# Credit valuation adjustment

**Credit valuation adjustment** (**CVA**) is the difference between the risk-free portfolio value and the true portfolio value that takes into account the possibility of a counterparty’s default. In other words, CVA is the market value of counterparty credit risk. This price depends on counterparty credit spreads as well as on the market risk factors that drive derivatives’ values and, therefore, exposure. CVA is one of a family of related valuation adjustments, collectively xVA; for further context here see Financial economics #Derivative pricing.

Unilateral CVA is given by the risk-neutral expectation of the discounted loss. The risk-neutral expectation can be written as

where is the maturity of the longest transaction in the portfolio, is the future value of one unit of the base currency invested today at the prevailing interest rate for maturity , is the fraction of the portfolio value that can be recovered in case of a default, is the time of default, is the exposure at time , and is the risk neutral probability of counterparty default between times and . These probabilities can be obtained from the term structure of credit default swap (CDS) spreads.

More generally CVA can refer to a few different concepts:

- The mathematical concept as defined above;
- A part of the regulatory Capital and RWA (Risk-weighted asset) calculation introduced under Basel 3;
- The CVA desk of an investment bank, whose purpose is to:
- hedge for possible losses due to counterparty default;
- hedge to reduce the amount of capital required under the CVA calculation of Basel 3;

- The "CVA charge". The hedging of the CVA desk has a cost associated to it, i.e. the bank has to buy the hedging instrument. This cost is then allocated to each business line of an investment bank (usually as a contra revenue). This allocated cost is called the "CVA Charge".

According to the Basel Committee on Banking Supervision's July 2015 consultation document regarding CVA calculations, if CVA is calculated using 100 timesteps with 10,000 scenarios per timestep, 1 million simulations are required to compute the value of CVA. Calculating CVA risk would require 250 daily market risk scenarios over the 12-month stress period. CVA has to be calculated for each market risk scenario, resulting in 250 million simulations. These calculations have to be repeated across 6 risk types and 5 liquidity horizons, resulting in potentially 8.75 billion simulations. ^{[1]}

## Exposure, independent of counterparty default

Assuming independence between exposure and counterparty’s credit quality greatly simplifies the analysis. Under this assumption this simplifies to

where is the risk-neutral discounted expected exposure (EE)

## Approximation

Full calculation of CVA is done via Monte-Carlo simulation of all risk factors which is very computationally demanding. There exists a simple approximation for CVA which consists in buying just one default protection (Credit Default Swap) for amount of NPV of netted set of derivatives for each counterparty.

## Function of the CVA desk and implications for technology

In the view of leading investment banks, CVA is essentially an activity carried out by both finance and a trading desk in the Front Office. Tier 1 banks either already generate counterparty EPE and ENE (expected positive/negative exposure) under the ownership of the CVA desk (although this often has another name) or plan to do so. Whilst a CVA platform is based on an exposure measurement platform, the requirements of an active CVA desk differ from those of a Risk Control group and it is not uncommon to see institutions use different systems for risk exposure management on one hand and CVA pricing and hedging on the other.

A good introduction can be found in a paper by Michael Pykhtin and Steven Zhu.^{[2]}
Karlsson et al. (2016) present a numerical efficient method for calculating expected exposure, potential future exposure and CVA for interest rate derivatives, in particular Bermudan swaptions.^{[3]}

## References

- ↑ Alvin Lee (17 August 2015). "The Triple Convergence Of Credit Valuation Adjustment (CVA)". Global Trading.
- ↑ A Guide to Modeling Counterparty Credit Risk, GARP Risk Review,July-August 2007 Related SSRN Research Paper
- ↑ Patrik Karlsson, Shashi Jain. and Cornelis W. Oosterlee. Counterparty Credit Exposures for Interest Rate Derivatives using the Stochastic Grid Bundling Method. Applied Mathematical Finance. Forthcoming 2016.