VaR: Mapping Positions to Risk Factors

While working with Value at Risk, we are often looking at calculating the P/L of individual positions, and then calculating the portfolio P/L based on the individual positions. However, there are a few problems in doing so.

First, it is not possible to model each and every position in the portfolio, as there could be many positions with different complexities or lacking historical data to support the data model. Second, the portfolios of these positions can become quite complex with many dependencies. For example, the covariance matrix of risk factors in all the instruments (say n instruments) can become extremely large. The number of volatilities (n), and data on correlations (n(n-1)/2) grows as n grows. This also poses challenge in terms of the computational power required to process the data.

The process can be significantly simplified by mapping these positions to a small set of risk factors. Instead of trying to calculate VaR for each instrument, what we need to do is to decompose these instruments into building blocks, or primitive instruments, which are further mapped to a small set of risk factors. The four common building blocks are: equity positions, zero-coupon bonds, futures/forwards, and spot foreign exchange.

The generic VaR mapping process is as follows:

  1. Choose a set of elementary risk factors and estimate their probability distributions
  2. Mapping: Decompose financial instruments into exposures on these risk factors. Each position is marked to market. Market value of each instrument is allocated to risk factors.
  3. Aggregate the exposure for all positions and build the distribution of P/L on portfolio. Arrive at the portfolio VaR.
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