Liquidation cost: why mark-to-market values are wrong

Equity portfolios are marked-to-market on the assumption that each share will recoup that amount of cash – but exiting large positions has a market impact, wiping out value. New research indicates this dynamic may be governed by a universal law. Laurie Carver reports

JP Bouchaud

It is the kind of story old traders tell around the campfire: it starts with a big, long position in a stock, the price of which starts to slide. The trader’s reaction is to sell, and sell in size, but the market is spooked and the depreciation only accelerates. The trader is desperate to exit, selling more stock – and the price drops again – and this continues until the bottom, or bankruptcy, is reached.

The phenomenon is as old as markets themselves. According to one near-contemporary – though unscientific – account of the bursting of the Dutch tulip bubble, trading was abandoned on the Haarlem tulip exchange on February 4, 1637 when investors looking to unload their inventory forced prices on successive trades to drop by 15%, then a further 25%, then 35%. It is not hard to draw parallels with the sudden evaporation of value in subprime mortgage-backed assets in 2007, as banks vainly sought buyers. These are extreme examples of an everyday effect – market impact, or an asset’s price movement due to order flows.

Practitioners have long known about this – and the consequent discrepancy between the mark-to-market value and the actual value that can be achieved during liquidation – but until now there has not been a convincing quantitative way to get hold of it. Indeed, finance theory tended to ignore the problem by assuming perfectly liquid, efficient, frictionless markets.

But new analysis by so-called econophysicists is shedding light on what is now being called liquidation value, and some believe the work could have profound consequences. “The new thing is that liquidity cost is very important to risk,” says Doyne Farmer, a professor at the Sante Fe Institute in New Mexico. “If regulators had valued the accounts of Lehman Brothers taking account of liquidity they would have seen their real level of leverage. The crisis might have been avoided,” he says.

The research also suggests mark-to-market values are significantly too high – it might reflect where the market is trading, but because there will be a market impact on liquidation, it does not reflect the actual realisable value of the position. For cash equities, the extent of the overvaluation for a large position in a single stock is likely to be measured in the low single digits, research suggests. Quants that have attempted to carry the same technique across to over-the-counter derivatives markets say the value of big, illiquid instruments could be overstated by as much as 20%.

If regulators had seen the accounts of Lehman Brothers or Long Term Capital Management under a liquidation value they might have stopped them. The crises might have been avoided

Work on liquidation value has been stimulated in part by improved access to data. “It started in the early 2000s,” says Fabrizio Lillo, a colleague of Farmer at the Santa Fe Institute and a professor at Scuola Normale, a university in Pisa. “By looking at the data from several markets, we could get a better sense of the order book, and a greater indication of how the price changes.” And the empirical evidence tells a story that is not the sort of thing you are supposed to find in economics – hard rules.

“What’s really intriguing is it seems to be a fundamental law,” says Jean-Phillipe Bouchaud, chairman and head of research at Paris-based hedge fund Capital Fund Management and a professor at École Polytechnique, also in Paris. “Some very illiquid markets may be a bit different – but we have a wide array of the more transparent markets all showing the same behaviour. The relationship seems quite robust, and may just be a fact of the way markets work.”

Quants have seen the price impact from liquidating positions – in different markets, across different sectors, with different execution strategies, and over different lengths of time – roughly track the square root of the volume of the asset traded, relative to daily turnover. Independent studies have found that same pattern in share price data from the New York, London, Paris and Tokyo stock exchanges, particular sectors such as financials, trading in large-cap and small-cap stocks, and on futures exchanges. Bouchaud has even observed it in his firm’s own trading in small- and large-tick futures contracts on a variety of underlyings (see figure 1), a correlation for which there is no a priori reason, he says.

Economists are disturbed by the idea of a fundamental law of price impact, according to Farmer at the Santa Fe Institute. “They always ask why we’re looking at the functional form of the price impact. They think it’s not an interesting problem, because they are so sure there are no laws in social sciences,” he says. “Well, it seems like there is a law in this case – and that makes science easier.”

One economist who does see value in seeking a functional form is Robert Engle, a professor of finance at New York University (NYU) and winner of the 2003 Nobel memorial prize for economic sciences. But he is sceptical of whether the square-root dependence is truly fundamental, suggesting it may be a result of market infrastructure rules, for instance. He doubts it will hold for less transparent OTC derivatives markets.

“There’s quite a lot of data supporting a square-root law to pretty good accuracy, at least for equities. But I don’t see why it should necessarily be a law of nature. Will we see the same shape of change if we switch from pennies to even smaller tick sizes? Suppose you look at credit default swap (CDS) data – would you expect to see it there?” he asks.

Capital Fund Management’s Bouchaud says he would not expect the law to hold across every traded market, but he at least believes there is an argument for why the form should arise in some markets. The significance of a square-root law is its concave shape (see figure 2). So, while larger-volume executions shift the price more, the rate at which they do so slows – for instance, the first half of a liquidation shifts the price more than the second. This is perhaps unintuitive, but researchers try to explain it through the concept of latent liquidity.

This is the extra liquidity outside the limit order book that becomes available when prices have moved far enough – representing investors that have decided to buy when a stock has fallen a certain amount, for example. Close to the current clearing price the latent liquidity is small, because such orders are more likely to be executed by market-makers instantly. Further away from the price at the top of the order book, more of these latent traders become active to take the other side of the trade, so there is less impact. If this additional trading volume grows in proportion to the bids and offers, at least close to the clearing price, this would create a square-root law in the impact, according to Bouchaud.

