## Abstract

In this article, the authors propose a comprehensive framework for simultaneously allocating assets among active, passive, and factor investments while accounting for the uncertainty in each of the sources of return and investor risk preferences toward them. The proposed model enables investors to overcome the often adopted approach of first deciding the asset allocation and subsequently allocating across active and factor strategies. The authors also highlight some business applications of the adopted approach, such as the construction of factor tilted portfolios and the substitution of low-cost factor strategies for higher-cost active portfolios.

**TOPICS:** Analysis of individual factors/risk premia, factor-based models, portfolio theory, portfolio construction

**Key Findings**

• Using an expected utility optimization model, the authors can simultaneously allocate assets among active, passive, and factor investments while accounting for investor risk preferences.

• Optimal asset allocation responds to changes in the level of risk aversion across systematic, alpha, and factor risk; expected factor-adjusted alpha levels; tracking errors; and predicted factor premiums.

• This approach allows for full customization of portfolios and brings to light many decisions that investors would otherwise make subconsciously.

When building single-period portfolios that include active, passive, and factor strategies, investors typically proceed in two consecutive steps. In the first step, investors use a mean–variance optimization (MVO) to determine the portfolio’s allocation among core asset classes. In the second step, investors determine the mix of active, passive, and factor strategies for each asset class. This second set of decisions is often somewhat arbitrary and ad hoc and has some important limitations.

The first limitation is that it ignores the correlation among active, passive, and factor strategies. Although the MVO step requires inputs for return, volatility, and correlation of core passive assets, the second step tends to assume that active, passive, and factor returns are independent of one another. This assumption of independence is questionable because, by construction, the returns of active strategies are dependent on passive benchmarks and factor premiums, as determined by their loadings in a factor style decomposition. The two-step approach also fails to account for the full range of investor risk preferences. Although it incorporates systematic (market) risk aversion, it ignores aversion toward active and factor risks. These risk preferences vary among investors, making one-size-fits-all solutions inefficient. The two-step approach inevitably yields portfolios that are ex-ante suboptimal for any given investor.

We address these limitations with a portfolio optimization approach based on expected utility. The approach requires that we make explicit forecasts for asset returns, factor premiums, active alpha, and the uncertainty associated with these return expectations. These forecasts yield a joint return distribution for different allocations of asset classes and investment strategies. We use this joint distribution to simultaneously optimize for active, passive, and factor strategies in light of investor risk aversion toward each of these distinct sources of uncertainty. We improve on the suboptimal two-step process with a one-step optimization.

We organize the rest of the article in four sections. In the first section, we review the current literature on active–passive–factors asset allocation. The subsequent section explains our methodology. In the next section, we present our results and our model’s sensitivity to a range of potential investor preferences. We then review several portfolio applications for this approach, including the construction of model portfolios based on factors and the substitution of low-cost factor strategies for higher-cost active management.

## LITERATURE REVIEW

In portfolio construction, risk aversion is commonly understood as aversion to systematic risk (i.e., risk aversion to risky assets vs. safe assets). However, investors also display aversion toward other dimensions of investment risk, such as alpha risk.^{1} The wide range of risk budgets—allowable allocation to active strategies—documented in institutional policy portfolios with similar asset allocations hints at these two dimensions of risk aversion. The two risk aversions can differ drastically from one investor to another. A growing body of research explores how to account for these different dimensions of risk and how to allocate between active and passive in a portfolio. Flood and Ramachandran (2000), for example, highlighted the importance of an investor’s alpha risk tolerance and the associated portfolio implications.

Waring et al. (2000) and Waring and Siegel (2003) discussed a framework for allocating between active and passive while explicitly accounting for active risk and an investor’s active risk preference. More recently, Buser (2015) adapted long-established portfolio optimization techniques to account for active risk. Conceptually, these studies extend the MVO as specified by Markowitz (1952, 1959)—which trades off systematic risk for systematic return—to alpha risk, thereby producing a three-dimensional frontier (i.e., efficient surface).^{2} The traditional MVO frontier can then be thought of as a special case of the efficient surface wherein an investor is extremely averse toward alpha risk.

The framework proposed by Waring et al. (2000) focused on the active–passive trade-off assuming quadratic utility. We build on this earlier work by incorporating multiasset decisions and factor risk aversion, which adds another dimension to the three-dimensional frontier.

