5 Results 35
5 Results 36
Broad Indices
Figure 5.3: 𝑝-values corresponding to the pairwise Wilcoxon singed-rank test
The results of the simulation of the broad indices are visualised in figure 5.3. The results again are - unsurprisingly - similar to the results obtained in subanalysis 1. The minimum-semivariance portfolio outperformed consistently in terms of returns, but the risk endured was significant to prevent the risk-adjusted returns (in terms of Sharpe ratios and Sortino ratios) to be significant. On the broad indices, the mean-variance portfolio achieved superior results thanks to lower out-of-sample risk.
Conclusion 6
M
ean-variance optimisation has been the most popular optimisation problem within the field of finance for nearly seven decades. That is not without a reason. Mean-variance optimisation is computationally efficient and applicable by a wide range of people without necessary knowledge of complicated mathematics.Unfortunately, it is to a certain extent the popularity of mean-variance optimisation that has caused Markowitz’
theory to receive criticism.
A large portion of that criticism is perhaps better called cynicism. However, there are considerable problematic aspects about mean-variance optimisation. The theory is perfect within her own assumptions but those assumptions barely hold in the real world.
The model’s inputs are hard to estimate with the accuracy needed for stable results. For instance, there is Markowitz’ curse, the fact that when diversification is needed the most, the allocated weights are least stable, is problematic. There have been made several suggestions how to circumvent the problems surrounding mean-variance optimisation. A recent one is hierarchical risk parity, an allocation method that uses machine learning algorithms to cluster assets before minimising volatility. As this thesis has shown, the weights allocated by hierarchical risk parity are more robust to slight changes in input, compared to mean-variance optimisation or minimum variance optimisation.
Furthermore, the definition of risk applied in mean-variance optimisation is not what an investor would perceiveas risk. Investors wish to mitigate losses, and downside risk measures such as semivariance suit that definition better than variance. Markowitz understood this as well - but computational constraints did not permit the development of semi-variance based optimisation until nearly a half century after Markowitz developed mean-variance optimisation.
Furthermore, mutliple researchers have provided significant proof that stock markets do not follow a Gaussian distribution, let alone that they follow a symmetric distribution. Asset returns are typically skewed and leptokurtic. Therefore, modelling them as if they are symmetric can lead to detrimental outcomes.
Whilst the mean-semivariance optimal portfolio in this thesis is only approximated, the minimum-semivariance portfolio can be found. This thesis has tested the performance of these portfolios can compared them to the performance of their ‘variance-based’ counterparts. The downside-risk based portfolios were consistently able to achieve higher returns than their counterparts. The results were the most obvious for the minimum-semivariance portfolio, which even was able to achieve lower volatility than the minimum-variance
6 Conclusion 38
portfolio, occasionally. Semivariance-based optimisation typically performed well in equity markets and less well in the broad indices, where there were more potential diversification benefits.
The way this thesis performed the analysis was done so to mitigate possible ‘cherry-picking’ of results. By selecting multiple samples to generate the portfolios upon, multiple indices upon which the analysis was performed, and further performing simulations, the probability of a false-positive was mitigated.
This thesis opted to answer the following research questions.
1. Do semivariance-based portfolios achieve higher cumulative and/or mean returns?
2. Do semivariance-based portfolios experience less severe maximum drawdowns?
3. Do semivariance-based portfolios achieve higher diversification ratios?
4. Do semivariance-based portfolios achieve higher Sharpe Ratios?
5. Do semivariance-based portfolios achieve lower Sortino Ratios?
Not all of these questions can be answered in full certitude, however, based on the findings, the conclusion is as follows.
1. Semivariance-based portfolios typically achieve higher returns than variance-based counterparts.
2. Semivariance-based portfolios typically achieve less extreme drawdowns than variance-based counter-parts.
3. Semivariance-based portfolios typically achieve similar levels of diversification than their variance-based counterparts.
4. Semivariance-based portfolios typically achieve similar or even higher Sharpe ratios than their counterparts, especially in equity markets.
5. Semivariance-based portfolios typically achieve higher Sortino Ratios than their counterparts.
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Appendix
1 Data
S&P 500: overview
Table 1:S&P 500 data description.
