Smart Management of Smart Beta Liquidity Which Liquidity Constraints Are Most Effective?
Anthony A. Renshaw and Frank Siu
Axioma, Inc.
October, 2014
© Copyright, Axioma, Inc. 2014 – All rights reserved
1. Introduction
Today’s historically low trading costs have played an important role in the emergence of “smart beta” and other factor products. Unlike cap-weighted products that require trading only to handle corporate actions, universe reconstitution, and fund cash flows, smart beta products require periodic rebalancing, e.g., trading, to maintain their mandated characteristics. Their current popularity is only possible because trading costs have become low enough such that smart beta performance can realistically compensate for the additional trading costs inherent to non-cap-weighting. Managing liquidity is therefore crucial to capturing smart beta or factor performance.
On the basis of simulated backtests, smart beta and other factor products often boast impressive track records. However, given the additional trading that occurs, can such advertised performance truly be realized once transaction costs are taken into account? One might expect these products to make a concerted effort to manage liquidity. However, explicit efforts to manage liquidity in existing smart beta and factor ETF and index products has been relatively modest.
In one common approach, which can be described as universe-driven liquidity, the trading costs and liquidity are managed solely by the choice of the universe of investible assets. An example of a universe-driven product is SPLV, the PowerShares S&P 500 Low Volatility ETF, which invests in the 100 assets from the S&P 500 with the lowest, realized annual volatility. All assets in the S&P 500 are assumed to be sufficiently liquid such that the fund executes whatever trades are needed at each rebalance to buy and hold the lowest 100.
A second common approach to manage liquidity, trading-driven liquidity, explicitly restricts trading at each rebalance, typically by limiting the turnover. Examples of trading-driven liquidity ETFs include USMV, the iShares MSCI USA Minimum Volatility ETF (round trip turnover limited to 20% at each rebalance) and LGLV, the SPDR Russell 1000® Low Volatility ETF (round trip turnover limited to 12% at each rebalance).
Until recently, there were no ETFs that employed asset-driven liquidity construction rules. This is an important case because even if a portfolio has very low turnover (e.g., satisfies trading-driven liquidity rules), the portfolio is illiquid if the turnover is from trading illiquid names. The presence of illiquid names is not the problem per se. The problem to be avoided is having to hold or trade them in large amounts.
In traditional portfolio construction, asset liquidity is managed by comparing each asset’s trade or holdings to its average daily volume. If the notional value the portfolio is N dollars, then the dollar amount held in the i-th asset is, where is the weight held in the i-th asset. If we are allowed to hold at most a fraction f1 of the (twenty-day) average daily volume traded in dollars of the i-th asset, denoted by , then the number of days needed to liquidate the i-th holding is . This leads to the well-known constraint
for each asset i (1)
for each asset, i, and some constant C.
Although this type of constraint is popular and a standard feature in commercial portfolio construction tools, such as Axioma PortfolioTM, it is not an attractive liquidity rule for an ETF or index because there is no good way to estimate what the notional value, N, might be. Unlike hedge funds, ETFs and other smart beta products must publically publish their portfolio construction methodology so that potential investors can understand the product. It is possible but onerous to change any published methodology, so every attempt is made to make the methodology as insensitive to time and events as possible. Portfolio construction rules based on a notional value of the product would be unattractive, since the hope is that the notional value will increase substantially over time.
Recently, Axioma and Stoxx worked in partnership to introduce a set of minimum variance indexes that incorporated a liquidity constraint on the portfolio that was independent of the notional value of the portfolio2. Instead of measuring liquidity with respect to the notional value of the portfolio, in this new approach the weighted average liquidation time of the portfolio3 was limited with respect to the weighted average liquidation time of the benchmark. Numerous backtests and studies4 confirmed that this form of constraint successfully prevented the portfolio construction methodology from taking illiquid positions.
