A Conceptual Framework For Analyzing Systemic Risk

The Cart Before The Horse

There has been a lot of chatter about the systemic risks posed by derivatives, particularly credit default swaps. Rather than offer any formal method of evaluating an enormously complicated question, pundits wield exclamation points and false inferences to distract from the glaring holes in their logic. Below I will not offer any definite answers to any questions about the systemic risks posed by derivatives. Rather, I will describe what I think is a reasonable and useful framework for analyzing systemic risks posed by derivatives. Unfortunately for some, this will involve the use of mathematics. And while the math used is fairly elementary, the concepts are not. This is especially true of the last section. That said, even if you do not fully understand the entirety of this article, one thing should be clear: questions about systemic risk are complex and anyone who gives declarative answers to such questions is almost certain to have no idea what they are talking about.

Risk Magnification And Syndication

As discussed here, derivatives operate by creating and allocating risks that did not exist before the two parties entered into the transaction. That is an unavoidable fact. Moreover, there is no physical limit to the notional amount of any given contract or the number of derivative contracts that parties can enter into. It is entirely up to them. That said, derivatives can be used to negate risks that parties were already exposed to in exchange for assuming other risks, thereby acting as a risk-switching/risk-transferring device. So, a corollary of these observations is that derivatives could be used to create unlimited amounts of risk but through that risk creation they could be used to negate an unlimited amount of risk that parties are already exposed to and thereby effectively “transfer” an unlimited amount of risk to those willing to be exposed to it.

Practically speaking, there is a limit to the amount of risk that can be created using derivatives. This limit exists for a very simple reason: the contracts are voluntary, and so if no one is willing to be exposed to a particular risk, it will not be created and assigned through a derivative. Like most market participants, derivatives traders are not in engaged in an altruistic endeavor. As a result, we should not expect them to engage in activities that they don’t expect to be profitable. Therefore, we can be reasonably certain that the derivatives market will create only as much risk as its participants expect to be profitable. Whether their expectations are correct is an entirely different matter, and any criticism on that front is not unique to derivatives traders. Rather, the problem of flawed expectations permeates all of human decision making.

Even if we ignore the practical limits to the creation of risk, derivatives allow for unlimited syndication of risk. That is, there is no smallest unit of risk that can be transferred. Consequently, any fixed amount of risk can be syndicated out to an arbitrarily large number of parties, thereby minimizing the probability that any individual market participant will fail as a result of that risk.

Finally, we should ask ourselves, what does the term systemic risk even mean? The only thing it can mean in the context of derivatives is that the obligations created by two parties will have an effect on at least one other third party. So, even assuming that derivatives create such a “problem,” how is this “problem” any different than that created by a landlord who plans to pay a contractor with the rent he receives from his tenants? It is not.

A Closer Look At Risk

As stated here, my own view is that risk is a concept that has two components: (i) the occurrence of an event and (ii) a magnitude associated with that event. This allows us to ask two questions: What is the probability of the event occurring? And if it occurs, what is the expected value of its associated magnitude? We say that P is exposed to a given risk if P expects to incur a gain/loss if the risk-event occurs. As is evident, under this rubric, that whole conversation above was grossly imprecise. But that’s ok. Its import is clear enough. From here on, however, we will tolerate no such imprecision.

Identifying And Defining Risks

Using the definition above, let’s try to define one of the risks that all parties who sold protection on ABC’s series I bonds through a CDS that calls for physical delivery are exposed to. This will allow us to begin to understand the systemic risk that such credit default swaps create. There is no hard rule about how to go about doing this. If we do a poor job of identifying and defining the relevant risks, we will have a poor understanding of those relevant risks. However, common sense tells us that any protection seller’s risk exposure is going to have something to do with triggering a payout under a CDS. So, let’s define the risk-event as any default on ABC series I bonds. For simplicities sake, let’s limit our definition of default to ABC’s failure to pay interest or principle. So, our risk-event is: ABC fails to pay interest or principle on any of its bonds. But what is our risk-magnitude? Since we are trying to define a risk that protection sellers are exposed to, our associated magnitude should be the basis upon which all payments by protection sellers are made. So, we will define the risk-magnitude as M=1 - \frac{P_d}{P} where P_d is the price of an ABC series I bond after the risk-event (default) occurs and P is the par value of an ABC series I bond. For example, if ABC’s series I bonds are trading at 30 cents on the dollar after default, M = .7 and a protection seller would have to payout 70 cents for every dollar of notional amount. The amount that bonds trade at after a default is called the recovery value.

