## Abstract

As smart beta investments in institutional portfolios have grown—along with the additional complexity introduced by multifactor approaches—there is an emerging need for guidance on how to allocate across the ever-increasing array of smart beta products. Smart beta and multifactor investments are exposed to a common subset of elementary smart betas, combined with more idiosyncratic residual exposures. Accounting for the incidental exposures to common factors as well as the idiosyncratic exposures is necessary in designing a well-diversified and efficient portfolio. Accordingly, this article develops a standard framework for investors to blend single-factor and multifactor smart beta within a total portfolio context. A case study demonstrates how the methodology can be applied to attain better portfolios.

**TOPICS:** Analysis of individual factors/risk premia, portfolio construction

There has been extraordinary growth in the use of smart beta funds by institutional investors, both large and small.^{1} Many investors have strategically allocated assets away from traditional index and active equity funds into multiple smart beta funds by balancing their exposures across widely understood ideas, such as value, small (size), momentum, quality, and low volatility. Furthermore, they have included more complex multifactor funds that combine multiple smart beta ideas into a single portfolio that arguably improves efficiency.^{2} It is unlikely that the evolution of these complex multifactor investments that we call collectively *advanced beta* has resulted in portfolios for which the cumulative exposure is known or in any way ideal. Because one of the important potential benefits of smart beta investing is enhanced transparency of exposures and risks, this is a key problem for its users. Now, with the ever-increasing allocation to advanced beta products, investors need guidance on how to construct an overall portfolio that improves the likelihood of attaining better investment outcomes.

The good news is that standard approaches to investment performance analysis and portfolio construction can be adapted to integrate multiple smart beta approaches with other assets. The first step is to understand the underlying exposures of each smart beta strategy and their correlations to other strategies. We then need to be explicit about our expectations for the performance of each strategy, recognizing that an analyst’s judgment is always required. Finally, procedures for portfolio construction, built from standard methods, can be used to optimize expected utility for the investor, based on our estimates and judgments.

With this approach, we are better equipped to answer the following questions:

• How do we discern whether a new smart beta fund may be a valuable addition to the portfolio?

• When does a multifactor strategy add value beyond a portfolio of simple factors?

• What is an efficient portfolio of smart betas and advanced betas?

• When should smart beta and multifactor portfolios be combined with traditional indexes?

• How does the investment policy affect the best allocation to advanced beta?

We begin with a framework to assess the potential value-add of advanced beta candidates and to establish forward-looking assumptions. This provides the essential information needed to understand the risk and benefits of combining exposures. Next, we define an objective function that leads to the principles for optimal combinations of smart betas and advanced betas. Finally, a case study demonstrates how the methodology can be applied to attain better portfolios.

## EVALUATING SMART BETA AND ADVANCED BETA PORTFOLIOS

*Smart beta* refers to rule-based long-only market exposures that are expected to yield risk-adjusted returns in excess of the returns available with market-capitalization-weighted indexes of standard asset classes. Advanced beta, also called *multifactor portfolios*, refers to portfolios that are constructed bottom up from several smart betas or other factors. Like any other active exposure, an individual smart beta or advanced beta strategy may outperform or underperform the benchmark during any period. Because of this variation, and the challenge of timing individual smart betas, a blended portfolio of smart betas makes good sense and has been recommended by most analysts. However, this does not mean that we should add *every* new smart beta to our portfolio.

We need a criterion to assess whether a new smart beta is likely to be a value-add for the portfolio. Satisfying a value-add criterion involves two questions. First, can we explain the performance of the candidate strategy via strategies that are already available to us? Second, to the extent that the candidate strategy does not duplicate what is already available, is the additional (residual) exposure attractive? Technically, we ask whether the candidate strategy displays evidence of a positive alpha after controlling for exposures from its normal portfolio.

