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A lot of people believe that diversification helps reduce non-market risk for a security and they build portfolios of assets in order to avoid "putting all eggs in one basket". Intermediate-level and sophisticated investors usually tend to do their own security analyses and select assets in accordance with their strategies (e.g. value or growth stocks). I have also noticed that a lot of writers on various web resources who specialize in building portfolios for specific purposes tend to wisely use fundamental analysis for security selection but do not pay enough attention to portfolio composition with respect to risk/reward considerations.

I am not a mathematician myself but I have had significant exposure to intermediate statistics at university and I know Excel quite well. Hence, I decided to use my skills to optimize a portfolio of securities with respect to certain constrains and investment goals. First of all, I limited the number of securities to 17. Quite many mutual funds and ETFs do hold more than this number (usually about 25-30 or even more depending on the size and scope of the fund) but academic research has shown that the benefits of adding another security to a portfolio that already holds about 15 assets are negligible (marginal utility reduces dramatically after a certain threshold has been reached). The following illustration summarizes this theory:

Secondly, I decided to use ETFs and closed-end funds traded on the exchange as the components of my portfolio. This enables me to reach a "second-degree" diversification, as these products are well-diversified themselves. The first criterion for the selected securities was that they must have been in existence for at least 10 years. This allowed me to assess their behavior over the full business cycle as well as see how they did during the financial crisis of 2008-2009. From a technical point of view, a larger sample (in this case, years) provides a more reliable output. Secondly, when such a list of ETFs or funds was available, I chose the ones that had the highest current yield and a sizeable capitalization (hundreds of millions of dollars at least). Stock Rover served as my screener for the most part.

The results of the structuring were remarkable: the portfolio yielded a small alpha over the broader S&P 500 at comparable volatility, while providing a decent dividend yield of roughly 4%. The weighted-average beta was below the market:

To remind, beta is the volatility relative to the overall market, while a portfolio's volatility is the standard deviation of returns with respect to itself. For optimization purposes, I used expected portfolio returns equal to weighted-average returns on assets within the portfolio: it makes sense to value recent returns higher than the returns at the inception of a fund.

Holdings

The portfolio holds funds invested in international equity markets, U.S. equity markets, high-yield debt, emerging markets debt, corporate debt (mainly U.S. corporations), precious metals, and US sovereign debt. More than half of the funds are invested in international markets (equity and debt). High-yield debt, while bearing the most volatility, provides the lion's share of the portfolio's dividends. Precious metals serve as a hedge during market downturns on par with U.S. Treasuries. Despite expected volatility, the portfolio in fact has had an insignificantly higher volatility of returns than S&P 500, while delivering alpha in 8 years out of the last eleven. This can be partially explained by the fact that assets held are uncorrelated among each other (see Covariance Matrix in the model). Now, let us have an overview of the holdings.

iShares MSCI Hong Kong Index Fund (NYSEMKT:ETF)

The exchange-traded fund invests in the Hong Kong market and yields a 3.0% dividend yield. It has a beta of 0.98 and a market capitalization of about $2.3B. The fund's adviser is Blackrock Fund Advisors.

iShares MSCI Emerging Markets Index (NYSEARCA:EEM)

The fund invests in global emerging markets and has a modest 2.13% dividend yield. The fund's beta is 1.29. This holding has the biggest market capitalization among other holdings in the portfolio: $38.2B. The ETF did very well during the recovery period, but has been slow during the last two years. The fund's adviser is Blackrock Fund Advisors.

iShares MSCI South Africa Index (NYSEARCA:EZA)

The ETF's investment geography is South Africa. The dividend yield provided by the fund is 2.58%, while the beta is 1.27. The market capitalization stands at a little over $500M. Morgan Stanley Capital International reviews the South African Index in which the ETF holds a representative sample. The fund's adviser is Blackrock Fund Advisors.

iShares MSCI Belgium Investable Market (NYSEARCA:EWK)

The fund invests in publicly-traded securities in the Belgian market (covering 99% of the investable securities) seeking price and yield performance. The current dividend yield is 4.62%. The fund's beta is 1.13 and the market capitalization is a little over $67M. The fund's adviser is Blackrock Fund Advisors.

iShares MSCI EAFE Index Fund (NYSEARCA:EFA)

