Random Portfolios in Finance This page is divided into the following sections: The Basics Evaluating Trading Strategies Performance Measurement Investment Mandates Additional Uses of Random Portfolios Discussion Products References The Basics We introduce random portfolios via the evaluation of fund manager performance. If we could compare an investment manager's portfolio to all of the portfolios that might have reasonably been chosen, then we would have the best possible measure of how well the manager did over the time period in question. In practice the number of potential portfolios will be astronomical. However, a random sample of those portfolios yields virtually the same answer, and is quite feasible with modern computers. The sophisticated term for this type of analysis is Monte Carlo simulation. This is essentially the familiar stock market dartboard game (sometimes monkeys are throwing the darts), but with a couple of important differences. The darts are not appropriately constrained. In particular, a dartboard portfolio is likely to have substantially higher volatility than a more reasoned portfolio -- thus giving the dartboard an unfair advantage when only returns are compared. The other difference is that only one dartboard portfolio is created -- a better use of random portfolios generates hundreds or thousands of portfolios. There are currently a few implementations of random portfolios that are more or less sophisticated, but they all rely on this simple, powerful idea. This is not new -- an early use was "program selected portfolios" by Dean LeBaron and colleagues at Batterymarch Financial Management in the 1970's. An even earlier use is described in an American Statistical Association speech by James Lorie in 1965 (any speech that starts with Mark Twain and ends in St. Tropez can't be all bad). We now turn to three important uses of random portfolios: testing trading strategies, assessing the skill exhibited by funds, and implementing investment mandates. Evaluating Trading Strategies Fund managers and potential fund managers should know that their investment process creates value. Random portfolios can increase the certainty that the process works. Techniques involving random portfolios can be used on an on-going basis for a live portfolio as well as in a backtest of a potential strategy. One of the biggest problems with developing trading strategies is the problem of data snooping. That is, the strategies will tend to look better than they really are. To see why, suppose that you tried 1000 trading strategies that were completely random. The one that performed best might look reasonably good. Hopefully an investment manager isn't going to be trying completely random strategies, but the selection bias will still exist. The use of random portfolios doesn't eliminate problems with data snooping, but it helps to control them. Details are given in the working paper "Random Portfolios for Evaluating Trading Strategies". Performance Measurement Investors, investment consultants and funds of funds all have the need to decide whether or not funds are exhibiting skill. There are three major approaches to this problem: benchmarks, peer groups and random portfolios. A fund is judged against a benchmark by comparing a series of returns from the fund with the corresponding returns for the benchmark. This method has a few problems -- a major one is the time it takes to decide that a good fund really is better than the benchmark. Since data from several time periods is needed for a single test, it takes several years if not a few decades. The power of these tests in the ideal setting is given in Burns (2007a). Another problem with benchmarks is that the difficulty of beating them is not constant. If the most heavily weighted assets in the benchmark happen to perform relatively well, then it will be hard to beat the benchmark. Conversely, if the most heavily weighted assets perform relatively poorly, then it will be easy to beat the benchmark. This phenomenon is often visible in tables of the percentage of funds beating their benchmark. Kothari and Warner (2001) discuss this. The second approach to performance measurement is to use peer groups. Here we find a number of funds that are similar to the fund in question and examine its performance relative to the peer funds. It may not be clear which funds should be considered peers. We would like a lot of peers in order to have more power for the test, but we also want to include only funds that are very similar. A fund has no real peers at all if the collection of its constraints is unique -- as is true of many hedge funds. We are meant to believe that if our fund of interest is better than 90% of its peers, then our fund's skill is roughly at the 90th percentile among its peers. This assumes that differences in skill dominate differences in luck. Such an assumption is unlikely to be justified. In particular if it is the case that no fund has skill (or all the funds have equal skill), then our fund is at the 90th percentile of luck. Burns (2007a) expands on this argument. Surz (2006, 2009) discusses additional problems with peer groups. To use random portfolios for performance measurement, generate a large number of random portfolios that satisfy the constraints on the fund. The performance of the fund is compared to the performance of the