Flirting with Models

The Research Library of Newfound Research

Category: Portfolio Construction Page 3 of 10

Ensembles and Rebalancing

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On February 24, 2020

In Portfolio Construction, Weekly Commentary

This post is available as a PDF download here.

Summary

  • While rebalancing studies typically focus on the combination of different asset classes, we evaluate a combination of two naïve trend-following strategies.
  • As expected, we find that a rebalanced fixed-mix of the two strategies generates a concave payoff profile.
  • More interestingly, deriving the optimal blend of the two strategies allows the rebalanced portfolio to out-perform either of the two underlying strategies.
  • While most rebalancing literature has focused on the benefits of combining asset classes, we believe this literature can be trivially extended to ensembles of strategies.

Two weeks ago, we wrote about the idea of payoff diversification.  The notion is fairly trivial, though we find it is often overlooked.  Put simply, any and all trading decisions – even something as trivial as rebalancing – create a “payoff profile.”  These profiles often fall into two categories: concave strategies that do well in stable environments is maintained and convex strategies that do better in the tails.

For example, we saw that rebalancing a 60/40 stock/bond portfolio earned a premium against a buy-and-hold approach when the spread between stock and bond returns remained narrow.  Conversely, when the spread in return between stocks and bonds was wide, rebalancing created a drag on returns.  This is a fairly trivial and obvious conclusion, but we believe it is important for investors to understand these impacts and why payoff is a meaningful axis of diversification.

In our prior study, we compared two different approaches to investing: strategic rebalancing and momentum investing.  In this (very brief) study, we want to demonstrate that these results are also applicable when applied to different variations of the same strategy.

Specifically, we will look at two long/short trend following strategies applied to broad U.S. equities.  When trend signals are positive, the strategy will be long U.S. equities and short the risk-free rate; when trend signals are negative the strategy will be short U.S. equities and long the risk-free rate.  We will use a simple time-series momentum signal.  The first model (“21D”) will evaluate trailing 21-day returns and hold for 1 day and the second model (“168D”) will evaluate trailing 168-day returns and holds for 14 days (with 14 overlapping portfolios).1  Both strategies implement a full skip day before allocating and assuming implementation at closing prices.

Source: Kenneth French Data Library.  Calculations by Newfound Research.  Returns are hypothetical and assume the reinvestment of all distributions.  Returns are gross of all fees, including, but not limited to, management fees, transaction fees, and taxes.  Past performance is not indicative of future results. 

So, what happens if we create a portfolio that holds both of these strategies, allocating 50% of our capital to each?  Readers of our prior note will likely be able to guess the answer easily: we create a concave payoff profile that depends upon the relative performance between the two strategies.  How, specifically, that concave shape manifests will be path dependent, but will also depend upon the rebalance frequency.  For example, below we plot the payoff profiles for the 50/50 blend rebalanced weekly and monthly.

Source: Kenneth French Data Library.  Calculations by Newfound Research.  Returns are hypothetical and assume the reinvestment of all distributions.  Returns are gross of all fees, including, but not limited to, management fees, transaction fees, and taxes.  Past performance is not indicative of future results. 

If we stop thinking of these as two strategies applied to the same asset and just think of them as two assets, the results are fairly standard and intuitive.  What is potentially appealing, however, is that the same literature and research that applies to the potential to create a rebalancing premium between assets can apply to a portfolio of strategies (whether a combination of distinct strategies, such as value and momentum, or an ensemble of the same strategy).

Below, we plot the annualized return of weekly rebalanced portfolios with different fixed-mix allocations to the 21D and 168D strategies.  We can see that the curve peaks at approximately 45%, suggesting that a 45% allocation to the 21D strategy and a 55% allocation to the 168D strategy actually maximizes the compound annualized growth rate of the portfolio.

If we follow the process of Dubikovsky and Susinno (2017)2 to derive the optimal blend of these two assets – using the benefit of hindsight to measure their annualized returns (7.28% and 7.61% respectively), volatility (17.55% and 17.97% respectively), and correlation (0.1318) – we derive an optimal weight of 45.33%.

Perhaps somewhat surprisingly, even if the correlation between these two strategies was 0.9, the optimal blend would still recommend about 10% to the 21D variation.  And, as extreme as it may seem, even if the annualized return of the 21D strategy was just 5.36% – a full 225 basis points below the 168D strategy – the optimal blend would still recommend about 10%.  Diversification can create interesting opportunities to harvest return; at least, in expectation.

And, as we would expect, if we have no view as to a difference in return or volatility between the two specifications, we would end up with a recommended allocation of 50% to each.

Conclusion

While most studies on rebalancing consider the potential benefits of combining assets, we believe that these benefits are trivially extended to strategies.  Not just different strategies, however, but even strategies of the same style.

In this brief note, we explore the payoff profile created by combining two naïve long/short trend following strategies applied to broad U.S. equities.  Unsurprisingly, rebalancing a simple mixture of the two specifications creates a concave payoff that generally profits when the spread between the two strategies is narrow and loses when the spread is wide.

More interestingly, however, we demonstrate that by rebalancing a fixed-mix of the two strategies, we can generate a return that is greater than either strategy individually.  We believe that this potential benefit of ensemble approaches has been mostly overlooked by existing literature and deserves further analysis.

 

 

Diversification with Portable Beta

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On February 18, 2020

In Craftsmanship, Portfolio Construction, Risk & Style Premia, Risk Management, Weekly Commentary

This post is available as a PDF download here.

Summary

  • A long/flat tactical equity strategy with a portable beta bond overlay – a tactical 90/60 portfolio – has many moving parts that can make attribution and analysis difficult.
  • By decomposing the strategy into its passive holdings (a 50/50 stock/bond portfolio and U.S. Treasury futures) and active long/short overlays (trend equity, bond carry, bond momentum, and bond value), we can explore the historical performance of each component and diversification benefits across each piece of the strategy.
  • Using a mean-variance framework, we are also able to construct an efficient frontier of the strategy components and assess the differences between the optimal portfolio and the tactical 90/60.
  • We find that the tactical 90/60 is relatively close to the optimal portfolio for its volatility level and that its drawdown risk profile is close to that of an unlevered 60/40 portfolio.
  • By utilizing a modest amount of leverage and pairing it will risk management in both equities and bonds, investors may be able to pursue capital efficiency and maximize portfolio returns while simultaneously managing risk.

