Nottingham University Business School

MSc Finance and Investment

Capital Market Analysis

Do returns on shares exhibit sufficient non-normality to warrant abandoning traditional mean-variance portfolio building procedures?

Roua Ioana DOBRE

Student ID: 4171076

1999 words

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All around us, concepts are always improved, paradigms always updated, views always

changed; so it should not surprise us that finance is not an exception. And yet, we are reluctant

to the new, we face great challenges when new theories are introduced, we go to great lengths

to avoid them and once they are finally accepted, we hit the wall when trying to apply them.

This process, usually more pompously explained, is called evolution. In finance, as in all other

applied fields, evolution is slowed down by the implementation of the new. So difficult is this

endeavour, that the very basis of investing, the portfolio building technique, is more than 50

years old. Of course, if the technique is good, why change it? However, this seems not to be the

case, as many researchers have deemed the traditional procedure unfitting as the basic

assumption shifts from normality to non-normality (Jondeau and Rockinger, 2005). What had

been thought of as normally distributed returns in financial markets has been discovered to be

non-normal as the formerly omitted outliers are beginning to be taken into account and crises

arise more often than normal. There is no universally accepted model to use instead, but there

are several alternatives upon which embracers of the new can begin building their models if

they haven’t already.

The times when extreme negative returns have shaken the normal distribution

assumption have actually been few, but sufficient to make researchers and practitioners alike

ponder on the implications. This is fortunate, as there is a great deal of evidence of non

normality to be considered, as show the many empirical studies conducted on this subject.

Chunhachinda et.al. (1996) perform the Wilk-Shapiro test and the Sen&Puri test to discover

that the world’s major 14 stock markets are not normally distributed and incorporate investors’

preference for skewness in a Polynomial Goal Programming model to determine the optimal

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portfolio. Asness et al (2001) identifiy serial correlation in hedge funds’ returns, Mashal and

Zeevi (2002) investigate the potential for extreme co-movements between financial assets by

directly testing the underlying dependence structure.

Extreme value theory provides an important set of tools and models by which joint

extreme behavior can be examined. Longin and Solnik (1995), Longin (1996), and Longin and

Solnik (2001) follow this approach in investigating large co-movements in international markets

and changes in correlation driven by the state of the market. Danielsson and de Vries (1997)

study tail index estimation via extreme statistics, while Starica (1999) develops extreme value

theory for conditional correlation models. The returns of hedge funds have been studied

carefully by Davies, et al (2006), Hayes (2006), Avramov, et al (2007), Hafner and Wallmeier

(2008) and a conclusion has been reached as to the non-normality of their returns. It is said that

since hedge funds and modern trading instruments have become such a great part of investors’

choices, observations of non-normality have increased. However, are these issues justification

enough for such a dramatic change of approach? Or can we “tweak” in some way the data or

the model in order to acknowledge non-normality or fix it back to an adjusted normality?

Abdullah and Hongtao (JP Morgan, 2009) have outlined the main ways in which non-normality

can be incorporated, namely unsmoothing serial correlation, modelling fat left tails using

extreme value theory and simulating the correlation breakdown with the help of copula theory.

They propose the use of Conditional Value at Risk (CVaR) as a better measure of downside risk.

This point of inflexion in financial theory has been inflicted by extensive losses suffered

by investors during several crises throughout modern history. The Mean-Variance portfolio

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building procedure was unable to predict these events, due to the assumption of normality

upon which it is constructed (Mandelbrot, 1963; Fama, 1963; Blattberg and Gonedes, 1974;

Kon, 1984; Longin, 1996). In addition, crashes appear to be more often occurrences than booms

(Fama, 1965; Simkowitz and Beedles, 1978; Singleton and Wingender, 1986; Peiro, 1999) but

not often enough to be statistically analysed, as a sound statistical analysis cannot be

constructed on a data set of maybe 20 observations altogether, scattered in 80 years’ time.

Subsequently, an abundant literature emerged, questioning the adequacy of the Mean-

Variance criterion proposed by Markowitz (1952) for allocating wealth. This apparent simplicity

by which Markowitz suggests investing should proceed is the reason why it underestimates

downside risk. When applying the Mean-Variance procedure, the basic principle is that clients

like return and dislike risk. However, this is greatly complicated when the assumption of

normally distributed returns fades. Positive skewness is preferred by investors, even trading

expected return for it, as shown by Arditti (1967) and Kraus and Litzenberger (1976). Scott and

Horvath (1980) say investors with consistent and rigid preferences for higher moments have

positive preferences for positive values of the odd moments and negative preferences for

negative values of the even moments. Not taking all this into account is a great danger, as is

ignoring the other manifestations of non-normality, as this accounts for greater downside risk

than predicted by the normal distribution upon which the Mean-Variance criterion is based.

Given all these problems that arise with the use of MVO, it is surprising that somehow, under

normal circumstances, the Mean-Variance-based portfolios still outperform the PGP-based.