Risk aversion and its implications for portfolio construction also extend to factors. Although research on factor investing is extensive, most focuses on standalone factor strategies rather than their role in a multiasset portfolio, as highlighted by Fisher and McDonald (2018). Dopfel and Lester (2018) focused on evaluating how smart betas and multifactor strategies can add value in portfolios. Their research followed a similar approach to Waring et al. (2000), but instead of looking at alpha risk, the authors focused on the portion of bottom-up multifactor portfolios that cannot be replicated through investing in more standard and readily investible factors. Bergeron, Kritzman, and Sivitsky (2018) showed instead how the approaches of asset allocation and factor investing can be combined to achieve a portfolio sensitive to the desired factor profile. Bellord et al. (2019) looked at the optimal blend between passive and active by considering expected shortfall risk in equity portfolios.

To our knowledge, there have been no concrete attempts to provide a holistic approach to optimally and simultaneously allocate among active, passive, and factors in a multiasset portfolio. Rao, Subramanian, and Melas (2018) come closest in their work on how to combine active, passive, and factors in one single solution, although they perform this optimization only with equities.

Our framework expands on current research by

1. separating the factor risk component from the systematic (asset class) risk and alpha risk components;

2. simulating return uncertainty across passive, factor, and alpha strategies; and

3. maximizing the expected utility of wealth across systematic, alpha, and factor risks to create a multiasset portfolio allocation in a single step.

^{3}

## OPTIMIZING A MULTIASSET PORTFOLIO WITH ACTIVE, PASSIVE, AND FACTORS

### Simulating Return Uncertainty across Passive and Factors

The first task in a multiasset optimization is to estimate the data-generating process for all asset classes to be considered in the analysis. Because this is a well-established topic, and many financial professionals use risk models to generate asset return distributions, we do not discuss this in detail. We nevertheless highlight a few important features of a well-designed asset return forecasting model:

1. the use of a probabilistic return forecasting framework (rather than a point forecasts approach);

2. sensitivity to initial conditions such as levels of interest rates, price–earnings ratios, and other asset valuations metrics;

3. the use of nonnormal probability distributions (i.e., those with fatter tails);

4. an accounting for both autocorrelation within an asset class and cross-asset return correlation; and

5. treatment of the important linkages between asset returns and the fundamental economic drivers (Davis et al. 2018).

In this article, we rely on a stochastic asset class forecasting model to generate asset return distributions and cross-correlations among assets (Davis et al. 2014).^{4}

### Simulating Return Uncertainty around Factor-Adjusted Alpha

Should an active fund manager be given credit for outperformance from systematic tilts toward factors, which attempt to harvest risk premiums over long periods of time? Access to factors can generally be gained at lower cost than typical active management fees. For instance, Bender, Hammond, and Mok (2014) showed that up to 80% of the alpha generated by US equity active managers can be explained by exposures to equity risk factors. Similarly, research done by AQR Capital Management (2017) and Roberts, Paradise, and Tidmore (2018) provides evidence that the majority of returns for active fixed-income managers are explained by exposure to high-yield securities, not market timing or security selection.

To disentangle skill from the return generated through exposure to systematic market and factor tilts, we follow a return-based ordinary least squares regression, similar to Sharpe (1992), Fama and French (1993), and Chin and Gupta (2017). The active manager skill is thus the factor-adjusted alpha, as illustrated in the following equation:

1where *r*_{i,t} refers to the total return of active manager *i* at time *t*; *r*_{M,t} is the benchmark total return; is the excess (to the market benchmark) factor return for factor *f*; β_{i} and correspond to the market beta and factor loading; and α_{i} is the factor-adjusted excess active return, or factor-adjusted alpha.

We use Equation 1 to decompose active returns into four components:

1. a systematic risk–market component (β

_{i}*r*_{M,t})2. a risk factor component

3. factor-adjusted alpha (α

_{i})4. an unexplained return variation around the benchmark or factor-adjusted error term (ε

_{i,t})

This decomposition of active total returns highlights the importance of accounting for all those sources of risk in the simultaneous active–passive-factor allocation problem.

