Adding a constraint to any investment management process can only reduce its efficacy. However, investors often have explicit or implicit constraints with which they are obliged to comply. When it comes to factor investing, portfolio constraints can have a devastating effect on the performance of multi-factor portfolios, in particular those based on the most simplistic approaches. Constraints may simply remove or limit significantly the anticipated exposure to factors and thus inhibit the capture of factor premiums.

An attractive strength of the more sophisticated diversified equity factor investing (DEFI) strategies which use portfolio optimisers for portfolio construction is their ability to efficiently minimise the impact of constraints on the capture of factor premiums. Moreover, in such strategies, the impact of constraints can be both measured and monitored. Here is a short overview of some adapted versions of DEFI strategies, by type of constraints.

Constraints are almost inevitable. Whether explicit or implicit, they have an impact of the performance of any investment management process. Here we discuss the impact of typical portfolio constraints on diversified equity multi-factor investing (DEFI) strategies.

#### Universe constraints

For most investors, the most important constraints arise from the structure imposed on their strategic asset allocation. Splitting the investment universe, whether by geography or by investment style, creates internal ‘walls’ which restrict the efficient use of the opportunity set that a diversified universe of securities offer. For example, how can you flee a euro crisis if you have a strategic euro equity strategic allocation? Factor investing strategies are more efficient when applied to large, diversified universes. In this respect, investing in a DEFI strategy applied to a global universe of stocks is more efficient than investing in a collection of regional DEFI strategies. The difference between these two methods of investing in factor strategies is that in the first case, when investing globally, arbitrage between countries and regions on the basis of their factor exposures is possible.

This is not possible in the second case when the opportunity set is limited to investing in a collection of regional strategies. According to our analysis, this constraint would roughly half the efficiency of the factor-based process. In addition, some regions, for example Europe, lend themselves well to DEFI strategies as they offer a diversity of markets and investors that seem conducive to the generation of larger factor premiums. There is therefore a cost to constraining the universe to a domestic market rather than investing across Europe in a broad sense.

#### Style constraints

Style pockets, with value and growth managers separated for instance, leads to similar inefficiencies. The opportunity set available to both such style managers is substantially reduced with the exposure to the value factor significantly constrained. When compared to the broader market capitalisation universe, the exposure to value is largely overweight in the case of the value manager, and largely underweight in the case of the growth manager. Choosing to overweight one or more factors in a multi-factor process is possible, and facilitates the attribution of a characteristic style to a fund, but it has a diversification cost.

#### Beta constraints

Aggressive and defensive buckets, separated by beta level, also create such internal ‘walls’. The natural beta for a DEFI strategy should be one. When this is the case, all excess returns generated from exposures to factors such as value, quality, momentum or low risk are uncorrelated from the market returns. For the investor, this offers the most attractive investment proposition. All excess return generated from the factor exposures add to performance and diversify risk. But, capturing factor premiums while targeting a beta of one is only possible with the more sophisticated DEFI strategies with the use of risk models to forecast beta and portfolio optimisers to efficiently set the portfolio beta to one while minimising the impact of constraints, such avoiding short selling of stocks. In this respect, capturing the low volatility factor premium, while targeting a portfolio beta of one is a challenge that only careful portfolio construction of DEFI strategies can meet.

#### Active risk constraints

Risk controls are the second largest type of constraints in investment strategies. The most obvious is perhaps the choice of an active risk budget, i.e. a tracking-error risk. Factor investing is based on the principle that exposure to factors such as value, quality, momentum or low volatility generate a premium. A low tracking-error risk budget will necessarily imply a low exposure to the factors and thus a harvesting of relatively modest premia from factors. The one advantage of low tracking-error risk is that the factor strategy will be hugging the benchmark index and thus is much less likely to be impacted by portfolio constraints. That means that the information ratio of the multi-factor strategy is likely to be higher at lower levels of tracking error.

Nevertheless, for the investor, collecting a much smaller premium from factors and thus low tracking-error risk remains a much less attractive investment proposition, even with a higher information ratio. The optimal tracking-error risk budget will be the highest level of tracking-error risk for which portfolio constraints are still not significantly impacting the collection of factor premiums.

