In the case of default losses, computations can vary widely. 

The models and the methodologies applied depend on :By order of importance

1. The data supplied.
2. The reports and type of analysis.
3. The assumptions defined in the Engine Specification or Terms and Conditions. 
   Terms and Conditions are always preset to produce the most conservative figures, but can always be overridden. 
   

The engine can either simulate default states from a credit curve , or it can simulate a Markov Chain of Rating Upgrades and Downgrades over the simulation horizons from one or multiple transition matrices

To trigger credit migration, the engine expects at least one Transition Matrix.
If other credit statistics, such as Survival rates, marginal or cumulative default probabilities are provided, then only default or non default states are observed. 
In the uncorrelated framework migration does not give way to the full cost (benefit) associated with downgrades (upgrades). Instead migration is used to assign the next rating state until it either hits a default or reaches the last simulation horizon.
To measure costs associated with Migration Risksvr™ expects a rating rank associated to each risky curve and a number of policies that stipulate negative forward spread rate collapse and spread calculations when spread rates are missing from a particular rating rank.

Risksvr™ can incorporate default correlations.
For this to work, the obligor specific risk OSR and the individual weights for each and every risk factor associated with the counterparty  are expected to be fed from an external system into the engine.
If weights are not supplied from an external system, Asset weights can be implied from he relative percentage of the positions associated with the obligors total holdings, or approximated with a selection of weights associated with selected factors (in essence a series of spectral decompositions).

If needed, Risksvr™ can generate weights by computing the individual exposure of each risk factor mapped to the trades belonging to the counterparty and the obligor specific risk provided. If no obligor specific risk is given, Risksvr™ can either derive the OSR from the firm's equity or apply a user defined value that is preset to .20 (20%) which is common practice in industry.

Risksvr™ can incorporate default correlations with Time-To-Default.  Instead of supplying or implying weights, we can incorporate obligor correlations through Time to Default. Time to default provides an elegant approach that blends both univariate probabilities of the asset's credit quality with the correlation across obligors /assets. 

Mapping Counterparty or Obligor Exposure To The Universe Of Risk Factors

Obligors as Groups of Counterparties: One to one or many relationship.

To model obligor Correlations, Risksvr assumed you will provide Obligors or at least Counterparties.

The Obligor binds one or more Counterparties through the Name. 
Each Obligor includes specific Risk (Obligor Specific Risk [OSR] or Firm specific Risk), which is a necessary component when running Correlated Defaults, and an associated Credit Curve Name. Each Credit Curve represents a series of default probabilities that evolve over the simulation horizons.

In order to compute the maximum reserve capital required as buffer for each obligor or counterparty, we model the return on a synthetic asset that represent the obligor’s weighted exposure to the universe of risk factors that are correlated to each other and a portion of idiosyncratic or specific risk that is independent from the other factors.

In its simplest and weakest form, this approach reduces to the CreditMetrics™ approach described in the Credit Metrics™ Technical document. However, this model does not limit itself to Three Equity indices and Country Mappings. Instead it provides a generic framework where any type of risk can be mapped to create a synthetic asset, be it interest rates, commodities foreign exchange, equity or any other asset whose returns can be measured.
 
In its most advanced form this model evolves as a Normalized Risk Factor Exposure Weighted Asset where funding costs of payables streams and investment benefits of receivable flows associated to spread curves are switched as the credit quality of the asset migrates from one rating rank to another over the simulation horizon.

Users can therefore decide to apply the mappings provided by the Dow Jones Global Equity™ Indices and fall back  on the results presented in the Credit Metrics™ Technical document or they can decide to go much further and narrow down on assets that represent much finer mappings.

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 .

 

 

The Mechanics of Rating Migration  and  Migration to Default

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 independent. 

This approach offers several advantages.../font>

• 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:

Pseudo Rating	Default	C	B	BB	BBB	A	AA	AAA
Rating Rank	0	1	2	3	4	5	6	7
Probability	0.02	0.04	0.06	0.08	0.12	0.2	0.4	0.08
Quantile / Cum. Prob.	0.02	0.06	0.12	0.2	0.32	0.52	0.92	1.00

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

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.

Counterparty Exposure and Proxy for Default

Equity prices are frequently used to proxy defaults, as they offer the advantage of higher liquidity than bonds and no recovery.

As with standard corporate and emerging market bonds, equity holders should model equities as a stream of dividends up to the Time of default. 

In this framework, equity valuation requires the sum of expected dividends times the forward probability of default [i.e. the marginal conditional default probability computed from the associated credit curve].

There is obviously no recovery since recovery is part of the liquidation process and equity is last in line when it comes to liquidation order. 

Time-To-Default Copula Engine

For details on Time-To-Default computaitons, click here

 

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