Stock Misplacement in Sized-Based Indices

////Stock Misplacement in Sized-Based Indices

Stock Misplacement in Sized-Based Indices

By | 2017-12-21T12:30:28+00:00 December 22nd, 2017|Guest Posts|2 Comments

There has been much discussion of the small-cap premium recently. Has its popularity cannibalized its utility? Are large firms preventing a level playing field? Will the small-cap premium exist going forward? This article does not address these concerns directly, but it does identify and analyze another potential issue that could diminish the utility of small cap exposures. In particular, I investigate how stock mispricing and misplacement could impact the performance of size-based indices.

I first provide some relevant background information and then I define and discuss the issue of security misplacement. In particular, I highlight potential implications security misplacement has for the performance of portfolios and indices involving small cap (SC) exposures. I then present a simulation model to capture this misplacement phenomenon and quantify its impact.

Figure 1: Security Misplacement

Source: Aaron Brask Capital

The Core Issue: Mispricing Causing Misplacement

I appreciate the behavioral side of finance and I’ve pitched a tent in the mispricing camp. I believe there is a certain level of chaos inherent in our markets. Indeed, there are many types of investors with different needs, emotions, models, etc. The inevitable flows from their competing perspectives create occasional, if not constant, mispricing in many securities. While it might be tempting to believe hedge funds and active market participants instantly jump on these opportunities and keep prices in line, in practice, there are limits constraining their ability to do so (constraints on short-selling and a popular J.M. Keynes quote come to mind).(1)

This is not to say one should not question why opportunities exist. Indeed, I have seen many hedge fund strategies blow up because there were seemingly obvious profit opportunities that turned out to be picking up pennies in front of a steamroller trades (see here for a piece for an in-depth discussion). I simply do not believe opportunities should be ignored if we have some reason to believe they are attractive and do not require more risk or transactional costs (even if we cannot specifically identify the root cause beyond the reasonable assumption that some opportunities are afforded by the general chaos inherent in our capital markets).

Just to be clear, my goal here is not to debate mispricing (it is just background for what follows). There is already much research on this topic and I believe the empirical evidence (e.g., from Alpha Architect and AQR) supports the notion that mispricing exists.

Figure 2: Mispricing versus Efficient Markets Illustration

Source: Aaron Brask Capital

So this leads me to the question, “So what if mispricing exists?” Even if we cannot necessarily pinpoint where it is (e.g., identifying specific stocks using bottom-up valuation analysis), understanding potential systematic implications of mispricing may allow investors to protect themselves from negative impacts.

For example, let us consider Research Affiliates and their concept of fundamental indexation.

Their basic premise is that a stock portfolio based on market capitalization weighting (MCW) effectively over-weights overpriced stocks and under-weights underpriced stocks. Their solution is to use fundamental weights (i.e., quantities from financial statements instead of market prices). This approach severs the link between the market’s opinion (which could be wrong) on price from the stock weights. Their research indicate a significant improvement in historical performance using their fundamental weight approach.

Note: Cliff Asness of AQR may have mocked the innovation claimed by fundamental indexation advocates (see these humorous slides), but I believe there is something special. Even if fundamental indexation is actually just one type of value investing (I would call it value-light), I believe it fulfills a need on behalf of investors who want market-proportioned portfolios without the mispricing biases. However, if relatively minor value-based adjustments are good, then larger value-based adjustments might be even better (e.g., the Quantitative Value approach for concentrated value exposure).

This article highlights another potential negative implication related to stock mispricing. In particular, some stocks will be inadvertently misplaced in different size-based indices or portfolios. For example, consider the companies in a large-cap (LC) index. It is possible some of them are overpriced to the extent they could be placed in the corresponding mid-cap (MC) index if they were fairly valued. Thus, this could bias the LC index with richly valued firms and impose a drag on performance.

Going to the other end of the spectrum, the opposite could be true for small-cap (SC) indices. They might naturally catch some of the underpriced MC stocks and benefit from higher returns (at least over the long-term).

