Reader Richard Kline is providing a mini-series that was prompted by an anonymous reader who had observed that a complex systems theory view might raise doubts about regulatory policy. Financial overseers believe that liquidity is always and ever good, but that view may be naive:
Perhaps a lesson to be learned here is that liquidity acts as an efficient conductor of risk. It doesn’t make risk go away, but moves it more quickly from one investment sector to another.
From a complex systems theory standpoint, this is exactly what you would do if you wanted to take a stable system and destabilize it.
One of the things that helps to enable non-linear behavior in a complex system is promiscuity of information (i.e., feedback loops but in a more generalized sense) across a wide scope of the system.
Richard, in a guest post last week, posed a question he wanted to take further:
To what extent have nonlinear processes promoted the Securitization Bubble, precipitated its collapse, or prolonged the resulting instabilities in the financial system?
After a background discussion, he presented five issues:
Does innovation require untrammeled information flow across social/ economic event spaces?
Is the crisis in securitized debt the result of a ‘black swan’ event?
Was the creation of the Securitization Bubble the result of nonlinear processes in the financial markets?
Is a financial event-space optimized for propagation desirable?
If not, what structure or process parameters might improve process outcomes?
His first two post elicited quite a few comments (including requests for more concrete examples, which he has taken to heart). Hopefully, this offering will elicit more reader discussion. Your observations very much appreciated.
Can nonlinear discontinuities destabilize the financial system, nationally or globally? Dynamical systems not only change their trajectory, they change their geometry. Nonlinear systems not only change their delta (roughly velocity), they change their dimension and state. To my observation, nonlinear shifts in the systemic organization of the top tier of the financial markets have neither principally caused the Securitization Bubble nor thus far precipitated its ongoing collapse; however, such shifts have exacerbated both phases. Risks going forward are far from negligible.
Before listing recent nonlinear dynamical shifts of note, a few criteria are in order. Many nonlinear events in pricing, resource allocation, and so on can be identified; large and small; brief and enduring. I focus here on conditions in major financial markets, in the US or the financial system as a whole. The dynamical nonlinearities suggested here all modified the systems in which the are active so that behavior or capital aligned differently in the ‘after’ state than in the ‘before.’ These shifts have been non-transitory; that is, if they are not necessarily ‘permanent’ they have endured since defining their present states. Some of these might best be called ‘events’ in that their transitions are short and continuous leaving an enduring changed state; others are better called ‘processes’ in that they are ongoing and give the appearance of self-amplifying feedforward or feedback loops dynamics. I do not attempt to model these occurrences mathematically so as to formally confirm their nonlinearity. However, these transformations all had short duration onsets with significant state changes; some clearly would best be simulated by logistic trajectories. Among salient dynamical transitions, probable or potential, in the financial system, I would include:
Asset monetization liquidity. Historically, many asset classes did not trade in bulk on capital markets, such as mortgages despite their large absolute capitalization, or for that matter gold bullion. For mortgages, their basic units were small, and varied widely in contractual form, risk, and face value. Mortgage investment required knowledgeable assessment and servicing staffs, and these contracts are subject to somewhat varying legal requirements in different jurisdictions. That state banks in China or Germany might buy loan portfolios from local originators from Arizona to Australia never occurred to anyone, and if it had their directors would have laughed and quickly departed. The process of wrapping masses of such assets into ‘bond like’ contracts which generated set payment streams ‘monetized’ them into a form readily tradable on national and even international capital markets.
This state change in the organizational order of the assets liquified them away from regional consumer and loan investment demand and toward the top of the capital markets where fee-throughput rather than asset retention was the profit opportunity. Volume surged enormously in a few years’ time, flooding global capital markets with dollar denominated security derivatives.
Potentially, liability dilution via asset securitization. Loss risk serves as a repellor which constrains behavior in capital markets. That is, behavioral trajectories veer away from loss potentials, and the more sharply the nearer they are too them. In addition, liability directly subject to litigation also serves as a repellor; prudent firms avoid or at least limit liability, profit potential notwithstanding. The design of asset backed securities has specifically served to dilute the liability of originators, underwriters, security raters, and security trustees from both derivative tranche purchasers and individual mortgage counterparties alike. Originators have the most exposure; many had trivial capital, and have closed their doors. Trustees have little liability so long as they follow the sercuritization contracts; underwriters have practically no liability; security raters will prove almost impossible to sue. The dilution of liability may have allowed the risk tolerance of all parties to this chain to shift signally closer to loss potential. This is a change in a behavioral state rather than in a market-determined quantity, but the possibility that a state change took place in mortgage lending in consequence of liability-shedding is significant.
