Analysis of Financial Time Series, Third Edition

7.1 Value at Risk

There are several types of risk in financial markets. Credit risk, operational risk, and market risk are the three main categories of financial risk. Value at risk (VaR) is mainly concerned with market risk, but the concept is also applicable to other types of risk. VaR is a single estimate of the amount by which an institution's position in a risk category could decline due to general market movements during a given holding period; see Duffie and Pan (1997) and Jorion (2006) for a general exposition of VaR. The measure can be used by financial institutions to assess their risks or by a regulatory committee to set margin requirements. In either case, VaR is used to ensure that the financial institutions can still be in business after a catastrophic event. From the viewpoint of a financial institution, VaR can be defined as the maximal loss of a financial position during a given time period for a given probability. In this view, one treats VaR as a measure of loss associated with a rare (or extraordinary) event under normal market conditions. Alternatively, from the viewpoint of a regulatory committee, VaR can be defined as the minimal loss under extraordinary market circumstances. Both definitions will lead to the same VaR measure, even though the concepts appear to be different.

In what follows, we define VaR under a probabilistic framework. Suppose that at the time index t we are interested in the risk of a financial position for the next ℓ periods. Let ΔV