The Reality Of Mortgage Modification

Also published on the Atlantic Monthly’s Business Channel.

Why A Decline In Home Prices Should Not Cause Defaults

It seems that we have taken as an axiom the idea that if the price of a home drops below the face value of the mortgage, the borrower will default on the mortgage. That sounds like a good rule, since it’s got prices dropping and people defaulting at the same time, so there’s a certain intuitive appeal to it. But in reality, it makes no sense. Either the borrower can afford the mortgage based on her income alone or not.  However, it does make sense if you also assume that the borrower intended to access the equity in her home before the maturity of the mortgage. That is, the home owner bought the home with the intention of either i) selling the home for a profit before maturity or ii) refinancing the mortgage at a higher principle amount.

If neither of these are true, then why would a homeowner default simply because the home they lived in dropped in value? She wouldn’t. She might be irritated that she paid too much for a home. Additionally, she might experience a diminution in her perception of her own wealth, which may change her consumption habits. But the fact remains that at the time of purchase, she thought her home was worth X. And she agreed to a clearly defined schedule of monthly payments over the life of the mortgage assuming a price of X. The fact that the value of her home suddenly drops below X has no impact on her ability to pay, unless she planned to access equity in the home to satisfy her payment obligations.  Annoyed as she might be, she could continue to make her mortgage payments as promised.  Thus, those mortgages which default due to a drop in home prices are the result of a failed attempt to access equity in the home, otherwise known as failed speculation.

In short, if a home drops in value, it does not affect the cash flows of the occupants so long as no one plans to access equity in the home. And so, the ability of a household to pay a mortgage is unaffected in that situation. This is in contrast to being fired, having a primary earner die, or divorce. These events have a direct impact on the ability of a household to pay its mortgage.

I am unaware of any proposal to date which offers assistance to households in need under such circumstances.

The Dismal Science Of Mortgage Modification

Simply put, available evidence suggests that mortgage modifications do not work.

[IMAGES REMOVED BY UST; SEE REPORT LINK BELOW]

The charts above are from a study conducted by the Office Of the Comptroller of the Currency. The full text is available here. As the charts above demonstrate, within 8 months, just under 60% of modified mortgages redefault. That is, the borrowers default under the modified agreement. If we look only at Subprime mortgages, just over 65% of modified mortgages redefault within 8 months. This may come as a surprise to some. But in my mind, it reaffirms the theory that many borrowers bought homes relying on their ability to i) sell the home for a profit or ii) refinance their mortgage. That is, it reaffirms the theory that many borrowers were unable to afford the homes they bought using their income alone, and were actually speculating that the value of their home would increase.

Morally Hazardous And Theoretically Dubious

Why should mortgages be adjusted at all? Well, one obvious reason to modify is that the terms of the mortgages are somehow unfair. That’s a fine reason. But when did they become unfair? Were they unfair from the outset? That seems unlikely given that both the borrower and the lender voluntarily agree to the terms of a mortgage. Although people like to fuss about option arm mortgages and the like, the reality is, it’s not that hard for a borrower to understand that her payments will increase at some point in the future. Either she can afford the increased payments or not. This will be clear from the outset of the mortgage.

So, it doesn’t seem like there’s much of a case for unfairness at the outset of the agreement. Well then, did the mortgage become unfair? Maybe. If so, since the terms didn’t change, it must be because the home dropped in value and therefore the borrower is now paying above the market price for the home. That does sound unfortunate. But who should bear the loss? Should the bank? The tax payer? How about the borrower? Well, the borrower explicitly agreed to bear the loss when she agreed to repay a fixed amount of money. That is, the borrower promised “to pay back X plus interest within 30 years.” This is in contrast to “I promise to pay back X plus interest within 30 years, unless the price of my home drops below X, in which case we’ll work something out.” Both are fine agreements. But the former is what borrowers actually agree to.

Not enforcing voluntary agreements leads to uncertainty. Uncertainty leads to inefficiency. This is because those who have agreements outstanding or would like to enter into other agreements cannot rely on the terms of those agreements. And so the value of such agreements decreases and the whole purpose of contracting is defeated. In a less abstract sense, uncertainty creates an environment in which it is impossible to plan and conduct business. As a result, this type of regulatory behavior undermines the availability of credit.

