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P2.T6.309. Default correlation, Malz sections 8.1 and 8.2

Kimberly D

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AIMs: Define default correlation for credit portfolios. Identify drawbacks in using the correlation-based credit portfolio framework. Assess the effects of correlation on a credit portfolio and its Credit VaR.



Questions:

309.1. Consider a pair of two speculative credits, rated BB and BB-, with default probabilities respectively of 2.00% and 3.00%. If their joint default probability is 0.40%, which is nearest to the implied default correlation?

a. Zero
b. 0.0830
c. 0.1424
d. 0.3750


309.2. According to Malz, each of the following is an important implication of default correlation in models of portfolio credit risk EXCEPT for:

a. Default correlation is hard to measure or estimate using historical default data
b. Default correlation exhibits too much sway on ("has a tremendous impact on") the credit portfolio's expected loss
c. Default correlations are small in magnitude such that an "optically" small correlation can have a rather large impact
d. The problem created by portfolio with (n) credits which require n*(n-1) pairwise correlations is often solved by assuming all pairwise correlations equal to a single parameter, but that parameter must be non-negative
(Source: Allan Malz, Financial Risk Management: Models, History, and Institutions (Hoboken, NJ: John Wiley & Sons, 2011))


309.3. Assume a portfolio with a total principal value of $1.0 billion divided into n = 20 positions where each position has a default probability of 1.0% and the positions are uncorrelated, as follows:
  • Portfolio value = $1,000,000,000
  • Number of positions (n) = 20; each position is $50.0 million
  • Each position's default probability (PD) = 1.0%
  • Default correlation = zero
  • Credit value at risk (CVaR) confidence level = 95.0%
We are interested in the 95.0% credit value at risk (CVaR) of the portfolio, where 95.0% CVaR = 95% unexpected loss (UL) - expected loss (EL). We can vary the granularity of the portfolio by increasing (n), or we can increase the default correlation, but in either case, we maintain the other assumptions; i.e., ceteris paribus. Each of the following is true EXCEPT for which is not?

a. The initial assumption (above) imply a 95% CVaR of $40.0 million
b. An increase in granularity, from n=20 to n=100 positions, will correspond to a DECREASE in the 95% CVaR
c. As granularity increases to a very large number of independent small positions (n --> ∞), the 95% CVaR tends toward $990.0 million
d. An increase in the default correlation corresponds to an DECREASE in the 95% CVaR

Answers:
 
If I am not mistaken the answers are as follows (I am almost sure about first two and not quite sure about last one):
309.1 c
309.2 b
309.3 c

But I`ve got a question: Malz tells us that as the number of positions increases the number of defaults also increases, but I don`t understand how he finds this number, for instance if we have 50 positions and 0,02 probability of default - the 95 percentile of defaults is equal to 3 - how Malz finds this value? Please explain if you can?
 

David Harper CFA FRM

David Harper CFA FRM
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Hi monte-carlo, It's an (inverse CDF) binomial; i.e., = BINOM.INV(n = 50, pd = 2%, confidence = 95%) = 3.
i.e.,
  • prob [0 defaults per binomial] + prob [1 defaults] + prob [2 defaults] = 92.16%
  • prob [0 defaults per binomial] + prob [1 defaults] + prob [2 defaults] + prob [3 defaults] = 98.22%, such that 95% quantile (function) is 3 defaults
 

ashanks

New Member
Hi David,

I have a doubt about 309.3, even after looking at references that I have access to. Much appreciated if you can give some clarification.
c] seems false, because believe the unexpected loss (UL at given level of significance) tends to expected loss (EL) as n->infinity, in the case that the correlation is zero.

However, to me d] also seems false. EL should be unaffected by correlations, while UL would increase with increasing default probability. So CVar (i.e. UL-EL) should inrease as well.

What am I missing?

Thank you!
 

David Harper CFA FRM

David Harper CFA FRM
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Hi ashanks,

Good point, but it matters (I think) whether the significance level less than or greater than the PD. In this question, 1% PD < 5% significance level. If we have 20 uncorrelated credits, answer (A) gives the 95% CVaR of $40 MM which is due to one (1) default. That is because, if uncorrelated:
  • The prob of zero defaults among 20 = 99%^20 = 81.79%; and
  • The prob of one default among 20 = BINOM.DIST(1,20,1%,false) = 16.52%; i.e., cumulatively that's 81.79% + 16.52% = 98.31%
As the correlation tends toward 1.0 the 81.79% probability increases toward 99.0% (i.e., at perfect correlation), such that as some point the 95% quantile shifts from one default to zero defaults and the CVaR will decrease from $50 - $10 = $40 MM to $0 - $10 = $-10 MM. (granularity is unchanged, ceteris paribus)

Note if PD > significance, then it's totally the other way; for example, if this question asked for a confidence with 99.5%, then 1% PD > 0.5 significance and:
  • n = 20, uncorrelated, 99.5% CVaR = $100 MM UL - $10 EL = $90; i.e., quantile at 2 defaults
  • n = 20, perfect correlation = 1.0, 99.5% CVaR = $1 BB - $10 MM EL = $990 MM.
I hope that explains, thanks,
 
