Efficient Asset Allocation for Individual Investors in the ETF World

Research has shown that investment success is largely driven by asset allocation (Brinson et. al (1991)). Although many investors choose exchange traded funds (ETFs) and mutual funds over individual stocks, most must make asset allocations because they hold multiple positions in their portfolios. One rule of thumb that is commonly used is to allocate 60 percent of the investable asset in stocks and 40 percent in bonds (Chaves et al. (2011), Brinson et al. (1991); and Ziemba (2013)). As ETFs have expanded beyond simply indexing equities or bonds to include indices based on currencies and commodities, it has become simple and affordable for investors to employ non-traditional asset classes to gain further diversification. In the aftermath of the 2008 global financial crisis, investors are looking to non-traditional investment alternatives to further diversify their portfolios. Simple allocation strategies such “60/40” are no longer valid to the extent they overlook the potential for further diversification. With a universe of some 2,792 ETFs available in 2021 (Bloomberg LP, 2021), investors have a wide variety of to choose from in gaining representation for many asset classes within their portfolios. The willingness of investors to use ETFs is reflected in the fact that ETF assets grew from less than one hundred billion in 2000 to seven trillion dollars in 2021 (Board of Governors of the Federal Reserve System US).

While many studies have shown that asset allocation is of more strategic importance than active security selection, few has examined efficient asset allocation within a portfolio made up exclusively of ETFs, This paper aims to create efficient portfolios using only ETFs and to compare the performance of these portfolios with naive allocation schemes. Producing efficient portfolios from EFTs alone is feasible given the acceptance of ETFs by the investor community and the proliferation of ETFs that make it possible to find an ETF for virtually any objective, focus, or asset class.

The motivation for this research study is to make useful and simple investment recommendations for individual investors, Although ETFs have received considerable attention among researchers and practitioners, there is little literature addressing asset allocation using ETFs. Only a few studies have used ETFs exclusively to provide allocation recommendations for individual investors. Our study attempts to fill this void in the literature by producing asset allocations among 34 ETFs using Markowitz’s mean-variance optimization technique along with Ledoit and Wolf’s (2004) modification for structured and unstructured covariance matrices, This study is a meaningful contribution to a very few research that seeks optimal asset allocations using ETFs exclusively. Our objective is to provide useful information for individual investors, particularly in the form of simple allocation recommendations, Individual investors often make arbitrary allocations among asset classes that result in poor diversification and higher management costs, At the same time, wealth management companies charge substantial fees for recommending efficient allocations, Efficient portfolios made up entirely of ETFs can provide exposure to broad indices for equities, bonds, as well as to non-traditional asset classes at a relatively low cost. Our expectation is portfolios that use an optimized allocation among several ETFs will provide individual investors with more effective diversification and potentially higher returns.

Based on ex-post analysis from 2012 to 2017, the top performing optimized portfolio produced an average annualized return comparable to that of the top performing ETF Similarly, another optimized portfolio produced a higher Sharpe ratio than 85% of the ETFs in our study. Nevertheless, when the same optimization methods is implemented using ex-ante returns, the optimized portfolios do not provide superior future performance relative to the ex-post analysis and also underperforms the best performing ETFs during the same time period. Further, these results are sensitive to the length of the historical time period used to determine the optimized allocation weights or the frequency of portfolio rebalancing. Using the modified shrinkage variance-covariance matrix to solve the issue of assigning excessive weight to certain ETFs (a general occurrence with the Markowitz optimization approach) improves the portfolio performance.

Section 2 of this paper provides an overview of the literature on ETFs and portfolio optimization, In section 3, the ETFs, the data sources, and the fund selection criteria are described with summary statistics. In section 4, the empirical model used to determine the optimal weights for the portfolio based on Markowitz Optimization is described. In section 5, the empirical results of optimized portfolio performance in comparison with the individual ETFs are presented. In addition, the performance of the optimized portfolio using the modified variance covariance matrix with shrinkage technique and average optimal weight are also evaluated, Concluding remarks are provided in section 6.

Asset allocation has often been cited as a more important factor than active security selection in contributing to investment success, Brinson et. al (1991) found that 91% of the variance in pension fund returns is explained by asset allocation. This result is also supported by Ibbotson and Kaplan (2000), who report that asset allocation alone explains 87.6% of mutual fund returns and 90.7% of pension fund returns, The literature addressing efficient allocation began with the seminal paper of Markowitz (1952), who demonstrated that allocating assets among indices provides superior return ex post, Over the years, many studies applied these principles to find an ideal portfolio allocation but with mixed results. An issue with the Markowitz model stems from its use of a historical variance covariance matrix to determine the efficient allocation. As a result, the Markowitz process often leads to so called “error maximization” which is reflected in the assignment of unusually high or low allocation weights driven by outliers in the data. Consequently, the weights produced are not stable and are very sensitive to the data.

