Navigating Portfolio Turnover with Autocorrelation Insights

This article outlines an approach that enhances the dialogue surrounding active portfolio management by presenting a solid analysis for managing turnover through insights into autocorrelation. It pushes the conventional narrative of portfolio optimization beyond just balancing risk and return to include essential operational factors like turnover. This provides portfolio managers with a more comprehensive strategy for managing portfolios efficiently and cost-effectively.

Many quantitative portfolio managers utilize signals to target excess returns, anticipating the signals will yield returns as predicted. As signal exposures shift with time and incoming data, managers must update their portfolio allocations to utilize this fresh information. This process naturally leads to portfolio turnover in response to specific alpha signals.

Transaction costs, which are closely tied to turnover rates, significantly affect portfolio performance. When comparing two signals with similar risk-return profiles, the preference leans towards the one with lower turnover, subsequently reducing transaction costs. This article also introduces a framework for integrating information from various signals, with a particular focus on the asset turnover implications in the construction of optimal portfolios.

Quantifying Turnover for One Signal

Accurately predicting portfolio turnover in response to a specific signal is therefore crucial for its successful performance post all costs. An insightful study by Qian, Sorensen, and Hua in 2007 introduced a formula to calculate turnover for active (long-short dollar-neutral) mean-variance optimized portfolios based on a quantitative factor.

Their analysis utilized (risk-adjusted) signal scores as expected return forecasts within a utility-optimization framework to determine optimal signal portfolios representations. Their findings indicated that portfolio turnover is influenced by two main factors: 1) the alpha signal's first order autocorrelation, denoted here as ρ, and 2) the portfolio's gross-leverage, L. Their derived formula approximating the two-sided turnover is:

The formula illustrates how gross-leverage impacts turnover in dollar-neutral long-short portfolios: a linear increase in both gross-leverage and turnover arises from simple rescaling of portfolio weights. Additionally, it is deduced that higher autocorrelation leads to better turnover rates, thus quantifying this highly intuitive insight.

It is revealing to highlight that in order to bound the asset turnover outlined in equation (1) by a maximum limit , a minimum limit for the autocorrelation ρ needs to be set, which can be expressed as follows:

Building upon this analysis, one could arrive at a similar conclusion by working directly with the time-series structure of the signal portfolio representations instead. Specifically, the turnover formula (1) can be directly utilized for the first order autocorrelation of the signal portfolio weights, bypassing the need for a specific optimization model as employed by Qian, Sorensen, and Hua to derive it. This is how it will be applied for the analysis that follows.

Combining Multiple Sources of Alpha

The approach discussed in the prior section is applicable when aiming to integrate data from various sources while managing turnover levels effectively. Specifically, controlling the first order autocorrelation of the integrated signal is essential for maintaining a predefined turnover threshold.

Consider J signals and their corresponding signal portfolios within a universe of n assets, where each portfolio is structured as a long-short dollar-neutral portfolio represented by a vector of weights of dimension n. We adopt a top-down approach by assessing signals via their portfolio representations. Thus, integrating multiple signals results in a mixed portfolio.

For each time period t the objective is to determine weights that sum up to one that constitute a mixed portfolio:

The combined portfolio is designed to optimize certain criteria, such as maximizing return or minimizing risk, while also managing first-order autocorrelation , which in turn controls turnover via the approximating formula (1).

It should be considered as a signal portfolio representation of a signal that combines the information of all J signals. By implementing a straightforward rescaling, its gross-leverage can be precisely set to L, ensuring consistency across all portfolio representations.

It's important to note that at time t, when determining the weights v, there is no access to signal scores and portfolios for the subsequent time t + 1. Therefore, to implement the constraint effectively, one might instead use from a previous period as the basis for the autocorrelation constraint. This approach aligns with managing turnover expectations through predictive measures of autocorrelation.

Application Example

We provide an illustrative example using three signals to demonstrate the preceding analysis.

Setup

We perform rolling time calculations to derive the evolution of optimal signal mixes in the following setup:

• Universe: Top 1,000 U.S. companies by market cap.

• Backtest Period: From June 30, 2010, to December 29, 2023, with calculations done quarterly.

• Objective: Maximize Information Ratio (Max IR) using an ex-ante variance forecast from a proprietary risk model for each period's optimization and historical returns forecasts with time window of 20 quarterly periods.

• Signal Weight Adjustment: Consecutive time period adjustments not exceeding 30%.

We consider three distinct signals classified according to their turnover rates or corresponding autocorrelations as SLOW, INTERMEDIATE (or INTERM.) and FAST. Specifically, each signal is represented through Fama-MacBeth regression portfolios, with exposure to Beta and Volatility risk factors neutralized. Each signal portfolio is scaled to maintain a gross-leverage of one (50% long, 50% short) to ensure comparable-in-time portfolio representations.

