Beta distributions
- Thread starter fashepard
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Hi Frank,
Right, CM follows Merton in assuming asset values are normally distributed, and that this random, normal variable can map to credit ratings.
Then the beta distribution is used for recovery/LGD. I would offer here that, IMO, unlike some of the other distributions which are scientifically purposeful (e.g., in Wilmott's extreme value theory, the GPD distribution mathematically follows from a peaks over threshold setup) the beta is "artful." Its application to recovery rates, to my knowledge, has no fundamental justification (I've read studies that try, but nothing that really succeeds). It is used primarily of convenience because (i) you can fit it to various scenarios/instruments with great flexibility, (ii) yet do so with only two params.
Here is previous thread citing CM justification:
"We can capture this wide uncertainty and the general shape of the recovery rate distribution – while staying within the bounds of 0% to 100% – by utilizing a beta distribution. Beta distributions are flexible as to their shape and can be fully specified by stating the desired mean and standard deviation.” (CreditMetrics Technical Doc p 80).”
On the member page, I have an XLS example of beta.
e.g., the green line (alpha = 2, beta = 4) I think would be typical of recovery on a junior debt [PDF peaks at lower recovery) and maybe something like a=2,b=2 for a senior note [PDF peaks at > 50% recovery]
I copy the key points from the prior thread here as, for exam purposes, deep insight into beta is not required:
Key points (for FRM):
* Recovery/LGD are notoriously hard to parameterize; data shows wide dispersion
* Empirically, most LGD models perform poorly
* Beta distribution often used (Portfolio Manager, PRT, CreditMetrics) primarily due to flexibility (takes many shapes) but cannot be binomial (two humped) which are observed
* Kernel modeling is non parameteric alternative
David
Right, CM follows Merton in assuming asset values are normally distributed, and that this random, normal variable can map to credit ratings.
Then the beta distribution is used for recovery/LGD. I would offer here that, IMO, unlike some of the other distributions which are scientifically purposeful (e.g., in Wilmott's extreme value theory, the GPD distribution mathematically follows from a peaks over threshold setup) the beta is "artful." Its application to recovery rates, to my knowledge, has no fundamental justification (I've read studies that try, but nothing that really succeeds). It is used primarily of convenience because (i) you can fit it to various scenarios/instruments with great flexibility, (ii) yet do so with only two params.
Here is previous thread citing CM justification:
"We can capture this wide uncertainty and the general shape of the recovery rate distribution – while staying within the bounds of 0% to 100% – by utilizing a beta distribution. Beta distributions are flexible as to their shape and can be fully specified by stating the desired mean and standard deviation.” (CreditMetrics Technical Doc p 80).”
On the member page, I have an XLS example of beta.
e.g., the green line (alpha = 2, beta = 4) I think would be typical of recovery on a junior debt [PDF peaks at lower recovery) and maybe something like a=2,b=2 for a senior note [PDF peaks at > 50% recovery]
I copy the key points from the prior thread here as, for exam purposes, deep insight into beta is not required:
Key points (for FRM):
* Recovery/LGD are notoriously hard to parameterize; data shows wide dispersion
* Empirically, most LGD models perform poorly
* Beta distribution often used (Portfolio Manager, PRT, CreditMetrics) primarily due to flexibility (takes many shapes) but cannot be binomial (two humped) which are observed
* Kernel modeling is non parameteric alternative
David
sipanivishal
Manager-Corporate Banking
Hi David,
What does a and b here represents,I mean is there any intrepretation to them just like N(d1) and N(d2) has in BS.
Thanks
Sipani
What does a and b here represents,I mean is there any intrepretation to them just like N(d1) and N(d2) has in BS.
Thanks
Sipani
Hi Sipani,
When I was playing with the XLS yesterday, I couldn't discern intuition for a or b. (I too was wondering). There are some rulesets (see under Shapes) but I couldn't glean anything natural, that i would be able to remember, out of them. Looked it up in my references, couldn't find anything.
This is how far i got yesterday:
* a>1 & b>1 make it unimodal; i.e., peak somewhere in the middle, not U-shaped
* mean of beta = a/(a+b)
* CreditMetrics uses unimodal, peak earlier for junior debt than senior debt
* So, if you use the first two rules above, I was able approximate the CreditMetrics distributions with: a>1, b>1 and lower mean for junior and higher mean for senior debt;
e.g., a = 2, beta = 4 implies mean of 2/6. Mean recovery of 33% and that looks near enough to CM's beta for a (junior) subordinated
e.g., a = 2, beta = 1.7 imples mean of 2/(2+1.7) = 55% recovery and that looks close to senior unsecured shape for CM
David
When I was playing with the XLS yesterday, I couldn't discern intuition for a or b. (I too was wondering). There are some rulesets (see under Shapes) but I couldn't glean anything natural, that i would be able to remember, out of them. Looked it up in my references, couldn't find anything.