The novelty of this approach underscored how scarce the existing research is on asset liquidity constraints for portfolio construction apart from notional-based rules, such as (1). The purpose of this paper is to review a range of asset-liquidity constraints that are independent of any notional value of the portfolio and could therefore be utilized in ETF and Index construction rules. Several constraints are examined in detail in terms of how they impact liquidity and performance (e.g., realized return and risk). The two most attractive constraints were:
- Limiting the weighted average liquidation time of the portfolio using the weighted average liquidation time of the benchmark, e.g., the approach employed by Stoxx.
- A variation of (1) in which the effective notional value, , is replaced with a statistic of the ADV distribution, such as the median ADV, the benchmark weighted average ADV, or the median benchmark holding liquidation time.
Both constraints can be applied with Axioma Portfolio. The first is a Limit Holdings constraint, while the second one uses the Limit Benchmark Deviation constraint.
2. An Illustrative Example
We illustrate liquidity issues in smart beta products using a prototypical portfolio construction example. We construct a minimum risk portfolio using the assets in the Russell 1000 Index as follows:
- Rebalance monthly from 12/31/01 to 5/30/14 (150 rebalancings).
- Assets held must belong to the Russell 1000 Index as of that date.
- Minimize the total risk of the portfolio as predicted by Axioma’s Fundamental Factor, Medium Horizon, US Equity risk model (AXUS3-MH).
- Long only, with maximum asset weight of 10%.
- Maximum round-trip turnover of 10%. This constraint is placed in Axioma’s Constraint Hierarchy in case of infeasibility (as occurs on some dates when the underlying benchmark is reconstituted). The first rebalancing is from cash and has a turnover of 100%.
- Minimum non-zero holding size of 3 bps.
Table 1 shows summary statistics for this optimized minimum risk portfolio. The performance of the minimum risk portfolio is, on paper, substantial. The annualized return is 50% greater than the benchmark (10.65% vs. 6.90%), while the annualized realized risk is less than 65% of the benchmark (15.11% vs 9.51%), leading to a Sharpe ratio of 1.12. The benchmark holds substantially more names but has about a tenth as much turnover.
Table 1. Summary statistics for the optimized minimum risk portfolio
using the Russell 1000 universe.
Fig. 1 shows the cumulative average names held and the distribution of the asset liquidation time, , as a function of ADV, in USD, for the benchmark (green) and for the minimum risk portfolio (red and blue), where we arbitrarily select N = $500 MM USD and f = 10%. The liquidation times have been grouped into ADV buckets, and the results are averaged over each of the 150 rebalancings. On the bar chart, the green bars represent the (equi-weighted) average liquidation time for all benchmark holdings for that ADV. The blue bars represent the average liquidation time for the minimum risk portfolio. The blue bars show the maximum liquidation time for the minimum risk portfolio.
Fig. 1. The cumulative names held (top) and the asset liquidity as a function of the ADV for the minimum risk portfolio (red and blue) and the Russell 1000 (green). Results averaged over each of the 150 rebalancings. On the lower chart, the blue bars are the maximum value; the red and green are average values. N = $500 MM and c = 10%.
The graph illustrates a number of important characteristics of liquidity. First, ADV ranges over more than five orders of magnitude5. The distribution of ADV is strongly skewed to the right and has a long left tail. Second, the “typical” (e.g., median) names held has an ADV of approximately $30 MM, and, for the minimum risk portfolio, would take a day or two to liquidate (for this N and f) on average; the maximum holding would take about 20 days to liquidate. This is significantly larger than the benchmark value, which is about 0.1 days.
Third, even though there are less than 5% of the names with ADV less than $1 MM, this small fraction of names would each take over 100 trading days to liquidate. In other words, this small fraction of names is over two orders of magnitude less liquid than the “typical” asset in the portfolio, and three orders of magnitude worse than the typical benchmark asset. We therefore suspect liquidity issues with the minimum risk portfolio.
