Hot off the press, a new paper by Guido Baltussen, Laurens Swinkels and Pim van Vliet at Dutch quant powerhouse, Robeco, covers global multi-asset factor premiums over an unprecedented sample of 217 years. We thought the topics and findings were important and timely enough to warrant a summary.

The new paper, titled “Global Factor Premiums” examines global equity indexes, 10-year government bond indexes, commodities and currency markets to understand how well the most pervasive, persistent, economically significant and investable style premia hold up on a very long out-of-sample dataset. Specifically, the authors study global multi-asset trend, momentum, value, carry, seasonality, and betting-against-beta (BAB) premia on monthly data back to 1799.

The authors focus exclusively on multi-asset factors since several authors have already published extensive research on factor persistence in individual securities. For example, Golez and Koudijs examined stock and bond returns back to 1629; Goetzmann and Huang analyzed stock momentum in Imperial Russia from 1865-1914; and Geczy and Samanov (2013) showed “Two Centuries of Price Return Momentum” in U.S. securities.

The novel contributions from this paper pertain to the following:

- Replication of seminal studies dealing with multi-asset premia over the period 1981 – 2011 using a uniform testing methodology to mitigate the potential impact of p-hacking from different implementation methodologies.
- The introduction of more rigorous statistical tests based on the Bayesian perspectives on p-values advocated by Cam Harvey.
- An application of uniform testing methodologies to examine factor premia over a very long out-of-sample dataset including the period 1799 – 1980 and 2012-2016.

Within each asset class the authors construct factor portfolios using the following uniform methodologies:

- Trend and momentum are defined as the 12-month-minus-1-month excess return. While this is consistent with traditional measures of cross-sectional momentum, “trend” as proposed by Moskowitz, Ooi and Pedersen; and Hurst, Ooi and Pedersen does not impose a skip month. The current authors make the case that a skip month is more conservative because it allows for a delay in data diffusion in very early periods.
- Value is defined as dividend yield for equity indices; real yield for bonds; 5-year reversal in spot prices for commodities; and absolute and relative purchasing power parity for currencies.
- Carry is simply the implied ex ante yield on each instrument.
- Return seasonality is computed as the mean excess return produced by a market in a certain month over the prior 20-year period.
- Betting-against-Beta is long the low-beta assets and short the high beta assets with positions neutralized for ex ante beta relative to the global asset class portfolio return.

For the purpose of the uniform tests, portfolios were formed at the end of each month. The trend portfolio takes long positions in markets where the trend is positive and vice versa. All other tests are cross-sectional, where the authors rank markets in each universe based on the factor measure and take a position equal to the rank minus its cross-sectional average. Note that this procedure is distinct from the methodology applied in many academic papers, which sort on target characteristics and take market-cap or equal-weight positions in securities that exceed certain quantiles in either direction. The authors note that their results hold in general for this alternative approach.

The authors find Sharpe ratios averaged 0.41 with uniform specification in the replication study. Half of the factor premiums were significant at the traditional 5% threshold, while 1/3 of the strategies were significant at the stricter 1% threshold. The multi-asset versions produced Sharpe ratios between 0.39 (BAB) and 1.15 (Carry).

Figure 1 below illustrates the Sharpe ratios observed from tests in the original papers (Panel A) versus results from the uniform replication tests (Panel B) performed by the authors. The grey dashed line shows a traditional α threshold of 5% (t-stat of 1.96) and the black dashed line shows a more conservative α threshold of 1% (t-stat of 3). Numbers above the bars represent Bayesian p-values using a 4:1 prior odds ratio, consistent with a threshold Cam Harvey classifies as “perhaps” sufficient to address p-hacking concerns^{1}. From our perspective, the Bayesian p-values are almost absurdly conservative, especially since many of the factors under discussion were documented long before the introduction of modern computing capabilities.

1 Bayesian p-value = -exp(1) x p-value x ln(p-value) x prior odds / (1+ (-exp(1) x p-value x ln(p-value) ) x prior odds)

Figure 1. Global factor returns: modern period

Panel A: Original documentation

Panel B: Replicating factors 1981-2011

Source: Baltussen, Swinkels and van Vliet (2019) “Global Factor Premiums”

From Figure 1 it’s clear that trend and carry are dominant factors in both the original and the replication samples, with statistical significance in excess of conservative thresholds in all asset categories. The uniform multi-asset versions of trend and carry produced very impressive Sharpe ratios of 1.09 and 1.15, respectively, with t-stats above 6. In addition, equity indexes and commodities exhibited strong momentum effects in both the original and replication samples.

Interestingly, the current authors found that value, seasonality and BAB showed only marginally significant effects in the replicating sample, even against the more tolerant frequentist thresholds (traditional p-values of 0.06, 0.05 and 0.04 respectively). The effects fell well below more conservative thresholds with respective Bayesian p-values of 0.64, 0.6 and 0.58. More troubling, these three well-known factors also failed to show Bayesian significance at the multi-asset level, even after accounting for the benefits of diversification between same factor returns across asset class categories. However, the authors did observe that:

The main purpose of this paper is to provide more robust and rigorous long-term evidence of the historical presence of global return factors,

utilizing their most simple or basic definitions as put forward in influential papers analyzing recent samples. In this light,this study does not examine smarter and possibly better definitions…

After discussing the results of the replication study over the modern period the authors applied the same factor analyses to their novel long-term dataset. New markets were introduced as data became available. The sample consisted of 13 global markets in 1800, increasing to 18 markets in 1822; 36 markets by 1870; 50 markets by 1914; and 66 markets by 1974. In 1999 the total number of markets declines from 68 to 63 because of the introduction of the Euro currency. The authors employed a number of methods to screen for bad data or outliers consistent with other similar studies.

Out-of-sample results were presented independent of the replicating sample (1800-1980 and 2012-2016) and also over the entire period from 1800-2016. Interestingly, the authors found an average Sharpe ratio across factors of 0.41 in the out-of-sample period, exactly consistent with what they found in the replicating sample from 1980-2012. However, t-values were much larger because of the larger sample, so that 19 of 24 combinations of factors and asset categories produced t-values above 3.

In addition to confirming the strong economic significance of trend and carry, the larger sample surfaced a highly significant seasonality effect. Return seasonality in government bonds and currencies was especially strong in the new data. They observed statistically significant momentum and value effects for three out of four assets. Momentum in commodities and value in currencies were notable exceptions, though the sample for commodity value only extended to 1968. Results for BAB were insignificant for all but the equity index category.

Figure 2. Statistical perspectives on global return factors, 1800-2016

Panel B: 1800-2016

Trend | Momentum | Value | Carry | Seasonality | BAB | ||

Equities | p-value | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |

Bayesian-p | 0.00 | 0.00 | 0.01 | 0.00 | 0.00 | 0.00 | |

BE-odds | >9,999 | >9,999 | 13.18 | >9,999 | >9,999 | >9,999 | |

Bonds | p-value | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.66 |

Bayesian-p | 0.00 | 0.00 | 0.01 | 0.00 | 0.00 | 0.75 | |

BE-odds | >9,999 | 1,040.34 | 16.36 | >9,999 | >9,999 | 0.06 | |

Commodities | p-value | 0.00 | 0.48 | 0.00 | 0.00 | 0.00 | 0.48 |

Bayesian-p | 0.00 | 0.79 | 0.00 | 0.05 | 0.00 | 0.79 | |

BE-odds | 4,424.81 | 0.05 | >9,999 | 3.78 | >9,999 | 0.05 | |

FX | p-value | 0.00 | 0.00 | 0.28 | 0.00 | 0.00 | 1.00 |

Bayesian-p | 0.00 | 0.00 | 0.79 | 0.00 | 0.00 | 1.00 | |

BE-odds | >9,999 | 1,090.15 | 0.05 | >9,999 | 79.16 | 0.00 | |

Multi Asset | p-value | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.01 |

Bayesian-p | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.23 | |

BE-odds | >9,999 | >9,999 | >9,999 | >9,999 | >9,999 | 0.63 |

Source: Baltussen, Swinkels and van Vliet (2019) “Global Factor Premiums”

The authors were aware that factor specification can play a material role in results, even over long sample horizons. They performed tests on a variety of methodological variations to test for robustness. For example, they removed the liquidity screen (increased Sharpe ratios); formed equal-weight tertile portfolios (no change); eliminated volatility scaling (over-weighted high volatility instruments and lowered aggregate Sharpe ratios); lagged signals by one month (small decay in most strategies but completely eliminated seasonality effect by construction); rebalanced quarterly (small decay in most strategies except value); and trimmed extreme returns (slight increase in performance). Overall the authors concluded that results were robust to alternative specifications.

Investors may be interested in whether the factors under consideration represented compensation for well-known economic or financial risk factors. The advantage of long-term data series is that it is easier to identify explanations that relate to relatively infrequent observations such as downside or macroeconomic risks.

The authors presented several comprehensive analyses to address common factor variation and sensitivity to known risks. They found low correlations across factors, and that most individual correlation coefficients between multi-asset factor series and each factor-asset-class series are also close to zero. The authors conclude that the 24 return factors are unique drivers of returns and share little common variation. This stands in contrast to other studies, such as Asness, Moskowitz and Pedersen (2013) “Value and Momentum Everywhere” which found that global value and momentum effects across assets and securities showed significant common covariance effects.

In spanning tests, the authors confirm the findings of Moskowitz, Ooi and Pedersen that trend returns subsume momentum returns. In other words, cross-sectional momentum effects are insignificant when controlling for trend effects. We would note that in a July 2017 paper “Cross-Sectional and Time-Series Tests of Return Predictability: What Is the Difference?” Goyal and Jegadeesh show that trend and momentum are not distinct sources of return, and that differences in returns stem from the time-varying net long exposure to risky assets invoked by trend strategies.

The new dataset provides a relatively large sample of 43 equity bear markets and 218 downside market states (where returns are below -1 standard deviation from the average). In contrast to other studies, the authors find that downside risk explains at best a part of the global factor returns, most notably carry. When beta is replaced by downside beta in the Fama and MacBeth regressions the authors find an insignificant cross-sectional risk premium of 0.05 percent (t-value = 0.26). As a result, the authors conclude that the factor premia are not explained by downside risk in the long-term sample.

The authors also calculate the contemporaneous annual factor returns for “good” and “bad” market states, and the return difference between the states, and perform several other regressions against common macroeconomic variables. They conclude, “In summary, our tests reveal very limited evidence of a link between macroeconomic risk and global return factors.”

The paper “Global Factor Premiums” analyzes well known return anomalies by employing long-term data not previously considered in the literature. They replicate seminal studies with a uniform methodology and introduce robust statistical tests that are resilient to p-hacking.

The authors find that trend and carry factors dominate in the replication studies with multi-asset Sharpe ratios of 1.09 and 1.15 respectively, exceeding even the strictest significance thresholds. Other factors exhibit inconsistent results.

In the extended dataset the authors observed highly significant results from trend, carry and seasonality premiums, bolstered by large sample sizes. Value was significant for most asset categories excepting currencies, and BAB produced significant results only for equity indexes.

Surprisingly, the authors did not surface a meaningful relationship between return premiums and major risk factors. The anomalies, while exhibiting extremely significant, pervasive, and persistent results across centuries of data, are still largely unexplained by contemporary theories of risk.

Finally, given that the premia have produced strong returns with low sensitivity to markets and traditional risk factors, allocations to risk factors have the potential to substantially expand the efficient frontier, and add value to most portfolios.

Welcome to ReSolve Asset Management’s 12 days of investment wisdom mini-series where we explore, from first principles, timeless investment wisdom that will help you maximize your long-term success and possibly change the way you approach the complex arena of investing altogether. From universe selection to portfolio construction, our aim is to offer you a comprehensive framework for a more thoughtful investment approach, to benefit yourself and your clients.

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**The Most Crucial First Question: Asset-Allocation or Security Selection?**

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**What True Diversification Really Is and How to Maximize it.**

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**The Two Fundamental Drivers that Determine All Economic Regimes **

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**Pushing the Diversification Frontier with True Factor Investing**

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**The Impact of Sequence of Returns Risk and How to Minimize it**

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**Factor Investing and the Pitfalls of Poor Strategy Construction**

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**How to think about your Alternative Sleeve in the Context of Getting the Most Bang for your Buck**

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**Why Financial Professionals and their Clients Need to Get Comfortable with Being Uncomfortable in the Coming Decade**

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**What It Means to be a True Systematic Manager and How to Spot the Lemon’s in your Lineup.**

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**How Thoughtful Portfolio Optimization Techniques can be a Total Game Changer for Portfolio Results**

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**How to Juice as Much Value as Possible from Trend Following – The Cheapest Factor Out There Today!**

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**Putting 11 days of Wisdom to work through a Multi-Asset Momentum Case Study**

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We’ve spent a great deal of time in past articles discussing the merits of portfolio optimization. In this article we will examine the merits and challenges of portfolio optimization in the context of one of the most challenging investment universes: Managed Futures.

Futures exhibit several features that make them challenging from a portfolio optimization perspective. In particular, there can be mathematical issues with large correlation matrices, and certain futures markets may exhibit high correlation in certain periods. However, for practitioners that are willing to make the effort, the extreme diversity offered by futures markets represents a lucrative opportunity to improve results through portfolio optimization.

Why are we convinced that diversity produces opportunity? We are motivated by the Fundamental Law of Active Management described by (Grinold 1989), which states that the risk-adjusted performance of a strategy is a mathematical function of skill and the square-root of breadth.

where IR is information ratio, IC is information coefficient and breadth is the number of *independent* bets placed by the manager. For our purpose we can substitute Sharpe ratio for information coefficient because we are focused on absolute performance, not performance relative to a benchmark. Information coefficient quantifies skill by measuring the correlation between a strategy’s signals and subsequent results.

Breadth is a more nebulous concept. Grinold described breadth as the number of securities times the number of trades. However, (Polakow and Gebbie 2006) raise the issue that, “The square root of N in mathematical statistics implies ‘independence’ amongst statistical units (here bets) rather than simply the notion of ‘separate bets’ as is most often implied” in the finance literature.

It is therefore insufficient to simply add more securities in an effort to increase breadth and expand one’s Sharpe ratio. Rather, investors must account for the fact that correlated securities are, by definition, not independent. This prompts questions about how to quantify breadth – the number of independent sources of risk or “bets” – in the presence of correlations.

All things equal investors seeking to improve results should seek to maximize the breadth that is available to them. We will show that traditional methods do a relatively poor job of maximizing breadth and diversification, and that a portfolio’s maximum potential can usually only be reached through optimization.

It’s illustrative to examine the number of independent bets that are expressed when portfolios are formed using traditional versus more advanced optimization methods. We will quantify the number of independent bets by taking the square of the Diversification Ratio of the portfolio.

(Choueifaty and Coignard 2008) showed that the Diversification Ratio of a portfolio is the ratio of the weighted sum of asset volatilities to the portfolio volatility after accounting for diversification.

This is intuitive because if all of the assets in the portfolio are correlated, the weighted sum of their volatilities would equal the portfolio volatility, and the Diversification Ratio would be 1. As the assets become less correlated, the portfolio volatility will decline due to diversification, while the weighted sum of constituent volatilities will remain the same, causing the ratio to rise. At the point where all assets are uncorrelated (zero pairwise correlations), every asset in the portfolio represents an independent source of risk.