Why market impact matters, of course, is that it creates a cost to trading – and this has been a big motivator for researchers. There have been simple models of price impact since the 1990s. A seminal 2005 paper in Risk by Robert Almgren, co-founder and head of research at algorithm-based brokerage Quantitative Brokers, studied the square-root form for market impact empirically (Risk July 2005, pages 57–62). Working as a consultant to Citigroup, Almgren used a data set showing the bank’s impact on US stock observations to develop his model. The fit to the data was far from perfect, but evidence has mounted for the square-root law as more data sets have been studied.

“My point of view is very pragmatic. I’m a broker, so what I really want to do is get an idea of the cost involved in executing a strategy for a client, and a functional form allows that,” he says.

Algorithmic and high-frequency trading firms such as Almgren’s try to minimise the cost by executing the trade over time to exploit the best liquidity conditions. While the quants say the dependence on volume seems fairly robust in the markets surveyed, the constant of proportionality it is multiplied by can vary from market to market, and from trader to trader. In the quants’ formalism, it is this constant that trading algorithms seek to minimise. But this can only take you so far, according to the Santa Fe Institute’s Farmer.

“The square-root dependence on volume means there’s a limit to how big you can get before your trading impact will wipe out your gains,” he says. “I think the coefficient can be made smaller, and good traders do that. But I think the difference between the best and worst will be at most a factor of two.”

“You can try to minimise costs, but if you’re trading large sizes the dominant costs come from impact,” says Capital Fund Management’s Bouchaud. “People worry about high-frequency traders picking up a basis point here and there, and they should, but it’s a bit of a red herring. The macro costs come from the true level of liquidity and algos are not going to change that.”

But there are other applications – and implications – for the research. Once the form for the impact is understood – depending on the volatility of the asset as well as the volume traded – it can be incorporated within discount factors, giving the expected value of a portfolio if it were actually liquidated. Traditional marking-to-market, which assumes all portfolios can be liquidated in their entirety at the current execution price, means valuing a portfolio as the sum of the values of its constituents. In contrast, liquidation value is non-linear.

The conclusion is that financial statements routinely overvalue assets – an issue that has been investigated by Bouchaud, as well as Farmer and colleagues at the Santa Fe Institute. To get rough numbers, a holding of 5% of the capitalisation of a typical large US stock, which accounts for 0.5% of the total market and has a daily volatility of 2%, will be overvalued by 3–6%, according to the quants’ formula.

“It’s not prohibitive on stocks, but it’s not negligible,” says Capital Fund Management’s Bouchaud. “For less-liquid instruments, the effects will be more pronounced. I’ve looked at CDS and collateralised debt obligation markets and come up with numbers like 10%, 20%. In the run-up to the 2008 crisis, people holding those structured instruments were forgetting they would have to liquidate – and potentially lose all their profits.”

The effect will be ramped up for leveraged positions, as well as OTC trades. When a trader takes on a highly leveraged position, there is the risk that the realised value of the asset will depreciate to a level that is lower than the funding liability, resulting in a net loss.

“Normally when you sell off, your leverage should go down,” says the Santa Fe Institute’s Farmer. “But if there is a price impact, it will typically outweigh the effect of lowering your position, and actually push it up.” According to the model, if there is sufficient latent liquidity to start lessening the impact, the leverage will steady and roll back over the course of the liquidation. But under certain conditions, there is a snowball effect that can drive the leverage to infinity (see figure 2). “At that point you’re bankrupt. For a mark-to-market position, you would only see it as you’re exiting the position, but by then you’re screwed.”

Despite its potential as a systemic risk indicator, however, implementing something as complex as liquidation value across the finance industry feels like a pipe dream to some researchers. The industry is struggling to adjust to new regulation as it is, accounting standard-setters would be unlikely to welcome an idea that would require market values to be adjusted by models, and it’s hard to imagine banks embracing something that would result in widespread writedowns.

“It’s potentially something regulators could use,” says NYU’s Engle. “But it would be hard to implement, both technically and politically. There’s still resistance even to marking-to-market some assets. If banks won’t mark sovereign bonds to market, they are not going to mark to liquidation cost.”

BOX: Multiple impacts
The idea of modelling the effect trading in one asset has on its price is hard enough. But when portfolios of multiple names need to be liquidated, it gets more complicated. The impact of selling a large amount of General Motors (GM) shares could send the Ford share price falling, too, with a feedback effect on GM. But the correlation between the two price impacts is hard enough to observe, never mind model.

“In reality, it can cost you more or less to liquidate two assets simultaneously than if you did it separately,” says Robert Almgren, co-founder and head of research at algorithm-based brokerage Quantitative Brokers in New York. “So you definitely need some kind of non-linear model with a cross-impact.”

Robert Engle, a professor of finance at New York University, agrees. “It’s hard to detect the cross-impact. I’ve looked for it and so have other people I work with. It’s just not easy to be sure what is joint impact and what is separate.”

Surprisingly, a particular case of joint impacts that may be easier to understand is in options markets. An option written on a stock will generally be delta-hedged with that stock, so the transaction costs of the underlying will affect the cost of hedging. Engle believes there is a good rule of thumb between the two.

“An interesting question is where the bid/offer spread comes from in options markets? It may not be that complicated – I think you can get pretty far just saying it’s the underlying’s spread times the delta,” he says.

However, from the point of view of traditional finance theory, there is a highly non-linear problem lurking here. If a trader hedging an option affects the underlying’s price, the future distribution is altered – and hence the probability of the option going into the money. This means that the risk-neutral expectation used to price the option is altered, which then changes the delta, in a feedback loop.

This problem is evidence that the fundamentals of quantitative finance need to be looked at, according to one quant at a European financial firm – including the sacred concept of the absence of arbitrage. “What is happening here is that in an illiquid market, arbitrage is subjective, not objective. Better arbitrageurs will be able to better manage their cost of trading. And the bid/offer spreads on options become an indicator of systemic risk.”

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