This post explains the Vasicek/Merton single factor model which is part of the Basel framework (IRB approach) and has been used to evaluate CDOs.
Imagine you loan money to a friend who will default with a probability of 1%. When it comes to paying back the loan you will either receive 100% (plus any interest) or 0%. That's pretty risky. So you figure you can improve your situation by making same size loans to n friends. (NB: You will need a license for doing so). In case they all default individually with a probability of p = 1%, the law of large numbers tells you that the more loans you make (increase n), the closer the average default rate (= portfolio default rate) will be to 1%. "The" central limit theorem states that the portfolio default rate will be normally distributed with a mean of 1% and a variance that goes to zero as n increases.
Now here is the result for n = 5000 loans and 100,000 simulations (portfolios):
Unfortunately, this is only true when default solely happens due to idiosyncratic reason such as illness or a divorce, i.e. when the default of one friend is not related to the default of another friend. But in case you make loans to colleagues from work this assumption won't be correct since bankruptcy of the company would turn most of your loans sour at the same time no matter how large (number of obligors) your loan portfolio actually is. In other words: Certain systemic risk can't be diversified away.
Assuming you really have lots of "similar" friends (p = 1%) and they are all pretty evenly distributed across the sectors the economy has to offer one could argue that defaults are actually only a function of this idiosyncratic risk (as before) and a single systemic risk factor reflecting the overall state of the economy. In case the economy does very well, hardly anybody will default (even an expensive divorce is not an issue) and in case the economy enters into a deep recession, the default rate goes north. The sensitivity of each obligor to this systemic factor and the correlation among the obligors is given by √ρ and ρ, respectively.
Result for n = 5000 loans and 100,000 simulations (portfolios):
The average portfolio default rate is not affected by an increase in correlation, but the higher the correlation the more likely extreme portfolio default rates become (good or bad). In case the correlation is one, we are back to a single obligor. Either nobody (0%) or everybody (100%) defaults.
The model (see comment section for details) is a useful starting point but since both systemic and idiosyncratic risk is assumed to be normally distributed and are connected via an uncertain correlation coefficient you could easily be on the wrong end of the trade. NB: That doesn't mean the Basel guys did a bad job. They had two parameters (correlation and systemic shock size) for calibrating the model.
Imagine you loan money to a friend who will default with a probability of 1%. When it comes to paying back the loan you will either receive 100% (plus any interest) or 0%. That's pretty risky. So you figure you can improve your situation by making same size loans to n friends. (NB: You will need a license for doing so). In case they all default individually with a probability of p = 1%, the law of large numbers tells you that the more loans you make (increase n), the closer the average default rate (= portfolio default rate) will be to 1%. "The" central limit theorem states that the portfolio default rate will be normally distributed with a mean of 1% and a variance that goes to zero as n increases.
Now here is the result for n = 5000 loans and 100,000 simulations (portfolios):
Unfortunately, this is only true when default solely happens due to idiosyncratic reason such as illness or a divorce, i.e. when the default of one friend is not related to the default of another friend. But in case you make loans to colleagues from work this assumption won't be correct since bankruptcy of the company would turn most of your loans sour at the same time no matter how large (number of obligors) your loan portfolio actually is. In other words: Certain systemic risk can't be diversified away.
Assuming you really have lots of "similar" friends (p = 1%) and they are all pretty evenly distributed across the sectors the economy has to offer one could argue that defaults are actually only a function of this idiosyncratic risk (as before) and a single systemic risk factor reflecting the overall state of the economy. In case the economy does very well, hardly anybody will default (even an expensive divorce is not an issue) and in case the economy enters into a deep recession, the default rate goes north. The sensitivity of each obligor to this systemic factor and the correlation among the obligors is given by √ρ and ρ, respectively.
Result for n = 5000 loans and 100,000 simulations (portfolios):
The average portfolio default rate is not affected by an increase in correlation, but the higher the correlation the more likely extreme portfolio default rates become (good or bad). In case the correlation is one, we are back to a single obligor. Either nobody (0%) or everybody (100%) defaults.
The model (see comment section for details) is a useful starting point but since both systemic and idiosyncratic risk is assumed to be normally distributed and are connected via an uncertain correlation coefficient you could easily be on the wrong end of the trade. NB: That doesn't mean the Basel guys did a bad job. They had two parameters (correlation and systemic shock size) for calibrating the model.