(Choueifaty, Froidure, and Reynier 2012) demonstrate that the number of independent risk factors expressed in a portfolio is equal to the square of the Diversification Ratio of the portfolio. Thus, we can find the number of independent risk factors in a portfolio as a function of the weights in each asset and the asset covariances, which allow us to calculate the portfolio volatility.

There are many ways to form futures portfolios. Futures are defined by their underlying exposures, which can range from extremely low volatility instruments like Japanese government bonds (JGB) to very high volatility commodities like natural gas. With such large differences in risk between futures contracts, few managers would choose to hold contracts in equal weight.

Perhaps the most common portfolio formation method among futures managers is to weight assets by the inverse of their volatility subject to a target risk. Contracts with low volatility would receive a larger capital allocation and vice versa. Exposure to each asset is calculated in the following way:

*w*_{i} = *σ*_{T}/*σ*_{i}

where *w*_{i} is the portfolio’s weight in market *i*, *σ*_{T} is the target volatility and *σ*_{i} is the estimated volatility of market *i*.

Consider a portfolio of JGB futures with an annualized volatility of 4% and natural gas futures with an annualized volatility of 50%. A manager targeting 10% annualized volatility from each instruments would hold 10%/4% = 250% of portfolio exposure in JGBs and 10%/50% = 20% in natural gas.

When we apply this equal volatility weighting methodology to a diversified universe of 47 futures markets across equities, fixed income, commodities and currencies we produce 6.85 independent sources of return.

The popular inverse volatility weighting method described above has the goal of dividing risk equally among diverse futures markets. Unfortunately, inverse volatility weighted portfolios are effective at diversifying risk *only* when all pairwise correlations between markets are equal (learn why in our seminal whitepaper, The Portfolio Optimization Machine: A General Framework for Portfolio Choice).

(Baltas 2015) observed that inverse volatility weighted portfolios are susceptible to major shifts in portfolio concentration as correlations change through time. This issue has become more important in the policy driven markets subsequent to the 2008 financial crisis. Figure 1 illustrates how pairwise correlations between futures markets shifted higher over the past decade.

Figure 1: **Rolling average annual daily pairwise correlations across 47 futures markets.**

*Source: Analysis by ReSolve Asset Management. Data from CSI.*

It is rarely the case that markets exhibit homogeneous correlations. Rather, segments of futures markets such as certain equity markets and some relatively fungible commodity markets like WTI and Brent crude tend to have high correlations while other markets have low or even negative correlations. Figure 2 illustrates just how diverse pairwise correlations can be across futures markets. On average the most correlated markets have had correlations of 0.56 while the least correlated markets have had correlations of -0.26.

Figure 2: **Rolling 95th and 5th percentile pairwise correlations across 47 futures markets.**

*Source: Analysis by ReSolve Asset Management. Data from CSI.*

Given that correlations exhibit such large dispersion, inverse volatility weighted portfolios will fail to produce optimally diversified portfolios. Methods for forming portfolios of futures that account for correlations require optimization^{1}.

The objective of portfolio optimization is to maximize the opportunity for diversification (i.e. maximize breadth) when correlations vary over a wide range. Some assets in our futures universe have strong positive correlations while others are negatively correlated. For example, over the past year Eurostoxx and DAX have experienced a correlation of 0.93 while British Pound and Gilt futures have experienced a correlation of -0.44. All things equal, assets with low or negative correlations relative to most other assets should earn a larger weight in portfolios.

There are several optimization methods to choose from. (Baltas 2015) proposed using a risk parity optimization, where all assets contribute the same target risk to the portfolio *after accounting for diversification*. The weights for the risk parity portfolio can be found using several methods. The following method was formulated by (Spinu 2013):

The advantage of risk parity optimization is that all assets with non-zero expected returns will earn non-zero weights. Thus, the portfolio will resemble what might be produced by a traditional inverse volatility weighted approach, and will be more inuitive in constitution.

However, the risk parity portfolio will not maximize portfolio diversification, and will not explicitly maximize the expected return of the portfolio with minimal risk. A risk parity weighted portfolio of 47 futures markets would produce 10.25 independent sources of return. This is a non-trivial improvement over the traditional method’s 6.85 bets.

As mentioned above, (Choueifaty, Froidure, and Reynier 2012) described why the square of the portfolio’s Diversification Ratio quantifies the number of independent bets available in a portfolio. We can solve for the portfolio weights that maximize the Diversification Ratio – and thus portfolio breadth – using a form of mean-variance optimization of the following form:

where *σ* and *Σ* reference a vector of volatilities, and the covariance matrix, respectively.

The optimization maximizes the ratio of weighted-average asset volatilities to portfolio volatility after accounting for diversification. When we solve for the most diversified portfolio of futures using this method we give rise to 13.01 independent sources of return.

Figure 3: **Number of independent bets from futures portfolios formed using different methods. Simulated results.**

*Source: Analysis by ReSolve Asset Management. Data from CSI. Simulated results.*

Let’s take a moment to understand the importance of the results in Figure 3. Greater breadth, in the form of independent bets, is a force-multiplier on Sharpe ratios.

Recall from our discussion of the Fundamental Law of Active Management above that expected Sharpe ratio is a function of . As such portfolios formed using the risk parity method should produce Sharpe ratios times higher than the Sharpe ratio of inverse volatility weighted portfolios. And mean-variance optimized portfolios can produce Sharpe ratios times higher. We will call this quantity the “Sharpe multiplier” *M*^{*}.

To put this in perspective, if traditional diversified managed futures strategies have Sharpe ratios of 1, moving from traditional formation methods to optimization-based methods could boost Sharpe ratios to 1.38. At 10% target volatility this boosts a 10% expected annualized excess return strategy to a 13.8% expected return. Over ten years, a $1 million portfolio would be expected to grow to $2.59million using the traditional portfolio inverse volatility weighting method, but it might grow to $3.64 million if portfolios were constructed to maximize diversification. That’s an extra $1million of wealth from applying exactly the same method to select securities, but more thoughtful portfolio construction.

Up to this point we have been calculating independent bets based on long-term average pairwise-complete correlations. Each correlation element is calculated based on the returns for each pair of futures since the inception of the shortest running futures contract.

Of course, not all futures contracts have data all the way back to 1988 and correlations are not stable through time. It is more useful to examine the true breadth – measured as number of independent bets – at each period based on point-in-time correlation estimates. Figure 4 tracks the number of independent bets produced by the three example portfolio formation methods at the end of each calendar year from 1988 through July 2018, derived from trailing 252-day (1-year) correlations.

Figure 4: **Rolling number of independent bets produced by different portfolio formation methods, smoothed by trailing 252-day average. Simulated results.**

*Source: Analysis by ReSolve Asset Management. Data from CSI. Simulated results.*

The correlation structure of our futures universe, and commensurate breadth, has fluctuated materially over the past thirty years. Breadth historically contracts when markets enter crisis periods, and expands when markets are functioning normally. In all cases however, portfolios that are optimized to maximize diversification produce considerably more breadth than traditional portfolios that ignore correlations altogether.

Notice that portfolios appear to produce greater breadth on average when they are regularly reconstituted to reflect point-in-time correlations. This is partly because the long-term average smoothes away the many different economic environments experienced by markets over the past three decades, each which produced its own diverse correlation structure. However, short-term sample correlation matrices also typically understate “true”” correlations for mathematical reasons, which are beyond the scope of this discussion. In practice, one should make adjustments to sample correlation matrices to account for sample biases^{2}.

Across rolling annual periods, traditional inverse volatility weighted futures portfolios produced an average of 10.28 independent bets, while risk parity and optimization weighted portfolios produced an average of 16.02 and 21.91 bets respectively.

Grinold’s Fundamental Law of Active Management implies that equally skilled managers should seek to maximize breadth to boost expected performance.

Managers of futures portfolios have traditionally employed naive portfolio formation methods that ignore information about correlations. These methods, which weight markets in proportion to the inverse of volatility, are optimally diversified only if markets all have equal pairwise correlations.

Figure 2 clearly shows that correlations deviate substantially from the assumption of equality over all historical periods. As a result, traditional portfolio formation methods will render overly concentrated portfolios most of the time.

Some authors have recently proposed risk parity weighting as a solution to the problem of diverse correlations. Risk parity seeks to form portfolios such that markets contribute equal volatility after accounting for diversification. We show that risk parity produces greater breadth than traditional methods on our futures universe. However, risk parity leaves a material amount of breadth on the table.

Portfolio optimization is the only way to extract the maximum amount of breadth when markets have diverse correlations. We show that optimization produces greater breadth than both traditional methods and risk parity at every time step over the past thirty years.

The Sharpe Multiplier quantifies the expected boost to strategy performance as a result of higher breadth. Figure 5 demonstrates that optimization-based methods provide a consistent boost to expected performance. On average, optimized portfolios may exceed the performance of portfolios constructed using traditional means by up to 50%.

Our analysis prompts at least one obvious question. If optimization has such great potential to improve performance, why do most futures managers avoid it? We’ll answer it in the next article in this series. To get the most out of the articles in this series, read The Portfolio Optimization Machine: A General Framework for Portfolio Choice.

Baltas, Nick. 2015. “Trend-Following, Risk-Parity and the Influence of Correlations.” https://ssrn.com/abstract=2673124.

Bun, Joël, Jean-Philippe Bouchaud, and Marc Potters. 2016. “Cleaning large correlation matrices: tools from random matrix theory.” https://arxiv.org/abs/1610.08104.

Choueifaty, Yves, and Yves Coignard. 2008. “Toward Maximum Diversification.” *Journal of Portfolio Management* 35 (1). http://www.tobam.fr/inc/uploads/2014/12/TOBAM-JoPM-Maximum-Div-2008.pdf: 40–51.

Choueifaty, Yves, Tristan Froidure, and Julien Reynier. 2012. “Properties of the Most Diversified Portfolio.” *Journal of Investment Strategies* 2 (2). http://www.qminitiative.org/UserFiles/files/FroidureSSRN-id1895459.pdf: 49–70.

Grinold, Richard. 1989. “The Fundamental Law of Active Management.” 15 (3). http://jpm.iijournals.com/content/15/3/30: 30–37.

Polakow, Daniel, and Tim Gebbie. 2006. “How many independent bets are there?” https://arxiv.org/pdf/physics/0601166v1.pdf.

Spinu, Florin. 2013. “An Algorithm for Computing Risk Parity Weights.” *SSRN*. https://ssrn.com/abstract=2297383.

- Given that correlations exhibit such large dispersion, inverse volatility weighted portfolios will fail to produce optimally diversified portfolios. Methods for forming portfolios of futures that account for correlations require optimization^[Many futures managers use heuristic methods to help enhance diversification. For example, they may form sub-portfolios of equity markets, fixed income, currencies, and commodities. Other managers divide commodities into sub-sectors like “energies”, “grains”, “metals” or “softs”. Markets within a sector are weighted by inverse volatility, and then sector portfolios are weighted in the final portfolio by inverse volatility. This may enhance diversification properties somewhat, but pairwise correlations can vary substantially within and across sectors, suggesting this approach provides only marginal benefits relative to naive inverse volatility weighting.↩
- See (Bun, Bouchaud, and Potters 2016) for an overview of biases in sample correlation matrices.↩

Our whitepaper “The Optimization Machine: A General Framework for Portfolio Choice” presented a logical framework for thinking about portfolio optimization given specific assumptions regarding expected relationships between risk and return. We explored the fundamental roots of common portfolio weighting mechanisms, such as market cap and equal weighting, and discussed the rationale for several risk-based optimizations, including Minimum Variance, Maximum Diversification, and Risk Parity.

For each approach to portfolio choice we examined the conditions that would render the choice mean-variance optimal. For example, market cap weighting is mean-variance optimal if returns are completely explained by CAPM beta, or in other words, if all investments have the same expected Treynor ratios. Minimum variance weighted portfolios are optimal if all investments have the same expected return, while Maximum Diversification weighted portfolios are optimal if investments have the same Sharpe ratios.

The Portfolio Optimization Machine framework prompts questions about how well academic theories about the relationships between risk and return explain what we observe in real life. While academics would have investors believe investments that exhibit higher risk should produce higher returns, we do not observe this relationship universally.

For instance, we show that both the Security Market Line, which expresses a relationship between return and stock beta, and the Capital Market Line, which plots returns against volatility, are either flat or *inverted* for both U.S. and international stocks over the historical sample. In other words, stock returns are either independent of, or inversely related to risk.

We also examined the returns to major asset classes, including global stocks, bonds, and commodities. For asset classes, there appears to be a positive relationship between risk and return, at least when returns are analyzed across different macroeconomic regimes. Normalized for inflation and growth environments, stocks and bonds appear to have equal Sharpe ratios in the historical sample.

The Sharpe ratio of diversified commodities has been about half of the Sharpe ratio observed for stocks and bonds since 1970 when conditioned on regime. However, we highlight that our analysis may produce bias against commodities, given that there were few regimes that would have been favorable to commodities in our historical sample. With such a small sample size, we believe it is premature to reject the hypothesis that commodity risk should be compensated at the same rate as risk from stocks and bonds.

Our whitepaper presented a great deal of theory, and offered guidance from history about the nature of the relationship between risk and return. Armed with this guidance, we can invoke the Optimization Machine decision tree to make an educated guess about optimal portfolio choice for different investment universes.

We show that the Optimization Machine is a helpful guide for optimal portfolio formation, but that the relative opportunity for optimal versus naive methods depends on size of the diversification opportunity relative to the number of assets in the investment universe. We instantiate a new term, the “Quality Ratio” to measure this quantity for any investment universe^{1}.

In this article we put the Optimization Machine framework to the test. Specifically, we make predictions using the Optimization Machine about which portfolio methods are theoretically optimal based on what we’ve learned about observed historical relationships between risk and return. Then we test these predictions by running simulations on several datasets.

We model our investigation on a well-known paper by (DeMiguel, Garlappi, and Uppal 2007) titled “Optimal Versus Naive Diversification: How Inefficient is the 1/N Portfolio Strategy?”, which discussed some of the major technical issues that complicate the use of portfolio optimization in practice. The authors also present the results of empirical tests of various portfolio optimization methods on several datasets to compare the performance of optimal versus naive approaches.

(DeMiguel, Garlappi, and Uppal 2007) run simulations on all-equity investment universes. To provide what may be more practical insights, we also run simulations on a universe of global asset classes that derive their returns from diverse risk sources, such as regional equity indexes, global bonds, and commodities.

Specifically, we evaluate the performance of naive versus optimized portfolios on the following data sets, which are all available at daily scale:

- 10 U.S. market-cap weighted industry portfolios from the Ken French data library
- 25 U.S. market-cap weighted equity factor portfolios sorted on size and book-to-market (i.e. value) from the Ken French data library
- 38 U.S. market-cap weighted sub-industry portfolios from the Ken French data library
- 49 U.S. market-cap weighted sub-industry portfolios from the Ken French data library
- 12 global asset classes from multiple sources
^{2}

We form portfolios at the end of each quarter, with a one day delay between calculating optimal portfolio weights and trading.

