portfolios
Summary
The document discusses how to estimate the covariance matrix needed to construct a Markowitz efficient frontier. It names maximum-likelihood estimation from historical returns as a standard approach, then describes alternatives: shrinkage of the covariance matrix, shrinkage of its inverse, and dynamic estimates built with GARCH models. The inverse matters because it shapes the efficient frontier.
The central caveat is that historical estimates may fail when the covariance structure changes. A forward-looking alternative described here estimates covariance from current option prices, on the premise that option prices incorporate market participants’ risk expectations. The document also recommends diversification as a way to limit the damage from inaccurate estimates, spurious correlations, or mistaken forecasts. It offers no empirical comparison of the methods and does not specify implementation details, so it presents a menu of approaches rather than evidence that one is consistently best.
Key ideas
- Historical maximum-likelihood estimates are one way to estimate the covariance matrix for a Markowitz portfolio.
- Shrinkage can be applied to the covariance matrix or its inverse, which helps determine the efficient frontier’s shape.
- Dynamic models such as GARCH can account for changing volatility and covariance patterns.
- Structural breaks can make estimates based on past returns unreliable.
- Option prices offer a forward-looking source of information for estimating risk, while diversification can reduce the effect of estimation errors.
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# Modern portfolio theory in practice # Modern portfolio theory in practice I am wondering about the Markowitz theory of portfolio construction in practice. Hence, if one wants to know the efficient frontier, what variances can one use. The only method that I can think is the MLE estimation from past data, but using past data is not effective in my opinion? What do you think? ## Answer by Stefan Voigt (score 2, accepted) https://quant.stackexchange.com/a/18171 In literature you'll find many approaches to compute the variance. As mentioned already, the standard ideas are to use MLE, Shrinkage on the Covariance Matrix (Ledoit, Wolf), Shrinkage on the inverse of the Covariance Matrix (Kourtis,Dotsis) which makes sense as in fact the inverse of the Covariance Matrix determines the shape of the efficient frontier. Incorporating dynamics into the estimation as for example with the complete battery of GARCH models is another approach. As you mentioned, relaying on past data is not always effective, as the possibility of structural breaks in the Covariance Matrix exist which would not be observable in past data. In order to overcome this problem there are also approach to use forward-looking data in order to estimate the Covariance Matrix. The core idea of this paper (Kempf) is to rely solely on current option prices when estimating the covariance matrix instead of using historical return information. They argue 'since option prices reflect the expectations of market participants about risk, this approach - unlike the backward-looking approaches used so far - is inherently forward looking' ## Answer by Dr.Raghnar (score 1) https://quant.stackexchange.com/a/18174 This is why Markowitz says that the diversification of the portfolio is always preferable. You have a lot of certain past data and some fallible speculations to evaluate the variance and expected return of a title. Inherently the best possible evaluation method, and there are several main ones, is not a foolproof inference. But, if you diversify your portfolio, you minimize the impact of bad assessment, including spurious correlations and of wrong speculation.
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