Apart from tax and transaction cost advantages, ETFs can be a useful security for those who have little time for analysis and wish to have some diversification. However, it may be interesting to see how using the top 10 holdings in an ETF, such as the Vanguard Total Stock Market ETF (VTI), would perform on an optimized basis.

The rationale for using an ETF is that it tracks an index. And if an index is thought of as the market portfolio, in the Modern Portfolio Theory sense of the word, then some may suggest that an efficient portfolio can be arrived at by putting some portion of your investible cash in this "market portfolio" and the rest in a risk free asset. But I have shown in my previous article ETFs, Efficiency & the Market Portfolio that this is often easier theorized than performed.

In this article, we will compare the performance of the Vanguard Total Stock Market ETF against the performance of its top 10 holdings optimized using a mean-variance algorithm.

The top 10 holdings in the Vanguard Total Stock Market ETF as at 27 July 2012 are as follows:AT&T (T), Apple (AAPL), Chevron Corp (CVX), Exxon Mobil Corp (XOM) , General Electric Co (GE), International Business Machines Corp (IBM), JPMorgan Chase & Co (JPM), Johnson & Johnson Co (JNJ), Microsoft Corp (MSFT), Procter & Gamble Co (PG). *Compare these with Berkshire Hathaway's holdings (BRK.A) (BRK.B)*

The resulting efficient frontier is shown below where volatility is the x axis and average return is the y-axis. The average return is the average of the daily moving annualized returns from inception or 3,650 days whichever is less. Note the actual performance of the ETF denoted by the white star symbol.

Going back in time and comparing the growth in portfolio value of the VTI against an optimal portfolio (constructed at that time from among its top 10 holdings) with volatility equivalent to an equal-weighted portfolio yields the following growth data.

Note that "Buy & Hold" means the optimal portfolio has not been re-balanced, "Rebalanced at Original" means the optimal portfolio was re-balanced without re-calculating the optimal weights, and "Rebalanced at Current" means the optimal portfolio was re-balanced after re-calculating the optimal weights at the time of each re-balance. Growth is defined as growth in the adjusted value every 365 days starting from 26 July 2005:

2006 | 2007 | 2008 | 2009 | 2010 | 2011 | 2012 | |

VTI | 4.5% | 19.1% | -12.3% | -20.2% | 16.2% | 25.5% | 0.7% |

Buy & Hold | 19.4% | 31.8% | -1.5% | -7.0% | 10.6% | 34.7% | 16.7% |

Rebalanced at Original | 19.4% | 30.3% | -2.5% | -7.6% | 3.1% | 29.7% | 2.8% |

Rebalanced at Current | 19.4% | 32.2% | -2.1% | -8.9% | 5.1% | 19.2% | 5.0% |

The optimal portfolio in 2005 that produced the growth data above is as follows: *Apple 8.0%, Chevron Corp 6.0%, Exxon Mobil Corp 43.6%, JPMorgan Chase & Co 23.7%, Procter & Gamble Co 18.7%*

## Testing the Assumption of Normality

The assumption of normality is an important consideration in optimization. The distribution of returns that went into the calculation of the optimized portfolio tested approximately normal i.e. about 68% of returns were within 1 standard deviation of the mean, 95% within 2 standard deviations and 99% within 3 standard deviations:

## Doing a Fair Comparison

You will notice that I compared the optimal portfolio (Green Dot) to the performance of an equal-weighted portfolio (Yellow Dot) in 2005. The equal-weighted portfolio had a volatility that was lower than the VTI (the white star) in 2005.

What if we calculated the optimal portfolio with same volatility as the VTI in 2005? The results turned out to be even more in favour of the optimal portfolio:

Irregardless of risk-free asset | 2006 | 2007 | 2008 | 2009 | 2010 | 2011 | 2012 |

Optimized Buy & Hold | 25% | 36.8% | -1.2% | -6.5% | 17.7% | 41.9% | 20.1% |

*Note that the top ten stocks of the VTI may vary through the years.*

## Risk Free Asset

Including a risk-free asset in the mean-variance calculations is equivalent to drawing a straight line from the risk-free asset to the tangent optimal portfolio on the efficient frontier.

In the above analysis, adding a risk-free proxy to the calculations did not make an impact. This is illustrated by the following graph where the iShares Barclays 1 to 3 Year Treasury Bond ETF (SHY) is included in the calculations and the optimal portfolio (yellow dot) is found to be on the part of the frontier where the risk free proxy has no impact (borrowing aside).

## Conclusion

The analysis is not meant to downplay the advantages of investing in an ETF. In fact, the ETF could also be part of the optimized portfolio at certain levels of volatility because of its diversification advantage. What the analysis does, however, point out is that it is altogether possible to better a return simply by using the right weights on the right securities.

An optimizer does not know which are "right" securities. But it can calculate a portfolio that is more efficient from any investment universe that you have proposed to it to be "right".

**Disclosure: **I have no positions in any stocks mentioned, and no plans to initiate any positions within the next 72 hours.