## Markowitz meanvariance portfolios are suboptimal

While the 4>a investor has a 4>a optimal portfolio described above, let's assume that the mean-variance investor is not aware of the 4>a paradigm and constructs a mean-variance optimal portfolio. We assume that the mean-variance investor operates under the same assumptions A1-A4 as the 4>a investor. Let ERq be the expected return and oq the standard deviation of the returns of a portfolio q. The mean-variance investor's optimal portfolio is:

M2(f) = arg minqegCTq subject to:

The mean-variance optimal portfolio can also be constructed by solving the obvious dual optimization problem of maximizing the expected return for a constrained risk level. One knows that, in the mean-variance paradigm, contrary to the 4>a paradigm, the mean-variance optimal portfolio is independent of any ETL tail probability specification.

The subscript 2 is used in rn2 as a reminder that when a = 2 you have the limiting Gaussian distribution member of the Stable distribution family, and in that case the mean-variance portfolio is optimal. Alternatively you can think of the subscript 2 as a reminder that the mean-variance optimal portfolio is a second-order optimal portfolio, i.e., an optimal portfolio based on only first and second moments.

Note that the mean-variance investor ends up with a different portfolio, i.e., a different set of portfolio weights with different risk versus return characteristics, than the 4>a investor.

The performance of the mean-variance portfolio, like that of the 4>a portfolio, is evaluated under the 4>a distributional model, i.e., its expected return and expected tail loss are computed under the 4>a distributional model. If in fact the distribution of the returns were exactly multivariate Gaussian (which they never are) then the 4>a investor and the mean-variance investor would end up with one and the same optimal portfolio. However, when the returns are non-Gaussian 4>a returns, the mean-variance portfolio is sub-optimal. This is because the 4>a investor constructs his/her optimal portfolio using the 4>a distribution model, whereas the mean-variance investor does not. Thus the mean-variance investor's frontier lies below and to the right of the 4>a efficient frontier, as shown in Figure 16, along with the mean-variance tangency portfolio T2 and mean-variance capital market line CML2.

As an example of the performance improvement achievable with the 4>a optimal portfolio approach, we computed the 4>a efficient frontier and the mean-variance frontier for a portfolio of 47 micro-cap stocks with the smallest alphas from the random selection of 182 micro-caps described above. The results are displayed in Figure 17. The results are based on 3,000 scenarios from the fitted 4>a multivariate distribution model based on two years of daily data during years 2000 and 2001. We note that, as is generally the case, each of the 47 stock returns has its own estimate Stable tail index ai,i = 1, 2,..., 47.

Here we have plotted values of TailRisk = e ■ SETL(e), for e = .01, as a natural decision theoretic risk measure, rather than SETL(e) itself. We note that over a large range of tail risk the 4>a efficient frontier dominates the mean-variance frontier by 14-20 bp daily!

We note that the 47 micro-caps with the smallest alphas used for this example have quite heavy tails as indicated by the boxplot of their estimated alphas, shown in Figure 18.

to Q

RETURN VERSUS RISK OF MICRO-CAP PORTFOLIOS Daily Returns of 47 Micro-Caps 2000-2001

-FinAnalytica Phi-Alpha

---Competitors' Markowitz a m cc

TAIL PROBABILITY = 1%

TAIL RISK

Figure 17

47 MICRO-CAPS WITH SMALLEST ALPHAS

1.10 1.15 1.20 1.25 1.30 1.35 1.40 1.45 ESTIMATED ALPHAS

### Figure 18

Here the median of the estimated alphas is 1.38, while the upper and lower quartiles are 1.43 and 1.28 respectively. Evidently there is a fair amount of information in the non-Gaussian tails of such micro-caps that can be exploited by the 4>a approach.

We also note that the gap between the 4>a efficient frontier and the Markowitz mean-variance frontier will decrease as the Stable tail index values a get closer to 2, i.e., as the multivariate distribution gets closer to a multivariate Normal distribution. This will be the case, for example, when moving from micro-cap and small-cap stocks to mid-cap and large cap stocks.

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