Combining standard and behavioral portfolio theories: a practical and intuitive approach

Abstract

A fully implementable portfolio model combining mental accounting and Black-Litterman to accommodate views on expected returns with multiple attitudes to risk

Notes

No potential conflict of interest was reported by the authors.

This work was started when the first author was at NEOMA Business School.

1 Also known as risk-aversion coefficient within mean–variance portfolio theory.

2 In BL, an arbitrary Sharpe Ratio (usually set to 0.5) is assumed to compute the market risk-aversion coefficient that is required to obtain the equilibrium vector of excess returns through reverse optimization.

3 Consequently, the utility function in BL is not conceived in absolute terms of total wealth as in mean-variance portfolio theory. Rather, it is conceived in terms of gains and losses relative to the benchmark portfolio, as is assumed in behavioral finance.

4 Besides eliminating the need to specify abstract confidence intervals around the views in units of variance, this method also save investors from having to set the original model’s controversial ‘weight-on-views’ parameter.

5 The authors propose a behavioral BL model in continuous-time, based on diffusion processes, that corrects for the effect that cognitive biases and psychological pitfalls may have on analysts and their views through a stress-test approach.

6 Investors can only specify a single level of risk-aversion, which is done in units of variance. Since variance carries little intuition, investors are prone to significant misspecification of their risk-averseness.

7 MA optimal portfolios always lie on the mean-variance efficient frontier.

8 In MA, normality is convenient to address the VaR-type constraints and provides closed-form or quasi closed-form solutions. Although this assumption is not strictly necessary in the optimization because only the first two moments of the distribution of returns are used, removing it will require some care as it will have an impact on the overall framework, from the inclusion of views via a non-normal extension of Black–Litterman to an optimization problem that may require to account for higher moments, and to the final asset allocation.

9 For reasons of simplicity, we treat losses as positive returns when running this algorithm.

10 An optimal is accomplished when the realized value exceeds the estimated quantile with an average frequency of (). This is how we know if the algorithm is producing accurate forecasts for . We do not expect to always achieve an optimal , but we expect it to be consistently close to .

11 The BL model uses excess returns to avoid the question of what the risk-free rate is, and whether it is actually risk-free. In addition, the excess return absorbs the stochasticity of the short rate.

12 The fund separation implies that the risky component of all subportfolios will have identical structures.

13 Das et al. (2010) show that imposing a short-selling constraint at the subportfolio level leads to only a minor loss of mean-variance efficiency. However, if the investor selects a well-balanced distribution of wealth among her mental accounts, typically the aggregated portfolio will not imply short positions. So, it might only be necessary to impose a short-selling constraint at the aggregate portfolio level, for which they provide a method for implementation.

14 We thank an anonymous referee for suggesting this point.

15 The omitted calculations, results, and MATLAB codes used are available upon request.

16 Throughout this implementation, we use a historical covariance matrix computed with the last monthly excess returns of the securities (from January 2004 to December 2013).

17 Consider the same order of securities from table 1.

Additional information

Funding

This work was supported by Région Champagne-Ardennes and the European Union through the RiskPerform grant.