In this section we study the portfolio choice problem analyzing the asymptotic behavior of data. In particular, we consider unbounded random portfolios of stable distributed returns, xr, that, with abuse of notation, we continue to call as portfolios of stable distributed returns.9
9 If logZiis stable distributed, thenZi=1+ziis log-stable distributed.
562 S. Ortobelli et al.
The recent crashes observed in the stock market showed that the stock returns are more volatile than those predicted by the models with finite variance of the asset returns. In the empirical financial literature, it is well documented that the asset returns have a distribution whose tail is heavier than that of the distributions with finite variance, i.e.,
P
|ri|> x
∼x−αiLi(x) asx→ ∞, (8) where 0< α <2 andLi(x)is a slowly varying function at infinity, i.e.,
xlim→∞
Li(cx)
Li(x) →1 for allc >0,
see Rachev and Mittnik (2000) and the references therein. In particular, in the data observed until now 1< α <2. The constrain 1< α <2 and the relation (8) imply that returnsri
admit finite mean and non-finite variance. The tail condition in (8) also implies that the vector of returnsr= [r1, . . . , rn]is in the domain of attraction of(α1, . . . , αn)-stable law.
That is, givenT i.i.d (independent and identically distributed) observations onr, namely r(t )=
r1(t ), . . . , rn(t )
, t=1,2, . . . , T , then, there exist normalizing constants
a(T )=
a1(T ), . . . , an(T )
∈R+n and b(T )=
b(T )1 , . . . , b(T )n
∈Rn, such that
T i=1
r1(i)
a1(T )+b(T )1 , . . . , T i=1
r1(i) a(T )n +b(T )n
−→d S(α1, . . . , αn) asT → ∞, (9)
whereS(α1, . . . , αn)is (α1, . . . , αn)-stable random variable. This convergence result is a consequence of the stationary behavior of returns and of the Central Limit Theorem for normalized sums of i.i.d. random variables which determines the domain of attraction of each stable law [see Zolotarev (1986)]. Therefore, any distribution in the domain of attraction of a specified stable distribution will have properties close to those of the stable distribution. The constantsaj(T )in (9) have the form
aj(T )=T1/αjLj(T ),
whereLj(T )are slowly varying functions asT → ∞.
Ch. 14: Portfolio Choice Theory with Non-Gaussian Distributed Returns 563
Each component ofS(α1, . . . , αn)=(s1, . . . , sn)has a Pareto–Lévy stable distribution, i.e., its characteristic function is given by
Φj(t)=
exp
−σjαj|t|αj
1−iβjsgn(t)tan π αj
2
+iàjt
ifαj =1, exp
−σj|t|
1+iβj
2
π sgn(t)log|t|
+iàjt
ifαj=1,
(10)
whereαj∈(0,2)is the so-called stable (tail) index ofsj,σ >0 is the scale (or dispersion) parameter,β∈ [−1,1]is a skewness parameter andàis a location parameter. Moreover, for every fixedα, the Pareto–Lévyα-stable law is aσ τ3(a)¯ class. Whenα >1 the location parameteràis the mean. However, there is a considerable debate in literature concerning the applicability ofα-stable distributions as they appear in Lévy’s central limit theorems.
A serious drawback of Lévy’s approach is that in practice one can never know whether the underlying distribution is heavy tailed, or just has a long but truncated tail. Limit theo- rems for stable laws are not robust with respect to truncation of the tail or with respect to any change from light to heavy tail, or conversely. Based on finite samples, one can never justify the specification of a particular tail behavior. Hence, one cannot justify the applica- bility of classical limit theorems in probability theory. Therefore, instead of relying on limit theorems, we can use the so-called pre-limit theorem which provides an approximation for distribution functions in case the number of observationT is “large” but not too “large”
[see Klebanov, Rachev and Szekely (2001), Klebanov, Rachev and Safarian (2000)]. In particular the “pre-limiting” approach helps to overcome the drawback of Lévy-type cen- tral limit theorems. As a matter of fact, we can assume that returns are bounded “far away”, say daily returns cannot be outside the interval[−0.5,0.5]. Thus, considering the empir- ical observation on asset returns, we can assume that the asset returns ri are truncated αi-stable distributed with support,[−0.5,0.5]. Even if the returns will be attracted by the CLT to the Gaussian law, pre-limit theorems show that for any reasonableT the truncated stable laws will be attracted to the stable laws. Therefore, it is plausible assuming that the vector of returns r= [r1, . . . , rn] is in the domain of attraction of a n-dimensional (α1, . . . , αn)-stable law.
