In my article entitled "In Search Of The Ultimate Efficient Portfolio Part 2", I showed that none of the 10 stock-based ETFs in a universe with 187 securities was on the efficient frontier, despite the fact some were geographically diverse.

Why weren't the ETFs on the efficient frontier? Is it possible to arrive at an optimal portfolio with the use of bonds and an ETF?

**ETFs and the Market Portfolio**

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, achieving the ultimate efficient portfolio with their use is easier said than done.

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 proxy and the rest in a risk free asset.

Let's look at an example.

The Guggenheim Russell Top 50 ETF (XLG) consists of the following stock holdings, with the first percentages being the original weights as at 6 July 2012 and the second percentages adjusted to maintain their relative proportions after omitting Phillip Morris International and Facebook due to their shorter histories. The weights reflect the proportionate market capitalizations of the various stocks :

Apple (AAPL) 9.20% 9.47%, Abbott Laboratories (ABT) 1.64% 1.69%, Amgen (AMGN) 0.93% 0.96%, Amazon.com (AMZN) 1.33% 1.37% American Express Co (AXP) 0.96% 0.99%, Bank of America Corp (BAC) 1.36% 1.40%, Bristol-Myers Squibb Co (BMY) 0.95% 0.98%, Berkshire Hathaway (BRK.B) 2.39% 2.46%, Citigroup (C) 1.27% 1.31%, Caterpillar (CAT) 0.91% 0.94%, Comcast Corp (CMCSA) 1.36% 1.40%, ConocoPhillips (COP) 1.13% 1.16%, Cisco Systems (CSCO) 1.46% 1.50%, CVS Caremark Corp (CVS) 0.99% 1.02%,

Chevron Corp (CVX) 3.37% 3.47%, Walt Disney Co (DIS) 1.39% 1.43%, General Electric Co (GE) 3.48% 3.58%, Google (GOOG) 2.49% 2.56%, Home Depot (HD) 1.29% 1.33%, International Business Machines Corp (IBM) 3.43% 3.53%, Intel Corp (INTC) 2.16% 2.22%, Johnson & Johnson (JNJ) 3.00% 3.09% JP Morgan Chase & Co (JPM) 2.11% 2.17%, Kraft Foods (KFT) 1.12% 1.15%, Coca Cola Co (KO) 2.46% 2.53%, McDonald's Corp (MCD) 1.46 1.50%,

3M Co (MMM) 1.00% 1.03%, Altria Group (MO) 1.15% 1.18%, Merck & Co (MRK) 2.04% 2.10%, Microsoft Corp (MSFT) 3.73% 3.84%, Oracle Corp (ORCL) 1.83% 1.88%, Occidental Petroleum Co (OXY) 1.13% 1.16%, PepsiCo (PEP) 1.77% 1.82%, Pfizer (PFE) 2.74% 2.82%, Procter & Gamble Co (PG) 2.71% 2.79%, QualComm (QCOM) 1.56% 1.61%, Schlumberger ltd (SLB) 1.42% 1.46%,

AT&T (T) 3.38% 3.48%, United Health Group (UNH) 0.93% 0.96%, Union Pacific Corp (UNP) 0.91% 0.94%, United Parcel Service (UPS) 0.93% 0.96%, U.S. Bancorp (USB) 0.99% 1.02%, United Technologies Corp (UTX) 1.11% 1.14%, Visa (V) 1.07% 1.10%, Verizon Communications (VZ) 2.06% 2.12%, Wells Fargo & Co (WFC) 2.62% 2.70%, Wal-Mart Stores (WMT) 1.94% 2.00%, Exxon Mobil Corp (XOM) 6.46% 6.65%

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.

The optimal portfolio with approximately the same volatility as XLG follows. Note that I demonstrated in previous articles the fact that you can easily include the risk free proxy in the mean-variance calculations. I have left this out in this discussion as the focus is on the risky portfolio :

*Amazon.com 1.4%, Amgen 8.6%, Apple 17.8%, Google 4.9%, McDonald's Corp 56.4%, Oracle Corp 1.8%, QualComm 1.9%, Union Pacific Corp 3.6%, Wal-Mart Stores 3.6%*

Going back in time and comparing the growth in portfolio value of the XLG against an optimal portfolio (constructed at that time) gives the following 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 the re-balance. Growth is from 1 Jan to 31 Dec in the respective years:

Growth in 2010 | Growth in 2011 | |

XLG | 7.59% | 4.75% |

Optimal (Buy & Hold) | 22.2% | 21.8% |

Optimal (Rebalanced at Original) | 22.2% | 21.0% |

Optimal (Rebalanced at Current) | 22.2% | 22.8% |

The results show that the ETF was not as efficient as the Optimal portfolio. So why wasn't XLG found on the efficient frontier? The answer is while ETFs track the major indices, the indices do not equal the market portfolio.

The market portfolio must by definition include every possible asset. Large as some of these indices are, they are not all-encompassing. Indeed, Roll has argued that the market portfolio is in fact impossible to calculate.

While the XLG weights are based on relative market capitalizations, the optimal portfolios get their weights from a mean-variance algorithm that looks at expected returns, standard deviations and correlations simultaneously.

Even if an all-encompassing ETF which included every possible asset did exist, only in an ideal situation, where the market was information efficient and in equilibrium, would it find itself sitting on the efficient frontier. Alas, such an ETF does not exist.

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