''Don't put all your eggs in one basket'' is a common saying that applies particularly well to investing. Thus, the concept of portfolio diversification exists and is generally accepted to be a good principle. But is it always and in every situation preferable to diversify one's investments? This Master's thesis explores this question in a restricted mathematical setting. In particular, we will examine the profit-and-loss distribution of a portfolio of investments using such probability distributions that produce extreme values more frequently than some other probability distributions. The theoretical restriction we place for this thesis is that the random variables modelling the profits and losses of individual investments are assumed to be independent and identically distributed.
The results of this Master's thesis are originally from Rustam Ibragimov's article Portfolio Diversification and Value at Risk Under Thick-Tailedness (2009). The main results concern two particular cases. The first main result concerns probability distributions which produce extreme values only moderately often. In the first case, we see that the accepted wisdom of portfolio diversification is proven to make sense. The second main result concerns probability distributions which can be considered to produce extreme values extremely often. In the second case, we see that the accepted wisdom of portfolio diversification is proven to increase the overall risk of the portfolio, and therefore it is preferable to not diversify one's investments in this extreme case.
In this Master's thesis we will first formally introduce and define heavy-tailed probability distributions as these probability distributions that produce extreme values much more frequently than some other probability distributions. Second, we will introduce and define particular important classes of probability distributions, most of which are heavy-tailed. Third, we will give a definition of portfolio diversification by utilizing a mathematical theory that concerns how to classify how far apart or close the components of a vector are from each other. Finally, we will use all the introduced concepts and theory to answer the question is portfolio diversification always preferable. The answer is that there are extreme situations where portfolio diversification is not preferable