This raises an interesting question: Is it possible mispricing is partially responsible for the SC premium?

Also, given that many value strategies rely on mispricing, to what extent do SC and value strategies overlap? These questions are the motivation for the simulation model and results I present in the next two sections.

Note: Pure value (i.e., shopping in the valuation trash can) might represent the logical extreme of the mispricing phenomenon highlighted above and has historically provided investors with significantly higher returns than standard market portfolios.

Simulation Framework

In order to highlight and quantify the impact of security mispricing on size-based exposures, I constructed a Monte-Carlo simulation with the following assumptions:

  • Universe of 300 stocks
  • Breakpoints for small, mid, and large at 100, 200, and 300, respectively
  • Sizes of stocks at inception approximated from S&P 500 distribution of market capitalization
  • Market prices follow fundamental performance (10% growth, but irrelevant)(2)
  • Reversion to mean (zero) for mispricing expected over three years
  • Mispricing lognormally distributed with 20% standard deviation (but I also scale it inversely with size)

I have not specified the actual fundamentals or valuation metrics underlying the model. That would be a slippery slope as one could argue what levels of valuations are appropriate for firms with different growth rates, risks, etc. I have only modeled the existence of mispricing. Of course, the fundamentals of different firms grow at different rates. In fact, one might argue SC firms should grow at higher rates than LC. However, this does not take away from the overall results as my assumption is that mispricing exists regardless of how one accounts for these other details.

Most of the assumptions above are generic but there are two parameters I have chosen: the level of volatility for mispricing (20%) and the period over which it is expected to revert (three years). There is nothing magical about these figures; I simply thought they were reasonable estimates. Moreover, the magnitude of turnover they generated in the simulations was on the same order of magnitude as historical levels of turnover realized in existing size-based indices.(3) This observation leads me to believe these assumptions are plausible. However, even if my volatility mispricing estimate of 20% was off, this would only impact the magnitude of the results – not the direction.

There are some interesting dynamics here. For example, more (less) mispricing increases (decreases) the amount of security misplacement and thus impacts performance more (less). This is fairly obvious in the LC (SC) portfolios where the impact is asymmetric and systematically negative (positive). However, it is also true in the MC portfolio. The (positive) impact of mispricing among larger stocks in this portfolio (where MC portfolio captures underpriced stocks from LC while migrating some overpriced stocks to LC) is naturally larger in magnitude than the negative impact of mispricing at the smaller end of the portfolio since the larger securities have larger weights. So the net impact of mispricing and misplacement is positive. Consequently, the net benefit is amplified (attenuated) with more (less) mispricing.

The model above assumes uniformity of mispricing across all sizes. In theory, there should be greater mispricing in SC stocks (e.g., less analyst coverage). This would balance out the impact asymmetry of mispricing on the larger versus smaller stocks – such as in the MC portfolio as discussed above. So I also ran simulations where mispricing was inversely scaled with size. Spoiler: Depending on how I scale mispricing, the net impact on MC portfolio performance could be positive or negative.

It should be noted this framework does not include micro-caps which could impose the analogous drag around the lower breakpoint for SC portfolios (i.e., I did not allow for companies to fall out of or migrate up into the SC portfolio at the smaller end). However, we can still observe the net impact on the MC portfolio and draw some conclusions about the mispricing dynamics in the context of size-based portfolios. In particular, the impact for the LC portfolio will always be asymmetric and systematically impose a negative performance bias from mispricing and misplacement (regardless of whether mispricing is scaled or not).

Note: I acknowledge this is not a perfect model. However, I believe it suffices in order to highlight and quantify (not necessarily precisely but at least reasonably) the potential impact of mispricing on size-based indices. As George Box (mathematician/statistician) is quoted saying, “All models are wrong but some are useful.”

Simulation Results

The simulation results are presented in the table below. It shows the results for the same lognormal distribution of mispricing (20% volatility) across all stocks – SC, MC, and LC (i.e., no scaling) and three scenarios for scaling mispricing inversely with size (i.e., mispricing is greater for smaller stocks).

