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# PFE

## Overview

In a typical OTC transaction, where there is no center-clearing party such as an established exchange market, deals are normally conducted between two parties in informal forms and with flexible terms. As such, there is an ineligible risk that one party may default on future payments that have positive value to the other. This risk is known as default risk (or counterparty risk), it is one type of credit risk.

Credit risk is difficult to measure as there are many factors involved, such as default probability, credit exposure and estimated recovery rate. Credit exposure is the net positive outstanding amount that will be lost in the event of default, it is the starting point to evaluate the cost of default event. There are numerous ways to calculate credit exposures, examples are current exposure, expected exposure, potential future exposure, worst case exposure, etc.

In this section, we talk about potential future exposure (PFE). Topics that we'll cover include PFE definition, PFE applications and PFE models.

## Definition

**Potential future exposure**(PFE) is the maximum amount of exposure that is expected to occur at future times at a given level of statistical confidence level. For example, 95% PFE is the level of exposure that will be exceeded at 5% probability. Mathematically, PFE can be informally defined as

- $ PFE(t) = \int_{-\infty}^{+\infty} max(x, 0)f(x) ds\! $

Since in practice we usually work with discrete quantities, the above formulation can be approximated in discrete representation as

- $ PFE(t) = \sum_{i=1}^N max(PortfolioValue(i), 0)\! $

The amount of PFE typically varies over time up until the end of the contract. PFE is most commonly computed using simulation methods, i.e. future portfolio values are simulated using carefully selected stochastic models, and a high percentile of the distribution is chosen as PFE at this future time point.

## PFE Applications

PFE is commonly used to approve trade limits, calculate credit risk and set economic or regulatory capital. Credit officers use PFE to set trade limits against credit lines. PFE is often used as an input to calculate credit risk. A dealer may use PFE to calculate economic capital to support the risk of a portfolio of counterparties.

## PFE Models

The estimation of PFEs requires well-chosen stochastic models and powerful simulation engine. For the simulation models, a good modeller needs to consider, among others, the following points:

- Powerful simulation engine is essential for computing future portfolio values: there could be many portfolios of assets to be simulated, a portfolio may contain thousands of assets, and each asset may require thousands of simulation paths in order to have more accurately estimated future values.
- Different stochastic models for different financial instruments and different markets. Examples are: lognormal models are often used in well-developed equity and major foreign exchange markets. Jump-diffusion models are normally used in the emerging equity or currency markets.
- Simulation horizon can affect the model choice. For example, jump-diffusion models are more appropirate for short-term simulations; in interest rate market, long-term simulation should normally include the mean-reversion feature to avoid unrealistic future interest rate senarios.
- Correlations among risk factors need to be considered. Risk factors might be correlated and their correlations may vary with time. Good correlation inputs can give more realistic and dependable future portfolio values.
- Calibrated parameters used as simulation inputs is important for generating future trajectories that are appropriate for the instruments to be evaluated. Parameters calibrated to historical data can be dated and is based on the idea that future values behave like the past. On the other hand, parameters calibrated to current option prices may not contain market participants' consensus view of future price behaviour.