random portfolios. For example, we state that the assets come from a given universe, there are 50 to 60 assets held at any one time, the maximum weight of an asset is 5%, and the volatility is no more than x%. A set of random portfolios are generated that obey these constraints. These provide the information necessary for a statistical test that the fund exhibited skill over the time period in question. The random portfolio technique is both more powerful and more fair than using a benchmark. Recall that when testing a fund versus a benchmark we had one difference in returns for each time period; we needed a number of time periods to form a test. With random portfolios we get a test for each time period. It seems paradoxical that we should be able to get more precise measures using randomness, but that is the case. It is more powerful because the random portfolios disclose market behavior. As Burns (2007a) demonstrates, the random portfolio technique can use the holdings of the fund at the beginning of the period to get even more precise performance measurement. Performance measurement with benchmarks and peer groups can be looked at from the perspective of random portfolio performance measurement. Using a benchmark is in a sense measuring the fund with only one random portfolio. It isn't random in the sense of a random sample, but it will be more random than we'd like to think. Key, though, is that there is only one benchmark instead of hundreds or thousands of random portfolios. Likewise, using a peer group is like using a small set of random portfolios. The catch, beside having a small number, is that we don't know what the skill level is of any of the peers. The measuring stick is blurry and we don't know what the units are. More details are in the working paper "Performance Measurement via Random Portfolios". Burns (2007b) discusses performance measurement in the slightly different setting of testing the recommendations of market commentators. Investment Mandates The purpose of a mandate is to try to get the fund manager to manage the money in the best interests of the investor. In other words, the mandate should encourage the fund manager to maximize the investor's utility. An important detail is that it is the utility of the investor's entire portfolio that should be maximized, not just the portion over which the fund manager has control. (This is discussed in more detail in the working paper "Portfolio Sharpening" .) Traditional mandates give the investment manager a benchmark and a maximum tracking error from the benchmark. This is wasteful in several respects. In virtually all cases the investor can get an index fund for the benchmark which has very low management fees. What's the advantage of hiring an active manager to run a fund that is extremely correlated to the index fund? If the manager doesn't outperform the benchmark by more than the extra management fees, then there is obviously no advantage at all. If the manager does have the skill to consistently beat the benchmark, then that skill could be put to much better use. A skilled fund manager should, in general, be able to achieve higher returns when the tracking error constraint is dropped. Often it is sensible for the investor to put some money into an index fund (which has very small management fees) and to put other money with an active manager. All else being equal, it is better for the active fund to have a low correlation with the index. This turns out to be the same as a large tracking error. Thus an unconstrained mandate can often be the best choice. There are unreasonable objections to unconstrained mandates that are basically, "We've always done it this way." There is also a seemingly reasonable objection to unconstrained mandates: how does the investor know if the fund manager is performing well? The answer, as we've just seen, is to use random portfolios to evaluate the performance. Random portfolios work equally well for performance measurement no matter what tracking error there is. However, random portfolios can have a more fundamental impact on investment mandates than just to improve performance measurement. Performance fees can be set using random portfolios. This can result in more value for the investor (investors only pay for what they get), and improved income for talented managers. More details are in the working paper "Performance Measurement via Random Portfolios". Additional Uses of Random Portfolios Additional uses of random portfolios that have been suggested include: • Evaluating the effect of constraints. That is, to use random portfolios to decide rationally what the constraint bounds should be. (Some discussion of this is in Portfolio Analysis with Random Portfolios. ) • Portfolio construction process attribution. • Assessing the quality of a risk model. This involves generating a number of believable portfolios and then comparing the risk model's prediction of volatility to the realized volatility for each generated portfolio. • Calculating a bid on a portfolio with unknown composition but known characteristics. Discussion Senior Consultant published some testimonials on PIPODs. While this is specifically about one implementation, most of the comments apply to random portfolios in general. Another indication of the usefulness of the idea is the number of times that it has been independently