## 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 pre-determine 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.

**TOPIC:** Portfolio construction

Asset allocation and factor investing are often seen as competing approaches to portfolio construction. Historically, most investors composed portfolios from broad asset classes, but now a growing constituency of investors is looking to factors as the building blocks for forming portfolios.

Those who choose to build portfolios from asset classes argue that they are easy to observe and are directly investable. Factors, by contrast, are not always readily observable, nor are they directly investable. Investors must identify combinations of assets that best track the behavior of factors, and because the mapping functions between assets and factors change through time, investors who choose to allocate to factors are exposed to an additional source of uncertainty beyond the uncertainty of the factor performance.

Investors who prefer to allocate to factors argue that asset classes are defined arbitrarily and do not capture the fundamental determinants of performance as effectively as factors do. In addition, some investors prefer to invest in factors because they believe that factors carry risk premiums that are not directly available from asset classes.

We propose a compromise. We argue that investors should continue to use asset classes as the building blocks for forming portfolios,^{1} but they should combine them in a way that balances their expected return and risk with adherence to a preferred factor profile. This approach preserves the benefit of investing in observable and directly accessible units, while enabling investors to capture preferred factor exposures.

## METHODOLOGY

The traditional approach for constructing portfolios, which was given to us by Harry Markowitz [1952], is to maximize expected utility as defined in Equation 1.

1In Equation (1), *E*(*U*) equals expected utility, μ_{p} equals portfolio expected return, λ_{RA} equals risk aversion, and equals portfolio variance. To proceed, we estimate the expected returns for all of the asset classes under consideration as well as the covariances between every pair of asset classes. We then vary the asset class weights for a given level of risk aversion so as to maximize expected utility.

In addition to the expected return and risk characteristics of these asset classes, we may have preferences for certain factor exposures. We could therefore define a factor profile as a weighted average of our preferred combination of factor exposures. Because factors are often measured in different units, it may be preferable to record their changes in value as logarithms, as shown in Equation 2.

2In Equation (2), *R*_{ln} equals the natural logarithm of the factor return, ln equals natural logarithm, *FV _{T}* equals the factor index value in this period, and

*FV*

_{T–1}equals the factor index value for one period prior.

Many factors, especially macroeconomic factors, have relatively stable values compared to asset classes. This is because macroeconomic factors are not traded. The prices investors assign to tradeable assets reflect their expectations about the level and uncertainty of future income. Macroeconomic variables reflect realized economic activity, or in the case of consensus estimates, expectations about the mean value of the macroeconomic variable. These values are, therefore, less volatile than asset prices. As a consequence, the covariances between asset classes and factors are relatively small, just as the covariances between high-risk and low-risk asset classes are small. Therefore, to build a portfolio that is reasonably sensitive to a comparatively stable factor profile, we need to rescale the factor returns by multiplying them by a constant. Otherwise, minimizing tracking error to something that is relatively stable is akin to minimizing total volatility. We discuss this issue more thoroughly in the Appendix.

Once we rescale the factor returns, we create a factor profile time series by taking a weighted average of the rescaled factor returns that accords with the relative importance we wish to assign to each factor. With this factor-profile time series, we estimate a second covariance matrix that is composed of the covariances between each asset class under consideration and the factor profile. The final step is simply to augment the portfolio construction objective function to include the covariances among the asset classes within the portfolio as well as the covariances between the asset classes and factor profile, as shown in Equation (3).

3Here, λ_{DFP} equals aversion to deviations from the factor profile, and ξ_{FP} equals deviations from the factor profile, which we define as the tracking error of the optimal portfolio to the factor profile.

This objective function is similar to an objective function proposed by Chow [1995], who demonstrated how to optimize a portfolio taking into consideration aversion to absolute risk as well as aversion to risk relative to a benchmark composed of assets from the portfolio’s opportunity set. In Chow’s setup, we need only specify a single covariance matrix of the assets because both the portfolios and the benchmark are formed from the same assets. His setup solves for absolute weights and relative weights simultaneously. In our setup, we specify a covariance matrix for the asset classes as well as the covariances of the asset classes with the factor profile.^{2}

Note that the objective function does not include a term for expected factor profile returns because we are not allocating to factors. We are allocating to asset classes, but in a way that is sensitive to how the asset classes co-vary with the factor profile. Moreover, our views about factor behavior are implicit in the weights we use to define the factor profile.