By fitting Equation 1 to the active manager’s historical return series data, we can estimate the expected level of factor-adjusted alpha, tracking error, and implicit active factor loadings for the portfolio optimization problem.^{5} Note that this fitting exercise is just one way to decompose total return to obtain the loadings used in the portfolio optimization. An investor can obtain equivalent parameters though manager oversight and due diligence or more subjective assessments of a manager’s skill.

Exhibit 1 shows the return decomposition for a real-world US equity active manager. The fund shows a strong factor-adjusted outperformance with a historical factor-adjusted alpha of 81 bps per year and a tracking error of roughly 4%. Thus, the fund manager has added value by security selection and timing, beyond traditional factor and market exposure. This factor-adjusted alpha is in addition to the excess return from exposure to the midcap factor and value factor.^{6}

To capture the uncertainty around the factor-adjusted alpha, we simulate a distribution around its expected value using the standard error of the residual in Equation 1 (i.e., the factor-adjusted tracking error). Specifically, we use Monte Carlo methods to simulate a nonnormal distribution (*t*-distribution).^{7} Although the return decomposition in Exhibit 1 corresponds to a single active manager, the same methodology could be used for a multimanager active strategy consisting of a predetermined mix of active funds.

### Maximizing Expected Utility of Wealth with Systematic, Alpha, and Factor Risks and Investor Preferences

Expected utility maximization approaches to asset allocation have been used for a long time. Adler and Kritzman (2007) and Sharpe (2007) provided compelling evidence on the benefits of adopting expected utility over mean–variance for portfolio optimization. We use a power utility function to model investors’ preference and attitude toward risk^{8}:

where γ is the relative risk aversion coefficient and *W _{T}* is the level of terminal wealth.

^{9}

We show in Appendix A that the total wealth at the end of the investment horizon *T* can be broken into three components:

where *R*_{p,T}, *R*_{f,T}, and *R*_{a,T} are cumulative returns at maturity *T* coming from systematic passive benchmarks, factor premiums, and factor-adjusted alpha exposures, respectively. Based on this decomposition of portfolio returns and to allow for different risk aversions for each source of risk, the single-period expected utility optimization problem can be defined as

where {*x _{i}*} represents the set of optimal weights.

*W*,

_{p}*W*, and

_{f}*W*are the wealth at maturity

_{a}*T*coming from systematic, factor, and factor-adjusted alpha exposures, respectively, such that , , and . Systematic, factor, and alpha risk aversions are represented by γ

_{p}, γ

_{f}, and γ

_{a}, respectively.

*C*and

*b*refer to the set of linear constraints. Additional details for the derivation of the objective function can be found in Appendix A.

We also consider specific portfolio constraints to express investor portfolio beliefs and preferences, such as upper and/or lower bounds exposures (e.g., real estate investment trusts may not be more than 10% of the total asset allocation; total credit exposure may not exceed 50% of total fixed-income allocation; home bias in US equities has to be at least 60% of the total equity allocation). Finally, we rely on a genetic algorithm to solve for the portfolio optimization problem such as the one specified by Equation 4.^{10}

## SENSITIVITY ANALYSIS AND APPLICATIONS

In this section, we explore the relationship between changes in the explicit model inputs—the three risk preference parameters and the estimated factor-adjusted alpha and tracking error—and changes in portfolio allocation.

We start with a look at each risk aversion’s impact on portfolio allocation, with all other inputs held constant. In Exhibit 2, Panel A, we show the impact of changing levels of systematic risk aversion (i.e., dislike for market volatility of passive asset classes) for a portfolio with only bonds and equities. As the level of risk aversion declines (i.e., risk tolerance increases), the allocation to equities rises.^{11}

In Exhibit 2, Panel B, we hold the systematic and factor risk aversion constant to examine the impact of changing levels of alpha risk aversion (i.e., dislike for dispersion of factor-adjusted alpha). As alpha risk aversion declines, the portfolio allocation to active equity strategies increases.

Finally, in Exhibit 2, Panel C, we hold the systematic and alpha risk aversion constant to examine the impact of changing levels of factor risk aversion. As factor risk aversion declines, the portfolio’s allocation to US equity explicit factors increases.