The exchange-traded fund invests in the MSCI EAFE Index developed by MSCI Inc. The Index holds stocks from Europe, Australasia, and Far East. The fund invests in a representative sample of stocks held in the Index. The market cap of the fund is above $53B. The fund yields 2.54% as of yesterday and has a beta of 1.2. The fund's adviser is Blackrock Fund Advisors.

iShares MSCI Germany Index Fund (NYSEARCA:EWG)

The fund seeks to provide investment returns based on the MSCI Germany Index, which measures the performance of the equity market in Germany. The fund has one of the lowest yields in our portfolio: 1.37%. The market cap is 6.34B, while the fund's beta is 1.58 (one of the highest in group). Clearly, this fund in weighted towards capital appreciation. The fund's adviser is Blackrock Fund Advisors.

iShares MSCI Mexico Inv. Mt. Index. (NYSEARCA:EWW)

The fund invests in the same stocks as the MSCI Mexico IMI 25/50 Index. The market capitalization is almost $2.6B and the fund's beta is 1.34 (one of the highest in group). The dividend yield is a modest 2.01%. The fund's adviser is Blackrock Fund Advisors.

iShares Barclays 20+ Year Treasury Bond (NYSEARCA:TLT)

The fund tracks the investment returns of an index composed of US Treasurys with maturities exceeding 20+ years. The ETF has a market cap of $2.3B and a dividend yield of 3.18%. It has a negative beta of -0.35. This means that for a 1% increase in the broad equity market, the ETF is expected to decrease in price by 0.35%. The fund has a negative covariance with the entire portfolio and was one of the few ones that delivered positive returns in 2008 and 2011. BlackRock Fund Advisors serves as investment adviser of the Fund.

iShares Russell 2000 Value Index (NYSEARCA:IWN)

The ETF tracks investment returns of the broader Russel 2000 Value Index by investing in a representative sample of securities within the Index that have lower multiples and lower expected growth. The fund has a P/E of 8.3X (relatively low compared to S&P 500), a market cap of $5.7B, and a beta of 1.28. It also offers a small dividend yield of 1.78%. Together with QQQ Trust, the fund delivered large positive returns in 2013. BlackRock Fund Advisors serves as investment adviser of the Fund.

BLDRS Developed Markets 100 ADR (NASDAQ:ADRD)

The ETF tracks holdings of BNY Mellon Developed Markets 100 ADR Index. At least 95% of the Index's holdings are represented in the fund. The ETF offers a yield of 2.87% and has a beta of 1.18. It has a market capitalization of just over $61M. Invesco PowerShares Capital Management LLC is the sponsor of the fund and The Bank of New York Mellon is the trustee.

ETFS Metal Securities Australia Ltd (OTC:EAUSF)

ETFS Metal Securities Australia Limited is an Australia-based company engaged in listing and issue of commodity securities (CSAL Commodity Securities). The Company's portfolio of CSAL Commodity Securities contains Classic Commodity Securities. The Company may issue further classes of CSAL Commodity Securities which will be similar to those already offered. The market capitalization is $33.9M and the beta is 0.85. The company pays no dividends.

Credit Suisse High Yield Bond Fund (NYSEMKT:DHY)

The fund is a closed-end investment company that allocates its capital across a diverse population of high-yield securities below investment-grade quality from U.S. issuers in many industries. The fund pays distributions monthly and currently offers a yield of 9.3%. The market cap is $302M. At least 80% of assets consist of fixed-income securities, while up to 20% can be comprised of other types of investments. The fund's investment adviser is Credit Suisse Asset Management, LLC.

PowerShares QQQ Trust, Series 1 (NASDAQ:QQQ)

The trust holds all of the component securities of the Nasdaq-100 Index. It offers a dividend yield of 1.01% and has a beta of 0.97. The trust's market capitalization is almost $46B. The trust posted the highest return in the portfolio of 30% this year. Invesco PowerShares Capital Management, LLC is the sponsor of the trust and The Bank of New York Mellon is the trustee.

BlackRock Corporate High Yield Fund VI (NYSE:HYT)

BlackRock Corporate High Yield Fund VI, Inc. is a diversified closed-end management investment company. The fund's primary investment goal is to provide high current income by investing in low-grade or unrated fixed-income securities. Capital appreciation is a secondary objective. The fund has a market capitalization of about $430M and has a beta of 0.61. The current yield is almost 8%. BlackRock Advisors, LLC is the trust's investment adviser. Its sub-adviser is BlackRock Financial Management, Inc.

