LO 45.5: Evaluate the tradeoffs involved in setting the threshold level when applying the GP distribution.
The Gnedenko-Pickands-Balkema-deHaan (GPBdH) theorem says that as u gets large, the distribution Fu(x) converges to a generalized Pareto distribution (GPD), such that:
if ^ exp 1 exp
if = 0
The distribution is defined for the following regions:
x > 0 for 6, > 0 and 0 < x < (3/6, for 6, VaR]. Because it gives an insight into the distribution of the size of losses greater than the VaR, it has become a popular measure to report along with VaR.
The expression for VaR using POT parameters is given as follows:
where: u = threshold (in percentage terms) n = number of observations N = number of observations that exceed threshold
The expected shortfall can then be defined as:
Example: Compute VaR and expected shortfall given POT estimates
Assume the following observed parameter values:
(3 = 0.75. = 0.25.
u = 1%. . Nu/n = 5%. Compute the 1% VaR in percentage terms and the corresponding expected shortfall measure.
VaR 1 + 0.25[0.05 0.75 1
, 0.99 1 0.99
0.75-0.25×1 2.486 ———– h—————— = 3.981% 1-0.25
2018 Kaplan, Inc.
Topic 45 Cross Reference to GARP Assigned Reading – Dowd, Chapter 7
G e n e r a l
i z e d E x t r e m e V a l u e a n d P e a k s – O v e r -T h r e s h o l d