One Man’s Garbage Is Another Man’s Glory

When one party to a derivative makes a payment, the other receives it. That seems simple enough. But it follows that if we consider only those payments made under the derivative contract itself, the net position of the two parties is unchanged over the life of the agreement. That is, derivatives create zero-sum games and simply shift and reallocate money that already existed between the two parties. So in continuing with our example above, it follows that we’ve also defined a risk that buyers of protection on ABC series I bonds are exposed to. However, protection buyers have positive exposure to that risk. That is, if ABC defaults, protection buyers receive money.

Exposure To Risk And Settlement Flow Analysis

If our concept of exposure is to have any real economic significance, it must take into account the concept of netting. So, we define the exposure of P_i to the risk-event defined above as the product of (i) the net notional amount of all credit default swaps naming ABC series I bonds as a reference obligation to which P_i is a counterparty, which we will call N_i, and (ii) M. The net notional amount is simply the difference between the total notional amount of protection bought and the total notional amount of protection sold by P_i. So, if P_i is a net seller of protection, N_i will be negative and therefore its exposure, N_i \cdot M, will be either negative or zero.

Because the payments between the two counterparties of each derivative net to zero, it follows that the sum of all net notional amounts is always zero. That is, if there are k market participants, \sum_{i=1}^kN_i = 0. The total notional amount of the entire market is given by N_T = \frac{1}{2} \sum_{i=1}^k|N_i|. This is the figure that is most often reported by the media. As is evident, it is impossible to determine the economic significance of this number without first knowing the structure of the market. That is, we must know how much is owed and to whom. However, after we have this information, we can choose different recovery values and then calculate each party’s exposure. This would enable us to determine how much cash each participant would have to set aside for a default at various recovery values (simply calculate each party’s exposure at the various recovery values).

Let’s consider a concrete example. In the diagram below, an edge coming from a participant represents protection sold by that participant and consequently an incoming edge represents protection bought by that participant. The amounts written beside these edges represent the notional amount of protection bought/sold. The amounts written beside the nodes represent the net notional amounts.


In the example above, D is a dealer and his net notional amount is zero, and therefore his exposure to the risk-event is 0 \cdot M = 0 . As is evident, we can vary the recovery value to determine what each market participant’s exposure would be in that case. We could then consider other risk-events that occur in conjunction with any given risk-event. For example, we could consider the conjunctive risk-event “ABC defaults and B fails to pay under any CDS” (in which case D’s exposure would not be zero) or any other variation that addresses meaningful concerns. For now, we will focus on our single event risk for explanatory purposes. But even if we restrict ourselves to single event risks, there’s more to a CDS than just default. Collateral will move through the above system dynamically throughout the lives of the contracts. In order to understand how we can analyze the systemic risks posed by the dynamic shifting of collateral, we must first examine what it is that causes collateral to be posted under a CDS.

We’re In The Money

CDS contracts come in and out of the money to a party based on the price of protection. If a party is out of money, the typical market practice is to require that party to post collateral. For example, if I bought protection at a price of 50bp, and suddenly the price jumps to 100bp, I’m in the money and my counterparty is out of the money. Thus, my counterparty will be required to post collateral. We can view the price of protection as providing an implied probability of default. Exactly how this is done is not important. But it should be clear that there is a connection between the cost of protecting debt and the probability of default on that debt (the higher the probability the higher the cost). Thus, as the implied probability of default changes over the life of the agreement, collateral will change hands.

Collateral Flow Analysis

In the previous sections, we assumed that the risk-event was certain to occur and then calculated the exposures based on an assumed recovery value. So, in effect, we were asking “what happens when parties settle their contracts at a given recovery value?” But what if we want to consider what happens before any default actually occurs? That is, what if we want to consider “what happens if the probability of default is p?” Because collateral will be posted as the price of protection changes over the life of the agreement and the price of protection provides an implied probability of default, it follows that the answer to this question should have something to do with the flow of collateral.

Continuing with the ABC bond example above, we can examine how collateral will move through the system by asking two questions: (i) what is the implied probability of the risk-event (ABC’s default) occurring and (ii) what is the expected value of the risk-magnitude (the basis upon which collateral payments are made). As discussed above, the implied probability of default will change over the life of the agreement, which will in turn affect the flow of collateral in the system. Since our goal in this section is to test the system’s behavior at different implied probabilities of default, the expected value of our risk-magnitude should be a function of an assumed implied probability of default. So, we define the expected value of our risk-magnitude as M_e = p^* \cdot M where p^* is our assumed implied probability of default and M is defined as it is above. It follows that this analysis will break CDS contracts into categories according to the price at which they were entered into. That is, you can’t ask how much something changed without first knowing what it was to begin with.