### A Factor Model Based on Elementary Smart Betas

A practical way to answer these questions with respect to new smart beta candidates is first to define *elementary smart betas* (ESBs) as a small set of representations of smart beta that are readily investible and represent the factors that are most strongly supported by long consensus of financial research. The ESB factors are defined as the returns of an associated smart beta index in excess of the broad market index returns. The excess return of any advanced beta candidate strategy then may be represented as a mix of asset class, smart beta, and residual exposures, as follows:

where is the excess return of the advanced beta candidate over the market index; *f*_{b} is the factor weight for market index exposure associated with advanced beta; is the market index (benchmark) return; *f*_{esb,i} are factor weights for the *i*th ESB; is the excess return of the *i*th ESB factor; and is the residual return. The tilde notation, as in , refers to a random variable where the mean is *r*_{b} and the variance is . The factor weights are estimated by returns-based style analysis (constrained regression) overlaid with the analyst’s judgment.^{3} A strategy with nonzero *f*_{b} is referred to as *leveraged* (*f*_{b} > 0) or *deleveraged* (*f*_{b} < 0). We would tentatively accept a smart beta candidate in the presence of a sufficiently positive alpha, but we otherwise reject the candidate.

### Choice of Elementary Smart Betas

We initially define the elementary smart betas as the Fama–French–Carhart set—value, small (size), and momentum—which has been researched and practiced over decades. We use the MSCI All Country World Index (ACWI) versions of these three factors in this article because these indexes are reasonable proxies for relatively low-cost, low-tracking-error, investible replications of the SB indexes.^{4} Also, confirming many prior studies, note that the value, small, and momentum active returns are uncorrelated, with value and momentum negatively correlated, supporting them as good initial choices for elementary smart betas for this sort of factor model.^{5} In practice, we have found that using these three ESB factors is a reasonable approach for establishing a normal portfolio, or *effective asset mix*, for a wide range of smart beta and multifactor portfolios. Should the list of elementary smart betas be expanded to include popular smart betas like quality and low volatility? Possibly, but the evidence suggests that quality and low volatility can be approximated as combinations of the primary ESBs in the context of a four-factor model.^{6} We leave open the possibility that the set of ESBs may be redefined or expanded in the future, as we develop more experience with smart beta.

### Selection Criteria and Model Assumptions

This factor model has two key properties: It serves to evaluate new smart beta strategies and to provide a foundation for blending several strategies. First, the model highlights that, to be justified, a new strategy should provide exposures beyond a simple combination of what we already have readily available; otherwise, it cannot add value. A candidate strategy may add value only if we expect it to yield a materially positive alpha relative to other strategies already available to us. Our expectation may be based either on historical or subjective estimates, a theme that we explore further in our case study. A positive expected alpha is not itself sufficient to justify investment in an advanced beta; the alpha must also be attractive relative to its residual risk and within an overall active risk budget. This is analogous to a traditional active manager having a convincing expected information ratio after controlling for its effective portfolio mix. Second, an investor who is considering new smart beta strategies most likely already has a firm belief in the advantages of the elementary smart betas, including an understanding of the range of expected premiums and volatility of ESBs. Therefore, the choice of a small set of investible elementary smart beta factors provides a logical foundation for portfolio construction.

Setting assumptions involves first establishing expected return and covariance estimates for the ESBs. For candidate strategies, we also need to estimate factor weights and residual risk and return. The latter involves subjective estimates akin to evaluating an equity manager’s expected alpha—not an easy task. The factor model based on ESBs provides a structure for understanding how combinations of smart beta and multifactor strategies accumulate in the portfolio and what changes in portfolio weights might improve overall performance. Later, in the case study section of this article, a fully worked example will show some reasonable assumptions that are established as guidance. Of course, we may be wrong about our factor weights (model uncertainty), and we may be wrong about our estimates of the residual returns (parameter uncertainty); nevertheless, this ESB factor model helps reduce ambiguity in candidate strategies and can lead to potentially more robust conclusions.^{7}

