Robeco logo

Important information

The Robeco Capital Growth Funds have not been registered under the United States Investment Company Act of 1940, as amended, nor or the United States Securities Act of 1933, as amended. None of the shares may be offered or sold, directly or indirectly in the United States or to any U.S. Person (within the meaning of Regulation S promulgated under the Securities Act of 1933, as amended (the “Securities Act”)). Furthermore, Robeco Institutional Asset Management B.V. (Robeco) does not provide investment advisory services, or hold itself out as providing investment advisory services, in the United States or to any U.S. Person (within the meaning of Regulation S promulgated under the Securities Act).

This website is intended for use only by non-U.S. Persons outside of the United States (within the meaning of Regulation S promulgated under the Securities Act who are professional investors, or professional fiduciaries representing such non-U.S. Person investors. By clicking “I Agree” on our website disclaimer and accessing the information on this website, including any subdomain thereof, you are certifying and agreeing to the following: (i) you have read, understood and agree to this disclaimer, (ii) you have informed yourself of any applicable legal restrictions and represent that by accessing the information contained on this website, you are not in violation of, and will not be causing Robeco or any of its affiliated entities or issuers to violate, any applicable laws and, as a result, you are legally authorized to access such information on behalf of yourself and any underlying investment advisory client, (iii) you understand and acknowledge that certain information presented herein relates to securities that have not been registered under the Securities Act, and may be offered or sold only outside the United States and only to, or for the account or benefit of, non-U.S. Persons (within the meaning of Regulation S under the Securities Act), (iv) you are, or are a discretionary investment adviser representing, a non-U.S. Person (within the meaning of Regulation S under the Securities Act) located outside of the United States and (v) you are, or are a discretionary investment adviser representing, a professional non-retail investor. Access to this website has been limited so that it shall not constitute directed selling efforts (as defined in Regulation S under the Securities Act) in the United States and so that it shall not be deemed to constitute Robeco holding itself out generally to the public in the U.S. as an investment adviser. Nothing contained herein constitutes an offer to sell securities or solicitation of an offer to purchase any securities in any jurisdiction. We reserve the right to deny access to any visitor, including, but not limited to, those visitors with IP addresses residing in the United States.

This website has been carefully prepared by Robeco. The information contained in this publication is based upon sources of information believed to be reliable. Robeco is not answerable for the accuracy or completeness of the facts, opinions, expectations and results referred to therein. Whilst every care has been taken in the preparation of this website, we do not accept any responsibility for damage of any kind resulting from incorrect or incomplete information. This website is subject to change without notice. The value of the investments may fluctuate. Past performance is no guarantee of future results. If the currency in which the past performance is displayed differs from the currency of the country in which you reside, then you should be aware that due to exchange rate fluctuations the performance shown may increase or decrease if converted into your local currency. For investment professional use only. Not for use by the general public.

I Disagree

03-19-2024 · Research

Covariance rhapsody: A reality check for evaluating risk models

Portfolio management is all about trading off expected return and risk. The key ingredient to measuring and managing portfolio risk is the variance-covariance (VCV1) matrix which needs to be estimated for the given investment universe.

    Authors

  • Maarten Jansen - Researcher

    Maarten Jansen

    Researcher

  • Harald Lohre - Head of Quant Equity Research

    Harald Lohre

    Head of Quant Equity Research

At its core, the VCV informs about assets’ riskiness and their inter-dependencies, as measured by their variance and covariance, respectively. The natural candidate to use is the sample covariance matrix; however, this estimator is prone to error and not suitable when the number of assets under consideration is large, as is often the case when optimizing equity portfolios. To this end the academic literature proposes myriad alternative VCV estimators to address these limitations. But how can we best evaluate the practical relevance of a given VCV estimator?

In a recent research paper, quant researchers Clint Howard, Maarten Jansen, Harald Lohre, and M. Sipke Dom set out to answer this question, putting a wide range of alternative VCV estimators to the practical test. Importantly, they challenge the common academic practice of evaluating the relevance of novel VCV estimators using the unconstrained global minimum variance (GMV) portfolio. Indeed, when validating VCV estimators based on the ex-post volatility of this portfolio, the researchers confirm the academic backing for considering shrinkage and covariance dynamics in modelling the VCV for equity portfolio construction.