Portable beta strategies seek to enhance returns by overlaying an existing portfolio strategy with complementary exposure to diversifying asset classes and strategies. In overlaying exposure on an existing portfolio strategy, portable beta strategies seek to make every invested dollar work harder. This idea can create “capital efficiency” for investors, freeing up dollars in an investor’s portfolio to invest in other asset classes or investment opportunities.

At Newfound, we focus on managing risk. Trend following – or absolute momentum – is a key approach we employ do this, especially in equities. Trend equity strategies are a class of strategies that aim to harvest the long-term benefits of the equity risk premium while managing downside risk through the application of trend following.

We wrote previously how a trend equity strategy can be decomposed into passive and active components in order to isolate different contributors to performance. There is more than one way to do this, but in the most symmetric formulation, a “long/flat” trend equity strategy (one that that either holds equities or cash; i.e. does not short equities) can be thought of as a 100% passive allocation to a 50/50 portfolio of stocks and cash plus a 50% overlay allocation to a long/short trend equity strategy that can move between fully short and fully long equities. This overlay component is portable beta.

We have also written previously about how a portable beta overlay of bonds can be beneficial to trend equity strategies – or even passive equity investments, for that matter. For example, 95% of a portfolio could be invested in a trend equity strategy, and the remaining 5% could be set aside as collateral to initiate a 60% overlay to 10-year U.S. Treasury futures. This approximates a 60/40 portfolio that is leveraged by 50%

Source: Newfound. Allocations are hypothetical and for illustrative purposes only.

Since this bond investment introduces interest rate risk, we have proposed ways to manage risk in this specific sleeve using factors such as value, carry, and momentum. By treating these factors as fully tactical long/short portfolios themselves, if we hold them in equal weight, we can also break down the tactical U.S. Treasury futures overlay into active and passive components, with a 30% passive position in U.S. Treasury futures and 10% in each of the factor-based strategies.

Source: Newfound. Allocations are hypothetical and for illustrative purposes only.

When each overlay is fully invested, the portfolio will hold 95% stocks, 5% cash, and 60% U.S. Treasury futures. When all the overlays are fully short, the strategy will be fully invested in cash with no bond overlay position.

While the strategy has not changed at all with this slicing and dicing, we now have a framework to explore the historical contributions of the active and passive components and the potential diversification benefits that they offer.

Diversification Among Components

For the passive portfolio 50/50 stock/cash, we will use a blend of the Vanguard Total U.S. stock market ETF (VTI) and the iShares Short-term Treasury Bond ETF (SHV) with Kenneth French data for market returns and the risk-free rate prior to ETF inception.

For the active L/S Trend Equity portfolio, we will use a long/short version of the Newfound U.S. Trend Equity Index.

The passive 10-year U.S. Treasury futures is the continuous futures contract with a proxy of the 10-year constant maturity Treasury index minus the cash index used before inception (January 2000). The active long/short bond factors can be found on the U.S. Treasuries section of our quantitative signals dashboard, which is updated frequently.

All data starts at the common inception point in May 1957.

As a technical side note, we must acknowledge that a constant maturity 10-year U.S. Treasury index minus a cash index will not precisely match the returns of 10-year U.S. Treasury futures. The specification of the futures contracts state that the seller of such a contract has the right to deliver any U.S. Treasury bond with maturity between 6.5 and 10 years. In other words, buyers of this contract are implicitly selling an option, knowing that the seller of the contract will likely choose the cheapest bond to deliver upon maturity (referred to as the “cheapest to deliver”). Based upon the specification and current interest rate levels, that current cheapest to deliver bond tends to have a maturity of 6.5 years.

This has a few implications. First, when you buy U.S. Treasury futures, you are selling optionality. Finance 101 will teach you that optionality has value, and therefore you would expect to earn some premium for selling it. Second, the duration profile between our proxy index and 10-year U.S. Treasury futures has meaningfully diverged in the recent decade. Finally, the roll yield harvested by the index and the futures will also diverge, which can have a non-trivial impact upon returns.

Nevertheless, we believe that for the purposes of this study, the proxy index is sufficient for broad, directional attribution and understanding.

Source: Kenneth French Data Library, Federal Reserve Bank of St. Louis, Tiingo, Stevens Futures. Calculations by Newfound Research. Data is from May 1957 to January 2020. Returns are hypothetical and assume the reinvestment of all distributions. Returns are gross of all fees, including, but not limited to, management fees, transaction fees, and taxes. You cannot invest directly in an index and unmanaged index returns do not reflect any fees, expenses or sales charges. Past performance is not indicative of future results. 

The 50/50 Stock/Cash portfolio is the only long-only holding. While the returns are lower for all the other strategies, we must keep in mind that they are all overlays that can add to the 50/50 portfolio rather than simply de-risk and cannibalize its return.

This is especially true since these overlay strategies have exhibited low correlation to the 50/50 portfolio.

The table below shows the full period correlation of monthly returns for all the portfolio components. The equity and bond sub-correlation matrices are outlined to highlight the internal diversification.

Source: Kenneth French Data Library, Federal Reserve Bank of St. Louis, Tiingo, Stevens Futures. Calculations by Newfound Research. Data is from May 1957 to January 2020. Returns are hypothetical and assume the reinvestment of all distributions. Returns are gross of all fees, including, but not limited to, management fees, transaction fees, and taxes. You cannot invest directly in an index and unmanaged index returns do not reflect any fees, expenses or sales charges. Past performance is not indicative of future results. 

Not only do all of the overlays have low correlation to the 50/50 portfolio, but they generally exhibit low cross-correlations. Of the overlays, the L/S bond carry and L/S bond momentum strategies have the highest correlation (0.57), and the L/S bond carry and passive bond overlay have the next highest correlation (0.47).

The bond strategies have also exhibited low correlation to the equity strategies. This results in good performance, both absolute and risk-adjusted, relative to a benchmark 60/40 portfolio and a benchmark passive 90/60 portfolio.

