Model Short-Comings

Markowitz’ mean-variance model might seem reasonable from a theoretical point of view, as it is easy to understand and clearly presents the important concepts of risk and return. However, Richard Michaud (1989) discusses the practical problems in his article The Markowitz Optimization Enigma: Is ‘Optimized’

Optimal?. He claims that that the model often leads to irrelevant optimal portfolios and reviews a number of disadvantages using the model, which likely appears with the applied data of this thesis, particularly due to their high correlation. Two of the most important disadvantages are stated and addressed below⁵⁸. 5.3.2.1 First Disadvantage: Investment Instability

We estimated expected return based on historic data in chapter 4. Regardless of estimation methods, exact estimates of future returns, variances, or covariance are, however, subject to estimation errors, and some would even argue they are impossible in the first place. Michaud and Black and Litterman (1992)

58 Michaud (1989): p. 35

56 support this claim, stating that the optimization process inherent in the model maximizes errors⁵⁹. The mean-variance model overweight assets with high expected return and negative correlation and underweight those with low expected return and positive correlation⁶⁰. These assets are according to Michaud (1989), those that are most prone to be subject to estimation errors.

This argument appears to be somewhat contradictory. The reason for investors to estimate high return on assets should be that they believe the asset will return well in the future. It then seems reasonable to appreciate that the model provides an overweight to these assets in the portfolio. The purpose of repositioning the portfolio is based on the presumption that returns change over time and are therefore constituted by the CAPM calculated on a monthly basis. Hence, the return estimate applied in the mean-variance model for e.g. Technology are higher in mid-2002 before the bursting of the dot.com bubble than in mid-2008 during the financial crisis. Return estimates during the course of dynamic asset allocation thus change, but investment opportunities display high covariance suggesting they follow similar return patterns. The mean-variance model will then have difficulties in selecting investment opportunities that trend differently. Hence, it is more likely to select to select investment opportunities based on return estimates. These estimates change between months, and so is their portfolio position likely to as well. One advantage and disadvantage follows. If one investment opportunity possess substantially higher expected active return than other, the model is likely to provide substantial portfolio weight to that sector, which it maintains until portfolio reweighting is initiated again. On the other hand, if that sector is indeed the only investment opportunity expected to outperform the benchmark upon portfolio repositioning the model should allocate a correspondingly large amount funds to that sector. The only problem with this scenario is that if the model allocates all or most funds into a single or a few investment opportunities in the first period the portfolio is likely to sustain losses in the next period as positive returns are likely to be followed by negative returns as a result of return stationarity among investment opportunities. This issue however, refers to market timing and should therefore be addressed accordingly.

5.3.2.2 Second Disadvantage: Concentration

A second disadvantage concerns the use of constraints to the portfolio. In order to place all investment funds in the portfolio we imposed the constraints of no short sales and no financial gearing meaning

59 Black, Litterman (1992): p. 34

60 Michaud (1989): p. 36

57 investment positions of each sector spans between 0% and 100% of the portfolio. Unfortunately, when using the mean-variance model to help optimize the critical allocation decision, the unreasonable nature of the yielded result has proved dissatisfying. When no constraints are imposed, the model tends to ordain large short and long positions in some sectors, and when constraints are imposed, the model often prescribed corner solutions with zero weights in many sectors as well as large weights in assets with low levels of capitalization that have high expected returns and low correlations towards other assets. Such scenario also applies with the investment opportunities. Figure 5.2 and 5.3 illustrates an example of asset allocation by the mean-variance model with and without constraints.

Figure 5.2 Short Sale Restricted Sector Allocation

Restricting minimum sector position to zero

Source: Own Creation

When short sale and financial gearing are imposed, minimum sector positions are zero, but dynamic asset allocation allocates all funds to Technology. The information ratio, 58,20%, on the left hand side of the figure, is quite attractive, but the model demands high portfolio beta, 4,63, to produce positive residual return. Restricting the portfolio beta to maximum 1,0 results in the following allocation.