### Active–Passive Decisions

Exhibit 3 provides a portfolio-level demonstration of the relationship between changes in the model inputs and portfolio allocations. What is the impact of lowering factor-adjusted alpha expectations, for example? Exhibit 3 shows two portfolios (portfolio A and portfolio B) with identical inputs except for the expected factor-adjusted alpha (81 bps vs. 11 bps per annum, respectively). As expected, the lower factor-adjusted alpha results in a reduced allocation to the US active fund, from 30% to 16%.

By comparing portfolios C and D, we see the impact of changes in tracking error. As the factor-adjusted tracking error declines from 4.03% to 2.00%, the allocation to the active fund increases from 13% to 26%, all else being equal. By virtue of the tracking error being reduced, the manager’s excess return per unit of risk (i.e., information ratio) has increased from 0.2 (81 bps/403 bps) to 0.40 (81 bps/200 bps).

Another important insight from the analysis in Exhibit 3 is that the allocations to passive asset classes and active strategies are not independent of one another. For instance, portfolio C’s allocation to equities is 4 percentage points higher than portfolio D’s because of the lower tracking error in portfolio D’s active equity strategy. Interactions among alpha risk contributions, active manager factor tilts, and other sources of portfolio volatility may lead to shifts in the overall asset class allocation. Two-step sequential approaches overlook these risk budgeting interactions by assuming independence among these features, leading to ex ante suboptimal solutions.

Thus, explicit views on risk preferences and characteristics of active funds (or a collection of active funds), such as alpha expectations, tracking error, and asset and factor premiums, can significantly affect asset allocation in a multiasset portfolio. Although some investors may not take these variables into consideration, their portfolio construction decisions include implicit assumptions about these variables without any calibration of how extreme those assumptions may be. Our approach enables investors to think through these assumptions and their implications for asset allocation explicitly.

We now explore practical applications for this approach to portfolio construction. We look first at using this approach to find lower-cost substitutes for high-cost active strategies and then to build factor model portfolios.

### High-Cost Manager Substitution

To what extent can a high-cost active strategy be replaced by some combination of lower-cost style factors? To answer this question, we consider an investor with low aversion to factors—in other words, an investor willing to bear factor risk in the portfolio.

Suppose that, initially, an investor can only achieve a desired factor exposure through an active investment. In this hypothetical situation, factor investment vehicles, such as factor exchange-traded funds (ETFs), are not readily available to this investor. Portfolio E in Exhibit 3 exemplifies a scenario in which this investor can only invest in a US equities active fund that has value and midcap tilts (see Exhibit 1), along with a high tracking error, and limited factor-adjusted expected alpha (11 bps after costs). Because this active fund is the only way for the investor to obtain the desired value and midcap factor exposure, the allocation to this manager is 20%.

If factor investment vehicles become available, the investor can build portfolio F. Portfolio F has the same inputs as portfolio E, but its investment opportunity set includes value and midcap passive factor strategies (e.g., factor ETFs). The active allocation drops from 20% in portfolio E to only 4% in portfolio F, and the total factor exposure increases to 33%. The implicit factor exposure with an active allocation in portfolio E is replaced with the desired explicit indexed factor allocation. This substitution offers an investor the opportunity to replace a potentially high-cost active fund with limited factor-adjusted alpha with lower-cost factor exposure.

### Factor Tilted Portfolios

Although factor investing is not new, the number of factor mutual funds and ETFs offered to investors has grown exponentially. Yet, deciding which factors to include in a portfolio and how to weight them is a complex challenge.

Consider an investor who wants no active exposure but is willing to consider factor tilts. By specifying different levels of systematic and factor risk aversion, our approach can construct portfolios with factor tilts that are optimized for these risk aversions and for the forecasted risk–return characteristics of each factor and the correlations among them.

In Exhibit 4, we load the model with US equity style factors that are generally considered to hold a premium over the long term: value, small-cap, momentum, low-volatility, and quality.

Portfolio G shows the optimized multifactor model portfolio with medium levels of systematic and factor risk aversion. We first notice that the totality of the passive US equity exposure is explained by factors. This is mainly because we are considering long-only factors that hold positive expected premiums over our investment horizon. Furthermore, the optimal allocation among factors is not homogeneous: The momentum factor has a 17% weight, whereas the quality factor has a weight of 2%. As a comparison, portfolio H has the same equity–bond mix and asset allocation. Rather than an optimized factor allocation, however, Portfolio H holds the five US equity factors in equal weights (39%/5 = 7.8%).