Hi ashanks,

Good point, but it matters (I think) whether the significance level less than or greater than the PD. In this question, 1% PD < 5% significance level. If we have 20 uncorrelated credits, answer (A) gives the 95% CVaR of $40 MM which is due to one (1) default. That is because, if uncorrelated:
  • The prob of zero defaults among 20 = 99%^20 = 81.79%; and
  • The prob of one default among 20 = BINOM.DIST(1,20,1%,false) = 16.52%; i.e., cumulatively that's 81.79% + 16.52% = 98.31%
As the correlation tends toward 1.0 the 81.79% probability increases toward 99.0% (i.e., at perfect correlation), such that as some point the 95% quantile shifts from one default to zero defaults and the CVaR will decrease from $50 - $10 = $40 MM to $0 - $10 = $-10 MM. (granularity is unchanged, ceteris paribus)

Note if PD > significance, then it's totally the other way; for example, if this question asked for a confidence with 99.5%, then 1% PD > 0.5 significance and:
  • n = 20, uncorrelated, 99.5% CVaR = $100 MM UL - $10 EL = $90; i.e., quantile at 2 defaults
  • n = 20, perfect correlation = 1.0, 99.5% CVaR = $1 BB - $10 MM EL = $990 MM.
I hope that explains, thanks,

Hi, this explanation was very helpful. I am missing one portion of the explanation, apologies if implied or written elsewhere.

"As the correlation tends toward 1.0 the 81.79% probability increases toward 99.0% (i.e., at perfect correlation)". Why or how does correlation = 1 imply probability of 0 defaults increases to 99% from 81.79%? Formulaic,

Correlation = 0: The prob of zero defaults among 20 = 99%^20 = 81.79% --> makes sense binomial
Correlation = 1: The prob of zero defaults among 20 = 99% ; can you help to derive this?

Is it because correlation = 1, implies n = 1 for any n and using n = 1 in binomial would result in 0 defaults 99% of the time?
Any help is greatly appreciated. Thanks!
 
Last edited:
n=20
each position default probability = 1%
correlation = 1

if one position defaults (probability of this is 1%) then every other would default as well because they are perfectly correlated

if one position does not default (probability of this is 99%) then every other would survive as well because they are perfectly correlated

just figure this out a moment ago. I guess I'm right :)
 

David Harper CFA FRM

David Harper CFA FRM
Staff member
Subscriber
@irenab Yes, that is correct. In the unrealistic scenario of perfect default correlation, there are only two outcomes: all 20 positions default, or none defaults. In this way, from a default perspective, the portfolio effectively reduces to one credit (as all the positions act the same). Thanks,
 

southeuro

Member
Hi David,

Hope you're doing alright - me panicking a bit with 4 weeks to go and having only (and mostly) covered credit and market risk books.. Am hoping the other books are somewhat less intense..

I understand the following -- please correct me if I make a mistake in those points:

- as PD increases CVAR increases
- as n increases CVAR decreases (because UL "gets erased", and we're left with EL)
- incidentally EL is not affected by granularity or PD
- Comparison of PD with significance level determines the loss in extreme quantile: i.e. if PD > significance level then we're hit with max loss, if PD < significance, no loss (this, given correlation=1 ofc) -- logical so far.

one thing I am having difficulty with:
Malz shows (through a figure) the exact number of defaults as n increases (logically it should increase, but by how much?) -- any way to calculate this for a given nth percentile?
related to this, and with regards to 309.3 (D) you provide a table which includes EL, value per position and UL (obtained by value per position * # of defaults).. Are we supposed to be able to calculate # of defaults somehow for the exam -- otherwise I don't see how we can obtain UL?

many thanks!
 

David Harper CFA FRM

David Harper CFA FRM
Staff member
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Hi @kik92 In the answer thread, I think we've established that my 309.3(D) is incorrect; i.e., it is not (necessarily) a true statement. I haven't revised it yet .... The answer thread has quite a bit of discussion https://www.bionicturtle.com/forum/...on-malz-sections-8-1-and-8-2.6955/#post-34592 e.g.,
Thomas wrote:
I am not quite convinced on 3(D). An increase of correlation appears to imply that the loans start behaving like a less granular portfolio. We have established that a less granular portfolio has a higher CVaR. Apart from the slightly pathological example of pi < confidence level and correlation = 1, the examples also seem to imply that higher correlations increase CVaR?

And I replied:
Hi @Thomas Obitz You are correct! Silly me. Actually, AlokS raised it earlier but I unfairly dismissed it by referencing Malz' "pathological example" and mine is a variation on those conditions: pi < correlation and illustrating with correlation = 1.0. But that's a really bad conclusion. In general, increases in default correlation increase the unexpected loss and, therefore, the CVaR. 309.3.D still needs a revision. Apologies for the incorrect 309.D. It is currently NOT a (necessarily) true statement.
 
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