But even if we do not accept that voluntary agreements should be enforced for reasons of efficiency, mortgages represent some of the most clear and unambiguous promises to repay an obligation imaginable. The fact that a borrower was betting that home prices would rise should not excuse them from their obligations. There are some situations where human decency and compassion could justify a readjustment of terms and socializing the resultant losses. For example, the death of a primary earner or an act of war or terrorism. But making a bad guess about future home prices is not an act that warrants anyone’s sympathy, let alone the socialization of the losses that follow.

The Elephant In The Room

This notion that Subprime borrowers were victimized as a result of some fraudulent wizardry perpetuated by Wall Street is utter nonsense. Whether securitized assets performed as promised to investors is Wall Street’s problem. Whether people pay their mortgages falls squarely on the shoulder of the borrower. Despite this, we are spending billions of public dollars, at a time when money is scarce and desperately needed, on a program that i) is demonstrably ineffective at achieving its stated goals (helping homeowners avoid foreclosure) and ii) rewards poor decision making and imprudent borrowing. Given the gravity of the moment, a greater failure is difficult to imagine. But then again, we live in uncertain times, so my imagination might prove inadequate.

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Credit Default Swaps And Mortgage Backed Securities

Like Your Grandsire In Alibaster

In this article, I will apply my usual dispassionate analysis to the role that credit default swaps play in the world of Mortgage Backed Securities (MBSs). We will take a brief look at the interactions between the issuance of mortgages, MBSs, and how the concept of loss plays out in the context of derivatives and mortgages. Then we will explore how the expectations of the parties to a lender/borrower relationship differ from that of a protection seller/buyer relationship and how credit default swaps, by allowing markets to express a negative view of mortgage default risk, facilitate price correction and mitigate net losses. This is done by applying the concepts in my previous article, The Demand For Risk And A Macroeconomic Theory of Credit Default Swaps: Part 2, to the context of credit default swaps on MBSs. This article can be considered a more concrete application of the concepts in that article, which will hopefully clear up some of the confusion in that article’s comment section.

The Path Of Funds In the MBS Market

Mortgage backed securities allow investors to gain exposure to the housing market by taking on credit risk linked to a pool of mortgages. Although the underlying mortgages are originated by banks, the existence of investor demand for MBSs allows the originators to effectively pass the mortgages off to the investors and pocket a fee. Thus, the greater the demand for MBSs, the greater the total value of mortgages that originators will issue and ultimately pass off to investors. So, the originators might front the money for the mortgages in many cases, but the effective path of funds is from the investors, to the originators, and onto the borrower. As a result, investors in MBSs are the effective lenders in this arrangement, since they bear the credit risk of the mortgages.

This market structure also has an effect on the interest rates charged on the underlying mortgages. As investor demand for MBSs increases, the amount of cash available for mortgages will increase, pushing the interest rates charged on the underlying mortgages down as originators compete for borrowers.

Loss In The Context Of Derivatives And Mortgages

I often note that derivatives cannot create net losses in an economy. That is, they simply transfer money between two parties. If one party loses X, the other gains X, so the net loss between the two parties is zero. For more on this, go here. This is not the case with a mortgage. The lender gives money to the borrower, who then spends this money on a home. Assume that a lender and borrower entered into a mortgage and that before maturity the value of the home falls, prompting the borrower to default on its mortgage. Further assume that the lender forecloses on the property, selling it at a loss. Since the buyer receives none of the foreclosure proceeds, the buyer can be viewed as either neutral or incurring a loss, since at least some of the borrower’s mortgage payments went towards equity ownership and not just occupancy. It follows that there is a loss to the lender and either no change in or a loss to the borrower and therefore a net loss. This demonstrates what we have all recently learned: poorly underwritten mortgages can create net losses.

Net Losses And Efficiency

You can argue that even in the case that both parties to an agreement incur losses, the net loss to the economy is zero, since the cash transferred under the agreement was not destroyed but merely moved through the economy to market participants that are not a party to the agreement. That is, if you expand the number of parties to a sufficient degree, all transactions will net to zero. While this must be the case, it misses an essential point: I am using net losses to bilateral agreements as a proxy for inefficient allocation of capital. That is, both parties expected to benefit from the agreement, yet both lost money, which implies that neither benefited from the agreement. For example, in the case of a mortgage, the borrower expects to pay off the mortgage but benefit from the use and eventual ownership or sale of the home. The lender expects to profit from the interest paid on the mortgage. When both of these expectations fail, I take this as implying that the initial agreement was an inefficient allocation of capital. This might not always be the case and depends on how you define efficiency. But as a general rule, it is my opinion that net losses to a bilateral agreement are a reasonable proxy for inefficient allocation of capital.

