Managing liquidity with portfolio staleness

The Markowitz (1952) mean–variance framework is at the core of modern asset allocation. In this setting, an investor optimally allocates wealth across risky assets caring only about the mean and the variance of portfolio returns. Although theoretically sound, the mean–variance framework has not reached widespread consensus in the finance industry. Among the causes of what Michaud (1989) defines “the Markowitz optimization enigma”, he identifies the fact that mean–variance portfolios ignore factors, such as liquidity, that are important for investment decisions. The relevance of liquidity in the asset allocation problem is widely recognized in the financial economics literature. For instance, Canner et al. (1997) show that portfolios recommended by four popular financial advisors are not on the mean–variance efficient frontier and consider assets marketability among the possible explanatory factors. Similarly, Ghysels and Pereira (2008), Hodrick and Moulton (2009) and Bazgour et al. (2016) find that fund managers are sensitive to liquidity while achieving their targets in the portfolio allocation decision. Despite these empirical evidences, few efforts have been devoted to understanding how investors can manage liquidity within the mean–variance framework.

Since the last financial crisis, both regulators and investors have expressed concerns about the worsening conditions of liquidity in the markets. First, the regulatory restructuring related to the traditional activities of banks and financial intermediaries tightened capital requirements, reducing their ability to supply liquidity. Funding constraints and higher margins can force liquidity suppliers to liquidate their positions in case of redemption pressures, thus raising market illiquidity (Gromb and Vayanos 2002; Chiu et al. 2012). Second, the growth of the ETF industry is raising concerns for a possible liquidity crisis.¹ In case of investor redemption, passive (index-tracking) funds following similar strategies can induce coordinated liquidations of securities (Scharfstein and Stein 1990), depressing market prices and thus leading to “liquidity spirals” (Brunnermeier and Pedersen 2008; Hameed et al. 2010). These premises warrant the development of a methodological framework to manage portfolio liquidity.

In this paper, we consider an investor who uses the minimum-variance framework² to allocate funds across stocks, and pursuits a volatility timing strategy (Fleming et al. 2001, 2003). The portfolio is rebalanced on a daily basis, and the solution of the optimization problem is a sequence of optimal weights that vary exclusively according to changes in the conditional covariances of the assets held in the portfolio. Using this framework, we aim to answer the following questions: how can we include liquidity in the dynamic portfolio optimization problem? How can we measure portfolio liquidity? Can we consistently generate portfolio liquidity gains selecting portfolios that are close to the minimum-variance portfolio?

The traditional approach to portfolio construction in the presence of illiquid assets entails the modeling of the transaction costs and the solution of a dynamic portfolio strategy. Some examples are given by Constantinides (1986), Gârleanu (2009) and Gârleanu and Pedersen (2013). A similar route is adopted in the literature on asset pricing with transaction costs (Amihud and Mendelson 1986; Acharya and Pedersen 2005; Vayanos 1998; Vayanos and Vila 1999) and in that on optimal trade execution (Bertsimas and Lo 1998; Almgren and Chriss 2000). Modeling and estimating the transaction costs is a difficult task. Commissions fees, bid–ask spreads, funding costs, price-impact are all relevant components of the transaction costs that may not be directly available to the investor. For this reason, a set of simplifying assumptions are typically imposed on the transaction cost function, and the existence of a closed-form or numerically tractable solution crucially relies on these assumptions.

The approach adopted in this paper is deliberately different. Our goal is to include the liquidity dimension in the standard Markowitz framework without explicitly modeling the transaction costs. To be more specific, we look for a synthetic and easy-to-compute measure which reflects the main determinants of portfolio illiquidity. In doing so, we try to conciliate the need for sufficiently liquid portfolios with parsimony and computational efficiency.

Bandi et al. (2017) show that asset prices observed at high-frequency update less frequently than what assumed in standard continuous time models of asset pricing. They introduce an economic indicator, named idle time, that estimates the probability of observing a zero return (or a stale price, whence the nomenclature adopted in this manuscript) at a given sampling frequency. Idle time provides information on the extent of liquidity of an asset, and it is related to both transaction costs and absence of volume (Bandi et al. 2020). We extend this notion to a portfolio of assets by defining the concept of portfolio staleness: on a given day, portfolio staleness is computed as the weighted average of assets’ idle times, using the corresponding portfolio weights. This measure has several attractive features: (a) is easy to compute, as idle time solely requires data on transaction prices to be implemented; (b) has a clear economic interpretation, as idle time can be regarded as an illiquidity proxy within a model of price formation with transaction costs and asymmetric information; (c) being idle time a probability, it naturally ranges between zero and one, allowing to easily compare and rank portfolios.