Optimization process does not perform well out of sample, Therefore, alternatives and solutions are used in the literature to overcome these issues. One common alternative is to use naïve or simplistic approaches as an alternative to portfolio optimization. Some of these approaches are 60/40 equity/ bond portfolio mentioned earlier and equal weighting. For example, Jacobs, et.al (2014) shows that simple heuristic allocations offer substantial benefits and often produce better results than classical Markowitz’s optimization. There are many more studies show that the naïve portfolio allocation techniques outperform the mean-variance optimization (See: Chaves et al. (2011) and DeMiguel et al. (2009)).

The second branch of approaches attempt to fix issues with relying on historical variance-covariance matrix that leads to unstable estimates. Bayesian methods and factor models are used to overcome these problems. Recently, robust optimization technique is also used to solve the same problems. See Kolm et al. (2014) for a more thorough review of the literature. Bayesian methods allows a researcher or investor to input his/her view into parameter estimation, With this input, the estimation becomes more robust. Many different versions of Bayesian models are used in the literature, All these models essentially combine a structured covariance matrix with a sample covariance matrix. Putting all the weight in the sample covariance matrix would lead to an unbiased estimation with potentially large estimation error, On the other hand, putting all the weight in a structured matrix would minimize the estimation error but increase the estimation bios potentially due to misspecification of the structured matrix, Single index models, for example, capital asset pricing model (CAPM) is an example of a model with a highly structured covariance, The other extreme would be regular Markowitz optimization with an unstructured covariance. Also, sample covariance model could be considered a multifactor model considering each asset in the sample as a unique factor, A significant part of the literature searches for the sweet spot between these two extremes (Marakbi 2016), Examples of such models are Treynor and Black (1973) and Black and Litterman (1992) models, All Bayesian or shrinkage models differ because of the different weights chosen between the structured and unstructured covariance matrixes (also called shrinkage constant) and the composition of the structured covariance matrix, For example, Frost and Savarino (1986) obtained a better out of sample performance using Bayesian approach to optimization, Ledoit and Wolf (2003 and 2004) offers a widely accepted shrinkage methodology, Our paper also utilizes this approach, The weight put into structured matrix is called the shrinkage constant, This approach helps to overcome the error maximization and the invertibility of variance covariance matrix, In other words, the number of assets could be more than the number of periods, Out of the sample performance with the shrinkage approach is better than the performance obtained from using only sample covariance in optimization process.

Minimum-Variance Portfolio (MVP) strategies are studied by many papers, It may not be desirable by many investors to have a low-risk portfolio (since usually the return will be lower also) initially. However, estimation problems we mentioned above is a lesser problem with a global minimum portfolio, In other words, theoretically, minimum variance portfolio may have a low return expectation, but in practice its return could be better than even a tangency portfolio, (Clarke et al. (2011), Frahm (2010), and Richardson (1989)).

There are some papers which study the portfolio allocation using exclusively mutual funds like our approach in this paper limiting the allocation to ETFs, Given the longer history of mutual funds, these results would provide a good perspective of the potential of portfolios constructed exclusively with ETFs.

Pastor and Stambaugh (2002) used equity mutual funds to create better portfolios, The study finds that investing in active mutual funds along with passive indexes could increase the Sharpe ratio of the portfolio, Louton and Saraoglu (2008) also used exclusively mutual funds to find out how many mutual funds is necessary for a well-diversified fund, Their findings suggest that 10 to 12 mutual funds are required to get reduction of most of the risk and drawn down in the portfolio, They did not offer an allocation strategy, Moreno and Rodríguez (2013) find that most of the actively managed mutual funds are not well diversified. The study used several optimization techniques to minimize the idiosyncratic risk, The optimal portfolio that minimizes the idiosyncratic risk had a good out of sample performance and this allocation provided the best alpha for the overall portfolio, Saraoglu and Detzler (2002) recommends using the analytic hierarchy process (AHP) to make asset allocation decisions using mutual funds, The study does not offer an out of sample performance result.

Using 36 Swiss ETFs, Milonas and Rompotis (2006) find that ETFs underperform their underlying indices in terms of both risk and return, Miffre (2007) compares the performance of country-specific ETFs with that of open or closed-end country funds and finds that ETFs are superior due to lower costs, lower tracking error, and being more tax efficient.