Expectations Of Lender/Borrower vs. Protection Seller/Buyer

As mentioned above, under a mortgage, the lender expects to benefit from the interest paid on the mortgage while the borrower expects to benefit from the use and eventual ownership or sale of the home. Implicit in the expectations of both parties is that the mortgage will be repaid. Economically, the lender is long on the mortgage. That is, the lender gains if the mortgage is fully repaid. Although application of the concepts of long and short to the borrower’s position is awkward at best, the borrower is certainly not short on the mortgage. That is, in general, the borrower does not gain if he fails to repay the mortgage. He might however mitigate his losses by defaulting and declaring bankruptcy. That said, the takeaway is that both the lender and the borrower expect the mortgage to be repaid. So, if we consider only lenders and borrowers, there are no participants with a true short position in the market. Thus, price, which in this case is an interest rate, will be determined by participants with similar positive expectations and incentives. Anyone with a negative view of the market has no role to play and therefore no effect on price.

This is not the case with credit default swaps (CDSs) referencing MBSs. In such a CDS, the protection seller is long on the MBS and therefore long on the underlying mortgages, and the protection buyer is short. That is, if the MBS pays out, the protection seller gains on the swap; and if the MBS defaults, the protection buyer gains on the swap. Thus, through the CDS, the two parties express opposing expectations of the performance of the MBS. Thus, the CDS market provides an opportunity to express a negative view of mortgage default risk.

The Effect Of Synthetic Instruments On “Real” Instruments

As mentioned above, the CDS market provides a method of shorting MBSs. But how does that effect the price of MBSs and ultimately interest rates? As described here, the cash flows of any bond, including MBSs, can be synthesized using Treasuries and CDSs. Using this technique, a fully funded synthetic bond consists of the long end of a CDS and a Treasury. The spread that the synthetic instrument pays over the risk free rate is determined by the price of protection that the CDS pays the investor (who in this case is the protection seller). One consequence of this is that there are opportunities for arbitrage between the market for real bonds and CDSs if the two markets don’t reach an equilibrium, removing any opportunity for arbitrage. Because this opportunity for arbitrage is rather obvious, we assume that it cannot persist. That is, as the price of protection on MBSs increases, the spread over the risk free rate paid by MBSs should widen, and visa versa. Thus, as the demand for protection on MBSs increases, we would expect the interest rates paid by MBSs to increase, thereby increasing the interest rates on mortgages. Thus, those with a negative view of MBS default risk can raise the cost of funds on mortgages by buying protection through CDSs on MBSs, thereby inadvertently “correcting” what they view as underpriced default risk.

In addition to the no-obvious-arbitrage argument outlined above, we can consider how the existence of synthetic MBSs affects the supply of comparable investments, and thereby interest rates. As mentioned above, any MBS can be synthesized using CDSs and Treasuries (when the synthetic MBS is unfunded or partially funded, it consists of CDSs and other investments, not Treasuries). Thus, investors will have a choice between investing in real MBSs or synthetic MBSs. And as explained above, the price of each should come to an equilibrium that excludes any opportunity for obvious arbitrage between the two investments. Thus, we would expect at least some investors to be indifferent between the two.

Depending on whether the synthetics are fully funded or not, the principle investment will go to the Treasuries market or back into the capital markets respectively. Note that synthetic MBSs can exist only when there is a protection buyer for the CDS that comprises part of the synthetic. That is, only when interest rates on MBSs drop low enough, along with the price of protection on MBSs, will protection buyers enter CDS contracts. So when protection buyers think that interest rates on MBSs are too low to reflect the actual probability of default, their desire to profit from this will facilitate the issuance of synthetic MBSs, thereby diverting cash from the mortgage market and into either Treasuries or other areas of the capital markets. Thus, the existence of CDSs operates as a safety valve on the issuance of MBSs. When interest rates sink too low, synthetics will be issued, diverting cash away from the mortgage market.

Synthetic CDOs, Ratings, And Super Senior Tranches: Part 2

Bait And Switch

My apologies, but this is going to be a three part article.  I have come to the conclusion that each topic warrants separate treatment. In this article, I will discuss the rating of CDO tranches. In the next, I will discuss the rating of Synthetic CDOs and those illusive “Super Senior” tranches.

