Reply to Professor Loistl

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Markowitz, Harry M. (1959). Portfolio selection: Efficient diversification of investments. New Haven: Yale University Press. Markowitz, Harry M. (2012). The great confusion concerning MPT, AESTIMATIO. The IEB International Journal of Finance, 4, 8–27. Markowitz, Harry M. (2014). Mean-variance approximations to expected utility. European Journal of Operational Research, 234, 346–355. Markowitz, Harry M. Bibliographical. Nobelprize.org. Nobel Media AB 2014. http:// www.nobelprize.org/nobel_prizes/economic-sciences/laureates/1990/markowitzbio.htmlWeb. Accessed 26.03.15. McFadden, Daniel (2001). Economic choices. The American Economic Review, 91, 351– 378. Merton, & Robert, C. (1972). An analytical derivation of the efficient portfolio frontier. Journal of Financial and Quantitative Analysis, 7, 1851–1872. Morgenstern, Oscar. (1979). Some reflectons on utility, in: Allais and Hagen (Eds.), l.c. (pp. 175-183). Myerson, Roger. (2015). Probability models for economic decisions, excerpts from Chapter 3: Utility theory with constant risk tolerance. home.uchicago.edu/ rmyerson/teaching/util206.pdf. Accessed 30.04.15.

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Reply to Professor Loistl There are many points in Professor Loistl’s comment on Markowitz (2014) upon which he and I agree. But there is one specific issue—a fraction of a sentence in Loistl (1976)—on which Professor Loistl and I disagree sharply. The sentence fragment asserts that “mean-variance approximation is not a good approximation of the expected utility at all.” Professor Loistl says that I should have quoted the entire sentence in which this fragment appears, namely: Looking at the figures, we are forced to an ambivalent conclusion: the mean-variance approximation is not a good approximation of the expected utility at all; however, it is more exact than a Taylor’s series expansion including higher terms of any order.” The fact that Loistl finds mean-variance approximations to be not as bad as Taylor series approximations with higher terms, is small consolation for Loistl’s judgment that mean-variance approximations are not good. Markowitz (2014) is a “stand alone” version of Chapter 2 of Markowitz and Blay (2014). The latter is the first of four planned volumes on Risk-Return Analysis: The Theory and Practice of Rational Investing. These four volumes will expand on the four chapters of Part IV of Markowitz (1959), which contain my fundamental assumptions—my justification for the practical use of mean-variance analysis. Specifically, I never assumed that return distributions are normal (Gaussian). Rather I accepted (and still accept) the von Neumann and Morgenstern (1944) and Savage (1954) conclusion that a rational decision maker would maximize expected utility (EU) using probability beliefs (PB) where objective probabilities are not known, updating these beliefs according to Bayes rule as evidence accumulates. Some, including Professor Loistl, do not accept the expected utility maximum as the touchstone of rational choice. His current piece argues that other reasons can be given for justifying the practical use of mean-variance optimization (MVO) and is quite supportive of MVO in practice. I thank him for that, but the fact is that I still hold to the expected utility maxim. Chapter 1 of Markowitz and Blay explains why. Chapter 2 of Markowitz and Blay, the one essentially reproduced as Markowitz (2014), surveys an extensive literature on meanvariance approximations to expected utility. For the most part this literature is highly supportive of MV approximation to EU. I return below to the question of why Professor Loistl (1976) came to such a different conclusion than most of the rest of us. Chapter 3 of Markowitz and Blay reports the results of research on six different methods of estimating the geometric mean of a return distribution from estimates of its arithmetic mean and variance. This

is connected to the logarithmic utility function by the formula

log(1 + g) = E log(1 + R )