Symbol Country Industry Name Sector Source
MMM United States Specialty Industrial Machinery 3M Company Industrials Yahoo! Finance
ABT United States Medical Devices Abbott Laboratories Healthcare Yahoo! Finance
ABBV United States Drug Manufacturers—General AbbVie Inc. Healthcare Yahoo! Finance
ABMD United States Medical Devices Abiomed, Inc. Healthcare Yahoo! Finance
ACN Ireland Information Technology Services Accenture plc Technology Yahoo! Finance
ATVI United States Electronic Gaming & Multimedia Activision Blizzard, Inc. Communication Services Yahoo! Finance
ADBE United States Software—Infrastructure Adobe Inc. Technology Yahoo! Finance
AMD United States Semiconductors Advanced Micro Devices, Inc. Technology Yahoo! Finance
AAP United States Specialty Retail Advance Auto Parts, Inc. Consumer Cyclical Yahoo! Finance
AES United States Utilities—Diversified The AES Corporation Utilities Yahoo! Finance
AFL United States Insurance—Life Aflac Incorporated Financial Services Yahoo! Finance
A United States Diagnostics & Research Agilent Technologies, Inc. Healthcare Yahoo! Finance
APD United States Chemicals Air Products and Chemicals, Inc. Basic Materials Yahoo! Finance
AKAM United States Software—Infrastructure Akamai Technologies, Inc. Technology Yahoo! Finance
ALK United States Airlines Alaska Air Group, Inc. Industrials Yahoo! Finance
ALB United States Specialty Chemicals Albemarle Corporation Basic Materials Yahoo! Finance
ARE United States REIT—Office Alexandria Real Estate Equities, Inc. Real Estate Yahoo! Finance
ALXN United States Biotechnology Alexion Pharmaceuticals, Inc. Healthcare Yahoo! Finance
ALGN United States Medical Devices Align Technology, Inc. Healthcare Yahoo! Finance
ALLE Ireland Security & Protection Services Allegion plc Industrials Yahoo! Finance
LNT United States Utilities—Regulated Electric Alliant Energy Corporation Utilities Yahoo! Finance
ALL United States Insurance—Property & Casualty The Allstate Corporation Financial Services Yahoo! Finance GOOGLUnited States Internet Content & Information Alphabet Inc. Communication Services Yahoo! Finance GOOG United States Internet Content & Information Alphabet Inc. Communication Services Yahoo! Finance
MO United States Tobacco Altria Group, Inc. Consumer Defensive Yahoo! Finance
AMZN United States Internet Retail Amazon.com, Inc. Consumer Cyclical Yahoo! Finance
AMCR Switzerland Packaging & Containers Amcor plc Consumer Cyclical Yahoo! Finance
AEE United States Utilities—Regulated Electric Ameren Corporation Utilities Yahoo! Finance
AAL United States Airlines American Airlines Group Inc. Industrials Yahoo! Finance
AEP United States Utilities—Regulated Electric American Electric Power Company, Inc. Utilities Yahoo! Finance
AXP United States Credit Services American Express Company Financial Services Yahoo! Finance
AIG United States Insurance—Diversified American International Group, Inc. Financial Services Yahoo! Finance
AMT United States REIT—Specialty American Tower Corporation (REIT) Real Estate Yahoo! Finance
AWK United States Utilities—Regulated Water American Water Works Company, Inc. Utilities Yahoo! Finance
AMP United States Asset Management Ameriprise Financial, Inc. Financial Services Yahoo! Finance
ABC United States Medical Distribution AmerisourceBergen Corporation Healthcare Yahoo! Finance
AME United States Specialty Industrial Machinery AMETEK, Inc. Industrials Yahoo! Finance
AMGN United States Drug Manufacturers—General Amgen Inc. Healthcare Yahoo! Finance
APH United States Electronic Components Amphenol Corporation Technology Yahoo! Finance
ADI United States Semiconductors Analog Devices, Inc. Technology Yahoo! Finance
ANSS United States Software—Application ANSYS, Inc. Technology Yahoo! Finance
ANTM United States Healthcare Plans Anthem, Inc. Healthcare Yahoo! Finance
AON Ireland Insurance Brokers Aon plc Financial Services Yahoo! Finance
AOS United States Specialty Industrial Machinery A. O. Smith Corporation Industrials Yahoo! Finance
APA United States Oil & Gas E&P APA Corporation Energy Yahoo! Finance
AAPL United States Consumer Electronics Apple Inc. Technology Yahoo! Finance
AMAT United States Semiconductor Equipment & Materials Applied Materials, Inc. Technology Yahoo! Finance
APTV Ireland Auto Parts Aptiv PLC Consumer Cyclical Yahoo! Finance
ADM United States Farm Products Archer-Daniels-Midland Company Consumer Defensive Yahoo! Finance
ANET United States Computer Hardware Arista Networks, Inc. Technology Yahoo! Finance
AJG United States Insurance Brokers Arthur J. Gallagher & Co. Financial Services Yahoo! Finance
AIZ United States Insurance—Specialty Assurant, Inc. Financial Services Yahoo! Finance
T United States Telecom Services AT&T Inc. Communication Services Yahoo! Finance
ATO United States Utilities—Regulated Gas Atmos Energy Corporation Utilities Yahoo! Finance
ADSK United States Software—Application Autodesk, Inc. Technology Yahoo! Finance
Continued on next page
Table 1:S&P 500 data description.