conceived. Given the strength of the idea, it is curious why random portfolios are not in common use -- at least for performance measurement. The technology acceptance model states that the uptake of a technology is mainly determined by its perceived usefulness and its perceived ease-of-use. Perhaps the main problem for random portfolios has been a lack of any perception at all. They have largely been ignored, so a large part of the financial community have not even heard of them. Until recently availability has been a significant issue, that is, ease-of-use was extremely poor. At this point, though, there are enough choices that at least one of the implementations should be suitable for any application. A key characteristic that should help acceptance is that random portfolios are not necessarily a replacement of alternative methods -- they can be used in addition to current methods. Admittedly the use of random portfolios is slightly more complicated than some current methods of performance measurement, but not very much more. Given their dramatic superiority, this should be a small price to pay. Even naively generating random portfolios can be useful. Examples of this include Mikkelsen (2001); Kritzman and Page (2003) and Assoé, L'Her and Plante (2004). Kothari and Warner (2001) show that benchmarking against an index is problematic, and their technique involves random portfolios. There is a tutorial on statistical bootstrapping that briefly discusses the relationship between random portfolios and resampling procedures. Products The following products were created independently of each other, and only POP is associated with Burns Statistics. POP Portfolio Construction Suite from Burns Statistics. This has a wide range of constraints, including the very important one of limiting the volatility of the portfolios. PODs and PIPODs from PPCA Inc. SimIAn from Hewitt Bacon & Woodrow References Assoé, Kodjovi, Jean-François L'Her and Jean-François Plante (2004). "Is There Really a Hierarchy in Investment Choice?" http://www.hec.ca/cref/pdf/c-04-15e.pdf Bridgeland, Sally (2001). "Process attribution -- a new way to measure skill in portfolio construction" Journal of Asset Management. Burns, Patrick (2003). Does my beta look big in this? (pdf) Burns, Patrick (2004). Performance measurement via random portfolios (pdf) Burns, Patrick (2006). Random portfolios for evaluating trading strategies (pdf) Burns, Patrick (2006). Portfolio analysis with random portfolios (pdf of annotated presentation slides) Burns, P. (2006). "Random Portfolios for Performance Measurement" in Optimisation, Econometric and Financial Analysis E. Kontoghiorghes and C. Gatu, editors. Springer. Burns, P. (2007a). "Bullseye" Professional Investor March issue. A very similar version is available as Dart to the Heart Burns, P. (2007b). Cramer vs. Pseudo-Cramer Daniel, G., D. Sornette and P. Wohrmann (2008). "Look-Ahead Benchmark Bias in Portfolio Performance Evaluation" working paper at SSRN Dawson, Richard and Richard Young (2003). "Nearly-uniformly distributed, stochastically generated portfolios" in Advances in Portfolio Construction and Implementation edited by Stephen Satchell and Alan Scowcroft. Butterworth-Heinemann. Elton, E. J., M. J. Gruber, S. J. Brown and W. N. Goetzmann (2003). Modern Portfolio Theory and Investment Analysis, Sixth Edition (Chapter 24, Evaluation of Portfolio Performance). Kothari, S. P. and Jerold Warner (2001). "Evaluating Mutual Fund Performance" Journal of Finance working paper at SSRN Kritzman, Mark and Sébastien Page (2003). "The Hierarchy of Investment Choice" Journal of Portfolio Management 29, number 4, pages 11-23. Lisi, Francesco (2009). "Dicing with the Market: Randomized Procedures for Evaluation of Mutual Funds". University of Padova working paper. http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1375730 Mikkelsen, Hans (2001). "The Relation Between Expected Return and Beta: A Random Resampling Approach" SSRN papers Surz, Ron (1994). "Portfolio Opportunity Distributions: An Innovation in Performance Evaluation" Journal of Investing. Surz, Ron (1996). "Portfolio Opportunity Distributions: A Solution to the Problems with Benchmarks and Peer Groups" Journal of Performance Measurement. Surz, Ron (1997). "Global Performance Evaluation and Equity Style: Introducing Portfolio Opportunity Distributions" in Handbook of Equity Style Management. Frank Fabozzi Associates. Surz, Ron (2004). "'Hedge Funds Have Alpha' is A Hypothesis Worth Testing" Albourne Village library. Surz, Ron (2005). "Testing the Hypothesis 'Hedge Fund Performance is Good'" Journal of Wealth Management. Spring issue. Surz, Ron (2006). "A Fresh Look at Investment Performance Evaluation: Unifying Best Practice to Improve Timeliness and Reliability" Journal of Portfolio Management Summer issue. Surz, Ron (2007). "Accurate Benchmarking is Gone But Not Forgotten: The Imperative Need to Get Back to Basics" Journal of Performance Measurement. Vol. 11, No. 3, Spring, pp 34-43. Surz, Ron (2009). "A Handicap of the Investment Performance Horserace" Published as "Handicap in the Investment Performance Horserace" in Advisor Perspectives 2009 April 28. Go to Burns Statistics Home. Direct access to this page is http://www.burns-stat.com/pages/Finance/random_portfolios.html First Version: 2004 December 30 Last Modified: 2009 June 04