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On the White Board - May 2010
May 15, 2010
Mark to Liquidity: quantifying portfolio liquidity risk
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Liquidity risk is famous for being an elusive concept. We all know that it appears in different contexts, and we all know that these aspects are interconnected. We have a good understanding of some of them, taken separately, but it’s certainly more challenging to capture the whole picture. In the economics and finance research literature, liquidity risk has been extensively studied in the following specific meanings:
- Liquidity premium: how much of an asset risk premium is explained by its illiquidity
- Asset liquidity indicators: what (observable or unobservable) characteristics of an asset can be used for ranking assets according to their liquidity
- Market impact: what prices we get when trading different sizes of an asset ? And how the market will move after the trade?
- Global liquidity indices: what global market indicators may be taken as representative of overall capital market liquidity?
For instance, we can resort to robust and well tested methods to confirm quantitatively our intuition that a blue chip stock is more liquid than a SME one, or that selling a large amount of T-Bonds we suffer a smaller liquidation cost than selling an equivalent amount of emerging market govies.
However, what has been missing for long time is a risk management instrument meant to probe and quantify what are the effects of liquidity risk on a given holding. Basic questions such as “How much is the liquidity risk of a portfolio?”, “How can I decompose it?”, “What are the sources of each component?”, “How can I reduce it?”, “What are the factors that drive it?”, “What if the market moves?”, “What if I buy or sell something?”, “What would have it been like in September '08?”, “What happens if I am forced to liquidate one quarter of my portfolio?”, “How is my VaR affected by liquidity risk?”, are still difficult to answer quantitatively, and there’s not even consensus on how these questions should be addressed exactly.
In other words, risk managers have long been looking for a “model for portfolio liquidity,” namely an instrument able to provide a precise figure for portfolio liquidity risk as a function of the market variables that explain and drive it. Portfolio liquidity has always been managed qualitatively, on a judgmental basis. But this is not enough when we come to the questions raised above, where a quantitative modeling of liquidity risk, taking into account portfolio effects and linking them to market conditions, is in order.
Mark-to-Liquidity (MtL) is a framework for portfolio liquidity aimed at putting such basic questions on a firm theoretical ground and providing quantitative answers to all of them. The first question to face is “what exactly is portfolio liquidity risk?” In other words, we need a clear definition of portfolio liquidity risk in the first place.
In MtL, this is defined as the implicit cost faced by a portfolio subject to liquidity or risk constraints in an illiquid market environment. By “constraints,” we mean the list of commitments that a portfolio must be prepared to, such as liquidations, redemptions, cash needs for payments, risk limits from investment policies or trading limits, margin requirements and so on. The investment rationale of every portfolio entails a list of constraints: we call this the “Liquidity Policy” (LP). By “in an illiquid market”, we mean “always”. No market is perfectly liquid and to some different extent size effects are always relevant when we buy or sell some asset.
In MtL it is the combination of portfolio constraints (what we call endogenous liquidity risk) and market illiquidity (exogenous liquidity risk) that generates portfolio liquidity risk, because it is the risk of being forced to trade in a market where trading itself is a cost. In the stylized case when we assume either no constraints (the portfolio can be held arbitrarily long) or no liquidation costs whatsoever, then we will have no liquidity risk on the portfolio.
The strategy behind MtL is not then inventing a “liquidity risk measure,” but rather changing the very definition of “portfolio value,” by taking properly into account potential liquidation costs due to the commitment to a given LP. It is really nothing but a more intransigent way of marking the portfolio to market, where however we do not overlook the liquidity structure of the market and the constraints of the portfolio.
In doing this, we naturally discover features that account for portfolio liquidity phenomena that have always eluded formalization, such as:
- Granularity effect: the more fragmented your positions are, the better it is from a purely liquidity perspective (namely leaving aside correlations),
- Added value of liquidity: a cash injection reduces liquidity risk,
- Size effects: when we scale a portfolio, we do not necessarily scale its value or liquidity risk correspondingly,
- Dependence of liquidity risk not only on the assets in the portfolio but on the set of constraints it is subject to.