## BLENDING ADVANCED BETAS

The objective of investing in smart betas, multifactor strategies, and other advanced betas is to improve overall portfolio performance in terms of expected return and risk. The Technical Appendix provides a framework for optimizing expected utility of returns that produces mean–variance-efficient allocations to advanced betas in the total portfolio context. Here, we simplify the narrative by considering a single asset class (e.g., global equity) with investments in several ESBs but with only a single advanced beta. We also simplify by assuming that elementary smart betas are uncorrelated, assuming advanced beta residuals are uncorrelated with ESBs, and ignoring budget constraints. The total return of the portfolio is

2The total return is composed of the benchmark return (the first component), the direct exposure to the elementary smart betas (the second component), and additional exposures associated with advanced beta (the next three components derived from the factor model in Equation (1)). Allocations to advanced beta modify exposure to the asset class (if strategies are leveraged or deleveraged), add incidental exposure to the ESBs, and add exposure to advanced beta residuals. The expected return and variance, with rearrangement, are

3 4where a_{ab} and are the residual return and variance of advanced beta. This is a simplified version of the vector and matrix relationships (Equations (A-1) and (A-2)) for expected return and variance in the Technical Appendix.

### Optimal Allocations: Total Return

We assume that investment policy is fixed but that our holdings of ESBs and advanced beta are unconstrained. The optimal (mean–variance-efficient) holdings then are determined by the first-order conditions

5 6These results are analogous to the conditions in Equations (A-7) and (A-8) in the Technical Appendix. The big picture is that optimal allocation to elementary smart betas is driven by the return/variance ratio of the ESBs (the first term of Equation (5)), with an adjustment for the incidental exposure to ESBs caused by optimal holdings of advanced beta (the second term of Equation (5)). Similarly, the optimal allocation to advanced beta is driven by the return/variance ratio associated with the advanced beta residual and the market index exposure caused by advanced beta (the first term of Equation (6)). There is also an adjustment to account for the risk associated with leverage/deleverage effects on investment policy. Notably, the underlying style of the advanced beta in terms of elementary smart betas as determined in Equation (1) is irrelevant because any exposure to ESBs is fully adjusted in Equation (5) to attain optimal exposure to ESBs. In effect, we choose the optimal allocation to advanced betas, and then the ESB allocations provide a *completion portfolio* to ensure that the overall allocation to ESBs is also optimal.^{8}

### Optimal Allocations: Active Return

The preceding discussion centered on the problem of asset allocation in the total return space. A related problem is optimizing the expected utility of active returns to maximize . In this problem, we are concerned with the active return (in excess of the policy benchmark) and the active risk (tracking error relative to policy benchmark). Starting with Equation (2), we remove the policy component and proceed to determine first-order conditions for optimal allocations with this new objective. The unconstrained optimal holdings for the active problem are

7 8The result of Equation (7) is the same relationship as in Equation (5) for optimal holdings of ESBs, and the result of Equation (8) is a simplified relationship for optimal holding of advanced beta. The optimal holdings of advanced beta are again proportional to the ratio of expected return and risk associated with leverage and residual risk from advanced beta, but the term for the leverage/deleverage impact on investment policy is absent. This result of Equation (8) is analogous to Equation (A-10) in the Technical Appendix.

By varying an investor’s risk preferences (l), we can generate a range of efficient portfolios and compare various portfolios to this efficient set. Exhibit 1 demonstrates an example of the relationships and distinctions between portfolios of advanced betas and ESBs in the active efficient frontier space that includes the requirement that holdings be long only and a total holdings constraint. In this example, Portfolio 0, with no allocations to ESBs or advanced betas, is an index fund with no active return and risk. Portfolio 5 achieves the greatest possible expected active return given an active risk budget of 3%. The final portfolio can be thought of as the sum of Portfolio 1, the best achievable advanced beta exposures; Portfolio 3, the incidental deleveraging associated with those exposures; and Portfolio 4, the best portfolio of ESBs, adjusting for the incidental exposure in Portfolio 3. Portfolio 2 is the incidental ESB exposure from the allocation to advanced betas; the difference between Portfolio 4 (best ESBs) and Portfolio 2 (incidental ESBs) indicates the improvement in Portfolio 5 resulting from the completion portfolio.

