In Search Of The Ultimate Efficient Portfolio

by: Maurice Chia

You know what an optimized portfolio is. It calculates the best (optimal) return for a given volatility or equivalently the least volatility for a given return subject to certain assumptions.

Without some computing power, the calculations can be onerous. When noble-prize winner Markowitz first created the efficient frontier, it was thought that the same calculations that today take a fraction of a second to calculate would have taken him about 3 years to do by hand.

So if an optimized portfolio does what it says it does and with computing power now at our disposal, why don't we try to push the whole stock universe into an optimizer to see what comes out.

Although computers are now more powerful than ever, the calculations required are still daunting if one were to consider a universe as large as, say, all stocks on the NYSE.

With a universe of only 10 stocks, the combinations to consider would be 1013. But this grows exponentially. With just 25 stocks presenting 33,554,406 combinations, imagine what 3796 stocks would entail.

Furthermore, even if you calculated all the correlations between any 2 stocks in a big universe, you would not be able to derive the frontier without looking at all the individual returns and volatilities at the same time.

To get an optimized portfolio, you have to look at return, volatility, and correlation of all the stocks in the universe at the same time.

Pushing the Envelope

We took 117 stocks (see the Appendix at the bottom) and pushed them through our optimizer which ran on an old Core 2 1.4 GHz Solo processor.

The number of combinations (of 2 stocks, 3 stocks, 4 stocks etc.) to be analyzed: 166,153,499,473,114,350,000,000,000,000,000,000

The resultant efficient frontier is shown below where the opportunity cost is the gap in terms of average annualized return between a portfolio with equal amounts in each of the 117 stocks and one with the same volatility on the efficient frontier.

The optimal computations sized down the 117 stocks to the following 14 stocks (all of which are assumed to be going-concerns):

RF Micro Devices 1.6%, Yahoo! 0.8%, Citrix Systems 0.1%, Duke Realty Corp 2.5%, EMC Corp 8.4%, Intuitive Surgical 8.4%, Alliance Resource Partners 29.1%, Altera Corp 0.6%, Amgen 7.2%, Coach 7.7%, Microsoft Corp 3.3%, Quest Diagnostics 7.0%, Southern Copper Corp 4.4%, Southern Co 18.5%.

Going back in time to June 15,2005 and performing the same analysis yielded the following performance of the optimal portfolio (blue bars) over 6 years. It is interesting to note that the portfolio yielded a positive return during the 2008/2009 downturn whereas a portfolio that consisted of equal amounts in each stock (the red bars) did not perform as well.


It is possible to passively invest if you have the right mix of stocks. But passive investing does not mean blind investing. You will still need to check if the stocks that constitute your portfolio continue to abide by the assumptions of the model.

Do the consensus estimates support the stocks that have been included into the optimal portfolio? They do.

What if we included more stocks into the universe for analysis? Can we find an even better portfolio? That will be the subject of my next article.


The following 117 stocks formed the asset universe and were chosen at random:

AK Steel Holding Corp (AKS), Abbott Laboratories (ABT), Advanced Micro Devices (AMD), Air Products and Chemicals (APD), Albemarie Corp (ALB), Allegheny Technologies (ATI), Alliance Resource Partners (ARLP), Altera Corp (ALTR), Altria Group (MO), American Electric Power Co (AEP), Amgen (AMGN), Apple (AAPL), Ashland (ASH), Automatic Data Processing (ADP), Baker Hughes (BHI), iShares Barclays 1-3 year Treasury Bond (SHY), Brinker International (EAT), CA (CA), CSX Corp (CSX), Capital One Financial Corp (COF), Celestica (CLS), Chevron Corp (CVX), Citrix Systems (CTXS), Coach (COH), Comerica (CMA), Consolidated Edison (ED), Cooper Industries (CBE), Corning (GLW), Crane Company (CR), Crown Castle International Corp (CCI), D.R. Horton (DHI), Dominion Resources (D), Dover Corp (DOV), Duke Realty Corp (DRE), E.I.du Pont de Nemours and Co (DD), EMC Corp (EMC), E*TRADE Financial Corp (ETFC), Eastman Chemical Co (EMN), Exelon Corp (EXC), Fifth Third Bancorp (FITB), First Horizon National Corp (FHN), Flextronics International Ltd (FLEX), Ford Motor Co (F), General Dynamics Corp (GD), General Electric Co (GE), Harley-Davidson (HOG), Honeywell International (HON), Hub Group (HUBG), Hudson City Bancorp (HCBK), Intel Corporation (INTC), International Business Machines Corp (IBM), International Game Technology (IGT), Intuitive Surgical (ISRG), Jacobs Engineering Group (JEC), Kansas City Southern (KSU), KeyCorp (KEY), Kimberly-Clark Corp (KMB), LM Ericsson Telephone Co (ERIC), LSI Corp (LSI), Lam Research Corp (LRCX), Legg Mason (LM), MeadWestvaco Corp (MWV), MicroSoft Corp (MSFT), New York Community Bancorp (NYB), NextEra Energy (NEE), Norfolk Southern Corp (NSC), Novartis AG (NVS), Occidental Petroleum Corp (OXY), Owens Illinois (OI), Parametric Technology Corp (PMTC), Parker Hannifin Corp (PH), Peabody Energy Corp (BTU), Polaris Industries (PII), Polycom (PLCM), Popular (BPOP), Praxair (PX), Quest Diagnostics (DGX), RF Micro Devices (RFMD), Rpc (RES), Regions Financial Corp (RF), Robert Half International (RHI), Rockwell Automation (ROK), SAP AG (SAP), STMicroelectronics NV (STM), Schlumberger Ltd (SLB), Southern Copper Corp (SCCO), St. Jude Medical (STJ), Stanley Black & Decker (SWK), Starbucks Corp (SBUX), Steel Dynamics (STLD), Stryker Corp (SYK), Suntrust Banks (STI), Sunoco Logistics Partners L.P. (SXL), Susquehanna Bancshares (SUSQ), Symantec Corp (SYMC), Synovus Financial Corp (SNV), T Rowe Price Group (TROW), TCF Financial Corp (TCB), Teradyne (TER), Texas Instruments (TXN), Textron (TXT), The Boeing Co (BA), Chubb Corp (CB), Southern Co (SO), Timken Co (TKR), The Travelers Companies (TRV), Total System Services (TSS), United Rentals (URI), United Technologies Corp (UTX), Verizon Communications (VZ), WW Grainger (GWW), Western Digital Corp (WDC), Woodward (WWD), Xerox Corp (XRX), Yahoo! (YHOO), Zions Bancorporation (ZION), iGATE Corp (IGTE)

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