## Saturday 11 January 2014

### One Plus One Equals Two-and-a-Half

If 2 investments both return 5%, can a portfolio made up of these 2 investments return more than 5%? In theory, it should not be possible, but in reality, it is possible to wring out some extra return from these 2 investments. How is this possible? Through portfolio rebalancing.

Modern Portfolio Theory states that if 2 investments are not perfectly correlated, a portfolio made up of these 2 investments should result in lower risk. However, it does not mention that extra returns might be obtained from this portfolio. To prove that this is possible, we consider the returns and standard deviations of 5 indices over the last 25 years since 1988, as follows:

 Index Returns St Dev STI 5.0% 22.9% DJIA 8.5% 14.4% HSI 9.3% 25.7% Nikkei -1.6% 21.5% FTSE 5.2% 14.6%

Table 1: Risks and Returns of 5 Component Indices

A portfolio of 2 of these indices is constructed, each with 50% weightage (e.g. 50% STI and 50% DJIA). A total of 10 such portfolios are thus constructed. The portfolios are rebalanced whenever the proportion of the indices deviate from the original weightage by 5%. The risks and returns of these 10 portfolios are assessed and compared to their component indices. Of the 10 portfolios, 2 of them actually exhibited higher return while 4 of them exhibited lower risk compared to both their component indices. Please see the tables below for the correlation among the indices, and the returns and risks of the 10 portfolios.

 STI DJIA HSI Nikkei FTSE STI 40.3% 54.6% 21.3% 34.0% DJIA 40.3% 38.2% 22.6% 59.7% HSI 54.6% 38.2% 16.1% 34.9% Nikkei 21.3% 22.6% 16.1% 21.0% FTSE 34.0% 59.7% 34.9% 21.0%

Table 2: Correlation (R-square) Among 5 Indices

 STI DJIA HSI Nikkei FTSE Self 5.0% 8.5% 9.3% -1.6% 5.2% STI 5.0% 7.3% 7.6% 2.3% 5.5% DJIA 8.5% 9.6% 3.9% 7.0% HSI 9.3% 4.6% 7.7% Nikkei -1.6% 2.2% FTSE 5.2%

Table 3: Return of All Portfolios Made Up of 2 Indices

 STI DJIA HSI Nikkei FTSE Self 22.9% 14.4% 25.7% 21.5% 14.6% STI 22.9% 17.0% 22.7% 19.0% 16.8% DJIA 14.4% 18.2% 15.5% 13.7% HSI 25.7% 19.8% 18.2% Nikkei 21.5% 15.5% FTSE 14.6%

Table 4: Risk of All Portfolios Made Up of 2 Indices

From Table 3, a portfolio of 50% STI and 50% FTSE returned an average of 5.5%, higher than STI's 5.0% and FTSE's 5.2%. Similarly, a portfolio of 50% DJIA and 50% HSI returned an average of 9.6%, higher than DJIA's 8.5% and HSI's 9.3%.

From Table 4, a portfolio of 50% STI and 50% HSI has a standard deviation of 22.7%, lower than STI's 22.9% and HSI's 25.7%. Similar observations can be seen from the portfolios of STI-Nikkei, DJIA-FTSE and HSI-Nikkei.

In total, 6 out of 10 portfolios exhibit either higher return or lower risk compared to their component indices. However, none of these portfolios exhibit both higher return AND lower risk.

The reason why a portfolio can beat the returns of its component indices is because when a particular index is performing poorly, the index that is performing better is sold and the proceeds reinvested into the poorer-performing index through portfolio rebalancing. When the poorer-performing index rebounds, the portfolio achieves better return compared to either of its component indices.

Having said the above, the portfolios do not beat both component indices all the time. A plot of the returns of a \$10,000 investment in the portfolio and its component indices indicate outperformance only during certain periods of time. See the charts below for the STI-FTSE and DJIA-HSI portfolios.

 Figure 1: Performance of Portfolio of 50% STI and 50% FTSE

 Figure 2: Performance of Portfolio of 50% DJIA and 50% HSI

Nevertheless, it cannot be denied that portfolio rebalancing provides benefits to investors either in terms of higher return or lower risk in majority of the cases. Portfolio rebalancing should be considered as part of any investor's arsenal of investment tools.

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