More closely related to our paper, Ma, MacLean, Xu, and Zhao (2011) employ a regime-switching risk factor in determining that sector ETFs allocations perform better than naïve allocation strategies. Furthermore, DiLellioa and Stanley (2011) state a similar conclusion after comparing several ETF strategies with the Standard and Poor’s 500 index as well as other benchmarks and finding that the ETF strategies outperform the benchmarks. Agrrawal (2013) demonstrates that multi asset class ETF portfolios dominate stock only investment options. Utilizing neural networking models, Zhao, Stasinakis, Sermpinis, & Shi (2018) are able to improve the portfolio efficiency of three ETFs compared to traditional mean-variance optimization.

Hlawitschka and Tucker (2008) used exclusively ETFs to test whether stock selection has any merit in addition to optimal asset allocation. The study finds a justification for investors to choose active stock selection over asset allocation strategies.

This study uses monthly price data obtained from Bloomberg for the period beginning August 2007 and ending December 2017. Because portfolio optimization is typically based on 60 monthly observations, it was necessary to limit the number of ETFs in this study, There are too many ETFs with similar objectives. In fact, some of the ETFs dominates others if it compared in most important categories (see Brown et.al. (2021)), The optimization process computes the ETF weights based on the sample covariance. For the solution, the inverse of the sample covariance matrix is necessary. The sample covariance matrix could be singular if the number of observations is less than the number of variables. In other words, there will not be a solution and the weights can’t be computed, Therefore, the number of ETFs used needs to be under 60 which is the number of observations used in the estimation, However, by modifying the covariance matrix, this issue can be resolved, For example, Moore-Penrose inverse can be used to solve this problem (Pappas, Kiriakopoulos, and Kaimakamis (2010)). In choosing a subsetfor our study from the 2,792 or so U.S. ETFs, we applied three screening criteria. First, the largest ETFs by market cap were selected because they are popular and meaningful from an investor’s perspective. Second, this study includes the ETFs with the longest possible history. The final criteria implemented is to include ETFs from as many asset categories as possible to improve the diversification potential of our portfolios. Where possible at least one ETF is included from each category within an asset class. For example, if two ETFs tracked the performance of the S&P 500 index, the larger of the two is included in the study and the other is eliminated. By applying the three criteria, we selected 34 ETFs for our study, Table 1 provides information on key characteristics of the 34 ETFs included in the study, Among 34 ETFs under investigation, SPDR S&P 500 ETF Trust (SPY) is the largest ETF in the US with more than 420 billion dollars in assets (in December 2021) and has been in the market for the longest period of time, for about 27 years. The ETF with the smallest market capitalization is the United States Oil Fund (USO) which tracks U,S. crude oil prices, Currently, the USO has a total asset value of about 2 billion dollars. The fund with the shortest history in our sample, Vanguard FTSE Developed Markets ETF (VEA) has been around for 10 years. Despite its short life, VEA is among the top four ETFs in our study in terms of market capitalization, VEA targets the performance of the FTSE Developed All Cap ex US Index and tracks stocks in developed market other than the U.S.

Beginning in June 2007 and ending in December 2017, a total of 126 monthly prices were obtained for each of the study’s 34 ETFs, Using the monthly price observations (P_t,i) of each ETE, the monthly returns for a given ETF i on day t are calculated, r_i,t = In(P_t,i/P_t-1,i). As shown in Table 2, the best performing ETF during the study period with an average annualized return of 11, 4 percent is QQQ (Powershares Trust Series). QQQ is a U.S. large cap growth ETF tracking the Nasdaq 100 index with total assets of approximately 58 billion dollars. Of the 34 ETFs under investigation, the worst performing was USO (United States Oil Fund) with -15.2 percent annualized return. USO, an oil commodity fund, is the smallest ETF in the study. Among the study ETFs, SHV (iShares Short Treasury Bond) has the least risk with a standard deviation (0.33 percent) and the lowest positive average annual return of (0.06 percent). SHV invests in 1-3 year U.S. Treasury securities has assets of approximately 11 billion dollars. SLV and USO were the riskiest ETFs in the study with standard deviations of approximately 34.00 percent. SLV (iShares Silver Trust) tracks the performance of silver as a precious metal investment, Note that USO has one of the highest standard deviations (33.77 percent) yet produced the lowest annualized average return (-18,5 percent) among the study ETFs.

