Credit Exposures

Credit Exposures Against What Counterparty, Master Agreement or Accounts ?

Most simplistic systems model exposure against a Counterparty. More advanced engines use the notion of master agreements.

Risksvr™ takes the analysis a step further and handles counterparty related information through accounts.
Accounts encompass the binding relationship between buying and selling parties and thus extends beyond positive / negative mark-to-market accumulation, collateral provisions etc, and condense the legal repercussions of payment delinquency and default.

The account is therefore defined by the party engaged in the relationship, the ISDA netting policy,, the collateral posted by the party and it's Home office or any party that will act as a sponsor for the party, the country, the rating and the recovery rate and volatility.

Terminology:

V(t)	:	is the value of a portfolio at time t.
E(t)	:	is the credit exposure at time t and is computed as Max{V(t),0}.
f	:	is the pdf (probability density function)
F	:	is the cdf (cumulative Density Function) of V(t).
T	:	is the instrument's time to maturity or expiration date.
Exposure at any time t can be zero.

Credit Exposure Calculation

To compute Credit Exposure and related statistics, we need to know what will be the cost of replacing our positions if the counterparty defaults on us.
In theory, credit exposures are computed over the entire simulation horizon.

In practice and since we are only interested in the amount that is positive to the receiving side,
we compute credit exposure by taking the sum of all positive values or replacement costs of the asset over discrete simulation horizons.

Netting - ISDA Master Agreement provisions

Both current and future potential exposures are affected by netting, if both parties have signed an ISDA master agreement.

A netting agreement specifies whether or not two or more trades should be allowed to offset each other.
If netting is active, the gross replacement cost of all positive values give way to the Maximum between 0 and the sum of all netted assets long and short that are held against the party in the account.

Posted Collateral

Both current and future potential exposure take into account collateral posted by the party.

Mean, Peak Potential Future Exposure Calculation

Expected Future Exposure � Mean Exposure, decreases over time

Potential Future Exposure (Mean Exposure + 1 (one) standard deviation) 84.3%

User Defined Potential Future Exposure Mean Exposure + confidence multiplier * std)

The current exposure is the cost of replacement of the exposure(s) today

The expected exposure at time t is the mean exposure at time t, and is given by

Mean Exposure is computed by accumulating the simulated position(s) values over each time horizons divided by the number of simulations.

The average exposure is the mean of the expected exposure, over time, applying a time-based discount factor:

The maximum exposure at time t is the maximum percentile of the exposure at time t.

For Credit Exposure, the industry standard is 95% (which corresponds to a 5% percentile) so that:

Maximum Exposure

The Maximum Exposure is computed as the Maximum amount computed over all the simulations horizons.

Peak Exposure

The peak exposure is the maximum value of the maximum exposure over the entire simulation horizon.
In other words it is the maximum of the 5% (tail) percentile.

Expected Future Exposure

The Expected Future Exposure at each future time node is the Mean of the portfolio's simulated values

Potential Future Exposure

The Potential Future Exposure at each time step is the Mean plus the portfolio's standard deviations times the confidence interval.

Maximum Total Potential Exposure

The Maximum Total Potential Exposure is the time when the Potential Future Exposure is highest, if normal distribution is assumed, which might be a mistake, it can be proven maximum value is reached at time/3.

It is however important to understand distribution is rarely normal. Indeed, Credit exposure are NOT computed with a zero mean. (the value is akin to Earning at Risk) and due to the long simulation horizon, Mean Reversion can play an important role.

Future Potential Exposure

is the difference between the Current Exposure and the Maximum Total Potential Exposure.
This is the measure of the additional exposure you can expect to incur between the Start Date and time T

Worst Case Exposure

Worst Case Exposure defines the probability of the contract over time.

Exposure Volatility or Quantiles

The Simple Approach: Exposure Volatility.

Exposure volatility is computed by taking the sum of squares of the difference between each simulated Exposure and the mean exposure (or zero in the "mean-zero" framework) divided by the number of simulations.

The Exposure Volatility is then multiplied by the confidence multiplier in order to reach the desired confidence band. (i.e. we multiply by 1.654 to obtain a 95% confidence, 1.99 to obtain a 97.5 % confidence, etc.).

This simplification is similar to parametric Value-at-Risk. Some implementations simplify further by approximating prix volatility by adjusting the underlying market risk factor volatility instead of performing a full revaluation!

The Real Approach: Credit Exposure Tails, buckets and Shortfall Probability.

A more precise Exposure is computed by computing the exact distribution through buckets and quantiles.

The only drawback with this approach lies in the cumbersome configuration.
The quality of results depend on exposure bucket size and widths, which in turn vary at each level of the drilldown.

There are fortunately a number of elegant numerical solutions to automate this procedure. Quantile accumulation is therefore used when sharper results are expected.

Note:

Expected and average exposure will be positive for typical securities.
Maximum exposure can be zero, and more rarely, peak exposure can be zero.
Expected exposure can be greater than maximum exposure.
Average exposure can be greater than peak exposure.

This is because none of the measures incorporate default probabilities.

In particular, average exposure only uses the discount factor weights.

Since default likelihood is an increasing function of time, an exposure well into the distant future might represent more risk than the same size exposure in the near future.

This suggests an extension of the average exposure function to include defaults with the credit default curve.