Value at Risk (VaR) models are used in many financial
applications. Their goal is to quantify the profit or loss of a
portfolio which could occur in the near future. The uncertainty of
the development of a portfolio is expressed in a ``forecast
distribution''
for period
.
is the conditional distribution of the random variable
,
which represents the possible profits and losses of a portfolio in
the following periods up to date
, and
stands
for the information in the available historical data up to date
. An estimator for this distribution is given by the forecast
model. Consequently the possible conditional distributions of
come from a parameter class
. The
finite-dimensional parameter
is typically estimated
from
historical return observations at time
, that is
approximately the trading days in a year. Letting
stand for this estimator then
can be approximated with
.
An important example of
is the Delta-Normal
Model, RiskMetrics (1996). In
this model we assume that the portfolio is made up of
linear
(or linearized) instruments with market values
and that the combined conditional distribution of the log
returns of the underlying
given the information up to time
is a multivariate normal
distribution, i.e.,
 |
(16.2) |
where
is the (conditional) covariance matrix of the
random vector
. We consider first a single position
, which is made up of
shares of a single
security with an actual market price
. With
we represent the exposure of
this position at time
, that is its value given
. The
conditional distribution of the changes to the security's value
is approximately:
with
Here we
have used the Taylor approximation
 |
(16.4) |
The generalization to a portfolio that is made up of
shares of
(linear) instruments is quite
obvious. Let
be the
-dimensional exposure vector at time
 |
(16.5) |
is the change in the value of the portfolio. For a single position
the conditional distribution of
given the information
is approximately equal to the conditional
distribution of
In the framework of Delta-Normal models this distribution belongs
to the family
N |
(16.6) |
with
The goal of the VaR
analysis is to approximate the parameter
and thus to approximate the forecast distribution of
.
Now consider the problem of estimating the forecast distribution
from the view point of the following model's assumptions. The
change in the value of the portfolio is assumed to be of the form
where
is i.i.d. N(0,1) distributed random variable,
is the exposure vector at time
and
is the (conditional) covariance matrix of the
vector
of the log returns. We combine the last
realizations of
from
the log return vector with a
matrix
. From these observations we
calculate two estimators from
; first the naive RMA,
i.e., rectangular moving average:
 |
(16.9) |
Since the expected value of the vector of returns
is zero
according to the Delta-Normal model, this is exactly the empirical
covariance matrix. The second so called EMA estimator, i.e., exponentially moving average, is based on an idea from
Taylor (1986) and uses an exponential weighting scheme.
Define for
a log return vector is exponentially weighted over time and a
matrix is constructed from this, then
is
estimated with
 |
(16.10) |
This normalization makes sense, since the sum
for
converges to
, thus the RMA estimator is the boundary case of
the EMA estimator. Both estimators can be substituted in
(15.7) and (15.8), and we obtain with

N
an approximation of the forecast distribution, i.e., the
conditional distribution of
. It should be noted that the
Bundesanstalt für Finanzdienstleistungsaufsicht
(
www.bafin.de
) currently dictates the
RMA technique.
The Value at Risk VaR is
determined for a given level
by
 |
(16.11) |
and estimated with
 |
(16.12) |
Here
represent the distribution function
of
,
. The quality of the forecast is of
particular interest in judging the VaR technique. It can be
empirically checked using the realized values
. In the event that the model assumptions, for
example, (15.7) and (15.8), are correct for the form
of the forecast's distribution, then the sample
should have independent uniformly distributed random
values over the interval
and
approximately independent identically uniformly
distributed random values. Then the ability of the forecasts
distribution to fit the data is satisfied.