This section is for examining the impact of estimation risk on the return distribution of a portfolio in a lognormal-securities market.3 Assume there are (N + 1) securities in the market, which includes N risky securities and a riskless asset. Let the vector of the observed security holding period returns, r = [r1; :::; rN]T follow a multivariate lognormal distribution with the logarithmic mean vector, = [ 1; :::; N]T and the logarithmic covariance matrix . That is, the vector of logarithmic holding period returns, R = [R1; :::; RN]Twhere Ri = ln ri is multivariate normally distributed with E(R) = and V ar(R) = . Then, the logarithmic holding period return on a portfolio of (N + 1) securities Rp including a riskless asset is normally distributed, with Rp = x0Rf + xTR where x0 is the investment proportion of the riskless asset and xT = [x1; :::; xN] denotes the vector of investment weights of N risky securities, and Rf = ln r0 is the logarithmic holding period return on a riskless asset. Since the true means and covariance matrix (the …rst two moments) are rarely known in practice, they are determined by sample estimates given below:
^p = x0Rf + xT^ (1.1)
^2p = xT^ x (1.2)
2Cairns (2000)[20] explicitly speci…es that “uncertainty” arises from three principal sources and hence contributes to three categories of related studies.
3Henceforth, returns and ri, used in this study are mainly referred to holding period returns (or called gross returns). Bawa and Chakrin (1979, p.48)[10] also use the same idea. A lognormal-securities market means that the return of the individual security in the market follows a lognormal distribution. Analo-gously, in a normal-securities market, the return of the individual security follows a normal distribution.
where ^ and ^ are, respectively, the vector of sample means of individual securities and the sample covariance matrix. The respective elements in ^ and ^ ar:
^i Ri = PT t=1
Ri;t = T (1.3)
^2i;j Si;j= PT t=1
(Ri;t ^i) Rj;t ^j = (T 1) (1.4)
where T is the total number of observations. The parameter estimates given in (1:1) and (1:2) have not taken the e¤ect of estimation risk into account. The Bayesian method can be used to incorporate estimation risk into the portfolio selection process. This can be done by integrating the posterior distribution of a portfolio’s return over the parameter space ( ; ). This yields the predictive distribution of returns which does not involve any unknown parameter. This distribution depends only on sample estimates and prior information. Under the assumptions of multivariate normality and di¤use prior information, Bawa, Brown and Klein in 1979 (henceforth BBK 1979) have shown that when estimation risk is taken into account, the predictive distribution of portfolio returns Rp should be a Student-t distribution with (T N ) degrees of freedom rather than a multivariate normal distribution. The mean and variance of this predictive distribution for Rp are, therefore, given by:
p = x0Rf + xT^ (2.1)
p = xT x= cxT^ x = c^2p (2.2)
where = c ^ , and c = 1 +T1 TTN12 > 1; T > (N 2). Therefore, the e¤ect of estimation risk increases the scale parameter ^2pby a constant proportion c which depends on sample size T and the number of securities N . When sample size T gets smaller or the number of securities N increases, estimation risk is higher as designed by a larger value of c in (2:2) since the facts that (@c=@T ) < 0 and (@c=@N ) > 0 are held in BBK.
Statistically, it is well-known that the sample estimator ^ cannot be an unbiased estimator of under small sample size T . Thus a large value of c in (2:2) can e¤ectively eliminate the in‡uence of the biased information on the prediction of the true parameters.
Hence, the factor c provides a coe¢ cient-adjustment mechanism under the small sample size scenario and lowers estimation risk. Similarly, the number of securities N in the factor c serves the analogous function as above, since when the number of securities N gets larger, the factor c becomes larger as well. It stems from the fact that when the number of securities is numerous, the number of parameters to be estimated gets larger as well, and thereby increasing the di¢ culty of a precise estimation for these moments4. Therefore, estimation risk becomes larger for a larger number of securities N , and thus a larger value of c is required to provide a coe¢ cient-adjustment mechanism to make estimated parameters not be greatly a¤ected by biased sample estimators.
