# CVA - Where does the default probability (PD) come from?

Some authors use CDS from the market to derive the implied default probability (from a risk-neutral point of view).

I wonder: how exactly does a CDS reflect counterparty risk?

Let me put an example: let's say I would like to price a derivative instrument (rate) that I am negotiating with counterparty $$A$$. I need to include the CVA (counterparty credit risk). Should I look for the CDS of $$A$$ in the market? I thought CDS reflect the debt issuer default risk, but maybe it also reflects counterparty credit risk.

Can someone clarify this please?

• A good question in fact. Typically standard CDS contracts (when yours truly last traded them) had 3 default triggers: Bankruptcy, Restructuring, Failure to Pay. None of these encompass failure to perform under Swap Obligations. This latter constitutes a '4th trigger', which may be documented for certain bespoke contracts however standard CDS would only be a proxy for this 'event'. Regardless, standard terms implied default probabilities is how CVAs generally got marked, in the absence of any observable market in '4th trigger' CDS. – Mehness Nov 1 '18 at 18:04
• Fyi - a simple derivative like an interest rate swap is not Borrowed Money. This is the standard term for the Obligation Category under which a Credit Event can be triggered (all capitalised terms are legally defined). Borrowed Money is a Bond / Loan, deposit or drawn down portion of a credit facility. Not an interest rate swap. So, as stated, standard terms CDS will not trigger if a counterparty refuses to pay a bank under an IRS, but honours coupons to bondholders. Now there are layers of documentation that mean that the global ISDA might collapse with implications (which haven't thought... – Mehness Nov 3 '18 at 18:10
• ..through yet), but just worth making the point that CVAs are hedged with CDS, however this is under a prevailing assumption, nothing more, that they are a good proxy. This is why as mentioned in comments it is tough to get regulatory recognition for vanilla CDS language-documented hedge trades (at least was when present dude last involved). If someone currently involved can set the record straight on reg treatment of contingent CDS hedge trades would be great! enough spiel I think but like I say these definitional details are very poorly socialised in math finance in general – Mehness Nov 3 '18 at 18:15

"Debt issuer default risk" and "counterparty risk" are very similar. From Risk magazine:

Counterparty Risk

The risk that a counterparty to a transaction or contract will default (fail to perform) on its obligation under the contract. Counterparty risk is not limited to credit risk (the risk that the counterparty cannot fulfill its contractual obligations for payment) but may also result from other problems associated with a counterparty unwilling to honor the contract.

In practice, it is implicitly assumed the material part of CVA can be captured by CDS prices. Quantifying other sources of counterparty risk would be particularly challenging due to the absence of legal precedents (bear in mind that the CDS market is only 25 years old).

Now suppose you have entered into some interest rate derivative contract with your counterparty $$A$$. The contract has maturity $$T$$ and its value at $$t is given by $$V(t)$$. Your exposure $$E(t)$$ at time $$t$$ is equal to the positive value of the contract: $$E(t)=\max(0,V(t))$$ CVA is the price of the risk of default of your counterparty thus you need to look at the future expected exposure under the risk-neutral measure. Enter the risk-neutral discounted expected exposure $$EE^Q(t,u)$$ for a time $$t: $$EE^Q(t,u) = E^Q_t[D(t,u)E(u)]$$ where $$Q$$ is the risk-neutral measure and $$D(t,u)$$ the discount factor from $$u$$ to $$t$$. This corresponds to the risk-neutral value at time $$t$$ of the future value to you of the derivative contract, namely what you stand to lose if your counterparty defaults (or is "unwilling to honor the contract"). The default can occur at any time $$u$$ between $$t$$ and $$T$$, hence CVA for counterparty $$A$$ at time $$t$$ is equal to (assuming a recovery rate $$\text{Rec}$$): $$\text{CVA}_A(t)=(1-\text{Rec})\int_t^TEE^Q(t,u)\color{blue}{\text{d}P_t^Q(u)}$$ where $$\text{d}P_t^Q(u)$$ is the risk-neutral probability of default of $$A$$ on the infinitesimal time interval $$[u,u+\text{d}u]$$, conditional on the current ($$t$$) information.

