The need for large numbers of Monte Carlo trials can be reduced through use of variance reduction techniques. While these are beyond the scope of this book, Glasserman, Heidelberger, and Shahabuddin (2000) provide an introduction to them in the context of value-at-risk estimation.
Most Monte Carlo approaches are based on the multivariate normal and lognormal distributions, though there has been some limited work out- side this framework. Recently, Hull and White (1998) and Hosking, Bonti, and Siegel (2000) have suggested approaches that involve simulating multi- variate normal random vectors and then transforming them to have mar- ginal distributions that better fit the known properties of changes in financial market factors. The heavy reliance on the normal and lognormal distributions seems to be driven by two considerations. First, these distribu- tions are convenient and tractable. Second, while distributions with fat tails relative to the normal are available, the data provide little guidance about which fat-tailed distribution should be selected. The whole issue is how much probability should be in the tails, but almost by definition there have been very few realizations in the tails. Thus, it is difficult to resolve this issue by looking at the data, and the designer has little basis to select and/or parameterize a particular fat-tailed distribution.
The use of Monte Carlo simulation to evaluate the risk of time- and path-dependent portfolio strategies, such as that underlying LaVaR is pushed by Dembo, Aziz, Rosen, and Zerbs (2000) under the rubric Mark- to-Future. An alternative approach to adjusting for liquidity is described by Berkowitz (2000).
APPENDIX: SIMULATING MULTIVARIATE
such that cov[xi, xj] iiij. To do this, we use the fact that if
is a vector of independent standard normal random variables (that is, var[ei] 1 and cov[ei, ej] 0 for i j) and A is a K K matrix, then the vector of random variables x given by the product x Ae has a covariance matrix given by the matrix product , where is the transpose of A.
Given this fact, we need to pick A such that ⌺, that is, such that A ⌺1/2 is the square root of the matrix ⌺. There is always at least one square root (and generally more than one) whenever ⌺ is a legitimate cova- riance matrix, so that there always exists an appropriate square root A.
Thus, to generate a vector of random variables x with covariance matrix ⌺, we first generate a vector of independent standard normal random variables e and then construct x as x ⌺1/2e. The elements of the vector x have covariance matrix ⌺, but they do not yet correspond to the changes in the market factors because they have expected values of zero. However, the changes in the market factors can be constructed simply by adding to each component of x the expected change in the corresponding market factor.
The remaining issues are how to generate the ei, and how to construct the square root A ⌺1/2. A realization of a standard normal pseudo-random variable can be constructed by applying the inverse of the cumulative standard normal distribution function to a pseudo-random variable uniformly distrib- uted on the interval [0,1]. That is, if F–1 denotes the inverse of the cumulative standard normal distribution function and u is a uniform pseudo-random vari- able, then ei F–1(u) is a standard normal pseudo-random variable. How- ever, this approach is not the most efficient; other approaches are discussed in Chapter 13 of Johnson, Kotz, and Balakrishnan (1994). Many software pack- ages include both functions to generate uniform pseudo-random variables and the inverse of the cumulative standard normal distribution function. Alterna- tively, computer codes to perform these computations are available in many software libraries and in standard references (e.g., Press, Teukolsky, Vetterling, and Flannery 1992). The square root ⌺1/2 is typically computed as the Cholesky decomposition of the covariance matrix, which is described in stan- dard references (e.g., Press, Teukolsky, Vetterling, and Flannery 1992) and is available in many statistical packages and software libraries.
=
e e1
... ei
... eK
=
= = ≠ ×
=
AA′ A′
AA′ =
=
=
=
=
7
105
Using Factor Models to Compute the VaR of Equity Portfolios
The example equity portfolio discussed in Chapters 2, 3, and 5 was carefully constructed, so that the portfolio returns depended only on the changes in the S&P 500 and FT-SE 100 stock market indexes. In particular, it was assumed that the portfolio of U.S. equities was so well-diversified that its returns could be assumed perfectly correlated with percentage changes in the S&P 500 index, and the other positions in the portfolio consisted of index futures and options.
Actual portfolios include instruments that are neither perfectly nor even highly correlated with one of the popular market indexes. This raises the question of what should be used as the underlying market factor or factors that affect the value of the portfolio. It is impractical to treat all of the securities prices as mar- ket factors, because actual equity portfolios may include hundreds, or even thousands, of stocks. Rather, one needs to find a limited number of market fac- tors that explain most of the variability in the value of the portfolio.
Factor models of equity (and other) returns provide a solution to this problem. Such models express the return on a security or portfolio of securi- ties in terms of the changes in a set of common factors and a residual or idio- syncratic component. For example, the return on the ith security, denoted ri, might be expressed as
(7.1) where f1, f2, . . . , fK denote the changes in K common factors, denote the factor loadings of the ith security on the K fac- tors, is a constant component of the return of the ith security, and is a residual or idiosyncratic component of the return. The number of factors K is usually relatively small, ranging from as few as one in simple single-index mod- els to as many as 50. A common, and even standard, assumption is that the residuals are independent across securities, that is, is independent of for Factor models are used for a variety of different purposes and differ in
ri = ␣i+i1f1+i2f2+. . . +iKfK+i,
i1, i2, . . . , iK
␣i i
i j
i≠j.
their choice of factors and methodologies for estimating the factor loadings. In some models, the factors consist of the returns on stock market indexes or portfolios (e.g., broad-based market indexes, industry indexes, or the returns on portfolios of small- or large-capitalization stocks), in others they consist of macroeconomic factors, such as unexpected changes in industrial production or inflation, and in others they consist of purely statistical factors (the principal components) extracted from the covariance matrix of returns.
Factor models provide a very intuitive way of thinking about the risk of a portfolio. If the changes in the factors (the fks) in equation (7.1) are the returns on portfolios, then (7.1) can be loosely interpreted as saying that one dollar invested in the ith security behaves like a portfolio of i1 dollars invested in the first portfolio, i2 dollars in the second portfolio, . . . , and iK dollars in the Kth portfolio. If the fks are not the returns on portfolios, then equation (7.1) can be interpreted as saying that one dollar invested in the ith security behaves like a portfolio of ik dollars invested in a portfolio with returns that are perfectly cor- related with changes in the kth factor. In addition, there is a constant component of the return ␣i and an idiosyncratic component i, which is small if the factors explain most of the variation in the portfolio return. If one ignores the residual
i, then equation (7.1) corresponds to the mapping performed in earlier chap- ters. That is, the factor model maps the ith security onto the common factors.