In this section we examine and compare the stable sub-Gaussian assumption with the nor- mal distributional one. Thus, we implicitly assume that returns belong to aσ τ2(m, σ )class wheremis the mean andσ is either the scale parameter of stable distributions or the stan- dard deviation of normal distributions.
In a recent work Ortobelli, Rachev and Schwartz (2002) compare the stable non- Gaussian assumption and the normal one by analyzing optimal allocations between a risk- less return and a benchmark index. Three different indexes have been taken into consid- eration: CAC40, DAX30 and S&P 500. Their analysis has indicated that either the heavy tails of data or a greater centralization of data around the mean can have a significant impact on the approximation of the investors’ choices. However, the stable non-Gaussian allocation is generally more risk preserving than the normal one. Precisely, the stable ap- proach considers a further component of risk which is due to the fat tails of the return
Ch. 14: Portfolio Choice Theory with Non-Gaussian Distributed Returns 575
distributions. This fact does not surprise us excessively. As a matter of fact, also Mehra and Prescott’s empirical analysis (1985) underlines that asset pricing puzzles can be justi- fied thinking of people much more risk averse. Clearly, we do not believe that the equity premium puzzle can be explained only considering the sub-Gaussian stable distribution instead of the Gaussian one. However, we believe that the distributional differences be- tween the data and the classic model used in finance can help to understand asset pricing puzzles. This conjecture is partly confirmed by assuming the stable distributions in place of the Gaussian one [see, for example, Kocherlakota’s (1997) test on CCAPM with heavy- tailed pricing errors].
Next, we extend Ortobelli, Rachev and Schwartz’s comparison to the multivariate case.
This comparison is formally and theoretically different from the previous one because here the benchmark index is given by the market portfolio which generally will change, if the distributional assumptions change too. Thus, as a consequence of Roll (1977, 1978, 1979a, b), Dybvig and Ross’ (1985a, b) analysis, we observe that:
(a) an investor, who fits the return distributions with a joint α1-stable sub-Gaussian dis- tribution, will consider as inefficient the choice of another investor who fits the return distributions with a jointα2-stable sub-Gaussian distribution withα1=α2; and (b) the stable CAPM is still subject of some of the criticism already addressed to the
classical one.
Nevertheless, it seems that the stable case explains better the empirical data. This is the main reason why here we interpret and analyze the different behavior between the investor who fits the data with joint stable sub-Gaussian distribution and the investor who fits the data with the joint normal distribution.
4.1. An optimal allocation problem
First, we consider the optimal allocation among 24 assets: 23 of those assets are risky assets with returnsr= [r1, r2, . . . , r23]and the 24th is riskfree with annual rate 6%. We analyze the portfolio choice problems when short sales are allowed and when short sales are not allowed. In view of this comparison, we discuss and study the differences in portfolio choice problems without examining them so as to choose one of the two assumptions (Gaussian or sub-Gaussian).14
In our comparison we use daily data taken from 23 international risky indexes valued in USD and quoted from January 1995 to January 1998. In the analysis proposed we first consider the maximum likelihood estimation of the stable parameters and of the Gaussian ones for every risky asset. Thus, Tables 1 and 2 assembles the approximating parameters obtained from using the program STABLE.15
In order to compare the different stable sub-Gaussian joint distributions and the joint normal distributions for the asset returns, we assume that the vector r is sub-Gaussian
14On this topic, recent studies [see Ortobelli et al. (2001), Ortobelli, Huber and Schwartz (2002)] have shown that sub-Gaussian multivariate models present a superior performance with respect to the mean–variance model.
15See Nolan (1997) and the web site www.ca.american.edu/∼jpnolan.