Assume that P_i entered into CDS contracts at m_i different prices. For example, he entered into four contracts at 20 bp and eight contracts at 50bp, and no others. In this case, m_i = 2. For each P_i, assign an arbitrary ordering, (c_{i,1}, ... , c_{i,m_i}), to the sets of contracts that were entered into at different prices by P_i. In the example where m_i = 2, we could let c_{i,1} be the set of eight contracts entered into at 50bp and let c_{i,2} be the set of four contracts entered into at 20 bp. Each of these sets will have a net notional amount and an implied probability of default (since each is categorized by price). Define n_{i,j} as the net notional amount of the contracts in c_{i,j} and p_{i,j} as the implied probability of default of the contracts in c_{i,j} for each 1 \leq j \leq m_i. We define the expected exposure of P_i as:

EX_i = M_e \cdot \sum_{j = 1}^{m_i}\left(\frac{p^* - p_{i,j}}{1 - p_{i,j}} \cdot n_{i,j}\right) .

Note that when p^* = 1,

EX_i = M \cdot \sum_{j = 1}^{m_i}\left(\frac{1 - p_{i,j}}{1 - p_{i,j}} \cdot n_{i,j}\right) = M \cdot N_i .

That is, this is a generalized version of the settlement analysis above, and when we assume that default is certain, collateral flow analysis reduces to settlement flow analysis.

So What Does That Awful Formula Tell Us?

A participant’s expected exposure is a reasonable estimate for the amount of collateral that will be posted or received by that participant at an assumed implied probability of default. The exact amount of collateral that will be posted or received under any contract will be determined by the terms of that contract. As a result, our model is approximate and not exact. However, the direction that collateral moves in our model is exact. That is, if a party’s expected exposure is negative, it will not receive collateral, and if it is positive, it will not post collateral. It also shows that even if a party is completely hedged in the event of a default, it is possible that it is not completely hedged to posting collateral. That is, even if it bought and sold the same notional amount of protection, it could have done so at different prices.


20 thoughts on “A Conceptual Framework For Analyzing Systemic Risk

  1. “Finally, we should ask ourselves, what does the term systemic risk even mean? The only thing it can mean in the context of derivatives is that the obligations created by two parties will have an effect on at least one other third party.”

    I think that’s a bit too broad. I like Arnold Kling’s definition.


    It’s not just that it affects at least one other party. With systemic risk the effect has a self-sustaining component. If every single swap affected at least one other party but tended to dampen without affecting any others, we wouldn’t have systemic risk.

  2. Hi Joe,

    You’ve incorporated a time component which is not present in my definition. If you take a closer look at the article, I think you’ll find that I allude to repeated trials, e.g., Default + B fails to pay out. That would have a self-replicating effect. I just wanted to build a foundational definition.

  3. Hi Charles,

    I find your post incomplete, because you don’t draw any connection between posting collateral and systemic risk. Does posting collateral increase or decrease the systemic risk of CDS?

    With all the recent complaints about mark-to-market accounting and the possibility that current market losses will never be realized, it seems obvious that collateral requirements of CDS are capable of forcing firms into a position of illiquidity and then bankruptcy — even though the CDS they write never realize a loss. This would imply that from a systemic point of view capital requirements are a better way of dealing with contracts like CDS than collateral, because capital requirements are less likely to precipitate a liquidity crisis for a firm — and if CDS can be moved into the held to maturity section of the balance sheet, the firm is entirely protected from bankruptcy triggered by a mark to market loss on CDS that is never realized.

    I guess my question is: Is there an implicit efficient markets assumption underlying the collateral posting regime that is currently in place for CDS? Maybe systemic risk should be analyzed under two assumptions: if the efficient markets assumption is correct and if it is not.

  4. Hi ACC,

    “I find your post incomplete, because you don’t draw any connection between posting collateral and systemic risk. Does posting collateral increase or decrease the systemic risk of CDS?”

    I stated at the outset and the title indicates that this is a conceptual framework for answering such questions. There is no way of answering your question without first creating such a framework and then doing empirical research. I have only done the former since I have done no empirical research, therefore I cannot come to any conclusions.