Depending on the investor’s risk preferences (l), the investor may choose a higher or lower active risk budget than Portfolio 5; in practice, it is common to set an active risk budget for each asset class. If a lower risk budget is chosen, then the optimal Portfolio is a combination of Portfolio 0 and Portfolio 5 and falls on the straight line between the two portfolios. These portfolios include an allocation to index funds along with a proportionate reduction in the allocation to the optimal blend of ESBs and advanced betas, and they dominate Portfolios 1–4. If a higher risk budget is desired, then the optimal Portfolio lies on the active efficient frontier to the right of Portfolio 5, where the holdings and long-only constraint are binding. Portfolios to the right of Portfolio 5, with no index allocations, have higher expected return than Portfolio 5 but with a decreasing information ratio as active risk increases.

### Impact of Leveraged or Deleveraged Advanced Beta

Advanced betas, and many well-known smart betas, as described in Equation (1), are frequently found to have factor weights with respect to the underlying market index. For example, low-volatility smart betas—and advanced betas built upon them—typically have a large negative factor weight with respect to the equity index (*f*_{b} < 0), and this may reduce the desirability of the associated advanced beta. Allocations to smart betas with this feature consume a large amount of active risk budget; in the total portfolio context, they effectively change the exposures set by investment policy. If this exposure is well understood, then the impact can be reversed fairly cheaply via derivatives, by compensating adjustments to the strategic asset allocation or by demanding strategies without leverage. A further benefit of neutralizing the leverage issue is that, in this case, the general problem of optimizing the allocations to active betas in the total portfolio context can be simplified to the active-only case, as demonstrated in the Technical Appendix. In our case study, we will investigate the impact of leverage adjustment on total portfolio performance further.

## CASE STUDY

An example will help demonstrate the principles and practicality of blending advanced betas and smart betas. We consider an institutional investor that is seeking to improve investment performance of the total portfolio via advanced beta, having already established an investment policy of 70% equities and 30% bonds (other asset classes are ignored for simplicity). We will set out the assumptions and then show the optimal holdings of advanced beta within the equities portfolio as described in the prior section, but relying on the more complete framework from the Technical Appendix.

### Assumptions

We need to establish assumptions about asset class, elementary smart beta, and advanced beta expected returns and covariance. Exhibit 2 shows the historical return and risk for asset classes and ESBs and the assumed forward-looking estimates.^{9} The forecasted risk is rounded from the historical data, and the forecasted return is based upon an assumed forward-looking Sharpe ratio. The Sharpe ratios for the ESBs are assumed to be equal across smart betas but lower than the asset class ratios. The estimates are intended to be conservative; they are lower than historical experience, except in the case of the value premium, which performed unusually poorly in the backtest period compared with its long-term average. Exhibit 3 shows the historical estimates for correlations. Except for the correlation between value and momentum, no estimates were statistically significant during the period. Our forward-looking assumptions are that the value–momentum correlation is -0.5, and the correlation between bonds and equities is +0.2. Other correlations between ESBs are set to +0.2 to provide conservatively higher portfolio risk estimates.

The assumptions for advanced beta strategies are based on fitting the factor model in Equation (1) plus the analyst’s judgment. We study four hypothetical candidate advanced beta strategies, characterized in Exhibit 4. The factor weights in these samples have been chosen to be broadly representative of some types of commercially available strategies, based on our own returns-based style analysis and judgment, and reflecting what we should know from our due diligence on each multifactor strategy. For example, the first candidate is a variation on a value strategy, with larger small-cap exposure than found in the MSCI ACWI Value Index; the second represents a more extreme small-cap tilt and tries to capture factor exposures not included in our set of elementary smart beta (such as quality); the third is a more straightforward quality approach; and the fourth is a representation of a low-volatility approach.

