Abstract
The aim of this paper is to provide empirical evidence on the statistical distributions of returns on 32 UK sector indices as well as the FTSE-All and the FTSE-100 indices. These data are modelled for several holding periods, ranging from one day to one quarter, using symmetric stable Paretian distributions and their characteristic exponents are estimated. Numerical results suggest that both short and long horizon returns are non-normal and that deviation from normality is stronger for short horizon returns, with the exception of few sectors. In sum, these results suggest that asset pricing and risk management models, among others, should be modified to take into account departures form normality.
Notes
We do not provide summary statistics for different return horizons to save space. However, all results are available upon request.
Various procedures for estimating the parameters α, β, δ, and c have been proposed in the literature such as the maximum likelihood approach (DuMouchel, Citation1971, Citation1973; Chen, Citation1991), Hill estimator (Hill, Citation1975), the method-of-moments approach (Press, Citation1972), the regression-type estimation approach (Koutrouvelis, Citation1980, Citation1981), tail estimation (Rachev and Mittnik, Citation1996), and the quantile approach (Fama and Roll, Citation1968, Citation1971; McCulloch, 1986), among others. See also de Haan and Perreira (Citation1999). However, for f ∈ [0.95, 0.97], the estimator we use is robust against variations in the value of α (Fama and Roll, Citation1968, Citation1971).
Mandelbrot
,
B
.
1963
.
The variation of certain speculative prices
.
Journal of Business
,
36
:
394
–
419
.
Fama
,
EF
.
1965
.
The behavior of stock market prices
.
Journal of Business
,
38
:
34
–
105
.
Officer
,
R
.
1972
.
The distribution of stock returns
.
Journal of the American Statistical Association
,
67
:
807
–
12
.
Rachev
,
S
and
Mittnik
,
S
.
1989
.
Stable distributions for asset returns
.
Applied Mathematics Letters
,
2
:
301
–
4
.
McCulloh
JH
1996
Financial applications of stable distributions
Handbook of Statistics
14
(Eds) G. S. Maddala and C. R. Rao
pp. 393–425
McDonald
JB
1996
Probability distributions for financial models, in
Handbook of Statistics
14
(Eds) G. S. Maddala and C. R. Rao
pp. 427–61
Rachev
S
Mittnik
S
2002
Stable Paretian Models in Finance, John Wiley & Sons Ltd.
Chichester
DuMouchel
W
1971
Stable distributions in statistical inference
PhD dissertation
University of Ann-Arbor
MI
DuMouchel
,
W
.
1973
.
On the asymptotic normality of the maximum-likelihood estimate when sampling from a stable distribution
.
Annals of Statistics
,
3
:
948
–
57
.
Chen
Y
1991
Distributions for asset returns
PhD dissertation
Stony Brook University, Department of Economics
Hill
,
BM
.
1975
.
A simple general approach to inference about the tail of distribution
.
Annals of Statistics
,
3
:
1163
–
74
.
Press
,
SJ
.
1972
.
Estimation of univariate and multivariate stable distributions
.
Journal of the American Statistical Association
,
67
:
842
–
6
.
Koutrouvelis
,
IA
.
1980
.
Regression-type estimation of the parameters of stable laws
.
Journal of the American Statistical Association
,
75
:
918
–
28
.
Koutrouvelis
,
IA
.
1981
.
An iterative procedure for the estimation of the parameters of the stable law
.
Communications in Statistics-Simulation and Computation
,
10
:
17
–
28
.
Rachev
,
S
and
Mittnik
,
S
.
1996
.
Tail estimation of the stable index α
.
Applied Mathematics Letters
,
9
:
53
–
6
.
Fama
,
EF
and
Roll
,
R
.
1968
.
Some properties of symmetric stable distributions
.
Journal of the American Statistical Association
,
63
:
817
–
36
.
Fama
,
EF
and
Roll
,
R
.
1971
.
Parameter estimates for symmetric stable distributions
.
Journal of the American Statistical Association
,
66
:
331
–
8
.
de Haan
,
L
and
Pereira
,
TH
.
1999
.
Estimating the index of a stable distribution
.
Statistics & Probability Letters
,
41
:
39
–
55
.
Fama
,
EF
and
Roll
,
R
.
1968
.
Some properties of symmetric stable distributions
.
Journal of the American Statistical Association
,
63
:
817
–
36
.
Fama
,
EF
and
Roll
,
R
.
1971
.
Parameter estimates for symmetric stable distributions
.
Journal of the American Statistical Association
,
66
:
331
–
8
.