EMPIRICAL AND THEORETICAL ATTACKS on the Markowitz mean-variance, portfolio theory have given impetus to the investigation of moments of higher order than the variance. Denoting w as an investor's wealth, and i his incomes (a random variable), if the investors utility function, U, is solely dependent on the sum of his wealth and income we can define his utility as U = U(i + w). The investor's rate of return on investment of w is defined as f = i/w and his utility may be given by U = U(rw + w). Letting jt = E(w + rw), representing the expected value on investment, expanding U in the familiar Taylor series, and taking the expected value of both sides gives: U2(tL) c2 +00U Ai) 1 E (U) = U(A) + 2y a2 + Ei=3 Ti! U,(A)(1where Un denotes the n-th derivative derivative of U and ,ij is the i-th central moment. In order to analytically study the expected utility of return on investment w, one must define the relevant moments of the distribution of return and the sign of the coefficient of each relevant moment. The mean and the variance can completely describe E(U) generally for only certain distributions of return such as the Normal, the Uniform and the Binomial distributions. If (i) the distribution of returns to a portfolio is asymmetric, (ii) the investor's utility function is of higher order than the quadratic, and (iii) the mean and variance do not completely determine the distribution, then the third and perhaps higher moments and the sign of their coefficients must be considered.1 0 For higher moments than the variance two questions may be asked: can the direction of preference (the sign of the coefficients in (1)) for each moment be determined on some a-priori grounds and if so, what is the preference direction for each moment? For the third moment the first question has received considerable attention (See: Arditti [1], Jean [4], Arditti and Levy [2], Kraus and Litzenberger [6], Levy and Sarnat [7], and Simkowitz and Beadles [10]). Arditti [1], and Kraus and Litzenberger [6] have implied a positive preference direction for the third moment (U3(w) > 0) in answer to the second question as regards skewness (normalized third moment). Other conditions such as "clientele effects" have been posited as indications of investor preferences for the third moment. The fourth and higher moments have received relatively little attention apparently because of the difficulty of dealing with the fourth and higher moments. Kaplanski [5] has indicated that it is difficult to discern just what the fourth * College of Business, Bradley University 'If any of these three conditions does not hold, then the mean and the variance are sufficient.
As a teacher for more years than I care to mention and a reviewer of several texts I am concerned with the treatment of intra-period compounding in finance books. We know that the effective rate of return is the compound sum of the rates for each period and the average of these rates is correctly specified by the geometric mean of those rates. The geometric mean also correctly specifies the intra-periodic rate of compound/discount given an annual compound/discount rate, r . Current textbooks in financial management [ l , 2, 61, investments [5] and financial institutions [3, 41' indicate that the appropriate intraperiod compoundldiscount rate is rlk: k is the number of sub periods. Two problems arise from this treatment. First, rates are not normalized:etc.; students should always be thinking in terms of effective rates of return so that correct comparisons can be made. Second, it is inconsistent to use the effective rate for multiperiod analysis and not for intra-period analysis (a distinction with no particular financial significance); it is incorrect to use (1 + r) in multiperiod compounding/discounting when k > 1 . The appropriate rate is {(I + r ) ( l ' k ) } -I.In introductory materials an important distinction must be made in the identification of intraperiod rates. In those cases where interest is paid on the basis of principal plus accumulated interest the usual treatment is correct as in Brigham [ l , pp. 80-821. In those cases where interest is not paid on an accumulated basis (i.e., bonds, preferred stock, etc.) the treatment is incorrect.For example, carrying the presentation in [ 1, pp. 80-821 to the determination of the annual interest on a bond, r, as in [l, pp. 106-71 we find a correct interest rate of 10 percent on a 14-year, 15 percent coupon, $1000 par value bond paying interest annually with a price of $1368.31 . I Were the same bond to pay * Bradley University 'This example was selected at the author's convenience. It should be pointed out that the author of the example is in plentiful and august company in his treatment of this topic. 116
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