Expanding the Efficient Frontier with Value and Momentum Strategies

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Expanding the Efficient Frontier with Value and Momentum Strategies

Awww…modern portfolio theory…that feel-good construct I teach to all of my graduate-level finance students each year. Simply input 1) a vector of expected returns and 2) a covariance matrix into your computer, and voilà, you have your optimal portfolio weights.

Like all things viewed with the benefit of hindsight, it sounds so easy, but the underlying theory and assumptions earned Harry Markowitz a Nobel Prize for his original paper on Portfolio Selection. And even though we sometimes rant on why you should be skeptical of complex practical applications of modern portfolio theory, the concepts and ideas that underlie it are critical for building successful investment programs. And that includes knowing when to use and when not to use certain ideas.

Bottom line: One should only discount certain concepts if you understand them. Unfortunately, many investors discount concepts because they don’t understand them.

The big takeway, or “punchline” if you will, from modern portfolio theory is the so-called mean-variance frontier (MV-frontier, also called the efficient frontier). The MV frontier simply maps out all the best expected combinations of risk/reward an investor can achieve by playing with the weights among the assets under investigation. Another important concept is the “tangency portfolio.” This portfolio is identified by the blue “-” in the chart below and sits on the CAL (capital allocation line). The tangency portfolio is the portfolio with the highest possible Sharpe ratio when used in combination with a risk-free asset (e.g., 90-day T-Bill). For those of you who haven’t taken an investment management course in a while, the CAL represents all combinations of risk-free rate and the tangency portfolio. These are “optimal” portfolios because there is no possible way to achieve a higher risk/reward in expectation. The optimal allocation weights for a 100% risk investor (i.e., no allocation to risk-free bonds) is the tangency portfolio.

The chart below includes the “IVY 5” assets from 1991 to 2014 and maps the efficient frontier using historical values for return/covariance:

  • SP500 = SP500 Total Return Index
  • EAFE = MSCI EAFE Total Return Index
  • REIT = FTSE NAREIT All Equity REITS Total Return Index
  • GSCI = GSCI Index
  • LTR = Merrill Lynch 7-10 year Government Bond Index

Results are gross of management fee and transaction costs and for illustrative purposes only. These are simulated performance results and do not reflect the returns an investor would actually achieve. All returns are total returns and include the reinvestment of distributions (e.g., dividends).

Over the past 20+ years, the commercial real estate market (REIT) has the highest absolute return, but with a relatively high standard deviation. Domestic equities (SP500) and international equities (EAFE) have relatively high returns and standard deviations. Long-term bonds (LTR) have the lowest standard deviation, but have relatively lower returns. Commodities (GSCI) perform the worst, with a high standard deviation and a low return.

The Mean-variance frontier (labeled: MV frontier) describes various optimal combinations of the 5 risky assets using historical data. “Optimal” combinations means that for a given risk level, the expected return is maximized. One can always create portfolios that would lie under the MV frontier, but they are considered sub-optimal, because there are better risk/reward combinations on the MV frontier.

Expanding the Efficient Frontier with Value and Momentum Strategies_1

Click to enlarge. 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.

Modern portfolio “works” in the sense that an investor is generally better off when they combine risky assets in a smart way and exploit the benefits of diversification (i.e., the individual asset risk/reward are all worse than the portfolios on the efficient frontier).

Expanding the Mean Variance Frontier with “Value”

In an ideal world, we could find portfolios that would expand the MV frontier and create opportunities with higher expected returns for a given level of risk. Perhaps counter-intuitively, adding highly volatile assets sometimes expands the MV frontier, if the volatility associated with the portfolio being added is unrelated to the other assets already included in the portfolio. An example of this situation is when one adds concentrated “factor” portfolios to a broader diversified portfolio. On a stand-alone basis, these concentrated exposures appear ugly, but in a portfolio context, they are much more interesting.

We explore this concept further by replacing the generic passive equity exposures (i.e., SP500, EAFE) in the IVY 5 with generic active value exposures. Thus, instead of exposure to the S&P 500 and EAFE, we now consider exposure to a simple “value” factor in domestic and international equity markets. We seek to identify how adding “value” to a diversified portfolio affects the efficient frontier:

  • SP500 –> VAL 10 = Ken French Top Decile Value (Value-weighted portfolio)
  • EAFE –> IVAL 5 = International Top Quintile Value (Average Top 3 market cap value-weight quintiles)

The chart below highlights the results. [At little later, we will examine in detail specifically how the addition of momentum affects the efficient frontier. For now, we choose to examine how our chosen active “value” allocations differ from the passive allocations, and the general effects this has had on the efficient frontier.]