Source: Kenneth French Data Library, Federal Reserve Bank of St. Louis, Tiingo, Stevens Futures. Calculations by Newfound Research. Data is from May 1957 to January 2020. Returns are hypothetical and assume the reinvestment of all distributions. Returns are gross of all fees, including, but not limited to, management fees, transaction fees, and taxes. You cannot invest directly in an index and unmanaged index returns do not reflect any fees, expenses or sales charges. Past performance is not indicative of future results. 

Finding the Optimal Blend

Up to this point, we have only considered the fixed allocations to each of the active and passive strategies outlined at the beginning. But these may not be the optimal holdings.

Using a block-bootstrap method to simulate returns, we can utilize mean-variance optimization to determine the optimal portfolios for given volatility levels.1 This yields a resampled historical realized efficient frontier.

Source: Kenneth French Data Library, Federal Reserve Bank of St. Louis, Tiingo, Stevens Futures. Calculations by Newfound Research. Data is from May 1957 to January 2020. Returns are hypothetical and assume the reinvestment of all distributions. Returns are gross of all fees, including, but not limited to, management fees, transaction fees, and taxes. You cannot invest directly in an index and unmanaged index returns do not reflect any fees, expenses or sales charges. Past performance is not indicative of future results. 

Plotting the benchmark 60/40, benchmark 90/60, and the tactical 90/60 on this efficient frontier, we see that the tactical 90/60 lies very close to the frontier at about 11.5% volatility. The allocations for the frontier are shown below.

 

Source: Kenneth French Data Library, Federal Reserve Bank of St. Louis, Tiingo, Stevens Futures. Calculations by Newfound Research. Data is from May 1957 to January 2020. Returns are hypothetical and assume the reinvestment of all distributions. Returns are gross of all fees, including, but not limited to, management fees, transaction fees, and taxes. You cannot invest directly in an index and unmanaged index returns do not reflect any fees, expenses or sales charges. Past performance is not indicative of future results. 

As expected, the lower volatility portfolios hold more cash and the high volatility portfolios hold more equity. For the 9% volatility level, these two allocations match, leading to the full allocation to a 50/50 stock/cash blend as in the tactical 90/60.

The passive allocation to the Treasury futures peaks at about 60%, while the L/S bond factor allocations are generally between 5% and 20% with more emphasis on Value and typically equal emphasis on Carry and Momentum.

The allocations in the point along the efficient frontier that matches the tactical 90/60 portfolio’s volatility are shown below.

Source: Kenneth French Data Library, Federal Reserve Bank of St. Louis, Tiingo, Stevens Futures. Calculations by Newfound Research. Data is from May 1957 to January 2020. Returns are hypothetical and assume the reinvestment of all distributions. Returns are gross of all fees, including, but not limited to, management fees, transaction fees, and taxes. You cannot invest directly in an index and unmanaged index returns do not reflect any fees, expenses or sales charges. Past performance is not indicative of future results. 

In this portfolio, we see a higher allocation to passive equities, a smaller position in the tactical equity L/S, and a larger position in passive Treasury futures. However, given the resampled nature of the process, these allocations are not wildly far away from the tactical 90/60.

The differences in the allocations are borne out in the Ulcer Index risk metric, which quantifies the severity and duration of drawdowns.

Source: Kenneth French Data Library, Federal Reserve Bank of St. Louis, Tiingo, Stevens Futures. Calculations by Newfound Research. Data is from May 1957 to January 2020. Returns are hypothetical and assume the reinvestment of all distributions. Returns are gross of all fees, including, but not limited to, management fees, transaction fees, and taxes. You cannot invest directly in an index and unmanaged index returns do not reflect any fees, expenses or sales charges. Past performance is not indicative of future results. 

The efficient frontier portfolio has a lower Ulcer Index than that of the tactical 90/60 even though their returns and volatility are similar. However, the Ulcer index of the tactical 90/60 is very close to that of the benchmark 60/40.

These differences are likely due to the larger allocation to the tactical equity long/short which can experience whipsaws (e.g. in October 1987), the lower allocation to passive U.S. equities, and the lower allocation to the Treasury overlay.

In an uncertain future, there can be significant risk in relying too much on the past, but having this framework can be useful for gaining a deeper understanding of which market environments benefit or hurt each component within the portfolio and how they diversify each other when held together.

Conclusion

In this research note, we explored diversification in a long/flat tactical equity strategy with a portable beta bond overlay. By decomposing the strategy into its passive holdings (50/50 stock/bond portfolio and U.S. Treasury futures) and active long/short overlays (trend equity, bond carry, bond momentum, and bond value), we found that each of the overlays has historically exhibited low correlation to the passive portfolios and low cross-correlations to each other. Combining all of these strategies using a tactical 90/60 portfolio has led to strong performance on both an absolute and risk-adjusted basis.

Using these strategy components, we constructed an efficient frontier of portfolios and also found that the “intuitive” tactical 90/60 portfolio that we have used in much of our portable beta research is close to the optimal portfolio for its volatility level. While this does not guarantee that this portfolio will be optimal over any given time period, it does provide evidence for the robustness of the multi-factor risk-managed approach.

Utilizing portable beta strategies can be an effective way for investors to pursue capital efficiency and maximize portfolio returns while simultaneously managing risk. While leverage can introduce risks of its own, relying on diversification and robust risk-management methods (e.g. trend following) can mitigate the risk of large losses.

The fear of using leverage and derivatives may be an uphill battle for investors, and there are a few operational burdens to overcome, but when used appropriately, these tools can make portfolios work harder and lead to more flexibility for allocating to additional opportunities.

If you are interested in learning how Newfound applies the concepts of tactical portable beta to its mandates, please reach out (info@thinknewfound.com).

Payoff Diversification

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On February 10, 2020

In Portfolio Construction, Risk Management, Weekly Commentary

This post is available as a PDF download here.

Summary

  • At Newfound, we adopt a holistic view of diversification that encompasses not only what we invest in, but also how and when we make those investment decisions.
  • In this three-dimensional perspective, what is correlation-based, how is payoff-based, and when is opportunity-based.
  • In this piece, we provide an example of what we mean by payoff-based diversification, using a simple strategically rebalanced portfolio and a naïve momentum strategy.
  • We find that the strategically rebalanced portfolio exhibits a payoff structure that is concave in nature whereas the momentum-based approach exhibits a convex profile.
  • By combining the two approaches – being careful in how we size positions – we can develop a portfolio that is less sensitive to the co-movement of underlying assets.