(1)

where g is the so-called “geometric mean” of the return distribution R. (Actually, 1+g is the geometric mean of 1+R; but the established terminology is more convenient.) This is of practical importance, because the inputs and the output of MV analysis must be arithmetic means not geometric means: since the arithmetic mean of a weighted sum is the weighted sum of arithmetic means; and this is not true of the geometric mean. But we need to at least approximate the geometric mean of a portfolio, because in the long run almost surely one gets the geometric mean not the arithmetic mean. The six ways considered in Chapter 3 of Markowitz and Blay of approximating g from the arithmetic mean and variance of R include (a) two proposed in Markowitz (1959), (b) the convenient rule of thumb

g = E − 12 V

(2)

where E and V are portfolio expected value and variance; and (c) three others that have been proposed. The six proposals were evaluated in terms of how well they would have approximated g from the E and V of series from two different databases, namely (1) returns on commonly used asset classes, and (2) real returns of the equity markets of 16 countries during the 101 years from 1900 to 2000 as reported by Dimson, Marsh and Staunton (2002). The results were, again, quite supportive of MV approximation to EU—in stark contrast to Loistl’s conclusion. Example Table 1 of the present paper, like Table 1 of Markowitz (2014), is a copy of Table 2 in Chapter 6 of Markowitz (1959, p. 121). The first column of Table 1 presents portfolio return R, from R = −0.5 to R = 0.5; the second column of Table 1 presents ln(1 + R); the third column presents the quadratic

qz (R) = R − 12 R2

(3)

As Markowitz (2014) notes, For returns R between a 30 percent loss and a 40 percent gain on the portfolio-as-a-whole there is little difference between ln(1 + R)and qz . At R = −0.30 (a thirty percent loss) ln(1 + R) = −0.36 whereas

Reply / European Journal of Operational Research 247 (2015) 680–681

as −20, −10, … 30, 40. He then compares U (y)with different Taylor ¯ for series approximations, including the quadratic, fit with X0 = y, y = −20, y = −10, … , y = 40, confining the comparisons to y > 0 for the logarithmic utility function. In other words, Loistl represents a 30 percent gain by R =30 rather than R = 0.3. Loistl’s conclusion—that the quadratic approximation fits poorly for these values of y—is consistent with the Markowitz (1959) observation that a quadratic fits well for returns in the interval −0.3 to 0.4.

Table 1 Comparison of ln(1 + R) and R – (1/2)R2 . R

ln(1 + R)

R – (1/2)R2

−.50 −.40 −.30 −.20 −.10 +.00 +.10 +.20 +.30 +.40 +.50

−.69 −.51 −.36 −.22 −.11 .00 .10 .18 .26 .34 .41

−.63 −.48 −.35 −.22 −.11 .00 .10 .18 .26 .32 .38

By way of confirmation that I did not misinterpret the numbers in Loistl’s Table 1, I note that the Loistl quadratic example, his Equation (7a), is

U (y ) = y − y2 /2000

the quadratic is −0.35. At R = 0.40 (a forty percent gain) ln(1 + R) = 0.34 whereas the quadratic is 0.32. Between these two values, i.e., for R = −0.20 to 0.30, the approximation equals the log utility function to the two-places shown. Even at a forty percent loss or a fifty percent gain, the difference is noticeable but not great: −0.51 vs −0.48 in the one case; 0.41 vs. 0.38 in the other. Thus Markowitz (1959) concludes that for choice among return distributions which are mostly within the range of a thirty percent loss to a forty percent gain on the portfolio-as-a-whole, and do not fall outside this range “too far, too often,” the E[ln(1 + R)] maximizer will almost maximize expected utility by an appropriate choice from the mean-variance efficient frontier. For example, if the eight returns, R = −0.3, −0.2, … , 0.4 were equally likely, then the arithmetic mean of R would be 5 percent, ER = 0.05, the geometric mean would be 0.024, and the approximation computed from qz would be 0.023. I consider that “not bad” as opposed to “not good.”1 The approximation qz is the quadratic Taylor expansion about return R = 0, no gain, no loss. The other mean-variance approximation proposed in Markowitz (1959) was the Taylor expansion about expected return R = E, which reduces to

1 EU ∼ = U (E ) + U  (E )V 2

(4a)

When U = ln(1+R), (4a) is

1 E ln(1 + R) ∼ = ln(1 + E ) − V/(1 + E )2 2

(4b)