576 S. Ortobelli et al.
α-stable distributed, withα=αk,k=1,2 whereα1=1.7488 represents the average of the indexes of stability andα2=1.8856 represents the maximum of the indexes of stability (see Table 2).16Moreover, when in the following tables we consider the index of stability α=2,we implicitly assume that the returns are jointly normal distributed. Thus, every portfolio of risky assets is stable distributed in the following way:
xr=d Sαk(σxr, βxr, mxr),
whereαk is one of the considered index of stabilityk=1,2, σxr =(xQkx)1/2 is the respective scale parameter,Qk= [Rij/2]kis the dispersion matrix, withk=1,2, βxr=0 is the skewness parameter, andmxr represents the mean ofxr. Observe that the matrix Qk is estimated with the method defined in the previous section and thus it depends on the index of stabilityαk fork=1,2.As observed previously, the rate of convergence of the empirical matrixQk to the unknown matrixQk will be faster, ifpis as large as possible.
In our estimations we usep1=1.7 (relative to α1=1.7488)andp2=1.8 (relative to α2=1.8856).
We assume the investors wish to maximize the following utility functional:
U (W )=E(W )−cEW−E(W )q
, (33)
wherecandqare positive real numbers,W=λz0+(1−λ)x¯ris the return on the portfo- lio,z0is the risk-free asset return, and
¯
xr= rQ−k1(à−z0e) eQ−k1à−eQ−k1ez0
is the tangent portfolio of returns (25). With reference to the allocation problem (33), we observe:
(1) Problem (33) is equivalent to the following maximization of the utility functional aE(W )−bEW−E(W )q
, (34)
assumingc=b/a in (33) for every a, b >0. Thus,E(|W−E(W )|q)represents a particular risk measure of portfolio loss, which satisfies (under the opportune stand- ardization) the main characteristics of the typical dispersion measures. Solving the op- timal allocation problem (33), the investor implicitly maximizes the expected mean of the increment wealthaW as well as minimizes the individual riskbE(|W−E(W )|q).
(2) Furthermore, whenq =2, the maximization of utility functional (33) motivates the mean–variance approach in terms of preference relations.
16We consider different indexes of stability, in order to value the effects of heavy-tailedness on the portfolio selection problems.
Ch. 14: Portfolio Choice Theory with Non-Gaussian Distributed Returns 577
SupposeX dominatesY in the sense of R–S. SinceE(X)=E(Y )andf (x)=c|x− E(X)|qis a concave utility function, for everyq∈ [1, α), it follows that:
U (X)=E(X)−cEX−E(X)q
U (Y ), ∀q∈ [1, α).
The above inequality implies that every risk averse investor with utility functional (34) should choose a portfolioW=λz0+(1−λ)x¯rthat maximizes the utility functional (33) for some realλand someq∈ [1, α).
We know that forλ=1, all the portfolio returnsW=λz0+(1−λ)x¯r admits stable distribution
Sαk
|1−λ|σx¯r,0, λz0+(1−λ)mx¯r
, k=1,2,
andW=z0whenλ=1. Now, in order to solve the asset allocation problem maxλ E(W )−cEW−E(W )q
,
notice first that, for allq∈ [1, α)and 1< α <2, we get U (W )=E(W )−cEW−E(W )q
=λz0+(1−λ)mx¯r−c
H (α,0, q)q
|1−λ|qσxq¯r, where
H (α,0, q)q
= 2q−16(1−q/α) q∞
0 u−q−1sin2udu
[see Samorodnitsky and Taqqu (1994), Hardin (1984)]. The above relation analyzes the stable non-Gaussian case. When the vectorradmits a joint normal distribution (i.e.,α=2), then for allq >0,
U (W )=E(W )−cEW−E(W )q
=λz0+(1−λ)mx¯r−c2q/26((q+1)/2)
√π |1−λ|qσxq¯r.
Hence, the real optimal solution of the problem in the important caseq∈(1, α), is given by
λ¯=1−sgn(1− ¯λ)
sgn(1− ¯λ)(mx¯r−z0) qcσxq¯rV (α,0, q)
1/(q−1)
(35)
578 S. Ortobelli et al.
and
x=(1− ¯λ)x,¯ (36)
wherex¯is given by (25) and
V (α,0, q)=
H (α,0, q)q
in the stable case(1< α <2), 2q/26((q+1)/2)
√π in the normal case(α=2).
