[R] MLE Estimation in Extreme Value Approach to VaR (using Frechet distribution)

Preetam Pal lordpreetam at gmail.com
Fri Sep 23 00:40:33 CEST 2016

Hi R-Users,

I am trying to estimate 95%-le VaR (Value-at-Risk) of a portfolio using
Extreme Value Theory. In particular, I'll use the Frechet distribution
(heavy left tail),

I have data on percentage returns ( R_t) for T = 5000 past dates. This data
has been divided into g = 50 non-overlapping periods of size n = 100 each
and we compute the minimum return r_i over each period (i = 1,2,3,....,50)

Firstly, I need to estimate, by maximum likelihood approach, the 3 unknown
parameters:  a (scale), b (shift) and alpha = -1/k (tail index)
 Interpretation: *(r_i - b)/a*  converges to the *Frechet distribution*,
which is given by: *F*(x) = 1 - exp[ -( 1+kx )^(1/k) ]*

 The likelihood (to be maximized wrt a,b and k ) is given by: L = f(r_1) *
f(r_2) *......*f(r_g),
 where *f(r_i)  =  (1/a) * [ 1 + k*m_i ]^(-1+ 1/k) * exp[- ( 1 +
k*m_i)^(1/k) ]*  i = 1,2,3,.....g
 Here, as a short-hand, I have used m_i = (r_i - b)/a

My question is: this ML-estimation by differentiating L is going to be
extremely messy and the data may be poorly-conditioned (eg, the returns
data may be positive, negative and of very small magnitude [~ 10^(-5) to
10^(-3) ].)
Wanted your help in performing this estimation process efficiently.

As a wrap, the 95%-le VaR would finally come to *VaR = b - (a/k) * [ 1 -
{-n*log(0.95)}^k ]*, but of course, I need to plug in the estimated a,b and
k values here.

Any help will be sincerely appreciated.
(For details, you can use Section 7.5.2 & 7.6 of '*Analysis of Financial
Time Series*' by Ruey S.Tsay -2nd edition)

- -
Preetam Pal
M-Stat 2nd Year,                                             Room No. N-114
Statistics Division,                                           C.V.Raman
Indian Statistical Institute,                                 B.H.O.S.

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