## Abstract

The authors propose a general framework referred to as Black–Litterman–Bayes (BLB) for constructing optimal portfolios for factor-based investing. In the spirit of the classical Black–Litterman model, the framework allows for the incorporation of investor views and priors on factor risk premiums, including data-driven and benchmark priors. Computationally efficient closed-form formulas are provided for the (posterior) expected returns and return covariance matrix that result from integrating factor views into an arbitrage pricing theory multi-factor model. In a step-by-step procedure, the authors show how to build the prior and incorporate the factor views, demonstrating in a realistic empirical example and using a number of well-known cross-sectional US equity factors, that the BLB approach can add value to mean–variance-optimal multi-factor risk premium portfolios.

**TOPICS:** Factor-based models, portfolio construction, portfolio theory

**Key Findings**

▪ The authors propose a general framework referred to as Black–Litterman–Bayes (BLB) for constructing optimal portfolios for factor-based investing.

▪ The framework allows for the incorporation of investor views and priors on factor risk premiums, including data-driven and benchmark priors.

▪ The authors provide computationally efficient closed-form formulas for the (posterior) expected returns and return covariance matrix.

▪ In a realistic empirical example, using a number of well-known cross-sectional US equity factors, they demonstrate that the BLB approach can add value to mean–variance-optimal multi-factor risk premium portfolios.

Risk premiums and smart beta investing have attracted considerable attention from institutional and individual investors for their simplicity, transparency, and low cost. Johnson (2018) estimated that at the end of 2018 almost 1,500 smart beta products were traded on the public exchanges alone, amounting to about $800 billion of assets under management worldwide. These funds represent a hybrid of active and passive investing, often deploying rules-based investment strategies and portfolio construction methodologies. Nevertheless, the main goal of these investment vehicles is to outperform the market, just like that of active approaches. For this purpose, the majority of these funds rely on combining well-known risk premiums into a portfolio, perhaps by a simple equally weighted approach or more sophisticated portfolio optimization schemes. Numerous academic studies provide support for this form of factor investing, confirming the presence of factor premiums in different asset classes, domestically and internationally, and across various themes, such as betting against beta, carry, low volatility, momentum, quality, return seasonality, size, and value (see, e.g., Fama and French 1993; Carhart 1997; Subrahmanyam 2010; Moskowitz, Ooi, and Pedersen 2012; Asness, Moskowitz, and Pedersen 2013; Ang 2014; Frazzini and Pedersen 2014; Keloharju, Linnainmaa, and Nyberg 2016; Koijen et al. 2018; and Baltussen, Swinkels, and Van Vliet 2019).

Although a number of studies have addressed portfolio construction for risk premiums (see, e.g., Bass, Gladstone, and Ang 2017; Bergeron, Kritzman, and Sivitsky 2018: Dopfel and Lester 2018; Bender, Le Sun, and Thomas 2018; and Aliaga-Diaz et al. 2020), most of the work in this area has focused on factor models from the frequentist’s perspective.

First introduced in two Goldman Sachs Fixed Income Research Notes in the early 1990s (Black and Litterman 1990, 1991b) and later published in academic journals (Black and Litterman, 1991a, 1992), perhaps the most well-known Bayesian portfolio construction methodology is the Black–Litterman (BL) model. The purpose of BL is to improve the inputs in portfolio construction. A key insight of Black and Litterman is that the unreasonable and unstable portfolio holdings frequently encountered in classical mean–variance optimization (MVO) is the result of feeding noisy and inconsistent risk and return forecasts into an optimizer. They realized that risk and return forecasts have to be consistent with one another to be effective in optimization. Their solution resulted in the BL model, which combines the market-implied views with investor views, producing consistent risk and return forecasts (see, e.g., Litterman and He 1999; Satchell and Scowcroft 2000; Fabozzi, Focardi, and Kolm 2006, 2010; Fabozzi et al. 2007; and Meucci 2009 for detailed expositions and discussions of the BL model). The BL method itself is often described as “being Bayesian,” but the original authors do not elaborate directly on its connections with Bayesian statistics. Kolm and Ritter (2017) introduced the Black–Litterman–Bayes (BLB) model, a most general Bayesian portfolio optimization procedure. They elucidated the direct link between BL and Bayesian regression, highlighting its relationship to Bayesian networks (Pearl 2014). In addition, they provided a detailed Bayesian treatment of views on factor risk premiums in the context of multifactor models. Specifically, they showed that the arbitrage pricing theory (APT) of Ross (1976), today a cornerstone in the practice of quantitative investing, has a natural Bayesian extension. They derived the associated BL optimal factor portfolio and discussed two common types of priors in these settings, benchmark-optimal and data-driven priors.