Another metric that is sometimes used to assess liquidity is the average liquidation time (in days) for an entire portfolio of K assets. Assuming, the weighted average portfolio liquidation time6 is
(2)
This average can be changed into average fulfillment times by replacing the term with , where is the portfolio weight before rebalancing.
Table 2. The average, maximum asset liquidation time and average portfolio liquidation and fulfillment times for the Russell 1000 and the minimum risk portfolio with no liquidity constraints over the time series of rebalancings. N = $500 MM and c = 10%. June is the annual Russell reconstitution.
Table 2 shows the average (across the times series of 150 rebalancings) maximum asset liquidation time (in days) for the Russell 1000 (0.66 days) and the minimum risk portfolio (136) days. The ratio of these two statistics is 207. Also shown are the average portfolio liquidation times and fulfillment times. The fulfillment times are also shown using just the numbers for the rebalancing at the end of June—that is when Russell performs its annual reconstitution of its indexes, which would be expected to be a more meaningful measure of the Russell 1000 fulfillment time.
The ratios are very similar:
- Maximum asset liquidation time = 207
- Weighted portfolio liquidation time = 211
- Weighted fulfillment time, June only = 275
These ratios correspond approximately to the difference in the liquidity shown in Fig. 1 on the left of the chart, e.g., the illiquid holdings, and could serve as a good measure of the relative illiquidity of the minimum risk portfolio.
3. Different Liquidity Rules
In this section, we evaluate four different liquidity constraints that are independent of notional value—parameterized by the constants, l, , g —and compare the relative performance of the four constraints.
However, before proceeding, we make a few general observations.
First, in the analysis so far, we have arbitrarily chosen values for N and N / f. This is inconvenient since we often have no good idea what values might be appropriate. However, we can choose any reasonable statistic of ADV—its median, the benchmark weighted average, etc.—to normalize the statistics and analysis. The normalized liquidation and fulfillment times then no longer correspond to days to liquidate or fulfill a portfolio, but the relative liquidity of the portfolio is identical. In this paper, we use the median value of ADV across all benchmark asset with non-zero ADV.
Second, liquidity can be managed with respect to the portfolio holdings, , or, alternatively, with respect to the rebalance trades, . All constraints considered can be equally well formulated in terms of either holdings or trades. However, even though one might anticipate that managing trade liquidity would be more relevant and common, in fact, there is a bias towards managing holdings liquidity since this is the more conservative approach (the trade is never larger than holdings) and it avoids the not-uncommon occurrence of a current holding having its ADV suddenly vanish, leading to infinite illiquidity. Nevertheless, even though we illustrate liquidity constraints below with holdings liquidity rather than trading liquidity, the results and conclusions apply equally well to trading liquidity constraints and it should not be concluded that trading liquidity is less important than holding liquidity.
Third, in some of the liquidity constraints considered, we formulate the constraint with respect to a benchmark. Of course, we do not need a benchmark to apply a liquidity constraints considered. In fact, a priori, a benchmark may or may not be a good measure of liquidity. There are highly liquid benchmarks, such as the S & P 500, and somewhat illiquid benchmarks, such as the Russell 2000. The Russell 2000 is able to hold some hard-to-trade stocks because it is rebalanced only once a year. For most smart beta products, rebalancing is likely to occur more frequently, so the sensitivity to liquidity is likely to be more acute than in the benchmark.
Nevertheless, even if the underlying benchmark is not particularly liquid, formulating a liquidity constraint with respect to the benchmark is extremely convenient and easy to calibrate. For the constraints shown here, the benchmark is convenient but not necessary.
Fourth, almost all smart beta construction procedures manage turnover as an essential component in managing liquidity. Although we do not emphasize turnover here, that does not imply that turnover is not important.
We now list four different candidate liquidity constraints that will be examined, parameterized by the constants , l, , g.
