RiskMetrics™ —Technical Document

74 Chapter 4. Statistical and probability foundations
where R t
is an NxN time-dependent correlation matrix. The variance of each return, σ it ,
and the
correlation between returns, ρ ij,
t
, are a function of time. The property that the distribution of
returns is normal given a time dependent mean and correlation matrix assumes that returns follow
a conditional normal distribution—conditional on time. Notice that in Eq. [4.54] we excluded term
µ i
. As will be discussed in more detail in Section 5.3.1.1, the mean return represented by µ i
is set
to zero.
In Appendix A we propose a set of statistical tests to assess whether observed financial returns follow
a conditional normal distribution. In Appendix B we discuss alternative distributions that
relax the normality assumption.
2
4.7 Summary
In this chapter, we presented the statistical and probability assumptions on the evolution and distribution
of financial returns in some simple models. This discussion served as background to the
specification of the assumptions behind the RiskMetrics VaR methodology.
In review, this chapter covered the following subjects. The chapter began by outlining a simple
version of the VaR calculation. We then:
• Defined absolute price change, relative price change, log price change, and returns.
• Showed the importance of understanding the use of different price change definitions.
• Established that RiskMetrics measures changes in portfolio value in terms of continuouslycompounded
returns.
• Introduced temporal aggregation and cross-section aggregation to show the implications of
working with relative and log returns.
• Introduced the random walk model for: 29
– Single-price assets
– Fixed income instruments
• Found evidence that contradicts the assumption that returns are IID (identically and independently)
normal. In reality, continuously compounded returns are:
– Not identical over time. (The variance of the return distribution changes over time)
– Not statistically independent of each other over time. (Evidence of autocorrelation between
return series and within a return series.)
• Explained the properties of the normal distribution, and, lastly,
• Presented the RiskMetrics model as a modified random walk that assumes that returns are
conditionally normally distributed.
29 While the random walk model serves as the basis for many popular models of returns in finance, another class of
models that has received considerable attention lately is based on the phenomenon of long-range dependence.
Briefly, such models are built on the notion that observations recorded in the distant past are correlated to observations
in the distant future. (See Campbell, et. al (1995) for a review of long-range dependence models.)
RiskMetrics —Technical Document
Fourth Edition