I have been building on the theoretical underpinnings of portfolio optimization in the last few articles that I wrote. In my last article, I looked at 7 portfolios that spanned the volatility range of a sample of the Harvard Endowment Portfolio. The purpose of the article was to illustrate certain lessons on the relationship between risk, return, and investment horizon.

Assuming that the data that goes into the calculations obey the important assumption of normality or near-normality, an optimizer can be a powerful tool to help you make an informed decision as you go about designing your portfolio.

Now, let's apply these lessons to the following all-ETF portfolio.

## The ETF Portfolio

The following portfolio has been selected across a spectrum of ETFs. Each and every ETF here is negatively correlated to at least one other ETF within this universe of investments.

- BLDRS Emerging Markets 50 ADR Index ETF (ADRE)
- iShares Barclays 1-3 Year Treasury Bond ETF (SHY)
- First Trust IPOX Index ETF (FPX)
- Market Vectors Gold Miners ETF (GDX)
- PowerShares DB Agriculture ETF (DBA)
- PowerShares DB Gold ETF (DGL)
- PowerShares DB Precious Metals ETF (DBP)
- PowerShares DB Silver ETF (DBS)
- CurrencyShares Australian Dollar Trust (FXA)
- CurrencyShares Japanese Yen Trust (FXY)
- CurrencyShares Swedish Krona Trust (FXS)
- CurrencyShares Swiss Franc Trust (FXF)
- SPDR Wilshire REIT ETF (RWR)
- SPDR S&P 500 Trust ETF (SPY)
- Utilities Select Sector SPDR ETF (XLU)
- Vanguard Health Care VIPERs ETF (VHT)
- iShares Barclays 20+ Year Treasury Bond ETF (TLT)
- iShares Dow Jones US Real Estate (IYR)
- iShares Barclays 7-10 Year Treasury Bond ETF (IEF)
- iShares Barclays Short Treasury Bond ETF (SHV)
- iShares Goldman Sachs Technology Index ETF (IGM)
- iShares Goldman Sachs Natural Resources ETF (IGE)

## Negative Correlations

*click images to enlarge*

The graph above shows the correlation between each pair of the 22 ETFs above over the period 13 Feb 2008 to 17 Jan 2012. The 3D graph has been tilted upwards for you to appreciate the number of pairs of securities with negative correlation.

The graph above is the corresponding efficient frontier. It is "efficient" in the sense that every portfolio on the red line has the maximum expected annualized return for any given volatility as well as the minimum volatility for a given expected return. It is called a "frontier" because no other combination of weights within this portfolio of 22 ETFs is more efficient. In other words, no portfolio can exist above the red line.

## Efficient Within Each Risk Category

The 7 personalized efficient frontiers below represent a numerical integration between attitudes to risk and the risk-return characteristics of this given set of 22 ETFs which forms the investment universe. Investors can match their goals using these 7 frontiers accordingly. Risk here is defined as volatility.

The Low Risk, Medium Risk, and High Risk categories are simply the lowest one-third, middle one-third, and highest one-third of securities in terms of their volatility. For example, in Profile 3 below, every portfolio on the white frontier is efficient but must constitute 50% Low Risk, 40% Medium Risk, and 10% High Risk securities.

## Profile 1 - 100% Low Risk Securities

The graph immediately above shows the Profile 1 efficient frontier (white line). The efficient portfolio selected for study is SHY: 62.3%, FXY: 27.5%, VHT: 5.8%, SHV: 4.3%.

The graphs immediately below show the safety of the selected portfolio where the probability of a negative annualized return over a 5-year holding period for the selected portfolio is negligible. This portfolio with an emphasis on Treasury is suitable for goals that mature in 1 year's time.

## Profile 2 - 70% Low Risk, 30% Medium Risk

The graph immediately above shows a Profile 2 efficient frontier (white line). A model efficient portfolio here is DBP:5.2%, FXA:6.4%, SPY:8.7%, XLU:9.6%, IYR:0.1%, FXY:58.7%, IEF:11.3%.

The graphs immediately below show the safety of the portfolio selected where the probability of a negative annualized return over a 5-year holding period for the selected portfolio is also negligible. This portfolio with an emphasis on Currency is suitable for goals that mature in 2 year's time.

## Profile 3 - 50% Low Risk, 40% Medium Risk, 10% High Risk

The graph immediately above shows a Profile 3 efficient frontier (white line). A model efficient portfolio here is ADRE:10.0%, DBP:30.2%, SPY:5.6%, XLU:1.4%, IGE:2.8%, FXY:50.0%.