Symbol Country Industry Name Sector Source
ADP United States Staffing & Employment Services Automatic Data Processing, Inc. Industrials Yahoo! Finance
AZO United States Specialty Retail AutoZone, Inc. Consumer Cyclical Yahoo! Finance
AVB United States REIT—Residential AvalonBay Communities, Inc. Real Estate Yahoo! Finance
AVY United States Business Equipment & Supplies Avery Dennison Corporation Industrials Yahoo! Finance
BKR United States Oil & Gas Equipment & Services Baker Hughes Company Energy Yahoo! Finance
BLL United States Packaging & Containers Ball Corporation Consumer Cyclical Yahoo! Finance
BAC United States Banks—Diversified Bank of America Corporation Financial Services Yahoo! Finance
BK United States Asset Management The Bank of New York Mellon Corporation Financial Services Yahoo! Finance BAX United States Medical Instruments & Supplies Baxter International Inc. Healthcare Yahoo! Finance BDX United States Medical Instruments & Supplies Becton, Dickinson and Company Healthcare Yahoo! Finance BRK.B United States Insurance—Diversified Berkshire Hathaway Inc. Financial Services Yahoo! Finance
BBY United States Specialty Retail Best Buy Co., Inc. Consumer Cyclical Yahoo! Finance
BIO United States Medical Devices Bio-Rad Laboratories, Inc. Healthcare Yahoo! Finance
BIIB United States Drug Manufacturers—General Biogen Inc. Healthcare Yahoo! Finance
BLK United States Asset Management BlackRock, Inc. Financial Services Yahoo! Finance
BA United States Aerospace & Defense The Boeing Company Industrials Yahoo! Finance
BKNG United States Travel Services Booking Holdings Inc. Consumer Cyclical Yahoo! Finance
BWA United States Auto Parts BorgWarner Inc. Consumer Cyclical Yahoo! Finance
BXP United States REIT—Office Boston Properties, Inc. Real Estate Yahoo! Finance
BSX Boston Scientific Corporation Yahoo! Finance
BMY United States Drug Manufacturers—General Bristol-Myers Squibb Company Healthcare Yahoo! Finance
AVGO United States Semiconductors Broadcom Inc. Technology Yahoo! Finance
BR United States Information Technology Services Broadridge Financial Solutions, Inc. Technology Yahoo! Finance BF.B United States Beverages—Wineries & Distilleries Brown-Forman Corporation Consumer Defensive Yahoo! Finance CHRW United States Integrated Freight & Logistics C.H. Robinson Worldwide, Inc. Industrials Yahoo! Finance
COG United States Oil & Gas E&P Cabot Oil & Gas Corporation Energy Yahoo! Finance
CDNS United States Software—Application Cadence Design Systems, Inc. Technology Yahoo! Finance
CZR United States Resorts & Casinos Caesars Entertainment, Inc. Consumer Cyclical Yahoo! Finance
CPB United States Packaged Foods Campbell Soup Company Consumer Defensive Yahoo! Finance
COF United States Credit Services Capital One Financial Corporation Financial Services Yahoo! Finance
CAH United States Medical Distribution Cardinal Health, Inc. Healthcare Yahoo! Finance
KMX United States Auto & Truck Dealerships CarMax, Inc. Consumer Cyclical Yahoo! Finance
CCL United States Travel Services Carnival Corporation & plc Consumer Cyclical Yahoo! Finance
CARR United States Building Products & Equipment Carrier Global Corporation Industrials Yahoo! Finance CTLT United States Drug Manufacturers—Specialty & Generic Catalent, Inc. Healthcare Yahoo! Finance CAT United States Farm & Heavy Construction Machinery Caterpillar Inc. Industrials Yahoo! Finance CBOE United States Financial Data & Stock Exchanges Cboe Global Markets, Inc. Financial Services Yahoo! Finance
CBRE United States Real Estate Services CBRE Group, Inc. Real Estate Yahoo! Finance
