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Mensuel : Edition de mai 2009
Rubrique : Finance/Economie
Titre : Recherche en Finance: Integrating liquidity risk in portfolio choice
Article : As the recent crisis unfolded, liquidity risk became a major topic. Actually, liquidity is generally an issue during stress situations. Decision variables that were independent, or supposed to be independent, suddenly become highly correlated. This phenomenon is known as the “phase locking effect”.

The notion stems from biology and was originally documented in biological systems where the automatic synchronization of the flickering of Southeast Asian fireflies was described. Recent research has suggested that the “phase-locking behaviour” can be modelled with a dummy variable added to factor models such as APT or CAPM. The dummy is supposed to catch return variations due to systemic shocks. The systemic shock dummy can then be used to distinguish between unconditional and conditional correlations, where the latter denote the correlations during stress situations. Those conditional correlations should actually be used to implement stress tests. Incidentally, note that even though unconditional correlations can be more or less zero, the conditional correlation can be quite close to unity.

All this sheds light on the importance of liquidity risk in portfolio optimization. Still, attempts to integrate liquidity risk in the portfolio optimization framework have been meagre. Indeed, the standard portfolio optimization approach, namely mean-variance, does not address liquidity problems. Nevertheless, over the last years a few authors have tried to develop a portfolio optimization framework with liquidity risk. In order to integrate liquidity problems in an optimization framework a liquidity metric is warranted. Before constructing such a metric we need to understand what a liquid asset actually is. Typically, liquidity means that assets can be traded quickly in large quantities and with little price impact. Based on this description, Lo et al. (2003) try to integrate liquidity in the standard portfolio setting. A few variables are used to construct the above-mentioned liquidity metric. First, they use trading volume as an asset is more liquid when it is traded frequently and at higher quantities. However, in order to compare volume to outstanding shares, turnover (trading volume divided by outstanding shares) is used as an indicator (Lo and Wang (2000)). Second, transaction costs such as bid-ask spreads are considered. Large bid-ask spreads are at least partly attributable to liquidity risks as market makers demand premiums for making markets in illiquid securities. Finally, Market capitalization, namely the market value of total outstanding equity, has also been suggested as a proxy for liquidity.

The liquidity indicators are constructed by aggregating daily data to a monthly timeframe. The indicators are then normalized so that their respective values vary between 0 and 1. The liquidity metric of the portfolio can then be constructed by evaluating the weighted average of the liquidities of the securities in the portfolio. Systematic liquidity effects can be integrated via portfolio weighted liquidity matrixes. The off-diagonal liquidity metrics then indicate cross-liquidity effects between the different assets. In principle, it is then possible to construct the portfolio along the three following dimensions: return, variance and liquidity. However, the liquidity parameter in the function that should be optimized is not defined. It is well known that mean-variance is determined from a Taylor expansion around the expected return and that this leads to the specific risk aversion parameter. The latter can then be calibrated for different investor profiles. Such an approach, however, has not been developed yet as far as liquidity is concerned. The analytical construction of a kind of “liquidity aversion” is a topic of current research.

There are three methods to integrate liquidity. First, liquidity filters may be used. Basically, only assets that have a minimum level of liquidity are included in the portfolio. The standard mean-variance optimization can then be applied to the selected universe of assets. Second, liquidity can be imposed as an additional constraint in the standard mean-variance optimization program. Finally, “liquidity aversion” can be directly formulated in a kind on Mean-Variance-Liquidity objective function. The latter approach seems the most straightforward, even though, as was discussed in the last paragraph, it is not theoretically defined. Mean-Variance-Liquidity Efficient Frontiers can now be constructed. The above-mentioned authors analyse the movements of the standard efficiency frontiers as the liquidity constraint varies. Of course, liquidity constraints tend to decrease the set of mean-variance tradeoffs. By analyzing the movements of the efficiency frontier the authors also determine the tangency portfolios. Basically, liquidity constraints increase the volatility and decrease the expected return of the tangency portfolio, hence inducing lower Sharpe ratios. The optimal portfolio for a respective investor will, of course, depend on his risk aversion and preference for liquidity.