#### Long-only constraint

Factor premiums are typically measured by calculating the excess return of a portfolio invested in stocks positively exposed to a given factor relative to the performance of a portfolio invested in stocks negatively exposed to the same factor. For the quality factor, for example, that typically implies measuring the excess return of a portfolio invested in the most profitable stocks against the performance of a portfolio invested in the least profitable stocks. In order to capture those excess returns, the factor premium, investors should then invest in a beta neutral long-short strategy that buys the portfolio invested in stocks positively exposed to a given factor and short-sells the portfolio invested in stocks negatively exposed to the same factor.

However, most investors do not want to invest in such strategies: they are simply not interested or cannot sell stocks short. If they invest in the market capitalisation index plus such a long-short factor portfolio then they earn the returns accruing to the market capitalisation index plus the factor premiums.

However, most investors prefer to invest in one single long-only portfolio that is exposed to factors. Such a portfolio can only partially capture factor premiums, typically the factor premium component generated by the stocks with a positive exposure to the factor. The factor premium component arising from short selling the stocks with a negative exposure to the factor is not captured by long-only investors.

However, our research demonstrated that for most factors the portion of the factor premium generated by stocks with negative factor exposures is generally, at best, comparable in size to the portion of the factor premium generated by the stocks positively exposed to the same factor. In addition, the correlation of the two components of the factor premium tends to be very high. As a result, we find that long-only DEFI strategies are almost as efficient as DEFI strategies incorporating short positions. The long-only constraint has a much smaller impact in the efficacy of capturing factor premiums than is generally perceived.

#### Turnover constraints

Turnover is a further subject of concern for risk managers because it represents a certain cost both in financial and operational terms. This is why many popular factor strategies are low turnover strategies. However, the optimal turnover is a trade-off between the higher cost of higher turnover and the lower efficacy in capturing factor premiums from reducing the turnover. One should trade more for as long as it is profitable after costs.

For a factor strategy, we find that the optimal level of turnover is actually quite high, typically above 100% one way per year. Market impact can be kept low by cleverly implementing the strategy in smaller doses while keeping turnover high. The capacity of DEFI strategies with such high levels of turnover is thus high for typical stock universes of development countries. Nevertheless, turnover in DEFI strategies can be reduced for cautious investors by increasing turnover aversion. However, lower expected returns should be expected from reducing the turnover.

#### ESG constraints

The third family of constraints that factor investing can encompass are extra-financial criteria, such as environmental, social & governance (ESG), low carbon or Islamic investment constraints. As with universe bucketing, excluding part of the investable universe on the basis of extra-financial criteria involves a cost in terms of a smaller opportunity set. While the debate about whether or not such exclusions add value, it is clear that DEFI strategies using a portfolio optimiser to minimise the impact of constraints are the optimal strategies when it comes to handling this issue. They are in fact the optimal way to best accommodate such exclusions, be they sector-based, such as Islamic, or more widespread, such as ESG. Efficient reduction of carbon emissions by as much as 50%, when compared to the market capitalisation indices, can be efficiently achieved (we addressed this issue in 'How to integrate a low-carbon objective into factor investing'.

#### Specific constraints such as tax, liquidity or solvency ratios

Some regulations apply time-dependent tax constraints, for instance by varying the level of tax according to the holding period: long-term investors being taxed less than short-term speculators. In a DEFI strategy it is possible to use the portfolio optimiser to privilege the sale of stocks purchased longer ago rather than the sale of more recent purchases so as to minimise the impact of taxation on the capture of factor premiums. The same goes for specific liquidity or solvency ratios applied to some institutional investors: DEFI strategies are superior thanks to the use of portfolio optimisers to minimise the impact of constraints.

#### Conclusion

Overall, DEFI strategies are the most efficient investment strategies to capture factor premiums thanks to the use of optimisers to accommodate constraints in a way that efficiently minimises their impact and allows for the monitoring and measuring of their impact. And it is not only the most typical portfolio constraints investors impose themselves but also tax constraints or constraints based on extra-financial measures such as ESG or carbon emissions that can be handled efficiently.

Please note that this article may contain technical language. For this reason, it is not recommended to readers without professional investment experience.