Derivative swap proliferation. From modest dollar volumes in 2000, derivative swap positions outstanding have soared to double-digit trillions of dollars. Putatively, the positions offset, counterparty risks notwithstanding, with only a small slice turning over at any given time. Too much confidence should not be placed in that best case scenario given that upfront fee generation by swap sellers rather than payout collection by buyers has driven this market, and that obligations are hedged rather than reserved. That said, the offsetting nature of many derivative hedges may be demonstrating near-term stability. Since the onset of the Securitization crisis in July 08, some swap sellers, notably the monolines and some insurance firms, have lost billions, but these numbers are far smaller than the losses of ABS holders. Furthermore, hedge funds have been active swap sellers to raise investment capital for years, despite which few hedge funds to this point have blown up and bankrupted outright. Over all however, this colossal, short onset shift in the volume of claims written into derivative swaps represents an increase in dimension with potential nonlinear changes in the risk exposure of swap contractees whose potential outcomes are impossible to assess given the opacity of the over the counter derivatives trade.
Bifurcation in the asset backed commercial paper markets in August 07. Up to July of 08, banks, their conduits, and other bank-like entities could issue extremely large volumes of very short term debt instruments to fund ongoing investment. In August 08, there was an equilibrium reset in the commercial paper markets after which very few institutions, regardless or size or reputation, could find further buyers for such debt. Full stop. Even funders in quite different industries found high resistance to such debt offerings. No massive debt failure or withdrawal of a market participant precipitated this shift; the transition was collective and diffused. No liquidity efforts by the public authorities since have sufficed to revive this market nine months later. Leaving aside the reasons for this shift, it represents a straightforward bifurcation catastrophe between two stable states in this market: it is now switched off.
Potentially, turbulence in commodity futures markets. From the Autumn of 07, and moreso from January 08, the normal relation of futures prices to market demand has decoupled for many commodities. Some markets such as that in oil have internal price rise trends due to changes in supply-demand equilibrium, though even there price rises seem well above trend. Others such as gold have had clearly speculative inputs. Agricultural markets have seen price spikes inexplicable by petroleum product cost increases or other constraints such that option hedging for producers doesn’t pay, grain elevators won’t commit to prices, and market surplus countries in some grains have embargoed exports of domestic stocks. The trend change is unclear and the time frame is yet too short to suggest the dynamic shape of the disturbance, although hot money flows, whether for hedging other positions or outright speculation may be involved. If the latter case is so, this would be an example of stress in one unstable system—derivative swap markets—rebalancing as stress in a linked and less dampened system. Regardless, this is an example of short onset nonlinear turbulence not obviously derived from endogenous (internal) trends in a system. Turbulence is indicative of pervasive changes in the system involved often preceding a radical state change, in this case the global commodities markets.
For comparison, here are a few speculative examples of larger scale nonlinear dynamic shifts which could occur in the present international financial system:
Default cascade from a derivative swap mismatch. The failure of any one swap contract, even if large, is not inherently a threat. The collective vesting of an entire source class of swaps—risk correlation—or the failure of a major counterparty intermediary—catastrophic loss of dimension—are very real potential concerns. The fact that derivative swaps have been purchased to buffer risk widely in the financial markets suggests that a local failure in the swap markets might entail a wide propagation of sudden risk realization. That’s bad, in case you were wondering.
Flight from the dollar as a reserve currency. The dollar is a very powerful attractor for capital upon which the structure of US financial markets depends. Flight from the dollar is the same systemically as the dampening of this attractor. Capital flows reversing out of the US and demand withering internally would be salient systemic trajectory changes from the decline of the dollar attractor. Since the latter processes are occurring at elevated if not severe levels, we may be within the ‘even horizon’ of such a shift, where small further changes can have disproportionate outcomes. Gently, gently, boys . . . .