The authors also apply their approach to Hedge Fund indexes. In that case, illiquidity is measured with the Ljung-Box Q statistic which measures the autocorrelation of the return series. With that respect, it is well-known that funds that exhibit high autocorrelation of returns are invested in illiquid assets. This stems from the fact that those funds have to calculate their NAV by valuing non-liquid assets. This leads to a smoothing of the return series. Hedge Fund returns are smoother than standard Mutual Fund returns and the volatility warranted by their risk exposure. There exist two potential explanations for this potential fact. Basically, Hedge Funds take exposure in illiquid assets and/or are cheating investors.

The issue of smoothing when assets are illiquid is well-known as “stale-pricing”. As some assets in the fund are illiquid it is difficult to evaluate the NAV and interpolations between prices are used to evaluate the current NAV, which leads to smoothing and actually boosts the Sharpe ratio. Autocorrelation tests on the return series can be used to detect illiquid exposures of funds. With that respect, Bollen and Pool (2006) argue that the most likely reason for return smoothing is fraud in order to boost the Sharpe ratio. Different liquidity metrics can now be imposed on a portfolio invested in different Hedge Fund indexes. As the liquidity constraint is increased, the allocation between the different Hedge Fund strategies changes drastically, indicating that the strategies are asymmetrically exposed to illiquidity problems. The only strategy which is robust to changes in the liquidity constraint is Equity Market Neutral. Indeed, during the simulation exercise the weight of this strategy is 50% of the portfolio, whatever the liquidity constraint. This seems to indicate that Equity Market Neutral is a more liquid strategy.

The above-mentioned methodology is certainly interesting and future research results will be interesting to follow. Still, there is an alternative research agenda that is currently being developed, which focuses on the integration of liquidity problems directly in the risk measure. The so-called liquidity adjusted Value at Risk is well-known. However, as the VaR is not sub-additive it cannot be used as an operational tool to optimize portfolios unless return distributions do not deviate too much from normality. This is the reason why coherent risk measures (Artzner et al. (1999), Acerbi and Tasche (2002)) should be used in such an optimization process. However, coherent risk measures are not consistent with liquidity problems and this is the reason why Föllmer and Schied (2002) developed so-called convex risk measures, basically coherent risk measures that integrate liquidity problems. The development of a portfolio optimization approach consistent with such convex risk measures is a topic of research on which we will focus in the Research page next month.

Dr. Michel Verlaine
Associate Professor of Finance
ICN Business School
Expert for IFB Audit and Advisory Services
Michel.verlaine@icn-groupe.fr

ICN Business School conjointement avec IFB Audit and Advisory offrent des programmes de Formation Continue sur les sujets traités dans cette rubrique. Pour de plus amples détails contacter l’auteur.

www.icn-groupe.fr
www.ifb-group.com

References

- Acerbi, C. and Tasche, D. (2002) “On the coherence of expected shortfall”, Journal of Banking and Finance, Vol. 26, p. 1487-1503.
- Artzner, P., Delbaen, F., Eber, J.-M. and Heath D. (1999) “Coherent Measures of Risk”, Mathematical Finance, Vol. 9, p. 203-228.
- Bollen, N. and Pool, K. (2007). “Do hedge fund managers misreport returns? Evidence from the pooled distribution”, Owen Graduate School of Management, working paper.
- Föllmer, H. and Schied, A. (2002) "Convex Measures of Risk and Trading Constraints", Finance and Stochastics 6, 429-447.
- Lo, A., Petrov, C. and Wierzbicki, M. (2003) “It’s 11 PM-Do You Know Where Your Liquidity Is? The Mean-Variance-Liquidity Frontier”, Journal of Investment Management, 1, 55-93.
- Lo, A. and Wang, J. (2000) “Trading Volume: Definitions, Data Analysis, and Implications of Portfolio Theory”, Review of Financial Studies 13, 257-300.
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