In order to express a multi-parameter choice in portfolio selection theory coherent with the empirical evidence and consistent with the expected utility maximization, we need the asymptotic distributional assumptionconsisting in:
(1) (Heavy tailedness assumption) Portfolios xr are unbounded random variables belonging toLpwith 1< p2 and the return vectorr= [r1, . . . , rn]is in the domain of attraction of(α1, . . . , αn)-stable law (1< αi 2, i=1, . . . , n). The assumption 1< αi 2 is supported by increasing empirical results as shown by Mandelbrot (1963a, b, 1967), Fama (1963, 1965, b), Mittnik, Rachev and Paolella (1997), Rachev and Mittnik (2000).
(2) (Consistency with the expected utility maximization) The distributions of the portfolio returnsxrbelong to the sameσ τk(a)¯ class of distribution functions.
564 S. Ortobelli et al.
Under these assumptions, as for Theorem 2, we obtain an admissible frontier for non- satiable and non-satiable risk averse investors.
A simpler way to express the asymptotic behavior of data consists in considering every portfolio in the domain of attraction of a Pareto–Lévyαstable distribution withα >1.
Given that, we implicitly assume that all optimal choices are identified by four parameters of the underlined stable law. Therefore, every portfolioxrcan be well approximated by a stable distribution, i.e., we can assume:
xr+(1−xe)z0 d
=Sα(x)
σ (x), β(x), à(x)
, (11)
wherez0is the riskless return,α(x)∈(min1inαi,2)is the index of stability,αj>1 is the index of stability of thej-th asset return,σ (x)is the scale parameter,à(x)=xE(r)+ (1−xe)z0is the mean andβ(x)is the skewness parameter. Properties ofσ τ4(a)¯ class are verified with this parameterization, so according to Theorem 2 every risk averse investor will choose a portfolio weight, solution of the following constrained problem
minx σ (x) subject to xE(r)+(1−xe)z0=m,
(12) β(x)=β∗,
α(x)=α∗
for somem,β∗,α∗. In this case, we are not able to find a closed form of the efficient frontier because we do not know a priori the joint distribution of the asset returns. In or- der to overcome this problem, we could consider another admissible parameterization of stable distribution for problem (11). For example, we can prove that the meanà(x)= xE(r)+(1−xe)z0, the scale parameters(x)=E(|xr−xE(r)|)and the fundamental ratios
ρ1(x)=E(|xr−xE(r)|q1)
(s(x))q1 and ρ2(x)=E((xr−xE(r))q2) (s(x))q2 ,
whereq1, q2∈(1,min1inαi);represent a parameterization which verifies the properties ofσ τ4(a)¯ class.10In fact, first observe thatρ1(x)andρ2(x)do not depend onà(x)and σ (x)because
xr−xE(r)q1=d σ (x)q1Sα(x)(1, β(x),0)q1,
10The symbologyxtstands for sgn(x)|x|t.
Ch. 14: Portfolio Choice Theory with Non-Gaussian Distributed Returns 565
and also
xr−xE(r)q2 d
=σ (x)q2
Sα(x)(1, β(x),0)q2.