To obtain estimates of default probabilities, you can extract information from the CDS market for counterparty $$A$$. To understand this, consider the following toy example: assume the risk-free rate is constant equal to $$r$$ and there exists a contract with a $$1$$-year maturity which pays $$1\times(1-\text{Rec})$$ if counterparty $$A$$ defaults within the year in exchange for a CDS premium $$s_A$$. By risk-neutral theory: \begin{align} s_A &= E^Q[e^{-r}(1-\text{Rec})1_{\{A \text{ defaults within 1 year\}}}] \\[3pt] & = e^{-r}(1-\text{Rec})\color{blue}{P^Q(A \text{ defaults within 1 year})} \end{align} Thus: $$\color{blue}{P^Q(A \text{ defaults within 1 year})}=e^r\frac{s_A}{1-\text{Rec}}$$ You observe that CDS premiums encapsulate expected default information. You could also consider that distributional information $$P^Q(\cdot)$$ contained by CDS premiums is not limited to default but includes any failure to pay: this would probably depend on the design of the CDS contract and its payment triggers.

Anyway, in practice to price CVA we only look at the probability information contained by CDSs and assume it captures any material event that might result in no payment.

 In practice you also need to consider which CDS you choose: different CDSs might have different contractual designs and definitions of "default", thus some CDSs might not be relevant to quantify the probability of default/no payment/etc. for a particular derivative contract $$V(t)$$. See comments from @Mehness.

References

Pykhtin, M. and Zhu, S. (2007). "A Guide to Modelling Counterparty Credit Risk", GARP Risk Review.

• Hi - legally it is not necessarily true that an FTP will trigger under a failure to perform under a swap obligation. Market participants for the most part don't really worry about this too much and make the assumption in your final sentence, however for example if trying legally to document a contingent CDS for a bespoke trade, you have to insert additional triggering language. This comment would have been current as of the 2013 ISDA definitions FYI. – Mehness Nov 1 '18 at 18:47
• this gleaned from experience putting together such trades with legal assistance from members of ISDAs Credit Events Determination Committee. However just mentioning for strict correctness, obviously this is less interesting than the pricing above and an annoying detail. Sadly a vanilla CDS is not necessarily a hedge for a counterparty that fails to make a payment under a swap confirm - this may not trigger an Event of Default (don't have the 2013 defs to hand else could point to the relevant clauses). – Mehness Nov 1 '18 at 18:56
• I see, I am not that familiar with the legal details of CDS and how "credit events" are rigorously defined. From a CVA perspective, of course you will extract distributional information from the CDS which is most relevant to the trade you are entering with your counterparty and assume this information is sufficient. This gets all the more convoluted than you sometimes need to price CVA on a netting set basis and the CDS must be representative of all the trades contained in that netting set. – Daneel Olivaw Nov 1 '18 at 19:01
• Sure - which is why no one bothers about this too much - effectively treating standard triggers as subsuming failure to pay a swap coupon stream. EXCEPT when a CVA trader might want to get regulatory capital relief for a counterparty specific derivative MTM-contingent CDS hedge. A vanilla CDS contract against the counterparty, whose payoff is linked to the portfolio of swaps vs that counterparty, may not be seen by a regulator as hedging the cprty risk if it doesn't have the right trigger (even if in this case the payoff is rates/inflation based etc...). So this can be material to a CVA desk. – Mehness Nov 1 '18 at 19:34
• But as I say, these are details that don't matter to much for day to day XVA risk management, it's only if eg you are trying to get Reg Cap recognition for a hedge trade, or if a counterparty stops paying under a swap but the CVA hedge CDSs don't trigger for some arcane lack of completeness in legal terms so you wear the swap MTM loss unhedged, that it matters. At least this was the state of play not super long ago... – Mehness Nov 1 '18 at 19:38