Benchmarking mean-variance portfolios using a shortage function: the choice of direction vector affects rankings!

Abstract

In addition to its use in data envelopment analysis models, the shortage function has been proposed as a tool to gauge performance in multi-moment portfolio models. An open issue is how the choice of direction vector affects the efficiency measurement, especially when some of the data can be negative and, from a practical point of view, whether and how the resulting league tables are affected. This paper illustrates empirically how the choice of direction vector affects the relative ranking of mean-variance portfolios. This result is relevant to all frontier-based applications, especially those where some of the data can be naturally negative.

Keywords:

• shortage function
• efficient frontier
• mean-variance efficiency

Acknowledgements

We thank a referee for the most helpful comments. The usual disclaimer applies.

Notes

1 Similar functions define duality relations in consumption theory (see Luenberger, 1995).

2 In principle, the same issues could be illustrated using geometric reconstructions of MVS efficient portfolio sets (see Kerstens et al, 2011a), but this would come at a substantial computational cost and the visual interpretation would be less straightforward.

3 Although recently quite some progress has been made in visualising properties of DEA frontiers: see, eg, Førsund et al (2009).

4 In finance, norms are used in a variety of contexts (see, eg, the definition of coherent risk measures in Jarrow and Purnanandam, 2005), but—to the best of our knowledge—never to appraise portfolio performance.

5 This set of admissible portfolios can be modified to include additional constraints that can be written as linear functions of asset weights (eg, transaction costs): see Briec et al (2004). Briec and Kerstens (2010) also consider the cases of a risk-free asset and shorting.

6 To obtain a positive relation between efficiency and the shortage function, one could consider taking the negative of the current definition. However, we prefer to stay in line with current practice in the literature.

7 Briec (1997) introduces a position-dependent direction vector in the efficiency literature. The article by, for example, Chambers et al (1998) opts for a fixed direction vector. Also Blackorby and Donaldson (1980) choose a fixed direction vector with unit coordinates when developing absolute inequality measures.

8 Briec et al (2004) demonstrate that, due to dual relations between shortage function and mean-variance utility function, the shadow prices associated with the shortage function can yield information about investors risk aversion.

9 Recall that the Euclidean length of a vector v=(v_M, v_V) is given by Note that instead of Euclidean length, other choices of norms could equally well be considered.

10 Normally, the arrows in the case of a fixed direction vector and a unit length fixed direction vector would just differ in length. This would hardly be noticeable on a separate figure. To save space, we therefore refer to the same figure.

11 As a matter of fact, as long as a portfolio model contains an even moment (variance, kurtosis, …, ie, all observed values in this dimension being strictly positive), this proportional interpretation can be maintained.

12 Both the initial database and the proposed selection are available upon simple request from the authors.

13 This figure exploits the fact that the shortage function is continuous in the portfolio weight vector (x) whenever the direction vector does not contain any zero component (see Briec and Kerstens, 2010: Proposition 2.4).

14 A bit similar to Briec and Lesourd (1999) who prove that some Hölder normed distance function is dual to the profit function.