Again, one would expect that the optimal allocation was different because the constant V (α,0, q)and the matrixQare different in the stable sub-Gaussian and in the normal case.
4.2. Stable versus normal optimal allocation: a first comparison
We analyze the differences in optimal allocations with reference to problem (33) when the investor chooses:
(1) joint normal distribution, or,
(2) jointαk stable sub-Gaussian distribution (k=1,2)whereα1=1.7488,α2=1.8856 as a model for the asset returns in his/her portfolio. Under these distinctive assumptions, the investors with utility functional (33) have different information about the distributional behavior of data. In particular, we examine the different market portfolio composition and the different investor’s wealth allocation in the riskless asset.
First, when short sales are allowed and when short sales are not allowed, we examine optimal allocation among the riskless return and 23 index-daily returns: DAX 30, DAX 100 Performance, CAC 40, FTSE all share, FTSE 100, FTSE actuaries 350, Reuters Commodi- ties, Nikkei 225 simple average, Nikkei 300 weighted stock average, Nikkei 300 simple stock average, Nikkei 500, Nikkei 225 stock average, Nikkei 300, Brent Crude Physical, Brent current month, Corn No 2 Yellow cents, Coffee Brazilian, Dow Jones Futures 1, Dow Jones Commodities, Dow Jones Industrials, Fuel Oil No 2, Goldman Sachs Commodity, S&P 500. We use the riskless return 6% p.a.
Using the estimated daily index parameters, we can compute the dispersion matrixes and the approximating “market” portfolios. The dispersion matrixQ is given by either the variance–covariance matrix (in the normal case) or the matrixQk (in the stable cases) which depends on the index of stabilityαk fork=1,2 (α1=1.7488 andα2=1.8856).
Therefore, as shown by Tables 3, 4, the market portfolio weights
¯
x= Q−1(à−z0e) eQ−1à−eQ−1ez0
change under the different distributional assumptions. We observe that the market portfolio composition does not change excessively when we use either the asymmetric estimator (20) and (21) of matrixQk or the symmetric one(Qk+(Qk))/2.However, using daily
Ch. 14: Portfolio Choice Theory with Non-Gaussian Distributed Returns 579
data the elements of the dispersion matrixes are of orders 10−6. Thus the approximation in using data could be determinant to express elements of the matrixes. In particular, Table 3 presents the market portfolio weights when we consider all 23 asset returns and short sales are allowed. Table 4 gives the market portfolio weights when no short sales are allowed.
Under this constraint, we value the market portfolio weights in terms of the risky portfolio compositions which maximize the extended Sharpe ratio, i.e., the market portfolio weights are the solution of the following optimization problem
maxx
E(xr)−z0
σxr
, xe=1,
xi0, i=1, . . . , n.
In this case the optimal allocation is reduced only among the four risky assets: DAX 100 Performance, FTSE all share, Nikkei 300 weighted stock average, Dow Jones Industrials and the riskless one. As argued by Roll (1977, 1978), Dybvig and Ross (1985a), differ- ent market portfolios imply a completely different security market line analysis. Thus, the approach which takes into account short sales presents more opportunities of earning than the approach with no short sales constraint. Therefore, it dominates the other approaches.
Besides, if the returns are jointlyαk stable sub-Gaussian distributed (for some determined k=1,2), then the Gaussian approach is inefficient. Since, in general, efficient and ineffi- cient portfolios can plot above and below the “real” security market line.
The analysis of Tables 3 and 4 points out that the composition of the market portfolio is strictly linked to the index of stability. In fact, we see that the allocation of the market portfolio in each asset component is generally monotone with respect of the stability index.