    Capital requirements are a terrible idea. Collateral already fills the role. The only difference between collateral and capital requirements is that the amount of capital that will be set aside in the latter case will be dictated by a regulator that is not as familiar with the market as they entities they will be regulating.

    As for you last comment, you’ve put the cart before the horse. First you examine the amounts of collateral the market requires. Then you can ask whether or not is efficient.

  5. Hi Charles,

    What makes your post incomplete is that you don’t propose hypotheses that connect your collateral model to systemic risk. So it’s not clear to me what hypotheses you want an empiricist to use your model of collateral to test.

    Your claim that collateral fills the same role as capital requirements is hard for me to understand. Capital requirements would have to be met at the moment that a counterparty entered into a CDS and would not necessarily change with the market price of the CDS, thus they are not in any way equivalent.

    It’s not clear to me why you are assuming that once a careful analysis of systemic risk is complete, capital requirements can not possibly play any stabilizing role. One potential hypothesis is that capital requirements support a collateral regime by giving firms reason to believe that any regulated counterparty would have the wherewithal to post collateral.

    To be honest, the fact that you are so sure about the conclusions of your research (“Capital requirements are a terrible idea.”) throws into doubt the very premise of your project.

  6. “Every swap dealer wants to get paid. Do you really think they need the help of a regulator in making sure that they do in fact get paid? Of course not. It’s a ridiculous notion.”

    Charles, recent historical experience proves demonstrably that this view is false — unless, of course, the dealers all owe Lehman’s estate money — which would be a very interesting state of affairs in and of itself given that they precipitated Lehman’s liquidity crisis.

  7. And I should add that AIG is a better example of the failure of self-regulation than Lehman (not to mention Bear Stearns). I’m beginning to wonder whether even the US government can afford to meet AIG’s collateral requirements.

  8. ACC,

    I don’t state a hypothesis because that is what an empiricist would do. I’ve given an example of a methodology that could be used to test a hypothesis. An empiricist could ask, “given the market structure I have determined that exists, how would it respond to a given assumed probability of default?”

    Capital requirements exist to ensure payment. Collateral exists to ensure payment. Thus, they serve equivalent functions. However, like you pointed out, capital requirements have no connection to the expected value of the payment. As such, they are a hamfisted regulatory version of collateral.

    You are conflating two sets of questions: those about systemic risk and collateral; and those about the superiority of collateral to capital requirements. My position on the latter has never changed. Capital requirements imposed on swap dealers make no sense and will drain liquidity in the market.

    Every swap dealer wants to get paid. Do you really think they need the help of a regulator in making sure that they do in fact get paid? Of course not. It’s a ridiculous notion.

  9. Acc,

    Recent events provide no evidence that capital requirements would be superior to collateral requirements. First, the CDS market has handled itself brilliantly despite all the turmoil. Second, as a conceptual matter, even if the CDS market hadn’t done so, as stated above, capital requirements and collateral are different methods of ensuring payment and so any problems that capital requirements would address have already been addressed by collateral. (In my opinion it is the collateral feature that has allowed the CDS market to function well because payments are made gradually over the life of the agreement, which mitigates losses in the event of a counterparty’s failure.)

    Third, who are the notable failures in the CDS market? The monolines, which were already subject to capital requirements. So, if recent events suggest anything, it is that capital requirements are not as effective as collateral, which is to be expected since collateral is rooted in the economic reality of a transaction whereas capital requirements are imposed by fiat.

    Finally, you boldly conclude without data or theory that the CDS market “precipitated” Lehmans’ liquidity crisis.

  10. From my Mac’s built-in dictionary:

    Systemic: of or relating to a system, esp. as opposed to a particular part : the disease is localized rather than systemic.

    When we refer to the capital markets as a system, how do we define the scope / boundary of the system?

    In the case of mortgage backed securities, the system extends into the high street including personal financial products.

    The major banks (I assume) have a ‘risk’ system which encompasses their entire business, i.e. all products. Each of these pools of risk don’t represent the ‘system’ though, only a portion of the whole.

    My perspective would be that the ‘system’ is so large that we humans would find it hard to model as an entire system without an effort to include nearly all finanical activity in the world into one database, and therefore be a near impossible task.

    The regulatory approach is to divide up the major market areas and put some constraints in place to limit the potential damage in that area, even if these constraints may or may not form part of an integrated whole model covering the entire “system”.

