# Factor investing optimization

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In the case of a full factor approach, the portfolio is constructed with attractive factors. Figure: Traditional portfolio versus the factor investing approach Source: Robeco Investment Solutions Factor-optimization approach A method of implementation according to which a portfolio is fully allocated to factors.

Quantitative investing: invisible layers surface to deliver attractive returns. Related insights more insights. Forecasting stock crash risk with machine learning. Both sector- and factor-based portfolios contain only US stocks. The 16 total return indices are downloaded from Bloomberg on a monthly basis for the longest jointly available period from January to November The dataset includes monthly return observations for each index.

For implementing the portfolio optimization strategies, we need up to five years 60 months for estimating the optimization inputs average returns and the covariance matrix. Three more years of data are required in the Black—Litterman optimization for determining the reliability of return estimates. Therefore, our out-of-sample evaluation period ranges from to and includes months. From the large number of factors discussed in the literature, only a few factors are investable at low cost, for instance, through ETFs.

Tables 2 and 3 display the statistical properties of the sector and factor indices, respectively. In general, the mean returns for the factor and the sector indices are similar. For the factor indices, the returns range from 0. For the sector indices, the minimum and the maximum values are, 0.

The standard deviations of returns reveal that, on average, the factor indices have lower risk with 4. One of the most important statistical aspects directly affecting the risk and therefore the performance of an optimized portfolio is the correlation structure among the assets.

In general, the group of assets with the lower correlation offers better diversification opportunities, which, however, does not necessarily mean higher performance. Both, maximal and minimal correlation coefficients in the factor indices are higher than for the sector group. With respect to the optimization process, our insights from the descriptive statistical analysis are twofold. First, the correlations between factor indices are noticeably higher than between sectors, offering lower diversification opportunities.

Second, average monthly returns were slightly higher for factor indices than for sectors, indicating higher portfolio returns. Therefore, it is essential to employ the different portfolio optimization algorithms to provide empirical evidence whether sectors or factors provide the better trade-off between risk and return and therefore are the superior building blocks for an optimal investment strategy. We structure this section as follows. Section 5. In this section, we report the full period results by providing the following performance results: a full period analysis of Sharpe ratios, b sensitivity analysis of changing constraints, c risk and return analysis, d multifactor performance analysis and e multifactor alpha differences between both strategies.

Full period analysis of Sharpe ratios Table 4. In Table 4 , we report and compare the pairwise Sharpe ratios of the sector- and factor-optimized portfolios. For the risk—return optimization approaches, we employ four different estimation window lengths as model inputs. This optimization framework weighting or optimization algorithm, return forecast and constraints we apply to both factor- and sector-based portfolios.

The initial results presented in Table 4 reveal that for all reported pair results, which are the differences between sector and factor portfolios, the factor allocations generate higher Sharpe ratios. Due to the relatively short history of the indices, however, the Sharpe ratio differences are statistically significant only for some portfolio pairs according to the Opdyke test.

Still, the Sharpe ratio differences are economically relevant and the information that factor portfolios dominate sector portfolios in all analyzed cases is a very strong outcome in favor of factor allocations. We next focus on the effects of different weight constraints columns 1 to 6 in Table 4 , respectively.

The factor performance is relatively stable for different optimization constraints, whereas the performance of the sector allocations tends to deteriorate with more relaxed optimization restrictions. In contrast, the lowest performance difference occurs for the minimum variance approach, although factor portfolios outperform sector allocations.

This is a rather surprising result, given that the average correlations between sectors were lower than between factor indices. The performance difference for the minimum variance approach gets even smaller when we further relax the optimization restrictions.

The analysis of the risk and return structure of the portfolios in the next section provides further insights for explaining these results. Separating Sharpe ratios into their two basic components return and risk offers some additional insights into the source of the performance differences between sector and factor allocations.

We present the annualized mean returns as well as the risk standard deviation, volatility of the optimized portfolios in Table 5. The benefits of the factor allocation originate not only from larger returns Table 5 , Panel A but also from lower portfolio volatility Table 5 , Panel B. In all cases, regardless of the allocation environment optimization algorithm, window length for optimization inputs and weight constraints , the factor portfolios reveal lower or equal risk as the sector portfolios.

These outcomes are in accordance with the observation in Bender, et al. There is only one exception, in which the sector allocation provides a marginal lower volatility: the minimum variance portfolio with narrow weight restrictions.

For the properties of the optimization methods, we identify some interesting details for the factor allocations. Similar to the Sharpe ratios, the risk profile of the factor portfolios closely relates to the constraints. Widening the level of restrictions columns 3 to 6 substantially increases the volatility of the sector portfolios, whereas the volatility of the factor portfolios remains at low levels.