In the MtL framework, the value of a portfolio is no more just a “shopping list sum,” but a richer set of information, which contains liquidity effects also.
We give a brief technical introduction to the framework below.
The basic framework
In an illiquid market, two notions of mark-to-market arise naturally,. We call them ‘best MtM’ (U) and ‘cost to exit’ (L).
We call U(p) the value we obtain by marking all long positions of a portfolio p to best available bid and all short positions to best ask. It may not be a size-sensitive notion of MtM, because the best ask and bid may be available only for small trades, but at least it takes into account the bid offer –spread.
We call L(p) instead the liquidation value of our portfolio, namely the amount of money we obtain by exiting our portfolio completely. In this case, we really need to explore all the structure of available bids and offers up to the size we need to liquidate. In general, it is clear that this is a much more prudential estimate and in fact L(p) < U(p)
U(p) will be an appropriate value for the portfolio only if it is not subject to any constraint. On the other hand, L(p) will be appropriate only when the portfolio has the extreme commitment to be liquidated completely. In normal cases, L(p) is too conservative a valuation and U(p) too optimistic. A realistic portfolio LP will be something in between “you need not be prepared to liquidate anything” and “you must be prepared to liquidate everything.” For instance, a mutual fund is implicitly subject to a LP where there is a commitment to liquidate some fraction (say 30%) of the assets under management (because of possible redemptions) still preserving the risk profile imposed by the investment policy (by risk limits, leverage limits, concentration limits,…).
To compute the value of this portfolio in the MTL framework, we solve an optimization problem, by finding the portfolio q* which:
- minimizes the liquidation costs, and
- is constrained so:
- q* can be obtained from p by trading at current market conditions
- q* satisfies the LP
In doing this, we are doing exactly what the portfolio manager in our example would do when facing the need to meet the redemptions while maintaining the risk profile of the portfolio. He would act in such a way as to minimize the liquidation costs.
After we have found the best portfolio q*, we simply say that the value of our original portfolio p is simply given by U(q*).
In formulas
where r is the portfolio liquidated from p to obtain q and L(r) are the proceedings from the liquidation.
As we see, the value of a portfolio VLP(p) in the MtL framework depends explicitly on the LP the portfolio is subject to. This means that the very same portfolio p has two different values in the hands of two different investors, and it will be higher for the investor who needs to face less stringent constraints.
U(p) and L(p) turn out to be really two extreme cases of VLP(p), the highest and the lowest possible, associated to extreme types of constraints.
Another surprising property of VLP(p) is that it is no longer linear in p. The sum of two portfolios is not worth the sum of their values in general, and a portfolio two times bigger is not necessarily worth twice as much. What may seem puzzling at first sight in fact fits our intuition regarding portfolio liquidity risk. It is exactly the nonlinearity of the portfolio value function that captures the liquidity effects that we were never able to quantify in formulas.
For instance, it is clear that when we improve the granularity of a portfolio--that is when we reduce the concentration in any given asset—we expect a reduction of liquidity risk. To improve granularity, we can for instance blend two portfolios, and we expect a measurable liquidity benefit. This is in fact what we obtain with MtL because VLP(p) can be shown to be a concave function
In other words, the value of the blend is higher than the blend of the original values. Another important fact captured by the nonlinearity of the value function is the additional value of liquidity. If we add a certain amount C of liquid cash (dollars) to a portfolio p, we certainly reduce its liquidity risk because we may prevent the liquidation of illiquid assets to meet the constraints. In MtL this is reflected in the fact that
where we see that the portfolio value increases by more than just C, because of the liquidity effect of adding cash.
References
Acerbi, C. and Scandolo G. (2008) “Liquidity Risk Theory and Coherent Measures of Risk”, Quantitative Finance , 8, 7, 681-692
Acerbi, C. (2009) “Portfolio Theory in Illiquid Markets”, in “Pillar II in the New Basel Accord”, ed. A. Resti, RISK books.