Thus, as a consequence of Property 1.2.17 in Samorodnisky and Taqqu (1994) ρ1(x)=E(|xr−xE(r)|q1)
(s(x))q1
=K 6(1−q1/α(x))cos(arctan(β(x)tan(π α(x)/2))q1/α(x)) (6(1−1/α(x))cos(arctan(β(x)tan(π α(x)/2))1/α(x)))q1,
whereK is a constant that depends only onq1. Hence, for every q1∈(1,min1inαi) and for every fixedβ(x),ρ1(x)is a decreasing function ofα(x)on the existence inter- val. Moreover, ρ1(x) is an even function of β(x) and it decreases in |β(x)| for fixed α(x)∈(min1inαi,2). Instead, ρ2(x) is an increasing odd function of β for every q2∈(1,min1inαi)and for every fixedα(x)∈(min1inαi,2). These relations im- ply that ρ1(x)andρ2(x)uniquely determinateα(x)andβ(x). Then, under the assump- tion (11), every risk averse investor will choose a portfolio weight, solution of the following constrained problem
minx Exr−xE(r) subject to xE(r)+(1−xe)z0=m,
(13) E(|xr−xE(r)|q1)
(s(x))q1 =ρ1, E((xr−xE(r))q2)
(s(x))q2 =ρ2
for some m,ρ1, ρ2. Differently from problem (12), problem (13) does not require the knowledge of the joint distribution of asset returns but it is still computationally too com- plex. Generally, in order to identify the efficient frontier and reduce the number of para- meters, we assume that αi =α for alli=1, . . . , n. Observe that stable distributions are stable with respect to summation of i.i.d. random stable variables and the vector of returns r= [r1, . . . , rn] isα-stable distributed withα >1 if and only if all linear combinations are stable [see Samordinsky and Taqqu (1994, Theorems 2.1.2 and 2.1.5)]. In this case the joint characteristic function of returns is given by
Φr(t)=exp
−
Sn
|ts|α
1−i sgn(ts)tan π α
2
γ (ds)+ità
,
where α is the index of stability, γ (ds) is the spectral measure concentrated on Sn= {s∈Rn| s =1}.
566 S. Ortobelli et al.
Thus, when the vector of returns isαstable distributed (withα >1), every portfolio xr+(1−xe)z0(except the riskless return, i.e.,x=0) is distributed as
xr+(1−xe)z0 d
=Sα
σ (x), β(x), à(x) , where
à(x)=xE(r)+(1−xe)z0, σ (x)=
Sn
|xs|αγ (ds) 1/α
and β(x)=
Sn|xs|αsgn(xs)γ (ds) (σ (x))α
are respectively the mean, the scale parameter and the skewness parameter of the portfolio xr+(1−xe)z0. Under this distributional assumption, every risk averse investor will choose a portfolio weight, solution of the following constrained problem
minx σ (x) subject to
xE(r)+(1−xe)z0=m, (14)
β(x)=β∗
for somem andβ∗. In order to determine estimates of the scale parameter and of the skewness parameter, we can consider the tail estimator for the index of stabilityαand the estimator for the spectral measureγ (ds)proposed by Rachev and Xin (1993) and Cheng and Rachev (1995). However, even if the estimates of the scale parameter and the skew- ness parameter are computationally feasible, they require numerical calculations. Thus, model (14) does not present an easy applicability from an empirical point of view. Sim- ilarly to problem (13), we can fixq < α and propose a different representation based on the moments type constrains. Therefore, instead of model (14), we obtain the following constrained problem
minx Exr−xE(r) subject to
xE(r)+(1−xe)z0=m, (15)
E((xr−xE(r))q2) (E(|xr−xE(r)|))q2 =ρ2
for somemandρ2. Optimization problems (15) and (13) can be used in a more general setting than optimization problems (12), (14). In fact, a priori other classes of distribution functions (not only stable distributions) for returns uniquely determined by the parameters m(x),s(x),ρ1(x)andρ2(x)could exist. Next, in order to overcome the intrinsic difficul- ties of the problems (12)–(14) and (15), we analyze different fund separation models that consider the asymptotic distributional assumption.
