130 Optimizing Optimization
to generate a distribution with some given marginals and a given dependence
structure.^18
Step 3. In order to estimate future returns valued at time T 1, we first
estimate a model ARMA(1,1)-GARCH(1,1) for the residuals of the factor
model (5.2). That is, we consider the empirical residuals:
erˆˆit it, i ˆij jtf
j,,,
αβ
114
∑and then we estimate the parameters g i,0 , g i (^) ,1 , h i (^) ,1 , k i (^) ,0 , k i (^) ,1 , p i (^) ,1 for all
I 1, ... ,30 of the ARMA(1,1)-GARCH(1,1):
ˆˆ,,, ,, ,
,,
egge hq q
qvz
vk
it i i it i i t i t
it it it
it
,
,
,
01111
2
iiiit iitkv pq
tT
,,, 011 ,,;
2
11
2
130 1
i ,..., ; ,...,. (5.6)
Moreover , as for the factor innovation, we approximate with α (^) j -stable distri-
bution Sαi(, , )σβμiii for any i 1, ... ,30 the empirical standardized innovations
zqvˆit,,, ˆit/ it, where the innovations qegge hqˆi t,,,,, ,, ˆˆit i 01111 i iti i t.
Then, we can generate S scenarios α (^) j -stable distributed for the standardized
innovations zs SiT
s
,
() ,,
1 1...^ and from Equation (5.6) we get S possible sce-
narios for the residuals evzsSiT(),s 1 iT, (^1) iT(),s 1 1,...,. Therefore, combin-
ing Step 2 with the estimation of future residuals from factor model (5.2), we
get S possible scenarios of returns:
,
()
, ,
()
,
rfˆ ˆ e() sS,,....
iT
s
iijjt
s
j
it
s
1
1
14
αβ∑ 1
(5.7)
The procedure illustrated here permits one to generate S scenarios at time
T 1 of the vector of returns.
5.4 The portfolio selection problem
Suppose we have a frictionless market in which no short selling is allowed
and all investors act as price takers. The classical portfolio selection problem
18 See, among others, Rachev et al. (2005) , Sun et al. (2008a) , Biglova et al. (2008) , and
Cherubini, Luciano, and Vecchiato (2004) for the definition of some classical copula used in
finance literature.