Estimating Global Investor Views with Reverse Optimization

By Two Sigma Client Solutions Team on September 21, 2020

The Two Sigma Client Solutions team uses reverse optimization to build forward-looking return estimates for major asset classes and compare their forecasted returns to their historical realized returns.

“It is difficult to make predictions, especially about the future.”¹ Yet predictions are an unavoidable necessity for any investor seeking to position their portfolio for 2020 and beyond. Any portfolio weighting decision should account for expectations of risk and return.

This leaves us with two sets of predictions: risk (correlation and volatility) and return. Risk is relatively easy; return is harder. Risk tends to be strongly persistent through time, and long-term risk is predictable using even very simple models.² While we believe that portfolio construction should rely heavily on risk inputs and diversification, the need for return forecasts remains unavoidable.³ So this still leaves us as investors with the considerably difficult task of estimating future returns.

Since any method of portfolio construction requires either explicit or implicit return assumptions, we prefer to make them explicit, so these assumptions can be examined and deemed sensible before driving critical investment decisions. What options do we as investors have at our disposal in determining our long-term return expectations? Differing asset definitions, data sets, and methodological details can result in a vast diversity of individual models and estimates, but we think most boil down to one of the following estimation archetypes:

  1. Estimating long-run return premia from historical data, as done in our whitepaper Forecasting Factor Returns.⁴
  2. Building up forecasts from current valuations, combining estimates for sub-components of longer-term returns such as income and growth expectations.
  3. Harnessing the wisdom of crowds, by surveying or estimating the average forward-looking views of market participants.

This Street View delves into a particular quantitative example of the third method: harnessing the wisdom of crowds. By applying a concept called reverse optimization to the worldwide investable asset portfolio, we show how long-term factor and asset class returns can be estimated from the aggregated allocation decisions of all investors.

We believe this exercise can help investors in a couple of ways. First, the implied return estimates for key risk factors and asset classes derived from global market allocations can be used as an input when formulating your own capital market assumptions, a key input into asset allocation decisions. Second, while we apply this reverse optimization methodology to the global market portfolio, an investor could perform this analysis on their own
portfolio to see what their own implied expectations are for future returns based on their existing asset allocation.

We’ll begin the Street View by setting up the reverse optimization problem and defining one of its key inputs: the global investable market portfolio. In the next section, we will uncover the outputs from the reverse optimization and translate them into a more usable format by establishing a bottoms-up “anchor” return estimate for global equity. Finally, we will estimate the long-term implied returns for several asset classes and see how they compare to realized returns over the past decade.

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  1. An unfortunately unattributable witticism, most likely translated from the proceedings of the Danish Parliament, or Folketing (see
  2. For a more concrete explanation of why we consider risk to be much more predictable than returns, please see our brief digression in Appendix 1.
  3. Even risk-based approaches to portfolio construction such as minimum volatility or risk parity require return assumptions to drive the portfolio weights. Minimum volatility, in particular, is equivalent to mean-variance optimization where one assumes all assets have equal expected return, while risk parity implicitly assumes a more complex relationship where higher risk assets (as measured by both variance and their covariance with other assets) have proportionally higher expected return. Finally, at a minimum, we believe investors should understand the risks in their portfolio and whether there are expected long-term positive return premia associated with those risks. This is a key step because it is not true that all risks come hand in hand with positive excess returns, as mentioned in “Risk Without Return” (Nigro, 2019).
  4. Forecasting Factor Returns” (Duncombe, Nigro, and Kay, 2019)

This article is not an endorsement by Two Sigma of the papers discussed, their viewpoints or the companies discussed. The views expressed above reflect those of the authors and are not necessarily the views of Two Sigma Investments, LP or any of its affiliates (collectively, “Two Sigma”). The information presented above is only for informational and educational purposes and is not an offer to sell or the solicitation of an offer to buy any securities or other instruments. Additionally, the above information is not intended to provide, and should not be relied upon for investment, accounting, legal or tax advice. Two Sigma makes no representations, express or implied, regarding the accuracy or completeness of this information, and the reader accepts all risks in relying on the above information for any purpose whatsoever. Click here for other important disclaimers and disclosures.