The authors of several related studies have explored BL and more general Bayesian approaches to portfolio construction, including Jones, Lim, and Zangari (2007); Zhou (2009); Avramov and Zhou (2010); Figelman (2017); Rubesam and Hwang (2019); and Melas et al. (2019).

In this article, we explore the BLB model for factor portfolios and factor views, one of the models introduced by Kolm and Ritter (2017). In particular, we take as a premise that some asset managers and hedge funds employ systematic investment processes in which the principal goal is to optimally harvest various factor premiums. In other words, these managers base their investment decisions on the view that certain factor risk premiums are likely to deliver risk-adjusted returns that are attractive or cannot be obtained through the capital asset pricing model (CAPM) market portfolio alone.

In the original BL framework, all views were on portfolios of securities. Factors are not portfolios; rather, they are unobservable random variables that represent sources of risk that are common to all securities trading in the market. Hence, the BL model is not directly applicable to views about factor performance. However, an extension of the same method used by Black and Litterman allows a portfolio manager to express views on factors directly.

In this article, we make several contributions. First, we show how to incorporate views on factor risk premiums into APT models. Second, we provide concrete closed-form formulas for the BLB expected returns and covariances that result from integrating an investor’s factor views into an APT model. Third, with a realistic empirical example, we demonstrate how the incorporation of factor views can add value to multifactor risk premium portfolios. Specifically, we describe a step-by-step procedure to build the prior and incorporate the factor views and demonstrate that, in the event the views are prescient, the BLB portfolio strategy outperforms an analogous MVO strategy in which the latter uses the views to build expected returns but does not benefit from the BLB prior or likelihood.

## THE BLACK–LITTERMAN MODEL

The BL model blends market-implied expected returns with investor views, taking into account the investor’s confidence about these inputs (Black and Litterman 1991a, 1992). In other words, the so-called BL expected returns are a confidence-weighted average of the market-implied expected returns and the investor’s views.

We consider a market with *n* securities whose returns **r**_{t} are multivariate and normally distributed with an expected return vector, **μ**, and return covariance matrix, **Σ**; that is

A mean–variance investor will choose portfolio holdings, **h** = (*h*_{1}, …, *h _{n}*)′, by solving the MVO problem

where μ_{p} := 𝔼[**h**′**r**_{t}] = **h**′**μ** is the expected portfolio return, is the portfolio variance, and λ is the investor’s risk aversion parameter that determines the trade-off between expected portfolio return and risk. The portfolio holdings **h** are in units of dollars, or whatever currency the investor is using. By solving Equation 2, we obtain the optimal holdings

In the BL model, investors can have their own views. A view might be that “the German equity market will outperform a capitalization-weighted basket of the rest of the European equity markets by 5%” (Litterman and He 1999). To express this view, we let **p** = (*p*_{1}, …, *p _{n}*)’ denote the portfolio that is long one unit of the DAX index and short a one-unit basket that holds each of the other major European indexes (e.g., UKX, CAC40, AEX) in proportion to their respective aggregate market capitalizations so that Σ

_{i}

*p*= 0. Denoting the expected return of this portfolio by

_{i}*q*= 0.05, we can express the investor’s view as 𝔼[

**p**′

**r**] =

*q*, where

**r**is the vector of security returns over the considered time horizon. If the investor has

*k*such views

the portfolios **p**_{i} are more conveniently arranged as rows of the *k* × *n* matrix, **P**. Then the investor’s views can be compactly expressed as

where **q** = (*q*_{1}, …, *q _{k}*)′. However, the expectations (Equation 4) alone do not convey how confident investors may be in their views. For this purpose, in the BL model, the investor specifies a level of confidence via the

*k*×

*k*matrix

**Ω**such that

In other words, an investor’s views are represented as the (1) mean and (2) covariance matrix of linear combinations of the expected returns **μ**. This information is partial and indirect because the views are statements about portfolio returns rather than about individual security returns directly. Furthermore, this information is noisy, with the noise **ε _{q}** assumed to be multivariate normally distributed, because the future is uncertain.