Sector Position Covariance Vector Expected Return Beta Portfolio Expected Return 5,53%

Basic Materials 0,00 -5,16 0,01 0,39 Benchmark Expected Return 1,47%

Consumer Goods 0,00 -21,95 0,01 0,14

Consumer Services 0,00 -16,34 0,01 0,56 Expected Residual Return 4,06%

Finance 0,00 8,43 0,01 0,35 Tracking Error 6,98%

Healthcare 0,00 -18,55 0,01 -0,24 Information Ratio 58,20%

Industrials 0,00 24,87 0,02 1,24

Oil & Gas 0,00 2,89 0,02 0,80

Technology 1,00 13,64 0,06 4,63

Telecommunications 0,00 9,42 0,03 1,71

Utilities 0,00 5,72 0,01 -0,16

Sum 1,00 2,96 4,63

Denmark 1,47% 1,06

58 Figure 5.3 Short Sale and Beta Restricted Sector Allocation

Restricting minimum sector position to zero Restricting portfolio beta to maximum 1,0

Source: Own Creation

Now the asset distribution is more diversified. The model selects sectors with both high and low beta values, but favors low beta sectors as these are given the larger positions. Restricting beta decreases the tracking error, and correspondingly the residual return, hence the information ratio, which is now much lower than without the beta restriction. Although the portfolio undertakes less market risk (beta equals 1,00) than the benchmark (beta equal 1,06) and acquires expected positive residual return, the asset allocation is not expected to add value to the portfolio as achieving positive active return by a negative active beta, is considered an act of luck rather than skill.

Even with the restriction of the portfolio beta the portfolios are highly sensitive to errors in their input data – expected return estimates, beta and covariance estimates. The mean-variance model provides excessive weight to assets with large expected returns or low beta. A small change in expected return on

Sector Position Covariance Vector Expected Return Beta Portfolio Expected Return 1,87%

Basic Materials 0,00 -5,16 0,01 0,39 Benchmark Expected Return 1,47%

Consumer Goods 0,00 -21,95 0,01 0,14

Consumer Services 0,00 -16,34 0,01 0,56 Expected Residual Return 0,40%

Finance 0,08 8,43 0,01 0,35 Tracking Error 3,63%

Healthcare 0,00 -18,55 0,01 -0,24 Information Ratio 11,14%

Industrials 0,04 24,87 0,02 1,24

Oil & Gas 0,13 2,89 0,02 0,80

Technology 0,04 13,64 0,06 4,63

Telecommunications 0,39 9,42 0,03 1,71

Utilities 0,31 5,72 0,01 -0,16

Sum 1,00 2,96 1,00

Denmark 1,47% 1,06

59 one asset might generate a radically different portfolio. According to Michaud this mainly depends on an ill conditioned covariance matrix, which is exemplified in insufficient historic data⁶¹.

An easy but rather primitive approach to avoid portfolio corner solutions and guarantee diversification is the implementation of the condition of a maximum portfolio sector position. This adjustment carries both advantages and drawbacks. Assuming that sectors are likely to show realized return fluctuations over short time periods, and depending on the frequency on portfolio repositioning, the investor may be unfortunate to conduct reposition during a month where realized returns are low or negative while they are extraordinary high in both prior and succeeding months. This sector will then accordingly be given zero or low asset weight during months of high returns and thus bypasses profitable short-term investment opportunities. This issue concerns market timing and in order to cope with it the investor needs to ensure that he captures these extraordinary high returns with at least a proportion of his portfolio a limitation of portfolio proportions is imposed, forcing the mean-variance model to conduct dynamic asset allocation to a broader number of investment opportunities. However, by ensuring a limited proportion of each investment opportunity also assumes a high risk as the months between the reweighting months might as well yield low or negative realized returns. Figure 5.4 extends figure 5.2 and 5.3 by imposing the restriction maximum 20% portfolio weight. Thus, running the mean-variance model in Excel two different conditions for the portfolio were tested: a maximum 20% restricted portfolio and an unrestricted portfolio. The performance of both portfolios is analyzed in the context of active portfolio management in chapter 6.

61 Michaud (1989): p. 35

60 Figure 5.4 Maximum Asset Position Allocation

Restricting maximum sector position to 20%

Source: Own Creation

The model does not manage to increase expected residual return from figure 5.3 but actually results in a lower information ratio, given the restriction of beta is imposed. The issue of market timing determines whether allocating few investment opportunities into the portfolio is a more attractive investment strategy as opposed to ensuring diversification, as it determines the period of time the investor must hold each portfolio, and the longer the time frame the more likely the rational investor is to hold the diversified portfolio.