Simulation Approach (Equal Weightings) compared to Simulation Approach (Unequal

Reviewing the above results it is shown that both of the above approaches improved upon the results found by previous literature. In chapter 2 it was revealed that numerous studies inferred that an optimally diversified portfolio should consist of between 10 and 30 shares (Elton & Gruber, 1977; Neu-Ner & Firer, 1997). Both the simulation approach with equal weightings and the simulation approach with unequal weightings indicated that the optimally diversified portfolio should consist of 15 shares or less. This reinforces the first respective hypothesis – the optimally diversified portfolio is shown to consist of between 10 and 30 shares. The second hypothesis however, was not supported from the above analysis. This hypothesis stated that by allowing the weights on each share to vary, the optimal portfolio should be enhanced and further diversification benefits unlocked. This proved not to be the case, the first approach solved for a portfolio consisting of 10 shares that yielded a variance of 0.307. This is notably superior to the portfolio solved for by the second approach; this portfolio consisted of 15 shares and yielded a variance of 0.744. The first approach also solved for this superior portfolio using a lesser number of simulations and in a quicker amount of time. However, it should be noted however that a result achieved on a portfolio where all of the shares are required to be weighted equally, should act as the lower bound of the potential best portfolio that could be established if the weights were allowed to vary. This is due to the fact that adding further flexibility to the methodology should only improve the results; if an equally weighted portfolio is indeed the optimal portfolio, using an approach where the weights on each share were allowed to vary would merely solve for the optimal portfolio which in this case would be an equally weighted portfolio.

The results found are not in agreement with the above argument. This could be due to the limited computing power available to the simulations. As previously discussed, allowing the weights on each share to vary exponentially increases the number of potential portfolios that can be constructed. It can be assumed that if sufficient computing power was available or if the simulation was allowed to run for a vastly increased amount of time, an improved result may have been established. This theory explains why varying the weights on each share as opposed to maintaining an equally weighted portfolio may have led to inferior results. This is supported by the results – the first valid trial number when the weights were equal was solved for in 46 seconds and found on the second simulation. When the weights were allowed to vary the first valid trial number was

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found after 1 minute and 42 seconds – this was simulation number 533. Furthermore, 99.98% of the trials run in the equally weighted simulation were valid but merely 19.29% of the trials were valid when the weightings on each share were allowed to vary. In summary, the above comparison reinforces the first hypothesis that an optimal portfolio does indeed consist of between 10 to 30 shares. The second hypothesis is not however proven – allowing the weights to vary does not in fact lead to an enhanced performance in solving for the optimal portfolio.

A graphic illustration of the results achieved from the traditional simulation approach (equally weighted and unequally weighted) when the portfolio was restricted to incrementally consist of a number of shares, ranging from 1 to 30 is shown in Figure 44. It can be seen from the figure that the expected trend (the minimum variance portfolio initially begins higher for a lower number of shares and dramatically decreases up to a point where including additional shares to the portfolio adds only a slight diversification benefit) is observed more noticeably in the simulation where the weights have been restricted to be equal. The trend of the equally weighted approach above is unstable (in some cases adding a share in fact increases monthly variance). This instability is exacerbated when the weightings allowed on each share are variable. It is expected that both trends should display a stronger trend if the simulations were allowed to run for longer (albeit potentially excessive) amounts of time, as mentioned previously. The reasoning for the weaker trend in the instance with unequally weighted constituents is discussed previously, and can be said to be due to the increased quantum of potential portfolios. Furthermore, the figure reiterates the findings mentioned earlier - allowing the weights to vary does not provide additional benefits to diversification. It is shown however, that the minimum variance portfolio contains less than 30 shares, demonstrating alignment to the findings of Neu-Ner and Firer (1997) and those of Statman (1987), with a slight improvement as fewer shares were deemed necessary to be employed (between 10 and 15), amongst other previous authors as discussed in Chapter 2. The analysis now

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continues to further answer the first research question through a similar comparative analysis, now aimed at the results achieved from the genetic programming approach.

Figure 44: Traditional Simulation Approach: Comparison of equally weighted and unequally weighted results

4.4.2 Genetic Programming Approach (Equal Weightings) compared to Genetic