Ch. 14: Portfolio Choice Theory with Non-Gaussian Distributed Returns 567
3.1. The sub-Gaussian stable model
Assume the vector of returnsr= [r1, . . . , rn] is sub-Gaussianα-stable distributed with 1< α <2. Then, the characteristic function ofrhas the following form
Φr(t)=E
exp(itr)
=exp
−(tQt)α/2+ità
, (16)
whereQ= [Ri,j/2]is a positive definite(nìn)-matrix,à=E(r)is the mean vector, and γ (ds)is the spectral measure with support concentrated onSn= {s∈Rn| s =1}. The termRi,j is defined by
Ri,j
2 = [˜ri,r˜j]α˜rj2α−α, (17)
wherer˜j=rj−àjare the centralized return, the covariation[˜ri,r˜j]αbetween two jointly symmetric stable random variablesr˜i andr˜j is given by
[˜ri,r˜j]α=
S2
si|sj|α−1sgn(sj)γ (ds), in particular, ˜rjα =(
S2|sj|αγ (ds))1/α =([˜rj,r˜j]α)1/α. Here the spectral measure γ (ds)has support on the unit circleS2.
This model can be considered as a special case of Owen–Rabinovitch’s elliptical model [see Owen and Rabinovitch (1983)]. However, no estimation procedure of the model pa- rameters is given in the elliptical models with non-finite variance. In our approach we use (16) and (17) to provide a statistical estimator of the stable efficient frontier. To estimate the efficient frontier for returns given by (16), we need to consider an estimator for the mean vectoràand an estimator for the dispersion matrixQ. The estimator ofàis given by the vectoràˆ of sample averages. Using Lemma 2.7.16 in Samorodnitsky and Taqqu (1994) we can write for everypsuch that 1< p < α
[˜ri,r˜j]α
˜rjαα =E(r˜ir˜jp−1)
E(|˜rj|p) , (18)
where the scale parameterσj can be written˜rjα=σj. It can be approximated by the moment method suggested by Samorodnitsky and Taqqu (1994) (Property 1.2.17) in the caseβ=0
σjp= ˜rjpα=E(|˜rj|p)p+∞
0 u−p−1sin2udu
2p−16(1−p/α) . (19)
It follows Ri,j
2 =σj2E(z˜iz˜p−j 1)
E(|˜zj|p) =σj2−pp+∞
0 u−p−1sin2udu 2p−16(1−p/α) E
˜ ziz˜jp−1
.
568 S. Ortobelli et al.
The above suggests the following estimatorQ= [Ri,j/2]for the entries of the unknown covariation matrixQ
Ri,j
2 = ˆσj2−pp+∞
0 u−p−1sin2udu 2p−16(1−p/α)
1 N
N k=1
˜ z(k)i
˜
z(k)j p−1, (20)
where theσj2is estimated as follows
ˆ
σj2=Rj,j
2 =
1 N
N
k=1|˜rj(k)|pp+∞
0 u−p−1sin2udu 2p−16(1−p/α)
2/p
. (21)
The moment estimator makes most sense for each fixedp∈(1, α). The rate of conver- gence of the empirical matrixQ= [Ri,j/2]to the unknown matrixQ(to be estimated), will be faster, ifpis as large as possible, see Rachev (1991).
Now, let us recall that our portfolio satisfies the relation xr=d Sα(σxr, βxr, mxr)
and furthermore,W=z0whenx=0, otherwise W=xr+(1−xe)z0 d
=Sα(σxr, βxr, mW), whereαis the index of stability,σxr =√
xQxis the scale (dispersion) parameter,βxr=0 is the skewness parameter andmW=xE(r)+(1−xe)z0is the mean ofW. In particular, every sub-Gaussianα-stable family is a particularσ τ2(m, σ )class.
In view of what stated before, when the returnsr= [r1, . . . , rn]are jointly sub-Gaussian α-stable distributed, every risk averse investor will choose an optimal portfolio among all portfolio solutions of the following optimization problem:
minx xQx subject to xà+(1−xe)z0=mW (22) for some given meanmW, whereW =xr+(1−xe)z0. Thus, every optimal portfolio that maximizes a given concave utility functionu,
maxx E u
xr+(1−xe)z0 belongs to the mean–dispersion frontier
σ=
m−z0
(à−ez0)Q−1(à−ez0) ifmz0, z0−m
(à−ez0)Q−1(à−ez0) ifm < z0,
(23)
Ch. 14: Portfolio Choice Theory with Non-Gaussian Distributed Returns 569
whereà=E(r); m=xà+(1−xe)z0; e= [1, . . . ,1]; andσ2=xQx. Besides, the optimal portfolio weightsxsatisfy the following relation:
x=Q−1(à−z0e) m−z0
(à−ez0)Q−1(à−ez0). (24)
Note that (23) and (24) have the same forms as the mean–variance frontier. In particu- lar, they assume a more general form for nonnecessarily symmetric dispersion matrixQ.
