Which Asset Allocation Weights Work the Best?

Which Asset Allocation Weights Work the Best?

October 27, 2015 Tactical Asset Allocation Research
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Okay, we’re sold on a closet-indexing approach to the markets. Now we’re investigating a variety of smart-beta products available in the market that weigh a large portfolio of stocks with some algorithm. But a natural question arises when trying to pick smart beta ETFs:

What is the optimal method to weigh an index?

Everyone seems to have a story these days for the “best” way to weigh an index. In this study we look at simple ways to weigh a large-cap stock index using prices only. [1]

We enter 4 different weighting schemes in our asset allocation weight horse race:

  • Equal-weight (EW):  Each stock given a weight of 1/250.
  • Momentum weight:  Each stock’s momentum is measured over past 12 months. Momentum score (1+momentum) calculated for top 250 firms.  The momentum weight is given by momentum score divided by sum of all momentum scores.  Higher momentum score = higher weight.
  • Volatility weight: Simple risk-parity technique applied to the top 250 firms, ignoring covariance between firms.  The volatility weights constructed using the past 12-month idiosyncratic volatility of daily returns (regressed against the value weight market return). Higher volatility = lower weight (explained in detail below).
  • Value-weight (VW): Calculated as market cap divided by total market cap of the top 250 largest firms.

Bottom-line up front: Low volatility worked the best on a risk-adjusted basis over the past 87 years. However, low volatility, was closely followed by momentum, equal-weighting, and value-weighting, respectively. Across the board, results are similar.

Let’s dig into the details on the different strategies we test in this study.

Asset Allocation Weight Approaches — List of the Strategies and Details

Universe: largest 250 domestic stocks every month (99.5% correlated w/S&P 500 from 1927-2014).

4 weighting schemes:

  • Equal-weight (EW)
  • Momentum weight
  • Volatility weight (see below)
  • Value-weight (VW)

Rebalance: monthly

Why use a 250 Stock Universe?

We use the top 250 stocks for the following reason:

  • In the 1920s and 1930s, there were fewer stocks (around 500 total). We want to focus our results on larger firms, so we pick the top 250 stocks.
  • Our goal is to identify a weighting scheme that works for a large pool of capital. We want to minimize the impact of poor liquidity and frictional costs, which can be substantial.

In the end, our choice of a 250 stock universe doesn’t matter that much:

  • From 7/1/1927 – 12/31/2014, the value-weight returns to the top 250 stocks each month is 99.5% correlated with the S&P 500 returns (value-weight index). Summary statistics shown below:
portfolio weighting approach_why 250 stock universe
The results are hypothetical results and are NOT an indicator of future results and do NOT represent returns that any investor actually attained. Indexes are unmanaged, do not reflect management or trading fees, and one cannot invest directly in an index. Additional information regarding the construction of these results is available upon request.

Asset Allocation Weighting Scheme Detail — Volatility Weight

To construct volatility weights, we estimate the volatility (σ_i ) of all available stocks (using data up to month t-1) and set the portfolio weight in asset class i to:

Volatility weight construction_1

We estimate σ_(t,i ) as the 12-month idiosyncratic volatility of daily returns (regressed against the value weight market return).  The number k_t is the same for all stocks and controls the amount of leverage of the volatility weighting portfolio.  The unlevered volatility weights (we only study unlevered returns in this report) are obtained by setting:

Volatility weight construction_2

This portfolio over-weights less volatile assets and under-weights more volatile assets.

Example: Let’s say that we have 3 stocks A, B, and C which each have volatilities 5%, 10%, and 20% respectively (this corresponds to σ above).  Then,

vol weight example

Volatility weight construction_3

So stock C, with the highest volatility, gets the lowest weight at 1/7; while stock A with the lowest volatility gets the highest weight at 4/7.