In a lognormal-securities market, one may state that a portfolio’s logarithmic holding period return Rp is the logarithm of the geometric average of holding period returns on (N +1) securities. That is, Rp= ln rp where rp = Ni=0rixi is the portfolio’s holding period
4See page 6 of DeMiguel, Garlappi, and Uppal (2009)[24] for related discussions.
return in a lognormal-securities market5. As mentioned above, the predictive distribution of Rpunder the consideration of estimation risk is a Student-t distribution with mean and variance given by (2:1) and (2:2). This result provides a means of deriving the predictive distribution of the portfolio’s holding period return rp in a lognormal-securities market.
To do it, we …rst introduce the predictive distribution of Rp, a Student-t distribution with (T N ) degrees of freedom:
f (Rpj p; h; v) = predic-tive distribution of Rp in (3) incorporates estimation risk. Following von Neumann-Morgenstern-Savage axioms, the investor’s optimal portfolio under estimation risk in a normal-securities market is given by the solution to the following maximization:
M axx : E [U (Rp)] = Z 1
1
U (Rp) f (Rpj p; h; v) dRp (4) where U (Rp) is the investor’s utility function of portfolio return in a normal-securities market. Based on the maximization problem (4) with the appropriate transformation, that is, Rp = ln rp or rp = eRp, the investor’s optimal portfolio under the consideration of estimation risk in a lognormal-securities market can be represented by the solution to the maximization problem given below:
M axx : E [U (rp)] = Z 1
0
U (rp) g (rpj p; h; v) drp (5)
where g (rpj p; h; v) is identi…ed as the predictive distribution of the portfolio return rpin a lognormal-securities market. Using (5) and the transformation, Rp = ln rp, the predictive distribution of the portfolio holding period return rp in a lognormal-securities market can be shown to be a log-Student-t distribution with (T N ) degrees of freedom, and the predictive distribution g (rpj p; h; v) is derived by applying the Jacobian transformation method to this log-Student-t distribution, which is easily shown that g (rpj p; h; v) = f (Rpj p; h; v) jJj where jJj is the Jacobian of transformation with its value to be rp1. Hence, the probability density function of rp is given below:
g (rpj p; h; v) =
Notice that because the value …eld of Rp is between 1 and 1, then under the relation of rp = eRp, the corresponding value …eld of rp is between 0 and 1. In addition, the pre-dictive distribution of rp in (5) and (6) does not depend on any unknown parameters, but
5This is given by ln rp = Rp= Ni=0ln rixi = ln (rx00 ::: rNxN) = ln Ni=0rixi . The holding period return riin a lognormal-securities market is also expressed in terms of gross return, thus the logarithmic holding period return Ri(= lnri)is also called lognormal gross return in this study. In addition, Rpfollows a Student-t distribution implies that rpis a log-Student-t distribution inferentially.
only relates to known parameter values such as p, p2,v, and h. With the known proba-bility distribution g (rpj p; h; v), and then the optimal portfolio under estimation risk in a lognormal-securities market can seem to be derived accordingly from (5) by maximizing the investor’s expected utility function of portfolio returns. However, something tricky happens during the optimization process. By observing from the value …eld of rp, which ranges from zero to in…nite, it implies that there may not exist …nite moments for rp. In other words, the moments of rp may explode during the integration process. This fact implies that the probability density function, g (rpj p; h; v), is not available statistically, and thus the maximization of (5) cannot be done in a straightforward manner6.
An e¤ective method to solve the arduous problem of in…nite moments of a log-Student-t dislog-Student-tribulog-Student-tion is log-Student-to apply log-Student-the quadralog-Student-tic approximalog-Student-tion, which is proposed by Ohlson (1977). Ohlson’s (1977) quadratic approximation can deal with the mean-variance opti-mization problem under the situation where means and variances of returns are in…nite.
The method allows investors to directly approximate their expected utility functions of portfolio’s returns in a lognormal-securities market and then derive the corresponding asymptotic result for optimal portfolio selection without the need of the probability den-sity function g (rpj p; h; v) during the optimization process. The way to apply Ohlson’s quadratic approximation for constructing optimal portfolios is shown in the next section.