Addressing the Actual vs. the Theoretical Optimised Portfolio

93 The table clarifies and gathers the outputs from the optimisations conducted in this section. The foundations of the optimisations; minimising variance or maximising return, affect the outcomes.

Different restrictions imposed (the two allocation restrictions and - in portfolio 2c - the equality in variance) imply different outcomes both in relation to the performances of the portfolios and the proportion of allocation to equity. The portfolio showing the lowest performance is actually the actual portfolio with a Sharpe ratio of 0,11. As concluded in this chapter, this means that it is possible to hold better performing portfolios than the sector does. The highest performing portfolio is portfolio 2c with a Sharpe ratio of 0,54. Compared to the remaining optimised portfolios, this shows higher allocations in equity than portfolio 1a and 1b and lower equity ratio than portfolio 2a and 2b with its level of 39,86%. This comparison between the different theoretical optimised portfolios should not be taken literally, since different restrictions are imposed to them individually, making them incomparable.

The next section addresses different factors relevant to comment on in relation to the analysis conducted so far.

94 are selected. However, this does not necessarily mean that portfolios estimated from theoretical modelling are comparable to actual portfolio compositions and performances.

The two points of interest will be elaborated upon in parallel. It is acknowledged that many factors have an impact on the question of basis for comparison and the diverging performances.

Only some of these are included.

(1) At first, it is found important to appoint the inference of the statistical tests on the data obtained for use in the optimisation process. See section 4.2 Statistical Analysis of Data for the Theoretical Optimisations. It was concluded that the data exhibits weak or non normality and is not randomly distributed. These facts affect the outcomes of the optimised portfolios. The Black-Litterman model assumes random and normally distributed returns. The consequences of the fact that this assumption is not fulfilled imply that results are biased and hence not valid, referring to section 4.2 Statistical Analysis of Data for the Theoretical Optimisations. This affects the basis of comparison of the optimised and actual portfolio. If the estimates extracted from the theoretical optimised portfolio are not true a comparison to real thus valid values is not appropriate.

Moreover, as a consequence of the less accurate estimates in the optimised portfolio, this affects the outcomes, including the performance. Therefore, a part of the divergence between the performances of the two portfolios is also explained by this fact.

(2) Another condition important to address, is how the covariance is applied in the Black-Litterman model and how that affects the outcome of the modelling. Speaking in practical model terms, when implementing the investor’s opinions into the model, only the expected returns are affected, while the volatility remains the same without any “opinion adjustment”. As the covariance matrix is included in the calculation of optimised weights, the unchanged covariance matrix put together with the opinion-adjusted expected returns possibly result in misleading optimised portfolios. From portfolio 2c it is evident that, on portfolio level, diversification is low and further investments in bonds are not diversified across this subgroup of assets. This is assessed partly due to changes in expected return and maintained levels of volatility. As an example, corporate bonds have a relatively high opinion-adjusted expected return at the retained low volatility. This combined with the fact that the bond markets mutually are highly correlated imply that all investments are made in this group of bonds.

95 Alternatively, a covariance matrix could be estimated according to the investor’s opinions, however, it demands advanced knowledge within the capital markets to estimate these in a proper way. Further, the theoretical model setup (Benninga, 2008) ascribes the use of historical data with regards to the covariance, why this is done in this analysis with the awareness of the drawbacks in relation to the factor appointed here and the use of historical data to predict the future. It is assessed that a part of the divergence between the outcomes of the actual and theoretical optimised portfolio stems from the absent “opinion adjustment”.

(3) The comparability of the theoretical and actual portfolios is relevant to question with respect to the time span over which data is obtained. The actual portfolio is conducted by use of five year data; 2006-2010. The reason for this is addressed in section 2.3 The Actual Portfolio of the Danish Pension Fund Sector. The optimal portfolio is conducted on data covering 10 years;

2001-2010. This is rationalised by a wish to achieve highest possible robustness in data, see section 4.1.2 Information on Data for the Theoretical Optimisations. This difference in number of years is assessed not necessarily a reason why the returns diverge from each other. Yet, the difference in number of years is criticised with respect to the basis of comparing the portfolios.

(4) Another critique of the appropriateness of comparing a theoretical model with a real world portfolio is the question of the validity in the theoretical model applied to conclude on the performance of the pension fund sector (see section 1.5.2 Data Review). The appropriateness of comparing “two worlds” is found reasonable to question. Nonetheless, since there is no other alternative than applying theory to the real world, it is done anyway with the acknowledgement of the problems attached to this.

(5) The theoretical model uses indices as asset opportunities in the portfolio. It is argued that both the divergence in the returns on the two portfolios as well as the comparability between them is affected by this. First of all, it is only possible to depict the real portfolio of the pension fund sector to some extent. The use of indices provides a rather simplified picture. In reality the portfolios are managed actively. For example, investments in foreign equity are assessed not carried out in a global index only, but rather, at least to some degree, divided into the different markets that are assessed beneficial to invest in from the pension fund’s point of view. For this

96 reason it is the performance of the individual investments, rather than the index’s performance, that subscribes the returns achieved on the actual investments carried out by the sector.

It should be noted that, some types of investments carried out by the pension fund sector can be approximated to the performances of indices since many investors benchmark this type of investments up against indices. Secondly, as addressed earlier not all real investment opportunities have been possible to include in the model - for example investments in property.

This has been taken into account in estimating the actual portfolio D - The Actual Portfolio. Yet, it is not possible to extract all “noise” why the two portfolios, ceteris paribus, are less comparable with respect to this factor.

All in all, having analysed the theoretical opportunities in heightening the performance of the Danish pension fund sector, it is clear that some factors, beside the solvency requirement and the high risk-management with respect to investments, explain the divergence between the theoretical optimal portfolio and the actual portfolio held by the sector. Further, there are circumstances, which make the comparison between the theoretical portfolio and the actual one inappropriate. These refer partly to the model applied for the purpose and partly to other practicalities, which constraint the comparison potential.

 Comparing theoretical optimised portfolios to actual portfolios is questionable.

 The violation of the statistical assumption affects the basis of comparison of the two portfolios negatively and it explains a part of the divergent performances.

 The absent “opinion adjustment” of the volatility in returns affect the divergence of the outcomes of the two portfolios.

 The difference in the number of years which the two portfolios are estimated from makes the comparability of them misleading.

 The use of indices in estimating the theoretical based portfolio can both explain divergence between the two portfolios’ outcomes and affect the comparison.

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