Is Markowitz Optimization Passé?
Introduction
60 years ago financial economist Harry Markowitz mathematically formulated the MeanVariance Efficient Frontier and a quantitative method that could help investors maximize riskadjusted return. But is Markowitz Optimization actually practical for investors?
In this blog I will introduce Markowitz’s model of the Efficient Frontier, identify several problems with Markowitz Optimization, and then empirically test the model’s ability to generate active return. I will demonstrate the Efficient Frontier using MATLAB and then backtest a strategy where an investor allocates a portfolio using the weights suggested by the minimum variance portfolio over the past N days, holds the portfolio for x days, then rebalances. I will repeat the experiment with different values of N with an equity portfolio and a bond portfolio, and compare the actively managed portfolios to their relevant benchmarks.
The Efficient Frontier
The Efficient Frontier is the set of all risky portfolios for a given universe of assets where an investor cannot increase return or decrease risk (standard deviation) by changing his portfolio weights. A portfolio is “efficient” if it has the best possible expected return for a given level of risk, or likewise the lowest possible risk for a given level of return. Markowitz specified that return is the average return for each asset, and risk is the standard deviation of each asset over a given period. Let’s say we are optimizing 3 risky assets, with μ (average daily return) = [0.0092 0.0955 0.2311], σ (daily standard deviation) = [0.7603 0.9575 1.3833], and the correlation matrix = [1.0000 0.6289 0.5297; 0.6289 1.0000 0.3826; 0.5297 0.3826 1.0000]. If we scatter plot 1000 fully invested (Σw_{i} = 1), randomly weighted portfolios with only long positions possible, it will look like this:
As you can see, the scatter plot seems to be bounded by a (approximate) horizontal parabola, sometimes referred to as the “Markowitz Bullet.” Graphically, the Efficient Frontier is the upper bound of the bullet, seen here in blue:
The Efficient Frontier can be found by hand using Lagrange Multipliers, but it is very tedious and not at all practical. It is more easily found by using software packages (in my case, MATLAB).
Problems with the Efficient Frontier
The Efficient Frontier is backwardlooking. That is, the expected return vector, expected risk vector, and correlation matrix are calculated from past return history. If a rational investor is only concerned with how her portfolio will perform rather than how it could have performed, then Markowitz Optimization is not useful. We can somewhat resolve this issue by using past risks and returns as our forecasts for future risks and returns, but as Black and Litterman (1992) pointed out, past returns are not at all a good forecasts for future returns.
Additionally, the weights suggested for the minimum variance portfolio (or a portfolio with a target risk or return, or maximizes the Sharpe Ratio) will change drastically with only slight changes in the expected return or risk. Black and Litterman give the example of a portfolio consisting of international indices:
“In a realtime setting, these slight changes to return and risk will likely lead an optimized portfolio to suggest significant changes to portfolio weights. These significant changes would be impractical for an investor constrained by commission and market liquidity.”
Is Markowitz Optimization still useful?
To answer this question I will backtest a portfolio management strategy where a hypothetical investor finds the minimumvariance portfolio for a set of ETF’s over the last N days, allocates cash to the portfolio weights, holds the portfolio for x days, then repeats the process. The set of ETF’s will remain the same over the entire backtest, and I will compare the performance of the actively managed portfolio against a benchmark. I will repeat the backtest with a bond portfolio and a stock portfolio and with different values of N.
The Setup
I will look at how the strategy performs in 2 asset classes: stocks and bonds. For each asset class I will use iShares ETF’s that reflect different sectors that could be held or different types and maturities of bonds.
For equities, my benchmark will be an iShares ETF tracking the S&P 500, IVV, and my portfolio will consist of 10 iShares ETFs that represent the 10 Dow Jones sectors:
IYE – Dow Jones U.S. Energy Sector Index Fund
IYF – Dow Jones U.S. Financials Sector Index Fund
IYH – Dow Jones U.S. Healthcare Sector Index Fund
IYJ – Dow Jones U.S. Industrial Goods Sector Index Fund
IYM – Dow Jones U.S. Basic Materials Goods Sector Index Fund
IYW – Dow Jones U.S. Technology Sector Index Fund
IYZ – Dow Jones U.S. Telecommunications Sector Index Fund
IDU – Dow Jones U.S. Utilities Sector Index Fund
IYC – Dow Jones U.S. Consumer Services Sector Index Fund
IYK – Dow Jones U.S. Consumer Goods Sector Index Fund
For bonds, my benchmark is an ETF that tracks the Barclays Aggregate Bond Index, AGG, and several iShares ETFs that track the components of the index:
LQD – iShares IBoxx Investment Grade Corporate Bond Fund
SHY – iShares Barclays 13 Year Treasury Bond Fund
TLT – iShares Barclays 20+ Year Treasury Bond Fund
MUB – S&P National AMTFree Municipal Bond Fund
MBB – Barclays MBS Bond Fund
TIP – Barclays TIP Bond Fund
For the backtest, I will find the minimum variance portfolio over the last N trading days, allocate my portfolio to the suggested weights, hold this portfolio for 5 trading days, and then repeat the process. My portfolio will be fully invested, with each weight constrained between 1.5 and 1.5:
Min σ
s.t. Σw_{i} = 1 & 1.5 < w_{i} < 1.5
I will evaluate performance by looking at the portfolio’s and benchmark’s average daily return and standard deviation, compute the portfolio’s Alpha and Beta(CAPM r_{p} » α + βr_{m}), and compare an approximation of the portfolio’s Sharpe Ratio (r_{p}/σ_{p}) to the benchmark’s approximate Sharpe Ratio. I will see how performance changes with different values of N by repeating the backtest with N=250, 200, 150, 100, 50, 25 for both equities and bonds, while holding x steady at 5 trading days.
The Data
The data comes exclusively from Yahoo Finance, via the Datafeed Toolbox in MATLAB. Data comes in the form of a time series, and spans from the inception date of the ETFs to Friday June 8^{th}, 2012. For equities, the data spans from January 1^{st}, 2001, to June 8^{th}, 2012. For bonds, the data spans from January 1^{st}, 2008, to June 8^{th}, 2012.
The Results
The optimized equity and bond portfolios generally outperformed their relative benchmarks. The two proceeding tables show the performance evaluation metrics mentioned previously for each portfolio:
Equity Table 