Core Results

  • Results are gross, no fees are included
  • All returns are total returns and include the reinvestment of distributions (e.g., dividends)
  • Portfolios are rebalanced monthly
  • Legend
    • Top 250 EW = Equal-weighting the largest 250 firms
    • Top 250 Vol = Volatility-weighting the largest 250 firms
    • Top 250 Mom = Momentum-weighting the largest 250 firms
    • Top 250 VW= Value-weighting the largest 250 firms
    • SP500 = S&P 500 Total Return Index

Hypothetical performance results have many inherent limitations, some of which, but not all, are described in the disclosures at the end of this document. No representation is being made that any fund or account will or is likely to achieve profits or losses similar to those shown herein. In fact, there are frequently sharp differences between hypothetical performance results and the actual results subsequently realized by any particular trading program. Indexes are unmanaged, do not reflect management or trading fees, and one cannot invest directly in an index.

Monthly Rebalance — Full Sample: 7/1/1927 – 12/31/2014

  • Over the entire time period, Volatility and Momentum weighting have the highest performance (CAGR, Sharpe, Sortino ratios).
  • Equal-weighting, Volatility-weighting techniques and Momentum-weighting all beat the standard value-weighting technique.
four weighting schemes summary statistics
The results are hypothetical results and are NOT an indicator of future results and do NOT represent returns that any investor actually attained. Indexes are unmanaged, do not reflect management or trading fees, and one cannot invest directly in an index. Additional information regarding the construction of these results is available upon request.

Monthly Rebalance — First Half: 7/1/1927 – 12/31/1971

  • Over the first half of the sample, Volatility and Momentum weighting have the highest performance (CAGR, Sharpe, Sortino ratios).
  • Equal-weighting, Volatility-weighting techniques and Momentum-weighting all beat the standard value-weighting technique.
four weighting schemes summary statistics_first half
The results are hypothetical results and are NOT an indicator of future results and do NOT represent returns that any investor actually attained. Indexes are unmanaged, do not reflect management or trading fees, and one cannot invest directly in an index. Additional information regarding the construction of these results is available upon request.

Monthly Rebalance — Second Half: 1/1/1972 – 12/31/2014

  • Over the second half of the sample, Volatility and Momentum weighting have the highest performance (CAGR, Sharpe, Sortino ratios).
  • Equal-weighting, Volatility-weighting techniques and Momentum-weighting all beat the standard value-weighting technique.
four weighting schemes summary statistics_first half
The results are hypothetical results and are NOT an indicator of future results and do NOT represent returns that any investor actually attained. Indexes are unmanaged, do not reflect management or trading fees, and one cannot invest directly in an index. Additional information regarding the construction of these results is available upon request.

And the winner of our asset allocation weight horse race is…

  • Volatility weighting has the highest Sharpe and Sortino Ratio in all periods and has the highest risk-adjusted returns.
  • Momentum weighting has the highest CAGR in all periods, but follows volatility weighting on a risk-adjusted basis.
  • Equal weighting works better than the value weight index, but the effect is marginal.
Winner of weighting horse race
The results are hypothetical results and are NOT an indicator of future results and do NOT represent returns that any investor actually attained. Indexes are unmanaged, do not reflect management or trading fees, and one cannot invest directly in an index. Additional information regarding the construction of these results is available upon request.

Summary:

Our 3 alternative weighting techniques all have better CAGR, Sharpe and Sortino ratios, compared to the standard value-weighting technique. However, we must point out that investors must account for trading costs and taxes — a value weight index can minimize both. If an investor decides to use an alternative index weighting technique, they may want to consider using ETFs to accomplish this goal.

Endnotes:

1. We do not study fundamental stock weighting in this particular study (e.g., RAFI, we’ll share that research in the future).

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Please remember that past performance is not an indicator of future results. Please read our full disclaimer. 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. This material has been provided to you solely for information and educational purposes and does not constitute an offer or solicitation of an offer or any advice or recommendation to purchase any securities or other financial instruments and may not be construed as such. The factual information set forth herein has been obtained or derived from sources believed by the author and Alpha Architect to be reliable but it is not necessarily all-inclusive and is not guaranteed as to its accuracy and is not to be regarded as a representation or warranty, express or implied, as to the information’s accuracy or completeness, nor should the attached information serve as the basis of any investment decision. No part of this material may be reproduced in any form, or referred to in any other publication, without express written permission from Alpha Architect.