Building upon these considerations, our first contribution is to propose a tractable framework where the liquidity dimension is integrated into the portfolio selection problem. This is done by imposing an additional constraint (henceforth referred to as staleness constraint) on the minimum-variance portfolio optimization that limits the degree of portfolio staleness. The intuition behind this approach is straightforward: a portfolio with lower staleness places larger weights on assets with lower idle time. To the extent that idle time proxies illiquidity, the new portfolio is expected to give larger weights to liquid assets than the standard minimum-variance portfolio. On the other side, a very tight upper bound on the level of staleness for the assets included in the allocation will unavoidably result in a less-diversified portfolio. These considerations naturally lead to wonder to what extent such a staleness constrained asset allocation delivers a more liquid portfolio. Several definitions and measures of liquidity are available for individual securities, but the literature offers little guidance toward defining and assessing portfolio liquidity. Here, we adopt the economically motivated measure of portfolio liquidity of Pastor et al. (2017) to evaluate the degree of liquidity of an asset allocation strategy. Pastor et al. (2017) argue that the liquidity of a portfolio should depend not only on the liquidities of the stocks held in the portfolio, but also on the degree of portfolio diversification. The intuition underlying this argument is that a more diversified portfolio can lead to lower trading costs.³ Similarly, when the total value of the assets in the portfolio relative to the market capitalization of the assets is large, trading is more expensive. They formalize this intuition in a simple economic setting and derive the following portfolio liquidity measure:where N is the number of assets held in the portfolio, \(\omega _i\) is the portfolio weight of stock i, and \(m_i\) denotes the weight of asset i in a market-cap-weighted benchmark portfolio. Note that \({\mathbb {L}}\) takes values between 0 and 1. It can be shown that the portfolio with lowest \({\mathbb {L}}\) is the one fully invested in the stock with smallest market capitalization, and that the portfolio with largest \({\mathbb {L}}\) is the one coinciding with the benchmark portfolio, for which \({\mathbb {L}}=1\). In addition, Pastor et al. (2017) show that \({\mathbb {L}}\) factorizes into the product of two components: (1) the average of individual liquidities, defined as a function of the stock market capitalization; (2) a measure of the degree of diversification, related to the number of stocks held in the portfolio and to the Herfindahl index of portfolio weights.

Our second contribution is to show empirically that an appropriate choice of the staleness constraint leads to significant liquidity gains over the standard minimum-variance portfolio, as quantified by the measure of Pastor et al. (2017). Moreover, we show that gains in portfolio liquidity are much larger than the unavoidable increase in portfolio variance. Noteworthy, the risk profile of the staleness constrained portfolio is found to be very close to that of the minimum-variance portfolio. This result is somewhat consistent with the evidence in Canner et al. (1997) that investors do not choose portfolios on the efficient frontier, but are not too far from it.

Finally, our third contribution is to show how the proposed asset allocation strategy can be implemented in practice in an out-of-sample exercise. Given that idle time significantly changes over time, the portfolio staleness constraint must be adapted dynamically to the evolving market conditions. We show that this can be done with a simple rule based on forecasts of portfolio staleness. As before, we find that staleness constrained portfolios are significantly more liquid than minimum-variance portfolios, and that the liquidity-risk tradeoff is definitely in favor of portfolio liquidity.

Our paper is closely related to the works of Michaud (1989) and Lo et al. (2003), but it substantially differs in at least three aspects. First, we work in a dynamic setting, as the portfolio is rebalanced on a daily basis; second, we use an economically motivated measure of portfolio liquidity (i.e. portfolio staleness) to assess the degree of liquidity of the constrained portfolio; third, we show how to set the portfolio staleness constraint to obtain out-of-sample liquidity gains. Our paper is also related to the literature on shrinkage of portfolio weights (Jagannathan and Ma 2003; DeMiguel et al. 2009). However, instead of reducing the uncertainty on the estimated covariance matrix, our aim is to shrink portfolio weights towards more liquid assets.

As a robustness check, we replicate our analysis using, in place of portfolio staleness, the weighted average of inverse daily trading volume and the weighted average of daily bid–ask spreads. We show that bounds on these illiquidity measures provide liquidity-volatility profiles similar to those obtained through portfolio staleness. However, we discuss why portfolio staleness is more suited within the portfolio optimization framework.

The paper is organized as follows: Sect. 2 introduces the methodology and describes the staleness constrained portfolio; Sect. 3 provides details on the econometric modeling and forecasting of portfolio staleness and realized covariances; Sect. 4 reports the results of the empirical analysis and provides a comparison with common liquidity proxies; Sect. 5 concludes. Supplementary analyses are relegated to “Appendix”.