If we were mainly interested in managing unconditional performance, we would specify low aversion to deviations from the factor profile. If, instead, we were primarily interested in tracking a factor profile, we would specify high aversion to deviations from the factor profile.

To summarize:

1. We estimate the expected returns and covariances of the asset classes in which we wish to invest.

2. We identify factors to which we seek exposure, either positive or negative.

3. We create factor time series by recording changes in the factor values measured in log units and rescaling them to render the portfolio reasonably sensitive to changes in factor values.

4. We define a factor profile by calculating a weighted average of the rescaled factor returns in accordance with our factor preferences.

5. We estimate the covariances between the asset classes and the factor profile.

6. We solve for a factor-sensitive optimal portfolio by maximizing an expanded objective function that incorporates both aversion to absolute risk and aversion to deviations from the factor profile.

## CASE STUDY

Let’s suppose we wish to allocate across seven asset classes: U.S. equities, foreign developed-market equities, emerging market equities, Treasury bonds, U.S. corporate bonds, commodities, and cash equivalents. Suppose also that we wish to form a portfolio that co-varies with inflation and real gross domestic product (GDP) growth. Exhibit 1 shows the expected returns, standard deviations, and correlations of these asset classes, along with the factor standard deviations and correlations with the asset classes.^{3}

The expected returns are estimated as equilibrium returns—that is, the risk-free return plus the product of the betas of the asset classes and their excess returns relative to a broadly diversified portfolio designed to represent the market. We assume a risk-free return of 3.5% and an expected Sharpe ratio of the market portfolio of 0.4.^{4} The standard deviations and correlations are estimated from quarterly returns starting January 1990 and ending June 2017. The inflation and GDP growth values are based on consensus estimates covering the same period.

We have selected two macro factors: GDP growth and inflation. Both are measured as changes in their respective 12-month consensus forecasts, using the Consensus Economics survey data available since 1989. We do this because we assume that changes in expectations affect asset prices. We multiply the factors by a scalar (75) to scale the factor volatility so that the factor covariances to assets have a meaningful impact in the optimization.

Exhibit 2 shows an optimal portfolio for an investor with moderate risk aversion who is indifferent to the factor exposures of these asset classes.^{5}

Now let’s suppose we are concerned about the prospect of *stagflation*, in which the economy experiences stagnant growth coupled with high inflation. We do not feel confident in conditioning the expected returns, standard deviations, and correlations explicitly; rather, we simply want a portfolio whose returns will covary positively with inflation and negatively with economic growth because we are concerned that inflation will be higher than expected and economic growth forecasts will contract against expectations.

If we were equally confident in our views about inflation and economic growth, we might create a factor profile that comprised a positive 100% weight to inflation and a negative 100% weight to economic growth. Suppose, however, that we were twice as confident about our expectation for inflation as we were for GDP growth. In this case, we would create a factor profile that had a positive 100% weight to inflation and a negative 50% weight to economic growth. Exhibit 3 shows the correlations of these factor profiles to the asset classes that compose our investment opportunity set.

In the case in which we are equally confident in our views about inflation and economic growth, given the correlations shown in Exhibit 3, we should expect the optimal portfolio to be tilted less toward equities and U.S. corporate bonds and more toward commodities and cash equivalents. The impact on Treasury bonds should be neutral because inflation surprises would be negative for Treasury bonds, but negative growth surprises would be positive. In the case in which we are twice as confident in our view for inflation compared to economic growth, we should expect an additional tilt away from Treasury bonds.