We assume reasonable idiosyncratic risk based on our examples, that the residuals are uncorrelated, and an equal information ratio for each advanced beta strategy. We therefore obtain a forward-looking estimate of expected residual return by multiplying the information ratio by the residual risk. Nothing in the methodology prevents us from making other assumptions, based on our study of candidate strategies. For example, an analyst could choose to assume different information ratios based on knowledge of the advanced beta strategy or manager.

### Optimal Holdings and Performance

We shall assume that an active risk budget of 3.0% at the asset class level is set for the exposures associated with allocations to smart beta and advanced beta. Exhibit 5 shows the holdings and performance for the optimal (mean–variance-efficient) portfolios that were determined from a standard optimizer. Using only elementary smart betas (column 1), the portfolio has an expected active return of 1.4%. The portfolio allocates most heavily to value and then momentum and least heavily to small. To understand why this allocation makes sense, recall that we assume equal expected information ratios across the smart betas, so the allocations are driven primarily by risk. Value and momentum are negatively correlated, so allocating more to this pair of smart betas has a strong diversification benefit. In balancing between these two, value is given a higher allocation because it has less risk than momentum. Small also has a place in the portfolio, as it is uncorrelated with value and momentum, and it has an intermediate level of risk. Because of diversification benefits, the smart beta–only portfolio attains an expected information ratio of 0.48, whereas the component smart betas have standalone information ratios of only 0.25.

The portfolio with advanced beta only (column 2, with no direct allocation to ESBs) makes allocations to all four strategies. The allocation across advanced betas is driven by the attractiveness of the residual return and risk, but also by the effective asset mixes, including (in advanced betas 3 and 4) some deleveraging. In our framework, deleveraging consumes some of the active risk budget but also reduces risk in the total portfolio. The performance of the advanced-beta-only portfolio is slightly higher (expected active return of 1.5% compared with 1.4%) than the smart-beta-only portfolio based on our choices of advanced betas and incidental ESBs but without allowing for direct adjustments in ESBs.

Exhibit 5 also presents the results for combinations of both smart beta and advanced beta. The result indicated by “without leverage adjustment” (column 3) refers to the case in which the cumulative leverage/deleverage implicit in advanced betas is permissible in the total portfolio. In this case, the holdings of advanced betas effectively move the total portfolio asset allocation away from investment policy, and this presents a governance issue for the chief investment officer. The result indicated by “with leverage adjustment” (column 4) refers to the case in which the leverage/deleverage implicit in advanced betas is corrected to ensure that the total portfolio asset allocation is congruent with investment policy. This leverage adjustment may be accomplished in different ways—by the investor via derivatives or readjustment of the asset allocation to precisely compensate, or by requiring the investment manager to deliver a product that is on benchmark without any need for adjustment. Allowing for both advanced betas and direct investment in smart betas, the expected active return improves appreciably to 1.8% (without leverage adjustment) or 2.1% (with leverage adjustment). The expected active return/risk ratio is 0.61 (without leverage adjustment) or 0.68 (with leverage adjustment), compared with 0.51 for the smart-beta-only portfolio and 0.48 for the advanced-beta-only portfolio. In the case of leverage adjustment (B), the allocations to advanced betas 3 and 4 are increased compared with the case without leverage adjustment (A) because the penalty associated with deleverage is neutralized. In either case, by blending in the advanced beta, we gain additional return and diversification benefits across the advanced beta residual returns, and we are able to adjust the allocation to ESBs via a completion portfolio. The allocations to advanced betas add significant incidental exposures to value and size but not to momentum; therefore, a larger, direct allocation to momentum is desirable as part of a completion portfolio. Considering both the direct and indirect allocations to smart beta, the portfolio with advanced beta implies net allocations to smart betas that move toward the optimal ESB allocations.^{10}

### Impact on Total Portfolio

Exhibit 5 also shows the total return and total risk of the three portfolios, when combined with the policy portfolio, based on an active risk budget of 3% within the equity portfolio. It is possible to increase or decrease the active risk budget, as discussed in the previous section. The investor’s decision should be based on a trade-off between what is needed to attain desired outcomes (total return and risk) and the investor’s confidence in his or her own skill in selecting strategies that may outperform.