The optimized factor model portfolio shows a higher allocation to momentum (+9%) and lower weighting to value, small-cap, and quality in particular (−6%) compared with the equally weighted equivalent. In addition, portfolio G shows a higher total expected return and volatility. Similarly, if we compare the risk-adjusted expected return for the two portfolios, we find that the optimized factor model portfolio has a Sharpe ratio of 0.31, whereas the equally weighted factor portfolio has a Sharpe ratio of 0.25.

## CONCLUSIONS

In this article we propose a comprehensive framework for simultaneously allocating assets among active, passive, and factor investments while accounting for the uncertainty in each of the sources of return and investor risk preferences toward them. Building on previous research on alpha risk, our expected utility portfolio optimization incorporates factor risk aversion explicitly into the asset allocation problem. Therefore, our suggested approach allows for investor preferences in different risk dimensions and helps investors recognize many of the choices they make implicitly.

The quantitative framework proposed in this article can have multiple research and business applications. In an advisory relationship, it allows for full customization of portfolios and reveals many decisions that investors would otherwise make subconsciously when choosing ad hoc portfolio allocations. In other words, the input requirements for our model enable a conversation with the investor about the conscious and explicit choices that must be made and are critical to the portfolio, such as setting realistic alpha expectations for the active strategies under consideration, selecting the best estimates for the associated alpha risk, and even self-reflecting on aversion to alpha and factor risk.

## ADDITIONAL READING

**Asset Allocation and Factor Investing: An Integrated Approach**

Alain Bergeron, Mark Kritzman, and Gleb Sivitsky

*The Journal of Portfolio Management*

**https://jpm.pm-research.com/content/44/4/32**

**ABSTRACT:** *The traditional approach to asset allocation is to identify the optimal allocation to broad asset classes based on assumptions for expected returns and risk. Increasingly, though, investors are choosing to stratify their portfolios into factors rather than traditional asset classes. Many investors, especially those who have qualitative views about macroeconomic factors, do not typically optimize exposure to factors based on explicit assumptions for return and risk. Instead, they predetermine the appropriate factor exposures based on fundamental views and then identify combinations of assets that they perceive to capture the targeted factor exposures most effectively. Both approaches have their benefits and limitations. The authors propose a framework that integrates traditional asset allocation with factor investing in a way that emphasizes the benefits of each approach while helping to overcome their respective limitations.*

**On the Optimal Mix of Active and Passive Investments**

Stephen A. Buser

*The Journal of Portfolio Management*

**https://jpm.pm-research.com/content/41/4/91**

**ABSTRACT:** *A modified version of the appraisal ratio developed by Treynor and Black in 1973 identifies a unique mix of active and passive investments that is optimal for every investor, regardless of his or her preferences for risk and return. This result suggests that financial advisors can consider recommending specific dollar amounts for clients to allocate to active investment strategies, rather than simply recommending qualitative assessments regarding such opportunities. The article identifies corresponding implications for services that provide estimates of analytic measures that are used to evaluate active investments, as well as for institutions that manage active portfolios on behalf of individuals.*

**Using Prime Alpha to Separate Skill from Luck in Fixed-Income Strategies**

Andrew Chin and Piyush Gupta

*The Journal of Investing*

**https://joi.pm-research.com/content/26/2/102**

**ABSTRACT:** *We assess the existence and persistence of manager skill using a unique and broad database of institutional manager returns within the U.S. fixed-income universe. We define prime alpha as the residual return after adjusting for the main fixed-income factors (credit and duration) and find that it is persistent across time. Prime alpha is a strong indicator of manager skill because managers with high prime alpha tend to continue to exhibit high prime alpha. We also show evidence that raw manager returns are not persistent because they are influenced by the cyclical nature of factor returns.*

## APPENDIX A

### OBJECTIVE FUNCTION DERIVATION

We considered an investor facing the portfolio choice problem of investing their wealth *W*_{0}, over an investment horizon *T*:

where *R _{t}* corresponds to the total return of the multiasset portfolio at time

*t*defined as

where *x _{i}* and

*r*are the portfolio weights and relative total returns for each asset class

_{i}*i*, and superscripts

*p*,

*f*, and

*a*refer to passive, factors, and active, respectively. The market benchmark return is represented by ; is the excess (to the market benchmark) factor return for factor

*f*; β

_{i}and

*L*correspond to the market beta and factor loading for each asset class, respectively; and α

_{i}_{i}is the factor-adjusted excess active return.

