Portfolio One – The Minimum Variance Portfolio

The theoretical optimisations conducted in this section are based on minimising the variance in the optimised portfolios. The reasoning behind this is to focus on the risk contained in the portfolios since it is recognised that the Danish pension fund sector highly focuses on risk in its investment strategy.

Firstly, it is found interesting to investigate, how much return the sector is possible to achieve if it wants to minimise risk as much as possible and if no legal restrictions are enforced. The

61 At GMO's homepage it is stated that these forward-looking statements are based upon reasonable beliefs of GMO and are no guarantee for future performance (GMO, 2011).

62 In practical terms the variance is minimised in the modelling. This could equally well have been the standard deviation. The outcome of all models included in this thesis is disclosed in terms of the standard deviation. This is done to make it consistent with the modern portfolio theory which primary takes its outset in the standard deviation as the measure of volatility.

82 calculations are based on matrix algebra following the Black-Litterman theory (see appendix C - The Theoretical Optimised Portfolios for the detailed modelling).

Calculating the covariance matrix of returns is a part of the optimisation process applying the Black-Litterman model. It is found relevant to make a short note on this. With the adjusted foreign equity market, the covariance between the asset classes follows in table 4.5.2.1.

The covariance matrix shows the covariance between the different asset markets. The diagonal represents the variances of the individual markets. The covariance between Danish equity and both foreign equity, government bonds and mortgage bonds is negative. This means that while returns on Danish equities are above their expected value, the other asset classes’ returns tend to be below their expected returns. Negative covariance has positive effect on diversification opportunities. All other markets vary positively with each other. In the optimisation process, the covariance matrix, based on historical data, stays unchanged regardless of the opinions implemented on the future in the different market returns. It is assessed that this affects the outcomes of the theoretical optimal portfolios. This will be elaborated upon in section 4.6 Addressing the Actual vs. the Theoretical Optimised Portfolio.

The next step in the optimisation process is to calculate the inverse covariance matrix and identify the opinion-adjusted expected excess returns on the different asset markets⁶³. These are

63 The opinion-adjusted expected return vector is depicted in appendix C - The Theoretical Optimised Portfolios.

Also the excess returns give an idea of the level of the opinion adjusted expected returns.

Covariance Matrix Yearly

Danish Equity

Foreign Equity

Government Bonds

Mortgage Bonds

Inflation-Linked Bonds

Corporate Bonds

Danish Equity 0,03779 -0,00103 -0,00199 -0,00091 0,00065 0,00011

Foreign Equity -0,00103 0,02378 0,00070 0,00069 0,00097 0,00048

Government Bonds -0,00199 0,00070 0,00161 0,00119 0,00124 0,00091

Mortgage Bonds -0,00091 0,00069 0,00119 0,00134 0,00114 0,00090

Inflation-Linked Bonds 0,00065 0,00097 0,00124 0,00114 0,00585 0,00098

Corporate Bonds 0,00011 0,00048 0,00091 0,00090 0,00098 0,00100

Source: Own contribution

Table 4.5.2.1 Covariance Matrix

83 used in calculating the optimal weights. Given a risk-aversion level of one, the minimum variance portfolio is given by:

This matrix shows the order of the assets in the following matrices

378,83 7,83 1.045,18 50,25 118,47 929,67

7,83 513,28 47,71 179,81 40,38 0,65

1.045,18 48,71 26.262,22 15.808,00 1.123,85 8.694,30 50,25 179,81 15.808,00 34.356,06 662,32 15.896,09 118,47 40,38 1.123,85 662,32 2.590,95 900,41 929,67 0,65 8.694,30 15.896,09 900,41 35.380,52

,

0,38 0,340,01 0,090,09 0,14

·

%

%

%

%

%

%

3,281,94 11,84 13,96

0 68,99

∑^-1 is the inverse covariance matrix and is the opinion-adjusted excess returns⁶⁴. is the optimal portfolio weight vector representing the optimal allocations in the different asset classes in the minimum variance portfolio. Before reaching the optimal weights, different restrictions are imposed in the optimisation.

In the Excel setup (see appendix C - The Theoretical Optimised Portfolios), these restrictions are implemented by use of the Solver add-in to reach the optimal portfolio weights. In this example minimising the portfolio variance with a constraint on the individual weights to be at least zero and restricting the weights to sum to one in total (since short sales are not allowed, see section 1.5.4 Specific Comments to the Theoretical Optimisation Model) are imposed. These two restrictions are imposed in all optimisation and will not be mentioned when the other restrictions, imposed to the different optimisations, are outlined. In the following optimisations, the Solver is equivalently applied to estimate the optimal portfolios.

84 In table 4.5.2.2, the asset allocation constituting the above optimal portfolio, the expected return on the portfolio, the standard deviation and the performance measurement in terms of the Sharpe ratio of the minimum variance optimised portfolio are illustrated. Also the actual portfolio held by the Danish pension fund sector is shown for comparison (see section 2.3 The Actual Portfolio of the Danish Pension Fund Sector for estimation of the actual portfolio⁶⁵).