I. Exclude Low ADV Names
The simplest liquidity constraint is to exclude names with an ADV less than some prescribed amount:
Hold Only If , i = 1, …, K (3)
which will cutoff the low liquidity holdings in Fig. 1. can be specified in terms of currency (local or numeraire) or shares, or cutoffs in both currency and shares can be used. One would almost certainly want to exclude any names with vanishing ADV, so this constraint is usually present for very low ADV names.
II. Limit Asset Weights Using the Benchmark Weights
Let the benchmark weights be denoted as . We can limit the liquidity of each holding to be at most a constant l times its liquidity in the benchmark using
, i = 1, …, K (4)
where l is a constant to be determined and is required for the solution to be feasible. This constraint can be applied even if no ADV information is available.
MSCI’s Minimum Volatility Indices Methodology7 applies a variation of this constraint in which the maximum asset weight is the lower of 1.5% or 20 times the weight in the benchmark.
III. Limit Asset Liquidation Time
In the prior art, the most common liquidity constraint is to restrict each asset’s liquidation time, . Here, we restrict each asset’s liquidity by replacing any notion of the portfolio size with the median ADV
, i = 1, …, K (5)
where b is a constant to be determined. At any rebalance, the right hand side is a simply a constant, and the median ADV could be absorbed into the constant b. However, when running a backtest over a number of rebalances, including median ADV allows the constraint to be effective with a single constant b.
IV. Limit Portfolio Weighted Average Liquidation Time
Finally, we restrict the weighted average liquidation time of the portfolio to be less than a constant, g, times the weighted average liquidation time of the benchmark.
(6)
Trade-Off Curves
Next, we plot some curves comparing the relative performance of the four different constraints.
Fig. 2. Realized return vs. realized risk for the different parameter frontiers for the minimum risk portfolio: (green), l (turquoise), (blue), and g (red).
Fig. 2 shows the traditional efficient frontier parameters, realized return and risk, for each of the four constraints: (green), l (turquoise), (blue), and g (red). At one end of the spectrum, as each of the constraints becomes less binding, the performance converges towards the unconstrained minimum risk portfolio results with the highest return and lowest risk. At the other end of the spectrum, the constraint becomes more binding, the return decreases, and the risk increases, although at different rates.
Fig. 2 clearly illustrates a negative relationship between liquidity and theoretical performance of minimum risk strategies. Any constraint that improves liquidity decreases realized return and increases realized risk. This result underscores the importance of investigating the implementability of these strategies in practice.
The constraint has a notably non-smooth and non-monotonic frontier. In general, such behavior is undesirable. is already frequently used but with very small values corresponding to the crowded points near the unconstrained performance results. It appears, however, that use of with larger cutoff values would not be recommended.
For the other relatively smooth constraints, has the highest returns for the least constrained solutions (risk greater than 11%), while g has the best returns for risk greater than 11%. The l constraint has more performance degradation than either b or g. Since we anticipate that the best range of parameters to use these liquidity constraints will only perturb the original solution slightly, the b constraint appears to be the most effective for such small values.
Fig. 3. Sharpe ratio vs. average maximum normalized asset liquidation time, , for the different parameter frontiers for the minimum risk portfolio: (green), l (turquoise), (blue), and g (red).
Fig. 3 shows the trade-off in Sharpe ratio versus the average maximum, normalized asset liquidation time, . Once again, the b constraint has the most reduction in illiquidity with the least reduction in Sharpe ratio.
Fig. 4. Sharpe ratio vs. normalized weighted average portfolio liquidation time,, for the different parameter frontiers for the minimum risk portfolio: (green), l (turquoise), (blue), and g (red).
Fig. 4 shows the trade-off in Sharpe ratio versus the normalized weighted average portfolio liquidation time, . Close to the original minimum risk solution, the b constraint is most effective (say, for portfolio liquidation greater than 0.04). For more binding constraints, the b and g constraints are essentially indistinguishable.
Fig. 5. Sharpe ratio vs. the average number of names held for the different parameter frontiers for the minimum risk portfolio: (green), l (turquoise), (blue), and g (red).
