Measuring Market Risk - Reserve Bank of India

MEASURING MARKET RISK 29
(i.e. the inverse of log-transformation) of a real number x is anti-log(x) = e x ;
sometimes denoted by exp(x).
4
It may be noted that the simple HS method corresponds to δ =1, where each
of the past k returns is assigned a constant weight 1/k.
5
Conventionally, μ t+1|t
is considered to be zero, though one can model the
return process to have estimates of time-varying/conditional means.
6
The symmetric hyperbolic distribution is a special case of generalized
hyperbolic distribution which depends on six parameters. For a discussion
of hyperbolic distribution, generalized and symmetric, one may see Bauer
(2000).
7
For more discussions on fitting symmetric hyperbolic distribution, one may
see the papers referred by Bauer (2000), such as, Eberlein and Keller (1995).
8
See, also, Gujarati (1995) for a discussion on the issues relating to Jarque-
Bera (1987) test for normality.
9
In this case the null hypothesis of zero skewness could be rejected only at
10% or higher level of significance.
10
For the sake of brevity, we present VaR estimates only for one day. But we
have noticed the similar pattern in other days in our database also.
Select References
Artzner, Philippe, Freddy Delbaen, Jean-Marc Eber and David Heath
(1999), Coherent Measures of Risk, Mathematical Finance, Vol. 9, No. 3
(July), pp. 203-28.
Baillie, R. T., Bollerslev, T. and Mikkelsen, H. O. (1996a), "Fractionally
Integrated Generalized Autoregressive Conditional Heteroskedasticity",
Journal of Econometrics, 74, 3–30.
Basel Committee (1988), International Convergence of Capital
Measurement and Capital Standards - Basel Capital Accord, Bank for
International Settlements.
Basel Committee (1996a), Amendment to the Capital Accord to Incorporate
Market Risks, Bank for International Settlements.
Basel Committee (1996b), Supervisory Framework for the Use of
‘Backtesting’ in Conjunction with Internal Models Approach to Market
Risk, Bank for International Settlements.
Bauer, Christian (2000), “Value at Risk Using Hyperbolic Distributions”,
Journal of Economics and Business, Vol. 52, pp. 455-67.