From 4.5.2.2 it is seen, that the minimum variance portfolio achieves a higher return of 4,78% at a lower volatility, compared to the actual portfolio of the Danish pension fund sector achieving a return of 3,75%. The minimum variance portfolio has a Sharpe ratio of 0,52, whereas the actual portfolio has a Sharpe ratio of 0,11. In line with the properties of the Sharpe ratio, it is concluded that the performance of the minimum variance portfolio, without any legal restrictions imposed, is higher than the actual portfolio (for further information see section 3.4 Performance Measures).

Investigating the equity ratio, the optimised portfolio allocates 5,22% in equity. This is much lower than the 19,68% in the actual portfolio. The high allocation of investments in mortgage bonds is surprising intuitively. Since the aim of the optimisation is to obtain an optimal portfolio with the lowest possible variance, it is clear that much is allocated in bonds. Because of a high

65 The asset class ”others” which appears in the actual portfolio in section 2.3.3 is split across the remaining asset classes to enable a comparison, see section 4.5.1 First Part of the Optimisation Processes.

Asset Class

Danish Equity 5,25%

Foreign Equity 14,43%

Government Bonds 19,02%

Mortgage Bonds 40,54%

Inflation-Linked Bonds 8,56%

Corporate Bonds 12,20%

Total 100,00%

Expected Return 3,75%

Standard deviation 5,25%

Sharpe ratio 0,11

Actual Portfolio Asset Class Optimal Portfolio

Danish Equity 3,28%

Foreign Equity 1,94%

Government Bonds 11,84%

Mortgage Bonds 13,96%

Inflation-Linked Bonds 0,00%

Corporate Bonds 68,99%

Total 100,00%

Expected return 4,78%

Standard deviation 3,05%

Sharpe ratio 0,52

Portfolio 1a: Minimum variance

Source: Own contribution

Table 4.5.2.2 Portfolio 1a

85 correlation between the different bond markets, much of the investments in bonds are place in mortgage bonds, which show the highest expected returns. This implies a less diversified portfolio. The correlation matrix is provided in appendix J – Correlation Matrix.

The legal restrictions, which limit the investment possibilities of the Danish pension fund sector, are of course incorporated in the actual portfolio. The actual portfolio depicts the real investments carried out by the sector and through that the conditions under which these are undertaken. The optimised portfolio, 1a, does not include these restrictions. It is found relevant to include legal restrictions in the theoretical optimisation model to obtain an as true and fair picture of the investment environment as possible.

The next step in the analysis is to find the optimised portfolio, implementing two allocation restrictions, regulated by Danish legislation. As stated in section 1.5.4 Specific Comments to the Theoretical Optimisation Model, the legislation possible to implement in the theoretical optimisations refers to the two allocation restrictions. These dictate maximum 70% of the total investments allocated in equity and maximum 40% in mortgage bonds⁶⁶. Table 4.5.2.3 illustrates the optimised minimum variance portfolio subjected to the two allocation restrictions.

66 The actual portfolio allocation in mortgage bonds actually exceed the 40% restricted by law. This is reasoned by the fact that the 5,20% invested in “others” are split across the remaining asset classes, referring to footnote 67.

When investigating the actual portfolio in section 2.3, the 5,20% invested in “others” is separated in reality.

Portfolio 1b: Minimum variance with legal restric.

Asset Class Optimal Portfolio

Danish Equity 3,28%

Foreign Equity 1,94%

Government Bonds 11,84%

Mortgage Bonds 13,96%

Inflation-Linked Bonds 0,00%

Corporate Bonds 68,99%

Total 100,00%

Expected return 4,78%

Standard deviation 3,05%

Sharpe ratio 0,52 Source: Own contribution

Table 4.5.2.3 Portfolio 1b

86 Including the two allocation restrictions does not affect the outcome of the optimisation. Table 4.5.2.3 shows the exact same outcome as the prior optimised portfolio. Looking at table 4.5.2.2 (portfolio 1a) it is evident that the two allocation restrictions are already obeyed before implementing them into the theoretical optimisation. Since this is the only feature included in this portfolio, obviously the same result will emerge.

From an intuitive point of view and not with the data input taken into account, the restrictions would imply more investments in less risky assets following the 70% limit in equity if this restriction was not obeyed before imposing it. According to modern portfolio theory, higher diversification in the portfolio, following an allocation change in the total investment, would lead to lower volatility at the same level of expected return. On the other hand, if investments are re-allocated to low risk assets, only as a result of the restrictions imposed, the expected return will probably decline together with the volatility since assets with low volatility has a lower return from a theoretical point of view. The theoretical gain from diversification in such case will be reduced caused by the restrictions imposed. Therefore, both lower volatility and return is expected from such scenario.

In document Benefitting Pension Contributors – An Analysis of the Danish Pension Fund Sector (Sider 87-92)