The Art of Credit Exposures

Keywords: Credit Exposures, Mean Exposure Maximum Exposure, Peak Exposure, Potential future Exposure, Average Potential Future Exposure,Future Expected Exposure, Current Exposure, Potential Future Exposure, Exposure Shortfall, Exposure Buckets, Netting Agreement, Collateral Requirement, Counterparty Accounts, Receivable - Payable, Replacement Cost

Mean, Peak Potential Future Exposure Calculation

Credit Exposures Against What: Counterparty, Master Agreement or Accounts

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

Risksvr goes one step further and handles counterparty related information through one or multiple Accounts.

The Account defines the relationship between the receiving and paying parties involved in one or multiple trades.

In many exotic trades a position has multiple exposures or legs and each leg is paid by one party and received by the other.

The account is therefore defined by a paying and receiving party, the collateral posted by each party, the currency and country risk, the rating, and recovery rank (stochastic recovery is disabled by default, since it will reduces exposure), counterparty limits (the payer and receiver limit) and netting agreement policies.

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.

Credit Exposure Computation

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.

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

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% so that:

The peak exposure is the maximum value of the maximum exposure over time:

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

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

Other statistics can be aggregated as we simulate over time.

We can therefore compute Maximum/Peak Credit Exposure, by computing the Maximum of the Value of the Contract over time.
We can also compute the Potential Exposure by computing the Variance and subsequently the volatility of the positions with regards to the market value.

Both current and future potential exposures are affected by netting and collateral
provisions. (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 gives way to the Maximum between 0 and the sum of all nettable assets (long and short) that are held against the party in the account.

Mean Exposure

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

Exposure Volatility from Sum of Powers or Quantiles

The Standard or Easy Approach: Exposure Moments.

Exposure Volatility is computed by taking the Sum of Squares of the difference between each simulated Exposure and the mean exposure (or 0 if in the "mean-zero" framework assumptions divided by the number of simulations-1.

As with standard VaR results, the Exposure Volatility is then multiplied by the confidence multiplier in order to reach the desired confidence interval. (i.e. 1.654 or 1.99 to obtain a 95% or 97.5 confidence figure.)

The Advanced Approach: Exposure Shortfall or Tail..

A more precise Exposure can be computed by creating distribution aggregation buckets and quantile accumulation.

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

There are fortunately a number of numerical solutions to automate this procedure.
Quantile accumulation is therefore used when better precision is required or when a second opinion is needed.

Why reinvent the Wheel ?

Risksvr� can readily provide standalone or integrated credit exposure modules, with position ageing, cash-account accumulation, exposure drilldown facilities.

The engine can be purchased either standalone with the required level of support, or as a one off C++ source code cut (or a series of subscriptions) at a very competitive price. You can purchase the framework, part of it or the entire application.

Source code purchase is for end users only. (NDA expected).

You can try Risksvr� online credit exposures here.

(Please ensure you have selected Credit or Market and Credit in the Analysis Section and that credit exposures are active (either tailed or moments)

Types of Exposure:

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)

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

� 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

Note:

t can be zero.

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.

However, 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.

Financial Instruments carry many risks.... and rewards for those who master them!