Portfolio Loss Versus Tranche Loss

In the previous article, we discussed how rating agencies model the expected losses on the portfolio of bonds underlying a CDO. The end result was a chart that plotted losses against a scale of probabilities. This chart purports to answer the question, “how likely is it that the portfolio will lose more than X?” So if our CDO has a single tranche, that is if the payment waterfall simply passes the cash flows onto investors, then this chart would presumably contain all the information we need about the default risks associated with the CDO. But payment waterfalls can be used to distribute default risk differently among different tranches. So, if our CDO has multiple tranches, then we need to know the payment priorities of each tranche before we can make any statements about the expected losses of any tranche. After we know the payment priorities, we will return to our chart and rate the tranches.

Subordination And Default Risk

Payment waterfalls can be used to distribute default risk among different tranches by imposing payment priorities on cash flows. But in the absence of payment priorities, cash flows are shared equally among investors. For example, if each of 10 investors had equal claims on an investment that generated $500, each investor would receive $50. Assuming each made the same initial investment, each would have equal gains/losses. However, by subordinating the rights of certain investors to others, we can insulate the senior investors. For example, continuing with our 10 investors, assume there are 2 tranches, A and B, where the A notes are paid only the first $500 generated by the investment and the B notes are paid the remainder. Assume that 5 investors hold A notes and that 5 investors hold B notes. If the investment generates only $500, the A investors will receive $100 each while the B investors will receive nothing. If however the investment generates $1,500 the A investors will receive $100 each and the B investors will receive $200 each. This is just one example. In reality, the payment waterfall can assign cash flows under any set of rules that the investors will agree to.

If the investment in the previous example is a portfolio of bonds with an expected total return of $1,000, then the payment waterfall insulates the A investors against the first $500 of loss. That is, even if the portfolio loses $500, the A investors will be fully paid. So, the net effect of the payment waterfall is to shift a fixed amount of default risk to the B investors.

Rating CDO Tranches

As a general rule, rating agencies define their various gradations of quality according to the probability of full payment of principal and interest as promised under the bonds. Assume that Joe’s Rating Agency defines their rating system as follows:

AAA rated bonds have at least a 99% probability of full payment of principal and interest;

AA rated bonds have at least a 95% probability of full payment of principal and interest;

A rated bonds have at least a 90% probability of full payment of principal and interest; and

Any bonds with less than a 90% probability of principal and interest are “Sub Investment Grade (SIG).”

Assume that the bonds underlying our CDO collectively promise to pay a total of $100 million in principal and interest over the life of the bonds. For simplicity’s sake, assume that the CDO investors will receive only one payment at maturity. Further, assume that we have conducted several hundred thousand simulations for our CDO and constructed the chart below:

It follows from the data in the chart that the probability that losses on the CDO will be less than or equal to: $35 million is 90%; $40 million is 95%; $65 million is 99%. We define the tranches as follows: tranche A is paid the lesser of (i) $35 million and (ii) the total return on the CDO pool (the green tranche);  tranche B is paid the lesser of (i) $25 million and (ii) the total return on the CDO pool less any amounts paid to tranche A (the yellow tranche); tranche C is paid the lesser of (i) $5 million and (ii) the total return on the CDO pool less any amounts paid to tranches A and B (the blue tranche); and tranche D is paid the lesser of (i) $35 million and (ii) the total return on the CDO pool less any amounts paid to tranches A, B, and C (the red tranche).

After some thought, you should realize that, according to Joe’s Ratings, tranche A is AAA; tranche B is AA; tranche C is A; and tranche D is SIG.

Synthetic CDOs, Ratings, And Super Senior Tranches: Part 1

Super Senioritis

I’ve been perusing the finance blogs lately and I’ve noticed a recent obsession with Synthetic CDOs, specifically the super senior tranches of these transactions. And so I felt it was necessary for me to chime in on the debate, by applying my usual toast-dry analysis to Synthetic CDOs for the second time. This is a huge topic that requires consideration of how Synthetic CDOs function, how they’re rated, and how tranches distribute risk among investors. As a result, I’ve decided to break the article into two parts. This first part deals with the basics of rating the assets contained in CDOs. The next will examine the application of ratings to tranches of CDOs, how that translates into the world of synthetic CDOs, and ultimately, culminate in a discussion of what are known as “super senior tranches.”