Exhibit 6 displays the range of mean–variance-efficient portfolios that are possible for the investor at various active risk budgets. On top of an ordinary efficient frontier for traditional asset classes, we have placed an *active tail* derived from the active efficient frontier introduced in the previous section. The base on the tail represents an investment policy with a 70–30 stock–bond allocation that is fully indexed (no smart beta or advanced beta). Points along the active tail represent optimal portfolios for various active risk budgets between 0% and 3% within the equity asset class. As we increase the active risk level, the expected return increases along the tail because of positive incremental returns from smart beta and advanced beta. Portfolios A (without leverage adjustment) and B (with leverage adjustment) are the optimal advanced beta and smart beta portfolios referred to in Exhibit 5.

A notable feature of the advanced beta tail without leverage adjustment is that it bends leftward at first, showing that the inclusion of advanced beta reduces total risk in this example. Risk reduction is caused by the allocation to advanced betas 3 and 4. Because of the deleveraging implicit in these strategies, we are effectively reducing the overall portfolio allocation to equities*.* With leverage adjustment, the total risk always increases as we increase the active risk budget. Nevertheless, all portfolios on the efficient frontiers with leverage adjustment (Active Tail B) have a higher total return at any given active risk budget than portfolios without leverage adjustment (Active Tail A). For example, Portfolio B (with leverage adjustment) has higher expected return than Portfolio A (without leverage adjustment), although both portfolios have the same active risk of 3.0%. Under our assumptions, there is an active risk budget constraint that effectively penalizes advanced beta strategies that deleverage the market return. As deleverage consumes the active risk budget while producing negative expected returns, Active Tail B is always preferred to Active Tail A. Ignoring the active budget constraint for the moment, we are able to attain the same return from Active Tail A with lower total risk than from Active Tail B; however, this is not a fair comparison because our active budget constraint implies that both active risk and total risk matter to the investor. Another perspective is that users of smart or advanced beta must think carefully about the market exposures generated by those strategies; otherwise, some strategies may have the unintended effect of changing investment policy indirectly.

This case study has demonstrated that it is practical to optimize exposures to advanced betas and smart betas in a total return context and that the resulting portfolios may have material benefits for investors based on our assumptions. If the reader is unconvinced by this, consider that the same methodology—with less favorable assumptions about advanced beta and smart beta—could be used to prove just the opposite.

## CONCLUSION

This article has presented a methodology for optimal allocation to smart beta and multifactor strategies (advanced betas) from an investor standpoint, based on standard principles of portfolio construction and setting explicit forward-looking return expectations that are familiar from other portfolio contexts. This contrasts with much of the current discussion of portfolios of smart betas based on historical returns and mechanical risk-budgeting strategies. A key innovation is to begin with a simple factor model for each advanced beta that includes exposure to investment policy factors, a limited set of elementary smart betas, and a residual return unique to the strategy.

Two principal findings about optimal blending emerge. First, allocations to elementary smart betas depend principally on the ratio of expected excess return to expected variance, adjusted to account for implicit holdings of similar factors elsewhere in the portfolio. Second, allocations to multifactor advanced betas should proceed similarly, with an adjustment for leverage/deleverage but without the need to adjust for any embedded holdings of elementary smart beta. Beyond guidance on the best allocations across smart beta and multifactor strategies, this process also informs us about the characteristics of advanced beta strategies that would be most beneficial for investor outcomes.

For investors concerned about active performance relative to investment policy benchmarks, strategies that embed significant deleveraging relative to the market (i.e., have a beta to the market substantially different from one), such as low-volatility strategies, should be approached with caution. Our framework brings into question the common practice of treating low volatility as another anomaly similar in principle to the other well-known anomalies. Not only does the low-volatility anomaly typically contribute higher active risk than the other anomalies, but its deleveraging exposure also detracts from expected returns (arguably by less than the capital asset pricing model prediction). Thus, any anomaly that embeds significant deleveraging seems more fit for exploitation by investors focused specifically on achieving lower total risk in their portfolios (by effectively deviating from investment policy) rather than providing an active return contribution.