VAL 10 and IVAL 5 are clearly much more volatile than the generic passive exposures. However, looking at volatility on an asset by asset basis disregards core principles of investment portfolio. We need to review the volatility of an asset in a portfolio context, and review how this asset expands/contracts the efficient frontier (again, we’ll see this more clearly in the chart at the bottom).

Expanding the Efficient Frontier with Value and Momentum Strategies_2

Click to enlarge. 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.

Expanding the Mean Variance Frontier with “Momentum”

We repeat the analysis above, but we replace the generic equity exposures with generic active momentum exposures, and keep access to REITs, GSCI and LTR in our IVY 5 portfolio.

  • SP500 –> MOM 10 = Ken French Top Decile Momentum (Value-weighted portfolio)
  • EAFE –> IMOM 5 = International Top Quintile Momentum (Average Top 3 market cap value-weight quintiles)

Similar to the value analysis, concentrated momentum portfolios are insanely more volatile than the generic exposures, but in a portfolio context they also expand the MV frontier.

Expanding the Efficient Frontier with Value and Momentum Strategies_3

Click to enlarge. 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.

What’s the MV Frontier look like when we add Value and Momentum?

In this last part, we add both value and momentum in our portfolios. So we have 7 portfolio assets to choose from.

  • SP500 –> VAL 10 and MOM 10
  • EAFE –> IVAL 5 and IMOM 5

We can see below how adding concentrated value and momentum exposures expands our portfolio opportunity set. We don’t tabulate the results, but if one includes even more concentrated value and momentum exposures (which on a stand alone basis are even more volatile than VAL/MOM 10 and IVAL/IMOM 10) the MV frontier expands even further. One can learn more about combining value and momentum here.

Expanding the Efficient Frontier with Value and Momentum Strategies_4

Click to enlarge. 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.


When considering an investment, one should examine that decision in the context of their current portfolio, not in isolation. And as we point out above, stand-alone value and momentum exposures–especially when they are concentrated–can look horrific and are considered “career killers” by asset allocators. But these same “career killer” exposures, when viewed through the lens of a diversified portfolio, can be extremely valuable. Of course, as is the case with any investment or asset allocation framework, we must always consider the FACTS, in addition to risk/return–fees, access, complexity, taxes, and search costs.

Good luck!

  • 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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  • This site provides NO information on our value ETFs or our momentum ETFs. Please refer to this site.

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About the Author:

After serving as a Captain in the United States Marine Corps, Dr. Gray earned a PhD, and worked as a finance professor at Drexel University. Dr. Gray’s interest in bridging the research gap between academia and industry led him to found Alpha Architect, an asset management that delivers affordable active exposures for tax-sensitive investors. Dr. Gray has published four books and a number of academic articles. Wes is a regular contributor to multiple industry outlets, to include the following: Wall Street Journal, Forbes, ETF.com, and the CFA Institute. Dr. Gray earned an MBA and a PhD in finance from the University of Chicago and graduated magna cum laude with a BS from The Wharton School of the University of Pennsylvania.


  1. Nicolas Flambeaux
    Nicolas December 22, 2015 at 12:24 pm

    Thanks for the article. New angles on MV frontiers are always welcome. Would you use that approach to adjust the theoretical 20 % par asset allocation in your robust asset allocation solution?

  2. Carl January 2, 2016 at 2:12 pm

    This was helpful, thank you. Mine is a meta- comment, about the time period of the analysis, because I’m curious about how things would appear in a short term analysis. For example, back testing done in 5-year slices. Here’s why.

    First, the behavioral reason. As you well know, it is difficult to get investors to think long term. Many may even consider a 5-year backtest model long term. Why not a less dogmatic answer for those impatient folks? One that is less what I tell my kids (“learn to think long term”) and more what they really want to know (“but what’s in Santa’s bag”)?

    (I’m trying to take an “invert, always invert” view on the notion, “let’s get everyone to focus on investing for the long term.”)

