Credit Part A

[humheehum]

Hi David,

Could you please explain the derivation of the Unexpected Loss formula - I simply don't understand where all the terms are coming from.

My guess is that this is based on the technical definition of UL i.e 1 s.d. event, so we are applying the variance formula to the terminal portfolio value. So E(V_h) = V - AE*LGD*PD, but I can't get beyond this.

Another confusion looking at the formula - why is EDF being multiplied by sigma LGD instead of sigma EDF.

Thanks
Ashim

 

[David Harper CFA FRM]

Hi Ashim,

I can't offer an intuition for the UL formula, sorry. Ong gives the derivation in Appendix A. You are correct in regards to the starting idea, it is variance of asset. I thought about trying to screencast this, but frankly, it will not be tested and it requires preliminary calculus to introduce it....Appendix A will not be comprehensible to many, I expect, but it is no matter: we need to understand the concept of UL (this is where is would focus; e.g., volatility of asset, deconstructs into RC) and the final formula only...David

 

[humheehum]

Thanks for the reply David. I will learn the formula.

Ashim

 

[humheehum]

Hi David,

On the screen cast on basket of default swaps (Hull) could you please explain why an increase in correlation leads to a lower value for a 1st or a 2nd to default.

I would think that if we have two situations- a 1st to default and 99th to default, then if correlation is zero between defaults- there would be a fair chance that the 1st to default would be triggered hence the CDS/value would be high, while for the 99th to default the CDS would be low as it unlikely to get 99 independent defaults.

If correlation increases then there would be no impact on the first to default, as given a large basket, you might still expect a default (however if i understood the screencast correctly - i think it implied that the default prob decreases). For the 99th event the CDS would increase as there now a higher cumulative prob of default.

Taken to an extreme if correlation is 1, then there is no difference between 1st to default and 99th to default?

Thanks for your help.

Regards,
Ashim

 

[humheehum]

Hi David - another question.

For the CDS valuation example, I don't see why do we have the accrual part - is it because the seller can invest the premium at the risk free rate? and why is that we have mid year time periods for the accrual, with the first accrual starting at 0.5 before the 1st premium at time period 1.

Thanks again
Ashim

 

[David Harper CFA FRM]

Hi Ashim,

In regard to default correlation, please see this thread.

To your example, 100 credits and they are independent (rho = 0), then the probability of a single default = 1 - p^100. So, I agree with you, with correlation = 0, the odds of a 1st-to-default getting triggered are very very high.

Now raise correlation to 1.0, as you say, the probability of a single default = 1-p. For a very large basket, as the correlation goes from 0 to 1.0, the probability of a single default decreases dramatically. For the junior tranches (or 1st,2nd,3rd-to default), increase in correlation decreases the probability of trigger (basket default) which lowers the spread (and makes the CDS less expensive).

So, I only disagree with your characterization of the junior (1st to default) but i agree with your characterization of the senior.

"Taken to an extreme if correlation is 1, then there is no difference between 1st to default and 99th to default?"
Yes, I think this may be the best way to look at this because, if we think about the shape of the distribution, as correlation increases, it is converging to match the bimodal "pattern" of a single obligor (e.g., 5% = 0, 95% probability = 1).

In regard to CDS valuation:

"Why is that we have mid year time periods for the accrual, with the first accrual starting at 0.5 before the 1st premium at time period 1."

Hull makes a simplyifying assumption to assume defaults at mid-year. An actual CDS valuation model would be more granular. This is purely to setup a problem that can be displayed easily as a spreadsheet!

Given his assumption of default only on June 30th, the accrual table assumes 0.5 of the annual CDS premium (the insurance premium) will be owed by the protection buyer to the seller. So, the 0.5 is part of the same simplifying assumption; I don't think we can read anything more into it...

"I don’t see why do we have the accrual part" - IMO, this is the hard part. Because we start with the natural idea of setting the PV of certain payments to equal the the PV of contingent payoff. So, we expect to PV only two cash flow series. The reason that we need to add the accrual is: to PV the cash flows only for the survival scenario is to include less than all (< 100%) of the outcomes. We PV the cash flows and probability-weight them, so we have to include all possible outcomes.

The survival payments are the likely payments, but there is the small chance of default (which is a mutually exclusive event), in which case, there will be in addition to the prior survival payments, an accrual payment.

If it helps, there is simplicity an unseen part to the other leg, too. What i mean is:

PV (payments by protection buyer) = PV (payoff by protection seller)
PV (probability-adjusted payments if survives + probability-adjusted payment if default) = PV (payoff if survives + payoff if defaults)
Note how the above has four terms where all mutually-exclusive outcomes are getting counted.


But there is no payoff if no default, so: PV(payoff if survives) = 0. And we are left with:

PV (probability-adjusted payments if survives + probability-adjusted payment if default) = PV (payoff if survives + payoff if defaults)

which equals

PV (payments + accruals) = PV (payoff)

David

 

[humheehum]

David - Many thanks for the explanation.

Some bits on Credit are still unclear, I will go through the notes and come back with any questions.

Regards,
Ashim