For reference, please note that returns would all be the same (assumed to be 10% here) in a perfectly efficient market simulation (i.e., no mispricing) by construction. Moreover, the total market portfolio in this simulation (all 300 stocks) averages 10% returns as well because the universe is static (i.e., no misplacement from outside the universe – such as micro-caps).

Figure 3: Impact of misplacement with and without scaling mispricing

  Large-cap Mid-cap Small-cap Equal-weight
No scaling 9.8% 10.8% 12.5% 11.3%
With scaling (10-30%) 9.8% 9.9% 13.5% 11.4%
With scaling (10-40%) 9.8% 9.4% 15.0% 12.1%
With scaling (20-40%) 9.6% 10.6% 16.6% 13.1%

Source: Aaron Brask Capital

As expected (by construction), LC performance fell short of the 10% market average in all of the simulations. As discussed above, this is due to it absorbing some overpriced stocks from the MC portfolio and losing underpriced stocks to the MC portfolio. On the other hand, the equal-weight (EW) portfolio outperformed in all instances and its outperformance increased as the overall level of mispricing increased (from the top of the table to bottom).

Note: At first, I thought the outperformance of the equal-weight over LC portfolios were too low. Indeed, Research Affiliates found that MCW imposed a performance drag of approximately 2%. However, I then realized I cannot compare the results directly to fundamental indexation because the simulation here only captures mispricing around the breakpoints – not across the entire portfolio.

It is also worth noting the impact of scaling on the MC portfolio. Depending on the scaling scenario, MC performance was either better or worse than the overall market. In cases where the relative scaling varied the most (from 10-30% or 3x and from 10-40% or 4x), the negative impact stemming from the lower breakpoint (absorbing overpriced from and losing underpriced to the SC portfolio) outweighed the benefit at the upper breakpoint with the LC portfolio. So the MC portfolio underperformed LC with the higher breadth of mispricing scaling but outperformed when scaling was less extreme.

The simulation was set up such that the SC portfolio did not suffer from these negatives issues associated with a lower breakpoint (i.e., assumed no micro-caps). So the significantly higher returns experienced by the SC portfolios should be taken with a grain of salt. In reality, they would have a lower breakpoint and it would impose a drag on performance. However, this does bring up another interesting consideration.

Another Interesting Result

Given our assumption that mispricing is lognormally distributed (which I think is reasonable), there is another factor that will determine the amount of performance impact from misplacement. In particular, the distribution of market capitalizations for the firms in the universe will be a significant determinant for how much misplacement occurs around breakpoints. Indeed, if the sizes of the firms around breakpoints are tightly clustered, then mispricing will be more likely to create more misplacement. If the sizes are more spread out, then the firms will be less likely to leapfrog across breakpoints due to mispricing.

This prompted me to investigate the distribution of (free-float adjusted) market capitalizations. The results I came up with are very interesting. I took the size of each company and divided it by the average size of the next 10 largest companies (then subtracted one). I then plotted the results across all companies in the iShares Total Market ETF. Interestingly, the shape was a smile. This indicates the largest and smallest firms were spread out (higher ratios) whereas many of the companies in the middle were closer together (on a percentage basis).

Figure 4: Dispersion of firm sizes across total market portfolio (free-float adjusted)

Source: Aaron Brask Capital

Curiosity got the best of me and I conducted the same analysis on the companies within the small, mid, and large iShares ETFs. This is where it gets more interesting. Each of these portfolios exhibited a similar smile-shaped output – again indicating the smallest and largest companies within each portfolio were more spread out.

But wait!

The distribution within the total market portfolio was not spread out except at the ends. So how could this be? If I were to tape these three distributions (for SC, NC, and LC) together, then I would have expected it to resemble the total market distribution as one big smile. However, that was not the case. It got me to thinking and I came up with two possible answers.