At Newfound, we embrace a holistic view of diversification that covers not just what we invest in, but also how and when we make those decisions.  What is the diversification most investors are well-versed in and covers traditional, correlation-based diversification between securities, assets, macroeconomic factors, and geographic regions.

We identify when as “opportunity diversification” because it captures the opportunities that are available when we make investment decisions.  This often goes overlooked in public markets (which is why we spend so much time writing about rebalance timing luck) but is well acknowledged in private markets where investors often allocate to multiple fund “vintages” to create diversification.

How is generally easy to understand, but sometimes difficult to visualize.  We call it “payoff diversification” to acknowledge that when viewed through he appropriate lens, every investment style creates a particular shape.  For example, when the return of a call option is plotted against the return of the underlying security, it generates a hockey-stick-like payoff profile.

In this short research note, we are going to demonstrate the payoff profiles of a strategically allocated portfolio and a naïve momentum strategy.  We will then show that by combining these two approaches we can create a portfolio that exhibits significantly less sensitivity to the co-movement of underlying assets.

The Payoff Profile of a Strategic Portfolio

Few investors consider a strategically allocated portfolio to be an active strategy.  And it isn’t; at least not until we introduce rebalancing.  Once we institute a process to systematically returning our drifted weights back to their original fixed mix, we create a strategy and a corresponding payoff profile.

But what does this payoff profile look like?  As an example, consider a U.S. 60/40 portfolio comprised of broad U.S. equities and a constant maturity 10-year U.S. Treasury index.  If equities out-perform bonds, our equity allocation will increase and our bond allocation will decrease.  If equities continue to out-perform bonds, we will benefit relative to our original policy weights.  Similarly, if equities under-perform bonds, then our relative equity allocation will decrease.  Again, should they continue to underperform, we are well positioned.

However, if we were to rebalance back to our original 60/40 allocation, we would eliminate the opportunity to benefit from the continuation of the relative performance.

On the other hand, consider the case where equities out-perform, our relative allocation to equities increases due to drift, and then equities subsequently under-perform.  Now allowing drift has hurt us and we would have been better off rebalancing.

We can visualize this relationship by plotting the return spread between stocks and bonds (x-axis) versus the return spread between a monthly-rebalanced portfolio and a buy-and-hold (drifted) approach (y-axis) over rolling 1-year periods.

Source: Kenneth French Data Library; Federal Reserve Bank of St. Louis.  Calculations by Newfound Research.  Returns are hypothetical and assume the reinvestment of all distributions.  Returns are gross of all fees, including, but not limited to, management fees, transaction fees, and taxes. The 60/40 portfolio is comprised of a 60% allocation to broad U.S. equities and a 40% allocation to a constant maturity 10-Year U.S. Treasury index.  The rebalanced variation is rebalanced at the end of each month whereas the buy-and-hold variation is allowed to drift over the 1-year period.  The 10-Year U.S. Treasuries index is a constant maturity index calculated by assuming a 10-year bond is purchased at the beginning of every month and sold at the end of that month to purchase a new bond at par at the beginning of the next month. You cannot invest directly in an index and unmanaged index returns do not reflect any fees, expenses or sales charges. Past performance is not indicative of future results. 

What we can see is a concave payoff function. When equities significantly out-perform bonds (far right side of the graph), the rebalanced portfolio under-performs the drifted portfolio.  Similarly, when bonds significantly out-perform equities (far left side of the graph), the rebalanced portfolio under-performs the drifted portfolio.  When the return spread between stocks and bonds is small– a case likely to be more indicative of mean-reversion than positive autocorrelation in the spread – we can see that rebalancing actually generates a positive return versus the drifted portfolio.

Those versed in options will note that this payoff looks incredibly similar to a 1-year strangle sold on the spread between stocks and bonds and struck at 0%.  The seller captures the premium when the realized spread remains small but loses money when the spread is more extreme.

The Payoff Profile of Naïve Momentum Following

We can now take the exact same approach to evaluating the payoff profile of a naïve momentum strategy.  Each month, the strategy will simply invest in either stocks or bonds based upon whichever had the highest trailing 12-month return

As this approach is explicitly trying to capture auto-correlation in the return spread between stocks and bonds, we would expect to see almost mirror behavior to the payoff profile we saw with strategic rebalancing.

Source: Kenneth French Data Library; Federal Reserve Bank of St. Louis.  Calculations by Newfound Research.  Returns are hypothetical and assume the reinvestment of all distributions.  Returns are gross of all fees, including, but not limited to, management fees, transaction fees, and taxes. The 60/40 portfolio is comprised of a 60% allocation to broad U.S. equities and a 40% allocation to a constant maturity 10-Year U.S. Treasury index.  The momentum portfolio is rebalanced monthly and selects the asset with the highest prior 12-month returns whereas the buy-and-hold variation is allowed to drift over the 1-year period.  The 10-Year U.S. Treasuries index is a constant maturity index calculated by assuming a 10-year bond is purchased at the beginning of every month and sold at the end of that month to purchase a new bond at par at the beginning of the next month. You cannot invest directly in an index and unmanaged index returns do not reflect any fees, expenses or sales charges. Past performance is not indicative of future results. 

While the profile may not be as tidy as before, we can see a convex payoff profile that tends to profit when the return spread is more extreme and lose money when the spread is narrow.  Again, those familiar with options will recognize this as similar to the payoff of a 1-year straddle based upon the return spread between stocks and bonds.  The buyer pays a premium but captures the spread when it is extreme.

Note, however, the scale of the y-axis.  Whereas the payoff profile for the rebalanced portfolio was between -3.0% and +2.0%, the payoff profile for this momentum approach is much larger, ranging between -30.0% and 40.0%.

Creating Payoff Diversification

We have seen that whether we strategically rebalance or adopt a momentum-based approach, both approaches create a payoff profile that is sensitive to the return spread in underlying assets.  But what if we do not want to take such a specific payoff bet?  One simple answer is diversification.

If we allocate to both the strategically rebalanced portfolio and the naïve momentum portfolio, we will realize both their payoff profiles simultaneously.  As their profiles are close mirrors of one another, we may be able to achieve a more neutral outcome.