^{12}

With further algebra, Equation A-1 and the set of Equations A-2 can be developed and rearranged such that the different risk components and sources of return (systematic, alpha, and factor) are grouped together:

A-3 A-4We can then rewrite Equation A-3 as

A-5Similar to alpha risk aversion, factor risk aversion applies to the relevant factor excess return in the portfolio. The portfolio choice problem consists then of finding optimal weights for each passive asset class, factor or active manager, or strategy in the portfolio. We can then express the expected utility optimization problem that we want to solve into the set of Equation 4.^{13}

## APPENDIX B

### CALIBRATING RISK AVERSION

Utility functions are a mathematical representation of investor preferences and attitudes toward risk. Even if we make the assumption that any investor’s preference is adequately modeled by a power utility function, economists and practitioners still need to figure out which level of the risk aversion coefficient (in our case γ) is the most appropriate for any given investor. Research focusing on estimating relative risk aversions has been conducted for more than 30 years, and, more recently, experiments and survey responses have also been used (e.g., Metrick 1995; Barsky et al. 1997).

Relative risk aversion coefficients do not have any intuitive economic meaning. Furthermore, risk aversion coefficients are ordinal numbers rather than cardinal. Thus, an investor who has twice the risk aversion of another will show a higher risk aversion coefficient but not necessarily one that is twice as big.

To our knowledge, literature on how to calibrate risk aversion for portfolio optimization studies is scarce. Liu and Xu (2010) showed that if in the special case of unconstrained optimization with quadratic utility, parametric return distributions, and where only two assets are available—a risky security and a risk-free asset—the relative risk aversion coefficient can be inferred from the equity–bond portfolio weight.

Although appealing, this approach would not work for our purposes. To calibrate risk aversion to any type of utility function, nonparametric return distributions, and level of portfolio optimization complexity (e.g., multiasset optimization with constraints), Liu and Xu (2010) proposed an “efficient-frontier approach.” This method creates multiple optimal portfolios using a fairly wide range of initial risk aversion coefficients. Investors would then identify those portfolios that meet their preferences based on a set of risk and performance statistics (e.g., expected return, volatility, Sharpe ratio, maximum drawdown) and keep adjusting the implied risk aversion until the optimal portfolio statistics converge to the desired set of portfolio risk and return metrics.

## APPENDIX C

### GENETIC ALGORITHMS IN PORTFOLIO OPTIMIZATION

Standard mean–variance portfolio optimization problems with linear constraints can be solved using quadratic programming. However, once nonlinear constraints, such as transaction costs or minimum lots in the portfolio (Lin, Li, and Li 2005), or higher moments (i.e., skewness and kurtosis) are captured by the distribution of the forecasts (Kshatriya and Prasanna 2017), the optimization problem becomes nonconvex and computationally unfeasible (e.g., NP-hard). In those instances, a derivative free approach, such as a genetic algorithm, can become very valuable.

Originally developed by Holland (1975), genetic algorithms are stochastic methods inspired by natural selection processes and are now widely used for constrained and unconstrained portfolio optimization problems. For instance, Rifki and Ono (2012) provided a comprehensive literature review on computational approaches to portfolio optimization using genetic algorithms. Compared with more traditional optimization methods, genetic algorithms present the advantage of requiring little or no knowledge of the problem (i.e., search space) at hand. In addition, they can perform well in large, complex, and multiobjective problems (Lin and Gen 2007) and are more likely to converge to the global (as opposed to local) optima of the problem. For these reasons, the optimized portfolio weights presented in our analysis are computed using a genetic algorithm.

## APPENDIX D

### US EQUITY STYLE FACTORS DEFINITION

Exhibit D1 below shows the criteria used to define and construct US equity style factors for the analysis reported in this article. Our methodology is not dependent on the definition of factors shown, and the approach we suggest can accommodate any other factor definition or benchmark.

