Been worth At risk

The VAr (of English Been worth At Risk , word for word: “value under risk”) is a concept used generally to measure the Risque of market of a wallet of financial instruments. It corresponds to the amount of losses which should be exceeded only with one probability given on a given temporal horizon.

History

This concept is originating in the sector of the Assurance. It was imported at the end of the Années 1980 on the financial markets with the the United States by the bank Bankers Trust and was popularized by the bank JP Morgan in 1993 and its service (free and public) Riskmetrics then adopted in an embryonic form by the Committee of Basle (Basle II) for the banks and Solvabilité II for the insurances.

Main features

The VAr of a wallet depends primarily on 3 parameters:

the Distribution of the results of the wallets. Often this distribution is supposed normal, but much of financial actors use historical distributions. The difficulty lies in the historical sample size: if it is too small, the probabilities of high losses are not very precise, and if it is too large, the temporal coherence of the results is lost (one compares noncomparable results);
the selected level of Confidence (95 or 99% in general). It is the probability that the possible losses of the wallet or the credit do not exceed Not been worth At Risk, by definition;
the temporal horizon selected. This parameter is very important because more the horizon is long plus the losses can be important. For example, for a normal distribution of the returns, it is necessary to multiply Been worth At Risk at one day by $\ sqrt {T}$ To have been worth At Risk on $t$ days.

Generally, the VAr gives an estimate of the losses which should not be exceeded except extreme event on a wallet which can be made up of various classes of credits. It gives in only one figure the amount at the risk of a wallet.

The VAr also makes it possible to analyze a wallet by extracting the credits which contribute more to the VAr, in other words the credits which add risk to the wallet. For example, the mVaR, or VAr marginal, can be defined like the difference of the VAr of the wallet with and without the credit considered.

Formal representation

Mathematically, the VAr is defined in an implicit way, starting from the distribution of the output of the credit considered over the period considered. Either $\ alpha$ a number between 0 and 1, and or $r$ the output carried out by the credit. The $VaR (\ alpha)$ is such as: $\ alpha=Pr (VaR . The VAr thus defined is the loss which has a probability \ alpha of being worse than the return of the wallet or credit. In other words it is the quantile 1- \ alpha of the distribution of the return of the wallet or the credit.$

External bonds

RiskGlossary.com
Investopedia.com. First part, Second part

Reference

pH. Jorion, " Been worth At Risk" , Mc Graw Hill

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