(DeMiguel, Garlappi, and Uppal 2007) tested a variety of portfolio formation methods including long-short and long-only versions of mean-variance and Minimum Variance optimizations. They also tested different types of shrinkage methods to manage estimation error.

We will follow a similar process, but we will impose long-only, sum-to-one constraints for all optimizations, and use rolling 252 day (i.e. one trading year) sample covariances without any shrinkage methods. The long-only constraint is in recognition of the fact that practitioners are aware of the instability of unconstrained optimization. We will address shrinkage methods in a later article when we discuss more robust optimization methods.

For our simulations, we will compare the performance of naive (equal weighted and market capitalization weighted) methods to portfolios formed using the following optimizations, all of which are long-only constrained (*w* > 0), with weights that sum to 1 ().

Note that all but one of the optimization descriptions below were described in our whitepaper on portfolio optimization, and are repeated here for convenience only. If you are familiar with the specifications and optimality equivalence conditions for these optimizations from the whitepaper you are encouraged to skip ahead to the description of the Hierarchical Minimum Variance optimization.

If all investments have the same expected return independent of risk, investors seeking maximum returns for minimum risk should concentrate exclusively on minimizing risk. This is the explicit objective of the minimum variance portfolio.

where *Σ* is the covariance matrix.

(Haugen and Baker 1991) proposed dispensing with any relationship between risk and return, at least for equities. Their paper was one of the first to demonstrate that stock returns are not well explained by beta. In fact, they observed a negative relationship between returns and volatility.

In the face of a spurious link between risk and return, (Haugen and Baker 1991) suggested that a regularly reconstituted long-only Minimum Variance portfolio might dominate the captitalization weighted portfolio for stocks.

(Choueifaty and Coignard 2008) proposed that markets are risk-efficient, such that investments will produce returns in proportion to their *total risk*, as measured by volatility. This differs from CAPM, which assumes returns are proportional to non-diversifiable (i.e. systematic) risk. Choueifaty et al. described their method as Maximum Diversification (Maximum Diversification), for reasons that will become clear below.

Consistent with the view that returns are directly proportional to volatility, the Maximum Diversification optimization substitutes asset volatilities for returns in a maximum Sharpe ratio optimization, taking the following form.

where *σ* and *Σ* reference a vector of volatilities, and the covariance matrix, respectively.

Note that the optimization seeks to maximize the ratio of the weighted average volatility of the portfolio’s constituents to total portfolio volatility. This is analagous to maximizing the weighted average return, when return is directly proportional to volatility.

An interesting implication, explored at length in a follow-on paper by (Choueifaty, Froidure, and Reynier 2012) is that the ratio maximized in the optimization function quantifies the amount of diversification in the portfolio. This is quite intuitive.

The volatility of a portfolio of perfectly correlated investments would be equal to the weighted sum of the volatilities of its constituents, because there is no opportunity for diversification. When assets are *im*perfectly correlated, the weighted average volatility becomes larger than the portfolio volatility in proportion to the amount of diversification that is available.

The Diversification Ratio, which is to be maximized, quantifies the degree to which the portfolio risk can be minimized through strategic placement of weights on diversifying (imperfectly correlated) assets.

Maximum Decorrelation described by (Christoffersen et al. 2010) is closely related to Minimum Variance and Maximum Diversification, but applies to the case where an investor believes all assets have similar returns and volatility, but heterogeneous correlations. It is a Minimum Variance optimization that is performed on the correlation matrix rather than the covariance matrix. W

Interestingly, when the weights derived from the Maximum Decorrelation optimization are divided through by their respective volatilities and re-standardized so they sum to 1, we retrieve the Maximum Diversification weights. Thus, the portfolio weights that maximize decorrelation will also maximize the Diversification Ratio when all assets have equal volatility and maximize the Sharpe ratio when all assets have equal risks and returns.

The Maximum Decorrelation portfolio is found by solving for:

where A is the correlation matrix.

Both the Minimum Variance and Maximum Diversification portfolios are mean-variance efficient under intuitive assumptions. Minimum Variance is efficient if assets have similar returns while Maximum Diversification is efficient if assets have similar Sharpe ratios. However, both methods have the drawback that they can be quite concentrated in a small number of assets. For example, the Minimum Variance portfolio will place disproportionate weight in the lowest volatility asset while the Maximum Diversification portfolio will concentrate in assets with high volatility and low covariance with the market. In fact, these optimizations may result in portfolios that hold just a small fraction of all available assets.

There are situations where this may not be preferable. Concentrated portfolios also may not accommodate large amounts of capital without high market impact costs. In addition, concentrated portfolios are more susceptible to mis-estimation of volatilities or correlations.

These issues prompted a search for heuristic optimizations that meet similar optimization objectives, but with less concentration. The equal weight and capitalization weight portfolios are common examples of this, but there are other methods that are compelling under different assumptions.

When investments have similar expected Sharpe ratios, and an investor cannot reliably estimate correlations (or we can assume correlations are homogeneous), the optimal portfolio would be weighted in proportion to the inverse of the assets’ volatilities. When investments have similar expected returns (independent of volatility) and unknown correlations, the Inverse Variance portfolio is mean-variance optimal. Note that the Inverse Volatility portfolio is consistent with the Maximum Diversification portfolio, and the Inverse Variance portfolio approximates a Minimum Variance portfolio, when all investments have identical pairwise correlations.

The weights for the inverse volatility and inverse variance portfolios are found by:

where *σ* is the vector of asset volatilities and *σ*^{2} is the vector of asset variances.

(Maillard, Roncalli, and Teiletche 2008) described the Equal Risk Contribution optimization, which is satisfied when all assets contribute the same volatility to the portfolio. It has been shown that the Equal Risk Contribution portfolio is a compelling balance between the objectives of the equal weight and Minimum Variance portfolios. It is also a close cousin to the Inverse Volatility portfolio, except that it is less vulnerable to the case where assets have vastly different correlations.

The weights for the Equal Risk Contribution Portfolio are found through the following convex optimization, as formulated by (Spinu 2013):

where *Σ* is the covariance matrix.

The Equal Risk Contribution portfolio will hold all assets in positive weight, and is mean-variance optimal when all assets are expected to contribute equal marginal Sharpe ratios (relative to the Equal Risk Contribution portfolio itself). Thus, optimality equivalence relies on the assumption that the Equal Risk Contribution portfolio is macro-efficient. It has been shown that the portfolio will have a volatility between that of the Minimum Variance Portfolio and the Equal Weight portfolio.

(Lopez de Prado 2016) proposed a novel portfolio construction method that he labeled “Hierarchical Risk Parity”. The stated purpose of this new method was to “address three major concerns of quadratic optimizers in general and Markowitz’s CLA^{3} in particular: Instability, concentration and underperformance.”

Aside from the well-known sensitivity of mean-variance optimization to errors in estimates of means, De Prado recognized that traditional optimizers are also vulnerable because they require the action of matrix inversion and determinants, which can be problematic when matrices are poorly conditioned. Matrices with high condition numbers are numerically unstable, and can lead to undesirably high loadings on economically insignificant factors.

(Lopez de Prado 2016) asserts that the correlation structure contains ordinal information, which can be exploited by organizing the assets into a hierarchy. The goal of Hierarchical Risk Parity is to translate/reorganize the covariance matrix such that it is as close as possible to a diagonal matrix, without altering the covariance estimates. The minimum variance portfolio of a diagonal matrix is the inverse variance portfolio. For this reason, we describe the method as Hierarchical Minimum Variance. We explain many of these concepts in much greater detail in a follow-on article^{4}.

We now proceed to discuss the results of a paper, “Optimal Versus Naive Diversification: How Inefficient is the 1/N Portfolio Strategy?” by (DeMiguel, Garlappi, and Uppal 2007), which is commonly cited to dismiss optimization based methods. According to the paper, the authors were motivated by a desire to “understand the conditions under which mean-variance optimal portfolio models can be expected to perform well even in the presence of estimation risk.” They emphasize that the purpose of their study “is not to advocate the use of the 1/N heuristic as an asset-allocation strategy, but merely to use it as a benchmark to assess the performance of various portfolio rules proposed in the literature.”

While we are committed to revisiting the analysis from (DeMiguel, Garlappi, and Uppal 2007), we question the generality of the paper for several important reasons. First, the authors chose to specify their models in ways that, while technically precise, violate most common-sense practices in portfolio management. In addition, they chose to run their empirical analyses on universes that are almost perfectly designed to confound optimization-based methods.

The specification issues relate primarily to the way the authors measure means and covariances for optimization. For example, they run simulations that form optimal portfolios monthly based on rolling 60- and 120-month estimation windows.

This is curious for a number of reasons. First, the authors do not cite evidence that investors use these estimation windows to form optimal portfolios in practice. Thus, there is no reason to believe their methodology represents a meaningful use case for optimization.

Second, the authors provide no evidence or theory for why estimates from 60 and 120 month windows should be informative about next months’ returns. In fact, they performed their analysis on equity portfolios, and there is evidence that equity portfolios are mean-reverting over long horizons.

The primary case for the existence of long term mean reversion was made in two papers published in 1988, one by (Poterba and Summers 1988), and the other published by (Fama and French 1988). These papers conclude that for period lengths of between 3 and 5 years (i.e. 36 an 60 months), long-term mean reversion was present in stock market returns between 1926 and 1985. Three-year returns showed a negative correlation of 25%, while 5-year returns showed a negative correlation of 40%.

If returns over the past 5-10 years are mean-reverting over the horizon chosen by (DeMiguel, Garlappi, and Uppal 2007) to estimate portfolio means, we shoud expect performance of optimal portfolios to be disappointing, as the return forecasts for portfolio optimization would be above average for periods that should actually produce below-average returns, and vice versa.

The authors also highlight that the estimation of covariances is confounded by sparseness issues on large universes. Specifically, the covariance matrix will be ill conditioned if the length of the estimation window is smaller than the dimension of the matrix. To find the optimal weights for 500 securities would require at least 500 data points per security. At monthly granularity, this would require 42 years of data, while we would need 10 years of weekly data.

However, the test datasets used in the paper are also available at daily granularity. At daily frequency, the covariance matrix is appropriately conditioned, and optimization can be performed on, 500 securities with less than two years of data. One is left to wonder why the authors used data at monthly frequency when daily data were available.

The investment universes used to compare the performance of naive versus optimal diversification methods seem poorly chosen based on the authors stated purpose to “understand the conditions under which mean-variance optimal portfolio models can be expected to perform well.” The authors conducted their analysis on investment universes composed exclusively of equity portfolios. Clearly, equity portfolios are dominated by a single source of risk, equity beta, and provide few opportunities for diversification.

(DeMiguel, Garlappi, and Uppal 2007) concede this issue directly in the paper:

… the 1/N rule performs well in the datasets we consider [because] we are using it to allocate wealth across

portfoliosof stocks rather than individual stocks. Because diversified portfolios have lower idiosyncratic volatility than individual assets, the loss from naive as opposed to optimal diversification is much smaller when allocating wealth across portfolios. Our simulations show that optimal diversification policies will dominate the 1/N rule only for very high levels of idiosyncratic volatility [Emphasis ours].

Idiosyncratic volatility is simply the volatility of the residuals after the asset returns are regressed on the dominant systematic risk factor. In the case of equity portfolios like the sector, industry and factor portfolios under investigation by (DeMiguel, Garlappi, and Uppal 2007), these are the residuals on equity beta. If most of the variance for the test universes is explained by equity beta, there will be very little idiosyncratic volatility, and very little opportunity for diversification.

One way to determine the amount of idiosyncratic risk in a universe of assets is to use Principal Component Analysis (PCA). PCA is a tool to identify the underlying independent (i.e. uncorrelated) sources of risk, or principal components, of the investments.

The results of PCA are eigenvalues, *λ*, which describe the amount of total variance explained by each principal component, and the eigenvectors *A*, which describe the sensitivities or “betas” of each asset to each principal component. There are always the same number of eigenvalues and eigenvectors as investments, so a universe of ten investments will be decomposed into ten eigenvectors with associated eigenvalues.

The principal components are ordered so that the first component *λ*_{1} is the one that explains the most variance. For a universe of equities, it is held that the first principal component represents market beta. Thus, the first eigenvalue quantifies the amount of total portfoio variance explained by market beta.

All of the other principal components represent directions of risk that are independent of market beta. So the total amount of idiosyncratic variance in a universe of assets is equal to 1 − *λ*_{1}.^{5}

We examined the amount of idiosyncratic risk available to provide diversification for each universe that we targeted for investigation in Figure 1. We also show a decomposition for an even more diverse universe of major futures markets to highlight the opportunity for diversification outside of conventional asset classes. The chart shows the amount of *idiosyncratic* risk available for diversification, so lower bars imply less diversification opportunity.

Figure 1: **Idiosyncratic risk in different investment universes.**

*Source: Calculations by ReSolve Asset Management. Data for industries and portfolios sorted on size and book-to-market from Ken French database. Data for country equity indices from Global Financial Data. Asset class data from S&P Dow Jones Indices. Futures data from CSI. Idiosyncratic risk is calculated as 1 – the proportion of total variance explained by the first principal component.*

You can see that about three-quarters of the variance in the industry and factor sort universes is explained by the first principal component, which represents U.S. equity beta. Just one quarter of the risk is idiosyncratic risk, which might be used to enhance diversification.

In contrast, about two-thirds and four-fifths of the risk in the asset class and futures universe, respectively, are derived from sources other than the first principal component. This leaves much more idiosyncratic variance for optimization methods to make best use of diversification opportunities.

To understand just how little opportunity for diversification there is in (DeMiguel, Garlappi, and Uppal 2007)’s choices of investment universes, we found it useful to quantify the number of uncorrelated sources of return (i.e. independent bets) that are available in each group of investments.

Recall that (Choueifaty and Coignard 2008) showed that the Diversification Ratio of a portfolio is the ratio of the weighted sum of asset volatilities to the portfolio volatility after accounting for diversification.

This is intuitive because, if all of the assets in the portfolio are correlated, the weighted sum of their volatilities would equal the portfolio volatility, and the Diversification Ratio would be 1. As the assets become less correlated, the portfolio volatility will decline due to diversification, while the weighted sum of constituent volatilities will remain the same, causing the ratio to rise. At the point where all assets are uncorrelated (zero pairwise correlations), every asset in the portfolio represents an independent bet.

Consider a universe of ten assets with homogeneous pairwise correlations. Figure 2 plots how the number of independent bets available declines as pairwise correlations rise from 0 to 1. Note when correlations are 0, there are 10 bets, as each asset is responding to its own source of risk. When correlations are 1, there is just 1 bet, since all assets are explained by the same source of risk.

Figure 2: **Number of independent bets expressed with an equally weighted portfolio of 10 assets with equal volatility as a function of average pairwise correlations.**

Source: ReSolve Asset Management. For illustrative purposes only.

(Choueifaty, Froidure, and Reynier 2012) demonstrate that the number of independent risk factors in a universe of assets is equal to the square of the Diversification Ratio of the Most Diversified Portfolio.

Taking this a step further, we can find the number of independent (i.e. uncorrelated) risk factors that are ultimately available within a universe of assets by first solving for the weights that satisfy the Most Diversified Portfolio. Then we take the square of the Diversification Ratio of this portfolio to produce the number of unique directions of risk if we maximize the diversification opportunity.