  • First, it is possible these mini-smiles are just the result of market volatility. Once an index rebalances, then the prices, and hence market capitalizations, will move around. This could cause more size disparity around the breakpoints.
  • The second idea that came to mind involves index rules. In particular, many index providers employ buffers around index breakpoints in order to mitigate unnecessary turnover. This could also keep some stocks from migrating across breakpoints and would also result in more size disparity near the breakpoints.

Figure 5: Dispersion of firm sizes across small, mid, and large portfolios (free-float adjusted)

Source: Aaron Brask Capital


My results were generated by a Monte-Carlo simulation and do not precisely capture all the dynamics (or real history) of our stock markets. However, I believe this is a fair market model for the purposes of illustrating the existence of potential mispricing impacts. Truth be told, the magnitude of my results were not as strong as I suspected before I conducted the analysis. In light of the minimal impact on the LC portfolio (20-40bps) and the variable impact on MC portfolios (teetering between positive and negative depending upon the scaling scenario), I am not convinced that mispricing and misplacement make up any significant portion of the SC premium. At the same time, I am not closing the door and am pleased to have uncovered the interesting smile phenomenon around index breakpoints.

  • The views and opinions expressed herein are those of the author and do not necessarily reflect the views of Alpha Architect, its affiliates or its employees. Our full disclosures are available here. Definitions of common statistics used in our analysis are available here (towards the bottom).
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References   [ + ]

1. See the limits of arbitrage piece here.
2. Uniform growth of the fundamentals does not impact the results since it has the same effect on SC, MC, and LC.
3. Turnover in my simulations was approximately 20-50% of the historical turnover in iShares sized-based ETFs.

About the Author:

Aaron Brask

Aaron is an accomplished Mathematical Finance PhD who runs his own registered investment advisor (RIA) – Aaron Brask Capital LLC. Before starting his own firm, Aaron’s last role was at Barclays Capital where he built and ran two global research teams focused on quantitative equity and derivatives strategies for the firm’s top clients. He has been published and quoted in major newspapers and magazines including the Wall Street Journal, Financial Times, Barron’s, and Risk magazine.

Aaron Brask Capital is an independent, fee-only registered investment advisor. That means I do not promote any particular products and cannot receive commissions from third parties. In addition to holding me to a fiduciary standard, this structure further removes monetary conflicts of interests and aligns my interests with those of my clients.

In terms of spirit, my firm embodies my own ethics, discipline, and expertise. In particular, my analytical background and experience working with some of the most affluent families around the globe have been critical in helping me formulate investment strategies that deliver performance and comfort to my clients. I continually strive to demonstrate my loyalty and value to my clients so they know their financial affairs are being handled with the care and expertise they deserve.


  1. J N December 26, 2017 at 9:45 am

    Thanks Aaron for another interesting post. I’m (confirmation-biasedly) pleased to see your simulation results agree with my prior, as I think the very idea that small cap portfolio contains underpriced mid-cap stocks and large cap portfolio contains overpriced mid-cap stocks is logically flawed. Think about this way – if we further divide the small cap bucket into more buckets, say SC1 to SC5 (largest size bucket within small cap), shouldn’t SC5 not only contain underpriced mid-cap stocks but also overpriced stocks from SC4? If we believe that’s true, then the “mispricing” in each size bucket should cancel out, as for each bucket, theoretically (a.k.a infinite bucketing) there should always be a size bucket larger where the underpriced stocks are absorbed from and a size bucket smaller where the overpriced stocks are absorbed from.

  2. Aaron Brask
    Aaron Brask December 26, 2017 at 8:51 pm

    Thanks for your comment. I would not necessarily call it a flaw in the logic, but an issue to investigate further. I did indeed try to address and comment on this specific point several times. In fact, the issue of balancing the positive versus negative impact (which does not necessarily net out to zero) for mid-caps (or any cap grouping other than LC) was the motivation for the section on ‘size dispersion’ where I discovered the ‘smiles’ in the data.

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