This is evident in the leftmost bar of the below figure that highlights a wide range of volatility outcomes for unconstrained GMV portfolios that differ only in the choice of underlying VCV estimator. Yet, these portfolios are often impractical due to their high leverage, concentration, turnover, and transaction costs. The researchers therefore investigated how the opportunity set for volatility improvement changes when making the GMV test portfolios more investable. Although long-only constrained GMV portfolios still allow for meaningful volatility improvements, their overly concentrated stock allocation calls for further constraining portfolio weights.

Figure 1: Volatility outcomes for test GMV portfolios

Figure 1: Volatility outcomes for test GMV portfolios

Source: Robeco 2024

Resorting to truly investible GMV portfolios, the researchers reveal a considerably reduced opportunity set for alternative VCV estimators. Similar findings hold for alternative risk-based portfolio construction approaches, such as risk parity portfolios that aim to maximize portfolio diversification.

These findings highlight the discrepancies between the optimal VCV matrix estimator across different portfolios, suggesting that what works best for an unconstrained GMV portfolio may not hold under more realistic conditions with significant investment constraints. Such realistic test portfolios suggest that the overall room for improvement from a given VCV estimator is limited. Nevertheless, statistically significant improvements can still be made under the right circumstances.

Read the full paper on SSRN.

Footnote

1From Investopedia: ‘Variance refers to the spread of a data set around its mean value, while a covariance refers to the measure of the directional relationship between two random variables… Variance is used by financial experts to measure an asset's volatility, while covariance describes two different investments' returns over a period of time when compared to different variables.’

Get the latest insights

Subscribe to our newsletter for investment updates and expert analysis.

Don’t miss out
Robeco

Robeco aims to enable its clients to achieve their financial and sustainability goals by providing superior investment returns and solutions.

Important information
The Robeco Capital Growth Funds have not been registered under the United States Investment Company Act of 1940, as amended, nor or the United States Securities Act of 1933, as amended. None of the shares may be offered or sold, directly or indirectly in the United States or to any U.S. Person (within the meaning of Regulation S promulgated under the Securities Act of 1933, as amended (the “Securities Act”)). Furthermore, Robeco Institutional Asset Management B.V. (Robeco) does not provide investment advisory services, or hold itself out as providing investment advisory services, in the United States or to any U.S. Person (within the meaning of Regulation S promulgated under the Securities Act).
This website is intended for use only by non-U.S. Persons outside of the United States (within the meaning of Regulation S promulgated under the Securities Act who are professional investors, or professional fiduciaries representing such non-U.S. Person investors. By clicking “I Agree” on our website disclaimer and accessing the information on this website, including any subdomain thereof, you are certifying and agreeing to the following: (i) you have read, understood and agree to this disclaimer, (ii) you have informed yourself of any applicable legal restrictions and represent that by accessing the information contained on this website, you are not in violation of, and will not be causing Robeco or any of its affiliated entities or issuers to violate, any applicable laws and, as a result, you are legally authorized to access such information on behalf of yourself and any underlying investment advisory client, (iii) you understand and acknowledge that certain information presented herein relates to securities that have not been registered under the Securities Act, and may be offered or sold only outside the United States and only to, or for the account or benefit of, non-U.S. Persons (within the meaning of Regulation S under the Securities Act), (iv) you are, or are a discretionary investment adviser representing, a non-U.S. Person (within the meaning of Regulation S under the Securities Act) located outside of the United States and (v) you are, or are a discretionary investment adviser representing, a professional non-retail investor.


Access to this website has been limited so that it shall not constitute directed selling efforts (as defined in Regulation S under the Securities Act) in the United States and so that it shall not be deemed to constitute Robeco holding itself out generally to the public in the U.S. as an investment adviser. Nothing contained herein constitutes an offer to sell securities or solicitation of an offer to purchase any securities in any jurisdiction. We reserve the right to deny access to any visitor, including, but not limited to, those visitors with IP addresses residing in the United States. This website has been carefully prepared by Robeco. The information contained in this publication is based upon sources of information believed to be reliable. Robeco is not answerable for the accuracy or completeness of the facts, opinions, expectations and results referred to therein. Whilst every care has been taken in the preparation of this website, we do not accept any responsibility for damage of any kind resulting from incorrect or incomplete information. This website is subject to change without notice. The value of the investments may fluctuate. Past performance is no guarantee of future results. If the currency in which the past performance is displayed differs from the currency of the country in which you reside, then you should be aware that due to exchange rate fluctuations the performance shown may increase or decrease if converted into your local currency. For investment professional use only. Not for use by the general public.