As reported in Markowitz and Blay, the approximation in (4b) is much better than the one based on qz in (3). Why Loistl gets a different result A substantial body of research reported in Markowitz (2014) finds that MV approximations are robust for a wide range of concave utility functions. So how did Loistl (1976) reach a different conclusion? According to Markowitz (2014), The reason Loistl reaches such a negative conclusion is as follows: Note that in Table 1 [in Markowitz (2014)] a six percent return is represented as R = 0.06, not as R = 6.0…. Loistl assumes that utility U is a function of a random variable y “where y is the investor’s wealth.” [p. 905]. He gives an example in which “Return in Percent” is listed

1

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The 0.023 is rounded up from 0.228 and the 0.024 is rounded down from 0.0242. The four place numbers would be closer if the distribution were unimodal, with a mode in the general area of the mean, rather than rectangular. The computation in the text did not use the rounded numbers in Table 1, but Xcel accuracy for ln(1 + R) and qz .

(5)

The entries in his Table (3) result by substituting, e.g. “−20” not −0.2 into this equation. I did not check, but I assume that the same holds true for Loistl’s other computations, including the meanvariance approximations to utility functions. Chapter 3 of Markowitz and Blay found that the MV approximation based on qz was the worst of the six tested. Nevertheless, as Table 1 here shows, even qz is a good approximation to the expected log (and therefore the geometric mean) as long as a return distribution is confined to returns between a 30 percent loss and a 40 percent gain. But the hypothetical distribution in the Loistl’s (1976) Tables 1 and 2, goes from a 20 percent loss to a 40 percent gain and, on its basis, Loistl concludes that “the mean-variance approximation is not a good approximation of utility at all.” Why is there such a radical difference in conclusions of the two analyses? What jumped out at me when I looked, and still jumps out at me, is that one treats a 20 percent loss as “−0.2” and the other as “−20.” In particular, the comparison in Table 1 (here and in Markowitz 2014) is impossible using the Loistl formulation, since the logarithm (as a real-valued function of a real argument) is not defined for negative numbers. It boils down to this: I say that mean-variance approximations to, e.g. E log(1+R) work “well enough for practical purposes” for the typical application of MV analysis for asset allocation. In general, I offer Markowitz and Blay Chapters 2 and 3 as evidence. In particular, when returns are confined to a range of 30 percent loss to 40 percent gain, I offer Table 1 here as an immediate and self-contained demonstration of the proximity of qz to log(1+R). Based on his own example, Loistl’s fragment of a sentence says the MV approximation is not good. The only explanation I can offer for this difference in conclusions is the differing representation of a “twenty percent loss.” References Dimson, E., Marsh, P., & Staunton, M. (2002). Triumph of the optimists. Princeton, NJ: Princeton University Press. Loistl, O. (1976). The erroneous approximation of expected utility by means of a Taylor’s series expansion: Analytic and computational results. American Economic Review, 66(5), 904–910. Markowitz, H. M. (1959). Portfolio selection: Efficient diversification of investments (2nd ed). New York: John Wiley & Sons. Markowitz, H. M. (2014). Mean-variance approximations to expected utility. European Journal of Operational Research, 234(2), 346–355. Markowitz, H. M., & Blay, K. A. (2014). Risk-return analysis: The theory and practice of rational investing: 1. New York: McGraw-Hill Education. Savage, L. J. (1954). The foundations of statistics (2nd revised ed). Dover, New York: John Wiley & Sons. Von Neumann, J., & Morgenstern, O. (1944). Theory of games and economic behavior (3rd ed.). Princeton, NJ: Princeton University Press.

Harry Markowitz Rady School of Management, University of California, San Diego (UCSD), San Diego, CA 92109, USA E-mail address: [email protected]

0377-2217/© 2015 Elsevier B.V. and Association of European Operational Research Societies (EURO) within the International Federation of Operational Research Societies (IFORS). All rights reserved. http://dx.doi.org/10.1016/j.ejor.2015.06.011 DOI of original article: 10.1016/j.ejor.2015.06.010