We apply this approach to calculate the number of independent sources of risk that are available to investors in each of our test universes. Using the full data set available for each universe, we solve for the weights of the Maximum Diversification portfolios, and calculate the square of the Diversification Ratios. We find that the 10 industry portfolios; 25 factor portfolios; 38 sub-industry portfolios; and 49 sub-industry portfolios produce 1.4, 1.9, 2.9, and 3.7 unique sources of risk, respectively. Results are summarized in Figure 3.

These are rather astonishing results. Across 10 industry portfolios, and 25 factor portfolios, there are less than 2 uncorrelated risk factors at play. When we expand to 36 and 49 sub-industries, we achieve less than 3 and 4 factors, respectively.

To put this in perspective, we also calculated the number of independent factors at play in our test universe of 12 asset classes, and find 5 independent bets. To take it one step further, we also analyzed the independent bets available to 48 major futures markets across equity indexes, bonds, and commodities, and found 13.4 uncorrelated risk factors.

Figure 3: **Number of independent risk factors present in the investment universe.**

*Source: Calculations by ReSolve Asset Management. Data for industries and portfolios sorted on size and book-to-market from Ken French database. Data for country equity indices from Global Financial Data. Asset class data from S&P Dow Jones Indices. Futures data from CSI. Number of independent bets is equal to the square of the Diversification Ratio of the Most Diversified Portfolio formed using pairwise complete correlations over the entire dataset.*

The Optimization Machine was created to help investors choose the most appropriate optimization for any investment universe given the properties of the investments and the investor’s beliefs. Specifically, the Optimization Machine Decision Tree leads investors to the portfolio formation method that is most likely to produce mean-variance optimal portfolios given active views on some or all of volatilities, correlations, and/or returns, and general relationships between risk and return, if any.

One of the most important qualities investors should investigate is the amount of diversification available relative to the number of assets. If the quantity of available diversification is small relative to the number of assets, the noise in the covariance matrix is likely to dominate the signal.

We’ll call the ratio of the number of independent bets to the number of assets in an investment universe the “Quality Ratio”. The “Quality Ratio” is a good proxy for the amount of diversification “signal to noise” in the investment universe. When the Quality Ratio is high we would expect optimization methods to dominate naive methods. When it is low, investors should expect only a very small boost in risk-adjusted performance from using more sophisticated techniques.

For example the Quality Ratio of the universe of 10 industry portfolios is 0.12 while the Quality Ratio of the universe of 49 sub-industries is 0.08. Compare these to the Quality Ratio of our asset class universe at 0.42.

Figure 4: **Quality Ratio: Number of independent bets / number of assets.**

*Source: Calculations by ReSolve Asset Management. Data for industries and portfolios sorted on size and book-to-market from Ken French database. Data for country equity indices from Global Financial Data. Asset class data from S&P Dow Jones Indices. Futures data from CSI. Number of independent bets is equal to the square of the Diversification Ratio of the Most Diversified Portfolio formed using pairwise complete correlations over the entire dataset. Quality ratio is number of independent bets / number of assets.*

The Quality Ratio helps inform expectations about how well optimization methods, in general, can compete against naive methods. For universes with low Quality Ratios, we would expect naive methods to dominate optimization, while universes with relatively high Quality Ratios are likely to benefit from optimal diversification.

Where a high Quality Ratio would prompt an investor to choose optimization, the next step is to choose the optimization method that is most likely to achieve mean-variance efficiency. The Optimization Decision Tree is a helpful guide, as it prompts questions about which portfolio parameters can be estimated, and the expected relationships between risk and return. The answers to these questions lead directly to an appropriate method of portfolio formation.

Most of the branches of the Optimization Decision Tree lead to heuristic optimizations that obviate the need to estimate individual asset returns by expressing returns as a function of different forms of risk. For example, Maximum Diversification optimization expresses the view that returns are directly and linearly proportional to volatility, while Minimum Variance optimization expresses the view that investments have the same expected return, regardless of risk. Thus, these optimizations do not require any estimates of means, and only require estimates of volatilities or covariances.

This is good, because (Chopra and Ziemba 1993) demonstrate that optimization is much more sensitive to errors in sample means than to errors in volatilities or covariances. The authors show that for investors with relatively high risk tolerances, errors in mean estimates are 22x as impactful as errors in estimates of covariances. For less risk tolerant investors the relative impact of errors in sample means rises to 56x that of errors in covariances. This doesn’t mean investors should always eschew optimizations with active views on returns; rather, that investors should take steps to minimize the error term in general. We discuss this concept at length in future articles.

For now, we will constrain our choices of optimization to common risk-based methods, such as Minimum Variance, Maximum Diversification, and Risk Parity. The optimizations are useful if we assume we can’t achieve any edge with better estimates of return. Later, we will explore how one might incorporate systematic active views, such as those rendered by popular factor strategies like momentum, value, and trend.

Let’s use the Optimization Machine to infer which portfolio formation method should produce the best results for each investment universe. To be specific, we want to forecast which optimization method is most likely to produce the highest Sharpe ratio.

Regardless which optimization is chosen, the the magnitude of outperformance for optimization relative to equal weighting will depend largely on the Quality Ratio of the investment universe. The industry and factor equity portfolios have low Quality Ratios, and should produce a marginal improvement over the equal weight approach. The asset class universe has a higher Quality Ratio, suggesting that we should see more substantial outperformance from optimization relative to equal weighting.

Recall from our paper, “The Optimization Machine: A General Framework for Portfolio Choice” that historically, the returns to stocks are either unrelated or inversely related to both beta and volatility. All risk based optimizations rely on either a positive relationship, or no relationship, between risk and return because an inverse relationship violates the foundational principles of financial economics (specifically rational utility theory), so we will assume the returns to stock portfolios of industries and factor sorts are all equal, and independent of risk.

Following the Portfolio Optimization Decision Tree, we see that the equal weight portfolio is mean-variance optimal if assets have the same expected returns, and if they have equal volatilities and correlations. The Minimum Variance portfolio is also mean-variance optimal if assets have the same expected returns, but the optimization also accounts for differences in expected volatilies and heterogeneous correlations.

Ex ante, the Minimum Variance portfolio should outperform the equal weight portfolio if covariances are heterogeneous (i.e. unequal), and the covariances observed over our estimation window (rolling 252 day returns) are reasonably good estimates of covariances over the holding period of the portfolio (one calendar quarter in our case).

It is also a useful exercise to consider which method is most likely to produce the *worst* results. Given that the empirical relationship between risk and return has been negative, we might expect optimizations that are optimal when the relationship is positive to produce the worst results. The Maximum Diversification optimization is specifically optimal when returns are directly proportional to volatility. It makes sense that this portfolio would lag the performance of the equal weight and Minimum Variance portfolios, which assume no relationship.

When performance is averaged across the four economic regimes described by combinations of inflation and growth shocks, stocks and bonds have equal historical Sharpe ratios^{6}. The historical Sharpe ratio for commodities is about half what was observed for stocks and bonds. However, given that our sample size consists of just a handful of regimes since 1970, we are reluctant to reject the practical assumption that the true Sharpe ratio of a portfolio of diversified commodities is consistent with that of stocks and bonds.

If we assume stocks, bonds, and commodities have similar Sharpe ratios the Optimization Machine Decision Tree suggests the mean-variance optimal portfolio can be found using the Maximum Diversification optimization. The Risk Parity portfolio should also perform well, as it is optimal when assets have equal marginal Sharpe ratios to the equal risk contribution portfolio.

The equal weight and Minimum Variance portfolios are likely to produce the weakest Sharpe ratios, because their associated optimality conditions are most likely to be violated. The major asset classes are generally uncorrelated, while the sub-classes (i.e. regional indexes) are more highly correlated with one another, so the universe should have heterogeneous correlations. In addition, bonds should have much lower volatility than other assets. Lastly, the individual asset returns should be far from equal, since the higher risk assets should have higher returns.

With our hypotheses in mind, let’s examine the results of simulations.

We run simulations on each of our target investment universes to compare the simulated performance of portfolios formed using naive and optimization based methods. Where volatility or covariance estimates are required for optimization, we use the past 252 days to form our estimates. We perform no shrinkage other than to constrain portfolios to be long-only with weights that sum to 100%. Portfolios are rebalanced quarterly.

For illustrative purposes, Figure 5 describes the growth of $1 for simulations on our universe of 25 portfolios sorted on price and book-to-market. Consistent with the ready availability of leverage, and for easy comparison, we have scaled each portfolio to the same ex-post volatility as the market-capitalization weighted portfolio^{7}. We assume annual leverage costs equal to the 3-month T-bill rate plus one percent. Scaled to equal volatility, portfolios formed using Minimum Variance have produced the best performance over the period 1927 – 2017.

Figure 5: **Growth of $1 for naive versus robust portfolio optimizations, 25 factor portfolios sorted on size and book-to-market, 1927 – 2018**

*Source: ReSolve Asset Management. Simulated results. Portfolios formed quarterly based on trailing 252 day returns for industries, factor portfolios, and monthly for asset classes. Results are gross of transaction related costs. For illustrative purposes only.*

Table 1 summarizes the Sharpe ratios of each optimization method applied to each universe.

Table 1: **Performance statistics: naive versus robust portfolio optimizations. Industry and factor simulations from 1927 – 2017. Asset class simulations from 1990 – 2017.**

10 Industries | 25 Factor Sorts | 38 Industries | 49 Industries | 12 Asset Classes | |
---|---|---|---|---|---|

Average of Asset Sharpe Ratios | 0.44 | 0.46 | 0.37 | 0.37 | |

Cap Weight | 0.44 | 0.44 | 0.44 | 0.44 | 0.68 |

Equal Weight | 0.50 | 0.51 | 0.49 | 0.50 | 0.56 |

Minimum Variance | 0.53 | 0.62 | 0.58 | 0.55 | 0.61 |

Maximum Diversification | 0.50 | 0.41 | 0.55 | 0.53 | 0.76 |

Hierarchical Minimum Variance | 0.51 | 0.56 | 0.55 | 0.55 | 0.66 |

Equal Risk Contribution | 0.51 | 0.54 | 0.54 | 0.54 | 0.71 |

Inverse Volatility | 0.51 | 0.54 | 0.52 | 0.52 | 0.69 |

Optimization Combo | 0.52 | 0.55 | 0.57 | 0.56 | 0.72 |

*Source: ReSolve Asset Management. Simulated results. Portfolios formed quarterly based on trailing 252 day returns for industries, factor portfolios, and monthly for asset classes. Results are gross of transaction related costs. For illustrative purposes only.*

**Stocks**

The first things to notice is that all methods outperformed the market cap weighted portfolio with a few notable exceptions: the Maximum Diversification portfolio underperformed the market cap weighted portfolio on the factor sort universe.

This should not be surprising. The market cap weighted portfolio is mean-variance optimal if returns to stocks are explained by their *β* to the market, so that stocks with higher *β* have commensurately higher returns. However, we showed in our whitepaper on portfolio optimization that investors are not sufficiently compensated for bearing extra risk in terms of market *β*. Thus, investors in the market cap weighted portfolio are bearing extra risk, which is not compensated.

The evidence confirmed our hypothesis that the Minimum Variance portfolio should produce the best risk-adjusted performance on the equity oriented universes. The Optimization Machine Decision Tree also indicated that the Maximum Diversification strategy should perform worst on the equity universes because of the flat (or even negative) empirical relationship between risk and return for stocks. Indeed, Maximum Diversification lagged the other optimizations in some simulations. However, it produced better results than Inverse Volatility and Equal Risk Contribution methods in many cases, and dominated equal weight portfolios for 38 and 49 industry simulations.

**Asset Classes**

Our belief that diversified asset classes should have equal long-term Sharpe ratios led us to hypothesize that the Maximum Diversification portfolio should dominate in the asset class universe. We expected the equal weight and Minimum Variance strategies to underperform. These predictions played out in simulation. The Equal Risk Contribution and Inverse Volatility weighted approaches were also competitive, which suggests the assumption of constant correlations may not be far from the mark.

In addition to publishing the results for each method of portfolio choice, we also published the results for a portfolio that averaged the weights at each period across all of the optimization strategies. For all universes except the factor sort universe, the unbiased average of all optimizations (including the least optimal strategy) outperformed the naive equal weight method.

While it’s true that the appropriate optimization based approaches produced better results than equal weighting for every universe, it’s useful to examine whether the results are statistically signficant. After all, the performance boosts observed for the best optimization methods are not very large.

To determine whether the results are economically meaningful or simply artifacts of randomness, we performed a block bootstrap test of Sharpe ratios. Specifically, we randomly sampled blocks of four quarters of returns (12 monthly returns for the asset class universe), with replacement, to create 10,000 potential return streams for each strategy.

Each draw contained a sample of equal weight returns alongside returns to the target optimal strategy, with the same random date index. Each sample was the same length as the original simulation.

We then compared the Sharpe ratio of each sample from equal weight returns to the Sharpe ratio of the sample of optimal weight returns. The values in Table 2 represent the proportion of samples where the Sharpe ratio for samples of equal weight returns exceeded the Sharpe ratio for samples of optimal strategy returns. As such, they are analagous to traditional p-values, where p is the probability that the optimal strategy outperformed due to random chance.

Table 2: **Pairwise probabilities that the Sharpe ratios of optimization based strategies are less than or equal to the Sharpe ratio of the equal weight strategy.**

10 Industries | 25 Factor Sorts | 38 Industries | 49 Industries | 12 Asset Classes | |
---|---|---|---|---|---|

Minimum Variance | 0.07 | 1.00 | 0.01 | 0.44 | 0.01 |

Maximum Diversification | 0.19 | 1.00 | 0.26 | 0.94 | 0.01 |

Hierarchical Minimum Variance | 0.01 | 0.00 | 0.00 | 0.00 | 0.07 |

Equal Risk Contribution | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |

Inverse Volatility | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |

Optimization Combo | 0.00 | 0.97 | 0.00 | 0.08 | 0.00 |

*Source: ReSolve Asset Management. Simulated results. Portfolios formed quarterly based on trailing 252 day returns for industries, factor portfolios, and monthly for asset classes. Results are gross of transaction related costs. For illustrative purposes only.*

This analysis yields some surprising results. While the Minimum Variance strategy produced the highest sample Sharpe ratio for all of the equity oriented universes, Risk Parity based methods like Equal Risk Contribution and Inverse Volatility were even more dominant from a statistical standpoint. The Hierarchical Minimum Variance approach also demonstrated a high degree of statistical robustness.

For the asset class universe, all but the Hierarchical Minimum Variance portfolio outperformed the equal weight portfolio on a statistically significant basis. And the Hierarchical Minimum Variance portfolio outperformed the equal weight portfolio 93% of the time. This further validates the importance of optimization when the universe of assets has diverse volatility and correlation features.

Risk Parity methods are more likely to dominate equal weight portfolios because they exhibit a smaller amount of active risk *relative to the equal weight portfolio*. However, while the Risk Parity portfolios might outperform the equal weight portfolios slightly more frequently on a relative basis, they are likely to more frequently *under*perform Minimum Variance and Maximum Diversification, for equity and asset class universes respectively, on an absolute basis.

Many investment professionals are under the misapprehension that portfolio optimization is too noisy to be of practical use. This myth is rooted in a few widely cited papers that purport to show that portfolio optimization fails to outperform naive methods.