Finally, there is currently a lively debate on whether factor strategies are best implemented through a series of single-factor portfolios or through a single, engineered, multifactor portfolio. The present framework identifies the complex portfolio trade-offs and detailed assumptions across factors that would be required to make that assessment, but it does not provide a final answer to the question. Nevertheless, we believe that the methodology can provide a useful basis for answering these questions and improving advanced beta implementation for both asset managers and asset owners.

## TECHNICAL APPENDIX

### Style Analysis of Advanced Beta

The normal portfolio for advanced beta strategies is estimated in terms of investment policy benchmark returns, elementary smart beta factors, and residual returns:

where

is the n

_{ab}× 1 vector of excess returns of advanced betathe (n

_{b}+ n_{esb}) × 1 vector of returns of asset classes and ESB factorsis the n

_{ab}× 1 vector of residual returns for advanced beta*F*= [*F*_{b}*F*] is the n_{esb}_{ab × (nb + nesb}) matrix of estimates defining the effective mix of market and ESB factors associated with advanced beta.

### Expected Return and Risk of Total Portfolio

Based on vectors of investment policy holdings *h _{b}*, direct investment in elementary smart beta

*h*, and advanced beta

_{esb}*h*

_{ab}, the effective total holdings of beta and smart beta are and . We then have the expected return and variance of the total portfolio:

where

*r*is the (n_{b}+ n_{esb}+ n_{ab}) × 1 vector of expected returns for beta, smart beta, and advanced beta residuals*V*_{b}is the n_{ac}× n_{ac}matrix of covariance of standard asset classes*V*_{esb}is the n_{esb}× n_{esb}matrix of covariance of elementary smart beta factors*V*_{abr}is the n_{ab}× n_{ab}matrix of covariance of advanced beta residuals

A diagonal block structure of *V* is suggested here to simplify the expansions in the next section and to aid interpretation of the optimization results. This assumption has been reasonably supported in case studies by observing very low cross-correlation between elementary smart beta and advanced beta exposures within asset class exposures.

### Objective Function

The investor seeks a portfolio that maximizes the expected utility of total return and total risk. We take the investor’s objective function to be .^{11} It is helpful to parse the objective function into components associated with policy, smart beta, and advanced beta. Expected total return and risk is then expanded from Equations (A-1) and (A-2) to

### Optimization

We consider investment policy to be fixed. The first-order conditions for the unconstrained optimization are

A-5 A-6The optimal holdings are then

A-7 A-8Equation (A-7) shows that the optimal holdings of ESBs are directly proportional to expected returns and inversely proportional to the contribution of ESBs to risk, adjusted for the ESB exposure present in our optimal holdings of advanced betas. The second set of conditions shows a similar direct relationship to returns and inverse relationship to contribution to variance. In this case, the returns include both the residual returns and the implied leveraged or deleveraged market return, and the contribution to variance is similarly adjusted. Holdings do not depend on ESB factors associated with advanced beta. There is also an adjustment term associated with leveraging caused by advanced beta. This latter term demonstrates that optimal allocations to advanced beta and smart beta depend on investment policy if *F _{b}* ≠ 0.

In the case of active return optimization, our objective is , and we modify Equations (A-3) and (A-4) by subtracting the investment policy term. The optimal active holdings are then

A-9 A-10Conditions in Equation (A-9) for optimal holdings of ESBs are the same as Equation (A-7) in the total return case, except for any changes in . Conditions in Equation (A-10) for the optimal holdings of advanced betas are simplified and do not include the leverage penalty term in Equation (A-8). Thus, if we have no leverage associated with advanced beta (*F _{b}* = 0), the total return problem and the active-only problem result in equivalent allocations under our assumptions.