We have to be careful, however, as to size the allocations appropriate.  Recall that the payoff profile of the strategically rebalanced portfolio was approximately 1/10th the size of the naïve momentum strategy.  For both profiles to contribute equally, we would want to allocate approximately 90% of our capital to the strategic rebalancing strategy and 10% of our capital to the momentum strategy.

Below we plot the payoff structure of such a mix.

Source: Kenneth French Data Library; Federal Reserve Bank of St. Louis.  Calculations by Newfound Research.  Returns are hypothetical and assume the reinvestment of all distributions.  Returns are gross of all fees, including, but not limited to, management fees, transaction fees, and taxes. The 60/40 portfolio is comprised of a 60% allocation to broad U.S. equities and a 40% allocation to a constant maturity 10-Year U.S. Treasury index.  The mixed portfolio is rebalanced monthly and is a 90% allocation to a rebalanced 60/40 and a 10% allocation to a naïve momentum strategy; whereas the buy-and-hold variation is allowed to drift over the 1-year period.  The 10-Year U.S. Treasuries index is a constant maturity index calculated by assuming a 10-year bond is purchased at the beginning of every month and sold at the end of that month to purchase a new bond at par at the beginning of the next month. You cannot invest directly in an index and unmanaged index returns do not reflect any fees, expenses or sales charges. Past performance is not indicative of future results. 

We can see that diversifying how we make decisions results in a payoff structure that is far more neutral to the co-movement of underlying securities in the portfolio.  The holy grail, of course, is not just to find strategies whose combination neutralizes sensitivity to the spread in returns, but actually creates a higher likelihood of positive outcomes in all environments.

Conclusion

In this research note, we aimed to provide greater insight into the idea of payoff diversification, the how in our what-how-when diversification framework.  To do so, we explored two simple examples: a strategically rebalanced 60/40 allocation and a naïve momentum strategy.

We found that the strategically rebalanced portfolio generates a payoff profile that is convex with respect to the spread in returns between stocks and bonds.  In general, the larger the spread, the more likely that rebalancing generates a negative return versus a buy-and-hold approach.  Conversely, the smaller the spread, the more likely that rebalancing generates a positive return.

The naïve momentum strategy – which simply bought the asset with the greatest prior 12-month returns – exhibited a convex profile.  When the return spread between stocks and bonds was large, the naïve momentum strategy was more likely to out-perform buy-and-hold.  Conversely, when the return spread was small, the naïve momentum strategy tended to under-perform.

Importantly, the magnitudes of the payoffs are significantly different, with the naïve momentum strategy generating returns nearly 10x larger than strategic rebalancing in the tails.  This difference has important implications for strategy sizing, and we find a portfolio mixture of 90% strategic rebalancing and 10% naïve momentum does a reasonably good job of neutralizing portfolio payoff sensitivity to the spread in stock and bond returns.

Re-specifying the Fama French 3-Factor Model

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On December 16, 2019

In Craftsmanship, Portfolio Construction, Risk & Style Premia, Risk Management, Value, Weekly Commentary

This post is available as a PDF download here.

Summary­

  • The Fama French three-factor model provides a powerful tool for assessing exposures to equity risk premia in investment strategies.
  • In this note, we explore alternative specifications of the value (HML) and size (SMB) factors using price-to-earnings, price-to-cash flow, and dividend yield.
  • Running factor regressions using these alternate specifications on a suite of value ETFs and Newfound’s Systematic Value strategy, lead to a wide array of results, both numerically and directionally.
  • While many investors consider the uncertainty of the parameter estimates from the regression using the three-factor model, most do not consider the uncertainty that comes from the assumption of how you construct the equity factors in the first place.
  • Understanding the additional uncertainty is crucial for manager and investors who must consider what risks they are trying to measure and control by using tools like factor regression and make sure their assumptions align with their goals.

In their 1992 paper, The Cross-Section of Expected Stock Returns, Eugene Fama and Kenneth French outlined their three-factor model to explain stock returns.

While the Capital Asset Pricing Model (CAPM) only describes asset returns in relation to their exposure to the market’s excess return through the stock’s beta and identifies any return beyond that as alpha, Fama and French’s three-factor model reattributed some of that supposed alpha to exposures to a value factor (High-minus-low or HML) based on returns stratified by price-to-book ratios and a size factor (small-minus-big or SMB) based on returns stratified by market capitalization.

This gave investors a tool to judge investment strategies based on the loadings to these risk factors. A manager with a seemingly high alpha may have simply been investing in value and small-cap stocks historically.

The notion of compensated risk premia has also opened the floodgate of many additional factors from other researchers (such as momentum, quality, low beta, etc.) and even two more factors from Fama and French (investment and profitability).

A richer factor universe opens up a wide realm of possibilities for analysis and attribution. However, setting further developments aside and going back to the original three-factor model, we would be remiss if we didn’t dive a bit further into its specification.

At the highest level, we agree with treating “value” and “size” as risk factors, but there is more than one way to skin a factor.

What is “value”?

Fama and French define it using the price-to-book ratio of a stock. This seems legitimate for a broad swath of stocks, especially those that are very capital intensive – such as energy, manufacturing, and financial firms – but what about industries that have structurally lower book values and may have other potential price drivers? For example, a technology company might have significant intangible intellectual property and some utility companies might employ leverage, which decreases their book value substantially.

To determine value in these sectors, we might utilize ratios that account for sales, dividends, or earnings. But then if we analyzed these strategies using the Fama French three-factor model as it is specified, we might misjudge the loading on the value factor.

“Size” seems more straightforward. Companies with low market capitalizations are small. However, when we consider how the size factor is defined based on the value factor, there might even be some differences in SMB using different value metrics.

In this commentary, we will explore what happens when we alter the definition of value for the value factor (and hence the size factor) and see how this affects factor regressions of a sample of value ETFs along with our Systematic Value strategy.

HML Factor Definitions

In the standard version of the Fama French 3-factor model, HML is constructed as a self-financing long/short portfolio using a 2×3 sort on size and value. The investment universe is split in half based on market capitalization and in three parts (30%/40%/30%) based on valuation, in this base case, price-to-book ratio.

