The team of FeiFei Li and Joseph Shim start from the top and look into which factors should be included in a multi-factor fund. Stepping slightly aside from the ongoing debates over how to define a factor, Research Affiliates stick to their knitting, opting for six factors as defined by the what they call the “robust academic literature”.

These are value, profitability, investment (both these last two characteristics associated with value according to the academics), size, low beta and momentum.

The team then moves on to constructing a viable long-only factor portfolio, starting with the large-cap universe of US stocks, except for the small size strategy. For each, Li and Shim use the top 30% based on NYSE breakpoints.

As an example, for value portfolio is constructed from stocks above the 70^{th} percentile on the exchange based on the book-to-market ratio. They then weight the selected stocks by capitalisation, except for low beta where they follow the weight by beta ranking as define din the literature. The portfolios are balanced annually each July with the exception of momentum and low beta which are rebalanced quarterly.

Going by these rules, Li and Shim show that the strategy delivered an annualised excess return of 1.86% over the study period between July 1973 and December 2018. Moreover, the correlations between the factors created a diversification benefit. They suggest that “at first glance”, the momentum and size factors appear to offer a substantial diversification benefit when added to a multi-factor strategy. But bringing implementation into the picture “requires more thoughtful analysis” given the reputation for high transaction costs for both the momentum and small size factors.

**Zeroing in on market impact**

To look into this area, Li and Shim suggest that the “implicit component of implementation cost” can be measured by the market impact of the trade, or the movement in a security’s price due to trading.

Li and Shim suggest the market impact of a portfolio rebalancing can be attributed to the following elements:

- Portfolio volume: this is the aggregate of median daily trading volume of all stocks in a portfolio. A strategy’s cost is inversely proportional to its portfolio volume; in other words, a strategy is more costly to implement if it requires holding illiquid stocks.
- Tilt: this is a measure of illiquidity that represents the degree to which portfolio weights deviate from the weights proportional to trading volume. The smallest possible value of tilt is 1.0, which is achieved by a volume- weighted portfolio, theoretically the most liquid combination of a given set of holdings.
- Turnover:: this is a measure of how frequently assets in a portfolio are bought and sold by the strategy over a 12-month period. In general, a strategy that requires a higher rate of trading incurs higher market impact costs.
- Turnover concentration: this is the degree to which trades are spread across the holdings in a portfolio. Highly concentrated trades are more costly to execute. A strategy that demands high liquidity from only a few names causes higher market impact than another strategy that spreads out trades across many holdings.

Using a model $10bn of assets under management, Li and Shim show that the average turnover across the six strategies is 61.6% and the average trading cost is 127 basis points.

**Integrating momentum and size**

The analysis shows that momentum, perhaps unsurprisingly, is the most expensive factor in terms of transactions costs, largely due to annualised one-way turnover of 159.5% (leading to a trading cost of 241 basis points).

But Li and Shim suggest that the benefits of implementing both momentum and size factors within a multi-factor strategy outweigh the negatives of cost (with momentum) and risk (associated with size).

They prove this by analysing four model portfolios:

- Portfolio 1: Value, low beta, profitability, and investment
- Portfolio 2: Four factors in portfolio 1 plus momentum
- Portfolio 3: Four factors in portfolio 1 plus size
- Portfolio 4: Four factors in portfolio 1 plus momentum and size

What they show is that diversification benefits outweigh the increased cost of implementation.

Li and Shim explain: “The addition of momentum helps lower tracking error and improves the IR because of negative or low positive correlations with other factors. These benefits are achieved without a large increase in implementation cost because offsetting trades across the factor strategies cancel each other out and because of the improved liquidity of the stocks held in a momentum strategy.”

The size factor, meanwhile, is “actually rather inexpensive to trade because of its relatively broad coverage and low turnover.” Adding it in combination thus can improve the performance and lower the tracking error of the multi-factor strategy given the low correlation of size with the other factors, resulting in a higher IR together with a reduction in trading cost.

Specifically, portfolio 4 here has the highest IR of the four portfolios, both before and after trading costs and with a barely lowered Sharpe ratio of 0.56.

Portfolio 4, which combines momentum and size with value, low beta, profitability, and investment, yields the highest IR of the four portfolios we analyse, both before and after trading costs, 0.64 and 0.57 (assuming $10B AUM), respectively, and only barely lowers the Sharpe ratio from 0.57 to 0.56.

**Concentrate please**

The next step is concentration levels. Li and Shim say that “a certain concentration” of stocks would intuitively appear necessary to extract the factor premium. They point to the academic literature which shows that quintile portfolios based on a specific signal shows a ‘monotonic’ pattern: that is, a portfolio selected according to the top 20% of stocks based on the underlying signal performs best, down through the quintiles.

After crunching the numbers, the analysis finds that a 25% concentration level offers the optimal mix of performance versus implementation costs. They note that the optimal cut-off might differ according to factor, but they caution that there is a lack of any reason why differing concentration levels should be considered and there would also be the risk of over-fitting.

**Research Affiliates: Alice’s Adventures in Factorland**

In concluding, Li and Shim suggest that understanding how to optimally combine factors in the presence of real world implementation costs is "critical for desirable investment outcomes".

“Neither extreme of maximizing paper portfolio performance, while ignoring the trading costs that reduce performance in practice, nor of focusing on low-cost implementation, while missing opportunities for better performance, will produce an optimal result for multi-factor smart beta investors,” they add.