Black and Litterman were motivated by the principle that, in the absence of any sort of investor views and constraints, the mean–variance-optimal portfolio holdings should be the global CAPM equilibrium portfolio, **h**_{eq}. Consequently, in the absence of any views, the investor’s model for security returns is given by Equation 1 with

where **π** is the global CAPM expected return in excess of the risk-free rate, and the inverse of the covariance matrix **C** represents the amount of confidence an investor has in the estimate of **π**. The distributional assumption (Equation 6) is referred to as the CAPM prior.

Given the security return distribution (Equation 1), the prior (Equation 6) and the investor’s views (Equations 5), the BL model yields the (posterior) expected returns

7and (posterior) return covariance matrix

8frequently referred to as the Black–Litterman expected returns and covariance matrix, respectively. Kolm and Ritter (2017) provided a full Bayesian derivation of the BL model.

## BLACK–LITTERMAN FOR FACTOR INVESTING

### Linear Factor Models

Taking APT (Ross 1976; Roll and Ross 1980) as a starting point, we consider a market in which the return in excess of the risk-free rate of the *i*th security on day *t* is given by

where *f _{t,j}*,

*j*= 1, …,

*k*are

*k*factors,

*x*

_{t,i,j}denotes the

*j*th factor loading of the

*i*th security, and ε

_{t,i}is the residual return (or idiosyncratic return) of the

*i*th security. We assume the residual returns ε

_{t,i}are white noise processes that are mutually uncorrelated with each other and across time and are uncorrelated with all factors; that is

Additionally, we assume for simplicity that the residual returns are normally distributed

10Typically, factor loadings *x _{t,i,j}* are known, exogenously given, and nonstochastic. Some examples of factors include size and value. For a size factor, the loadings are frequently some nonlinear transformation of the market capitalization of each stock, which are known and exogenous. For a value factor, the loadings are frequently taken to be some measure of the book value or earnings of each company, divided by market capitalization. The value factor is also often passed through a nonlinear transformation to normalize it cross sectionally. The factors

*f*

_{t,j},

*j*= 1, …,

*k*are unobservable (or hidden) random variables that collectively account for all co-movement of security returns. Because they cannot be observed directly, they must be obtained via statistical inference. In contrast to the residuals, different factors are not necessarily independent.

Using matrix/vector notation, we write Equations 9 and 10 as

11where **r**_{t} := (*r _{t}*

_{,1}, …,

*r*)′ is an

_{t,n}*n*-dimensional random vector of cross-sectional security returns in excess of the risk-free rate,

**X**

_{t}is the

*n*×

*k*matrix of factor loadings known before time

*t*,

**f**

_{t}is the

*k*-dimensional random vector process of factors,

**ε**

_{t}is the

*n*-dimensional random vector process of residual return, and

is the (strictly positive) *n* × *n* diagonal matrix of residual variances.

We note that **r**_{t} represents close-to-close total returns from *t* − 1 to *t*, when *t* denotes a day or, when more precision is required, the exact time of the equity market close on day *t*. The subscript *t* has the same meaning for the other random processes.

We assume the **f**_{t} process has finite first and second moments such that

where the elements of **μ _{f}** are referred to as the factor risk premiums. From Equations 11 and 13, we immediately obtain that the expected security returns and their covariance matrix are given by

where denotes the transpose of **X**_{t}. We note that Equation 15 and the positivity assumption in Equation 12 together imply that the security-level inverse exists. Classical MVO uses the security-level covariance matrix and expected returns. As is well known, the historical sample covariance matrix is a poor estimator and should not be used for MVO. For instance, if we consider a universe that is similar to the Russell 3000, then one has *n* = 3,000 securities and *n*(*n* + 1)/2 × 4,500,000 independent elements in the covariance matrix, which is too many independent parameters to estimate from daily security returns. In this situation, APT models are especially useful in that they provide the covariance matrix estimator (Equation 15) that, unlike the sample covariance matrix, leads to stable, well-diversified optimal portfolios (Ritter 2016).