The graphs immediately below show the safety of the portfolio selected where the probability of a negative annualized return over a 5-year holding period for the selected portfolio is also negligible. This portfolio with emphases on Currency & Precious Metals is also suitable for goals that mature in 2 year's time.

## Profile 4 - 30% Low Risk, 40% Medium Risk, 30% High Risk

The graph immediately above shows a Profile 4 efficient frontier (white line). A model efficient portfolio here is ADRE:27.9%, RWR:2.1%, DBP:40.0%, DGL:13.2%, FXY:16.8%.

The graphs immediately below show the safety of the portfolio selected where the probability of a negative annualized return over a 5-year holding period for the selected portfolio is about 0.05%. This portfolio with emphases on Emerging Market stocks and Precious Metals is suitable for goals that mature in 3 year's time.

## Profile 5 - 10% Low Risk, 40% Medium Risk, 50% High Risk

The graph immediately above shows a Profile 5 efficient frontier (white line). A model efficient portfolio here is ADRE:20.4%, DBS:29.6%, DBP:40.0%, DGL:10.0%.

The graphs immediately below show the safety of the portfolio selected where the probability of a negative annualized return over a 5-year holding period for the selected portfolio is about 0.75%. This portfolio with emphases on Silver, Emerging Market stocks and Precious Metals is suitable for goals that mature in 5 year's time.

## Profile 6 - 30% Medium Risk, 70% High Risk

The graph immediately above shows a Profile 6 efficient frontier (white line). A model efficient portfolio here is ADRE:12.6%, DBS:57.4%, DBP:30.0%.

The graphs immediately below show the safety of the portfolio selected where the probability of a negative annualized return over a 5-year holding period for the selected portfolio is about 2.33%. This portfolio with emphases on Silver and Precious Metals is suitable for goals that mature in 7 year's time.

## Profile 7 - 100% High Risk Securities

The graph immediately above shows a Profile 7 efficient frontier (white line). A model efficient portfolio here is ADRE:22.0%, DBS:78.0%.

The graphs immediately below show the safety of the portfolio selected where the probability of a negative annualized return over a 5-year holding period for the selected portfolio is about 3.31%. This portfolio with emphasis on Silver is suitable for goals that mature in 9 year's time.

## Taming your optimizer

1. These model efficient portfolios are by no means the ultimate. They are, however, mathematically optimized and the user can regard them as true as at the time of analysis. Only securities with more than 3 years of return data have been included in the analysis to improve stability.

2. The optimizer may leave a certain security out of the selection if it has found another security within the universe that offers a better advantage. For example, if 2 securities have the same expected return, the optimizer will choose the one that reduces overall portfolio volatility *more*.

Similarly, if 2 securities contribute in the same way to portfolio volatility, the one with the higher expected return will be chosen. Notice I said contribute to portfolio volatility and not just volatility. This is because a security contributes both its own volatility, as well as, the effect of its correlation to the rest of the mix in the portfolio.

3. In certain situations, 2 securities with almost identical characteristics (i.e. expected return, volatility, and correlation to all others in the mix) can be swapped without impacting the overall risk-return character of the portfolio.

Why would you even want to swap them? Because the optimizer does not have the benefit of insight and knowledge that you may be privy too.

4. The reason why a particular portfolio mix is suitable for goals that mature in a certain number of years time is because of time diversification. An investor should be *willing* to hold on to that portfolio for the number of years stated in order to take advantage of time that can reduce the probability of an annualized negative return. In this exercise, we matched goal maturity to when the probability of negative return was 1% or less.

Note however that being "willing to hold" is different from actually holding a security for the number of years indicated. Whether you continue to hold on to a security is dependent on whether it continues to obey the assumption of normality or near-normality.

5. Another purpose for goal matching is it illustrates the risk you take much better. While volatility, confidence intervals and even profiling can be abstract, waiting a number of years for something to happen (in this case, the probability of negative return to reach 1% or less) is easily related to.

## In Conclusion

With a well diversified universe, a portfolio can be created and optimized to cater to almost any level of risk tolerance that the investor has. In the portfolios above, the emphasis moves from Treasury products to Currency to Precious Metals with an appropriate proportion of emerging market stocks.

The process of designing an optimized portfolio will give you a bearing on the risk that you take. For example, you will notice that beyond a certain threshold, higher risk portfolios tend to have lower returns per unit of volatility.

But this is only a starting point. Monitoring your portfolio to make sure the assumptions behind the model are not violated and re-balancing, if needed, are important actions that will help your portfolio to grow steadily.

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