SPECIAL SECTION: MATHEMATICAL FINANCE CURRENT SCIENCE, VOL. 103, NO. 6, 25 SEPTEMBER 2012 666 *For correspondence. (e-mail: rlk@cmi.ac.in) Modelling in the spirit of Markowitz portfolio theory in a non-Gaussian world Rajeeva L. Karandikar 1, * and Tapen Sinha 2 1 Chennai Mathematical Institute, H1 Sipcot IT Park, Siruseri 603 103, India 2 Department of Actuarial Studies, Instituto Tecnologico Autonomo de Mexico (ITAM), Rio Hondo #1, Tizapan San Angel, Mexico City 01000, Mexico Most financial markets do not have rates of return that are Gaussian. Therefore, the Markowitz mean variance model produces outcomes that are not opti- mal. We provide a method of improving upon the Markowitz portfolio using value at risk and median as the decision-making criteria. Keywords: Financial markets, median, portfolio theory, value at risk. Introduction: a short history of portfolio theory PORTFOLIO diversification has been a theme for the ages. In the Merchant of Venice, William Shakespeare had An- tonio say: ‘My ventures are not in one bottom trusted, Nor to one place; nor is my whole estate Upon the fortune of this present year’ (source: http://shakespeare.mit. edu/merchant/merchant.1.1.html , quoted in ref. 1). Shake- speare wrote it during 1596–98. As Markowitz 1 noted, even Shakespeare knew about covariance at an intuitive level. A similar sentiment was echoed by R. L. Stevenson in Treasure Island (1883), where Long John Silver com- mented on where he keeps his wealth, ‘I puts it all away, some here, some there, and none too much anywheres…’ (source: http://www.cleavebooks.co.uk/grol/steven/island- 11.htm ). This is, of course, a classic example of diversifi- cation. Not all writers had the same belief about diversifica- tion. For example, Mark Twain had Pudd’nhead Wilson say: ‘Put all your eggs in the one basket and – watch that basket’ (Twain, M., 1893, chap. 15). Curiously, Twain was writing the novel to sell it to stave off bankruptcy. Measuring ‘average’ returns to value a lottery has been around for millennia. But it was Bernoulli 2 who found that decisions based on just the average return of a ran- dom variable lead to problems – famously – the St Petersburg paradox, where a decision-maker rejects a gamble with an infinitely large average payoff. Bernoulli 2 too made a remark about diversification: ‘… it is advis- able to divide goods which are exposed to some small danger into several portions rather than to risk them all together’. The concept of risk for one asset has been summarized as the standard deviation for several centuries. It has the advantage of being measured in the same units as the original variable. This measure of risk for evaluating sin- gle assets in isolation was suggested by Irving Fisher 3 . He even commented on the time it takes to compute the stan- dard deviation in different ways. Markowitz 4 , in the preamble of his famous paper noted the following: ‘We next consider the rule that the investor does (or should) consider the expected return a desirable thing and the variance of return an undesirable thing. This rule has many sound points, both as a maxim for, and hypothesis about, investment behavior.’ We can ask the following questions: Why not consider other measures of central tendency or other measures of variability? The justification is easy for using the mean as a measure of central tendency, if the random variable (here, the rate of return of a portfolio of assets) follows a symmetric distribution (with finite central moments). In that case, other measures of central tendency such as the mode and the median are the same as the mean. In addi- tion, in such cases, a variability measure like the semi- variance (variance is calculated using squared deviations from the mean, whereas semi-variance is calculated by taking squared deviation from any arbitrarily chosen value) is easy to calculate if the distribution happens to be normal. Once we permit skewed distribution, alterna- tives such as the median as a measure of central tendency and general quantile measures or specific measures such as the value at risk (VaR) or tail VaR may be more rea- sonable measures of risk. These are essentially measures of risk beyond (or below) a certain threshold. A brief overview of the Markowitz model Suppose we have n individual stocks in the portfolio. We can work out the return and risk of the portfolio for all combinations of different proportions invested in differ- ent individual stocks. It produces an ‘envelope’ that Markowitz called the ‘efficient set’. In Figure 1, the shaded area represents all possible combinations of indi- vidual stocks. The parabola represents the envelope.