The goal of this article was to illustrate how the Portfolio Optimization Machine is a useful framework to identify which optimization method should be most appropriate for a given investment universe. We used the Optimization Machine along with data and beliefs to form hypotheses about optimal portfolio choice for a variety of investment universes. Then we proceeded to test the hypotheses by simulating results on live data.

The choices invoked by the Portfolio Optimization Machine produced superior results. Both naive and optimal methods dominated the market cap weighted portfolio. Optimization based methods dominated naive equal weighted methods in most cases, except where an optimization expressed relationships between risk and return that were precisely converse to what was observed in the historical record. For example, Maximum Diversification expresses a positive relationship between return and volatility, while stocks have historically exhibited a flat, or perhaps even inverted relationship. We should therefore not be surprised to learn that Maximum Diversification underperformed the equal weight portfolio when applied in some equity oriented universes.

While optimization based methods rivaled the performance of naive methods for the cases investigated in this paper, we acknowledge that our test cases may not be representative of real-world challenges faced by many portfolio managers. Many problems of portfolio choice involve large numbers of securities, with high average correlations. Investors will also often demand constraints on sector risk, tracking error, factor exposures, and portfolio concentration. Other investors may run long/short portfolios, which introduce much higher degrees of instability.

We are sympathetic to the fact that most finance practitioners are not trained in numerical methods. While portfolio optmization is covered in the CFA and most MBA programs, the topic is limited to the most basic two-asset case of traditional mean-variance optimization with known means and covariances. Of course, this doesn’t resemble real world problems of portfolio choice in any real way.

In future articles we will explore more challenging problems involving lower quality investment universes with more typical constraints. We will dive more deeply into some of the mathematical challenges with optimization, and present novel solutions backed up by robust simulations. Later, we will describe how to incorporate dynamic active views on asset returns informed by systematic factors, which we call “Adaptive Asset Allocation.”

Bun, Joël, Jean-Philippe Bouchaud, and Marc Potters. 2016. “Cleaning large correlation matrices: tools from random matrix theory.” https://arxiv.org/abs/1610.08104.

Chopra, Vijay K., and William T. Ziemba. 1993. “The Effect of Errors in Means, Variances, and Covariances on Optimal Portfolio Choice.” *Journal of Portfolio Management* 19 (2): 6–11.

Choueifaty, Yves, and Yves Coignard. 2008. “Toward Maximum Diversification.” *Journal of Portfolio Management* 35 (1). http://www.tobam.fr/inc/uploads/2014/12/TOBAM-JoPM-Maximum-Div-2008.pdf: 40–51.

Choueifaty, Yves, Tristan Froidure, and Julien Reynier. 2012. “Properties of the Most Diversified Portfolio.” *Journal of Investment Strategies* 2 (2). http://www.qminitiative.org/UserFiles/files/FroidureSSRN-id1895459.pdf: 49–70.

Christoffersen, P., V. Errunza, K. Jacobs, and X. Jin. 2010. “Is the Potential for International Diversification Disappearing?” *Working Paper*. https://ssrn.com/abstract=1573345.

DeMiguel, Victor, Lorenzo Garlappi, and Raman Uppal. 2007. “Optimal Versus Naive Diversification: How Inefficient is the 1/N Portfolio Strategy?” http://faculty.london.edu/avmiguel/DeMiguel-Garlappi-Uppal-RFS.pdf: Oxford University Press.

Fama, Eugene, and Kenneth French. 1988. “Permanent and Temporary Components of Stock Prices.” *Journal of Political Economy* 96. https://teach.business.uq.edu.au/courses/FINM6905/files/module-2/readings/Fama: 246–73.

Haugen, R., and N. Baker. 1991. “The Efficient Market Inefficiency of Capitalization-Weighted Stock Portfolios.” *Journal of Portfolio Management* 17. http://dx.doi.org/10.3905/jpm.1991.409335: 35–40.

Lopez de Prado, Marcos. 2016. “Building Diversified Portfolios that Outperform Out of Sample.” *Journal of Portfolio Management* 42 (4): 59–69.

Maillard, Sebastien, Thierry Roncalli, and Jerome Teiletche. 2008. “On the properties of equally-weighted risk contributions portfolios.” http://www.thierry-roncalli.com/download/erc.pdf.

Poterba, James M., and Lawrence H. Summers. 1988. “Mean Reversion in Stock Prices: Evidence and Implications.” *Journal of Financial Economics* 22 (1). http://www.nber.org/papers/w2343: 27–59.

Spinu, Florin. 2013. “An Algorithm for Computing Risk Parity Weights.” *SSRN*. https://ssrn.com/abstract=2297383.

- This should not be confused with the Quality Ratio from Random Matrix Theory, which is the ratio of variables to number of independent samples. See (Bun, Bouchaud, and Potters 2016)↩
- SPY ETF extended with S&P 500 Index, VGK ETF extended with S&P Europe BMI, VPL ETF extended with S&P Asia Pacific BMI, VWO ETF extended with S&P Emerging BMI, IEF ETF extended with S&P US Treasury 7-10 Year TR Index, TLT ETF exteneded with S&P US Treasury Bond 20+ Year TR Index, LTPZ extended with S&P US TIPS 15+ Year Index and PIMCO Real Return Fund, BWX ETF extended with S&P/Citigroup International Treasury Ex-US TR Index and T. Rowe Price International Bond Fund, EMB ETF extended with PIMCO Emerging Markets Bond Fund and T. Rowe Price Emerging Markets Bond Fund, DBC ETF extended with Deutsche Bank Liquid Commodity Index, RWO ETF extended with Cohen & Steers Global Realty Shares,Inc. Class I, GLD ETF extended with spot gold in USD.↩
- Critical Line Algorithm↩
- For a full description of the method including Python code see this .↩
- We are concerned with the
*directions*of risk, rather than the*magnitudes*of risk. The direction of risk is endogenous to the investments while the magnitude of risk – the volatility – can be scaled up and down by changing the exposure. As a result, we focus our decomposition on the correlation matrix, rather than the covariance matrix.↩ - See our paper “The Optimization Machine: A General Framework for Portfolio Choice”↩
- “Market” factor plus the risk free rate from Ken French database.↩

Investors sometimes become concerned when they observe discrepancies in the performance of seemingly identical strategies with identical expected long-term performance.

In this article, which is a summary of a more comprehensive whitepaper (**download here**), we hope to alleviate these concerns by showing that short-term discrepancies will almost always arise when the same strategy is executed on different channels, like SMAs, UMAs, or mutual funds. Moreover, these discrepancies can be quite large in the short term without compromising the long-term expectations of the strategy.

For example, we offer AAA mandates using both futures and ETFs for a given volatility target. While they provide very similar underlying market exposures in the long term, over the short-term small differences in the underlying constituents of the ETFs versus futures, will produce short-term deviations.

Differences in trading costs and turnover preferences for each delivery channel also prompt us to execute AAA at different rebalance frequencies, and on different days of the week. Again, while the long-term returns for these mandates are statistically indistinguishable, performance over shorter horizons like a few months or years can be quite distinct due simply to how often and when these portfolios are rebalanced.

Consider two strategies that have precisely the same methodology, executed on the exact same investment universe. However they are executed with different rebalance frequencies between 1 and 20 days, and on different days of the month.

The strategies have returns that are statistically indistinguishable from one another over 25 years. How different could short-term performance really be if all that stood between them was the portfolio’s rebalancing frequency?

Perhaps surprisingly, the average difference between the best and worst strategy simulation was 7.2% over any given year.** The difference was larger than 10.9% five percent of the time.** Even when the strategies were tracking closely, we still observed a 4.2% dispersion between the best and worst performer. Figure 1 shows a few of the years where merely changing the rebalancing frequency led to large dispersions.

Source: ReSolve Asset Management. Each chart represents the growth of $1 for the worst and best performing strategy permutation each the calendar year. Simulated and hypothetical data. Past performance is no guarantee of future results.

The moral of the story is that investors have to either buy into the underlying process or not. Short-term performance is just noise. Investors need to trust (or not) that the fundamental drivers of the methodology (momentum/trend factors and diversification) are likely to prevail in the long-term, even in the face of random dispersion in the short-term.

Of course, this doesn’t just apply to AAA. It is axiomatic across any investment approach. The performance of the S&P500 would have been materially different from year-to-year if it added or removed companies at different times. How different would the index have been if the index committee had added Apple, Microsoft, or Amazon just 6-months earlier or later?

The ability to ignore short-term noise and focus on long-term evidence is what separates the willing losers from the alpha harvesters.

It is widely accepted among investment professionals that, while portfolio optimization has compelling theoretical merit, it is not useful in practice. Practitioners are concerned that optimization is an “error maximizing”^{1 }process fraught with insurmountable estimation issues. The abstract of an early academic critique of mean-variance optimization, (Michaud 1989) states:

The indifference of many investment practitioners to mean-variance optimization technology, despite its theoretical appeal, is understandable in many cases. The major problem with mean-variance optimization is its tendency to maximize the effects of errors in the input assumptions. Unconstrained mean-variance optimization can yield results that are inferior to those of simple equal-weighting schemes.

Many nay-sayers selectively quote the above passage as reason to dismiss optimization altogether. However, this same abstract continues with the following thoughts:

Nevertheless, mean-variance optimization is superior to many ad hoc techniques in terms of integration of portfolio objectives with client constraints and efficient use of information. Its practical value may be enhanced by the sophisticated adjustment of inputs and the imposition of constraints based on fundamental investment considerations and the importance of priors. The operating principle should be that, to the extent that reliable information is available, it should be included as part of the definition of the optimization procedure.

It’s clear that portfolio optimization is a powerful tool that must be used thoughtfully and responsibly. However, even the critics agree that optimization is the only mechanism to make best use of all the information available to investors.

In this paper, we will first build a theoretical framework that will enable us to determine the most appropriate method of portfolio construction for most situations. We’ll introduce the Portfolio Optimization Machine^{TM }and suggest how an investor might decide which type of optimization is most consistent with the qualities, beliefs, and assumptions he holds about the assets under consideration.

¹ Michaud (1989)

In his 1998 second edition of ** “Stocks for the Long Run^{1}”**, Jeremy Siegel added a chapter called “Technical Analysis and Investing with the Trend”, where he explored simple trend rules to time the U.S. stock market. In the chapter, Dr. Siegel revealed that the simple trend following strategy produced similar returns to a strategy of buying the index and re-investing dividends over the very long run, but with less portfolio volatility and smaller maximum peak-to-trough losses.

To this day, many novice investors and advisors make use of simple trend rules to try to time exposure to stock markets. The 200 day moving average that Dr. Siegel explored (and many other market timers and trend traders have been using for decades) is perhaps the most closely watched indicator. With the introduction of liquid ETFs tracking major equity indexes, it’s a simple matter for any investor to own stocks when the major indexes trade above this simple moving average, but cut and run when they break.

While novice investors typically stumble onto the concept of trend-following in the context of stock-market timing, professionals know that trend-following is not about using trends to time one or two individual markets. Modern professional trend-followers often trade dozens of futures markets across equities, bonds, currencies, commodities, and more obscure markets like carbon offsets.

In fact, professionals have long understood that the key to success with trend following, which most novice investors overlook, is **diversification**. In the preface to Michael Covel’s classic book,** “ Trend Following^{2} ”**, Larry Hite, one of the original

In my early days, there was only one guy I knew who seemed to have a winning track-record year after year. This fellow’s name was Jack Boyd. Jack was also the only guy I knew who traded lots of different markets. If you followed any

oneof Jack’s trades, you never really knew how you were going to do. But, if you were like me and actually countedallof his trades, you would have made about 20 percent a year. So, that got me more than a little curious about the idea of trading futures markets “across the board.” Although each individual market seemed risky, when you put them together, they tended to balance each other out and you were left with a nice return with less volatility.

Larry’s insight was that the only way to achieve consistent results is to trade markets “across the board”. At the time, Larry was referring to the Chicago Board of Trade, which housed trading for most major commodity futures. Now futures are traded on a wide variety of exchanges, and investors are no longer constrained to trading commodity futures. But the same lesson holds today as it did four decades ago when Larry Hite began his trading career. That is, if you trade just one market “you never really know how you are going to do”, but if you trade markets across the board, you have a good chance of earning “a nice return with less volatility”.

Many novice investors and advisors choose to apply trend following concepts to time stock indices. More than anything this probably reflects the public’s pre-occupation with stocks. But some analysts also argue that investors should focus on equities because they have the largest risk premium.

**This completely misses the point.**

The highest risk premium argument only holds for small investors who, despite overwhelming headwinds, insist on managing their own portfolios^{4}, and for investors who do not understand the Capital Market Line (CML).

Remember, the goal for most investors is to maximize their return at a level of risk that they can bear. Investors have many options to achieve different rates of return. Typically, as an investor seeks higher levels of return he is encouraged to take on greater exposure to the equity risk premium. However, this is not the only way to achieve higher returns.

Following on the work of Harry Markowitz and Jack Treynor, Bill Sharpe published a 1964 treatise titled ** “Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk^{5}”**, which described a method for investors to achieve higher returns without sacrificing diversification. He proposed that an investor with typical preferences regarding tradeoffs between risk and return would prefer to hold an efficient diversified portfolio at all times.

Investors who can tolerate higher risk in pursuit of higher returns would borrow money to invest in more units of this diversified portfolio. This is preferable to moving out the efficient frontier into portfolios with increasing concentration in stocks.

Consider Figure 1 which describes an efficient frontier (in dark blue) and CML (in red) derived from recently published asset return premia and correlation estimates from a major institution^{6}. Emerging Stocks are expected to produce the highest returns, while Foreign Bonds have the lowest expected return. The portfolio that is expected to produce the maximum return per unit of risk, the maximum Sharpe ratio portfolio, is highlighted with a red point on the chart, and is found at the point where the CML intersects the efficient frontier.

**Figure 1: Capital Market Line vs. Efficient Frontier**

*Source: ReSolve Asset Management. For illustrative purposes only.*

Let’s examine a case where a “Growth” investor wishes to maximize his rate of return given that he can tolerate 15% annual volatility, consistent with the historical risk character of a 70/30 global stock/high grade bond portfolio. Under typical conditions, this investor would be forced to push out the efficient frontier and take on a concentrated portfolio of equity markets. Figure 2 describes the composition of this portfolio.

Figure 2: **Mean-Variance Optimal Portfolio at 15% Target Volatility Along Efficient Frontier**

*Source: ReSolve Asset Management. For illustrative purposes only.*

Given the capital market assumptions above, to achieve the highest return possible at no more than 15% volatility would require that he hold 29.4% Long Treasuries, 18% Asian Stocks, 52.6% Emerging Stocks, since this is the most efficient portfolio at 15% volatility.

This portfolio would be expected to earn 4.8% annual excess return, highlighted with a gold point on the efficient frontier in Figure 1.

Now consider an investor who is liberated from the no-leverage tyranny imposed by the efficient frontier. He can choose to own the more diversified mean-variance optimal portfolio described in Figure 3, and borrow (with margin or futures or other derivatives) to purchase more units of the portfolio using leverage until he achieves his target volatility. Again given our working capital market assumptions, this portfolio would be dominated by Emerging Bonds, Long Treasuries, and Asian Stocks.