Comparing the mean returns presented in Table 5 Panel A suggests quite similar results relative to the Sharpe ratio and the risk results. In all reported cases, the factor-based portfolios yield larger or equal mean returns compared to the sector-based portfolios.

The relationship between the returns and the allocation constraints is similar to the one we observed when assessing the Sharpe ratio differences for both strategies. Extending the level of the allocation freedom increases the return differences between factor and sector portfolios.

Multifactor performance analysis: Fama—French factors Table 6. Next, we next analyze the performance of the optimized portfolios within a multifactor model framework. Table 6 Panel A depicts the multifactor alphas of the factor and sector portfolios along with the significance levels for the null hypothesis that alphas are different from zero. Table 6 Panel B contains the differences between alphas of factor and sector portfolios. In Table 6 , we observe positive multifactor alphas for all optimized portfolios.

For the long-only case, all portfolios have positive alphas with the vast majority being significantly larger than zero. More specifically, all factor portfolios have statistically significant positive multifactor alphas, while for sector portfolios many alphas are not significantly larger than zero. Moreover, for all risk—return optimization models we find that alphas are larger for factor portfolios compared to sector allocations.

In contrast, for risk-based allocations risk parity and minimum variance factor portfolios reveal lower multifactor alphas than sector portfolios. This finding is in line with our earlier evidence that sectors are less correlated than factors.

It also supports the conclusion of Briere and Szafarz that sector investing helps to reduce risks during crisis periods, while factor investing can boost returns during expansion periods. The highest alpha 1. In line with Bessler et al. Overall, the multifactor analysis confirms our finding that factor portfolios dominate sector portfolios at least for risk—return optimization models.

In Table 6 Panel B, we provide additional insights into the performance difference between sector and factor portfolios. Panel B provides the difference between the alpha quotients of both allocation strategies. The positive alpha differences suggest that factor portfolios have larger multifactor alphas than sector portfolios.

Comparing the three columns in Panel B, we observe that the alpha differences between the sector and factor strategies tend to become larger with wider investment restrictions, illustrating that the advantages of factor portfolios increase when relaxing investment constraints. However, due to our relatively short evaluation period, the differences in multifactor alphas between factor and sector portfolios are mostly statistically insignificant, yet economically relevant.

Next, we assess the potential outperformance of a dynamic factor timing strategy compared to a buy-and-hold factor portfolio. For this, we compute another set of multifactor alphas by regressing the returns of the optimized factor portfolios on its own components the single MSCI factors. If the optimized portfolio was a buy-and-hold portfolio of any initial weighting, the estimated alpha would be zero and the underlying assets would fully explain the portfolio where the regression coefficients would equal the respective factor weights.

In this case, the more passive the management of the underlying constituents, the closer to zero is the estimated alpha. In contrast, since the portfolios consist entirely of the same factors applied in the multifactor regression, the source of the alpha in the factor portfolios stems uniquely from the dynamic factor allocation. In summary, the comparison of the factor and sector alphas covers the difference between the two risk narratives described in Section 1.

In Table 7 , we present the results alpha from the multifactor performance analysis. For most of the optimized portfolios, we observe positive multifactor alphas. For the long-only portfolios, all portfolios show positive alphas with the vast majority being significantly larger than zero. This supports our conjecture that combining the factors in an optimized portfolio results in higher returns than the return of a static buy-and-hold factor portfolio. In contrast to the Fama—French factor analysis in section e , we find sector portfolios yielding higher multifactor alphas.

This might appear surprising, as it seems to contradict our earlier results. However, it is very reasonable to expect that the same underlying factors explain very well factor portfolios that consist of these factors. Particularly, factors can better explain returns of factor-optimized portfolios, based on the same underlying factors, rather than portfolios consisting of sectors.

Next, we analyze portfolio turnover of factor and sector allocations. Higher portfolio turnover is associated with larger transaction costs. Our analyses so far included 20 basis points of transaction costs, which is reasonable, given that we implement the sector and factor allocations with exchange-traded funds ETFs.

In this section, we analyze whether our assumption on transaction costs affects our results. Table 8 Panel A presents the portfolio turnover for all factor and sector portfolio. We find that for almost all allocation strategies, portfolio turnover is lower for factor compared to sector portfolio. Therefore, the relative advantage of factor compared to sector allocations increases with larger transaction costs. An important aspect of the performance analysis is an in-depth risk analysis.

For this, we analyze whether the performance difference between factor and sector portfolios is attributable to different levels of tail risk, such as maximum drawdown, or to the skewness and kurtosis of the return distribution. The first part of the risk analysis covers the comparison of the maximum drawdowns MDD that occurred during the full investment period in factor and sector portfolios.

The maximum drawdown represents the absolute losses between the highest peak and the subsequent lowest trough of the portfolio. Table 8 Panel B presents the maximum drawdowns for all strategies over the full evaluation period. Analyzing the results, we do not find a strong relationship between the MDD and the portfolio allocation strategy. The factor-based allocations have even lower drawdowns in most of the analyzed cases.

