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# Impact of Default Correlations on EL

#### VinayB

##### New Member
@David Harper CFA FRM - Hi David : While I understand that EL is linear and additive and is not impacted by Default correlations, Jonathan Golin's Chapter1 Second Edition, Page 24 mentions that "On a portfolio basis, a fifth variable, correlation between credit exposures within a credit portfolio, will also affect expected loss". How do I reconcile this statement with my understanding above of EL being not dependent on correlations.

Regards,
Vinay

#### bpdulog

##### Active Member
@David Harper CFA FRM - Hi David : While I understand that EL is linear and additive and is not impacted by Default correlations, Jonathan Golin's Chapter1 Second Edition, Page 24 mentions that "On a portfolio basis, a fifth variable, correlation between credit exposures within a credit portfolio, will also affect expected loss". How do I reconcile this statement with my understanding above of EL being not dependent on correlations.

Regards,
Vinay
Most likely you will be tested on EL for a single credit and not a portfolio of credits.

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @VinayB Great observation, I agree with you. Golin's statement here is not supported by the FRM (e.g., Ong Chapter 6). Correlation between PD and EAD impacts EL (because PD*EAD*LGD = EL is multiplicative), but correlations between exposures do not impact portfolio EL (because EL_1 + EL_2 + ... + EL_n = EL_portfolio is linearly additive).

@Nicole Seaman Because Golin is currently assigned, can I trouble you to reach out to the author(s) and let them know we think they have a mistake, I don't see an errata (http://www.wiley.com/WileyCDA/WileyTitle/productCd-0470821574.html)?

This must be the basis for the previous question at https://www.bionicturtle.com/forum/...tutions-and-sovereigns-golin.7648/#post-42758 ie..,
Hi @Arsalan Amin Do you happen to recall where your read that correlations impact EL, because I think it might be a misunderstanding? Default correlation does impact unexpected loss (UL) but does not impact expected loss (EL). EL is a mean, while UL is the function of variance, so another way to look at this is that default correlation does not impact the mean. Here is a simple example: two six-sided dice. As random variables, their expected value (the mean) is 7.0. This is true whether they are independent--as they generally are--or correlated. Consider a farcical situation where we glued the two dice together, so their correlation is perfect (i.e., possible rolls are 1-1, 2-2, 3-3, 4-4, 5-5, and 6-6). Compare the two scenarios (independent dice versus perfectly correlated dice)
Two independent 6-sided dice: mean = 7.0, variance = 5.83 = 2.92*2 and standard deviation = sqrt(5.83) = 2.42
Two perfectly correlated 6-sided dice: mean = 7.0, variance = 11.67 and standard deviation = sqrt(11.67) = 3.42; i.e., when the correlation goes from zero to 1.0, the variance doubles while the mean is unchanged.
This is similar with credits where default is modeled as a Bernoulli (i.e., either survive = 0 or default = 1). Using Malz' example (P2. Topic 6), imagine we have a $1.0 million portfolio with 100 equally-sized obligors (i.e., 100 obligors *$10,000 per). Further, let's say they all have PD = 2.0%. Finally, assume zero recovery. If they are independent (implies zero default correlation), the portfolio's expected loss = 2% * 100 * $10,000 =$20,000. As we increase default correlation, this expected loss is unaffected. Even with perfect default correlation, the portfolio EL is still \$20,000. However, as default correlation increases, unexpected loss does increases.

Finally, formula-wise, EL = PD * EAD * LGD, and portfolio EL is the weighted sum of individual EL. Default correlation does not enter the EL formulas, whereas it does show up in Portfolio UL. I hope that helps!

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#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
@Nicole Seaman Thanks (for liking) here is the full reference for Golin which we believe contains a mistake (the red statement simply isn't correct, or requires some qualification):

"[Chapter 2, 2nd edition:] Application of the Concept: To summarize, expected loss is fundamentally dependent upon four variables, with the period often assumed to be one year for the purposes of comparison and analysis. On a portfolio basis, a fifth variable, correlation between credit exposures within a credit portfolio, will also affect expected loss. The PD/ LGD/ EAD concept just described is extremely valuable as a way to understand and model credit risk. It will be developed at length in subsequent chapters. "-- Golin, Jonathan; Delhaise, Philippe. The Bank Credit Analysis Handbook: A Guide for Analysts, Bankers and Investors (Wiley Finance) (p. 24). Wiley. Kindle Edition.

#### Nicole Seaman

Staff member
Subscriber
@Nicole Seaman Thanks (for liking) here is the full reference for Golin which we believe contains a mistake (the red statement simply isn't correct, or requires some qualification):

"[Chapter 2, 2nd edition:] Application of the Concept: To summarize, expected loss is fundamentally dependent upon four variables, with the period often assumed to be one year for the purposes of comparison and analysis. On a portfolio basis, a fifth variable, correlation between credit exposures within a credit portfolio, will also affect expected loss. The PD/ LGD/ EAD concept just described is extremely valuable as a way to understand and model credit risk. It will be developed at length in subsequent chapters. "-- Golin, Jonathan; Delhaise, Philippe. The Bank Credit Analysis Handbook: A Guide for Analysts, Bankers and Investors (Wiley Finance) (p. 24). Wiley. Kindle Edition.

Nicole

#### VinayB

##### New Member
Thank you @David and Nicole!

Did we hear back from the author on this topic?

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @VinayB I see in our project manager that Nicole reached out to him way back on July 17th via linkedin but I don't see a reply (it must be a mistake, btw, correlation doesn't impact EL in the credit portfolio). Thanks,