Figure 3: **Maximum Sharpe Ratio Portfolio**

*Source: ReSolve Asset Management. For illustrative purposes only.*

Leverage is a foreign concept for many novice investors. But readers with an open mind will recognize that by using a prudent^{7} amount of leverage to invest in the maximum Sharpe ratio portfolio, they can now take advantage of investments that earn their premium during very different market environments, and from diverse economic regions.

The concentrated equity investor would only expect to earn attractive returns during periods of sustained economic growth, benign inflation, and abundant liquidity. However, the maximum Sharpe ratio portfolio in Figure 3 contains Treasury bonds, which are designed to benefit during deflationary growth shocks; emerging bonds, which would benefit from an increase in credit quality among emerging countries, as well as potential currency tailwinds; and equity markets from around the world.

Aside from the benefits of broader diversification, the choice to leverage the maximum Sharpe ratio portfolio in Figure 3 provides an opportunity to earn 6.45% excess return, a full 1.65 percentage points more per year than the investor can expect to achieve by moving out the frontier. This point is also highlighted on the CML in Figure 1.

To close the loop, investors who focus their trend trading on equity indexes because the equity risk premium is of a larger magnitude than other premia are missing the point. Under reasonable assumptions about the general relationship between risk and return, a diversified portfolio will produce considerably greater return *per unit of risk*. When scaled up the CML using prudent amounts of leverage, this leads to a higher absolute return, *period*, than a concentrated position in equities.

Of course, the benefits of diversification grow in proportion to the number of alternative sources of return that are available. Diversification is the key.

Notwithstanding the benefits of diversifying into uncorrelated markets outside of equities, many investors start their investment journey with a myopic focus on equities. Upon discovering the benefits of trend following, they often spend years applying the techniques to time a major stock index, such as the S&P 500 or their local market.

In this section we will continue to focus on equity market trend following. We’ll examine the performance of a representative trend-following strategy applied to fifteen global stock index futures.

First we’ll observe the performance of the trend strategy when applied to the individual markets. Then we’ll demonstrate the monstrous advantage that is available from trading a diversified strategy of all equity indexes. Finally, we’ll expand our horizon to include assets outside of equities, and unleash the true potential of diversified multi-asset trend-following.

We first examine the distribution of trend-following performance when applied to individual equity index futures, relative to the performance that can be achieved by trading a diversified basket of equity index futures. To keep things simple, we examine the performance of a simple moving-average trend trading strategy, based on a 200 day (~10 month) lookback horizon. The strategy will hold a market long when its price is above its moving average, and exit when price falls below.

Let’s first observe the growth profile of our toy trend strategy when it is applied to futures tracking several major equity markets around the world.

Figure 4 plots the growth trajectories for each equity index futures market. S&P 500 futures started trading in 1983, and other markets were introduced over time. To account for the fact that some markets have less time to compound (because they come into existence later), we bring each market into existence at the current level of the S&P 500 strategy. Thus, the terminal value for each strategy in Figure 4 offers a meaningful indication of relative performance.

Figure 4: **Growth of $1: Simple 200 Day Moving Average Long/Flat Trend Strategy Applied to Futures on Major Equity Indices**

*Source: Calcuations by ReSolve Asset Management. Data from CSI Data. Growth of $1 from executing long/flat 200-day moving-average trend strategies on equity index futures scaled to ex post 20% volatility.*

There is a large dispersion in results across markets. Japanese investors trading the TOPIX or the Nikkei fared much worse than Finnish or Swiss investors trading exactly the same strategy. In fact, the worst strategy grew $1 to just over $2 in 35 years, while the best strategy turned $1 into over $16.

Perhaps surprisingly, while the trend strategy improved risk-adjusted performance relative to a buy and hold strategy for the majority of equity indexes, investors in Commonwealth countries (UK, Canada and Australia) experienced lower absolute and risk-adjusted returns by following trends.

In our toy example, if the investor chooses to run a trend-following strategy on just one equity index, he has equity assets to choose from. Assuming the investor chose one equity index to trade at random, the best estimate of the investor’s performance is the median performance among all strategies.

Importantly, it is the median that matters to one investor who chooses just one strategy, not the mean, because the investor can live just one life, and has chosen not to take advantage of the law of large numbers^{8}.

What many investors miss is that, absent extremely confident views about which market will outperform in the future, investors are better off trading all of the markets at once.

Let’s consider a more humble investor that is focused on investing in equities, but cannot decide which market(s) to trade. Instead, he chooses to trade all of the equity index assets as part of one diversified strategy.

The “Portfolio” quantity in Figure 5 reflects the performance of this diversified strategy relative to the median strategy. Critically, the diversified strategy benefits from the fact that the returns from each of the individual strategies are not perfectly correlated. In other words, they diversify one another.

Figure 5 compares the Sharpe ratio of a diversified trend strategy, which trades all equity markets, against the median performance of trend strategies over the 15 equity index futures.

An investor choosing a market to trade at random would have expected to experience a Sharpe ratio of 0.45, while an investor who traded all markets as a diversified trend strategy would have achieved a Sharpe ratio of 0.76. The difference between the “Median” and the “Portfolio” performance is the bonus that an investor accrued from taking advantage of this diversification.

Astonishingly, investors who chose to diversify would have produced 1.69x the return per unit of risk relative to an investor trading one random market. This diversification bonus is so large that the “Portfolio” strategy surpasses the risk adjusted performance of all but one of the individual strategies^{9}. Which means an investor would have had to be better than 93% accurate in choosing which index to trade *in advance* in order to achieve better performance than one could generate from simply diversifying across all of them.

**For those focused on U.S. stocks, it’s worth noting that the diversified strategy dominated trend trading on the S&P 500, with a Sharpe ratio of 0.51, by almost 50%. ^{10}**

Figure 5: **Marginal Sharpe Contribution from Diversified Long/Flat Trend Trading Across Equity Markets vs. Median Performance by Individual Market**

*Source: Calcuations by ReSolve Asset Management. Data from CSI Data. “Median” is the median Sharpe ratio of long/flat 200-day moving-average trend strategies applied to individual equity index futures. “Portfolio” is the Sharpe ratio of a strategy that trades all of the equity index futures markets as one aggregate strategy. Diversification Bonus is the improvement in performance from trading all equity index futures as a diversified strategy instead of choosing one market to trade. Performance does not account for fees, transacation costs, or other factors which may impact performance. For illustrative purposes only.*

We mentioned above that the concept of diversification extends (obviously) to other asset classes besides equities. The fact is, it rarely pays to focus your efforts on any one market, in any asset class.

Let’s expand our domain of analysis to include six bond futures, seven currency futures, and twenty commodity futures. Figure 6 describes the diversification bonus from choosing to trade all markets in each category, rather than trading any single one.

The commodity asset category provides a particularly interesting case study. Long/flat trading based on a simple 200 day moving average has not been a particularly profitable strategy for individual commodity markets over the past few decades. The median Sharpe ratio for trend strategies across twenty commodities is just 0.24.

However, commodity markets are generally uncorrelated with one another. Which means that there is a large advantage to running even relatively ineffective strategies “across the board.” Incredibly, an investor would have achieved 2.29x the return per unit of risk relative to an investor trading any one random commodity market.

Figure 6: **Marginal Sharpe Contribution from Diversified Long/Flat Trend Trading Across Major Asset Categories vs. Median Performance by Individual Market Within Each Asset Category**

*Source: Calculations by ReSolve Asset Management. Data from CSI Data. “Median” is the median Sharpe ratio of long/flat 200-day moving-average trend strategies run on each individual market in each asset category. Markets are weighted using ex post inverse volatility. “Portfolio” is the Sharpe ratio of a strategy that trades all of the markets in the asset category as one aggregate strategy. “Diversification Bonus”” is the improvement in performance from trading all markets in the category instead of choosing one market to trade. Performance does not account for fees, transacation costs, or other factors which may impact performance. For illustrative purposes only.*

Now that we’ve quantified the diversification bonus for investors who are concentrated in any one asset category, we conclude by expanding the trend strategy to trade all assets from all categories at once. This is obviously where the rubber hits the road, since professional trend followers use all the instruments at their disposal to achieve the largest divesification bonus possible.

Figure 7 describes the gargantuan bonus available to investors who understand the power of combining trend-following with diversification across all major asset categories. It’s shocking to see that diversification alone can transform many independent strategies with low Sharpe ratios on their own into a diversified strategy with long-term performance that rivals even the most successful markets and hedge funds.

Figure 7: **Marginal Sharpe Contribution from Diversified Long/Flat Trend Trading Across All Markets vs. Median Performance by Individual Market**

*Source: Calculations by ReSolve Asset Management. Data from CSI Data. “Median” is the median Sharpe ratio of long/flat 200-day moving-average trend strategies run on each individual market across all asset categories. Markets are weighted using ex post inverse volatility. “Portfolio” is the Sharpe ratio of a strategy that trades all markets in all categories as one aggregate strategy. “Diversification Bonus”” is the improvement in performance from trading all markets as one aggregate strategy instead of choosing one asset to trade. Performance does not account for fees, transacation costs, or other factors which may impact performance. For illustrative purposes only.*

For natural reasons, many novice investors and advisors try to harness the power of trend following to trade their favorite equity index. But this misses the point. By trend-trading a single index investors are extremely vulnerable to the probability of choosing an equity market with low forward returns, unproductive trends, or both.

The true benefit of trend following is only realized when investors take advantage of the extreme liquidity and diversity of global futures markets to trade a wide range of markets across all major asset categories. Our analysis shows that an investor would have achieved more than double the risk-adjusted performance of a median equity trend strategy by trading a diversified strategy across many diverse markets.

Traditionally, many diversified futures funds were only availble to Qualified Investors. This barrier has lifted over the past few years with the introduction of liquid alternatives, as several private funds have been “converted” to traditional mutual funds. Catalyst and Rational Funds have been especially active in making top futures strategies available to everyone. Even better, gains on futures receive favourable tax treatment, and futures funds are often extremely capital efficient.

- Siegel, Jeremy J. “Stocks for the Long Run, 2nd ed.”, McGraw-Hill (1998)↩
- Covel, Michael W. “Trend Following: Learn to Make Millions in Up or Down Markets”, FT Press (2009)↩
- Schwager, Jack D. “Market Wizards”, HarperCollings (1989)↩
- Small investors may have skill and certain advantages (i.e. liquidity, mandate flexibility, portfolio agility, long-term thinking, etc.), but they also pay egregious fees for margin, typically cannot effectively access futures and swaps markets, and either a) pay onerous commissions or b) trade for “free” but get killed on execution because the broker sells their flow for rebates. Remember, the broker has to get paid somehow…↩
- Paper can be found here↩
- NOT official ReSolve estimates.↩
- Some readers might wonder about the term “prudent” referring to leverage. For context, the average leverage ratio for the S&P 500 is 1.7x since 1995, which means there is $1.7 of debt for ever $1 of equity.↩
- To gain a deeper understanding of the concept of ergodicity and the difference between ensemble mean and finite time mean, please see this article: https://aip.scitation.org/doi/full/10.1063/1.4940236 .↩
- I know, I know, you all would have known to trade Finnish index futures in advance, so this point is moot.↩
- Many investors have probably examined the performance of a 200 day moving average strategy on the S&P 500 using index data. It may surprise aspiring quants to learn that the exact same strategy run on the S&P 500 total return cash index produces a substantially higher Sharpe ratio over the same period. Please do not learn the wrong lesson! Given very small changes in the exact path of returns, we could just have easily have seen that the futures strategy outperformed the cash index strategy. This dispersion in performance highlights the extreme fragility of this type of strategy run on just one index, and the sensitivity to path dependence.↩

Last week we were jazzed to have Dr. Kathryn Kaminski deliver a comprehensive presentation on Managed Futures Trend Following: The Ultimate Diversifier, where she covered the role of convergent and divergent strategies, and introduced other important themes like:

- The role of managed futures in institutional portfolios
- Diversifiers vs. crisis alpha

- Return based style analysis to differentiate between managed futures funds, and build optimal portfolios
- Identifying and managing the drivers of managed futures returns

- Sources of “craftsmanship alpha” in managed futures
- The qualities to look for and what to avoid

Yes, post 2008 hasn’t been an easy period for Managed Futures. In all fairness, it has also been one of the easiest periods for Equity investors in the past 100 years or more.

There are several things to consider:

- Low interest rates are a drag for Managed Futures.If 80% or more of your assets are in T-bills and cash management accounts, this doesn’t help. In the 70s and even in the 80s Managed Futures had an 8% tail wind from interest on collateral. That would be nice right now.Put simply, low rates aren’t good for Managed Futures in pure carry terms. In addition, low rates may also create less movement in futures prices (Remember – the futures price includes the time value of money). Alex Greyserman and I examined this issue in our book. We considered the amount of market divergence in recent years and couldn’t find any evidence that prices are more efficient and less divergent than before. (See Chapter 5)
- No-crisis equals less divergence in prices equals less trend to capture. Hutchinson and O’Brien (2014) examine this issue in a paper entitled “Is this time different? Trend Following and Financial Crises” https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2375733

They show that when there is a sustained period without crises things are not so easy for Trend Following. I also have a whitepaper on “Crisis Alpha Everywhere” discussed in PIOnline that shows that it is the amount of crisis in different assets, not just equities, that matters. Equity crisis is a bigger driver, but it is crisis in general that drives divergence/dislocation that creates tradable trends.

http://www.pionline.com/article/20170330/ONLINE/170329884/crisis-alpha-everywhere

To put it more simply, there has been less crisis since 2008. There are some exceptions: 2014 was a rough year in commodities and currencies, CTAs did very well as we would expect. If there are no more serious crises in the next 10 years, I would expect this to be a headwind for Managed Futures returns.

Trend following should still be a main allocation for managed futures. The only difference is that they can consider a wide range of strategies as long as they don’t have them already. For example, if an investor buys a multi-strategy CTA this could help smooth some of the difficult periods for trend following. Since managed futures trade mostly in futures, the key strategies to consider are those that could provide risk premiums: things like carry, macro, short term strategies, and possibly volatility strategies.

One major red flag for me as a former allocator was manager discretion. If a trend following manager is changing their portfolios based on the heat of the moment and going against the system – that is not ideal. Trend following often works its best when things are uncomfortable for investors, and it is the “rule based” approach that lets the strategy take positions that aren’t always comfortable.

Take the example of oil in 2014. No macro trader wanted to short oil because it seemed impossible that the price could keep going down. A trend system should short oil because it is going down. If the manager is willing to come in and make a discretionary decision, this is going against the approach – put in the words of the talk – adding convergent to the divergent approach. This often can backfire.

Another red flag is style drift – if the manager is moving their strategy around too much they are most likely suffering from hindsight bias. A simple example would be speeding up or slowing down the trend system due to recent performance. When slow trend systems have worked, if they move their system to be slower they are chasing past performance, which is not a great plan.

Carry is a good fit for trend – as long as it isn’t too big and as long as it is multi-asset class. Macro strategies are also good because they have similar positions but they tend to get into positions at different times.

