In this section we propose a distributional analysis of the optimal portfolio choice problem amongn+1 assets:nof those assets are risky with gross returns1Z= [Z1, . . . , Zn], and the(n+1)-th asset has risk-free gross returnZ0. When unlimited short selling is allowed, every portfolio of gross returns is a linear combination of the constant riskless gross return Z0, and the risky returnsZi, i.e.:
x0Z0+ n
i=1
xiZi, (1)
where(x0, x)∈Rn+1,x∈Rn. Therefore, the distribution functions of all admissible in- vestments belong to a translation and scale invariant family2determined by a finite number of parameters.
Assume price taker agents have preferences depending only on the probability distribu- tion of terminal wealth. This assumption allows von Neumann–Morgenstern’s preferences (1953) over wealth or more generally Machina’s preferences (1982) over wealth but it precludes state dependent preferences.
Assume that the market faced by a decision maker comes from a standard model of per- fect market (no transaction costs, taxes, asymmetric information, or arbitrage opportunities and all securities are perfectly divisible) which may not be complete.
Thus, in order to classify the parametric portfolio distribution functions consistent with the expected utility maximization, we distinguish and analyze the differences in portfolio allocation when:
(1) institutional restrictions (no short sales, limited liability) are allowed; or, (2) unlimited short selling is allowed without penalty.
2.1. Portfolio choice with institutional restrictions
When limited liability and no short sales are allowed, portfolios of gross returns (i.e., xZ0 where Zi >0 and xi 0, ∀i) are positive random variables. Thus, we assume that the portfolios of gross returns are positive random variables belonging to a scale invari- ant family, denoted withσ τk+(a), that admits positive translations and it has the following¯ characteristics:
1 Generally, we assume the standard definition ofi-th gross return between timet and timet+1,Zi= (Pt+1,i+d[t ,t+1],i)/Pt ,i,wherePt ,i is the price of thei-th asset at timet andd[t ,t+1],ithe total amount of cash generated by the instrument betweent andt+1. We distinguish the definition of gross return (with the capital letter) from the definition of return denotedzi=Zi−1 (or the alternative definition of continuously compounded return ri=logZi).
2 Recall that a parametric familyof distribution functions istranslation invariantif whenever the distribution FX(x)=P (Xx)belongs to, then for everyt∈R,FX+t ∈ as well. Similarly, we say that a familyis scale invariantif whenever the distributionFXbelongs to, then for everyα >0,FαXbelongs toas well.
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(1) Every distributionFXbelonging toσ τk+(a)¯ is associated to a positive random variable Xand is identified fromkparameters
(mX, σX, a1,X, . . . , ak−2,X)∈A⊆Rk,
wheremXis the mean ofX,σXis the positive scale parameter ofX.3We assume that the classσ τk+(a)¯ isweakly determinedfrom its parametrization. That is, the equality
(mX, σX, a1,X, . . . , ak−2,X)=(mY, σY, a1,Y, . . . , ak−2,Y) implies thatFX d
=FY,but the converse is not necessarily true.
(2) For every admissible realt0, the distribution functionFX∈σ τk+(a)¯ has the same parameters asFX+t ∈σ τk+(a), except the mean and the dispersion measure. In partic-¯ ular, the applicationf (t)=σX+t is a nonincreasing continuous function.
(3) For every admissible positiveα, the distribution functionFX∈σ τk+(a)¯ has, the same parameters of the distributionFαX except for the mean that is αmX and the scale parameter that isασX (wheremX andσX are respectively the mean and the scale parameter of the random variableX).
When portfolios belong to aσ τk+(a)¯ class, we can identify stochastic dominance rela- tions4among portfolios and the following theorem holds.
Theorem 1. Assume all random admissible portfolios of gross returns belong to aσ τk+(a)¯ class. LetwZandyZbe a couple of portfolios respectively determined by the parameters
(mwZ, σwZ, a1,p, . . . , ak−2,p) and (myZ, σyZ, a1,p, . . . , ak−2,p).
Then, the following implications hold:
(1) SupposemσwZwZ =mσyyZ
Z,thenwZFSDyZif and only ifσwZ> σyZ.
(2) mσwZwZ mσyZyZ andσwZ σyZ with at least one inequality strict, implies wZ FSD yZ.
3 In our context we use the mean as location parameter but the analysis can be extended to translation invariant families which do not admit finite the first moment. Moreover, we recall Pitman’s seminal work (1939) on the estimation of location and scale parameters.