r_{p} 
r_{m} 
σ_{p} 
σ_{m} 
α 
β 
Sharpe_{p} 
Sharpe_{m} 

250 
0.0265 
0.0113 
1.3948 
1.3441 
0.0226 
0.34282 
0.01901 
0.00843 
200 
0.0281 
0.0139 
1.4198 
1.3403 
0.0236 
0.320293 
0.01978 
0.01037 
150 
0.0326 
0.0106 
1.3884 
1.3432 
0.029 
0.337017 
0.02345 
0.00788 
100 
0.0413 
0.008 
1.4182 
1.3385 
0.0387 
0.333005 
0.02915 
0.00595 
50 
0.0424 
0.0113 
1.3725 
1.3436 
0.0387 
0.32282 
0.03088 
0.00842 
25 
0.0657 
0.0062 
1.5024 
1.3461 
0.0638 
0.303283 
0.04371 
0.0046 
Bond Table 

r_{p} 
r_{m} 
σ_{p} 
σ_{m} 
α 
β 
Sharpe_{p} 
Sharpe_{m} 

250 
0.0093 
0.0211 
0.1184 
0.2767 
0.0089 
0.015512 
0.0783 
0.07633 
200 
0.0096 
0.0323 
0.1298 
0.3356 
0.0089 
0.021502 
0.07382 
0.09633 
150 
0.0096 
0.0253 
0.1308 
0.4479 
0.0098 
0.00773 
0.07377 
0.05657 
100 
0.0111 
0.0219 
0.1454 
0.4407 
0.0111 
0.00114 
0.07645 
0.04967 
50 
0.0103 
0.0218 
0.1416 
0.4399 
0.0103 
0.00229 
0.07241 
0.04963 
25 
0.0111 
0.0219 
0.1454 
0.4407 
0.0111 
0.00114 
0.07645 
0.04967 
For each N, the portfolios have positive excess return (α), meaning the portfolios outperform the market with respect to the projected return implied by β. The optimized portfolios also outperformed their benchmarks when looking at the Sharpe Ratios. r_{m}, σ_{m}, and Sharpe_{m} are different for each row because the number of backtested trading days recorded rises with smaller values of N, which, in the case of equities, includes a market crash that negatively affects the average daily return.
Graphically, the equity and bond portfolios for N=100 look like this:
From the above graphs that the cumulative return for the optimized equity portfolio significantly outperforms the market, yet the bond portfolio does not beat the benchmark in cumulative return. However, σ_{p} of the equity portfolio was higher than its benchmark for all tested values of N, and σ_{p }of the bond portfolio was significantly lower than its benchmark. For the riskaverse investor, the bond portfolio might be a better choice, as the investor gets a higher amount of return for each unit of risk, as demonstrated by the Sharpe Ratio.
Interestingly, as N decreases, α rises and β falls. β falls as N decreases because the suggested weights changes more significantly from holding period to holding period, making it less correlated with the market. We can see this graphically in the equities portfolio by plotting the weight for each ETF over time for N=250 and N=25:
Clearly the weights for the N=25 portfolio change significantly with each holding period, yet the weights for the N=250 portfolio also change more drastically than a large fund is capable of following. The N=25 portfolio weights also very frequently hit the position bounds 1.5 < w_{i} < 1.5, while the N=250 portfolio never hits the position bounds.
Conclusion
So is Markowitz Optimization still useful? The results of my backtest show that under ideal conditions, allocating to a past optimal portfolio seems to be a viable strategy. However, the ideal conditions are not at all realistic. In the backtest, I did not account for commission, the borrowing rate for margin and shorting, and imperfect liquidity. Clearly a more robust backtest would account for all of this. The short answer to the question is: no, probably not. It is not at all feasible for managers of large portfolios to make the drastic changes to allocation suggested by any value of N that I’ve tested here.
The results confirm Black and Litterman’s assertion that standard optimization methods result in portfolio weights that are extremely sensitive to small changes in the return vector and covariance matrix. The above graphs of weights show that small changes in the return vector and covariance matrix resulted in very significant swings in how the N=25 portfolio is allocated. The suggested portfolio weights have very profitable outcomes, but are not feasible for managers of large portfolios. Better alternatives might be the BlackLitterman asset allocation model or using econometric methods like ARMAGARCH to forecast return and covariance.
References
Black, F. and Litterman, R. (1992). Global Portfolio Optimization. Financial Analysts Journal, Vol. 48, No. 5. pp 2843.
Markowitz, H. (1952), Portfolio Selection, Journal of Finance, 7, pp. 7791.
Bodie, Z. and Kane, A. and Marcus, A. (2005). Investments, Fifth Edition. McGrawHill/Irwin.