Let us consider a portfolio of N assets. Let \(t=s_{n,0}<s_{n,1}<\cdots <s_{n,n}=t+1\) be a sampling partition of \(n+1\) points of the time interval \([t,t+1]\), which can be thought of as representing one trading day. Let us also denote by \(\Delta _{n,j}=s_{n,j}-s_{n,j-1}\) the lengths of the n sub-intervals and let \(s\in [t,t+1]\). The vector of logarithmic efficient price processes \({\mathbf {X}}_s=\left( X_{s}^{(1)},\ldots ,X_{s}^{(N)}\right) \) is assumed to follow a Brownian semimartingalewhere \(\varvec{\mu }_s\) is an N-dimensional drift process, \(\varvec{\sigma }_s\) is an \(N\times N\) matrix and \(\mathbf{W }_s\) is an N-dimensional standard Brownian motion. The assumptions on the sub-interval lengths \(\Delta _{n,j}\) and on the drift and diffusion coefficients \(\varvec{\mu }_s, \varvec{\sigma }_s\) are the same as in Bandi et al. (2017). Following Bandi et al. (2017, (2020), we assume that, due to illiquidity frictions, the observed (on the sampling partition) logarithmic price processes \(Y_{s_{n,j}}^{(i)}, i=1,\ldots ,N\), differ from the efficient logarithmic price paths. More precisely, \(Y_{s_{n,j}}^{(i)}\) is assumed to be driven by the following recursive equationwhere \(B_{j,n}^{(i)}\) is a triangular array of measurable Bernoulli variates so thatHere, \(p_i\in ]0,1]\) represents a random asymptotic probability, which we will address as “probability of stale price”. The Bernoulli variates are pairwise independent,⁴ that is, for all \(i_1\ne i_2, {\mathbb {P}}[B_{j,n}^{(i_1)}=a,B_{k,n}^{(i_2)}=b]={\mathbb {P}}[B_{j,n}^{(i_1)}=a]{\mathbb {P}}[B_{k,n}^{(i_2)}=b]\), for all \(j,k=1,\ldots ,n\). The key idea behind the modeling in Eq. (3) is that it allows for the possibility of no trade, hence capturing inhibition of the trading activity. As in Bandi et al. (2017), we further assume that the supremum of the number of consecutive flat trades, denoted by \(K_n^{(i)}\), diverges at a rate slower than the number of observationsfor \(i=1,\ldots ,N\). Such assumption is necessary for the development of the asymptotic theory of the idle time estimator described in Sect. 3.2; see Bandi et al. (2017). In what follows, we will denote by \(p_{i,t}\) the asymptotic probability of stale price of asset i on day t. Note that the above assumptions allow the Bernoulli variates to be correlated with the efficient price process, temporally dependent and non identical distributed.

Finally, it is worth mentioning that the relationship between the Bernoulli variates and the efficient price process can be understood in a context of a micro-founded model of price formation. We refer the interested reader to the model described in Bandi et al. (2017).

Let \(\varvec{\Sigma }_t=\int _{t}^{t+1}\varvec{\sigma }_s\cdot \varvec{\sigma }_s^{T}\,\hbox {d}s\) denote the \(N\times N\) integrated covariance matrix of the vector of efficient prices \({\mathbf {X}}\), on day t. Let \({\mathfrak {p}}_t=(p_{1,t},\ldots ,p_{N,t})'\) be the \(N\times 1\) vector of stale price probabilities. Denoting by \(\varvec{\omega }_t=(\omega _{1,t},\ldots ,\omega _{N,t})'\) the portfolio weights, we define the portfolio staleness on day t as the weighted average of stale price probabilitiesFor each day t, the investor solves the following quadratic optimization:where \(\mathbb {1}_N\) is the \(N\times N\) identity matrix, \({\varvec{0}}_N\) is an \(N\times 1\) vector of zeros, \(\varvec{\iota }\) is an \(N\times 1\) vector of ones and \({\overline{\text {S}}}_t\) is a cap on the portfolio staleness \(S_t\). The problem in equations (6) extends the classical minimum-variance (henceforth MV) Markowitz problem⁵ by including the linear constraint \(\text {S}_t \le {\overline{\text {S}}}_t\).

Since, as empirically proved in Bandi et al. (2020), asset staleness measures illiquidity (along several dimensions), it is natural to impose the constraint \(S_t\le {\overline{S}}_t\) for the purpose of putting larger weights on more liquid assets. As a consequence, the resulting portfolio composition is expected to be more liquid than that of the classical MV portfolio. We notice that the optimization problem (6) incorporates a liquidity constrain with the minimal set of input data: portfolio staleness \(S_t\) can be obtained as soon as transaction prices are available, as it happens for the variance–covariance matrix \(\varvec{\Sigma }_t\).