As a matter of fact, even ifQis a symmetric matrix (it is definite positive) the estima- tor proposed in the sub-Gaussian cases (21) and (22) generally is not necessarily sym- metric. Therefore, in some extreme cases we could obtain the inconsistent situation of stable distribution associated to a portfoliox with square scale parameter equal or lower than zero.11 Moreover, (24) exhibits the two fund separation property for both the sta- ble and the normal case but the matrix Qand the parameter have different meaning. In the normal case, Q is the variance–covariance matrix and σ is the standard deviation, while in the stable caseQis a dispersion matrix andσ is the scale (dispersion) parameter, σ=√
xQx. According to the two-fund separation property of the sub-Gaussianα-stable approach, we can assume that the market portfolio is equal to the risky tangent portfolio under the equilibrium conditions (as in the classical mean–variance Capital Asset Pricing Model (CAPM)). Therefore, every optimal portfolio can be seen as the linear combination between the market portfolio
¯
xr= rQ−1(à−z0e) eQ−1à−eQ−1ez0
, (25)
and the riskless asset return z0. Following the same arguments as in Sharpe, Lintner, Mossin’s mean–variance equilibrium model, the return of assetiis given by:
E(ri)=z0+βi,m
E(x¯r)−z0
, (26)
11Observe that for everyx∈Rn,we getxQx > 0 if and only if(Q+(Q))/2 is a definite positive matrix.
Thus, we can verify that(Q+(Q))/2 is definite positive in order to avoid stable portfoliosxzwith negative scale parameter estimators. Moreover, we observe that the symmetric matrix (Q+(Q))/2 is an alternative estimator of the dispersion matrixQwhose statistical properties have to be proved. In particular, if we want to simulate the vectorz˜= [˜z1, . . . ,z˜n]of the centredαstable sub-Gaussian return distributions, we generally use the dispersion matrixΣ=(Q+(Q))/2. As a matter of fact, we first generate the vectorG= [G1, . . . , Gn] of the joint Gaussian distributionG=N (0,Σ) using the Cholesky decomposition matrix. Then, the vector of returns [see Samorodnitsky and Taqqu (1994)] is given by:
˜ z=√
A G,
whereA=dSα/2(2(cos(π α/4))2/α,1,0)is anα/2 stable random variable independent of the Gaussian vectorG.
570 S. Ortobelli et al.
whereβi,m= ¯xQei/x¯Qx¯, witheithe vector with 1 in thei-th component and zero in all the other components. As a consequence of Ross’ necessary and sufficient conditions of two-fund separation [see Ross (1978a)], the above model admits the form
ri=ài+biY +εi, i=1, . . . , n,
whereài =E(ri), E(ε/Y )=0, ε=(ε1, ε2, . . . , εn), b= [b1, . . . , bn] and the vector bY+εis sub-Gaussianα-stable distributed with zero mean.
Hence, our sub-Gaussian α-stable version of CAPM is not much different from Gamrowski–Rachev’s (1999) version of the two-fund separationα-stable model. As a mat- ter of fact, Gamrowski and Rachev (1999) propose a generalization of Fama’s α-stable model (1965b) assumingri =ài +biY +εi,for everyi=1, . . . , n, whereεi andY are α-stable distributed andE(ε/Y )=0. In view of their assumptions,
E(ri)=z0+ ˜βi,m
E(x¯r)−z0
,
where
β˜i,m= 1 α ¯xr˜αα
∂ ¯xr˜αα
∂x¯i =[˜ri,x¯r˜]α
¯xr˜αα
.
Furthermore, the coefficient[˜ri,x¯r˜]α/ ¯xr˜αα can be estimated using the above formula (18).