4 Recall that the portfolioxZfirst order stochastically deaminates (FSD)yZif and only if for every increas- ing utility functionsu,E(u(xZ))E(u(yZ))and the inequality is strict for someu.EquivalentlyxZFSD yZ if and only ifP (xZt)P (yZt)for every real t and strictly for somet.Analogously, we say thatxZsecond order stochastically dominates (SSD)yZ, if and only if for every increasing, concave utility functionu,E(u(xZ))E(u(yZ))and the inequality is strict for someu. Equivalently,xZSSDyZ, if and only ift
−∞FxZ(v)dvt
−∞FyZ(v)dvfor every realtand strictly for somet[see, among others, Quirk and Saposnik (1962), Fishburn (1964), Hanoch and Levy (1969), Hadar and Russel (1969)]. We also say that xZRothschild Stiglitz stochastically dominates (R–S)yZ if and only if for every concave utility functions u,E(u(xZ))E(u(yZ))and the inequality is strict for someu.EquivalentlyxZR–SyZ if and only if E(xZ)=E(yZ)andxZ SSDyZ[see Rothschild and Stiglitz (1970)]. However, there exist many other stochastic orders used in Economics and Finance, see, among others, Levy (1992), Shaked and Shanthikumar (1994).
Ch. 14: Portfolio Choice Theory with Non-Gaussian Distributed Returns 555
(3) wZR–SyZ, if and only ifmwZ=myZandσwZ< σyZ.
(4) mσwZwZ mσyZyZ andmwZmyZ with at least one inequality strict, implieswZ SSD yZ.
(5) σwZσyZandwZSSDyZ, implieswZFSDyZ.
(6) mwZmyZ andσwZσyZ with at least one inequality strict, implieswZ SSD yZ.
The proofs of Theorem 1 and of the next results are given in the appendix.
Observe that there exist counterexamples to the converse of implications (2), (4), (5) and (6) in Theorem 1. Thus, in order to obtain the converse of these implications, we need additional hypotheses [see Ortobelli (2001)]. Theorem 1 stresses the limits of mean–
variance rule. In fact, suppose the portfolios of gross returns (without considering the risk- less gross return) belong to a σ τ2+(a)¯ class uniquely determined by the mean and the variance. Then, all non-satiable investors will choose portfolio solutions of the following constrained system
maxx xQx subject to E(xZ)
√xQx =h, wherexe=1, (2)
xi0, i=1, . . . , n,
for someh, wheree= [1, . . . ,1],Qis the variance–covariance matrix of the vector of gross returnsZ= [Z1, . . . , Zn]. Letσ∗be the maximum standard deviation of all admis- sible portfolios. Let us denote withà∗the portfolio mean of gross returns with maximum variance. As a consequence of Theorem 1 and Bawa’s results (1976), when the variance–
covariance matrixQis not singular andhvaries in the following interval:
à∗
σ∗ h
àZQ−1àZ=max
x
xàZ
√xQx, (3)
whereàZ is the mean of the vector of gross returnsZ, then the solutions of optimization problem (2) describe a set that contains the efficient frontier for agents with utility func- tions monotonically increasing in wealth. Moreover, under our assumptions, there exists a nonempty neighborhoodU of the global minimum variance portfolioZQ−1e/(eQ−1e) such that every admissible portfolio belonging toU (ZQ−1e/(eQ−1e))is not a solution of optimization problem (2).
With reference to the portfolio selection problem, recall that Markowitz (1952, 1987) and Tobin (1958, 1965) proposed the following selection rule for non-satiable risk averse investors: “From among a given set of investment alternatives (which includes the set of securities available in the market as well as all possible linear combinations of those basic securities), the admissible set of alternatives is obtained by discarding those investments with a lower mean and higher variance than a member of the given set”. On the basis of
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Fig. 1. The continuous curve represents efficient portfolios for non-satiable investors considering restrictions of nonnegative wealth;- - -dominated portfolios. The class of non-satiable investor’s optimal choices (which are contained in the arc ABC) is different from the class of risk averse investor’s optimal choices (which are contained in the arc DEAB), even if they generally have many common choices. Therefore, when we consider the risk averse non-satiable investor’s optimal choices, we obtain the feasible optimal portfolios (which are contained in the arc
AB) that are only a part of portfolios given by the classic Markowitz and Tobin’s rule (arc EAB).