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About the Author

Jack Vogel, Ph.D.

Jack Vogel, Ph.D., conducts research in empirical asset pricing and behavioral finance, and is a co-author of DIY FINANCIAL ADVISOR: A Simple Solution to Build and Protect Your Wealth. His dissertation investigates how behavioral biases affect the value anomaly. His academic background includes experience as an instructor and research assistant at Drexel University in both the Finance and Mathematics departments, as well as a Finance instructor at Villanova University. Dr. Vogel is currently a Managing Member of Alpha Architect, LLC, an SEC-Registered Investment Advisor, where he heads the research department and serves as the Chief Financial Officer. He has a PhD in Finance and a MS in Mathematics from Drexel University, and graduated summa cum laude with a BS in Mathematics and Education from The University of Scranton.


  • Lister

    Wonderful, thanks for sharing. Looking forward to the study of fundamental weighting, as I am invested in it via Schwab’s ETFs.

  • Jack Vogel, PhD

    Thanks!

  • dosh1965

    Very insightful read and comparative study. I understand that proper measurements of trading costs/impact and taxes are difficult to provide, but perhaps you can provide asset turnover numbers as a proxy for the different asset allocation methodologies?

  • Andrew Greene

    Does weighting by value (earnings yield not market cap), momentum, or a combined weighting from the two metrics in a small (around 50 stocks) value or multifactor portfolio outperform equal weighting? Seems to me that we should weight the stocks with better factor ratings more but maybe in a small portfolio like this the factors are not precise enough to tell the difference between the stock ranked number 1 and number 50.

  • Jack Vogel, PhD

    We did not look at weighting by earnings here, something we may study in the future.

  • barbados

    Hi Jack, Just wanted to bring up something questionable about risk parity. I refer to this article as the source of my concern regarding risk parity as a viable strategy: http://www.advisorperspectives.com/articles/2015/11/17/a-new-challenge-to-factor-based-investing/2

    According to the article “A prominent paper in the Financial Analysts Journal claimed to show that a risk-parity strategy outperformed a 60/40 portfolio using data going back to the 1920s. Kritzman said this result was invalidated when it was revealed that it contained interval error. Once the data was corrected to adjust for the fact that returns were serially correlated, the 60/40 portfolio was shown to outperform the risk-parity portfolio by the nearly same amount that the FAJ article had claimed it underperformed.”

    The interval error was described by the article in the following manner: “Interval error has been largely ignored by academics and practitioners, he said. It stems from the mathematical principle that risk increases in proportion to the square root of time. For example, when extending a monthly risk estimate to an annual estimate, one should multiply by the square root of 12. In practice, however, that process is unreliable, Kritzman said, because assets are often “auto-correlated” (returns in one period are correlated with prior periods). If autocorrelation is higher, then risk is greater over shorter time periods than what would be predicted mathematically using historical data.”

    Would love to hear your thoughts on this issue if possible. Thank you.

  • Love to have him share his methodology and results so the world can review and digest. It is easy to attack a peer-reviewed article from afar. Also, making a claim that academics doing research ignore auto-correlation is akin to saying that painters ignore paint. Not sure where that statement is coming from…

  • barbados

    Okay noted. Thanks Wesley.

  • ettore trucco

    I don’t understand the way to weight momentum: Momentum score (1+momentum) ???

  • Jack Vogel, PhD

    Yes, weight is (1+mom) / [sum of (1+mom)]. I hope that helps!

  • Hannibal Smith

    You guys are da bomb! This is the first time I’ve seen “value weight” defined. All this time I thought it was some vague method of weighting according to over/under-valuedness. Why not just call it the common parlance of “market-cap weight” and avoid the confusion?

  • depending on your audience, “value weight” is the common parlance. Although I agree 100% that “market-cap weighted” is a lot less confusing in the new world with “smart beta” funds that weight stocks by their valuations.

    I agree, confusing as hell…that’s why we define it up front…hopefully the reader can map it into their framework.