Required Reading

You are likely to struggle greatly with this article unless you have some familiarity with Synthetic CDOs. And because I am an unabashed self-promoter, I highly recommend you read my introductory article on Synthetic CDOs and my article on Tranches. If you’re going to read only one, then read the one on tranches.

Tranches And Structured Products

Payment waterfalls allow the risks of an investment to be allocated among different groups of investors, or tranches. For example, in the case of Mortgage Backed Securities, a fixed amount of prepayment risk can be allocated to one tranche by tailoring the rules in the payment waterfall to pass all prepayments of principal to that tranche. But there are risks beyond prepayment risk. One obvious example is default risk. In the MBS world, this is the risk that because of defaults in the underlying mortgages the cash flows from the mortgages backing the notes will be inadequate to make payments on those notes. Obviously, default risk will be a primary concern of any investor. The risk that you will not get paid is arguably paramount to all others. So, payment waterfalls have been developed to address this risk and tailor the distribution of default risk in a way that allows each investor to assume a desired default risk level. But before we can understand how investors distinguish between these different levels of default risk, we must understand how rating systems describe these different levels.

Rating Systems And Rating Agencies

You have undoubtedly heard terms such as “AAA rated” and “AA rated” thrown somewhere near names like S&P and Moody’s. It’s not necessary to become familiar with the peculiarities of each rating agency’s system to appreciate the general idea: the ranking of default risk. That is, the market wants a short-hand system that both describes the probability of default for a particular financial product and can be compared across a disparate class of financial products. So, rating agencies developed models and systems of ratings (using confusingly similar labels like “AAA,” “Baa,” etc.) that purport to do just that.

How CDO Ratings Work

Part 1: Past Performance And Correlation

The models that rating agencies use to produce their ratings are backward looking. That is, the first step in the process is to accumulate data about how financial products have behaved in the past. Rating agencies, and investors, will look to the past and produce charts like this:

They will note that in the past, of all bonds that Moody’s deemed Aaa, less than 1% of such bonds defaulted within 10 years of issuance. People then assume, quite reasonably, that this data provides probabilities of default across time for the various ratings. That is, they assume that if we wish to know the probability that a B rated bond will default in year three, we simply look to the above chart and discover that it is .1977 or 19.77%. Examination of this assumption is beyond the scope of this article. But for a great article on that topic (containing the above table and more!) go here.

A CDO is in essence a portfolio of bonds. So in order to model the cash flows of the portfolio, rating agencies turn to charts like the one above and examine the past performance of bonds similar to those in the portfolio. They also look at the correlation of default between the bonds in the CDO portfolio. Correlation, in this context, is a very precise term. And it’s impossible to do justice to the concept in a few sentences. That said, in layman’s terms, when considering the correlation of default between two bonds, rating agencies are looking for a connection between the bonds defaulting. That is, if bond 1 defaults, how does that change our expectation of the probability that bond 2 will default? Exactly how this is done is also well beyond the scope of this article. For those of you who are interested, you can read all about this and more here.

Part 2: Scenario Analysis

So after we have all of our data, we can begin to construct a chart of how likely a given level of loss is. This is done through scenario analysis. That is, the models are run hundreds of thousands of times (and possibly more) using different inputs. In each of these simulations, some bonds might “default.” That is, the model predicts that given a particular set of inputs, certain bonds will default. After each of these simulations, an amount of loss will be calculated, which is based on the estimated recovery values for the bonds in the pool that “defaulted” during that particular simulation. We can then ask, out of all of the simulations, how many times did the loss go above X? So if we ran our simulation 500,000 times, and if the loss was greater than $1 million in only 5,000 of these simulations, then we could say that the probability of the loss being greater than $1 million is .01, or 1%.

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 where is the price of an ABC series I bond after the risk-event (default) occurs and 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, 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 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 is a counterparty, which we will call , and (ii) . 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 . So, if is a net seller of protection, will be negative and therefore its exposure, , 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 market participants, . The total notional amount of the entire market is given by . 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 . 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 ?” 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 where is our assumed implied probability of default and 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 entered into CDS contracts at different prices. For example, he entered into four contracts at 20 bp and eight contracts at 50bp, and no others. In this case, . For each , assign an arbitrary ordering, , to the sets of contracts that were entered into at different prices by . In the example where , we could let be the set of eight contracts entered into at 50bp and let 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 as the net notional amount of the contracts in and as the implied probability of default of the contracts in for each . We define the expected exposure of as:

.

Note that when ,

.

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.