Using additional data from the Kenneth French Data Library and the same methodology, we will construct HML factors using sorts based on size and:

  • Price-to-earnings ratios
  • Price-to-cash flow ratios
  • Dividend yields

The common inception date for all the factors is June 1951.

The chart below shows the growth of each of the four value factor portfolios.

Source: Kenneth French Data Library. Calculations by Newfound Research. Past performance is not an indicator of future results. Performance is backtested and hypothetical. Performance figures are gross of all fees, including, but not limited to, manager fees, transaction costs, and taxes.  Performance assumes the reinvestment of all distributions. 

Over the entire time period – and for many shorter time horizons – the standard HML factor using price-to-book does not even have the most attractive returns. Price-to-earnings and price-to-cash flow often beat it out.

On the other hand, the HML factor formed using dividend yields doesn’t look so hot.

One of the reasons behind this is that the small, low dividend yield companies performed much better than the small companies that were ranked poorly by the other value factors. We can see this effect borne out in the SMB chart for each factor, as the SMB factor for dividend yield performed the best.

(Recall that we mentioned previously how the Fama French way of defining the size factor is dependent on which value metric we use.)

Source: Kenneth French Data Library. Calculations by Newfound Research. Past performance is not an indicator of future results. Performance is backtested and hypothetical. Performance figures are gross of all fees, including, but not limited to, manager fees, transaction costs, and taxes.  Performance assumes the reinvestment of all distributions.

Looking at the statistical significance of each factor through its t-statistic, we can see that Price-to-Earnings and Price-to-Cash Flow yielded higher significance for the HML factor than Price-to-Book. And those two along with Dividend Yield all eclipsed the Price-to-Book construction of the SMB factor.

T-Statistics for HML and SMB Using Various Value Metrics

 Price-to-BookDividend YieldPrice-to-EarningsPrice-to-Cash Flow
HML2.90.03.73.4
SMB1.02.41.61.9

Assuming that we do consider all metrics to be appropriate ways to assess the value of companies, even if possibly under different circumstances, how do different variants of the Fama French three-factor model change for each scenario with regression analysis?

The Impact on Factor Regressions

Using a sample of U.S. value ETFs and our Systematic Value strategy, we plot the loadings for the different versions of HML. The regressions are carried out using the trailing three years of monthly data ending on October 2019.

Source: Tiingo, Kenneth French Data Library. Calculations by Newfound Research. Past performance is not an indicator of future results. Returns represent live strategy results. Returns for the Newfound Systematic Value strategy are gross of all management fees and taxes, but net of execution fees.  Returns for ETFs included in study are gross of any management fees, but net of underlying ETF expense ratios.  Returns assume the reinvestment of all distributions.

For each different specification of HML, the differences in the loading between investments is generally directionally consistent. For instance, DVP has higher loadings than FTA for all forms of HML.

However, sometimes this is not the case.

VLUE looks more attractive than VTV based on price-to-cash flow but not dividend yield. FTA is roughly equivalent to QVAL in terms of loading when price-to-book is used for HML, but it varies wildly when other metrics are used.

The tightest range for the four models for any of the investments is 0.09 (PWV) and the widest is 0.52 (QVAL). When we factor in that these estimates each have their own uncertainty, distinguishing which investment has the better value characteristic is tough. Decisions are commonly made on much smaller differences.

We see similar dispersion in the SMB loadings for the various constructions.

Source: Tiingo, Kenneth French Data Library. Calculations by Newfound Research. Past performance is not an indicator of future results. Returns represent live strategy results. Returns for the Newfound Systematic Value strategy are gross of all management fees and taxes, but net of execution fees.  Returns for ETFs included in study are gross of any management fees, but net of underlying ETF expense ratios.  Returns assume the reinvestment of all distributions.

Many of these values are not statistically significant from zero, so someone who has a thorough understanding of uncertainty in regression would likely not draw a strict comparison between most of these investments.

However, one implication of this is that if a metric is chosen that does ascribe significant size exposure to one of these investments, an investor may make a decision based on not wanting to bear that risk in what they desire to be a large-cap investment.

Can We Blend Our Way Out?

One way we often mitigate model specification risk is by blending a number of models together into one.

By averaging all of our HML and SMB factors, respectively, we arrive at blended factors for the three-factor model.

Source: Tiingo, Kenneth French Data Library. Calculations by Newfound Research. Past performance is not an indicator of future results. Returns represent live strategy results. Returns for the Newfound Systematic Value strategy are gross of all management fees and taxes, but net of execution fees.  Returns for ETFs included in study are gross of any management fees, but net of underlying ETF expense ratios.  Returns assume the reinvestment of all distributions.

All of the investments now have HML loadings in the top of their range of the individual model loadings, and many (FTA, PWV, RPV, SPVU, VTV, and the Systematic Value strategy) have loadings to the blended HML factor that exceed the loadings for all of the individual models.

The opposite is the case for the blended SMB factor: the loadings are in the low-end of the range of the individual model loadings.

Source: Tiingo, Kenneth French Data Library. Calculations by Newfound Research. Past performance is not an indicator of future results. Returns represent live strategy results. Returns for the Newfound Systematic Value strategy are gross of all management fees and taxes, but net of execution fees.  Returns for ETFs included in study are gross of any management fees, but net of underlying ETF expense ratios.  Returns assume the reinvestment of all distributions.

So which is the correct method?

That’s a good question.

For some investments, it is situation-specific. If a strategy only uses price-to-earnings as its value metric, then putting it up against a three-factor model using the P/E ratio to construct the factors is appropriate for judging the efficacy of harvesting that factor.

However, if we are concerned more generally about the abstract concept of “value”, then the blended model may be the best way to go.

Conclusion

In this study, we have explored the impact of model specification for the value and size factor in the Fama French three-factor model.

We empirically tested this impact by designing a variety of HML and SMB factors based on three additional value metrics (price-to-earnings, price-to-cash flow, and dividend yield). These factors were constructed using the same rules as for the standard method using price-to-book ratios.

Each factor, with the possible exceptions of the dividend yield-based HML, has performance that could make it a legitimate specification for the three-factor model over the time that common data is available.

Running factor regressions using these alternate specifications on a suite of value ETFs and Newfound’s Systematic Value strategy, led to a wide array of results, both numerically and directionally.