For a global perspective, CRO is not really a new concept but in the US the term and concept is new. In 2012, one of the largest Swedish Pension funds discussed a similar objective and they built what they called a “protection portfolio” which has a similar mandate to the CRO mandates in the US. Pension funds in Europe, the Middle East, and Japan have been investing in Managed Futures with a similar objective for many years. US pensions have been somewhat newer to investing in Managed Futures.

How does CRO shift the conversation about Managed Futures? Historically Managed Futures has been bucketed in with “convergent” strategies in hedge funds. In this case, when compared with these strategies they came up short as a stand-alone investment. The Sharpe ratios are lower than the typical hedge fund.

On the other hand, from a “global” portfolio perspective they are one of the best global risk reducers. Hedge funds tend to have higher Sharpe ratios, but while they are generally uncorrelated they tend to correlate during periods of crisis, or when credit or liquidity risk are an issue. In this case, they are “locally optimal” in a hedge fund portfolio but not always globally optimal from the entire portfolio perspective.

A CRO strategic class is meant to put strategies like managed futures (in particular trend following) into a different bucket with a different strategic objective than hedge funds. Given this, investors have understand that if they are looking for risk mitigation they may make different choices than if they are looking for something that is just uncorrelated.

**No offence, but trend following on equity indexes is not good at all as a stand-alone.**

I would estimate the Sharpe ratio about 0.08 over long horizons (see our book on trend). This doesn’t mean you shouldn’t do it but it’s not generally that fun to do. It is a part of a trend system but I wouldn’t advise doing it alone. Timing seems to be more difficult in equities, traditionally trends on short rates and fixed income have had the best Sharpe ratios.

**If the S&P goes down, CTAs will tend to short the S&P, but it will depend on how quickly the S&P goes down and what happened prior to the S&P going down.**

Most CTAs who focus on longer-term trends may reduce their long positions but not start shorting until some time after the S&P goes down. This is why Q1 has been hard for CTAs even in an equity correction. Since the long term trend in equities was up up up, they were long equities, but when this trend reverted they lost on the correction and had to reduce equity positions.

**A key thing to remember is that CTAs tend to have long term signals it can take some time for them to go from long to short in any asset.**

Great question, I would suggest a 20% allocation but I do agree that it requires the right framing for an investor. The investor needs to see the investment as complementary to the remainder of the portfolio, not necessarily just crisis alpha. This is precisely why a CRO or strategic class for pensions is a big step for understanding managed futures. If they are less comfortable with hugging their peers or a benchmark, a lower allocation may be more appropriate. I often focus on also telling them with this investment it is important to not buy high and sell low. Many investors invest after a big “divergent event” and then they get frustrated that it doesn’t repeat. If they want to reduce an allocation, reduce after, not at the bottom. This is why framing the investment is so important.

Historically shorts do not preform as well as longs. This is simply because over the long run many assets with a risk premium tend to up-trend.

But since most investors hold long assets, being able to be short from time to time can have great complimentary properties. Without doing any analysis, I would suggest that the contribution is 80/20 or 70/30.

In our book, Alex and I look at the long bias in equities in Chapter 13 of our book using the equity bias factor. You can see there that most of the time it is better to have more long bias, but sometimes when it really matters being short can really add value.

I would suggest Robert Carver’s book “Systematic Trading” and Andreas Clenow’s book “Following the Trend.”

**The article below illustrates how capital efficiency, taxes, and fees impact portfolio choices given what we feel to be reasonable assumptions.**

However, we recognize that **you** may have different views on these variables.

That’s why we prepared a simple online application so that you can generate a bespoke report based on your own assumptions.

Returns to the simplest domestic capitalization weighted indexes have dominated virtually all active strategies over the nine years since the Global Financial Crisis. It’s not hard to understand why many investors have opted to eschew active strategies altogether, and instead have migrated *en masse* to the lowest cost index products. And for most investors, when considering traditional active long-only equity or bond mutual funds, it is prudent to place a high priority on fees. After all, most equity mutual-funds and smart-beta index products will have a correlation of 0.8 or higher to a traditional 60/40 portfolio^{1}. As a result, they are likely to produce only small marginal benefit in terms of portfolio efficiency.

However, the equation changes somewhat when investors start to consider allocations to alternatives.

Alternatives are constructed to capture returns from sources that are structurally uncorrelated with equities and bonds. Therefore, they may be expected to be uncorrelated to core portfolios, and substantially improve portfolio efficiency by increasing returns, reducing volatility, or both. However, many investors in alternatives evaluate these products using the same criteria that they use to compare traditional funds.

This is a mistake.

The evaluation of alternatives introduces an extra dimension into the equation that investors don’t need to think about with traditional equity funds. It’s the concept of capital efficiency.

Capital efficiency measures the amount of market exposure one can achieve with an investment per unit of capital invested. This has become a central theme for many institutional investors, who understand that they may face low expected returns on capital in their core portfolios over the next few years.

Capital efficiency can be best understood as “bang for your buck”.

Consider two funds, A and B, with similar expected Sharpe ratios, fees and taxes. In other words, the Funds are equally efficient. However, Fund A is run at 6% volatility while Fund B is run at 12%. This is possible in the world of alternatives because they often involve leverage or the use of derivatives like futures.

Given that Fund B runs at 2x the volatility of Fund A, Fund B should be expected to produce 2x the excess returns. In other words, an investor who carves out 20% of their portfolio to invest in liquid alternatives would gain 2x the marginal improvement in returns and Sharpe ratio on their portfolio by investing in Fund B instead of Fund A.

To help illustrate the point, imagine an investor owns a traditional portfolio consisting of 60% in a US equity index ETF and 40% in a bond index ETF. Acknowledging that expected returns are low for both stocks and bonds at the current time, the investor wishes to diversify with a 20% allocation to alternative investments. His objective is to raise his expected returns with minimal risk.

He evaluates his options and identifies three attractive funds:

- A market-neutral equity fund with an expected gross Sharpe ratio of \(1.1\), targeting 7% annualized volatility on up to 300% gross exposure, with a gross expense ratio of 2.24%, and 0 correlation with the current portfolio
- A managed futures mutual fund with an expected gross Sharpe ratio of \(1.1\), targeting 12% annualized volatility on up to 300% gross exposure, with a gross expense ratio of 2%, and 0 correlation with the current portfolio
- A GTAA ETF of ETFs with an expected gross Sharpe ratio of 0.8, expected
*long-term average*annualized volatility of 8.25% on a maximum of 100% gross exposure, with a gross expense ratio of 0.8%, and a correlation of 0.5 with the current portfolio.

Let’s dig a little deeper into how we arrived at these assumptions.

The market neutral fund is, by definition, hedged against market beta. Thus, it’s reasonable to assume it will have a consistent correlation of zero to the market. If the fund has equal risk exposure to five uncorrelated styles (say size, value, quality, investment, and momentum) with average Sharpe ratios of 0.5, the expected Sharpe ratio of a well-crafted portfolio is about \(0.5 \times \sqrt{5}=1.1\)^{2}.

A quick glance at larger equity market neutral funds shows that they tend to exhibit between 5% and 7% volatility, so we chose an estimate near the top of the range to give this option the benefit of the doubt (you’ll see why below). We used a fee estimate from a fund managed by a very large quantitative shop that is known for competitive fees in the alternative space.

Market-neutral funds typically turn-over greater than 100% of their portfolio per year, and all gains are taxable as ordinary income.

A managed futures fund typically allocates to a large basket of global asset classes across equities, bonds, currencies, and commodities, and sometimes to more esoteric markets like carbon credits.

A study by Hurst, Ooi and Pedersen (2017)^{3} found that a diversified trend-following strategy produced excess returns of 11% annualized, net of estimated trading costs, on volatility of 9.7% over 126 years from 1880-2016, for a Sharpe ratio of \(11\%/9.7\%=1.1\).

In its worst decade (1910-1919) the strategy produced net returns of 4.1%, while it produced over 20% annualized returns in its best decade (1970-1979).

Many managed futures strategies combine trend signals with carry signals, and this combination has improved the results to trend strategies in historical testing^{4}. When managed futures funds are properly constructed, 60% of gains on trading would typically be taxed as long-term capital gains, while 40% would be taxed as regular income.

The website Allocate Smartly has taken the time to replicate over forty global tactical asset allocation (GTAA) strategies using a common data set over similar time horizons.

We examined data from the site and found that the average strategy produced 8.6% annualized returns net estimated trading costs on 8.25% annualized volatility. Excess returns were 6.6% annualized after subtracting a 2% risk-free rate. The average of pairwise correlations for all strategies with a U.S. 60/40 portfolio was 0.5, and the strategies produced an average Sharpe ratio of 0.8.

A survey of GTAA ETFs listed on U.S. exchanges indicated a range of fees between about 0.8% and 1.7%, so we chose a fee estimate near the bottom of the range. In many cases, trading within Exchange Traded Funds does not produce taxable gains until the fund is sold, at which time an investor pays capital gains on the difference between sale and purchase prices.

How might the investor analyze the relative benefits of these funds to a portfolio over a 10-year evaluation horizon, given his objective to add a 20% allocation to these funds? For our analysis, we assume the core 60/40 allocation is held indefinitely (ignoring tax implications of periodic rebalancing), but the alternative fund is sold after 10 years.

Let’s assume the 60/40 portfolio has an expected net after-tax annualized return of 4% with average 12% annualized volatility. For the purpose of our analysis, it’s simpler to deal with *excess* returns, net of the expected risk-free rate^{5}. If we assume a 2% expected risk-free rate over the next 10 years, the 60/40 portfolio is expected to produce 2% *excess* annualized returns, with an expected Sharpe ratio of \((4\%-2\%)/12\%=0.17\). We assume zero fees on the core portfolio to reflect the costless options available to investors at Schwab, Fidelity, and other dealers, and the near-zero fees on many core ETFs.

We assume a tax rate of 39.6% on ordinary income and 23.8% on long-term capital gains^{6}. Table 1 summarizes the impact of adding a 20% allocation to each of our three hypothetical funds to the 60/40 portfolio, while Figure 1 illustrates the potential difference in ending portfolio value.

We present a detailed breakdown of the analysis that led to the values in Table 1 in an Appendix below.

Table 1: **Hypothetical Impact of Adding a 20% Alternative Fund Allocation to a Core 60/40 Allocation: Summary Statistics**

100% Core 60/40 | 80% Core + 20% Market Neutral Fund | 80% Core + 20% Managed Futures Fund | 80% Core + 20% GTAA ETF | |
---|---|---|---|---|

Weighted Average Fee | 0% | 0.45% | 0.4% | 0.16% |

Expected Net After-Tax Portfolio Return | 4% | 4.26% | 5.17% | 4.54% |

Expected Portfolio Volatility | 12% | 9.7% | 9.9% | 10.5% |

Expected Portfolio Net After-Tax Sharpe ratio | 0.17 | 0.23 | 0.32 | 0.24 |

$1 Grows To | 1.48 | 1.52 | 1.66 | 1.56 |

Source: Calculations by ReSolve Asset Management. For illustrative purposes only.

Figure 1: **Impact of Adding a 20% Alternative Fund Allocation to a Core 60/40 Allocation: Hypothetical Growth**

Source: Calculations by ReSolve Asset Management. For illustrative purposes only.

From Table 1 it’s clear that a 20% allocation to the alternative mutual funds adds up to 0.45% in weighted average portfolio fees, compared with just 0.16% in fees from the addition of the GTAA ETF. However, despite these higher fees and less advantageous tax status, a hypothetical 20% position in the managed futures fund increases expected wealth creation after 10 years by 1.37 times what would be expected from the core 60/40 portfolio on its own. This compares to improvements of just 1.08 times and 1.17 times the wealth creation that would be expected when capital is allocated to the market neutral fund and GTAA ETF, respectively.

**The key point is that the improvement in capital efficiency from managed futures overwhelms all other considerations.**

In addition, due to the low correlation between the managed futures fund and the core portfolio, the improvement in hypothetical returns is achieved with just slightly higher portfolio volatility than what we’d expect from an investment in the market neutral fund, and **less** volatility than an investment in the more highly correlated GTAA ETF.

As a result, an investment in the hypothetical managed futures fund could boost portfolio Sharpe ratio by 1.88x, representing a much larger boost than what one could achieve from the other funds.

The evaluation of alternatives introduces an extra dimension into the equation that investors don’t need to think about with traditional equity funds. Capital efficiency measures the amount of market exposure one is able to achieve per unit of capital invested. In other words, it quantifies “bang per buck” of an investment.

All things equal, it is often more advantageous to a portfolio’s performance, from an **after-fee, after-tax** perspective, to allocate capital within the alternative sleeve to strategies with low correlation and high target volatility, especially if the strategy trades futures. This is true **even if** these funds have higher fees and less favorable tax treatment. Focusing exclusively on fees and taxes misses the forest for the trees.

Note that the advantages of allocating to capital-efficient alternatives may be magnified through thoughtful “asset location”. In other words, investors with retirement accounts have the opportunity to hold less tax-efficient investments in IRAs or 401ks in order to defer or perhaps even eliminate the tax drag.

**The article above illustrates how capital efficiency, taxes, and fees impact portfolio choices given what we feel to be reasonable assumptions.**

However, we recognize that **you** may have different views on these variables.

That’s why we prepared a simple online application so that you can generate a bespoke report based on your own assumptions.

Just set your variables and generate your report – Go to the app now!

First, it’s critical to standardize the expected return across funds, after fees and taxes.

The market-neutral fund is expected to produce gross *excess* returns of \(1.1 \times 7\% = 7.7 \%\). Now let’s subtract fund fees of 2.24% for net returns of 5.46%. The fund returns are considered 100% ordinary income for tax purposes, so if we assume a tax rate of 39.6%, expected net after-tax return would be \(5.46\% \times (1-39.6\%)=3.3\%\). As a result, the new portfolio would be expected to produce an excess return of \(80\% \times 2\% + 20\% \times 3.3\%=2.26\%\) with annualized volatility of \(9.7\%\)^{7}, so that the expected net after-tax portfolio Sharpe ratio would be \(\frac{2.26\%}{9.7\%}=0.23\). All gains are crystallized each year.

The managed futures mutual fund is expected to produce gross *excess* returns of \(1.1 \times 12\% = 13.2 \%\) less fund fees of 2% for net returns of 11.2%. Since the fund trades futures, gains should be taxed at 60% long-term capital gains and 40% ordinary income. If we assume long-term gains are taxed at 23.8% and ordinary income is taxed at 39.6%, we can estimate a combined tax rate on gains of approximately \(60\%\times23.8\%+40\%\times39.6\%=30.12\%\), so expected net after-tax return would be \(11.2\% \times (1-30.12\%)=7.83\%\). As a result, the new portfolio would be expected to produce an excess return of \(80\% \times 2\% + 20\% \times 7.83\%=3.17\%\) with annualized volatility of \(9.9\%\)^{8}, so that the expected net after-tax portfolio Sharpe ratio would be \(\frac{3.17\%}{9.9\%}=0.32\). Again, all gains are crystallized each year.

Finally, the GTAA ETF is expected to produce gross *excess* returns of \(0.8 \times 8.25\% = 6.6 \%\). Now let’s subtract fund fees of 0.8% for net returns of \(5.8\%\). Because of the unique tax treatment of trading inside certain ETFs, we will assume the fund crystallizes zero capital gains each year.

