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Scores for managing the liquidity of corporate bonds portfolios

This topic will be discussed during the Amundi World Investment Forum



Key point

  • Liquidity is becoming a vital criterion for selecting bonds
  • Liquidity monitored and managed with bond scores



In this document we describe how we build scores for corporate bond issues that are updated every day on the basis of timely characteristics. We also describe how the scores are used by the fund managers in the construction of corporate bond portfolios.

1. Background

Since the level of market liquidity is dispersed and is generally low for corporate bonds, it is advisable to take liquidity into account when investing in such bonds. It is wise to weigh the effort it takes to wind or unwind positions into the bond selection process. To be able to do so the level of liquidity must be quantified in some way. To this end Barclays -among other data providers- has developed liquidity cost scores (LCS) for all members of their Global Corporate Bond index on a monthly basis since 2009, see Konstantinovsky et al. (2015).
The scores are consulted by the bond portfolio managers in Amundi, in particular when constructing index-tracking portfolios, for which liquidity is crucial. It is of interest for Amundi to develop its own scores, for two reasons: (i) an efficient portfolio management requires a monitoring of the liquidity conditions on a daily rather than monthly basis, and ultimately (ii) independence from data providers is preferred.
Barclays builds scores using a proprietary database constituted by their in-house trading desk. For bonds that are traded at least twice a month, the score corresponds simply to the average bid-ask spread over the month multiplied by the bond duration. For bonds that are traded less frequently, scores are derived through a modelling procedure. Scores are assigned to bonds as a function of their attributes, eight in total as listed in Table 1.




The specification of Barclays’ model, given in equation (1) in next section, is straightforward: the liquidity score of a bond is the sum of its attributes weighted by the respective model parameters. The parameters are calibrated onto the set of bonds that trade regularly for which bid-ask spreads are registered. Estimates are obtained through standard cross-sectional regression analysis. The model is then run for the lowly-traded bonds making the assumption that the same relationships between liquidity and bond attributes apply.
The Amundi liquidity scoring model that we have developed operates in a similar manner. A linear relationship is specified between attributes and liquidity, which is in its turn calibrated onto the Barclays liquidity scores that are available at the end of each month. The model is scores for all European and US corporate bonds in this way. The model is presented in next section equation (2). In section 3 is explained how the scores are being used in the investment process and in section 4 is described how we want to improve our scoring model. 

1 See Ben Dor et al. (2007).

2 The liquidity scoring model

2-1. Model specification 

Barclay's liquidity scoring model is specified as follows: 




Region & currency
We cover two regions: North-America and Europe. The rest of the world (±15% of the Global Investment Grade universe in 2015) is not part of our scoring scheme. And two currency categories are defined: USD and EUR, covering 95% of the universe under study. The miscellaneous 5% have been assigned to these categories by a simple rule: European currencies fall under the Euro and the rest under the dollar. In all, the two regions and two currencies set four different values for the intercept. It means that four average liquidity levels are set.
Compared to Barclays’ model we have replaced the intercept specified by the credit ratings and industrial sectors by one that is specified by region and currency denomination. We have done this for two reasons: (i) ratings change over time making the modelling framework unstable, which is undesirable and (ii) it is generally acknowledged that liquidity levels differ more over regions than over industry sectors, due to local legislation and other market-related issues. Figure 2 gives empirical evidence. It can be seen in the Figure that the frequency distributions are similar between the industrial sectors measured over the test sample and period.

Coupon & outstanding amount
It appears that relatively high-coupon bonds are more liquid than bonds with low payouts. Amihud and Mendelson (1991), who have studied this market phenomenon, argue that high coupons are popular for liability-matching purposes and therefore more exchanged. The outstanding amount is a valuation-independent measure of debt size, size being determinant for the liquidity level, as demonstrate Helwege et al. (2013) among others.