Mahalanobis - am 2011-03-03 15:00 - Rubrik: Finance
Mahalanobis meinte am 3. Mar, 15:02:
The Model
Assume every friend i earns a certain return Ri each year. When this return drops below a given cut-off level ci, he will default on the loan, i.e. the respective default probability is:
As we've said before, we want the return to be a function of the overall state of the economy and something that reflects idiosyncratic risk which is independent of the state of the economy.
The overall state of the economy is denoted by Y (= systemic factor) and the idiosyncratic shock by εi. The squared "beta", ρ which lies between 0 and 1, tells you the importance of the systemic factor relative to the idiosyncratic one.
Usually, we don't have any good empirical data to calibrate this model. So we make the following assumption:
- Everybody is exposed to systemic risk to the same extent, i.e. the correlation coefficient ρ is the same for everybody. (A kind of average...)
- Every individual defaults with the same probability pi = p and this unconditional default probability is known to us (i.e. the portfolio has to be somehow homogeneous, like "only BBB corporates")
- Y and εi follow a standard normal distribution.
And that's all we need! Now we can ask: Given that we know the unconditional default probability, p, and the correlation, ρ, how does the distribution of the portfolio default rate look like for a large portfolio (see blog post)?
Regulators are usually looking at this problem from a stress testing point of view. They ask: How many defaults would you have, in case Y turns out to be really bad, e.g. a 1 in a 100 years event (y = qnorm(0.01)= -2.33)?
What's so neat about the Vasicek model is that given Y, R is normally distributed with a variance that goes to zero as n goes to infinity! So given that you know Y and given your portfolio is very large, you can actually calculate your portfolio default rate given the unconditional default rate:
related items:
The Merton Model (Moody's KMV) : Video : David Harper, Bionic Turtle
Source:
Kreditrisikomessung, Henking et al., Springer, 2006
Kreditderivate und Kreditrisikomodelle, Martin et al., vieweg, 2006
* E(Ri) = 0, Var(Ri) = 1, and a linear combination of two independent standard normal distributed random follows a standard normal distributin as well.
Mahalanobis meinte am 3. Mar, 18:07:
R Code
n <- 5000t <- 100000
rho <- 0.3
p <- 0.01
port.def <- array(0,t)
for(i in 0:t)
{
y <- rnorm(1)
e <- rnorm(n)
r <- sqrt(rho)*y + sqrt(1-rho)*e
port.def[i] = length(r[r<qnorm(p)])/n
}
# Plot histogram
library(MASS)
truehist(port.def, xlim=c(0,0.1), xlab="portfolio default rate", main="Vasicek Distribution, p = 0.01 and rho = 0.3")
dsquared (guest) meinte am 4. Mar, 19:12:
Quants vs analysts!
Unfortunately, this is only true when default solely happens due to idiosyncratic reason such as illness or a divorce, i.e. when the default of one friend is not related to the default of another friend.This is why you quants will always need one or two streetwise neanderthals like me about, to remind you that there are all too many cases where the divorce of one set of friends is intimately related to the divorce of another set of friends!
Mahalanobis antwortete am 5. Mar, 21:04:
Set of friends
I've actually thought about that! I work for a pretty small organization but at least twice a year a woman goes on maternity leave because she got pregnant from a co-worker. And homicide statistics tell us that you are most likely to get killed by somebody you know.I often have to think of what a philosophy professor and old friend of mine once told me: You can have as many extramarital affairs as you like. This might even be good for your relationship. But keep one thing in mind: Never ever shit in your own backyard!
The only reason why quants would stop listening to streetwise analysts is that at one point they would have to admit that an educated guess beats a sophisticated model more often than not...
thank (guest) meinte am 21. May, 19:19:
The wig is widespread within the present day style. Therefore, there are many times more in a craze, much less a necessity. A report celebrities and pros work represent unique types transmit hair wigs at random for their interest in the mind of hair trend. The option of sporting a wig specific provision depends on the occasion. Personally, it reflects the taste of the person. Other factors as the availability of many wigs sorting depends on the color of skin with adult men and women and their character outside their preferences effect. Right listed here are some important points about this well known for understanding hairpiece