To investigate whether optimized ETF portfolios yield superior returns as compared to individual ETFs, we employed classical Markowitz optimization (1952) which identifies the minimum variance portfolio for a given return. First, the return data from the first 60 months was used to identify the optimal ETF allocations for 10 optimized portfolios, These 10 portfolios represent the entire efficient frontier for our 34 ETFs from the minimum risk-return portfolio (Portfolio 1) to the maximum risk-return portfolio (Portfolio 10), Second, month 61 returns are calculated for each of our ten optimized portfolios and the portfolios are ranked based on average annualized return from the maximum. Third, at the end of each month beginning with month 61, the ETF composition of each optimized portfolio is re-estimated using returns from a 60-month rolling period (the oldest price observation is removed and the most recent (from the prior month) is added). The returns on the ten newly optimized portfolios (based on the re-estimated ETF allocations) are then calculated and ranked for the subsequent month. The re-estimation process for the ETF allocations of each portfolio and the calculation of returns based on those re-estimated ETF allocations are repeated for each month until December 2017, a total of 65 times. Next, a model ETF allocation is determined for each of our ten portfolios by averaging the 65 monthly weights assigned to each ETF in each of the ten portfolios. Finally, the annualized average returns of the ten model portfolios over the 65-month period are compared with the average returns of individual ETFs during the same time period. Note that the replication of the efficient asset allocation for 65 times also allows us to examine if there are any persistent weights assigned to a certain asset class. See the Appendix to understand the methodology in more detail.

Based on the classical Markowitz optimization, the efficient allocation (Markowitz 1952) can be determined as follows:

The objective function is to minimize the risk with certain constraints.

Objective = Min(v’Σv)

Subject to 1v≥0v′r=μv′1=1

where

v = the vector of weights put in each ETFs in the portfolio.

r = the vector of ETF returns

μ= portfolio target return

Σ= covariance matrix of ETF returns

Using the Matlab optimization package, 10 portfolios are constructed on the efficient frontier formed by the 34 ETFs in the study, Because these 10 portfolios represent the whole efficient frontier, investors can choose a particular portfolio along the efficient frontier that suits their personal risk tolerance. The returns of the 10 optimized portfolios are also compared with naïve portfolio allocation strategies as well as the individual ETFs, The naive portfolio can be constructed in such a way that all 34 ETFs are equally weighted.

In this article, efficient asset allocation among ETFs are investigated based on the Markowitz’s portfolio optimization approach. Using the data from 2007 to 2017, we empirically examine whether investors can achieve a better portfolio performance with various risk and return levels by simply allocating investment among different ETFs, The optimized portfolios are compared with best performing individual ETFs during the same time period, To determine optimal weight, for each ETF composition in each of 10 efficient portfolios, we utilize 1-month rolling returns of past 60 months, past 40 months, and 60 months cumulative returns.

Markowitz’s portfolio optimization approach is known for putting too much weight on a certain asset class, We remedy this issue by implementing the modified variance-covariance matrix in optimization process, The true test of a portfolio allocation strategy is the out of sample ex-ante portfolio selection, We compared the performance of out of sample mean-variance efficient portfolios with the actual ETFs performance.

Overall findings suggest that optimization procedure using historical 60-months data provides an efficient portfolio performance that is comparable with individual top performing ETFs, On the ex-post basis, the top performing portfolio yields the highest average monthly returns of 15,6% with highest Sharpe ratio of 1.01, similar to investing 100% in QQQ ETF that emphasizes on the large-cap high growth stocks in the U.S. However, this efficient allocation using the Markowitz’s portfolio optimization approach, does not predict a superior future performance. In 2012-2017 period, this approach underperformed many ETFs, The Sharpe ratio of the top performing optimized portfolio of 0.5720 is less than half of the Sharpe ratio of top performing QQQ, The out of sample forecast of optimized portfolio is a little better when only more recent historical data is used in determining optimal weight because of quicker feedback. The performance is worse when portfolio is rebalanced annually instead of monthly. When a modified shrinkage covariance matrix was employed to reduce the problem of excessive weights assigned to some ETFs, the performance of optimized portfolio improved slightly by 0.2% in terms of returns and 0.004 in Sharpe ratio. Lastly, constructing efficient portfolios based on the average of optimized weights improves the returns of top performing portfolio by 370 basis points and increases Sharpe ratio significantly (+0.5). Efficient portfolios using the average of optimized weight performs better than 85% of the individual ETFs in terms of Sharpe ratio, In conclusion, the average weight approach to portfolio optimization represents the best overall strategy. This is the only out of sample portfolio optimization approach that was comparable to the best ETFs in our list in terms of mean-variance efficiency, Investors should consider using this technique as an investment strategy. While this study is limited to the extent to which Markowitz portfolio theory holds for ETFs, it has clear practical applications for financial professionals as a possible alternative to other portfolio allocation approaches. Additionally, the US-based research provides a useful platform for further research in the Australian market.