By looking at the equations in (6), it is clear that the proposed framework differs from the literature on portfolio construction with transaction costs (Constantinides 1986; Gârleanu 2009; Gârleanu and Pedersen 2013) in at least two aspects. First, portfolio illiquidity is captured by portfolio staleness to the same extent that the transaction costs of the individual assets are captured by idle time. Therefore, no explicit modelization of the multiple components of the transaction costs (commissions fees, bid–ask spreads, funding costs, price-impact) is required. Second, the optimization problem is a simple quadratic problem with linear constraints. It is thus readily implementable by financial institutions that aim to conciliate the request of lower trading costs with computational efficiency.

A more general framework would include a further constraint on portfolio expected returns. However, the latter are notoriously difficult to estimate and measurement errors can lead to overweight (underweight) securities with large (small) expected returns (Jorion 1985; Garlappi et al. 2006). Since our aim is to evaluate the improvement of portfolio liquidity for every unit of risk, we avoid to target expected returns. However, the methodology remains unchanged for mean–variance portfolios.

Given the information available at time t, the quadratic minimization problem (6), which returns the weights of the SC portfolio, can be solved based on one-step ahead forecasts of the integrated covariance matrix \({\widehat{\varvec{\Sigma }}}_{t+1}\) and of the vector of stale price probabilities \({\widehat{{\mathfrak {p}}}}_{t+1}\). In addition, the cap \({\overline{\text {S}}}_{t+1}\) that defines the staleness constraint must be chosen. In this section, we describe the forecasting models adopted to obtain the predictions \({\widehat{\varvec{\Sigma }}}_{t+1}\) and \({\widehat{{\mathfrak {p}}}}_{t+1}\) and show how to set a dynamic portfolio staleness constraint \({\overline{\text {S}}}_{t+1}\).

Our dataset consists of transactions prices of 50 NYSE assets and covers \(T=2244\) business days, from 03-01-2006 to 31-12-2014. The time resolution is the minute. We select trades occurring between 9:30 and 16:00, leading to 390 timestamps. Summary statistics for the assets in the sample are reported in Table 1. In order to assess the effectiveness of our methodology at different dimensions, we build three groups of 10, 25 and 50 assets, respectively. The stocks included in the 10- and 25-sized groups are randomly selected, while the 50-sized group includes all the assets in the dataset.

Both \(\text {IT}_{i,t}\) and \(\mathbf{RC }_t\) are computed at the same sampling frequency of 1 min.⁷ We perform both an in-sample and an out-of-sample analysis. The in-sample analysis is made on the whole dataset and is aimed to provide empirical evidence that SC portfolios are significantly more liquid than standard MV portfolios. Portfolio liquidity is assessed through the measure of Pastor et al. (2017) in Eq. (1). In particular, we use the weight of each asset i in the S&P500 to obtain a proxy for the weight \(m_i\) that appears in Eq. (1). The out-of-sample analysis is performed through a rolling window of 1000 days. Our dynamic HAR models are re-estimated on a daily basis and then used to construct a sequence of SC portfolios based on one-step-ahead forecasts of \(\mathbf{RC }_t\) and \(\text {IT}_{i,t}\). Similarly to the in-sample analysis, we show that out-of-sample SC portfolios feature larger liquidity. Moreover, we provide guidelines on how to set in practice the quantile level \(\alpha \) of the staleness constraint.

Building upon the interpretation of idle time as an illiquidity proxy, we introduced the concept of portfolio staleness and showed that it can be effectively used to manage liquidity in a standard minimum-variance framework. A staleness constrained portfolio leads to significant improvements in terms of individual asset liquidities and to lower price impact due to a better degree of diversification. Interestingly, the unavoidable deterioration of portfolio risk due to the staleness constraint is much less important than the corresponding liquidity gain. Using transaction data of a cross section of NYSE assets, we showed how to properly select the staleness constraints in a real-life out-of-sample portfolio choice problem.

The proposed approach can be viewed as an “econometric framework” to manage liquidity in a standard minimum-variance problem. At each time period, the user of our methodology computes one-step-ahead forecasts of the relevant econometric variables (covariances and idle times) in order to recover the portfolio weights. The optimization problem is solved in closed form without explicitly modeling the multiple components of the transaction costs (commissions fees, bid–ask spreads, funding costs, price-impact). In this perspective, the methodology differs substantially from the methods based on direct inclusion of transaction costs in the investor’s objective function, which lead to more complex dynamic programming optimization problems.