CDW United States Information Technology Services CDW Corporation Technology Yahoo! Finance
CE United States Chemicals Celanese Corporation Basic Materials Yahoo! Finance
CNC United States Healthcare Plans Centene Corporation Healthcare Yahoo! Finance
CNP United States Utilities—Regulated Electric CenterPoint Energy, Inc. Utilities Yahoo! Finance
CERN United States Health Information Services Cerner Corporation Healthcare Yahoo! Finance
CF United States Agricultural Inputs CF Industries Holdings, Inc. Basic Materials Yahoo! Finance
SCHW United States Capital Markets The Charles Schwab Corporation Financial Services Yahoo! Finance
CHTR United States Entertainment Charter Communications, Inc. Communication Services Yahoo! Finance
CVX United States Oil & Gas Integrated Chevron Corporation Energy Yahoo! Finance
CMG United States Restaurants Chipotle Mexican Grill, Inc. Consumer Cyclical Yahoo! Finance
CB Switzerland Insurance—Property & Casualty Chubb Limited Financial Services Yahoo! Finance
CHD United States Household & Personal Products Church & Dwight Co., Inc. Consumer Defensive Yahoo! Finance
CI United States Healthcare Plans Cigna Corporation Healthcare Yahoo! Finance
CINF United States Insurance—Property & Casualty Cincinnati Financial Corporation Financial Services Yahoo! Finance
CTAS United States Specialty Business Services Cintas Corporation Industrials Yahoo! Finance
CSCO United States Communication Equipment Cisco Systems, Inc. Technology Yahoo! Finance
C United States Banks—Diversified Citigroup Inc. Financial Services Yahoo! Finance
CFG United States Banks—Regional Citizens Financial Group, Inc. Financial Services Yahoo! Finance
CTXS United States Software—Application Citrix Systems, Inc. Technology Yahoo! Finance
CLX United States Household & Personal Products The Clorox Company Consumer Defensive Yahoo! Finance CME United States Financial Data & Stock Exchanges CME Group Inc. Financial Services Yahoo! Finance
CMS United States Utilities—Regulated Electric CMS Energy Corporation Utilities Yahoo! Finance
KO United States Beverages—Non-Alcoholic The Coca-Cola Company Consumer Defensive Yahoo! Finance
CTSH United States Information Technology Services Cognizant Technology Solutions Corporation Technology Yahoo! Finance CL United States Household & Personal Products Colgate-Palmolive Company Consumer Defensive Yahoo! Finance CMCSAUnited States Entertainment Comcast Corporation Communication Services Yahoo! Finance
CMA United States Banks—Regional Comerica Incorporated Financial Services Yahoo! Finance
CAG United States Packaged Foods Conagra Brands, Inc. Consumer Defensive Yahoo! Finance
COP United States Oil & Gas E&P ConocoPhillips Energy Yahoo! Finance
ED United States Utilities—Regulated Electric Consolidated Edison, Inc. Utilities Yahoo! Finance
STZ United States Beverages—Wineries & Distilleries Constellation Brands, Inc. Consumer Defensive Yahoo! Finance COO United States Medical Instruments & Supplies The Cooper Companies, Inc. Healthcare Yahoo! Finance
CPRT United States Specialty Business Services Copart, Inc. Industrials Yahoo! Finance
GLW United States Electronic Components Corning Incorporated Technology Yahoo! Finance
CTVA United States Agricultural Inputs Corteva, Inc. Basic Materials Yahoo! Finance
COST United States Discount Stores Costco Wholesale Corporation Consumer Defensive Yahoo! Finance
CCI United States REIT—Specialty Crown Castle International Corp. (REIT) Real Estate Yahoo! Finance
CSX United States Railroads CSX Corporation Industrials Yahoo! Finance
Continued on next page