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# CVA

#### Kavita.bhangdia

##### Active Member
Why is higher recovery rate means higher implied prob. Of default?
And if that is the case then changes to CVA will be net of increase in probability ofdefault and decrease in loss amount..

So will the final CVA lesser if the recovery amount is increased?

Thanks
Kavita

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#### Kavita.bhangdia

##### Active Member
Thanks Delo but chapter 10 says that recovery rate is negatively correlated with default prob...check page 95, chapter 10 Gregory, David's notes..

Thanks
Kavita

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @Kavita.bhangdia

@Delo 's link is relevant. On the one hand (above presumably, but more exactly Delo's link), Gregory's assertion is based on a narrow, mathematical application of the well-used approximation: λ = S/(1-R); i.e., if we hold the spread, S, constant, then conditional on a constant spread, if we increase recovery, R, then the hazard rate (conditional PD) increases. This is just a mathy way of saying: if the market charges you fixed S for credit risk, but the recovery is/goes higher (compared to baseline), the compensation must be due to increased default probability. Or just: Spread ~= PD*LGD; i.e., credit spread is compensation for PD and LGD. (nevermind this is also just an simplifying approximation!)

Gregory Chapter 10 is (i) an empirical observation and, more importantly, (ii) doesn't say "conditional on constant spread." Gregory Chapter 10:
Recovery values, like default probabilities, tend to show a significant variation over time, as illustrated in Figure 10.10. We can see further variation according to variables such as sector (Table 10.5). Recoveries also tend to be negatively correlated with default rates (e.g., see Hamilton et al., 2001). This negative correlation means that a high default rate will give rise to lower recovery values. Hence, the random nature of default probability and recovery over time coupled to the negative
... we'd expect the higher default rates and correlated lower recovery values to be associated with higher spreads. This is a good illustration of not getting too mesmerized by formulas and the difference between a model, which exist in an unrealistic setting, and empirical observations. I hope it helps, thanks!

#### Matthew Graves

##### Member
Subscriber
This is a classic example of mixing up observations on real-world relationships and variables in a pricing model.

In a pricing scenario, a lot of the data used for pricing instruments (e.g. curves) are themselves implied from observable prices in the market. Hazard rate curves can be obtained from CDS and bond quotes in the market with the addition of a recovery rate. However, the recovery rate is not (usually) directly observable in the market and a value must be assumed. Since the hazard rate curve obtained must re-price the constituent instruments correctly back to the observed market prices, the (assumed) recovery rate and implied PDs are effectively balancing each other. If you increase your assumed recovery rate and re-calculate the curve, the implied PDs from the new Hazard rate curve will also increase to compensate, and hence you still end up with the correct market prices for the curve constituents. The key point here is that the PDs obtained from the hazard rate curve are not necessarily accurate in a real world sense, they are simply pricing variables.

All of this is distinct from trends in real-world PDs which are observed through actual historical defaults on obligations.

#### Matthew Graves

##### Member
Subscriber
Not quite sure what you mean. The market price has baked into it the market expectations of probability of default and recovery.

#### Arka Bose

##### Active Member
Im sorry, my question was why wouldn't the market price adjust to increase in recovery rates. I mean this line of your 'If you increase your assumed recovery rate and re-calculate the curve'.

#### Matthew Graves

##### Member
Subscriber
If the market believes that the recovery rate (not directly observable in general) has increased then that would be reflected in the market prices.

What I'm talking about above is a situation where you're trying to obtain an estimate of the hazard rate curve (possibly for use in another pricing) from observed market prices. You need to provide a recovery rate estimate in order to get the hazard rate curve from the market prices.

Example:
Base case RR = 40%

If we increase the RR estimate to 50% and then re-calculate the hazard rate curve we will find that the PDs obtained from the curve will be higher than the base case.
If we decrease the RR estimate to 30% and then re-calculate the hazard rate curve we will find that the PDs obtained from the curve will be lower than the base case.

This is true because you're using the same observed market prices to calculate the curve in both cases. If you increase the amount recovered, the PD must also increase (and vice versa) in order for the instruments comprising the curve to price back to the observed market prices using the calculated curve and provided RR.

#### Arka Bose

##### Active Member
Thank you, so it is kind of sensitivity analysis rather than the actual mechanism which i was confused about.

#### Biju

##### Member
Hi David,
In the CVA study note it was mentioned that "increasing recovery rate increases the implied default probability but reduces the resulting loss."

Can you clarify why the default probability decreases with Recovery Rate? I thought this should be other way round..Is it because more recovery pressure forces default?

Thanks
Biju

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
@Biju I moved your (very good question which vexed me at first also ) to the previous thread where I think we figured it out; notice the link to where I shared Gregory's XLS that is the basis for his statement http://trtl.bz/0503-gregory-cva-2nd-chap12 You can see above that I believe "This is just a mathy way of saying: if the market charges you fixed S for credit risk, but the recovery is/goes higher (compared to baseline), the compensation must be due to increased default probability. " but I think Matthew Graves also makes a great point. I hope that helps!

#### FrmL2_Aspirant

##### Member
Hi @David Harper CFA FRM , lets say, for CVA calculation of 2 years, we have been provided hazard rate that is constant YOY, then we can simply calculate cva1 -> using [1-e^(1*lambda)] and cva2 -> [1-e^(2*lambda) minus 1-e^(1*lambda)], then CVA as a total of CVA1 and CVA2, is this way correct ?

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#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @FrmL2_Aspirant If lambda, λ, is the hazard rate then 1 - exp(-λ*T) is the cumulative default probability and the difference between the 1- and 2-year cumulative PDs, [1 - exp(-2*T)] - [1 - exp(-1*T)], is the unconditional default probability during the second year (as seen from today). I'm not sure how to connect these to CVA, except that unconditional PD is an input into CVA which is a time-weighted combination of EE*PD*LGD and it therefore a sophisticated time-average variation of the more familiar EL = exposure*PD*LGD. CVA is the subject of assigned Gregory 12. I hope that helps!

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