Assumptions

VaR is estimated using the expression
VaRóPpp 2.33 t (2.1)
where Ppp 2.33 is Daily-Earnings-at-Risk (DEaR), which describes the magnitude of
the daily losses on the portfolio at a probability of 99%; pp is the daily volatility
(standard deviation) of portfolio returns; t is the number of days, usually ten, over
which the VaR is estimated; p is usually estimated using the historical variance–
covariance. In the historical variance–covariance approach, the variances are
defined in
p2
t ó1
T
; T
ió1
e2
tñi (2.2)

where e is the residual returns (defined as actual returns minus the mean). On the
other hand, the ES approach is expressed as
p2
t óje2
tñ1ò(1ñj) ;
ê
ió1
jie2
tñi (2.3)
where 0OjO1 and is defined as the decay factor that attaches different weights over
the sample period of past squared residual returns. The ES approach attaches
greater weight to more recent observations than observations well in the past. The
implication of this is that recent shocks will have a greater impact on current volatility
than earlier ones. However, both the variance–covariance and the ES approaches
require strong assumptions regarding the distributional properties of security
returns.
The above volatility estimates relies on strong assumptions on the distributional
properties of security returns, i.e. they are independently and identically distributed
(i.i.d. thereafter). The identically distributed assumption ensures that the mean and
the variance of returns do not vary across time and conforms to a fixed probability
assumption. The independence assumption ensures that speculative price changes
are unrelated to each other at any point of time. These two conditions form the basis
of the random walk model. Where security returns are i.i.d. and the mean and the
variance of the distribution are known, inferences made regarding the potential
portfolio losses will be accurate and remain unchanged over a period of time. In these
circumstances, calculating portfolio VaR only requires one estimate, the standard
deviation of the change in the value of the portfolio. Stationarity in the mean and
variance implies that the likelihood of a specified loss will be the same for each day.
Hence, focusing on the distributional properties of security returns is of paramount
importance to the measurement of risk. In the next section, we examine whether
these assumptions are valid.