We state without proof that the inverse of Equation 15 can be obtained by the Woodbury matrix inversion lemma, given by

16where . The key to making computations (Equation 16) run quickly is to systematically avoid the bottleneck of creating and manipulating a full *n* × *n* matrix (as opposed to *n* × *k* or *k* × *k* matrixes, where *k* is much smaller than *n*). For example, in the Markowitz case, one should compute

observing that the right-hand side can be calculated without having to build a full *n* × *n* matrix.

### Black–Litterman for Factor Models

Leveraging the general BLB approach of Kolm and Ritter (2017), we describe a BL-type model for factor models of the kind in Equations 9 and 10. For simplicity, we treat **F**_{t} as known. Therefore, once **X**_{t},**D**_{t} are given, **Σ _{t}** is also known via Equation 15, just as in the original BL model. It follows that the source of uncertainty is in the expected factor premiums

**μ**. Next, we discuss two kinds of priors for factor premiums.

_{f}### Data-Driven Priors

If the **f**_{t} process is stationary, such that **μ _{f}** and

**f**

_{t}are approximately constant over time, then we can obtain a prior,

**π**, from a (Bayesian) time-series model of the factor returns

_{f}**f**

_{t}. One possible choice is the mean (or rolling mean) of a time series of the ordinary least squares estimates

More sophisticated approaches, such as that of hierarchical or mixed-effects models, can also be used (Gelman et al. 2003).

An advantage of data-driven priors is that they do not require a benchmark portfolio (such as the capitalization-weighted benchmark in the original BL model). Consequently, data-driven priors may be preferred in absolute return strategies in which the effective benchmark is cash. Below we provide an empirical example that uses a data-driven prior in a hypothetical absolute-return strategy.

### Benchmark Priors

If there is a benchmark portfolio, **h**_{B}, then close in spirit to the original BL model is to find a benchmark-optimal prior. For a general discussion of benchmark-optimal priors in this setting, we refer the reader to Kolm and Ritter (2017). For instance, we could use the prior

where **ξ** is a *k*-dimensional parameter vector, and **V** is a *k* × *k* matrix that is symmetrical and positive definite.

Kolm and Ritter (2017) showed that by combining this prior with the factor model of Equations 9 and 10, one obtains the a priori expected returns and return covariance matrix

20 21Here all quantities are assumed to be sampled at the same time *t*. However, for readability purposes, in this section, we omit the explicit time index, allowing the time dependence to be implicit. Hence, the a priori mean–variance-optimal portfolio is given by

From Equation 22, it is clear that unlike the original BL, any arbitrary benchmark portfolio cannot be realized as an a priori optimal portfolio. Those that can are necessarily of the form λ^{−1}**Σ**^{−1}**Π**, where **Π** is some linear combination of the columns of **X**. These are precisely the portfolios that are optimal with respect to a set of individual risk premiums that are in the factor model. Because not every possible portfolio is realizable as a priori optimal, the market portfolio may also not be optimal. Nevertheless, if (1) the market is in a CAPM equilibrium and (2) one of the columns of **X** contains the CAPM betas, then the individual risk premiums will be proportional to that column of **X** and the market portfolio is realizable as a priori optimal.

### Factor Views

We are free to express views on factor risk premiums in the same way we would express views in the original BL model. In particular, we can express views on any (linear) portfolios of risk premiums. However, it is less likely for portfolio managers to have views on linear combinations of factors. Perhaps the most common and parsimonious situation is one in which we have a view on each factor risk premium that is independent of our views on other factors. Using the value and momentum factors as an example, a portfolio manager might have two independent views: (1) a view on the value premium and (2) a view on the momentum premium. We can express views of this kind as

23where is a diagonal *k* × *k* matrix with all ω_{j} > 0.