However, while taxes on the market neutral and managed futures funds are crystallized in full each year, the ETF structure simply defers capital gains until the ETF is sold. Assuming the ETF is sold at the termination of our 10-year holding horizon, the investor would expect to pay 23.8% long-term gains tax on the compounded gains at termination. If we assume a net growth rate of \(5.8\%\) after fees, the fund should turn \(\$1\) into \(\$1.76\) after 10-years. The investor would pay 23.8% on the gain of \(\$0.76\), which reduces the net gain to \(\$0.76\times(1-23.8\%) = \$0.58\). Thus, the net expected annualized return actually works out to \((1+0.58)^{(1/10)}-1=4.7\%\).

Therefore, the new portfolio would be expected to produce an excess return of \(80\% \times 2\% + 20\% \times 4.7\%=2.54\%\) with annualized volatility of \(10.5\%\)^{9}, so that the expected net after-tax portfolio Sharpe ratio would be \(\frac{2.54\%}{10.5\%}=0.24\).

- Author’s estimates based on a small survey of U.S. equity mutual funds and smart-beta ETFs.↩
- The volatility of a portfolio with N assets and average pairwise correlations of \(\theta\) is \(s_p=\bar{s_a}\sqrt{1/N +\bar{\theta}}\). Note that for any \(\bar\theta<1\) portfolio volatility will be lower than the average volatility of the constituent assets, while portfolio expected return is constant, resulting in a higher Sharpe ratio.↩
- See https://dx.doi.org/10.2139/ssrn.2993026 Exhibit 1.↩
- For example the paper “Time-Series Momentum, Carry and Hedging Premium” by Molyboga, Qian, and He (2017) suggests that adding carry to trend improves the Sharpe ratio by 0.17. See https://dx.doi.org/10.2139/ssrn.3075650 .↩
- Note that we use annualized returns rather than average returns when translating between Sharpe ratios, returns and volatilities. There are some issues with this, but it was important to compare expected compound growth, which requires annualized returns. The use of arithmetic means rather than annualized (geometric) means should not have any material impact on the economic interpretation of the results.↩
- Assumes the highest federal tax bracket for ordinary income (~39.6%), with commensurate federal tax rate on capital gains and dividends and including Medicare Contribution Tax of 3.8% (total of ~23.8%). Source↩
- \(\sqrt{80\%^2\times12\%^2+20\%^2\times7\%^2+2\times80\%\times20\%\times12\%\times7\%\times0}=9.7\%\) .↩
- \(\sqrt{80\%^2\times12\%^2+20\%^2\times12\%^2+2\times80\%\times20\%\times12\%\times12\%\times0}=9.9\%\) .↩
- \(\sqrt{80\%^2\times12\%^2+20\%^2\times8.25\%^2+2\times80\%\times20\%\times12\%\times8.25\%\times0.5}=10.5\%\) .↩

Confidential and proprietary information. The contents hereof may not be reproduced or disseminated without the express written permission of ReSolve Asset Management Inc. (“ReSolve”). ReSolve is registered as an investment fund manager in Ontario and Newfoundland and Labrador, and as a portfolio manager and exempt market dealer in Ontario, Alberta, British Columbia and Newfoundland and Labrador. Additionally, ReSolve is an SEC registered investment adviser. ReSolve is also registered with the Commodity Futures Trading Commission as a commodity trading advisor and a Commodity Pool Operator. This registration is administered through the National Futures Association (“NFA”). Certain of ReSolve’s employees are registered with the NFA as Principals and/or Associated Persons of ReSolve if necessary or appropriate to perform their responsibilities.

PAST PERFORMANCE IS NOT NECESSARILY INDICATIVE OF FUTURE RESULTS.

The information contained herein are of an illustrative nature and for informational purposes only and does not constitute financial, investment, tax or legal advice. These materials reflect the opinion of ReSolve on the date of production and are subject to change at any time without notice due to various factors, including changing market conditions or tax laws. Where date is presented that is prepared by third parties, such information will be cited, and these sources have deemed to be reliable. Any links to third party websites are offered only for use at your own discretion. ReSolve is a separate and unaffiliated from any third parties listed herein and is not responsible for their products, services, policies or content of their website. All investments are subject to varying degrees of risk, and there can be no assurance that the future performance of any specific investment, investment strategy or precut referenced directly or indirectly in this website will be profitable, perform equally to any corresponding indicated historical performance level(s), or be suitable for your portfolio. Past performance is not an indicator of future results.

#### Calendar 2017 was an exemplary year for ReSolve strategies.

#### Strategies targeting long-term equity-like volatility produced almost double the return of U.S. stocks, with some mandates achieving over 40% growth.

#### Strong results are a function of proven process + relentless discipline + favourable conditions. We offer a comparative analysis to cement the point.

Statistics | Return | Volatility | Max Drawdown |
---|---|---|---|

AAA 16% | 42.37% | 16.30% | -5.72% |

AAA 12% | 29.57% | 11.63% | -4.40% |

AAA 8% | 17.67% | 7.62% | -3.08% |

AAA 8% Un-Levered | 15.47% | 5.73% | -1.78% |

Risk Parity 12% | 24.06% | 8.78% | -4.39% |

Risk Parity 6% Un-Levered | 12.66% | 4.44% | -2.13% |

Tactical Equity | 17.66% | 7.50% | -2.74% |

SOURCE: ReSolve Asset Management. PAST PERFORMANCE IS NOT NECESSARILY INDICATIVE OF FUTURE RESULTS. See disclaimer.

Calendar 2017 was an exemplary year for ReSolve strategies. Per Table 1, at a 12% annualized volatility target, consistent with the long-term risk of a traditional 60/40 global balanced portfolio^{1}, performance of Adaptive Asset Allocation (AAA) and Global Risk Parity (RP) mandates rivaled most global equity benchmarks^{2}. Adaptive Asset Allocation mandates targeting a volatility closer to the long-term profile of global equities (16%-20%) produced almost double the return of U.S. stocks, with some mandates achieving over 40% return.

Figure 1: Live geometrically linked net daily returns to ReSolve USD mandates in 2017. SOURCE: ReSolve Asset Management. See disclaimer.

Figure 1 compares the net total return trajectories of ReSolve’s Adaptive Asset Allocation, Risk Parity, and Tactical Equity mandates against global equities^{3}.

Long-time investors know that ReSolve takes an ensemble approach to strategy design to minimize the risk of specifying the exact wrong model for the current market environment. The current version of Adaptive Asset Allocation uses resampling to draw many combinations of lookback and weighting parameters to estimate momentum, correlations, and volatilities, each time we create an optimal portfolio. In addition, the Strategy uses four techniques to measure momentum and trend; three transforms of momentum data; and five portfolio optimization methods.

One of these methods is thematically consistent with the toy model we described in our seminal whitepaper, Adaptive Asset Allocation: A Primer (2012 rev 2015), where assets are ranked on average momentum, and the top half of assets are held in minimum variance weights. But we also apply forty other combinations of momentum measures, transforms, and optimization methods when we form portfolios. Final portfolios represent a thoughtful combination of all forty-one sub-strategies. While the long-term performance profiles of these sub-strategies are statistically indistinguishable, we observe fairly wide dispersion in performance from year-to-year.

Figure 2: Return trajectory of Adaptive Asset Allocation sub-strategies in 2017. Live aggregate strategy performance is highlighted in gold. SOURCE: ReSolve Asset Management. See disclaimer.

Figure 2 illustrates how each of the forty-one sub- strategies that comprise ReSolve’s Adaptive Asset Allocation approach performed this year, without any volatility targeting. The worst strategy produced less than 9%, while the best generated almost 21%. This is quite a range given that the strategies all seek to maximize momentum and trend while minimizing portfolio volatility. Meanwhile, the live aggregate strategy^{4} delivered net returns of 15.5%.

Some investors may be wondering why the unlevered version of Adaptive Asset Allocation lagged global stocks. After all, global equities produced one of the best trends ever in 2017. The reason is that ReSolve’s strategies, in contrast with many other ostensibly similar strategies, explicitly account for the fact that adding assets with lower expected Sharpe ratio, but low correlation to other portfolio assets, will improve the expected risk-adjusted performance of the portfolio.

Take the case of a two asset portfolio, where one asset has twice the expected Sharpe ratio of the other asset. If the two assets are perfectly correlated, an investor should simply choose to invest all her capital in the asset with the best expected performance. However, if the assets are uncorrelated, the investor is always better off, in terms of expected risk-adjusted performance, to invest in a combination of the two assets rather than simply investing all her funds in the asset with the best expected performance^{5}.

Figure 3: Optimal risk budget for a lower Sharpe asset given different correlation assumptions. SOURCE: ReSolve Asset Management.

Figure 3 considers the case of a two asset portfolio, and quantifies the risk budget^{6} that should be allocated to an asset with a lower expected Sharpe ratio (with the remainder in the higher Sharpe asset), to maximize the portfolio’s expected risk-adjusted performance. Given the example where one asset has half the expected Sharpe ratio of the other asset, the violet line in the chart shows that the most efficient portfolio would combine about 63% in the higher Sharpe asset with a 37% position in the lower Sharpe asset, when the correlation between the two assets is zero. Indeed, assets with negative Sharpe ratios may improve the efficiency of a portfolio given sufficient negative correlation. The light blue line tracks the optimal allocation to a lower Sharpe asset given a correlation of -0.5 with the higher Sharpe asset. You can see that an asset with a Sharpe ratio that is negative and 0.25 times as large as the higher Sharpe asset would still deserve a risk budget allocation of almost 30%!

Why does this matter? Many popular Global Tactical Asset Allocation strategies simply emphasize assets with favorable characteristics, which often leads to portfolios that are fully allocated to just one asset class. In years like 2017, this approach works well, because there are no trend reversals, and the best approach in retrospect was simply to hold 100% in emerging stocks all year. However, under conditions of uncertainty, portfolios that account for correlations will hold more diverse assets, which positions them to capture a smaller proportion of losses on reversals, and to transition into alternate asset classes more quickly.

This is one of the reasons why even a simple Adaptive Asset Allocation strategy is the top ranked strategy by Sharpe ratio, and second-best strategy by annualized total return, over the past twenty years (in simulation) according to a popular GTAA portal^{7}.

PERFORMANCE ATTRIBUTION ANALYSIS

The relatively strong performance of ReSolve’s strategies this year accrued from a confluence of factors. First, most global assets delivered positive returns on the year, and several markets produced spectacular growth. Indeed, the top performing assets produced their returns with low volatility and high Sharpe ratios, so that it was possible to amplify returns using the Capital Market Line. Second, there was a large disparity between the top and bottom- performing assets, which made it easy for our momentum indicators to separate the wheat from the chaffe. In addition, the best performing assets outperformed the worst performing assets very consistently all year, with few rank-reversals.

At the end of 2015, we produced a report to explain some of the reasons why ReSolve’s strategies had struggled that year. We described the necessary conditions for long-only active asset allocation strategies to thrive, and compared the conditions that prevailed in 2015 with conditions during other calendar years over the previous two decades. The analysis made clear just how extraordinarily challenging that year was relative to other years on a wide variety of dimensions. We thought that this year it would be interesting and instructive to contrast conditions in the current year with conditions that prevailed in calendar 2015, using the same analytical tools.

This exercise is useful because, while we have made incremental improvements to our strategies over the past few years, the fundamental mechanics that inform our strategies have not changed materially. By revisiting the metrics that we used to explain the unexciting performance in 2015, and demonstrating how current conditions would have predicted strong performance this year, it will become clear that results in any one year do not reflect on the value of our strategies, but rather reflect the role of luck on short term investment results.

To kick-off our analysis, let’s first examine just how amazing the current year was for equity investors. While investors in U.S. equities have enjoyed outstanding returns for the past nine years, 2017 rewarded investors from virtually every corner of the globe. Moreover, per Figure 4, equities produced their mouth-watering returns with vanishingly low monthly volatility, leading to stratospheric Sharpe ratios at monthly scale.

Figure 4: Rolling 12-month annualized Sharpe ratio. SOURCE: Global Financial Data from 1926 – 2015, CSI Data for VT ETF from 2015 – 2017. PAST PERFORMANCE IS NOT NECESSARILY INDICATIVE OF FUTURE RESULTS. See disclaimer.

In fact, 2017 was the only calendar year since 1926 that global equities offered investors positive returns in every single month^{8}.

Recall that ReSolve’s Adaptive Asset Allocation and Risk Parity mandates are long-only. These strategies will produce positive returns only if one or more markets are rising. In most years, this is not an issue, as there is almost always a strong bull market somewhere. During periods of strong global growth surprises, equity markets should produce good returns. In other years characterized by negative growth or outsized inflation shocks, other assets, like high grade bonds or commodities, should do well. But there are usually good opportunities for dynamic, long-only strategies to prosper, as evidenced by the top row of Figure 5.

Figure 5: Total returns to major asset classes in calendar years 2000 – 2017, sorted from highest to lowest. SOURCE: Calculations by ReSolve Asset Management. Data from CSI, S&P Dow Jones, Deutsche Bank. PAST PERFORMANCE IS NOT NECESSARILY INDICATIVE OF FUTURE RESULTS. See disclaimer.

Most years present good opportunities for long-only active asset allocation strategies to prosper, but some years, like 2001 and 2015, conspire to produce few profitable opportunites.

A historical case study will help cement the point. Imagine that on January 1st of each year, we knew in advance which major global asset classes9 would deliver returns above the group’s average, as well as positive returns, in the coming year. Of course, knowing the future in this way is impossible, but it helps to illustrate the available opportunity set for an active long-only global asset allocation strategy. Perhaps unsurprisingly, an investor with this type of perfect foresight – who allocated capital equally among the top half of positively performing assets each year – would have produced average annual returns of 21.5% since 1991.

Moreover, as can be seen in Figure 6, this strategy would have generated returns over 15% in 21 out of 27 years, and gains of more than 10% in 23 of the years. Fully 26% of years provided returns of greater than 25%! Any investor in his right mind would gleefully invest in this ‘crystal ball’ strategy…

**Continued in the report…**

*Footnotes*

*1. A portfolio consisting of 60% MSCI Global All-Cap World Index, 20% U.S. Aggregate, and 20% S&P/Citigroup International Treasury Ex-US total return indexes exhibited an annualized daily volatility of 11.7% since 1990.* * 2. The Vanguard Total U.S. Stock Market ETF (VTI) returned 21.28%, and the global Vanguard Total Market ETF (VT) returned 24.57%.* * 3. Total return index of the Vanguard Total Stock Market ETF (VT)* * 4. ReSolve Adaptive Asset Allocation: 8% Volatility (USD) unlevered mandate.* * 5. Note the repeated emphasis of the word expected. Obviously if an investor has perfect information about which asset will outperform in the coming period, they should invest all their capital in that asset, with maximum leverage. Unfortunately, markets rarely present investors with perfect information, so investors must act to optimize their expected future wealth under conditions of uncertainty, which supports diversification.* * 6 .If the assets have equal expected volatility then risk budget is the same as portfolio weight.* * 7. See the “Strategies” page in the member’s section at Allocate Smartly https://allocatesmartly.com* * 8. Actually, the twelve months ending November and December 2017 represent two of just sixteen twelve-month rolling periods in the past 1075* * months where global equities offered investors positive returns in every single month.*