Credit Risk and Concentration I am working with a UK credit-union and we are looking to build a model to assess our credit risk and changes to this over time. We have a number of loans to borrowers who each have a credit rating (lets say these are banded A, AA, AAA, B, BBB, etc...).  
The way I have proceeded so far is to get the history of defaults in each rating band and build a conjugate model (Gamma/Poisson) to measure the risk profile of each band. I can then use the expected default and variance to build up some measure of expected loss, risk profiles, confidence levels, etc.... 
The loans are to individual members and as the credit union is not tied to a specific organisation/industry I have treated the individual loans as independent and identically distributed with no strong, identifiable correlation. 
One factor that worries me is that some members might have abnormally higher loan balances than others. Say, for example I have 300 loans in category B that average £500, but I have 15 loans in category AAA that average £10,000. The exposure in each category is the same, but as the average loan per borrower is higher in one category, should I recognize in my model that a few defaults in this category will have a larger financial effect than a higher number of defaults in the other category? If so, how can I model this? Is a simple Poisson model acceptable for this?
I guess I'm asking how I should treat concentration factors when I'm currently assessing the risk as if its homogeneous with regard the borrowers and the amounts the borrow. 
Any help would be very welcome!
 A: In regulatory environment there are actually three parameters tied to credit risk: 
1) Exposure at default (EAD) which means nominal amount of money your institution is at risk to lose. You could add unused credit limits /lines there if you want. 
2) Loss given default (LGD) which means 1-recovery rate. Recovery rate depends on collateral etc..  
3) Probability of default (PD) which is historical or modeled probability for making default. Of course you must also define what default is..  
Expected loss (EL) at money units is following:  
EL=PD*EAD*LGD  
If you have parameter estimates then you can easily calculate reserves needed to cover expected losses from defaults. For extra reserves safeguarding against unexpected losses you need to have some idea of prob. dist. of losses. 
I have seen constructs where LGD is function of PD, such that higher PD will lead to smaller recovery rate.  
EL=PD*EAD*LGD(PD)
If you assume independence between these terms you can just get LGD and EAD numbers from your own financial data and model PD separately. 
A: It is possible to reflect the concentration credit risk in terms of Economic Capital using a modification of the Vasicek VaR model. The Vasicek model is the key model for credit risk under the Basel II framework and it assumes that the credit risk exposures are uniform. Nevertheless, it is possible to modify it by  applying a Monte Carlo method that randomly samples k exposures on the portfolio according to the expected default rate k/n. By repeating this procedure, one obtains a loss distribution with concentration. The following code does this job.
#-------------------------------------------------------------------------------

#Number of iterations used in Monte Carlo

N_iter <- 10000 #1000000

# Number of exposures

N <- 150

# Probability of Default

PD_Portfolio <- 0.05
LGD_Portfolio <- 0.6

#Creates a dummie sequence of Exposures at Default

set.seed(123)
u <- seq(1,N,1)
u1 <- pnorm(rnorm(u))

EAD<- 10^3/min(u1)*u1

#Correlation Factor (should be changed according to the class of exposures)

R <- 0.03*(1-exp(-35*PD))/(1-exp(-35))+0.16*(1-(1-exp(-35*PD))/(1-exp(-35)))

max_loss_R <- function(PD,s) {
  
  return(pnorm( R^(-0.5)*(sqrt(1-R)*qnorm(s)-qnorm(PD)) ))
  
}

# Tabulates the possible loss values

S_range <- seq(0,1,1/N)
N2 <- N+1
loss_tabulation <- cbind(0:N,S_range,max_loss_R(PD_Portfolio,S_range))
loss_tabulation_df <- as.data.frame(loss_tabulation)
colnames(loss_tabulation_df)<- c("#Defaults(D)","%Defaults(D/N)","#Defaults<=D")

#Random sample of a uniform distribution

vec_N_Defaults <- rep(0,N_iter)
Loss <- rep(0,N_iter)

for(i in 1:N_iter){

y <- runif(1)
vec_N_Defaults[i] <- LGD_Portfolio*which(abs(loss_tabulation_df[,3]-y)==min(abs(loss_tabulation_df[,3]-y)))
Loss[i] <- LGD_Portfolio*sum(EAD[sample(1:N,vec_N_Defaults[i],replace=F)])

}

vec_N_Defaults2 <- sort(vec_N_Defaults,decreasing=TRUE)

Loss2 <- sort(Loss,decreasing=TRUE)

#Estimates the Economic Capital factor K for the portfolio with no concentration

K_granular <- quantile(vec_N_Defaults2,0.999)/N

#Estimates the Economic Capital factor K for the portfolio with concentration

K_concentration <- quantile(Loss2,0.999)/sum(EAD)
```

A: Going by your comment: if you have a group of people with the same credit rating, and a total amount of X that you want to loan out, it is less risky to loan it to more people (where each person takes a smaller loan.)
There are many ways to get to this result, but my favourite is through the Kelly criterion. Even though the expected value (in terms of arithmetic expectation) is the same, you want to avoid the case where there's a small probability of a large drop in wealth. You should instead aim for the case where there's a large probability of a small drop in wealth.
The reason is that wealth compounds: it takes you just as long to go from 10 to 100 as it does from 100 to 1000. A drop from 100 down to 50 (even if unlikely) hurts that process much more than a drop from 100 to 95 (even if rather likely.)
For more details, the book The Kelly Capital Growth Investment Criterion is awesome.