Volume public intervention for alternative energy sources. Even though alternative energy sources are affordable in absolute terms, their higher cost relative to carbon sources suppresses the market for them, which in turn makes innovation climb the hill of steeper costs. The installed legacy base of carbon energy technologies is stacked against scale economies for alternatives as well. Whether by raising the cost of carbon, subsidizing that of alternatives, or both in combination, the cost slope relative disadvantages of innovation could be lowered. Nonlinear propagation of emergent technologies can result, i.e. rapid technology regime shift. This would be a change in dimension, potentially leading to a change in state. When alternatives can flow freely the potential for unintended consequences is high, though, as in the example of inappropriate usage of maize corn for ethanol production.
Given the examples listed here, it may seem that nonlinear dynamical changes in the financial system or its sub-markets are wholly undesirable; certainly they are disruptive, and often hazardous. The alternative energy example points out that such shifts can be desirable. For another example, mass childhood inoculation against common infections diseases has yielded systemic changes in survival rates and family relations most see as positive. The mutability and multivariate nature of systemic processes makes intervention difficult regardless. Moreover, systems, well . . . change.
Maintaining stability is not simply or even primarily a process of avoiding ill-conditioned or unstable areas or processes in systems as they are presently organized around known equilibria but more effectively one of monitoring and projecting the influences of changes in system parameters on the system as a whole. In particular, intervention to halt nonlinear shifts already begun is often ineffective or systemically destabilizing. Consider respectively efforts to increase commercial banking liquidity and mass loss transfers through derivative hedging in the present crisis. In my view, interventions to prevent major nonlinear changes in market and financial system states must occur earlier, when observable trajectories or values begin to diverge from established equilibrium ranges and before they are far-from-equilibrium. Velocity, for example, may be a desirable trait, and we may know where some dangerous curves are in a state space, but even so it is much easier to slow or prevent kinetic energy from accumulating in a moving system than to get it out non-catastrophically once it is there. A brief discussion of generic nonlinear dynamics for the financial system follows, with a specific discussion of connectivity, propagation, and intra-systemic stability coming later.
As mentioned, systems viewed as a parameter space with a time dimension—financial corporations, markets, specific financial instruments and their exchange webs, etc.—can change their dimension and their state. An example of a dimensional change might be a primary financial dealer opening or closing a LBO underwriting desk. Dimensional changes can be fractional, however. A good example was the creation of SIV conduits by major commercial banks, enterprises inextricably linked to sponsor banks and which themselves performed formally very similar activities but without the reserves or regulation of the banks. An example of a state change would be a large percentage shift in bank tier capital at any level from wholly owned debt to debt derivatives. Changes can be gradual or sudden; smooth or turbulent.
Systems not only change their internal order (state, dimension, velocity, etc.), they can also and even moreso change their external order. Deposit insurance in the US offers direct protection to small savers, and keeps consumer banks funded; more importantly, it promotes savings in the economy as a whole and largely dissipates bank run pressures within the financial system: the systemic impacts are greater than the local impacts even though the parameter applies at a micro level.
Such mutual modulation of sub- and supra-systems, and for that matter self-modulation within systems has a much underappreciated consequence, especially with regard to financial state spaces: typical probability tails do not follow toward their ends. Normal probability distributions can hold near to known equilibria with clear major parameters. Where dimensions are added to a system, though, or at values further from established equilibria, flat earth discontinuities are very real possibilities from changes in the larger system’s ‘covering order’ which invalidate probability tails assessed near to the equilibrium of a system’s internal order. Such ‘cover set’ changes needn’t even be chaotic to be transformational, such as atmospheric changes at high elevations shifting an ‘athletic state space.’ Assessments of standard deviations from known trajectories or equilibrium states push the firelight of observation out somewhat but do not well predict unsmooth or discontinuous contextual modulation since they only project intra-system order which itself may not hold at distant values. This is less an issue of all systems breaking down at some point than that all organizational stability is local and knowledge drawn from it is yokel; good, near to home; further, there may be dragons.
