In recent years, value-at-risk has been considered one of the best measures of risk by banks and other financial institutions. The Group of Thirty, the International Swaps and Derivatives Association, and the Derivatives Policy Group have all recommended the valuation of positions to the market and to assess financial risks using a VaR system. To define the VaR of a portfolio, we need to set a time horizon, usually short (one day to two weeks), and a high confidence level, 95% to 99%. For example, if the time horizon is one day, confidence level 99%, and VaR USD1 million, then this says that with a confidence level of 99%, the maximum loss over the next day is USD1 million. In other words, there is only 1% chance that the loss of the portfolio in one day, if any, will exceed USD1 million. A graphic definition is given in the figure. (The curve is the likelihood of the portfolio values at the end of the time horizon.)
There are basically three approaches to calculating the VaR of a portfolio: historical simulation, Monte-Carlo simulation and analytical methods. Historical simulation involves using historical changes in the market rates and prices to construct a distribution of potential future portfolio profits and losses, and then estimating the VaR from the distribution. More precisely, one first identifies the underlying risk factors and considers the behavior of these factors to obtain their percentage changes over a historical period of time. Next, these percentage changes to current rates or prices are used to get potential future values for the rates or prices. The portfolio is then evaluated according to these values. Finally, the mark-to-market profits and losses are used to estimate VaR at the given confidence level.
Rather than using the historical, observed changes in the market factors to generate hypothetical profits and losses, Monte-Carlo simulation generates a statistical distribution that is believed to adequately capture or approximate the possible changes in the market factors. For each risk factor, a distribution is chosen for its possible future values. Normal, lognormal, and jump-diffusion distributions are all possible candidates. Historical data is then used to determine the parameters of the chosen distributions. Potential future rates or prices are then generated randomly according to this distribution. The portfolio is then re-evaluated, and the VaR is obtained, as before.
Analytic methods also identify risk factors and produce a distribution for each factor. The parameters of the distributions are calibrated using historical data as well. Based on these numbers (variances and co-variances), one can calculate the standard deviation of the portfolio, and determine the appropriate VaR. If potential future values of the portfolio are assumed to have a normal distribution, then 95% VaR is equal to 1.65 times the standard deviation, and the 99% VaR is 2.33 times the standard deviation.
Each method has its own advantages and disadvantages. The major disadvantage of the two simulation methods is that one needs a sufficiently large number of simulations in order to predict the future values accurately, and, therefore, the time required can sometimes become unmanageable. The historical method suffers another problem it is not easy to perform a "what if" analysis to examine the effect of alternate assumptions. In contrast, the analytical method provides additional insight and is time-efficient, but it is not easy to explain to senior management. Furthermore, the results may not be reliable when options are involved.
The VaR of an option is not easy to calculate because its relationship to the underlying risk factor is usually non-linear. A consequence of this is that the option is not normally distributed, even if the underlying is normally distributed. Most analytical approaches try to approximate the relationship by Taylor's series. The simplest one, the so-called delta method, considers only the first-order approximation. It basically treats all the relationships as linear, and uses variances/co-variances to compute the standard deviation of the portfolio. The results can be fairly inaccurate if the linear approximations do not fit the relationships well. To remedy the situation, the (usually) more accurate delta-gamma method takes parabolic (second-order) approximations into consideration. It provides a better fit to the relationship, but it still makes the assumption that the possible values of the portfolio form a normal distribution. Recently, there have been attempts to correct this unrealistic assumption by considering higher moments of the distribution of the portfolio.
The analytical methods described above all try to express the VaR of an option in terms of its standard deviation. An alternative is to express the VaR of the option in terms of VaR of the underlying. For plain-vanilla call and put options, this approach admits simple, explicit and exact formulae. The idea can be generalized to provide algorithms for exotic options and options with multiple risk factors.
This week's Learning Curve was written byH. Gifford FongandKai-Ching LinofGifford Fong Associatesin Lafayette, Calif.