BlackRock Debt Strategy Fund, Inc. (NYSE:DSU)

BlackRock Debt Strategy Fund, Inc. is a diversified, closed-end management investment company. The fund seeks to provide current income by investing mainly in a diversified portfolio of the United States companies' debt instruments, including corporate loans that are rated in the lower rating categories or unrated debt instruments of comparable quality. It has a market capitalization of $441M and a beta of 0.7. Currently, it yields approximately 7.4% a year in distributions. The fund's investment adviser is BlackRock Advisors, LLC, an indirect, wholly-owned subsidiary of BlackRock, Inc. The Company's sub-adviser is BlackRock Financial Management, Inc.

Morgan Stanley Emerging Markets Debt Fund (NYSE:MSD)

The fund is a non-diversified, closed-end management investment company. Its primary objective is to provide investors with high current income and as a secondary goal, to seek capital appreciation, through investments primarily in debt securities of government and government-related issuers located in emerging markets. The market capitalization is $227M. The current yield is exactly 7.5% and the beta is 0.54, one of the lowest in the group. The fund's investment adviser and administrator is Morgan Stanley Investment Management Inc.

ING Groep, N.V. Perpetual Debt Securities

The debt is a subordinated perpetual debenture issued by ING Groep N.V., a Dutch bank, back in 2002. The overall size of the issue was $1B with a 7.20% coupon. The securities may be redeemed at any time during the next interest payment (this was the option kept by the bank after 2007). The price history is remarkably stable: the security traded near $25 per unit during 10 years out of 13, with the exception of 2008, 2009, and 2011. Currently, the market capitalization is at $555M and the beta is 1.64. Because the security trades above par, the annual dividend yield is slightly lower at 7%. The security's trustee is The Bank of New York Mellon.

Overall, the total asset allocation within the portfolio (at equal weights given to every asset class - vanilla option) can be illustrated in the following diagram:

(click to enlarge)

Analysis

After constructing a covariance matrix and using the wonderful Solver function, I created the following output consisting of various scenarios suitable to different investors:

(click to enlarge)

Scenario 1: Simplest Composition

The easiest way to construct a portfolio is to give equal weights for all holdings. Although investors may think that this diversification method decreases risk, data shows that it really is not the most effective way to decrease portfolio variance. On the other hand, when comparing to Risk-Minimized scenarios this portfolio shows the highest expected return, especially given that the current dividend yield for this portfolio is 4%.

Scenario 2: Long/Short

This portfolio is deemed to have the smallest variance on the list (in fact, zero. I am not sure if that holds on day-to-day basis but in the long term, the portfolio is expected to have on average zero volatility with respect to its own returns. Again, past results may not repeat in the future). Investors should also be cautious with the suggested exposure to gold and long-term Treasurys. However, the option of shorting may not be feasible to some investors.

Scenario 3: Optimal Return/Risk

This 8-piece portfolio (0% to 25% Weights) gives the highest return among the Risk-Minimized group, while operating at a reasonable risk level (still below the expected return). Also notice that, compared to the Long-Only Portfolio, it provides less exposure to gold and Treasurys and more towards US equities.

Scenario 4: Maximized Return/Modest Risk

The 5% to 15% Portfolio solution provides the highest return among the portfolios already mentioned, while maintaining a dividend yield very close to the Equal-Weights Portfolio (just below 4%). Also, for extra 42 basis points of risk, this portfolio has obtained more than 1% extra expected return when compared to the Simplest Composition solution. Here, the biggest weights are allocated towards the highest gainers: Nasdaq-tracking trust and a Mexico fund. Overall, the portfolio holds all securities from the list.

Scenario 5: Maximum Return

This portfolio provides a significantly higher expected return, while the risk, although being over 50%, is proportionally modest. The combination of large long-short bets is attracting but may not be feasible for retail investors (brokers may not allow to use the money received from short-selling to get into long positions due to margin restrictions). Either way, risk-embracing investors should consider this strategy. I will explore more viable options for this type of investor crowd in my future articles.

One can see how basic knowledge of statistics and Excel can cater to one's investment needs. Just because you chose the right assets does not mean you have the right portfolio for yourself.

Source: Using Quantitative Methods To Optimize Your Portfolio