Then the intuition suggests that the stable sub-Gaussian approaches take more into consid- eration the component of risk because of the fat tails. Recall that the tail behavior of every stable non-Gaussian distributionX=d Sα(σ, β, à), with 1< α <2, is given by
λ→+∞lim λαP (±X > λ)=Cα
1±β
2 σα, (37)
whereCα=(1−α)/(6(2−α)cosπ α2 ). Therefore, the fat tails of smaller stability indexes underline the risk of the loss component of every portfolio. In particular, under the diverse distributional assumption, we distinguish the different perception of risk in the market portfolio components. This issue can be easily analyzed in the market portfolio weights with reference to the 23 returns when no short sales are allowed. In fact, Table 2 shows that the index of stability of FTSE all share is greater than the other indexes of stabil- ity (of the assets DAX 100 Performance, Nikkei 300 weighted stock average, Dow Jones Industrials). Observe that in Table 4 the component of the FTSE all share in the market portfolio increases with the index of stability αk of the sub-Gaussian approach and the component of the other assets (DAX 100 Performance, Nikkei 300 weighted stock average,
580 S. Ortobelli et al.
Dow Jones Industrials) decreases with the index of stability. Thus, the market portfolios obtained under Gaussian and sub-Gaussian distributional hypotheses consider the risks due to heavy tails differently. On the other hand the mean of market portfolios decreases with the index of stability. However, if we accept the idea that the market portfolios represent in some sense the market behavior, then according to the classic mean–risk interpretation, an optimal portfolio that has a greater mean, it has also a greater risk. This fact appears clear enough when we consider and compare the dispersion measures
x¯kQjx¯k in every mean–risk plane for every market portfolio weights
¯
xk= Q−k1(à−z0e) eQ−k1à−eQ−k1ez0
,
for everykandj. Observe thatσ˜j,k =
¯
xkQjx¯kis the dispersion measure of market port- foliox¯kr considering theαj stable Paretian approach. Therefore, for every fixed mean–risk plane (i.e., for every fixedαj stable distributional approach) we can compare the market portfolio risk positions considering their risk position σ˜j,k (varying k). According to a mean–risk interpretation, we could observe that market portfolio with greater mean admits also a greater dispersion measureσ˜j,k in any mean–risk plane (see Tables 5 and 6).
As a consequence of relation (37) it follows that every stable non-Gaussian distribution X=d Sα(σ, β, à), with 1< α <2, admits
EX−E(X)q
<∞ forq < α and (38)
EX−E(X)q
= ∞ forqα. (39)
Hence, the weight of the risk measureE(|X−E(X)|q)in optimization problem (33) is generally greater for the investors who use the stable laws for asset returns whenqis quite close to the index of stabilityα.
In Tables 7, 8 we listed the optimal allocationλ¯ for the normal and the stable fit. Recall thatλ¯ is the optimal proportion of funds invested in the risk free asset which maximizes E(W )−cE(|W−E(W )|q), whereW =λz0+(1−λ)x¯r. We have chosenq =1.45 in Table 7 andq =1.55 in Table 8, so thatq is strictly less than all indexes of stability in the data set. On the other hand, we want to value and compare the different effects of q distant or closer to the stability parametersαk. For any given allocation problem, we remark in bold character and in italics respectively the greatest and the smallest allocation in the riskless asset. Both tables show the greatest diversity among the optimal allocations considering small risk aversion coefficientsc.Instead, the very risk averse investors assume a less risky position with every distributional hypothesis and the allocations in the riskless asset do not change very much.
As we see from these tables, whenq=1.45 andq=1.55 the investors who fit the data with the Gaussian approach generally assume a less risky position than the investors who fit the data with the sub-Gaussian approach. Thus, if the stable sub-Gaussian approximation
Ch. 14: Portfolio Choice Theory with Non-Gaussian Distributed Returns 581
presents greater performances than Gaussian one (as observed by many empirical analysis) the “stable investors” have more opportunities of earning than the “Gaussian investors”. In particular, the investors withα1=1.7488 stable sub-Gaussian approach invest less in the riskless asset than the investors who fit the data with the other approaches. However, if we considerq very closer toα1in optimization problem (33), then, as a consequence of (38) and (39), the investors who fit the data withα1stable sub-Gaussian approach assume a less risky position than the investors who fit the data with the Gaussian approach. In this case, the “stable investor” has a very risk preserving behavior because he prefers not allocating too much wealth in the risky asset. In this sense, intuition suggests that the stable approaches with lower indexes of stability generally are more risk preserving than those with greater indexes of stability because they consider the component of risk due to the fat tails of asset returns. Therefore, the stability index plays a strategic role in the stable optimal portfolio selection. Conversely,q in the above optimization problem can be an opportune measure of the magnitude to be given to the component of risk due to the heavy-tailedness of the asset returns. The importance given toqis intuitively linked to the conditions of the market in which the investor operates.
