Challenging Historical Data

75 Summing up, the foreign equity index imposes the biggest surprise with low volatility and negative average return. Government-, corporate- and mortgage bonds are not precisely located in risk-return space in accordance to what was expected.

To briefly comment on the performance of the different markets, it is seen in table 4.3.1 that the bond markets, except the inflation-linked bonds, exhibit the highest performance. These markets have the highest return per unit of overall risk. The Danish equity market is very volatile, why this market does not outperform the bond markets in spite of the fact that it exhibit highest return.

It has a Sharpe ratio of 0,30. From what has emerged with respect to the foreign equity market, it is not surprising that this exhibits a negative Sharpe ratio, -1,1, why this portfolio itself is not efficient, since it is located below the minimum variance portfolio, referring to section 3.2.1.1 The Efficient Frontier.

Having presented the individual investment opportunities in the portfolio each constituting individual non diversified portfolios, the analysis investigating the diversification possibilities now proceeds. Before turning to this matter, it is found necessary to comment on the challenges in using historical data in the theoretical portfolio optimisation problems.

76 When applying historical data, the range of period is important. The wider time range, the more cycles in economies the data contain. Some argue that this provides a more accurate picture of how economic factors come into existence. On the other hand, it is questioned if 30 year data says much about the current development. Today, the development in economies happens so fast that the historic cycles may not be that saying with respect to the cycles of newer dates. It is assessed that more often than prior, events or other things causing shock in economies happens.

Referring to section 4.1.2 Information on Data for the Theoretical Optimisations, it is seen that just over the last 10 years (2001-2010) several factors have implied different shock in the economies.

From the 10 year data in this research setup, investing in foreign equity on average has paid out a negative return. This makes the standard mean-variance optimisation model useless due to mathematical circumstances. When constructing the minimum variance portfolio, the model setup will place a large proportion of the investment in foreign equity. Such an investment, clearly, does not make sense at all.

It is important to state, that the historical data outcomes of the other assets are just as poor and misleading as the foreign equity index. The standard mean-variance model does not capture this, since none of the other assets show negative expected returns as foreign equity does⁵⁹. This, among other factors, indicates that the standard mean-variance model may not be applicable in solving optimisation problems. The Black-Litterman model is used in this thesis, referring to section 3.3 The Black-Litterman Optimisation Model. This theoretical model is selected to reach as well-authenticated outcomes as possible in the research setup. Among other factors, this namely allows for incorporating opinions to the expected returns, hence these are not based on the historical data.

The data set on foreign equity is changed and it is decided the only data input changed. It is acknowledged that this is not optimal, but a reasonable action due to the theoretical optimisation

59 The standard mean-variance optimisation model is not used in the analysis in this thesis, however see appendix I – Standard Mean-Variance Model to see how the negative returns affect the standard model.

77 model selected in this analysis. An argumentation will follow; For the purpose of the application of the Black-Litterman model, the implied equilibrium returns are the model’s outset. After applying the investor’s opinions of the development in the markets, the opinion-adjusted expected returns will serve as the return input for the continuation of the optimisation process.

The change of foreign equity is made in order to obtain an expected standard deviation, rather than the one the historical data defines. Obviously, changing the standard deviation also implies a change in the expected return. However, the return is adjusted for opinions in the optimisation process anyway, why the change has no effect on the outcome. The change in standard deviation will affect the analysis since the portfolio weights, the returns on the portfolio etc. are based on the covariance matrix.

Practically, an expected/mean return and a standard deviation are estimated from the rationale described below (a monthly mean return of 0,40% and a standard deviation of 4,45%). Then 119 random numbers have been drawn by use of the Excel add-in Data Analysis. These random numbers represents the returns on the foreign equity market from this point on in the analysis.

The choice of the specific values is addressed both from a theoretical and intuitive approach.

Theoretically, it is assessed that equities have higher risk than bonds. Furthermore, international diversification reduces the volatility in the portfolio hence it is reasonable, that, in this context, the foreign equity index is located more to the left in the risk-return space compared to the domestic, Danish equity index. This corresponds to what is seen in figure 4.3.1 Risk-Return Space in section 4.3 The Individual Asset Markets. Theory also assumes that equity pays a higher return than bonds, as a consequence of the higher risk implied. This speaks in favour of locating foreign equity with a higher return than bonds.

Seen from a more intuitive point of view it is found reasonable that Danish equity has outperformed the foreign index with regards to expected returns. In the 10 year period of interest (2001-2010) the global economy has been highly affected by the turmoil in the financial markets, among other factors. This has also affected the Danish economy hence the Danish equity markets, but not to the same extent. Denmark had a more stable and positive economic outlook

78 prior to the crisis; high employment, positive public finances etc, rather low debt to foreign countries etc. than many other countries in the world (Central Bank of Denmark, 2011b).

Several aspects should be mentioned with respect to the lower return on the foreign equity index.

As stated earlier, approximately 50% of the investments in the selected foreign equity index are placed in the USA. The US economy has been in recession in more periods over the last ten years, the country still experience high unemployment and the economic downturn has not turned around yet (Bloomberg, 2010). Approximately 10% of investments in the index have been allocated in Japan and United Kingdom respectively (see appendix C - The Theoretical Optimised Portfolios). It is known that especially United Kingdom has also been remarkably affected by the financial crisis.

Having addressed the improperness of using historical data as predictor of the future as well as conducted and justified a change in the data set, the next section contains the more in-depth quantitative analysis. Theoretical optimal portfolios are derived and they are compared to the actual portfolio held by the Danish pension fund sector in parallel. This enables an assessment of whether it is possible for the Danish pension fund sector to optimise its portfolio, to benefit pension contributors to a higher extent than it does.

4.5 Theoretical Portfolio Optimisation & Comparison to the Actual Portfolio