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# Gregory: Recovery impact on CVA

#### Marco.Musci

##### Member
Hello guys,

I have a doubt about Recovery impact on CVA as explained in Gregory page 142 : "increasing LGD reduces the risk-neutral default probability but increases the loss in the event of default". If mathematically I can understand it, I have some problem to understand the economic logic behind it, for me intuitively an increase of LGD should lead to an increase of the PD.
Can anyone help me to better understand this logic?

#### QuantMan2318

##### Active Member
Subscriber
Hi @Marco.Musci

My intuition is purely theoretical, but let me explain it in how I understood the metric

Gregory states that the implied default probability is provided by the following equation:

$F(u) = 1-\exp(-spread/(1-recovery)*u)$

And as you can observe from the above formula, you can see that as the recovery rate reduces - which implies that the LGD increases - the Probability of default falls as the recovery rate is in the denominator of the negative exp function which in turn is reduced from 1 for the PD

Gregory himself states that when you have a LGD of 50% and PD of 10% which gives rise to EL of 5% and this is equal to having LGD of 100% and PD of 5%, you can see that both the two instances are equal

1. The first Bond has a lower LGD and higher PD to equal 5% EL
2. The second Bond has a higher LGD and lower PD for the same EL

So we are looking at the same metric which are having two parameters and hence one reduces to compensate for the increase in the other

Hope this helps