### The Black–Litterman Expected Returns and Covariance Matrix for Factor Portfolios

By combining the factor model (Equations 9 and 10), the factor prior (Equation 19) and the investors factor views (Equation 23), Kolm and Ritter (2017) showed that the (posterior) expected returns and return covariance matrix are given by

24 25where all quantities are sampled at time *t*, and , and are the posterior hyperparameters. We emphasize that **Σ** is the covariance matrix of security returns from the factor model (Equation 15). Consequently, from the expected returns (Equation 24) and covariance matrix (Equation 25), we obtain the MVO portfolio

where

27 28Equations 26–28 represent the solution to BLB optimization in the context of factor models. We observe that the security-level risk premiums are linear combinations of the factors that form the columns of **X**. Intuitively, we can think of as a set of factor risk premiums adjusted to take account of the views. These adjustments give more weight to factors that have high prior mean–variance ratios, ξ_{i}/V_{ii}, and/or high expected return-uncertainty ratios, .

### Discussion

The original work of Black and Litterman was done under normality assumptions for the return distributions. This is mathematically convenient because the normal prior is a conjugate prior for the normal likelihood, so the posterior is also normal with all means and covariances obtainable in closed form. Nevertheless, it is of clear interest to investigate the applicability of similar methods beyond the normal distribution.

Chamberlain (1983) showed that under elliptically distributed return distributions, expected utility—for any concave utility function—is a function of only the portfolio’s return and variance. Elliptical distributions are a large family of distributions, including many fat-tailed distributions such as the Student’s *t*-distributions. In the context of BLB optimization, this implies that we are free to use any form for the prior and the views—including models in which the unknown parameter(s) need not simply be the mean return—as long as the final posterior distribution *p*(**r**_{t+1}|**q**) of the cross-sectional security return vector is a member of the elliptical family; then any risk-averse expected-utility maximizer will end up maximizing the posterior expected portfolio return minus a constant multiple of the posterior portfolio variance. Of course, with a different (e.g., nonnormal) model, the investor need not arrive at precisely the same expected return and variance formulas as did Black and Litterman. Additionally, if the prior is not conjugate to whatever likelihood is chosen, then computing the posterior moments may be difficult or even impossible.

One of the early successes of the Black–Litterman model was that it provided a form of regularization of the fundamentally unstable inversion of the historical sample covariance matrix. However, the Black–Litterman technique is not primarily meant as a regularization technique. It has that property as a by-product because its likelihood and prior tend to pull the portfolios toward known, reasonable portfolios.

Importantly, with common usages of APT, regularization is, strictly speaking, no longer necessary because the resulting covariance matrix is already stably invertible. For example, when the mean–variance portfolio is computed via Equation 17, the phenomenon of corner portfolios does not occur. The resulting portfolios are typically well diversified, as long as the factor model itself is reasonable.

We stress that, even with a stable covariance matrix, the BL procedure is still tremendously valuable—perhaps even more so, because even the classical BL model does still use **Σ** as an input, while being less sensitive to its instabilities. However, as compared to MVO, the BL model gives the portfolio manager additional degrees of freedom to control the portfolio, in the form of ω_{i}, which quantifies the uncertainty in the *i*th view. A key difference is that the ω_{i} parameters do not appear in the classical Markowitz formulation.

## AN EMPIRICAL CASE STUDY

In this section, we provide an empirical example of the BLB model for factor models and investigate whether our methodology provides reasonable empirical results. Specifically, we construct an instance of BLB model (Equations 26–28) suitable for quantitatively oriented institutional portfolio managers interested in systematically trading single-name US equities in a factor-based fashion.

Our universe of stocks is chosen with certain liquidity thresholds in place. In particular, stocks must be priced above $5, bid–offer spreads must be less than 100 bps, and they must trade at least 10,000 shares and $1 million of volume per day. We restrict attention to only common stock and only the most liquid share class of each company. This typically results in a universe of 1,800–2,000 stocks, but in the period around the 2008 financial crisis, the size of our universe briefly dropped to around 1,400 stocks.

Our factor model includes a Global Industry Classification Standard–based industry classification with 58 industries that are either single Global Industry Classification Standard subindustries or several closely related subindustries merged together. Our model also includes a set of continuous numerical factors known colloquially as *style factors* because they are thought to represent investment styles such as value, contrarian, or dividend capture.

Roughly speaking, *factor timing* is defined as attempting to predict which factors will have superior factor returns over a forthcoming time period, with the goal of monetizing those views in the event they materialize. Any such predictions may be naturally translated into views on factors or combinations of factors.