Age & time-to-maturity
As is generally recognized in the literature (see Bao et al., 2011; Edwards et al., 2007; Helwege et al., 2013), the age of the bond plays a role: the older a bond the less liquid. And the remaining lifespan adds additional information to the level of liquidity: the longer the timeto- maturity the less liquid.

Yield, price, DTS & peer status
The first three variables are price-dependent thus volatile over time. Including these variables in unprocessed format would make the model unstable as a result. To avoid that we smooth the variables taking five-working-day moving averages, so as to capture more stable yet timely levels. We reckon Barclays’ model would gain by processing their price and DTS variable likewise. The price-dependent variables turn out to have significant explanatory power in tests, even smoothed.

Peer status is defined by three possible configurations: a bond is the single issue of a company, or otherwise it can be either ‘on-the-run’, meaning the most recent issue, or else ‘off-the-run’. To give an idea, in April 2015 5% of the bonds were single, 83% off-the-run and 12% on-the-run. A difference in liquidity level is detected between the latter two in the literature, by Helwege et al. (2013) for instance, no comparison is made with the single status.


2-2. Model estimation


The model has been estimated by running cross-sectional regressions of the bond attributes on the Barclays liquidity scores. We have carried out forty such regressions, one per month between January 2012 and April 2015. All bonds belonging to the Barclays Global Corporate Bond Index and being domiciled in North America or Europe have participated. We have retained the variables that are statistically significant, intuitive and relatively stable over time. We have run the usual validity checks necessary for a linear regression model, in particular we have verified that there is no co-linearity issue. The parameter estimates as for April 2015 are given in Table 3.




We recall that the LCS have the opposite signs of the liquidity levels. A positive sensitivity to age for example, with a coefficient of 0.15, means that the older a bond is, the higher the liquidity cost so the less liquid it is. Yield, DTS and price go the same way: the higher their levels the lower the liquidity. The coupon and outstanding amount have the opposite impact (negative coefficients). So the bigger the total debt amount, the lower the cost so the more liquid the bond, which is intuitive.

Interestingly we find that bonds denominated in their local currency, i.e. European bonds in euros and American bonds in dollars -denoted EU, € and US, $- are significantly more liquid than those domiciled abroad. Within that European bonds seem slightly more liquid than US bonds, with a coefficient of -0.26 versus -0.24. In accordance with the literature we find that on-the-run bonds are more liquid than off-the-run bonds on average, with a coefficient of -0.18 compared to -0.15. However this relative result is overshadowed by the third category, the single bonds which are substantially less liquid than multiple issues, though we find no reporting of this phenomenon in the literature.

We establish the frequency distributions of the liquidity scores over the test sample and period. They are given in Figure 4. On the left are the Amundi scores, derived by means of model (2), and on the right those published by Barclays. 

The scores don’t appear to be normally distributed. A relatively small portion has high scores and are thus less liquid than average. Note that all Barclays scores are positive. This conditions must have been imposed, which indeed makes sense. We do the same: the few scores that end up negative in the estimation process are set to zero.
We make note of the camel-back shape, which is apparent in the distribution of the Barclays scores and is more pronounced in the Amundi scores. The explanation of this shape lies in the DTS variable, which on its turn possesses this atypical shape over the test period.