Next, we analyze the skewness of portfolio returns. In general, investors seek positive skewness since it translates in a higher likelihood of positive returns or positive outliers. The negative skewness, in contrast, represents in general the higher likelihood of occurrences on the negative side of the return distribution losses.

For brevity, the results for the skewness analysis is available in the online appendix. Our results suggest that all of our optimized portfolios reveal negative skewness, meaning that tail risks on the negative side left is higher than normally distributed returns. In general, there is no clear relationship between skewness and the allocation framework.

On the other hand, for the risk—return optimization strategies BL, MV, BS , the skewness is higher for factor portfolios than for sector portfolios in the vast majority of the cases. Overall, a higher skew risk cannot explain the higher returns of the factor portfolio as the skewness is even lower for most factor portfolios. Kurtosis characterizes the second key property of the return distributions.

Our results for the kurtosis analysis are available on the online appendix. In general, we find that all observed portfolios show leptokurtic distributions, meaning that the extreme return observations occur more frequently than expected for normally distributed returns. Similar to the skewness analysis, the results do not reveal a clear pattern how the skewness relates to factors or sectors. However, on average, and in most analyzed cases, the sector portfolios have a slightly lower excess kurtosis than factor portfolios.

However, as the excess kurtosis captures both, extreme positive and negative returns, the higher kurtosis of most factor portfolios might be due to periods of high factor returns. Our conclusion from the MDD and the overall risk analysis is that the higher returns and Sharpe ratios of the factor portfolios are not explainable with the higher tail risks of these portfolios.

Moreover, we do not find a clear difference between factor and sector portfolios with regard to tail risk measures. Following the analysis of the full period, we now investigate the performance for different optimization strategies and for different sub-periods. For this, we split the time series of the optimized portfolios in several sub-periods based on the state of the economy, where we distinguish between two different states: economic expansion and economic recession.

Based on NBER recession dummies, we divide the full sample into three sub-periods. The first sub-period spans the period from May to July and includes the global financial crisis. The second sub-period contains the subsequent recovery of the global economy and financial markets.

The results of the sub-period analysis, expressed as the difference between the Sharpe ratios of the factor- and the sector-based portfolios, we present in Table 9. We structure the results again in the same way as we did for the Sharpe ratios in Table 4. In each Panel of Table 9 , we present the results for one restriction, moving from strictest to moderate to lowest restrictions. The results suggest that this separates the dominance of one over the other strategy into two groups: During both crisis periods, the global financial crisis and the Covid crisis, sector portfolios outperformed factors portfolios, indicating that sector portfolios offer higher diversification potential particularly during crisis periods relative to factor portfolios.

This finding is in line with Briere and Szafarz who report that sector investing helps to reduce risks during crisis periods, while factor investing can boost returns during expansion periods. Consequently, during the long expansion period from August to February , factor allocation clearly outperformed sector allocations. While the sector allocations dominate in two out of three sub-periods, it is important to recognize that the two sub-periods in which sectors outperform are relatively short, spanning only 11 and 9 months.

In contrast, the second sub-period in which factor portfolios dominate stretches over more than 10 years. Therefore, for the full period we find factor portfolios clearly outperforming sector portfolios. Compared with the long-only case Panel A of Table 9 , we find similar results for the second and third sub-periods with factor portfolios dominating during the long second sub-period and sector portfolios outperforming during the short Covid crisis period.

However, for the first sub-period Global Financial Crisis , relaxing the weight restrictions inverses the results. While for long-only portfolios, sector portfolios dominated during the Global Financial Crisis, for portfolios with short positions, the factor portfolios also dominate during this period. The results for portfolios with the lowest restrictions we report in Table 9 Panel C. In general, the structure of the performance differences remains the same as for the moderate restrictions Panel B with only minor differences.

As in the case with moderate restrictions, factor portfolios dominate in the first and second sub-period. Only in the third sub-period Covid crisis , sector allocations achieved larger Sharpe ratios. One conclusion from the sub-period analyses is the observation that the performance differences between the sector- and factor-based allocations are not stable over time.

The relative performance of sector and factor portfolios seems to relate to the state of the economy. While overall and for the full period factor portfolios dominate sector portfolios, the latter seem to be beneficial during crisis periods. However, the available history of factor indices is still too short to allow robust conclusions. Adjusting the relative size of short and long positions, optimization algorithms and different estimation window length in the optimization process, affects the comparative results, but do not fundamentally change the pattern of the relative performance.

