A moment-free nonparametric quantity-of-quality approach to optimal portfolio selection: A role for endogenous shortfall and windfall boundaries?

Abstract

This article proposes a Quantity-of-Quality (QoQ) approach to optimal portfolio selection, which builds on the intuition of the widely applied h-index and e-index from the bibliometric literature. While moment-free and nonparametric, the method embraces quantity-of-high-quality returns and upside potential while simultaneously avoiding quantity-of-low-quality returns and downside risk. A non-standard measure of central tendency is also present, which functions in a way similar to a portfolio mean or median. The method delivers attractive and intuitively appealing results, and appears to be less susceptible to overfitting issues than the stylized Sharpe Ratio portfolio. The method is demonstrated with an established data set, and out-of-sample performance is gauged using training-holdout analysis in two distinct data sets. Because the proposed method uses a fundamentally different portfolio selection objective function than standard moment-based methods, the QoQ approach extracts information about the data-generating process that is perhaps overlooked or deemphasized by traditional moment-based methods, and as such may serve as a capable complement to standard moment-based portfolio selection criteria.

KEYWORDS:

• Nonparametric estimation
• moment-free decision models
• behavioural finance
• h-index
• e-index

Notes

Acknowledgement

The author is grateful to two anonymous referees that provided unusually careful comments and suggestions; the paper has surely benefitted from their earnestness.

Disclosure statement

No potential conflict of interest was reported by the author.

Notes

1 See also Haley and McGee (2006), Haley and Walker (2010), and Haley (2008, 2012).

2 Dozens of refinements and generalizations of the h-index now exist (e.g., Bornmann et al., 2011), among them the g-index (Egghe, 2006) and the e-index (Zhang, 2009). Numerous empirical studies of the h-index have also appeared (e.g., Cronin & Meho, 2006; Haley, 2013; Oppenheim, 2007), as well as studies regarding its variability across citation databases (Henzinger et al., 2009; Haley, 2014a).

3 This “history” is often truncated to a specific set of recent years; the exact interval varies by application.

4 The h-index is also commonly computed for individual scholars. In the scholar-level case, the ordered pairs are likewise (article rank, article citation count), but the universe of articles would be those published by a specific scholar across a variety of journals.

5 The “h-core” refers to the articles included in a scholar’s h-index. In the example depicted in Table 1, the h-index is 6, meaning that the 6 most highly cited articles comprise the h-core for each of the scholars. The 6 articles in the h-core collectively have 6² citations (or h² citations more generally). Typically, the articles included in the h-core have more than h² citations, the citations above and beyond the h² citations are called excess citations, and form the basis for the e-index.

6 Despite this general acceptance, the bibliometrics literature has many important cautions regarding the uninformed application of these methods; see, for example, Martin (2013), Wilhite and Fong (2012), Haley (2017c).

7 The 45-degree line serves an analogous purpose in the h-index and e-index.

8 The method could be framed instead around basis points, though the 45-degree line assumption would need to be adapted accordingly.

9 In cases where the sample size n is 100–200, the 45-degree line works well. In cases where the sample size is much larger or much smaller, the user might consider rescaling the slope by 100/n. This is a simple generalization that is akin to expressing the rank as percentile rank.

10 Instead of QoQ-m, one might consider using the midpoint of QoQ-h and QoQ-hh; see Haley (2018) for an exploration of this possibility in a non-portfolio selection environment.

11 This is a somewhat non-standard use of “excess return”, but it nonetheless captures the intuition of measuring returns in excess of the two points that section-off the two tail areas.

12 The gray shaded areas below the crosshatched areas could also be incorporated into the QoQ objective function, if such an extension was desired.

13 The number of candidate assets is often determined by the type of fund class the investor seeks to invest in; for example, large-cap value stocks. This field of assets may be pruned to include only those stocks that pass certain filters (e.g., based on fundamentals).

14 Shorting is not permitted in this formulation, but extensions that allow shorting might also be developed.

15 This suggests but does not necessarily guarantee that a global solution was found; global optima are notoriously difficult to confirm. The best global algorithms can do is broadly explore the domain and perturb local solutions until the algorithm’s convergence criteria are met within pre-specified tolerances.

16 The value for k used herein correspond to the number of assets in the Haley–Whiteman data set, from which a portfolio – optimal in the QoQ or SR sense – is created. More generally, the value for k will be determined within the asset manager’s context; for example, a large-cap value fund manager may (or may not) pre-filter all large-cap value stocks in some way, the remainder of which would be the “universe” of k asset from which she would build the portfolio. In short, k will be determined by the type and class of assets the fund manager is building the fund around; at times k may be very small (<10) other times much larger (>100). The QoQ mechanism can work for both such cases, and anything in between.

17 The Sharpe Ratio portfolio objective function is defined to be wμ/sqrt(w∑w^T), where w is a 1 × k vector of non-negative portfolio weights that sum to one, μ is the k × 1 vector of asset return means; and ∑ is the k × k covariance matrix of asset returns.

18 The term multiplicity means that one of the stock identification numbers (which were integers 1–500) appeared twice during the random number generation process. The corresponding stock was only used once; thus the goal of randomly selecting 12 stocks was, by luck of the draw, truncated to 11 stocks.