In the following example, we simulate a hypothetical portfolio manager who possesses definite skill at factor timing. We then illustrate how to use the factor-based BLB to construct a sequence of portfolios that successfully monetizes the portfolio manager’s factor-timing views.

The views are expressed through (1) the *k*-dimensional vector **q** representing the expected returns on the respective factors and (2) the *k* × *k* matrix **Ω** that represents the uncertainties in those views. Specifically, we take **Ω** to be diagonal, where and ω_{i} can be thought of as the width of a confidence interval around *q _{i}*, for each

*i*= 1, …,

*k*.

To simulate a manager who is skilled at factor timing, we assume that, at the very beginning of our historical data, this portfolio manager made a single call regarding the expected returns and uncertainties on a handful of factors and has no views on the remaining factors. In particular, the portfolio manager chooses **q** and **Ω** at the beginning of 2007 and remains steadfast in those views until the present time. With the benefit of hindsight, we simulate the performance of this hypothetical portfolio manager’s strategy, in which some of the factor bets turned out to be correct. We strongly stress that this study is meant to simulate what happens in the event the portfolio manager made the right call at the beginning, as a way of illustrating the efficacy of the BLB portfolio construction with factors. In this article, we neither simulate nor provide guidance on the mechanism of factor timing itself.

Specifically, we assume that the portfolio manager decided, at the beginning of the sample period (January 2007), to harvest the size (the small-cap effect) and liquidity premiums. In addition, the portfolio manager decided to assign negative premiums to firms with high short interest and a positive premium to a quality factor based on accounting metrics. The directions of these four factor bets align with well-known results from behavioral finance. In Exhibit 1, we depict the cumulative factor return performance of these four factors over the 2007–2020 sample period. Furthermore, we assume that the portfolio manager was prescient in deciding to take no explicit view on the well-known and much-studied value and momentum factors, neither of which performed well in the US equity market over the sample period of 2007–2020. We refer to this portfolio manager’s views as the *prescient views* because they benefit from having taken no explicit view on value and momentum (within the US equity market) over the period.

The procedure for our empirical study is as follows:

**1.**For each day*t*, construct the matrix of factor loadings**X**_{t}by loading the data as a pandas data frame, transforming the loadings, and applying the function**patsy.dmatrix**with the appropriate model formula. The transformation entails reshaping the raw loading (e.g., market cap) to be Gaussian and centered within the estimation universe.**2.**For each day*t*, estimate the factor returns*f*_{t,j}using ordinary least squares as .**3.**Choose an in-sample period for the factor covariance matrix. We choose 2007–2015 to be in sample with respect to the factor covariance,**F**.**4.**Estimate the factor covariance**F**as the sample covariance of with*t*restricted to the in-sample period. We separate the calculation of the factor covariance into volatility estimation and correlation estimation. The factor covariance is given by**F**=**SRS**, where**S**is a diagonal matrix containing the factor volatilities, and**R**is the factor correlation matrix. Note that**R**was estimated in Python by applying the method**numpy.corrcoef**to the coefficients estimated in a previous step. We mitigate the effect of estimation error in**F**^{−1}by applying a lower bound of 0.01 to the factor volatilities**S**.**5.**For a market-neutral study, we set our prior to**ξ**=**0**, and**V**=**SR**_{s}**S**, where**R**_{s}denotes the Ledoit–Wolf shrinkage estimator for the correlation matrix (see Ledoit and Wolf 2004). Under this prior, an all-cash benchmark portfolio is a priori optimal. Practitioners may wish to choose a prior under which a different benchmark, such as the S&P 500, is optimal; details of this procedure are given in the discussion of Equation 29.**6.**We set**q**and**Ω**to be consistent with the prescient views outlined in the preceding. Specifically, we specify one view per factor. For all industry and style factors not explicitly represented in the prescient views, we set*q*= 0 and ω_{i}_{i}= 0.05, representing no view on their expected return, with a high level of uncertainty. For the factors involved in the prescient views, we set*q*equal to −3 bps for size, −1 bp for liquidity, −0.5 bp for short interest, and +1 bp for quality. We also set a premium of +1 bp for the intercept factor._{i}**7.**For each*t*, produce forward-looking estimates of**D**_{t}using historical residual vols blended with implied vols.**8.**For each*t*, compute without expanding as a full*n*×*n*matrix by multiplying each*i*th row of**X**_{t}by . Note that**Z**_{t}has the same dimensions as**X**_{t}; both are*n*×*k*matrixes. Hence, as discussed previously,**Z**_{t}is easier to compute with than an*n*×*n*matrix.**9.**For each*t*, define the intermediate calculation result and compute**10.**Finally, compute the portfolio return .