While many investors consider the uncertainty of the parameter estimates from the regression using the three-factor model, most do not consider the uncertainty that comes from the assumption of how you construct the equity factors in the first place.

Understanding the additional uncertainty is crucial for decision-making. Managers and investors alike must consider what risks they are trying to measure and control by using tools like factor regression and make sure their assumptions align with their goals.

“Value” is in the eye of the beholder, and blind applications of two different value factors may lead to seeing double conclusions.

The Dumb (Timing) Luck of Smart Beta

By

On November 18, 2019

In Craftsmanship, Defensive, Momentum, Popular, Portfolio Construction, Risk & Style Premia, Value, Weekly Commentary

This post is available as a PDF download here.

Summary

  • In past research notes we have explored the impact of rebalance timing luck on strategic and tactical portfolios, even using our own Systematic Value methodology as a case study.
  • In this note, we generate empirical timing luck estimates for a variety of specifications for simplified value, momentum, low volatility, and quality style portfolios.
  • Relative results align nicely with intuition: higher concentration and less frequent rebalancing leads to increasing levels of realized timing luck.
  • For more reasonable specifications – e.g. 100 stock portfolios rebalanced semi-annually – timing luck ranges between 100 and 400 basis points depending upon the style under investigation, suggesting a significant risk of performance dispersion due only to when a portfolio is rebalanced and nothing else.
  • The large magnitude of timing luck suggests that any conclusions drawn from performance comparisons between smart beta ETFs or against a standard style index may be spurious.

We’ve written about the concept of rebalance timing luck a lot.  It’s a cowbell we’ve been beating for over half a decade, with our first article going back to August 7th, 2013.

As a reminder, rebalance timing luck is the performance dispersion that arises from the choice of a particular rebalance date (e.g. semi-annual rebalances that occur in June and December versus March and September).

We’ve empirically explored the impact of rebalance timing luck as it relates to strategic asset allocation, tactical asset allocation, and even used our own Systematic Value strategy as a case study for smart beta.  All of our results suggest that it has a highly non-trivial impact upon performance.

This summer we published a paper in the Journal of Index Investing that proposed a simple solution to the timing luck problem: diversification.  If, for example, we believe that our momentum portfolio should be rebalanced every quarter – perhaps as an optimal balance of cost and signal freshness – then we proposed splitting our capital across the three portfolios that spanned different three-month rebalance periods (e.g. JAN-APR-JUL-OCT, FEB-MAY-AUG-NOV, MAR-JUN-SEP-DEC).  This solution is referred to either as “tranching” or “overlapping portfolios.”

The paper also derived a formula for estimating timing luck ex-ante, with a simplified representation of:

Where L is the timing luck measure, T is turnover rate of the strategy, F is how many times per year the strategy rebalances, and S is the volatility of a long/short portfolio that captures the difference of what a strategy is currently invested in versus what it could be invested in if the portfolio was reconstructed at that point in time.

Without numbers, this equation still informs some general conclusions:

  • Higher turnover strategies have higher timing luck.
  • Strategies that rebalance more frequently have lower timing luck.
  • Strategies with a less constrained universe will have higher timing luck.

Bullet points 1 and 3 may seem similar but capture subtly different effects.  This is likely best illustrated with two examples on different extremes.  First consider a very high turnover strategy that trades within a universe of highly correlated securities.  Now consider a very low turnover strategy that is either 100% long or 100% short U.S. equities.  In the first case, the highly correlated nature of the universe means that differences in specific holdings may not matter as much, whereas in the second case the perfect inverse correlation means that small portfolio differences lead to meaningfully different performance.

L, in and of itself, is a bit tricky to interpret, but effectively attempts to capture the potential dispersion in performance between a particular rebalance implementation choice (e.g. JAN-APR-JUL-OCT) versus a timing-luck-neutral benchmark.

After half a decade, you’d would think we’ve spilled enough ink on this subject.

But given that just about every single major index still does not address this issue, and since our passion for the subject clearly verges on fever pitch, here comes some more cowbell.

Equity Style Portfolio Definitions

In this note, we will explore timing luck as it applies to four simplified smart beta portfolios based upon holdings of the S&P 500 from 2000-2019:

  • Value: Sort on earnings yield.
  • Momentum: Sort on prior 12-1 month returns.
  • Low Volatility: Sort on realized 12-month volatility.
  • Quality: Sort on average rank-score of ROE, accruals ratio, and leverage ratio.

Quality is a bit more complicated only because the quality factor has far less consistency in accepted definition.  Therefore, we adopted the signals utilized by the S&P 500 Quality Index.

For each of these equity styles, we construct portfolios that vary across two dimensions:

  • Number of Holdings: 50, 100, 150, 200, 250, 300, 350, and 400.
  • Frequency of Rebalance: Quarterly, Semi-Annually, and Annually.

For the different rebalance frequencies, we also generate portfolios that represent each possible rebalance variation of that mix.  For example, Momentum portfolios with 50 stocks that rebalance annually have 12 possible variations: a January rebalance, February rebalance, et cetera.  Similarly, there are 12 possible variations of Momentum portfolios with 100 stocks that rebalance annually.

By explicitly calculating the rebalance date variations of each Style x Holding x Frequency combination, we can construct an overlapping portfolios solution.  To estimate empirical annualized timing luck, we calculate the standard deviation of monthly return dispersion between the different rebalance date variations of the overlapping portfolio solution and annualize the result.

Empirical Timing Luck Results

Before looking at the results plotted below, we would encourage readers to hypothesize as to what they expect to see.  Perhaps not in absolute magnitude, but at least in relative magnitude.

For example, based upon our understanding of the variables affecting timing luck, would we expect an annually rebalanced portfolio to have more or less timing luck than a quarterly rebalanced one?

Should a more concentrated portfolio have more or less timing luck than a less concentrated variation?

Which factor has the greatest risk of exhibiting timing luck?

Source: Sharadar.  Calculations by Newfound Research.

To create a sense of scale across the styles, below we isolate the results for semi-annual rebalancing for each style and plot it.

Source: Sharadar.  Calculations by Newfound Research.

In relative terms, there is no great surprise in these results:

  • More frequent rebalancing limits the risk of portfolios changing significantly between rebalance dates, thereby decreasing the impact of timing luck.
  • More concentrated portfolios exhibit larger timing luck.
  • Faster-moving signals (e.g. momentum) tend to exhibit more timing luck than more stable, slower-moving signals (e.g. low volatility).