Theorem 1, we find that Markowitz–Tobin’s selection rule is not optimal for non-satiable risk averse investors. In this context it is necessary to underline that no short sales or limited liability restrictions are imposed in a market where no riskless return is allowed. As a consequence, all portfolios are random variables uniformly bounded from below.5As a matter of fact, Theorem 1 cannot be extended to nonpositive random variables. Markowitz, Tobin, Bawa and many other authors left behind this observation in their considerations using normal distributions for returns. They have considered as efficient the portfolios on the upper neighborhood of global minimum variance (EA in Figure 1) but the same portfolios whose domain is under this restriction are not all efficient. Therefore, we proved that Markowitz and Tobin selection rule cannot be optimal even when portfolios belong to a family uniquely determined from the mean and the variance. It is well known that a lower variance does not imply a better choice for a non-satiable risk averse investor [see, example, in Hanoch and Levy (1969)]. Moreover, in an opportune neighborhood of global minimum variance portfolio, optimal portfolios for non-satiable investors do not exist. However, when riskless borrowing or lending is allowed, the mean–variance rule provides a sharper decision which permits to derive the efficient set for decision making with increasing and concave utility functions. In fact, if riskless asset is allowed, the global
5 Recall that a random variableXisbounded from below(above) if there exists a realtsuch thatP (Xt)=0 (P (Xt)=0). Analogously, a parametric familyisuniformly bounded from below(above) if there exists a realtsuch that, for every random variableX∈ ,P (Xt)=0 (P (Xt)=0).
Ch. 14: Portfolio Choice Theory with Non-Gaussian Distributed Returns 557
minimum dispersion portfolio is the riskless asset itself. Thus, as shown by Levy and Kroll (1976) and Kroll and Levy (1979), the classification of the efficient frontiers given by the stochastic dominance analysis assumes a simpler form.
Under institutional restriction on the market (no short sales, limited liability), we can assume that the family of all admissible portfolios of gross returns xZ belongs to a scale invariant family which admits positive translations. If every distribution func- tion FX ∈ is associated to a positive random variable X uniquely determined6 by (mX, σX, p1,X, . . . , pk−2,X), wheremXis the mean,σXis the standard deviation and
pi,X=E((X−E(X))i+2)
σXi+2 fori=1, . . . , k−2
are the firstk−2 nontrivialfundamental ratios, then, the familyis a particularσ τk+(a).¯ Note fori=1 andi=2 thei-th fundamental ratios are respectively Pearson’s asymmetry and kurtosis coefficients of the random variable X. Thus, all risk averse investors will choose non-R–S stochastically dominated portfolios among the solutions of the following constrained optimization problem
minx xQx subject to E(xZ)=m, xe=1,
(4) E((xZ−E(xZ))i)
(xQx)i/2 =qi, i=3, . . . , k, xj0, j=1, . . . , n,
for somemandqi,i=3, . . . , k, wheree= [1, . . . ,1],Qis the variance–covariance matrix of the vector of gross returnsZ= [Z1, . . . , Zn]. Moreover, all non-satiable investors will choose portfolio weights, solutions of the following optimization problem
maxx xQx subject to E(xZ)
√xQx h, xe=1,
(5) E((xZ−E(xZ))i)
(xQx)i/2 =qi, i=3, . . . , k, xj0, j=1, . . . , n,
6 Recall that a class of distributions is uniquely determined from k parameters when the equality (mX, σX, a1,X, . . . , ak−2,X)=(mY, σY, a1,Y, . . . , ak−2,Y)implies the equality of the respective distributions, i.e.,FXd
=FY,and vice versa.
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for someqi,i=3, . . . , k, andhà∗/σ∗, whereσ∗ is the maximum standard deviation of all admissible portfolios andà∗is the mean of that portfolio of gross returns. Similarly, all non-satiable risk averse investors will choose portfolio weights among the solutions of the following optimization problem
maxx E(xZ) subject to E(xZ)
√xQx h, xe=1,
(6) E((xZ−E(xZ))i)
(xQx)i/2 =qi, i=3, . . . , k, xj0, j =1, . . . , n,
for someqi,i=3, . . . , k, andhàj∗/σj∗, whereàj∗ is the maximum mean of all pri- mary gross returns andσj∗is the standard deviation of that return. We obtain optimization problems analogous to (4), (5) and (6) when we consider the riskless asset. In this case, the mean is given byE(xZ)+(1−xe)Z0and we require that 0xe1 instead of requiringxe=1. Theorem 1 is used for positive random variables. However, the above results can be generalized to families of random variables, which are uniformly bounded from below. In fact, without loss of generality, we can consider a translation that makes all random variables positive.