Serial correlation defies the assumption of independent and identically distributed

returns, as one month’s return is influenced by the previous month’s return. This is found also

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as co-movements across international markets, which obviously greatly influence the success of

the strategies applied when constructing and optimising a portfolio. Fat left tails imply

observing negative returns in greater magnitude and with higher probability than assumed by

the normal distribution. Correlations under extreme conditions behave significantly different

than in normal conditions (Abdullah and Hongtao, 2009). “Even a cursory look at financial data

suggests that some time periods are riskier than others; that is, the expected value of the

magnitude of error terms at some times is greater than at others. Moreover, these risky times

are not scattered randomly across quarterly or annual data. Instead, there is a degree of

autocorrelation in the riskiness of financial returns” (Engle, 2011). It seems like mayhem at first

glance, and the fact that there are a lot of alternative models to the MVO does not help.

Perhaps this is a reflection of the fact that the data will never perfectly fit a theoretically

created model which by definition simplifies reality, so the investor has to choose the one that

in his opinion is close enough to reality. After all, he should only invest what he affords to lose,

so it should not bother him that once in a while he has to sit back and wait for the market to

rise again. He might even make a profit out of the crises, buying at small prices and waiting to

sell high.

A choice has to be made, as to whether the MVO approach can still be used, provided

an adjustment of the data, or another model has to be introduced, and if so, which one. I

believe the option should concern a trade-off between ease of use and goodness of fit. The

Mean-Variance Optimization would definitely be easy to continue using, because no shifting

costs would be involved in the process, only some adjustments to the data, as shown by

Abdullah and Hongtao (2009). However, the costs of continuing using MVO are high. Hu &

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Kercheval (2007) give a very straightforward illustration of this, portraying the fact that when

choosing the mean-variance and standard deviation as a measure of risk and operating under

the Normal distribution assumption, portfolios lie under the efficient frontier, losing a potential

increase of 20 to 30% for moderate level risk, given that the student-t distribution is a better fit

for the data. Furthermore, even when using 99% Expected Shortfall, but, for convenience still

assuming filtered returns are normal, sub-optimal portfolios are created (Hu & Kercheval,

2007). This means that not only the risk of negative returns is higher, but excluding the times of

financial crises, the investor still loses by not taking advantage of the existence of optimal

portfolios.

So, having reached the conclusion that applying the mean-variance building procedure is

no longer a viable option, academics have proposed an enormous stream of models to include

skewness, or (more rarely) higher order moments into portfolio theory. A variety of alternative

models have been offered, that fall under the category of mean-variance-skewness (MVS)

portfolio optimization. The majority of these models fail, however, to take into account all of

the objectives set, favouring one or two of the objectives in detriment of the other(s), an issue

MVO does not face, hence constantly outperforming in “normal” times.

Starting from specifications of the indirect MVS utility function, dual approaches search

for optimal portfolios via preference parameters reflecting attitudes towards risk and skewness

(Jondeau and Rockinger (2006) and Harvey, Liechty, Liechty, and Müller (2010) are recent

utility-based studies) (Briec, Kerstens, van de Woestyne, 2011). However, as much as the

authors of all the models proposed to replace the Mean-Variance portfolio optimization

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technique defend their course of action, no consensus seems to have emerged about a general

approach to multi-moment portfolio models.

It can be said that Goal Programming has been, and still is, the most widely used multi-objective

technique in management science because of its flexibility in handling decision-making problems with

several conflicting objectives and incomplete or imprecise information (Romero 1991; 2004; Chang

2007). There seems to be a preference for the Polynomial Goal Programming framework, but it is

nonetheless hard for practitioners to adopt due to its constant improvement by academics. The

advantage of the PGP framework is that it is general enough to accommodate investor desires for higher

moments skewness and kurtosis through preference parameters. “The exponents and coefficients

associated with the terms in the objective function are selected (by the decision maker) to reflect the

relative importance of satisfying goals as the corresponding deviational variables approach zero. These

powers and weights establish the marginal rates of substitution involved in the satisfaction of both inter-

level and intra-level goals. Each refinement of the objective function attempts to bring the resulting

solution closer to the decision maker’s true preference” (Deckro & Hebert, 1988). This is a time

consuming job. However, a decision is hard to reach by practitioners, as PGP often involves the solution

being a more complicated one, because the optimization of the 4 moments at the same time can be out

of reach. Hence, the investor is faced with a set of non-dominated solutions and a decision to make.

That decision can be further burdened by the possibility of mistakes made in the computations, which

amplified by higher moment formulae, can paint a different picture than that shown by reality, thus

cancelling the advantage of using PGP instead of MVO in the first place. Each refinement of the model

results in the increase in both accuracy and complexity of the objective function, along with rising the

costs of obtaining a solution.

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In conclusion, I believe that the evidence clearly suggests non-normality and the traditional

mean-variance portfolio building procedures does leave out important pieces of information that could

improve the outcome and avoid serious losses. However, there is a combination of factors which

prevent me from fully recommending shifting to PGP. When choosing the procedure, there is a trade-off

between simplicity and goodness of fit to reality. If MVO is chosen, the investor gets a simplified view of

the world and as such, extreme events are ignored but “normal” situations are better forecasted. If PGP

is chosen reality is better portrayed, but apart from the actual costs of shifting to a new model,

computational complexity and magnifying errors become weighing disadvantages. Bearing this in mind, I

am not wholeheartedly pro mean-variance, but cannot be against it either, because the alternatives are

not yet good enough to replace it. I do believe that evolution will do something about that. As W.

Edwards Deming said, “It is not necessary to change. Survival is not mandatory”.

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