Although the BLB method for factor models as presented here is not much more complicated than a standard MVO approach to the same problem, we believe that any added complexity should be justified through added performance. Therefore, to gauge the value-add of the portfolio construction technique, we also include results for an MVO baseline, which is proportional to where **μ**_{t} = **X**_{t}**q**. Note that the baseline includes **q** and hence also benefits from the prescient views, but it does not benefit from the additional Bayesian portfolio construction of BLB. Of course, for the baseline computation, we use Equation 16; hence, our method does not entail the (notoriously unstable) security-level sample covariance matrix.

In Exhibit 2, we show a comparison between the BLB and MVO approaches. We have scaled the two portfolios so that they realize the same volatility over the sample period. The BLB portfolio sequence achieves a Sharpe ratio of 1.16 whereas the MVO portfolio achieves about 0.9. There are several reasons for the difference, but perhaps the most important is the presence of **Ω** in the BLB portfolio. Furthermore, the BLB portfolio has a higher gross market value but achieves lower exposure to factors on which the manager has no views, in part because of the shrinkage toward **V**^{−1}**ξ**. In any case, this illustrates that the BLB method provides the manager with additional degrees of freedom that have the potential to reduce risk relative to MVO portfolios that also benefit from the same prescient factor bets.

The average turnover of the two strategies we consider is comparable: Each turns over an average of about 10%–15% of gross market value per day. Such high turnover is unsurprising because the portfolio formation process is unaware of costs. The turnover is mostly driven by changes in the factor exposures themselves and changes in the residual variance forecasts. This is likely too much turnover to be directly implementable, but the transaction cost–aware optimization that we discuss in the conclusions section would be implementable.

Although we shall not do so here, it is possible to run this procedure with a benchmark that is approximately the market-capitalization-weighted benchmark with holdings vector **h _{B}**. This is achieved by choosing a normal prior having the property that the benchmark portfolio is a priori optimal, meaning that the benchmark is optimal under the expected returns and covariance matrix described by Equations 20 and 21.

Mathematically, referring to Equation 22, this amounts to solving for **ξ** in the equation

Solving for **ξ** in this way might not be possible exactly, in which case one may solve for it in the least-squares sense.

## CONCLUSIONS

In this article, we proposed a general framework referred to as BLB for constructing optimal portfolios for factor-based investing. In the spirit of the classical Black–Litterman model, this framework allows for the incorporation of investor views and different priors on factor risk premiums, including data-driven and benchmark priors. We provided computationally efficient closed-form formulas for the (posterior) expected returns and return covariance matrix that result from integrating factor views into an APT multifactor model.

Finally, in a step-by-step procedure, we showed how to build the prior and incorporate the factor views, demonstrating in a realistic empirical example using a number of well-known cross-sectional US equity factors that the BLB approach can add value to mean–variance-optimal multifactor risk premium portfolios.

In closing, we make a few observations regarding application of these results in transaction cost–aware MVO. The original BL model and BLB extension with factor views that we have presented here share in common that the portfolio construction step is a standard mean–variance optimization. The BL and BLB methods are based on adjusting the posterior security return density (where only the first two posterior moments are needed, in the case of elliptical distributions). In particular, we denote by **h _{0}** the starting portfolio and

*c*(

**h**,

_{0}**h**) the anticipated cost (predicted before trading begins) to trade from

**h**→

_{0}**h**. For institutional asset managers,

*c*(

**h**,

_{0}**h**) is typically dominated by slippage costs due to temporary and permanent impact. Nevertheless, the optimization problem for BLB with costs is

where notation refers to Equations 24 and 25.

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