3 Managing the liquidity of a portfolio 

The individual bond scores can be aggregated within a bond portfolio to get an idea of the general liquidity level. It may be useful for investment managers to monitor this level for risk control purposes. More interestingly the liquidity level can be proactively managed by integrating the scores into the portfolio construction process. We describe how this can be done.
Corporate bond portfolios are habitually constructed via a stratification process. Rather than proceeding by a Markowitz mean-variance optimization laid out in the standard finance textbooks, it appears more suited for corporate bonds to deploy a selection process based on directly- observed characteristics. In the purpose to build an index-tracking portfolio, bonds are selected on the basis of their DTS contribution, which is defined as the weight in the index times the duration times the spread level, denoted as w·d·S. Gouzilh et al. ( 2014) describe the selection process, proceeding in two phases. The investment universe is first stratified, meaning that it is divided into subsamples that are most heterogeneous between each other and most homogeneous within. The intention is to separate out the various risk factors that play on the corporate markets by those samples. The second step is then to replicate the samples by a restricted number of bonds. We follow an iterative selection procedure whereby bonds are compared in terms of their individual- and collective DTS contribution.
Here is where the liquidity scores can be inserted. We define a liquidity variable as the inverse of the liquidity scores, that is LIQ = 10 – LCS, and augment the selection criterion by this variable. We carry out a simple multiplication, obtaining w·d·S·LIQ, and apply essentially the same selection procedure. The magnitude of the scores determines to what extent the liquidity criterion will play a role compared to the index-tracking objective. If they are scaled down so that the liquidity target becomes more stringent, the tracking error will mechanically increase, and vice versa. 
In back-tests over forty months, between 2012 and 2015, in the Euro zone we have studied how the two investment objectives interact, to what extent the tracking-error target can be achieved alongside the goal of increasing the liquidity level. We find that, interestingly, the liquidity situation of a portfolio can be improved greatly with little sacrifice in terms of tracking error. Figure 5 shows the test results for a portfolio containing 70 bonds on average over the period. When adding the liquidity criterion, the liquidity level goes up 20% on average (see Figure a), while the one-year rolling tracking error remains practically unchanged (see Figure b).



An explanation for the positive result can be found by looking at the initial selection criterion, the DTS contribution. Both duration and spread are unfavourable for the liquidity of a bond. The longer the duration and thus time-to-maturity and the higher the spread level, the less liquid the bond tends to be. Adding the liquidity scores must offset this bias towards illiquid bonds to a certain extent.
We find that the benefit of adding the liquidity criterion diminishes when the portfolio becomes less concentrated. For a portfolio containing 250 bonds on average the gain in liquidity is similar to the one containing 70 bonds, while the sacrifice in terms of tracking error becomes tangible. For a portfolio containing 450 bonds the ex post tracking error deteriorates from 0.23% without liquidity constraint to 0.30% with. Our interest for investment purposes lies in the concentrated portfolios though, where adding the liquidity criterion shows beneficial.

4 Ongoing research

We are aware2 that individual bond scores derived through cross-sectional regressions are effective for making relative comparisons across bonds, but less so for analysing liquidity levels over time. There is some time-series information incorporated in the Liquidity Cost Scores stemming from the fact that they are anchored on the bid-ask spreads registered over time by the Barclays trading desk. However this information is remote and incomplete. We consider to improve the time content of the scores by adding a term to our model that captures a general market condition. We are looking at a difference in spread level that can
be detected between those deduced from Credit Default Swaps and their underlying bonds. A price difference between a derivative instrument versus its underlying should not exist in theory and can in practice stem from market-related issues only, not in the least from a discrepancy in liquidity level.
We have run tests taking the average spread differences over the 125 members of the iTraxx index. Figure 6 compares the score levels we obtain over time. The Barclays scores (in black) appear to be stable over time, whereas the Amundi scores (in red) drop over the period. We don’t have an explanation for this at this stage. When incorporating the time component in the model the scores (in green) become more versatile on average yet conserve the same tendency as the scores without this component. In order to calm the volatility we have smoothed the time variable over a five-working-day time-window, as for the other pricedependent variables.

2 Guy Lodewyckx made the point.

















 Amundi has built its own liquidity scoring model
























Amundi liquidity scores based on ten bond characteristics































No notable difference in liquidity level on average over sectors






















Single issues substantially less liquid than multiple issues, wheter it be on-or -off-the-run










































Liquidity scores inserted into the objective function of the protfolio optimizer











Portfolio liquidity level can be improved at little cost
in terms of tracking error




Marielle de JONG, Head of Fixed Income Quantitative Research at Amundi
Arthur CLUET, Former Quant Research Trainee - Paris

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Scores for managing the liquidity of corporate bonds portfolios
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