Exhibit 4 validates our expectations. It presents the original optimal portfolio that is insensitive to factors along with two factor-sensitive portfolios based on the factor profiles shown in Exhibit 3. These factor-sensitive portfolios are based on the same risk aversion coefficient used to derive the original portfolio that is insensitive to factors.^{6}

Exhibit 5 shows factor-sensitive portfolios based on all of the same inputs, with the following exception: We set aversion to deviations from the factor profile equal to 0.1 instead of 0.2. The factor-sensitive portfolios in Exhibit 5 are more similar to the optimal portfolio that ignores factors. Moreover, they have higher expected returns than the factor-sensitive portfolios in Exhibit 4. As to be expected, however, they have higher standard deviations and higher factor profile deviations than the factor-sensitive portfolios in Exhibit 4.

Let’s assume we are interested in three additional economic scenarios: demand-led growth (100% weight on both Consumer Price Index [CPI] and GDP), recession (-100% on both CPI and GDP), and disinflationary growth (-100% to CPI, 100% to growth). Exhibit 6 shows how the optimal weights vary across these economic scenarios.

The stagflation scenario (100/-100) discussed earlier leads to more cash equivalents. This is because only commodities have a meaningful positive correlation with this factor profile, but the commodity unconditional risk premium (0.4%) is low relative to its risk (25%). The disinflationary growth scenario (-100/100) leads to a large reallocation of Treasury bonds to U.S. corporate bonds, with about 5% toward equities. The demand-led growth scenario (100/100) tilts the portfolio toward equities, in particular U.S. equities, U.S. corporate bonds, and commodities, at the expense of Treasury bonds. Finally, the recessionary scenario (-100/-100) leads to an allocation away from risky assets (equities, commodities, and U.S. corporate bonds) and toward Treasury bonds.

## PERFORMANCE

We expect that the conditioned portfolios will perform better than the unconditioned portfolio during periods that correspond to their respective factor profiles. For example, the stagflation portfolios should outperform when inflation is high and GDP is contracting or expanding at a slower-than-average pace.

To test our expectations, in Exhibit 7 we compare the performance of the unconditioned portfolio and the five factor-sensitive portfolios over the samples of returns that correspond to their factor profiles. We have defined the samples as quarterly periods in which both factors moved in the direction of the scenario. For example, for the 100/-100 scenario, we included all quarters in which inflation was higher than expected and GDP growth was lower than expected. Keep in mind that this performance is not out of sample. We merely wish to illustrate that factor-conditioned portfolios should perform better than an unconditioned portfolio when the factors behave as anticipated and assuming that we have faithfully captured the factor sensitivities of the asset classes.

As expected, the conditioned portfolios that are optimized based on the factor profiles produce the highest return for the periods corresponding to their factor profiles. Note that the portfolios optimized for negative economic conditions (lower growth than expected) have lower volatility as they become more defensive, whereas portfolios optimized for high inflation conditions have more commodities. For example, consider the stagflation scenarios. The factor-conditioned portfolios perform better than the unconditioned portfolio during the subsample of above-average inflation and below-average economic growth. Whereas the unconditioned portfolio returned 2.4% with a risk of 8.3% during the stagflation periods, the factor-conditioned portfolios had higher returns of 3.3% and 3.2%, respectively, for the two factor profiles with lower risk. This is precisely the result we would have hoped to achieve. If we had specified higher aversion to deviations from the factor profile, this outperformance would have been even greater. Another scenario that has interesting results is the recession scenario, corresponding to low inflation and low growth. The unconditioned portfolio has higher volatility during that period than any other scenario. Meanwhile, the conditioned portfolio cuts the volatility during that period in half, with a higher expected return. However, one should not attach too much meaning to the size of the difference. Investors can alter the impact by increasing or decreasing the factor profile tracking error aversion parameter.

Granted, these results are in sample, but they show that, to the extent we can anticipate factor behavior along with the covariation of asset classes with factors, we can produce superior factor-sensitive performance without investing in new factor portfolios. We do so merely by accounting for the factor sensitivities of the asset classes that already compose our opportunity set.

## SUMMARY

We presented a framework for integrating asset allocation with factor investing, which has the dual benefits of allocating to observable and directly investable asset classes in a manner that tracks a preferred factor profile.