2.2. Portfolio choice when unlimited short sales are allowed
In the last fifty years the researchers of portfolio choice theory often used unbounded ran- dom variables for portfolio of returns, typically: the Gaussian laws. They also used to study continuously compounded portfolio of returns, say xr=n
i=1xilogZi, where7 ri =logZi.
In particular, we assume that the distribution functions of portfolios belong to a transla- tion and scale invariant family denoted withσ τk(a)¯ with the following characteristics:
7 The continuously compounded portfolio of returnsxrrepresents an approximation to the portfolio of returns xz(i.e.,n
i=1xilogZi∼xz, wherezi=Zi−1). Thus, continuously compounded portfolio of returnsxrare equivalently identified and called portfolio of returns. However, observe that
XFSDY if and only if logXFSD logY , while
XSSDY implies logXSSD logY
but the converse is not necessarily true (you can find a simple counterexample with the log-normal class). Hence, when we study the optimal choices by considering the approximationn
i=1xilogZi∼xz, we find a set of choices that would be closer to the efficient set as well as the approximation would be right. (The approximation is good enough when we consider daily – or weekly – data in the empirical analysis).
Ch. 14: Portfolio Choice Theory with Non-Gaussian Distributed Returns 559
(1) Every distributionFXbelonging toσ τk(a)¯ is identified fromkparameters(mX, σX, a1,X, . . . , ak−2,X)∈A⊆RkwheremXis the mean ofX,σXis the positive scale para- meter ofX. We assume that the classσ τk(a)¯ isweakly determinedfrom its parameter- ization. That is the equality(mX, σX, a1,X, . . . , ak−2,X)=(mY, σY, a1,Y, . . . , ak−2,Y) implies thatFX=d FY but the converse is not necessarily true.
(2) For every admissible realt, the distribution functionFX∈σ τk(a)¯ has the same para- meters, except the mean, asFX+t ∈σ τk(a)¯ (the translated ofFX).
(3) For every admissible positiveα, the distribution functionFX∈σ τk(a)¯ has the same parameters of the distributionFαX∈σ τk(a)¯ except for the mean that isαmXand the scale parameter that isασX(wheremXandσXare respectively the mean and the scale parameter of the random variableX).
The random variables associated to the distribution functions of aσ τk(a)¯ class are not uniformly bounded from below because everyσ τk(a)¯ class is translation invariant. When portfolios belong to aσ τk(a)¯ class, we can identify a stochastic dominance relation among portfolios unbounded from below and the following theorem holds.
Theorem 2. Suppose the distribution functions of all random portfolios belong to the same classσ τk(a). Let¯ wr andyr be a couple of random portfolios unbounded from below respectively determined by the parameters
(mwr, σwr, a1,p, . . . , ak−2,p) and (myr, σyr, a1,p, . . . , ak−2,p).
Then, the following properties are equivalent
(1) E(wr)E(yr),σwr σyr with at least one inequality strict.
(2) wrSSDyrandyr=d wr−(E(wr)−E(yr))+εandE(ε/wr)=0.
As for Theorem 2, when all portfolios are random variables unbounded from below and their distribution functions belong to a σ τ2(a)¯ class, two portfoliosX and Y such that σX> σY andXSSDY cannot exist. On the contrary, when the random portfolios consid- ered in Theorem 2 are random variables bounded from below, we need further assumptions to get the above equivalence [see Ortobelli (2001)].
According to Theorem 2, it follows that when all portfolios are unbounded random vari- ables belonging to aσ τk(a)¯ class, it is easier to characterize their stochastic dominance properties. In this sense, the continuously compounded portfolios of returnsxr, are nat- ural candidates for a simpler stochastic dominance analysis.
Samuelson (1969), Samuelson and Merton (1975) were among the first to investigate the conditions for the mean–variance criterion to provide an approximate optimum. Cham- berlein (1983) has shown that when the riskless return is allowed, the families of elliptical distributions with finite variance are necessary and sufficient for the expected utility of fi- nal wealth to be a function only of the mean and the variance. Hence, when the portfolios
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are unbounded random variables8with distribution functions belonging to the same ellip- tical distribution family having finite variance, we can use Markowitz and Tobin’s rule to individuate the optimal portfolios. Similarly, assuming that:
(a) there is no riskless asset;
(b) the portfolios of returns are unbounded random variables;
(c) the lastn−1 components of the return random vector are elliptically distributed (with finite variance) conditional on the first component which has an arbitrary distribution with finite variance [see Chamberlein (1983)];
then, Markowitz and Tobin’s rule can be used to individuate the optimal portfolios. Thus, Theorems 1 and 2 underline a further limitation (the above point (b)) of the previous studies on this issue.
