Description of Equity Buffer Approach

Let , and denote the standardized returns of each risk factor’s asset class against which the counterparty is exposed (s): Interest Rates, Commodities, Foreign Exchange and Equities respectively.

As usual the asset’s return is expressed as the log of price changes of the asset.

The Return is then unitized or standardized/centralized to fit the standardized Normal distribution of mean zero and variance one.

Hence, for every Return R computed 
we compute the unitized or centralized return:
. i.e.

Now, lets denote C as the counterparty and  the return on the “portfolio” or synthetic asset of Counterparty C . The model assumes the decomposition of counterparty’s portfolio  in terms of each individual risk factor returns:

We seek to map the weight(s) of the counterparty’s individual exposure to risk factor(s), including his own idiosyncratic /specific risk OSR so that the weight sum’s up to 1. (i.e. 100%).
This ensures that the sum of returns for each individual risk factor, which are unitized to fit the standard Normal distribution with mean 0 and variance 1, will also be unitized or standardized to fit the normal distribution with mean zero and variance one

where are the weights associated with the systematic (or factor) risk, and  is the weight of the counterparty’s / obligor specific risk (OSR). The variable is a standard normal random variable with mean zero and variance one.

For each counterparty, the weights of each risk factor, including the counterparty / obligor specific risk is re-based so that the overall return fits the normal distribution with expected mean zero and variance one.

For example, if the obligor is exposed to two interest rate factors, , one Commodity factor , two foreign exchange factors, , and one equity risk factors, then his decomposition will become:

The standard deviation of the above sum of correlated returns is obtained either from the dot product of individual weighted returns or a correlation matrix  


or decomposed via cholesky technology accordingly.

In general, suppose the vector is represented by decomposition factors and . It is required that . To solve this we re-base the weights by applying.

where are standard normal variables and < is the correlation between .

The specific risk factor, , is a standard normal random variable related to the OSR .

Since the uncorrelated case is simpler and slightly more intuitive, we will begin by describing the independent "Markov" approach and then present the enhancements implemented to correlate migration. Finally, we will show how Time to Default can be used to leverage both approaches.

Univariate Uncorrelated Framework:

In the univariate or uncorrelated model each obligor is assumed to be indepenedent. 

This approach offers several advantages.

• It is intuitive.

• It is much simpler in terms of configuration since no correlations are required.

• It is conservative. 

• The univariate / uncorrelated approach will always return the Maximum Loss.

• It can be shown that equivalent 
  
  

advantage of this approach is  the transition matrix defines explicitly the probabilities of reaching a specific rating rank:
For each rating state the engine partitions the probability space according to the initial counterparty rating 

According to these probabilities, there are uniform quantiles, < , such that the Probability(< ) = < , for j=1,…,N-2 and Probability( ) = , Probability(< ) = . Where N is the number of rating categories plus default. 
In this instance we assume represents the state of default and < the highest rating rank.

If the uniform random draw generated by the engine falls within the first quantile  then the engine assumes a default has occurred and computes loss given default.

If the uniform random draw falls beyond the first quantile and migration is indeed active, then the engine will deduce the rating rank of the uniform random draw from the quantile in which it fell.

It will then use this rating rank as the next rating state until it either hits a state of default or reaches the final simulation horizon.
In the uncorrelated framework, migration is used to produce the next rating state only. No costs are associated with upgrades or downgrades. They only serve to determine the next rating until either final simulation horizon or default.

For Example: lets assume there are 7 ratings ranks where 1 is the state of default and the current counterparty  rating rank is 6. 
Lets assume the transition matrix row for a rating rank of 6 is:

If the uniform random draw is, say, 34 then the counterparty will be downgraded to rating rank 4

4::(0.02 +0.04+0.06+ 0.08+ 0.12 = 32)

5::(0.02 + 0.04+0.06+0.08 + 0.12 +0.2= 52)

The Correlated framework applies to the counterparty’s synthetic unitized return to simulate migration (as described above). Prior to simulation and if not fed from an external source, the engine computes the systematic and specific weights for each counterparty. Once in the simulation sequence, the engine draws a normal random variable to generate the specific risk factor Z and then performs the decomposition

The transition’s Matrix row that corresponds to the counterparty rating is then mapped into subintervals of the normal distribution from 1 to N where N is the highest rating defined and R1 is the state of default.

The engine therefore defines normal quantiles, , such that the Probability( ) = , for j=1,…,N-2 and Probability( , Probability( ) = . Let denote the cumulative normal function (CDF) :

The quantiles are then mapped into the same uniform partition used to simulate uncorrelated migrations [0,1] into intervals of lengths , by using the inverse of  i.e. , by mapping these probabilities the engine can check directly which probability interval the uniform deviates falls into the normal quantiles

and checks which one of these subintervals does fall into.
 The subintervals, starting from zero, are of lengths in sequence.