Autocorrelation and Turnover

The average cross-sectional autocorrelation for these signal portfolios are:

• SLOW: 94.2% average first order autocorrelation, leading to an average turnover of 30.61% over the analysis period.

• INTERM.: 64.02% average first order autocorrelation with an average turnover of 85.08% over the entire period.

• FAST: 12.79% average first order autocorrelation, resulting in the highest average turnover of 128.88%.

See Figure 1 for a detailed graph showing the turnover evolution over time for each signal.

Optimal Signal Mixes

We calculate the Max IR composite weights for SLOW, INTERM. and FAST, both with and without asset turnover constraints. Given the characteristics of these signals, an asset turnover of less than 30% is unlikely. Thus, to provide greater flexibility in signal combinations, we explore turnover limits ranging from 40% to 100%.

Table 1 presents ex-post statistics for both the initial set of signals and their optimized versions during the backtest period. In all cases, we standardize the gross-leverage to one for consistency in comparison. Notably, as the asset turnover limit is relaxed, the realized turnover and the realized information ratio increase concurrently.

Specifically, the Max IR optimal mix achieves the highest recorded ex-post information ratio at 0.3, but it also exhibits the highest turnover at 89.37%. Generally, all optimized mixed signals demonstrate an inverse relationship between turnover and IR improvements. Furthermore, the optimized signals outperform the equal weight composite signal in terms of either realized information ratio or realized turnover or, in the unconstrained and high asset turnover limit cases, both.

The asset turnover induced by each signal is illustrated in Figure 2. The impact of the asset turnover limit constraint is evidently showcased there. Specifically, signals with a lower turnover limit reduce turnover in phases where our reference signal mix, namely Max IR, leads to high turnover. Furthermore, turnover on each backtest date increases in direct proportion to the turnover limit. On instances when the Max IR composite signal triggers small turnover, signals with a high enough asset turnover limit appear to mirror it closely, as can be seen with the Max IR and Max IR with asset turnover set at 100%.

The following discussion details the changes in signal mix weights across different setups. When the asset turnover limit is set to 40% or 60%, a significant preference is shown for the SLOW signal over FAST and INTERM., as illustrated in Figures 3 and 4.

In these instances, the FAST signal is almost entirely absent from the mix. Despite SLOW generating the worst volatility of the three signals (see Table 1), it is favored due to its lower associated turnover. As we increase the turnover limit to 80%, there's a noticeable transition in weight from SLOW to INTERM., with a minor inclusion of FAST, as depicted in Figure 5.

Upon relaxing the asset turnover limit to 100%, there is a substantial shift in weight towards INTERM. and FAST, reducing the emphasis on SLOW (see Figure 6). This adjustment brings the structure of signal weights closer to the reference Max IR weights in Figure 7. A more significant difference between the two signal mixes is observed starting in 2020 (compare Figures 6 and 7); with no asset turnover limit, greater weight is allocated to FAST, whereas a set limit shifts more weight towards INTERM.

Conclusion

In summary, the analysis illustrates how blending different signals can be fine-tuned to adhere to a specified turnover threshold by controlling the first-order autocorrelation of the aggregate signal. The progression of signal mixes under different turnover constraints, as shown in the detailed figures and tables, provides a comprehensive view of how strategic turnover management can be effectively integrated into the investment process.

If you'd like to explore how to develop more robust and diversified portfolios by expertly combining optimal alpha signals, download our white papers: Optimal Signal Weights and Mixing Alpha Signals with Asset Turnover Control.

References

Qian, E., Sorensen, E.H., and Hua, R. 2007. "Information Horizon, Portfolio Turnover, and Optimal Alpha Models." The Journal of Portfolio Management, Vol. 34(1), pp. 27-40.

Ding, Z., Martin, R.D., and Yang, Ch. 2020. "Portfolio turnover when IC is time-varying." Journal of Asset Management, Vol. 21, pp-609-622.

Grinold, R. 2010. "Signal Weighting." The Journal of Portfolio Management, Vol. 36(4), pp. 24-34.

Svallin, J., Mitov, G., Margaritov, E., Radev, N., Gum, B., Bilarev, T., and Stefanov, I. 2023. "Optimal Signal Weights." FactSet Whitepaper, link

Svallin, J., Mitov, G, Radev, N., Bilarev, T., and Stefanov, I. 2024. "Mixing Alpha Signals with Asset Turnover Control." FactSet Whitepaper, link

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