What is perhaps the most surprising is the sheer magnitude of timing luck.  Consider that the S&P 500 Enhanced Value, Momentum, Low Volatility, and Quality portfolios all hold 100 securities and are rebalanced semi-annually.  Our study suggests that timing luck for such approaches may be as large as 2.5%, 4.4%, 1.1%, and 2.0% respectively.

But what does that really mean?  Consider the realized performance dispersion of different rebalance date variations of a Momentum portfolio that holds the top 100 securities in equal weight and is rebalanced on a semi-annual basis.

Source: Sharadar.  Calculations by Newfound Research.  Past performance is not an indicator of future results. Performance is backtested and hypothetical. Performance figures are gross of all fees, including, but not limited to, manager fees, transaction costs, and taxes.  Performance assumes the reinvestment of all distributions. 

The 4.4% estimate of annualized timing luck is a measure of dispersion between each underlying variation and the overlapping portfolio solution.  If we isolate two sub-portfolios and calculate rolling 12-month performance dispersion, we can see that the difference can be far larger, as one might exhibit positive timing luck while the other exhibits negative timing luck.  Below we do precisely this for the APR-OCT and MAY-NOV rebalance variations.

Source: Sharadar.  Calculations by Newfound Research.  Past performance is not an indicator of future results. Performance is backtested and hypothetical. Performance figures are gross of all fees, including, but not limited to, manager fees, transaction costs, and taxes.  Performance assumes the reinvestment of all distributions. 

In fact, since these variations are identical in every which way except for the date on which they rebalance, a portfolio that is long the APR-OCT variation and short the MAY-NOV variation would explicitly capture the effects of rebalance timing luck.  If we assume the rebalance timing luck realized by these two portfolios is independent (which our research suggests it is), then the volatility of this long/short is approximately the rebalance timing luck estimated above scaled by the square-root of two.

Derivation: For variations vi and vj and overlapping-portfolio solution V, then:

Thus, if we are comparing two identically-managed 100-stock momentum portfolios that rebalance semi-annually, our 95% confidence interval for performance dispersion due to timing luck is +/- 12.4% (2 x SQRT(2) x 4.4%).

Even for more diversified, lower turnover portfolios, this remains an issue.  Consider a 400-stock low-volatility portfolio that is rebalanced quarterly.  Empirical timing luck is still 0.5%, suggesting a 95% confidence interval of 1.4%.

S&P 500 Style Index Examples

One critique of the above analysis is that it is purely hypothetical: the portfolios studied above aren’t really those offered in the market today.

We will take our analysis one step further and replicate (to the best of our ability) the S&P 500 Enhanced Value, Momentum, Low Volatility, and Quality indices.  We then created different rebalance schedule variations.  Note that the S&P 500 Low Volatility index rebalances quarterly, so there are only three possible rebalance variations to compute.

Source: Sharadar.  Calculations by Newfound Research.  Past performance is not an indicator of future results. Performance is backtested and hypothetical. Performance figures are gross of all fees, including, but not limited to, manager fees, transaction costs, and taxes.  Performance assumes the reinvestment of all distributions. 

We see a meaningful dispersion in terminal wealth levels, even for the S&P 500 Low Volatility index, which appears at first glance in the graph to have little impact from timing luck.

Minimum Terminal Wealth

Maximum Terminal Wealth

Enhanced Value

$4.45

$5.45

Momentum

$3.07

$4.99

Low Volatility

$6.16

$6.41

Quality

$4.19

$5.25

 

We should further note that there does not appear to be one set of rebalance dates that does significantly better than the others.  For Value, FEB-AUG looks best while JUN-DEC looks the worst; for Momentum it’s almost precisely the opposite.

Furthermore, we can see that even seemingly closely related rebalances can have significant dispersion: consider MAY-NOV and JUN-DEC for Momentum. Here is a real doozy of a statistic: at one point, the MAY-NOV implementation for Momentum is down -50.3% while the JUN-DEC variation is down just -13.8%.

These differences are even more evident if we plot the annual returns for each strategy’s rebalance variations.   Note, in particular, the extreme differences in Value in 2009, Momentum in 2017, and Quality in 2003.

Source: Sharadar.  Calculations by Newfound Research.  Past performance is not an indicator of future results. Performance is backtested and hypothetical. Performance figures are gross of all fees, including, but not limited to, manager fees, transaction costs, and taxes.  Performance assumes the reinvestment of all distributions. 

Conclusion

In this study, we have explored the impact of rebalance timing luck on the results of smart beta / equity style portfolios.

We empirically tested this impact by designing a variety of portfolio specifications for four different equity styles (Value, Momentum, Low Volatility, and Quality).  The specifications varied by concentration as well as rebalance frequency.  We then constructed all possible rebalance variations of each specification to calculate the realized impact of rebalance timing luck over the test period (2000-2019).

In line with our mathematical model, we generally find that those strategies with higher turnover have higher timing luck and those that rebalance more frequently have less timing luck.

The sheer magnitude of timing luck, however, may come as a surprise to many.  For reasonably concentrated portfolios (100 stocks) with semi-annual rebalance frequencies (common in many index definitions), annual timing luck ranged from 1-to-4%, which translated to a 95% confidence interval in annual performance dispersion of about +/-1.5% to +/-12.5%.

The sheer magnitude of timing luck calls into question our ability to draw meaningful relative performance conclusions between two strategies.

We then explored more concrete examples, replicating the S&P 500 Enhanced Value, Momentum, Low Volatility, and Quality indices.  In line with expectations, we find that Momentum (a high turnover strategy) exhibits significantly higher realized timing luck than a lower turnover strategy rebalanced more frequently (i.e. Low Volatility).

For these four indices, the amount of rebalance timing luck leads to a staggering level of dispersion in realized terminal wealth.

“But Corey,” you say, “this only has to do with systematic factor managers, right?”

Consider that most of the major equity style benchmarks are managed with annual or semi-annual rebalance schedules.  Good luck to anyone trying to identify manager skill when your benchmark might be realizing hundreds of basis points of positive or negative performance luck a year.

 

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