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(ℓ) be the change in value of the underlying assets of the financial position from time t to t + ℓ and L(ℓ) be the associated loss function. These two quantities are measured in dollars and are random variables at the time index t. L(ℓ) is a positive or negative function of ΔV(ℓ) depending on the position being short or long. Denote the cumulative distribution function (CDF) of L(ℓ) by Fℓ(x). We define the VaR of a financial position over the time horizon ℓ with tail probability p as
From the definition, the probability that the position holder would encounter a loss greater than or equal to VaR over the time horizon ℓ is p. Alternatively, VaR can be interpreted as follows. With probability (1 − p), the potential loss encountered by the holder of the financial position over the time horizon ℓ is less than VaR.
The previous definition shows that VaR is concerned with the upper tail behavior of the loss CDF Fℓ(x). For any univariate CDF Fℓ(x) and probability q, such that 0 < q < 1, the quantity
is called the qth quantile of Fℓ(x), where inf denotes the smallest real number x satisfying Fℓ(x) ≥ q. If the random variable L(ℓ) of Fℓ(x) is continuous, then q = Pr[L(ℓ) ≤ xq].
If the CDF Fℓ(x) of Eq. (7.1) is known, then 1 − p = Pr[L(ℓ) < VaR] so that VaR is simply the (1 − p)th quantile of the CDF of the loss function L(ℓ) (i.e., VaR = x1−p). Sometimes, VaR is referred to as the upper pth quantile because p is the upper tail probability of the loss distribution. The CDF is unknown in practice, however. Studies of VaR are essentially concerned with estimation of the CDF and/or its quantile, especially the upper tail behavior of the loss CDF.
In real applications, calculation of VaR involves several factors:
1. The probability of interest p, such as p = 0.01 for risk management and p = 0.001 in stress testing.
2. The time horizon ℓ. It might be set by a regulatory committee, such as 1 day or 10 days for market risk and 1 year or 5 years for credit risk.
3. The frequency of the data, which might not be the same as the time horizon ℓ. Daily observations are often used in market risk analysis.
4. The CDF Fℓ(x) or its quantiles.
5. The amount of the financial position or the mark-to-market value of the portfolio.
Among these factors, the CDF Fℓ(x) is the focus of econometric modeling. Different methods for estimating the CDF give rise to different approaches to VaR calculation.
Remark
The definition of VaR in Eq. (7.1) is based on the upper tail of a loss function. For a long financial position, loss occurs when the returns are negative. Therefore, we shall use negative returns in data analysis for a long financial position. Furthermore, the VaR defined in Eq. (7.1) is in dollar amount. Since log returns correspond approximately to percentage changes in value of a financial asset, we use log returns rt in data analysis. The VaR calculated from the upper quantile of the distribution of rt+1 given information available at time t is therefore in percentage. The dollar amount of VaR is then the cash value of the financial position times the VaR of the log return series. That is, VaR = Value × VaR(of log returns). If necessary, one can also use the approximation VaR = Value × [exp(VaR of log returns) − 1]. □
Remark
VaR is a prediction concerning possible loss of a portfolio in a given time horizon. It should be computed using the predictive distribution of future returns of the financial position. For example, the VaR for a 1-day horizon of a portfolio using daily returns rt should be calculated using the predictive distribution of rt+1 given information available at time t. From a statistical viewpoint, predictive distribution takes into account the parameter uncertainty in a properly specified model. However, predictive distribution is hard to obtain, and most of the available methods for VaR calculation ignore the effects of parameter uncertainty. □
Remark
From the prior discussion, VaR is just a quantile of the loss function. It does not fully describe the upper tail behavior of the loss function. In practice, two assets may have the same VaR yet encounter different losses when the VaR is exceeded. Furthermore, the VaR does not satisfy the sub-additivity property, which states that a risk measure for two portfolios after they have been merged should be no greater than the sum of their risk measures before they were merged. Therefore, care must be exercised in using VaR to measure risk. We discuss the concept of expected shortfall later as an alternative to measuring risk. The expected shortfall is also known as the conditional value at risk (CVaR). □