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# Asset Allocation and Factor Investing

An Integrated Approach to Portfolio Construction

When it comes to portfolio construction, the debate of asset allocation versus factor investing can grow quite heated. Those who choose to build portfolios from asset classes argue that they are easy to observe and are easily investible — unlike factors. These investors also believe that portfolios from asset classes are more stable out of sample compared to portfolios composed from factors. 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 that allows investors to get the best of both approaches. By integrating asset allocation and factor investing, we can preserve the benefit of investing in observable and directly accessible units, while capturing our preferred factor exposures.

We start with the traditional approach to portfolio construction, known as mean-variance optimization, which was introduced by Harry Markowitz in 1952 through his modern portfolio theory. The equation for mean-variance optimization is as follows

You’ll notice that this equation does not consider factors. In order to do so, we need to expand the objective function to include an additional term that reflects

**aversion to deviating from a factor profile.**Of course, in order to do so, we need to build a factor profile. One of the challenges in building a factor profile is that factors are not always measured in the same unit, so instead we must record their changes as logarithms (shown in the equation below).

Another challenge is that many factors are relatively stable because, unlike assets, they are not traded and thus not subject to investor uncertainty. The problem with trying to build a portfolio that is sensitive to a particular factor profile is that the covariances between the factor returns and asset returns are going to be too low to capture any significant sensitivities. Therefore, to build a portfolio that is reasonability sensitive to a comparatively stable factor profile, we need to rescale the factor returns by multiplying them by a constant.

- 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 (positive or negative)
- 3.We create factor time series by recording changes in the factor values (measured in log logarithms) and rescaling them to create a portfolio that is 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 assets 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

To learn more about this approach, we recommend you read the full article on which this summary is based on,

**Asset Allocation and Factor Investing: An Integrated Approach**, published in the 2018 Quantitative Special Issue of The Journal of Portfolio Management.