Value-at-risk systems and their application in integrated risk management

Journal of the Academy of Business and Economics, March, 2004 by Hari P. Sharma, Dinesh K. Sharma, Julius A. Alade

3.2 VAR for Parametric Distribution

The VAR methods in this category are simple enough under the assumption of parametric distribution such as normal distribution. This parametric approach has often been referred to as the delta-normal method since normality is assumed. The credit of motivating parametric models of VAR goes to JP Morgan's Risk Metrics methodology for developing estimates of standard deviation and correlation among portfolio assets using an exponentially weighted average approach.

In order to design VAR system for parametric distribution, a normal deviate ([alpha]) is calculated as:

(3.2.1) -[alpha] = |[R.sup.*]|-[mu]/[sigma]

Where [mu] and [sigma] are translated from general distribution into standard normal distribution. |[R.sup.*]| is the cutoff return and is generally negative.

Associating the normal deviate ([alpha]) with [R.sup.*], Jorion (1996, 2000) shows that

(3.2.2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

From the equality in equation (3.1.8), the problem of VAR is equivalent to finding the deviate [alpha] such that the area to the left of it is equal to 1-c. This is done by turning to tables of the cumulative standard normal distribution function, that is, the area to the left of a standard normal variable with value equal to d:

(3.2.3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

For instance, at the 95% confidence level, 1-c = 5%. Therefore, the associated [alpha] corresponding to the lower 5% of the normal distribution is equal to 1.65. Jorion (1996, 2000) notes that equation (3.2.3) provides an illustrative linkage that shows that "VAR may be found in terms of portfolio value ([W.sup.*]), cutoff return ([R.sup.*]), or normal deviate ([alpha])." Therefore, VAR under the assumption of normality is:

(3.2.4) VA[R.sub.[mu]] = - [W.sub.0]([R.sup.*]-[mu]) = [W.sub.0] [alpha][sigma][square root of]/[DELTA]t

Where [W.sub.0] is defined as before to be initial portfolio value, [alpha] is the normal deviate associated with (1-c) and [sigma] the standard deviation of portfolio returns. To find the VAR of a portfolio, one needs to multiply the estimated [sigma] by the relevant percentile and initial investment. Obviously, under the assumption of normality, the only true unknown is the estimate of [sigma]. Therefore, the problem becomes one of forecasting the volatility and correlations between individual assets and subsequently portfolio volatility. When VAR is defined as an absolute dollar loss, the formula is as follows:

(3.2.4) VAR(Zero) = - [W.sub.0]([R.sup.*]) = [W.sub.0] ([alpha][sigma][square root of]/[DELTA]t - [mu] [DELTA]t)

4. VALUE AT RISK SYSTEMS

The calculation of value-at-risk (VAR) for large portfolios of complex derivative securities presents a tradeoff between speed and accuracy. The fastest methods rely on simplifying assumptions about changes in underlying risk factors and about how a portfolio value responds to these changes in the risk factors. Various methods are possible in computing VAR. These methods basically differ in distributional assumptions for the risk factors and linear versus full valuation, where linear valuation approximates the exposure to risk factors by a linear model. The simplest methods are the variance-covariance solution popularized by RiskMetrics[TM], and the delta-gamma approximations described by Britten-Jones and Schaefer (1999), Rouvinez (1997) and Wilson 1999. These rely on the assumption that a portfolio value changes linearly or quadratically with changes in market risk factors. One of the most difficult aspects of calculating VAR is selecting among the many types of VAR methodologies and their associated assumptions (Minnich, 1998). In this section, the focus is on the three classic methods: (1) variance-covariance matrix (delta and delta-gamma approaches), (2) historical simulation, and (3) Monte Carlo simulation.