## ENDNOTES

↵

^{1}In this article, we define alpha as the idiosyncratic return component that cannot be explained by market or factor risk exposures. Alpha is entirely explained by active manager’s security selection or market timing skill.↵

^{2}Chow (1995) proposed a similar approach to viewing the efficient frontier in a three-dimensional space, considering portfolio returns, policy risk, and active risk.↵

^{3}Waring et al. (2000) already introduced, in an expected utility maximization setting, the importance of finding an optimal asset allocation and active exposure all at once. Their study also recognized that most of the time, investors accomplish the task sequentially rather than simultaneously, which is suboptimal.↵

^{4}Although we rely on a proprietary risk model, our asset allocation framework is independent of the specific choice of model used to generate the asset return forecasts.↵

^{5}In this article, we focus on equity style factors only. We define our own style factors, constructing the historical series bottom-up using single-stocks data—see Appendix D for further details. However, the approach and methodology that we propose could be conducted by relying on already available equity factor benchmarks such as the MSCI USA Value Index or the Russell MidCap Index. In addition, our framework is applicable to any definition of factors and across different asset classes, including typical fixed-income factors (e.g., duration, credit) and factor replication for alternative strategies (e.g., hedge funds).↵

^{6}Based on the ordinary least squares regression of the historical active manager returns, the value factor loading is not statistically significant (*p*= 0.218). However, we want to keep it to highlight how our framework would work for investors who are willing to have a value factor exposure in their portfolio, either implicitly through an active manager exposure or explicitly through a passive factor investment.↵

^{7}This is done by first splitting historical active managers’ excess returns for each asset class (e.g., US equities, US bonds) into three different tracking error buckets and mapping low, medium, and high tracking errors, respectively. Then, for each bucket, the level of degrees of freedom used to generate*t*-distributed random numbers is inferred by fitting a*t*-distribution to the historical factor-adjusted active excess returns.↵

^{8}Power utility functions, like the one we use in our model, have the benefit of showing decreasing absolute risk aversion and constant relative risk aversion, which tends to be more consistent with investor behavior (Arrow 1971). Quadratic utility functions instead present increasing absolute risk aversion that does not fit with typical investor attitude toward investing (Borch 1978).↵

^{9}For further details on how to calibrate risk aversion coefficients based on investor preferences and attitude toward risk, refer to Appendix B.↵

^{10}See Appendix C for further details on genetics algorithms used for portfolio optimization.↵

^{11}Although we are using a wide range for the systematic risk aversions in Exhibit 2, Panel A, the equity allocation does not reach the 0% or 100% bounds. This is due to several reasons, such as the level of the forecasted expected returns, volatilities and correlations for the different asset classes over the 10-year investment horizon, and the time effects of returns’ predictability (Barberis 2000).↵

^{12}For simplicity, we consider in this article one active fund or strategy per asset class only, although the approach can be easily expanded to include multiple active strategies for each asset class.↵

^{13}Optimizing the utility function reported in the set of Equation 4 can be computationally heavy and time consuming. In addition, the optimization outputs can be sensitive to the set of assumptions used in the genetic algorithm (e.g., crossover fraction and number of generations). For this article, to make the optimization problem more computationally tractable, we further linearize and simplify our objective function (i.e., total expected utility) and recast it into .**Disclaimer**All investing is subject to risk, including the possible loss of the money you invest.

Please remember that all investments involve some risk. Be aware that fluctuations in the financial markets and other factors may cause declines in the value of your account. There is no guarantee that any particular asset allocation or mix of funds will meet your investment objectives or provide you with a given level of income.

Factor funds are subject to investment style risk, which is the chance that returns from the types of stocks in which the fund invests will trail returns from the stock market. Factor funds are subject to manager risk, which is the chance that poor security selection will cause the fund to underperform relevant benchmarks or other funds with a similar investment objective.

Investments in bonds are subject to interest rate, credit, and inflation risk.

Investments in stocks or bonds issued by non-US companies are subject to risks including country/regional risk and currency risk.

*ETF shares are not redeemable with the issuing fund other than in very large aggregations worth millions of dollars. Instead, investors must buy and sell ETF shares in the secondary market and hold those shares in a brokerage account. In doing so, the investor may incur brokerage commissions and may pay more than net asset value when buying and receive less than net asset value when selling.*

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