    Which takes me to the variety of investor demographics. There are younger folks who ought think long term, and family office / endowment folks who think multi-generationally. The under-served group (from the backtest timeframe point of view) are older, near or at retirement. Presumably they’re looking to preserve capital and generate some income stream within their lifetimes; their need is perhaps as simple as returns greater than or equal to inflation. Importantly, they can’t well tolerate a large drawdown if it requires a large percentage of their remaining lifespan to recover.

    (A 60 year old male’s life span in 2015 is 21 more years. A six year recovery period on a drawdown – or even in order to average returns out to better than inflation – represents 29% of his anticipated lifespan. The equivalent for a 40 year old investor is 38 years, and a six year recovery is 16% of his lifespan. Big difference.)

    Does this group need a completely different approach to an MV frontier? How might Alpha Architect’s “robust asset allocation” model appear to perform on 5-year backtest slices?

    If you’ve already addressed this, apologies and please point to the articles. If not, how might you modify a model? Or, might you conclude (ala AQR’s 2014 article, “Efficient frontier ‘theory’ for the long run,” http://www.aqr.com/cliffs-perspective/efficient-frontier-theory-for-the-long-run) that an efficient frontier that works in short term slices simply does not exist?

    If that was your hypothesis, what might be the implications? Both explicitly for the people I claim may be underserved, but also philosophically about models that need decades to overcome the jittery nature of their inputs?

    Thanks for tolerating the long comment… Happy New Year!

    • Wesley Gray, PhD January 3, 2016 at 5:43 pm

      Hey Carl,

      re investor psychology: Our mission is to empower investors through education because we think it is the only way to make folks better investors. Of course, we are operating in the face of an insane amount of psychology and ‘hard-wired’ issues that have been operating in markets for centuries. If someone said we were doomed for failure in our mission, I wouldn’t be able to disagree with the assessment, but that doesn’t mean we’re going to give up!

      re RAA: I think our RAA model mission is to capture the upside on global risk premiums, but the priority is preservation of capital (i.e., limit large drawdowns). In other words, we’re trying to generate higher returns and lower risk — a fool’s errand in many respects, but RAA is our best attempt based on our assessment of the evidence and market psychology. We wrote an entire book dedicated to the process in DIY Financial Advisor. Who the heck knows how it will all play out, but we got our thought process and design on paper — so at least you know what you’re getting. I think a 5-year horizon is reasonable for this sort of system and others like it (i.e., dual momentum). Whereas, for our long-only value and momentum strategies, a 5-year horizon gives you a good shot at survival, but the looming threat of huge equity-like drawdowns come with the territory.

      re Asness/future: As far as a system for 5-year slices…as I mentioned previously, I think tactical systems (i.e., trend-following/momentum elements) betting across global risk premiums (e.g., RAA and Dual Mom) are the best alternative to 2-3% returns in a buy-and-hold treasury bond fund that is tax-inefficient. But tactical systems that capture most of the upside, but side-step big drawdowns, may never play ball again…you just never know…Unfortunately, assets earn risk premiums because a lot of investors can’t handle the pain and anguish associated with the losses in the short-run. Most — if not all — strategies that generate smooth return patterns over relatively short horizons are arbitraged away quickly and/or have some sort of hidden risk (e.g., picking up pennies in front of a steamroller).

      Solution: What we need to do is vote out this whole “efficient market concept” and “market competition” element that plagues all of our portfolios! 🙂

  3. Peter June 22, 2017 at 12:44 pm

    Hi, great post, Dr. Gray! To what extent is minimum-variance optimization used within the products/services Alpha Architect offers? Any thoughts on the issue that variance is less adequate as a risk-measure when the returns of an asset are non-normal (as is certainly the case in crisis periods)? If you do incorporate optimization techniques that consider the higher-order moments of an asset’s return distribution, do you have a preference for any in particular? Mean-Modified VaR optimization? Thanks!

    • Wesley Gray, PhD June 23, 2017 at 10:13 am

      Even though I used to teach these techniques and had a whole course on it, we don’t use any optimization techniques in practice. I think the mental exercise of understanding how they work are good for building portfolios, but actually using these techniques in practice is riddled with land mines. We use simple risk parity for situations when we are pooling assets with materially different generic vol estimates, but otherwise we stick on equal-weight robustness focused pooling mechanisms. To deal with non-normal issues we use trend-following

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