    Whether collateral, capital or CCPs solve a problem in one market area, doesn’t make me sleep well at night due to the size and scale of the capital markets.

    Given the recent defaults, the processing of the credit events using the DTCC Trade Warehouse has been almost completely automated, so post-default processing is well automated.

    The existance of a CCP for CDS trades is likely to bring a new cost of OTC trading, that of the collateral/margin posting vs. a release of regulatory capital due to the CCP interposing itself.

    Which will be higher? Will this be a net change in cost to trade CDS contracts? I don’t think anyone really knows.

  11. First of all, I am a complete novice who is trying to wrap his mind around these issues. So, please be gentle.

    Your definition of risk seems to be limited when assessing systemic risk. It does not take into account the ability of individuals to absorb losses above certain magnitudes. For instance, consider stock portfolios designated for retirement by (A) a 30 year old and (B) a 60 year old. If a 30 year old takes a 50% hit, he has ample time to recove. If a 60 year takes a 50% hit, he has absorbed a worse insult to his system. While each party has taken the same hit, the retirement plan of the 60 year old has “blown up”.This does not seem to be reflected in your diagrams.

    Your article and the subsequent comments seem to be discussing leveraging of these risks in terms of capitalization and collaterization. I again apologize for my ignorance, but it seems as though many companies have leveraged their risks via derivatives. This has caused them to absorb insults that have “blown up” their systems.

    You acknowledge some level of interdependence with the ananlogy of of a developer relying on rent money to pay for a construction project. The difference is that those business models seem to be fairly well understood. The developer probably ran a credit check on his tennant and require first and last month rent up-front. I am not sure that people doing business with corporation, such as auto companies, understood to what extent tose companies relied on derivatives, or maybe leveraged deratives.

    I hope am making sense.

  12. Hi Catfish,

    You said that my definition of risk “does not take into account the ability of individuals to absorb losses above certain magnitudes.” Of course it does. First you calculate someone’s exposure, and then you ask if they have enough cash to cover it. That’s the whole point.

    Not sure what you’re getting at with your second comment.

    As for your third, a swap dealer will certainly look to the credit quality of his counterparty. If it’s dodgy, he’ll ask for collateral. As for other people doing business with companies that used derivatives, not sure what you’re basing that on or where you’re going with it. Do you have an article?

    You made sense, no worries.

  13. I suggest that you read the paper on low probability events by Melvin Hinich in the journal Macroeconomic Dynamics. Low probability states of nature can not be estimated from past data.

  14. i think its a bit of an elementary conclusion… i more robust treatment would involve discussion of how we should conceptually handel the fact that exposure as a function becomes piecewise or non-diff. BEFORE bankrupcy becomes an absolute certainty

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  18. Hi,

    Although I would agree with the gist of a lot of what you are saying I would take issue with the idea that the problems of derivatives are just like those of a landlord paying a contractor with rent. As I see it the important point is that the majority of derivaitves are traded by banks (+other financial institutions) – no point arguing about abstract ideas such as the neutrality of the instrument itself – they are designed by and traded by banks and that is innescapable. Derivatives plus banks are a systemic risk because banks have an order of magnitude of liabilities completely different to landlords. Banks in fact hold only promises to pay financed overwhelmingly by other promises – so widespread that the very meaning of money enters the equation. Money is not an objective thing – cash is not objective. As individuals for example we hold promises to pay from banks (current and savings accounts) which can be converted into promises to pay from the central bank (notes and coins). If one bank makes a loss (even if systematically offset elsewhere) large enough to force it to default on its liabilities then there is systemic risk. Bank defaults hit systems harder exactly because the structure relies on trust (a much used word at the moment without I feel enoguh thought as to what it really means) which takes time (and e.g. repeated transactions) to build. Put simply once savings of firms and individuals are lost they can’t be simply magically replaced.

    All that said I agree that there is a lot of noise still blaming derivatives for all manner of ills which has simply not been thought through. Equally I agree the CDS market did what it was supposed to do and that the spectre of counterparty risk to e.g. a Lehmans is overstated. The markets got spooked after LEH not because of the chain of CP risk (I think the economist put the loss per bank in the region of 200mUSD – leaving 10s billions losses due too something else) but because Bear got rescued, LEH didn’t and suddenly no-one knew who was too big to fail and who wasn’t. Trust in the last resort evaporated. The danger of course with the ‘derivatives are evil’ hysteria is new regulation which takes us in exactly the wrong direction because of this.

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