Now, we see that in the above sub-Gaussian symmetricα-stable modelx¯Qx¯= ¯xr˜2α andx¯Qei=12∂ ¯xr˜2α/∂x¯i. Thus, we get the equivalence between the coefficientβi,mof model (26) andβ˜i,mof Gamrowski–Rachev’s model, i.e.:
βi,m=x¯Qei
¯
xQx¯ = 1 σx¯r
∂σx¯r
∂x¯i =[˜ri,x¯r˜]α
¯xr˜αα = ˜βi,m, whereσx¯r is the scale parameter of market portfolio.
3.2. A three fund separation model in the domain of attraction of a stable law
Let us assume that the vectorr= [r1, . . . , rn]describes the following three-fund separat- ing stable model of security returns:
ri=ài+biY +εi, i=1, . . . , n, (27)
where the random vectorε=(ε1, ε2, . . . , εn) is independent fromY and follows a joint sub-Gaussianα1-stable distribution (1< α1<2 ), with zero mean and characteristic func- tion
Φε(t)=exp
−|tQt|α1/2 ,
Ch. 14: Portfolio Choice Theory with Non-Gaussian Distributed Returns 571
whereQis the definite positive dispersion matrix. On the other hand,Y=d Sα2(σY, βY,0) is α2-stable distributed random variable independent fromε,with 1< α2<2 and zero mean. Under these assumptions, the portfolios are in the domain of attraction of aαstable law with α=min(α1, α2) and belong to a σ τ3(a)¯ family. A testable case in which Y is α2-stable symmetric distributed (i.e.,βY =0), was recently studied by Gửtzenberger, Rachev and Schwartz (1999). WhenβY =0 andα1=α2, our model can lead to the two- fund separation Fama’s model. The characteristic function of the vector of returnsr = [r1, r2, . . . , rn]is given by:
Φr(t)=Φε(t)ΦY(tb)eità
=exp
−|tQt|α1/2− |tbσY|α2
1−iβYsgn(tb)tan π α2
2
+ità
, (28) whereb= [b1, . . . , bn]is the coefficient vector andà= [à1, . . . , àn]is the mean vector.
Next we shall estimate the parameter in model (27), (28). First, the estimator of àis given by the vectoràˆ of sample average. Then, we consider as factorY a centralized index return (for example the market portfolio (25) given by the above sub-Gaussian model).
Therefore, given the sequence of observationsY(k), we can estimate its stable parameters.
Observe that the random vectorεadmits a representation as a product of random variable V and Gaussian vectorG:
ε=V G.
V =√
A, whereAis anα1/2-stable subordinator, that is A=d Sα1/2
cos
π α1 4
2/α1
,1,0
;
G is a (n×1)-Gaussian vector with null mean and variance–covariance matrix Q and it is independent from A. We can generate values Ak, k=1, . . . , N, of Aindependent fromG. We address to Paulauskas and Rachev’s work (1999) the problem of generating such valuesAk. Using the centralizing returnsr˜j=rj−àj onY we write the following OLS estimators12forb= [b1, . . . , bn]andQ:
bˆi= N
k=1Y(k)r˜i(k)/Ak N
k=1(Y(k))2/Ak, i=1, . . . , n, and
Q= 1 N
N k=1
(r˜(k)− ˆbY(k))(r˜(k)− ˆbY(k))
Ak .
12For a discussion see Tokat, Rachev and Schwartz (2002).
572 S. Ortobelli et al.
The selection ofα1is a separate problem. A possible way to estimateα1is to consider the OLS estimatorb˜i=N
k=1Y(k)r˜i(k)/N
k=1(Y(k))2 and then to evaluate the sample resid- ualsε˜(k)= ˜r(k)− ˜bY(k). If these residuals are heavy tailed, one can take the tail expo- nent as an estimator forα1. The asymptotic properties of the above estimator can be de- rived arguing similarly with Paulauskas and Rachev (1999) and Gửtzenberger, Rachev and Schwartz (1999).
