Most applications of VaR are used to control for risk over short horizons and require a conditional
Value-at-Risk estimate that employs information up to time t to produce a VaR for some time
period t + h.
Definition 8.3 (Conditional Value-at-Risk). The conditional α Value-at-Risk is defined as
P r (rt +1 < −V aRt +1|t
|Ft
) = α (8.3)
where rt +1 =
Wt +1−Wt
Wt
is the time t + 1 return on a portfolio. Since t is an arbitrary measure of
time, t + 1 also refers to an arbitrary unit of time (day, two-weeks, 5 years, etc.)
Most conditional models for VaR forecast the density directly, although some only attempt to
estimate the required quantile of the time t + 1 return distribution. Five standard methods will
be presented in the order of the restrictiveness of the assumptions needed to justify the method,
from strongest to weakest.