## ENDNOTES

↵

^{1}Dimson, Marsh, and Staunton [2017] reported that the proportion of large-asset owners using at least five smart beta indexes has risen 10-fold, from 2% in 2014 to over 20% in 2016. Morningstar [2016] reported that, as of mid-2016, there were 1,123 exchange-traded products involving smart beta, with collective assets under management of approximately $550.5 billion worldwide. This does not include many smart beta assets in other formats. See also Kahn and Lemmon [2016].↵

^{2}Bender and Wang [2016] demonstrated the efficiency advantages of a bottom-up approach in multifactor portfolios compared with a combination of single-factor portfolios. However, Amenc et al. [2017] suggested that although factor score intensity improves, reflecting that improvement in higher performance of multifactor portfolios is more challenging.↵

^{3}See Sharpe [1988, 1992] for a description of returns-based style analysis, which is similar to a regression of monthly returns that is constrained to reflect a fully invested, long-only benchmark.↵

^{4}The Fama–French versions of the factors are not readily investible and, therefore, are not a relevant benchmark for comparison with other smart beta funds. Technically, the MSCI indexes are not investible either. We take the indexes, however, as reasonable proxies for relatively low-cost, low-tracking-error investible replications of the factors.↵

^{5}Based on monthly returns from 1997 to 2015, small and value SB have a correlation of +0.016; momentum and small SB have a correlation of +0.081; and value and momentum SB have a correlation of -0.497. The negative correlation between value and momentum factors has been well established, both within and across asset classes. Dimson, Marsh, and Staunton [2017] showed long-term evidence for risk premiums associated with these three smart betas. See also Fama and French [1992]; Carhart [1997]; and Asness, Moskowitz, and Pedersen [2013]. The three ESB factors are also uncorrelated with the ACWI and GBMI indexes used to represent stock and bond asset classes in this article.↵

^{6}Remembering that quality is negatively correlated with value and small, it is not surprising that a best-fit model (based on ACWI index data from 1999 to 2015) of a quality smart beta yields an unconstrained factor portfolio that would be significantly short on value and small ESBs and have a negative alpha. Constraining it to a long-only portfolio, we would find an effective asset mix of 12% cash, 75% ACWI index, and 13% momentum and a zero alpha. For the low-volatility smart beta, an unconstrained best-fit model (based on ACWI index data from 2002 to 2015) yields a factor portfolio that would be long on value and momentum and short on small ESBs and have a negative alpha. Constraining it to a long-only portfolio, we would find an effective asset mix of 34% cash, 13% ACWI index, 38% value, and 15% momentum and a negative alpha.↵

^{7}As Kim, Kim, and Fabozzi [2017] pointed out, “Unfortunately, although practitioners believe that robust optimization is also applicable in recent factor based smart beta strategies, there does not seem to be a noticeable attempt to apply it in the asset management industry.” They suggested that beginning with a factor model is the right approach.↵

^{8}Note that if λ coincides with optimal policy, , and then Equation (6) simplifies to . Alternatively, if we optimize*h*for a given λ, we obtain , a shift from current policy. We then can focus on pure alpha and residual risk, and Equation (6) simplifies to ._{b}↵

^{9}See J.P. Morgan [2015] for a comprehensive report on asset class assumptions. The estimates used in this article are similar to the assumptions found in the report, and the findings of this article are not too sensitive to variations in these estimates.↵

^{10}The effective allocation to elementary smart beta styles contributed by smart beta and advanced beta in this example (without leverage adjustment) in comparison to the ESB-only optimal allocations are value 55% versus 51%, small 17% versus 11%, and momentum 28% versus 38%.↵

^{11}The discussion of objective functions follows the development by Grinold and Kahn [2000] for active equity portfolios. The objective function may be modified based on choices for risk aversion associated with various risk components and the utilization of equilibrium conditions. As another example, Waring et al. [2000] focused on optimizing portfolios of managers within an asset class and allowed for risk aversion factors that differ for policy risk versus active risk.

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