5. Conclusions
Firstly, we study, analyze and discuss portfolio choice models depending on a finite num- ber of parameters. The distributional analysis presented permits to classify the admissible parametric families of returns. Moreover, by the interrelation between the parameters of each parametric family, we can order the portfolio choices using the basic principles of the stochastic dominance analysis. Thus, we can identify a dispersion measure which has some basic characteristics and represents the implicit measure of the return portfolio risk.
In view of the classification of parametric portfolio choices, that is alternative to Ross’
multiparameter one, we can distinguish the different efficient frontiers for investors who are non-satiable, risk averse or both (non-satiable and risk averse). In particular, we distin- guish further restrictions to the classic Markowitz–Tobin’s efficient frontier when no short sales are allowed. Besides, we can identify the optimization problems we have to solve in order to determine more accurate estimations of the investor’s optimal allocations. In this sense, the analysis presented represents a general theory and a unifying framework to understand the parametric distributional approach to the portfolio choice theory.
Secondly, we show a simple classification of the portfolio choices considering the as- ymptotic behavior of returns with heavy tailed distributions. As a matter of fact, when returns have a stationary behavior they are in the domain of attraction of a stable law.
Therefore, we present some examples of models in the domain of attraction of stable laws.
The first distributional model considered is the case of the sub-Gaussian stable distributed returns. It permits a mean risk analysis pretty similar to Markowitz–Tobin’s mean–variance one. In fact, this model admits the same analytical form for the efficient frontier but the pa- rameters have a different meaning in the two models. Thus, the most important difference is given by the way of estimating the parameters. In order to present heavy tailed models
582 S. Ortobelli et al.
that consider the asymmetry of returns, we study a three fund separation model where the portfolios are in the domain of attraction of an(α1, α2)stable law. Next, we analyze the case ofk+1 fund separation model with portfolios in the domain of attraction of an (α1, . . . , αk)stable law. In all models we explicate the efficient frontier for the risk averse investors. In this context, we have shown that if the stable optimal portfolio analysis is stable, our approach is theoretically and empirically possible. Indeed, this work should be viewed only as a starting point for new empirical and theoretical studies on the topic of optimal allocation.
Finally, the comparison made between the stable sub-Gaussian and the normal approach in terms of the allocation problems has indicated that the stable sub-Gaussian allocation is more risk preserving than the normal one and can give more opportunities of earning.
Precisely, the stable approach, differently from the normal one, considers the component of risk due to the fat tails. Therefore, we find that the tail behavior of sub-Gaussian and Gaussian approaches could imply substantial differences in the asset allocation. Taken into account that the stable approach is more adherent to the reality of the market, then, as argued by Gửtzenberger, Rachev and Schwartz (1999), we can obtain models that improve the performance measurements with the stable distributional assumption.
Acknowledgment
We are grateful to Stoyan Stoyanov and Boryana Racheva-Jotova (Sofia University and Bravo Risk Management Group, Santa Barbara)for the computational analysis and helpful comments.
Appendix A: Proofs
In order to prove the following results, we use some Hanoch and Levy’s results [in partic- ular see Theorems 3 and 4 in Hanoch and Levy (1969)].
Proof of Theorem 1: Implication 1. According to definition ofσ τk+(a)¯ family, it follows wZ
σwZ
=d yZ σyZ
because the two random variables have the same parameters. IfσwZ> σyZ, then for every t0
P (wZt)=P wZ
σwZ
t
σwZ
P
wZ σwZ
t
σyZ
=P (yZt)
and the above inequality is strict for somet. Conversely, ifwZFSDyZ, thenE(wZ) >
E(yZ).