We showed how to augment the classic Markowitz objective function to include a term for aversion to deviation from a factor profile. Our approach differs from the Chow [1995] approach to integrating absolute and relative performance because Chow required only a single covariance matrix. Our approach assumes investment in assets with sensitivities to noninvestable factors. We must, therefore, estimate a covariance matrix of assets as well as a covariance matrix that captures the sensitivities of the asset classes to the factors.^{7}

We illustrated this integrated approach with a case study that showed how the composition and performance of the optimal portfolios change as we change the composition of the factor profile and as we vary our degree of aversion to deviations from the factor profiles. We produced in-sample results showing that factor-sensitive portfolios performed better during regimes in which the factors behave in accordance with our expectations, compared both to their performance during the full sample and relative to the unconditioned portfolio during the factor-conditioned subsample. Whether this performance would prevail out of sample depends on the extent to which the historical covariation of the factors and asset classes persists in the future.

Finally, we wish to emphasize that this new methodology for integrating asset allocation and factor investing is very general. Although we illustrated it with a simple example based on several broad asset classes and two macroeconomic factors, it can be applied in a variety of settings, such as the construction of portfolios that are sensitive to security attributes such as value, size, and momentum; portfolios that are sensitive to unavailable asset classes; or portfolios that are sensitive to volatility regimes.

## APPENDIX

Factors typically have very low volatility compared to assets because, unlike assets, they are not traded. Therefore, optimizing a portfolio to capture factor exposures drives the portfolio toward cash. Increasing risk aversion to the factor profiles does not solve the problem; it simply drives the portfolio even closer to cash. Effectively, the optimizer attempts to capture low-volatility factor exposures by avoiding volatility. Exhibit A1 illustrates this by showing portfolio volatility for different factor profile aversion coefficients.

Exhibit A2 reinforces this point by showing how the portfolio’s cash exposure grows with increasing aversion to deviations to the factor profile.

Regardless of the factor profile, for a given factor aversion coefficient, all portfolios tend to be fairly similar in terms of volatility and exposure to cash. The higher the factor aversion, the more cash and the lower the expected standard deviation. Although increasing the factor profile aversion does increase the dispersion of volatilities among portfolios for different factor profiles, this is of second order importance.

We can overcome this problem by rescaling factor volatilities. We simply multiply factor returns by a chosen constant. Exhibit A3 shows the in-sample performance of the optimal portfolios based on various volatility multipliers. For example, for the 100/–100 portfolio, Exhibit A3 shows the realized return of that portfolio when inflation is above average and growth is below average. We keep constant both the absolute risk aversion and the factor aversion coefficients and only vary the factor return multiplier.

With a factor volatility multiplier of 1, the conditioned portfolios perform better than the unconditioned portfolios during periods of negative growth but not during periods of better-than-average growth. This is to be expected because cash performs well during negative growth surprises. However, for all of the other multipliers, the conditioned portfolios outperform the unconditioned portfolios. The exhibit shows that results are robust to the choice of multiplier as long as it is sufficiently large. The choice of multiplier becomes a calibration decision similar to the choice of factor profile aversion parameter.

## ENDNOTES

We wish to thank Richard Lindsey, Cel Kulasekaran, and Todd Mattina for helpful comments.

↵

^{1}Our view is that asset classes are more stable than factors out of sample because, unlike factors, they are not exposed to mapping error. For evidence supporting this claim, see Cocoma et al. [2017].↵

^{2}We could retain Chow’s setup by constructing a grand covariance matrix of assets and factors and then optimizing subject to appropriate constraints. We prefer our approach, however, because it is more direct and transparent.↵

^{3}The factor standard deviations and correlations are based on factor returns that are rescaled by a factor of 75.↵

^{4}Market capitalization weights for equity are taken from the World Bank based on the World Federation of Exchanges data. Market capitalization for fixed income uses the Bank of International Settlements data.↵

^{5}This portfolio is derived using Equation (1), assuming a risk aversion coefficient equal to 2.16, which has been calibrated to provide an expected return of 7%.↵

^{6}The factor-sensitive portfolios are derived using Equation (3), assuming a risk aversion coefficient equal to 2.16 and a coefficient for aversion to deviations from the factor profile equal to 0.2.↵

^{7}See endnote 2.

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