We can now find optimal portfolios when all returns are unbounded random variables uniquely determined by a finite number of moments. Thus, if short sale is allowed, all risk averse investors will choose non-R–S stochastically dominated portfolios that are solutions of the constrained optimization problem (4) without the constrainxj 0, j =1, . . . , n.
Similarly, we obtain optimal solutions for non-satiable investors maximizing the mean for some fixed central moments.
2.3. Relations with Ross’ multi-parameter models
Consider the problem of optimal allocation amongn+1 assets:nof those assets are risky with non-redundant returnsr= [r1, . . . , rn], and the(n+1)-th asset return isz0risk-free.
Then,we are interested in the cases of portfolio distributions belonging to aσ τk(a)¯ family withk < n. As argued by Ross (1978a), in order to reduce the variables of the portfolio choice problem, we have to assume some restrictions on the vectorε=(ε1, ε2, . . . , εn)in the following representation of the returns:
ri= q p=1
bi,p
Yp−E(Yp)
+εi, i=1, . . . , n, (7)
whereYi andεi are random variables andbi,j are scalars. Differently from Ross, we pro- pose to study the case where all random variablesxεbelong to aσ τs(a)¯ family. Then, the scale parameterσxε of random variablexε, has to verify the properties relatively to the σ τs(a)¯ class. Thus, consider the parameterization given by:
(1) the parameters of theσ τs(a)¯ family, and
(2) the parametersc∗j=xbã,j/σxε, forj=1, . . . , q.
This parameterization verifies the properties of aσ τk(a)¯ family with k=s+q. In fact, for every positive realαthe parameters ofαx(r−à)do not change except for the scale
8 The elliptical families with finite variance are symmetric around the mean and are not necessarily associated to unbounded random variables, see Ingersoll (1987), Owen and Rabinovitch (1983). Then, following Theorem 2 we need to specify when elliptically distributed random variables have to be unbounded.
Ch. 14: Portfolio Choice Theory with Non-Gaussian Distributed Returns 561
parameter that becomesασxε. Then, all portfolios belong to aσ τk(a)¯ family withk=s+q when the returns admit the form (7) and all admissible random variablesxεbelong to a σ τs(a)¯ family. Typical examples of this approach are the stable Paretian models presented in the next section.
Note that the above proposed moment analysis generalizes many of the three moment models presented in the last decades [see, for example, Kraus and Litzenberger (1976), In- gersoll (1987), Simaan (1993)]. Moreover, we do not need to require that portfolio returns verify the fund separation conditions as happened in the three moment models. Therefore, the above theorems represent a first classification of portfolio distribution functions which is alternative to those proposed from Ross (1976, 1978a). In fact, we underline the follow- ing differences from Ross’ models:
(1) We express necessary and sufficient conditions to identify optimal portfolios. We can derive the efficient frontiers solving a constrained optimization problem.
(2) We do not require the closure of the random law under addition.
(3) The above theorems are an unifying and generalizing extension of moment analysis in the portfolio selection theory. In particular, the previous analysis describes further restrictions in using Markowitz and Tobin’s selection rule as optimal portfolio selection rule.
(4) We express a portfolio choice theory dependent on a finite number of parameters con- sistent with expected utility maximization. We do not specify which parameters iden- tify the distribution functions of asset returns. We only require very general properties which determine the existence of a scale parameter and a shift parameter.
(5) The above results can be applied to every economic choice in uncertainty condi- tions when the distribution functions are weakly determined by a finite number of parameters and verify properties ofσ τk(a)¯ or ofσ τk+(a)¯ classes. Besides, this clas- sification of choices under uncertainty conditions implies a first classification of the admissible dispersion measures [see Ortobelli (2001), Giacometti and Ortobelli (2001)].
As it follows from the previous considerations, the models introduced here can be theo- retically improved and empirically tested. However, a more general theoretical and empir- ical analysis with further discussion, studies and comparison of the above models does not enter in the objective of this chapter and it will be the subject of future research.