Factor-Targeted Asset Allocation: A Reverse Optimization Approach

Abstract

We demonstrate that using a mean-variance portfolio to obtain implied factor risk premia can result in stable weights for a factor portfolio when assets’ expected returns follow a factor structure that is subject to pricing errors. We propose a methodology to construct asset portfolios based on these factor portfolio weights, taking into account the possibility of pricing errors. Our simulation shows that these “factor-targeted” portfolios have higher and more stable Sharpe ratios than traditional allocation methodologies in various scenarios involving expected return assumptions. Furthermore, while our factor-targeted portfolios exhibit similar Sharpe ratios to the mean-variance portfolio built using factors for high levels of pricing errors, the factor-targeted portfolios have more stable portfolio weights, which makes them more appealing in practice.

Keywords:

• factor investing
• factors
• portfolio allocation
• reverse optimization

PL Credits: 2.0:

Acknowledgements

The authors wish to thank Redouane Elkamhi (University of Toronto) for his valuable insights and vivid discussions on this research topic. The authors also wish to thank Michael Wissell for his support and encouragement on continuous innovation and research, which makes this article possible, and Jaylie Lee for her helpful comments.

Disclaimer

This article is for informational purposes only and should not be construed as legal, tax, investment, financial, or other advice. The views and opinions expressed here are those of the authors alone and do not necessary reflect the views of their employers and their affiliates.

Disclosure

No potential conflict of interest was reported by the author(s).

Notes

1 While the “true” underlying factors driving asset returns are unobservable to econometricians, recent studies by both investment managers and academics have agreed upon that macro factors (e.g., economic growth, real rate, and inflation) are important drivers of asset returns. Recent examples of articles that use similar sets of factors as the ones we employ include Bass et al. (2017), Bender et al. (2018), and Gladstone et al. (2021). Such macro factors can be replicated using portfolios of traded securities such as equities, real return bonds, commodities, break-even inflation, and credit.

2 In the special case where the assets’ expected returns are priced by a set of factors without errors, the implied factor returns of the mean-variance tangency portfolio are equal to the true factor premia.

3 Since the implied factor returns are consistent with estimates from a cross-sectional regression of assets’ expected returns on factor loadings, so are the factor returns’ errors.

4 It is well known that for a given set of factor weights, there might be many asset portfolios that have those exact factor exposures (for a discussion, see Greenberg et al. 2016).

5 However, our methodology is flexible and investors can use their preferred allocation rule for the target asset weights. They can also target their portfolio to different desired factor weights determined through other means.

6 For completeness, we note there is evidence that, in the absence of pricing errors, factor-based asset allocation is not superior to asset-class based asset allocation (e.g., Idzorek and Kowara 2013).

7 We also compared our RFT portfolios to other traditional portfolios like minimum volatility, equal risk contribution (Maillard et al. 2010), and equal weights portfolios that do not use expected returns as input. At higher levels of pricing errors, our RFT portfolios perform similarly to these portfolios. However, at lower levels of pricing errors, our method takes advantage of information embedded within assets expected returns and builds portfolios that outperform these traditional portfolios.

8 Elkamhi et al. (2021) provides a short discussion on the intuition of using traditional portfolios as the target asset weights.

9 Analysis on the volatility of asset weights is valuable for investors because it directly affects turnover and transaction costs, which are known to be important determinants when applying an allocation rule in practice.

10 The use of factor mimicking portfolios is common in practice (e.g., Bender et al. 2018; Greenberg et al. 2016) as they are tradable.

12 For example, see the capital market assumption surveys from Horizon Actuarial Services:

https://www.horizonactuarial.com/blog/2020-survey-of-capital-market-assumptions

13 The difficulties of estimating expected returns—which ultimately affect asset allocation—are also discussed in Black (1993).

14 It is worth noting that in Table 4, the reverse optimized factor weights for the traditional portfolios can be quite arbitrary as they are affected only by ${\Sigma }_a$ and the assets that are included in the optimization (we explore this point later in the results and discussions section).

15 This result is by design as it can be derived from Equation (14)(14) $$\mathrm{ }{w}_{ft,x}={\left[\gamma \mathrm{ }{B_i}^{\prime }{B}_i\mathrm{ }+\mathrm{ }\left(1-\gamma \right)I_{N\times N}\right]}^{-1}\mathrm{ }\left(\gamma \mathrm{ }{B_i}^{\prime }{w}_{f,tp}^{\mathrm{*}}+(1-\gamma )\mathrm{ }{\overline{w}}_x\right),\mathrm{ }\gamma \mathrm{ }=\mathrm{ }0.999$$(14) .

16 We thank for the editor and two anonymous referees for this suggestion.

17 Note for example for a pricing error of 2%, the portfolio that is long RFT MinVol and short the MVO on factors exhibits a positive average information ratio but a probability of the IR being greater than zero is 44%. In this case, this means that the distribution of information ratios across the 5,000 simulation is skewed.

18 The second sub portfolio does not have any reverse-optimized factor weights because by definition $B^{\prime }{\epsilon }^{*}= 0.$

Additional information

Notes on contributors

Jacky S. H. Lee

Jacky S.H. Lee is Senior Managing Director, Total Portfolio at Healthcare of Ontario Pension Plan Trust Fund in Toronto, Ontario, Canada.

Marco Salerno

Marco Salerno is Principal, Total Portfolio at Healthcare of Ontario Pension Plan Trust Fund in Toronto, Ontario, Canada.