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.
Most applications of VaR are used to control for risk over short horizons and require a conditionalValue-at-Risk estimate that employs information up to time t to produce a VaR for some timeperiod t + h.Definition 8.3 (Conditional Value-at-Risk). The conditional α Value-at-Risk is defined asP r (rt +1 < −V aRt +1|t|Ft) = α (8.3)where rt +1 =Wt +1−WtWtis the time t + 1 return on a portfolio. Since t is an arbitrary measure oftime, 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 toestimate the required quantile of the time t + 1 return distribution. Five standard methods willbe presented in the order of the restrictiveness of the assumptions needed to justify the method,from strongest to weakest.
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