In this study, we compare the performance of two different low-cost asset allocation strategies, one building on investable factors and the other one on investable sectors, both via ETFs. We extent the earlier research of Briere and Szafarz in different directions. While Briere and Szafarz build on Fama—French factors that are not directly investable, we focus on investable factor and sector indices and in addition analyze a variety of different out-of-sample investment strategies.

For the entire investment period between May and November , we find that factor portfolios provide a superior performance relative to sector portfolios. Even though the differences are not always statistically significant, which might be due to the relatively short evaluation period, the results are economically relevant with substantial Sharpe ratio differences. The results are consistent among all analyzed asset allocation strategies and estimation window length for the input parameters.

The reason for the results become more evident when we analyze the performance differences in more detail. The factor portfolios generate higher average returns with lower risk volatility. Our analysis of tail risk measures such as skewness or kurtosis of returns or the maximum drawdown reveals that factor portfolios do not exhibit larger levels of tail risk. Therefore, higher risk cannot explain the superior performance of factor portfolios.

Moreover, we find that for almost all allocation strategies, portfolio turnover is lower for factor compared to sector portfolios. Hence, for larger transaction costs the relative benefits of factor compared to sector allocations become even more pronounced.

Consistent with the Sharpe ratio results, we find for all risk—return optimization models that factor portfolios provide higher multifactor alphas compared to sector portfolios. Only for risk-based allocations, sector portfolios partially provide larger multifactor alphas than factor portfolios.

This result is in line with our finding that sectors reveal a lower correlation structure and hence a higher diversification potential than long-only factors. To analyze the potential benefits of factor timing, we regress factor portfolios on the same six MSCI long-only factors, which we employ as assets in the factor portfolios. We find that risk—return optimization as well as risk-based allocations add value compared to a buy-and-hold factor strategy.

This finding is in line with Briere and Szafarz who reported that sector investing reduces risks during crisis periods, while factor investing can boost returns during expansion periods. However, for the full sample, we find a clear outperformance of factor portfolios. Overall, factor indices offer an attractive investment universe and are already investable for instance via ETFs. For further research, it might be interesting to investigate whether combining sectors and factors in a single portfolio adds additional value.

Briere and Szafarz propose blended portfolios, which combine the diversification benefits of sector investing particularly during crisis periods with the risk premiums of factor investing. Analyzing the benefits of blended portfolios, we leave for further research. Kim et al. They conclude that despite the ambiguity between factors and assets, incorporating the factor models in the investment process might improve the investment strategy.

Ang et al. Critical views on the factor allocations approach have also been stated by Asness , Asness et al. See Anderson et al. Coqueret summarizes the most important reasons for the higher attention. Second, in crisis-periods investors tend to buy lower-risk products. Finally, since Black as well as Haugen and Heins introduced the so-called low-volatility paradox, a large body of literature concludes that lower-risk assets do not necessarily perform worse than their higher-risk counterparts do.

Mean—variance relies on the critical assumption that asset returns are normally distributed. Therefore, it is sensible to apply the mean—variance framework for portfolio optimization even if asset returns are non-normal, as long as they are symmetric. For a detailed exposition see Bessler et al. For an application, see Bessler et al. Earlier studies use similar values ranging from 0. Diversified or Concentrated Factor Tilts?.

The Journal of Portfolio Management 64— Article Google Scholar. Anderson, Robert M. Bianchi, and Lisa R. Financial Analysts Journal 75— Ang, Andrew. Book Google Scholar. Ang, Andrew, and Knut N. Arnott, Robert D. Working Paper. Asness, Clifford S. Moskowitz, and Lasse H. Value and Momentum Everywhere. The Journal of Finance — Journal of Portfolio Management 1—6. Journal of Portfolio Management 72— Benchmarks as Limits to Arbitrage.

Understanding the Low-Volatility Anomaly. Financial Analysts Journal 40— Benartzi, Shlomo, and Richard H. The American Economic Review 79— Portfolio of Risk Premia. A New Approach to Diversification. The Journal of Portfolio Management 17— Multi-asset portfolio optimization and out-of-sample performance. The European Journal of Finance 23 1 : 1— Bessler, Wolfgang, and Dominik Wolff.

Journal of Banking and Finance 1— Black, Fischer. Capital market equilibrium with restricted borrowing. The Journal of Business 45 3 : — Black, Fischer, and Robert Litterman. Global Portfolio Optimization. Financial Analysts Journal 28— Briere, Marie, and Ariane Szafarz.

Finance Research Letters 33 : 1—5. Google Scholar. Carhart, Mark M. On Persistence in Mutual Fund Performance. The Journal of Finance 52 1 : 57— Coqueret, Guillaume. Diversified minimum-variance portfolios. Annals of Finance — Optimal versus naive diversification. The Review of Financial Studies — Optimal Timing and Tilting of Equity Factors.

The Financial Analysts Journal 75 4 : 84— The Journal of Portfolio Management, 47 2 : 9—

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