Estimating Returns and Covariances for Portfolio Construction
Summary
The document discusses how portfolio managers obtain expected returns and the variance-covariance inputs used in portfolio theory. It describes historical performance as a common basis for estimating variances and covariances, while expected returns may be informed by models such as CAPM or by investment research and valuation work. It also cautions that ARIMA-style time-series forecasts are generally viewed by the contributors as weak tools for long-horizon return prediction, though this is presented as practical opinion rather than a demonstrated result.
Portfolio construction combines return views with risk estimates and chosen objectives, such as targeting return or allocating across asset classes and industries. The answers suggest methods and further reading, including Black-Litterman, momentum approaches, and quantitative portfolio-management references. They note that forecasting returns is difficult, while finding combinations with lower variance may be more tractable. The discussion is introductory and offers no specific estimation procedure, dataset, or comparative evidence; its claims about forecasting limits and relative difficulty should not be treated as universal conclusions.
Key ideas
- Historical asset-class performance can inform variance and covariance estimates.
- Expected returns may come from asset-pricing models, research, or valuation forecasts.
- The contributors express skepticism about ARIMA for long-term return prediction, without proving that it cannot work.
- Portfolio construction combines return views and risk estimates under chosen objectives.
- Black-Litterman and momentum-based approaches are named as directions for further study.
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Full text
# Basic question on Portfolio Theory
# Basic question on Portfolio Theory
I was revising my stuff about portfolio theory and I noticed that every single time, expected return and corresponding variance or covariance are given! (not calculating ourselves). So I'm just wondering if there is a way to figure out those values statistically. I think, forecasting using autoregressive model or ARIMA is plausible, however, it seems to be very difficult and not useful when we decide to choose a portfolio for a long term investment.
It will be very thankful if somebody recommends me related textbook or leaves a brief answer.
Thank you
## Answer by Sergey Bushmanov (score 4, accepted)
https://quant.stackexchange.com/a/21177
- Theoretically speaking (as it's done in financial textbooks at b-school level), variance and covariance are calculated on historical performance of asset classes, forward looking returns are CAPM calculated returns.
- ARIMA. Practically speaking, ARIMA is useless for predicting long term returns (or portfolio management if you wish). Why? A short answer is that all info from historical prices already incorporated in the current price of an asset (EMH). I haven't seen a theoretical proof of this statement, but this seems to be true (same for using ARIMA for predicting short term).
- Practical Portfolio management. You need to have a "correct" prediction of future asset returns ("market", "below market", "above market" or "sell", "hold", "buy" or sometimes "conviction buy" as GS puts it, based on your research and models. DCF eg). Then you try to combine those assets into portfolios according to some targets: absolute return, asset classes, industries etc, optimizing risk/return based on your forecasts and history.
Competing on predicting returns is hard: some compare this business to throwing darts (both journalists and well respected analysts like Albert Edwards in his "The dangers of DCF"). Putting several assets together to produce a combination with lowest possible variance seems to be a bit easier. Cliff Azness' AQR Management is in this business.
You may read more about modern [quantitative] portfolio management in these books:
- Pfaff, Financial Risk Modelling and Portfolio Optimization with R. Examples with R code.
- Antti Ilmanen, Expected Returns: An Investor's Guide to Harvesting Market Rewards. Book shows how history is incorporated into portfolio management practically.
- Richard R. Lindsey (editor), How I Became a Quant: Insights from 25 of Wall Street's Elite. This book is not a how-to on portfolio management. Rather, it's narrative of successful quants, some of them being portfolio managers. I bought this book because I was interested in what Cliff Azness doing at his firm.
Hope this helps in answering OP's questions...
## Answer by Richi Wa (score 5)
https://quant.stackexchange.com/a/21171
Of course estimating expected returns is the very core of portfolio management. Finding a useful covariance matrix too. To find both fills a book. So I first thought about closing the question. But it is a chance to discuss today's approaches.
A nice approach that is very up-to-date where mementum investing seems very fashionable is the following: Momentum and Markowitz: A Golden Combination
Also apply some Black-Litterman procedure you learn a lot about this topic.
ARMA-like approaches for financial markets does not seem to be a promising approach (see e.g. "What can be forecast" in the book by Athanasopoulos and Hyndman.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.