Fig. 5 shows the trade-off in Sharpe ratio versus average number of names held. With the exception of , all the constraints increase the number of names held. If the portfolio is highly sensitive to the number of names held, it may make sense to manage liquidity with the constraint. Otherwise, the constraint that increases the number of names held is g8, and the second most is . However, since the horizontal axis is logarithmic, the difference in these two constraints is large. One should expect many more names when applying g than when applying b.
Fig. 6. Sharpe ratio vs. the average two-way turnover for the different parameter frontiers for the minimum risk portfolio: (green), l (turquoise), (blue), and g (red).
Fig. 6 shows the trade-off in Sharpe ratio versus the average two-way turnover. Here we see a clear distinction between, which generally increases turnover as it becomes more binding, and g, which generally decreases turnover as it becomes more binding.
Fig. 7. Sharpe ratio vs. the average predicted beta for the different parameter frontiers for the minimum risk portfolio: (green), l (turquoise), (blue), and g (red).
Finally, Fig. 7 shows the Sharpe ratio versus the average predicted beta by Axioma medium horizon, fundamental factor US equity risk model, AXUS3-MH. All constraints increase predicted beta as they become more binding, with and g having the most increase and generally being indistinguishable.
Table 3. Performance statistics for the minimum risk portfolio for the four different constraints with parameters chosen such that the realized risk is approximately 10.5%.
Table 3 shows the backtest performance statistics for the four different constraints with the parameter of each constraint chosen such that the realized risk is approximately 10.5%. These results are consistent with the trends illustrated previously, specifically:
- The non-smooth constraints, , has the most error in matching the realized risk of 10.5%.
- The largest return is for b (9.42%). For this particular set of parameters, b appears to be the most attractive constraint in many respects.
- As expected, the number of names for g (325) is much higher than any of the other constraints (133 to 178). Using g may only be appropriate if the number of names can be large.
- The increase in predicted beta is largest for g (0.605).
- The turnover changes are modest except for .
- As expected, b and g have the shortest asset and portfolio liquidation times.
Although not presented here, the same constraints were tested with other factor tilts, such as Value-Momentum. The results were similar: there is a negative relationship between liquidity and performance.
4. Recommendations
The results lead us to propose the following recommendations for managing liquidity with minimal degradation of the factor performance. The most effective constraints to manage liquidity are
, i = 1, …, K
and
Both of these constraints are intuitive, easy to calibrate, and straightforward to implement using modern portfolio construction software, such as Axioma PortfolioTM. The first constraint impacts performance less than the second, but the difference is small. The second can lead to a large number of names held. In the first constraint, can be replaced with any reasonable statistic based on ADV, such as the benchmark weighted average ADV, the average ADV, (even one), etc.
In addition, it often make sense to use a minimum allowable ADV constraint:
Hold Only If , i = 1, …, K
but only with small values of . In the present study, a value of
was appropriate.
5. References
Axioma Portfolio is a trademark of Axioma, Inc.
Russell 1000 is a registered trademark of Russell Investments, Inc. Copyright © Russell Investments 2014. All rights reserved.
1. f is referred to as the participation rate.
2. See http://www.stoxx.com/download/indices/rulebooks/stoxx_indexguide.pdf, Chapter 16.1, published August 2014.
3. Or a subset of the portfolio.
4. See F. Siu, “Tradability of Minimum Variance Portfolios”, Axioma Research report, June 2014.
5. The average ADV for the assets increased by approximately a factor of three from 2002 to 2009, but has been relatively steady since 2009.
6. The equi-weighted, average portfolio liquidation time was also investigated but found to be a far less effective statistic.
7. See http://www.msci.com/eqb/methodology/meth_docs/MSCI_Minimum_Volatility_
Methodology_Jan12.pdf, published January 2012.
8. This is not surprising since this constraint has a form very similar to a limit on the minimum number of effective names using the inverse Herfindahl Index.