How can one contain probability error when assessing a systemic trajectory in time, then? One approach is to evaluate a system in relation to its cover set dimension. Projecting a system’s variables and order alone compounds errors of self-reference, a problem that is endemic in short-term financial models, or at least in how they are commonly discussed. It is insufficient to project the probabilities of a system without projecting the probabilities of its context. This is less because the ‘system level’ perspective is wrong, and the covering system context is ‘right’ than because explicitly comparing levels of order adds parallax to analysis and ideally spotlights errors of self-reference or inadequate dimensional reference. As an example, SoCal condos evaluated as a system had ceaseless price appreciation, strong buyer demand, floods of re-fi liquidity, and voracious capital market support. From the higher level perspective of price-to-income and price-to-rent, the SoCal condo system appeared far from long-term equilibria. In fact, capital market demand provided the entire support for the price rise: viewed at that level, the demand vector shows as entirely reversed, with alpha capitalists paying higher and higher prices for future consumer cash flow with SoCal condo buyers catching a tailwind from what was for them an over the horizon silent auction.
Describe the system from the perspective of the parameter space, not the parameter space from the perspective of the system as is so often done. As a very crude analogy, to gauge the SoCal condo market, weigh the mortgage market; for the mortgage market, weigh the capital market; for the capital market, weigh the primary dealer-central bank structure; for the financial structure, weigh currency, production, and trade; for the valuation level, weigh resource supply-demand and national economy liquidity-growth. Perspective one level up is essential; two levels up is advisable. The optimal approach is to take the observed system as a ‘middle level’ and evaluate the immediate higher and lower level systems. So for SoCal condos again, one looks up at the mortgage market, and down at the buyer asset level of income history, debt, regional job market, and so on. This is basic, but it bears repeating given that few in the instance of the SoCal condo market appear to have assessed anything other than their own level. Analysis of system risk and opportunity must be explicitly multi-level to capture mutual modulation between different levels of order. This sounds a deal of work, but without it one is buying blind.
It is well-known that for complex systems small influences can have very large consequences. Indeed, for such contexts the logical operation of large-large often fails for systems, that a large effect necessarily requires a large cause. Small changes can have disproportionate propagation if a system is uncorrelated at the time; these are ‘initial conditions outcomes,’ of which founder effects are an important class. MS-DOS gained a founder effect. Small changes can carry disproportionate weight if they occur near a critical threshold value for a system parameter whether central or not; these are circuit breaker effects. When the Fed by quarter point steps crossed the threshold of negative real rates after 2001, leaving money to sit safe ceased to be a viable money management option, and the mad dash for a few basis points of spread return was on. Small changes can amplify their impact if such changes occur far from systemic equilibrium; these are tipping point effects. Oil at $135 a barrel may be a tipping point for mid-term rebalancing of present energy source distribution.
This is not to say that individual financial markets are especially prone to veering into far-from-equilibrium chaos, or to so-called ‘butterfly effects’ whereby small initial differences of opportunity might compound relentlessly into destabilizing financial hurricanes. Rather, the reverse is observably true: the core equilibria in many financial markets are deep and stable basins of attraction which constrain routine behaviors within close limits and dampen compounding or chaotic trajectories. This is notably true for trading exchanges. Markets do have powerful internal attractors such as interest rates, term obligation, historical price trajectories, known-risk premia, and the like. Markets do have hard parameters such as loan collateral, outside audits, regulatory restraints or firewalls, set and binding contract boilerplate, criminal and community penalties for fraud, and so on. Very few behaviors are as closely observed by as many competent observers as are market and financial interactions, so the general quality of information is not mythical if not perfect.
The greatest stability factors in markets in my view, however, are their buffers against loss, since a certain amount of loss is highly probable. Some buffers are time tested best practices such as margin calls, capital reserves, diversification, and reinsurance to diffuse loss. These have often been strengthened by regulation, and the lender of last resort function and deposit insurance functions are also regulatory buffers. Historically, one of the best factors of market stability is among the least known, that the large volume of market participants relative to the value and volume of the markets makes for many nodes which disperse loss risk widely; if some lose everything, many lose little, and some or many gain. . . .
It is worth noting that over the last fifteen years, virtually every stabilizing factor or function just mentioned has been tampered with in the US financial system. If markets are by design and yet remain relatively stable against external shocks, their inherent vulnerability to internal combustion is only aggravated here by the largely intentional corrosion of essential safeguards.
