MPF_RRFI - Lecture 05
Historical Simulation and Extreme Value Theory
(Chapter 13)
Methodology
one-day VaR, confidence 99\%
indenify risk factors
calculate last 500 daily returns
99\% VaR is the 5th worst outcome
for ES calculate average of losses that are worse than VaR (the worst four losses)
Stressed VaR find a historical period of 251 days (250 consecutive returns) where the
VaR was the worst
Accuracy of VaR
standard error of historical VaR
where -th percentile of the distribution is estimated as , is the number of observations,
is probability density function
normal distribution is not the best assumption (fat tails), alternative e.g. Pareto distribution
Extreme Value Theory
describe the science of estimating the tails of a distribution
used to improve VaR or ES estimates when we are interested in a very high confidence level
it is a way of smoothing and extrapolating the tails of an empirical distribution
Model-Building Approach (Chapter 14)
variance-covariance approach
in some cases faster than historical simulation approach
extension of Martowitz portfolio theory (normality assumptions)
The Basic Methodology
⟹
√
1
f(x)
(1 − q)q
n
q x n
f(x)
it uses VaR and ES equations from Chapter 12 (equation 12.1 and 12.2)
we need to estimate the standard deviation
using the assuption that changes are normally distributed calculate the selected percentile
for multi-asset portfolio we need to know the correlation calculate standard deviation
based on portfolio theory
Generalization
assumption that the change in the value of the portfolio is linearly related to proportional
changes in the risk factors so that
where is the dollar change in the value of the whole portfolio in one day, is the
proportional change in the ith risk factor in one day, is a variation of the delta risk measure
standard deviation for assets
Extensions of the Basic Procedure
stressed VaR and ES
non-normal distributions
Monte Carlo simulations for non-linear portfolios
V aR = μ + σN
−1
(X)
ES = μ + σ
e
−Y
2
/2
√2π(1 − X)
⟹
ΔP =
n
∑
i=1
δi
Δxi
ΔP Δxi
δi
n
σP
=




⎷
n
∑
i=1
n
∑
j=1
ρij
δi
δj
σi
σj