# CDS - Accumulated Default Risk?

Say I issue insurance contracts covering fire damage in personal households. Fires occur with a probability of $x$% in a household (and they obviously occur independently from one another). If the potential repair after a fire costs $P$ \$, then the fair value of such an insurance contract would be$x\%\cdot P\$.

Now with CDS we can't really apply this principle, because if the reference entity defaults, then everybody that holds a CDS on this entity will get paid out, i.e. the risk accumulates.

Am I right in assuming that this is indeed a real issue? If not, what is wrong with my reasoning. But if so, how is this issue circumvented in practice?

Most, but far from all, companies maintain a relatively steady debt load. When a bond matures, they fund its principal payout with a new bond.

Sometimes companies do take on more and more debt, meaning that CDS protection sold during earlier times of small debt loads becomes more valuable (and underpriced, from the point of view of the protection seller). Each increase in debt load is effectively a change in capital structure, which is an important and under-modeled part of quantitative finance.

Homeowners insurance runs some similar risks. A homeowner might significantly increase the value of his or her possessions while remaining under the same policy.

I originally thought you were asking about a key difference between CDS and fire insurance. Namely, there is not much risk of many homes burning all at once (modulo the occasional wildfire in the western USA). Corporate bankruptcy, in contrast, tends to happen in bunches.

The lack of independence is handled with copula models, where the event probabilities are translated to the probability of some quadrant of multidimensional space. Typically this quadrant is evaluated under gaussian or student-t distributions, where correlation is easy to define and compute.

There has been much debate in recent years about whether these models are truly capable of dealing with small quadrants, which correspond to tail events.

• It doesn't seem to me that the question was about correlation of defaults of multiple entities but about multiple CDS on a single entity. – AFK Aug 11 '14 at 22:43
• I agree, and will add a section on capital structure. – Brian B Aug 12 '14 at 10:58

There is no issue here.

In the world of insurance:

• An actuarial value is placed on the insurance by examining the probability of the fire event based on historically similar events and other input data.
• The event probability (per annum) times the expected size of the loss, plus some extra safety premium and costs, determines the annual premium that must be paid.
• The insurance company builds a reserve against this risk to cover itself in the event that a fire occurs.
• There is a distribution around the mean loss so that the losses in one year may be more than the expected loss. However insurance companies know about the law of large numbers which says that as the number of similar independent contracts written becomes large, deviations from the mean will become less likely. So they seek to write more contracts.
• They will also be required by Insurance Solvency requirements to hold an extra cash buffer against a large deviation in the number of fires.

The world of CDS is quite different:

• Every CDS trade has a buyer and a seller so a CDS moves credit risk - it does not create it
• The price of the CDS is determined by a market not by historical data.
• Market participants do pay attention to the historical performance of similar companies, or companies with similar ratings, but this is just one ingredient out of many inputs.
• Market prices respond to market news about the CDS reference entity
• Most market participants can hedge their position easily
• Most of the largest holders of CDS are dealers who are hedged.
• Those who do not hedge are investors who sell protection in order to earn a risk-adjusted return. They should be able to survive a credit event.

If a reference credit does default, then all contracts linked to that reference entity will be triggered. Many of the market players are hedged and so have no net loss. Those who are not hedged will make a gain or loss depending on their position. However, across the market the summed gains and losses will offset exactly.

For many of these reasons, the worlds of insurance and derivatives are quite different.