In order to determine portfolios R–S non-dominated when unlimited short selling is allowed, we have to minimize the scale parameter σW =√
xQx for some fixed mean mW =xà+(1−xe)z0 and b˜ =xb/√
xQx. Alternatively, as shown by Ortobelli, Rachev and Schwartz (2002), we can obtain these portfolios from the solution of the fol- lowing quadratic programming problem:
minx xQx subject to xà+(1−xe)z0=mW, xb=b∗
(29)
for somemW andb∗. Thus, under our assumptions, every portfolio that maximizes the expected value of a given concave utility functionu,
maxx E u(xr)
belongs to the following frontier (1−λ2−λ3)z0+λ2
rQ−1(à−z0e) eQ−1(à−z0e)+λ3
rQ−1b
eQ−1b (30)
spanned by the riskless returnz0, and the two risky portfolios u(1)=rQ−1(à−z0e)
eQ−1(à−z0e) and u(2)=rQ−1b eQ−1b.
Observe in (28) that whenα=α1=α2>1, every portfolioxris anα-stable distribu- tion and satisfies the relation
W=(1−xe)z0+xr=d Sα
σxr, βxr, (1−xe)z0+mxr
andW=z0whenx=0, where
σxαr=(xQx)α/2+ |xbσY|α, βxr=|xbσY|αsgn(xb)βY
σxαr , mxr=xE(r).
Hence, this jointlyα-stable model is a fund separation model whose solutions are given by the optimization problem (14) and these solutions satisfy the quadratic programming problem (29).
Ch. 14: Portfolio Choice Theory with Non-Gaussian Distributed Returns 573
3.3. Ak+1fund separation model in the domain of attraction of a stable law
As empirical studies show in the stable case one of the most severe restrictions of perfor- mance measurement and asset pricing is the assumption of a common index of stability for all assets – individual securities and portfolio alike. It is well understood that asset returns are not normally distributed. We also know that the return distributions do not have the same index of stability. However, under the assumption that returns have different indexes of stability, it is not generally possible to find a closed form to the efficient frontier.
Generalizing the above model instead, we get the followingk+1 fund separation model [for details onkfund separation models see Ross (1978a)]:
ri=ài+bi,1Y1+ ã ã ã +bi,k−1Yk−1+εi, i=1, . . . , n. (31) Here, nk2, the vector ε=(ε1, ε2, . . . , εn) is independent fromY1, . . . , Yk−1 and follows a joint sub-Gaussian symmetric αk-stable distribution with 1< αk <2, zero mean and characteristic function Φε(t)=exp(−|tQt|αk/2), and the random variables Yj =d Sαj(σYj, βYj,0), j =1, . . . , k−1, are mutually independent13 αj-stable distrib- uted with 1< αj <2 and zero mean. Under these assumptions, the portfolios belong to a σ τk+1(a)¯ class. If we need to insure the separation obtained in situations where the above model degenerates into ap-fund separation model withp < k+1, we require the rank condition [see Ross (1978a)]. In order to determine portfolios R–S non-dominated when unlimited short selling is allowed, we have to minimize the scale parameterσW=√
xQx for some fixed meanmW=xà+(1−xe)z0andb˜j=xbã,j/√
xQx,j =1, . . . , k−1.
Alternatively, as shown by Ortobelli, Rachev and Schwartz (2002), we can obtain these portfolios from the solution of the following quadratic programming problem:
minx xQx subject to xà+(1−xe)z0=mW, xbã,j=cj, j=1, . . . , k−1.
(32)
13In order to estimate the parameters, we need to know the joint law of the vector(Y1, . . . , Yk−1). Therefore, we assume independent random variablesYj, j=1, . . . , k−1.Then the characteristic function of the vector of returnsr= [r1, . . . , rn]is given by
Φr(t)=Φε(t) k−1
j=1
ΦYj(tbã,j)eità.
Under this additional assumption, we can approximate all parameters of any optimal portfolio using a similar procedure of the previous three fund separation model. However, if we assume a given joint(α1, . . . , αk−1) stable law for the vector(Y1, . . . , Yk−1), we can generally determine estimators of the parameters studying the characteristics of the multivariate stable law.