Assuming (falsely but for the sake of argument) that economic behaviors remain well within the stable basins near to market and system equilibria, there are still three kinds of behavior which by themselves could pose sudden, nonlinear, systemic instabilities. Two strike me as unlikely; one, the reverse. As is well-known, complex systems can generate emergent properties or sub-systems, states of order not projectable from the components or design of the generative system. From my perspective, genuine emergence is quite rare in market behavior; it is difficult to even think of a good example. The yen carry trade due to prolonged and artificially low interest rates in Japan might be a partial instance. One hypothetical reason for a relative lack of emergent behavior in financial markets if that observation holds, aside from the comparative robustness of markets, is that emergent opportunities for profit in the bazaar of ten thousand eyes are quickly spotted and ‘captured’ by existing participants or strategies while ones for loss are swiftly publicized and contained. Emergent behaviors are perhaps ‘re-correlated’ to the financial system before they cohere real teeth and momentum. If, that is, they appear sufficiently different from known behaviors to be distinguished by experienced participants.
It is less remarked on though very common for complex systems to generate resonance effects, system wide oscillations in amplitude, concentration, or both. Particularly in systems with diffusion amongst many nodes, like say markets. The regular, ‘noisy’ wriggles of exchange prices are very probably resonance functions in the main, ditzy ad hoc ‘explanations’ by daily and weekly commentators notwithstanding. Resonance effects in markets are something to which the quant side boys have likely devoted an immoderate store of propellor-head iterations, and if they have found a way to consistently profit from these via ultra large positions they are not about to tell the world. In principal, resonance effects in markets or the financial system could ratchet on themselves to produce destructive swings in amplitude, to ‘frequency cancel’ more normal movements leading to participant misreadings of markets, or move erratically ruining market predicability. In fact, such ‘resonance run’ outcomes seem to be of little effect. One reason perhaps is that market behaviors have their own very well described oscillations over the duration of hours and weeks. Systemic robustness again may capture resonance more than be disrupted by it, and so more a little more forward or back, up or down than otherwise—but in comprehensible directions. Another reason is that markets do not trade 24/7/365;. Markets close nightly at least, whereas futures trading does not have the same effect even as it carries an ‘echo’ of the market to which the futures map. The quickest way to damp resonance is to shut of the machine. This quality should not be underestimated.
Yves here. Even though only a subset of institutional investors, program trading was a big enough resonance effect to cause the 1987 crash. And the remedy? Forced trading halts.
By far the greatest risk to markets and especially to the financial system in my view lies with correlation risks. Discussions of these prompted this series to begin with, and specific kinds of correlation risks will be the focus of the next post. Markets stay near to equilibrium because correlation is dampened by dissimilar behaviors. There are many participants who do many different things, or even do the same thing but in slightly different ways at slightly different times. Indeed, many actions in the financial system are countervailing: somebody buys, somebody sells; somebody loses, somebody gains. If and when they do the same thing or begin to converge in time, systemic equilibrium changes; moreover, the velocity or amplitude of what is correlated can more easily scale in a nonlinear fashion in the process. If everyone in California turns their air conditioning on full at the same time, they’ll all soon be in a sweat.
As one example for now, the earlier hedge funds made boatloads of money by mapping small, diffuse opportunities for high-leverage arbitrage. Once many hedgies began applying the same strategies at the same time, profitability vanished; too many mouths and not enough pie. Hedgies turned their leverage more nearly to old fashioned speculation, with hedge offsets. “We’re well conditioned, right?” However, if they have collectively correlated their hedge offsets they may shift large positions simultaneously to the same point elsewhere in the financial system. It is possible that some of the surge in commodities prices from the middle of 08 is the result of such a correlated bet by hedgies and hedge-like concerns. Correlation is much to be feared, and the financial system as a whole is under-conditioned for it. In part, this is because not only is the financial system quite large in relation to any one investment strategy or position, said system has historically been significantly compartmentalized into discrete investment foci, some of which had very low short-term liquidity. No longer: that’s the problem . . . .
In sum, while most fear sudden turbulent nonlinear changes, these are rare as the inherent ‘stickiness’ of financial behavior near to equilibria and the diffuse nature of market participants makes far-from equilibrium states difficult to achieve for markets specifically, if somewhat less difficult for the financial system as a whole. It is the slow, smooth aggregation of imbalance which kills markets most of the time in my view. Such gradual transitions to disequilibrium take time to balloon their impacts given the inertia embodied in existing market mechanisms. Despite the fact that some such instabilities can be localized in focus, even these may have system-wide impacts, typically through dimension collapse or correlation threshold state changes. Nonlinear events tend to happen only after perfectly observable trajectories extend either well beyond system core equilibrium horizons, after known systemic safeguards are disregarded, or both: it’s the comrade-betrayer not the assassin-stranger that gets us; the swine flu pigeon, not the black swan.