This is how far i got yesterday:
* a>1 & b>1 make it unimodal; i.e., peak somewhere in the middle, not U-shaped
* mean of beta = a/(a+b)
* CreditMetrics uses unimodal, peak earlier for junior debt than senior debt
* So, if you use the first two rules above, I was able approximate the CreditMetrics distributions with: a>1, b>1 and lower mean for junior and higher mean for senior debt;
e.g., a = 2, beta = 4 implies mean of 2/6. Mean recovery of 33% and that looks near enough to CM's beta for a (junior) subordinated
e.g., a = 2, beta = 1.7 imples mean of 2/(2+1.7) = 55% recovery and that looks close to senior unsecured shape for CM
David
...in case it's interesting, here is the CreditMetrics technical document and their example beta distributions...David
http://learn.bionicturtle.com/images/forum/cm_72.png
http://learn.bionicturtle.com/images/forum/cm_72.png
For today's screencast on beta distribution for LGD, I learned that in the beta distribution:
alpha is the "center" or "steepness of the hump," and
beta is shape or "tail fatness" (source: Michael Ong, Chapter 8).
David
alpha is the "center" or "steepness of the hump," and
beta is shape or "tail fatness" (source: Michael Ong, Chapter 8).
David
best_seller249
New Member
Hi David,
What is the real meaning of creating a beta distribution for recovery rate? I mean that alpha and beta are computed from average recovery rate and standard deviation of the data set and then pdf will be created. We can create a graph which illustrates distribution of recovery rate. However, after all, we only take the mean recovery rate to incorporate into expected loss model right? So apart from visualizing a good graphic, is there any application or intuition behind this consideration? Many thanks
What is the real meaning of creating a beta distribution for recovery rate? I mean that alpha and beta are computed from average recovery rate and standard deviation of the data set and then pdf will be created. We can create a graph which illustrates distribution of recovery rate. However, after all, we only take the mean recovery rate to incorporate into expected loss model right? So apart from visualizing a good graphic, is there any application or intuition behind this consideration? Many thanks
Hi @best_seller249
Not to my knowledge. I only really know what deServigny says in Chapter 4 (previously FRM assigned) about the beta distribution and why it's useful for recovery/LGD:
Not to my knowledge. I only really know what deServigny says in Chapter 4 (previously FRM assigned) about the beta distribution and why it's useful for recovery/LGD:
- Bounded at [0,1] corresponding to 0% to 100% recovery/loss
- Flexible; e.g., can be symmetrical
- Only requires two params ("parsimonious")
- Cannot be bimodal
- Cannot cope with "point masses" at 0 or 100%
hi @David Harper CFA FRM CIPM just wanna check if we are required to know the exponential, beta, gamma and Weibull distribution for Nov14's exam?
Hi @tosuhn
My Part 1 advice here still looks good @ https://forum.bionicturtle.com/threads/p1-focus-review-3rd-of-8-quantitative.6156/
e.g., AIM: Describe the key properties of the following distributions: uniform distribution, Bernoulli distribution, Binomial distribution, Poisson distribution, normal distribution, lognormal distribution, Chi-squared distribution, Student’s t, and F-distributions, and identify common occurrences of each distribution
For Part 2, I'd guess you only care about exponential (AIM: Explain the relationship between exponential and Poisson distributions) and beta (as above, due to its commonality). Weibull has flirted in and out of the syllabus (as a generalized exponential with added functionality) but I don't *think* it's ever appeared. Similarly gamma distribution, I just don't perceive it to be on the "realistic" syllabus, FWIW. I hope that helps!
My Part 1 advice here still looks good @ https://forum.bionicturtle.com/threads/p1-focus-review-3rd-of-8-quantitative.6156/
e.g., AIM: Describe the key properties of the following distributions: uniform distribution, Bernoulli distribution, Binomial distribution, Poisson distribution, normal distribution, lognormal distribution, Chi-squared distribution, Student’s t, and F-distributions, and identify common occurrences of each distribution
For Part 2, I'd guess you only care about exponential (AIM: Explain the relationship between exponential and Poisson distributions) and beta (as above, due to its commonality). Weibull has flirted in and out of the syllabus (as a generalized exponential with added functionality) but I don't *think* it's ever appeared. Similarly gamma distribution, I just don't perceive it to be on the "realistic" syllabus, FWIW. I hope that helps!
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