Value at Risk

Overview



Value at risk is common risk model. It measures the amount of money lost in a worst case scenario over a given time interval. In order to use the model, one needs to specify a threshold, which is specified as fraction or perecentage, call it {% \alpha %}. The VAR model would report the dollar loss in the scenario where only 1 - {% \alpha %} % of the scenarios are worse.

As an example, if {% \alpha %} is specified at the 99% level, VAR would report the dollar loss of the scenario where only 1% of the scenarios are worse.

Illustration



Portfolio Distribution



The challenging part of constructing a VAR model is to obtain the portfolio distribution. The method that is used to obtain the distribution catagorizes the model in the following:

  • Parametric models are models where the porfolio distribution is assumed to be of a given mathematical form, with only a set of parameters needed to specify it completely. The canonical example is the assumption that the portfio value is normally distributed.
  • Monte Carlo VAR models are a natural extension of parametric models. The monte carlo method also assumes distributions for the assets in the portfolio, which are typically fit by a set of parameters. However, the modeler need not know how to sum these distributions to get the portfolio distribution analytically. As long as one can simulate the individual assets, a modeler can siulate the portfolio values and then calculate the VAR.
  • Historical Simulation

Scaling VAR



The definition of a VAR measure nees to specify a time frame. That is, it determines the possible amount of loss at a certain confidence interval, over a given time frame. Often, it is necessary to specify a different time frame from that for which the measure was calculated. If the resources are available, the measure could just be re-run at the new time frame, however, there are assumptions that can be made that make scaling apossilbe without having to re-run the measure.

As an example, for an asset (or portfolio) that follows i.i.d (independent and identically distributed) normal returns, the variance is proportional to time. That is, if we know the variance of the portfolio over the time span of 1 day, then the variance of the portfolio over a 10 day time frame is 10 times the 1 day variance. (see geometric brownian motion).

(see Alexander pg. 21)

VAR Risk Types



Value at Risk is a measure that applies to a portfolio as a whole, regardless of the types of assets within the portfolio. However, the methods used to calculate VAR are often dependent on the assets within the portfolio. When a portfolio consists of multiple asset types, it is sometimes possible to compute the VAR separately for each asset class and then to aggregate the results.



  • Market VAR: is the risk to a portfolio due to changes in the prices of the underlying assets. Measuring market var is generally a similar exercise for geometric brownian motion assets such as equities, foreign exchange, and commodities.

    Market risk for a fixed income portfolio refers to the risk that arises from changes in market interest rates.
  • Credit VAR: is the risk to a portfolio due to the risk of the fixed income assets in the portfolio defaulting.

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