Monkeys with darts

The idea of assigning likelihoods to portfolio manager performance is nothing new. It involves no more that randomly generating portfolio allocations that comply with the mandate terms, and comparing manager performance with these. The problem, of course, is with the sheer scale of the numbers of portfolios that need to be generated to assure representativeness.

There are 1.73 x 10³⁰ unique portfolios containing one or more of a universe of just 100 securities, and that would just be the presence or equal weights case. Stratified sampling techniques will help, but a 100,000 sample size will, almost surely, miss all portfolios containing 28 or fewer securities, and similarly, all portfolios containing 72 or more securities. We might be unconcerned by the absence of the highly concentrated portfolios, the 28 or fewer, given their diversification characteristics, but those with holdings of 72 or more constitute the majority of real world portfolios. These diverse portfolios are just 6.26 x 10^-6 of the total, but they are 8 x 10²⁴in number.Notwithstanding these problems, we can gain insight into the most common method of portfolio asset allocation, benchmark relative security weightings. We show, as with figure 1 (top image) a small selection of benchmark-relative random portfolio weightings, together with their average and the benchmark (market capitalisation) weights. These portfolios are long only and cannot borrow. A benchmark-relative long weight in one security is financed by relative underweight positions in other securities. In many ways, this is the epitome of the traditional DB pension fund mandate.The portfolio allocations under this randomisation are eminently plausible with allocations ranging from (almost) nothing to around 250% of the market capitalisation. Few managed portfolios ever exceed such weighting ranges. The results, though, are surprising. Figure 2 (inset) shows the returns of the individual securities and the benchmark for an arbitrary year, together with 15 random portfolios.The benchmark return is -1.14% and the individual stocks have returns ranging from – 67% to +99%. The random portfolios are all rather tightly clustered around the benchmark, even though they may have active weights as high as 150% of the benchmark weight. There are some that outperform the benchmark, and some that don’t, but this out/underperformance is never substantial when we consider the range of the individual security returns. It appears that benchmark relative asset allocation exploits less than 10% of the available opportunity set, the investible universe. In many regards, it is a very safe bet for the fund manager, but most unlikely to shine. It takes highly undiversified portfolios to achieve high levels of spanning of the opportunity set. Equal weight portfolios of 10 securities achieve just 50% and the whole set is spanned by the extremes of two security portfolios. Given the magnitude of the dispersion of individual security returns, diversification appears very far from a costless benefit when viewed in this light.It is perhaps worth understanding the degree of dispersion of portfolios using equivalent equal weight portfolios. This is the inverse of the GINI coefficient, which is equal to the number of securities when these are equally weighted. The benchmark weights used in the earlier illustrations, though spread across all one hundred securities, are equivalent to a thirty-three security, equal weight portfolio. The randomisations of this benchmark portfolio may be much more concentrated, in some extreme cases equivalent to as few as ten security portfolios. Of course, the effect that portfolio concentration has on performance depends critically upon the distribution of security returns. Portfolio concentration is perhaps best thought of as a form of gearing, capable of producing more extreme outcomes, both positive and negative.It should be understood that identical returns may be produced by radically different portfolio asset allocations. This is illustrated below (in this diagram, negative allocations are simply used for clarity of exposition). In fact, the only points of agreement among these two allocations occur when both exclude a security entirely. The portfolios are radically different from the perspective of dispersion; A is equivalent to a 23 security equal weight portfolio, and B equivalent to 48.4 securities.Obviously, monkeys using the dart selection technology would tend not to produce market capitalisation based portfolios; on average, theirs would be expected to be uniform, or equal weight. It is interesting that in this case, when the equal weight portfolio returns 3.14%, very, very few of the randomised benchmark-relative portfolios outperform it. The use of benchmarks as fund management crutches seems suspect in yet another way.Con Keating is head of research at BrightonRock Group