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4

It is actually that you forgot your $1 - R$ in formula (2) :) The index survival curve is defined similarly to the tranche's : $Q\left(t\right) = 1 - \mathbb{E} \left[L\left(t\right)\right] = 1 - \left(1 - R\right)\mathbb{P}\left(\tau < t\right)$. Hence, your formula for the 0-100 tranche survival curve does coincide with the index'. That history of loss ...

2

I assume that you calculate ECL in the context of IFRS9 -correct? market practice often follows the following appraoch: estimate a TTC PD/LGD (TTC = through the cycle). This corresponds to your lifetime estimate (e.g. one marginal PD value for each year of the life of your exposure) in the average of the economic cycle. But for IFRS9 provisioning you have ...

1

I think what you are calling marginal PD is simply the intra year PD. PD usually refers to the 1-year default probability, so if the default time is denoted by $\tau$ then $$PD = \Bbb P (\tau \leq 1 \ \text{year}).$$ What you are refering to as marginal PD is the probability that you default within a shorter period of time, e.g. one month ($n = 12$) or ...

1

Failing an analytic answer I would use Newton's method as a quick and dirty numerical iterator: You can rearrange to: $$(1-\rho) \left (\Phi^{-1}(LGD.DR) - \Phi^{-1}(DR) \right ) + \Phi^{-1}(PD) - \Phi^{-1}(PD.LGD) = 0$$ which given you have fixed DR, LGD and $\rho$ is essentially: $$K + \Phi^{-1}(PD) - \Phi^{-1}(PD.LGD) = f(PD) = 0$$ Newton's ...

1

You are building a model - the question you are asking is a trade off between accuracy and complexity. If the accuracy only improves in a minor capacity and the extension is considered complex you can ask the question "is it really necessary"? If the accuracy greatly improves then whether the extension is considered complex or not I suspect that for ...

1

There is a bit of stuff here so I will incrementally add details if there are any questions. The analysis is pretty much done as in Appendix K from Hull, J. (2012). Risk management and financial institutions,+ Web Site (Vol. 733). John Wiley & Sons. Python imports: import numpy as np import pandas as pd from scipy.optimize import brentq TABLE ...

1

Yes you can use implied default intensities to price Bonds if you have a quoted CDS for the issuer of the Bond or for an issuer with roughly the same characteristics even tough that's not the best thing to do. You can also avoid the whole CDS dilema with a repo on the security itself, and in this case counterparty risk is fully taken by the repoer... @Edit:...

1

All that you have highlighted is well posed. It makes sense to use conditional PD values applied to the remaining balance. In fact, the expected loss is indeed a measure used to understand how much exposure one has to the default of the counterparty. Therefore, if a counterparty has paid back a part of the debt, then the only default risk exposure is on ...

1

EAD can also be higher than credit limit because of adding the costs of collection activities, noting that these can take a long time. As you mention, the credit limits will tend to have been maxed out, but also missed payments and accruing interest may have increased the exposure before the point in time at which the default is established (definitional ...

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