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normality. Parameters can thus be estimated by (quasi) maximum likelihood estimation,
yielding a consistent estimate of the conditional variance process. The
day-ahead forecast is then given by:
b 2 T +1jT = b! + br 2 T + b b 2 T
(A.5)
As in the EWMA model discussed previously, given the GARCH estimates b t
and the resultant standardized innovations, b t = r t =b t ; the ‘robust’one-day VaR-
GARCH forecast is determined as:
V aR ;T +1 (GARCH) = Q (b t )b T +1jT : (A.6)
with Q (b t ) denoting the -quantile of the empirical distribution.
C. VaR Extreme Value Theory
This method can be seen as a parametric re…nement of the previous approaches.
Essentially, the procedure requires the characterization of the tail behavior of the
set of i.i.d. innovations t in the return process. To circumvent the problem that t
is not observable directly, the estimated residuals b t = r t =b t can be used instead,
with b t determined according to some volatility model, such as those discussed
previously. Since GARCH estimates tend to outperform any other procedure, we
estimate the empirical process b t on the basis of the GARCH(1,1) estimates as
discussed above.
The rest of the procedure is described as follows. Given the series b t ; the total
sample period is divided into B = 740 blocks of length l = 5 observations to record
the maximum value of each block (i.e., the maximum loss in the period), say m b ,
b = 1; :::; B; in a time-series process. The Extreme Value Theory suggests …tting
the Generalized Extreme Value distribution (GEV, also known as Fisher–Tippett
distribution) to this series. The GEV arises as the limit distribution of properly
normalized maxima of a sequence of i.i.d. random variables, and is characterized
by the density function
f (z b ; 1 ; 2 ; 3 ) =
1 1=3 1
n
[1 +
3 z b ] exp
2
[1 + 3 z b ]
1= 3
o
(A.7)
if z b > 1; where z b = (m b 1 ) = 2 denotes the standardized variable. The
(unknown) parameters characterize the shape ( 1 ), scale ( 2 ) and location ( 3 ) of
the distribution and can be estimated consistently by di¤erent methods, such as
maximum-likelihood. The importance of this approach is that by inverting this
distribution (with the unknown parameters replaced by their consistent estimates),
we can go from the asymptotic GEV distribution of maxima to the distribution of
the observations themselves and obtain a closed-form expression for the unconditional
VaR of b t given ; namely,
"
b
Q (b t ) = b 2
3 1
b 1
log 1
# b1
1
l
(A.8)
Finally, as in the EWMA and GARCH approaches, we generate the one-day ahead
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