5. Risk management
VaR is a common tool used in assessing and managing the risk of a portfolio of securities. Many of the methods used to calculate VaR are based on the assump-
tion that security returns are conditionally normally distributed. Differences be- tween these methods often reflect differences in the modeling of the security
returns’ unconditional means and, in particular, differences in the modeling of their covariance structure.
Consider a simple portfolio of two securities where the weight w is placed in
1
Ž .
security 1 and the weight w 1 y w
is placed in security 2. Assume that the
2 1
security variances are s
2
and s
2
, respectively, and that for short return horizons
1 2
the mean returns are themselves negligible. As before, we let r denote the correlation between the security returns.
If these parameters are known and the security returns are assumed to be bivariate normally distributed then it is straightforward to assess the percentiles of
this portfolio’s return. In particular, the portfolio variance is given by: s
2
s w
2
s
2
q w
2
s
2
q 2 w w rs s
P 1
1 2
2 1
2 1
2
and the VaR at the a significance level for an initial investment of I is then I s z
P a
w x
where z is that point of the standard normal distribution Z for which P Z z
a a
s a. For a prespecified a and a known initial investment I , the VaR is then the square root of a linear function of r. It follows then that the specification of the
correlation r plays a key role in assessing VaR. To better see this, consider a special case in which we invest equal amounts in
two securities having equal variance s
2
. In this case the portfolio variance reduces to
1 1
2 2
s s s q
r .
P
ž
2 2
Taking the base case as r s 0, we see that the resultant VaR is
1 I z s
,
a
2 while for r 0, VaR is given by
1 1
I z s q
r .
a P
2 2
Hence the ratio of the VaR in the general case to the base case is given by 1
q r . For example, compared to r s 0, this ratio is 31.6 when r s 0.1, 50.0 when r
s 0.25 and 70.7 when r s 0.50.
Fig. 3. Germany–UK 95 VaR ratios.
The behavior of this ratio for varying values of r allows us to assess the effects of stochastic correlation on portfolio risk. Using an estimated 95 confidence
band for r, Fig. 3 plots the corresponding band of upper and lower estimated VaR ratios for the Germany–UK return series through time. As can clearly be seen,
sharp movements in estimated correlation are translated into sharp movements in estimated VaR. Furthermore, the inherent uncertainty surrounding r clearly
translates into uncertainty surrounding VaR. Risk management techniques that ignore the stochastic component of correlation are likely to provide erroneous risk
assessments.
6. Conclusions
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