Robert Rosen. 1985. Anticipatory Systems.
Ralph H. Abraham and Christopher D. Shaw. 1992. Dynamics: The Geometry of Behavior.
[Rosen’s study has a terrific idea-to-sentence ratio. Abraham and Shaw’s text promotes visual understanding of primary dynamical processes. Fine websites exist for both bodies of work.]
Government policy also plays a role in promoting such shifts. Take the first point above of asset monetisation – the process by which illiquid assets are made more liquid.
As Minsky argues in his ‘Stabilizing an Unstable economy’,(he does not use the word ‘non linear’ anywhere though) an important role the Fed has played since the 1960s is to widen the range of assets it will accept either directly or indirectly through the discount window.
When a crisis hits and threatens the a number of financial institutions, the Fed has often reacted in this way, offering to accept securities which were behind some of the problems in the first place. Thus the recent actions such as the TAF etc are not something which are substantially different from what the Fed has done in earlier decades and in response to earlier crises.
There is obviously a moral hazard problem here. As the number of securities conferred with ‘legitimacy’ by the Fed expands, market participants are effectively insured to some extent against the losses they would have to take.
Couple this with the argument made by Raghuram Rajan that in todays world, one of the major sources of market alpha is the deliberate creation of more and more illiquid instruments. This is because as market instruments become more liquid, standardised and widely traded, (in part due to fed actions maybe…though rajan doesnt talk about this), banks earn less of a spread from developing and selling them. Thus it is CDOs today. but tomorrow it is something else which is also a radical new ‘financial innovation’ which also of course, happens to be highly illiquid, and highly complex.
This puts in perspective the recommendations made by many to force CDS trading onto exchanges. Once the returns from CDS origination fall,banks and traders will merely look elsewhere for better returns. the problems dont lie with CDS per se.
Yves,please edit this post,the use of “of” for off,the use of “too” for to,and especially the references to events in 2008 which occured in 2007 detract from this interesting piece.
“The installed legacy base of carbon energy technologies is stacked against scale economies for alternatives as well. Whether by raising the cost of carbon, subsidizing that of alternatives, or both in combination, the cost slope relative disadvantages of innovation could be lowered. Nonlinear propagation of emergent technologies can result, i.e. rapid technology regime shift…”
Good points, Richard.
So Moe Gamble, I’m all for it, but care has to be taken. It would be east to push great costs onto existing fixed power producers; an ‘unintended’ but very real consequence. That issue has come up before in comments at NC, but Germany’s present experience with solar power shows that there are losers as well as winners when the cost slope is shifted.
This will be true and more, unintended modulation, regarding remedies to structural problems in the financial system, but there, too, we should try.
A lot of these structured finance products were intentionally designed to be illiquid, as the risk-return concept requires that one must take greater risks to earn greater returns. Lack of liquidity in markets for a particular product is one way of increasing the risk of said product…supplying the lure of reward that can be sold to potential investors. The issue is really how many buyers of these derivatives really knew what they were doing.
So Scott, I question whether what you say was the case, that ABS were _intentionally_ structured to be illiquid. Unless you have specific inside information on this issue. Until the markets for securitized assemblages locked up in July 07, even many of the worst products could be turned over with no more than a trivial discount.
I do believe, now, that the tranching of aggregated mortgage and loan pools was done in a way to intentionally _and artificially_ create spread differentials and seniority tiers in asset classes for which this made no inherent sense. The goal there, to me, seems to be much what you infer, that buyers would flock to artificially induced ‘advantage potentials’ mapped to assets which said buyers otherwise might have stayed away from. Whether ASBs were or were not liquid in a downstream market for acquirers may have been an issue largely of indifference to the underwriting fee gorgers who produced said (in)securities; juicing demand and clearing the stuff off their own books would seem to me to be the principal goals of the underwriters. I don’t know that tranching is an issue of nonlinear dynamics, but I do hope that becomes labeled as a bad business practice for many if not all securitity classes.
To anon of 10:17, you are right, and indeed I saw some of the same glitches—after the post had gone up. *sigh* I’ve always needed an editor, as we see again . . . .