Investment Portfolio Analysis: Sortino Ratio
Introduction
“After the stock market crash (in 1987), they rewarded two theoreticians, Harry Markowitz and William Sharpe, who built beautifully Platonic models on a Gaussian base, contributing to what is called Modern Portfolio Theory. Simply, if you remove their Gaussian assumptions and treat prices as scalable, you are left with hot air. The Nobel Committee could have tested the Sharpe and Markowitz models — they work like quack remedies sold on the Internet — but nobody in Stockholm seems to have thought about it”
- Nassim Nicholas Taleb, The Black Swan (2007)
Prior to the advent of calculus, the ancient Greeks used “the method of exhaustion” to find the area of a circle. Using this method, they circumscribed polygons about the circle and increased the number of sides until the area of the polygon visually resembled that of the circle. Unsurprisingly, this method was time-consuming, inaccurate, and exhausting. But it wasn’t until 500 BC that Eudoxus used exhaustion to develop the now commonplace formula for the area of a circle: A = π * r².
Similarly, the modern financial community has seen its fair share of methodological advancements. In the mid 20th century, economists began developing and implementing state-of-the-art theories and strategies that were considered to be modestly effective at evaluating risk and predicting return. Despite such advances, however, these methods too had their flaws. As stated by Nassim Nicholas Taleb, “[I]f you remove their Gaussian assumptions and treat prices as scalable, you are left with hot air.” It wasn’t until the early 1990s that the progressive sector of the financial community began to factor in material psychological assessments of risk and reward when constructing investment portfolios.
We are taught how to balance portfolios to hedge risk and account for loss aversion, but we aren’t taught how we can combine these two concepts when making investment decisions. One explanation is that we have yet to develop a formula that can entirely eliminate uncertainty because the possibility of a Black Swan event is unpredictable. Still, the potential consequences of our errors should be weighed more heavily than our estimated projections.
We have made great progress since the days of “the method of exhaustion”, but our methods are still not infallible, and will likely never be.
The sections of this report will consist of the following:
- Detailing the origins of Modern Portfolio Theory and Post Modern Portfolio Theory.
- Utilizing concepts presented in Post Modern Portfolio Theory like the Sortino ratio to create forecasts.
- Performing an analysis and portfolio optimization of three risk profiles: risk-averse, risk-neutral, and risk-seeking.
- Detailing the result of our simulations and providing the most optimized portfolio combination for each risk profile according to the Sortino ratio.
- Developing ideas for expanding portfolio optimization methodologies.
Theory
Modern Portfolio Theory
Modern Portfolio Theory (MPT) was introduced in 1952 by economist Harry Markowitz. According to Investopedia, “Modern portfolio theory (MPT) is a theory on how risk-averse investors can construct portfolios to maximize expected return based on a given level of market risk” (Chen, 2021). MPT introduced two concepts that are integral to portfolio optimization. The first concept is diversification: the idea that owning different types of financial assets is less risky than owning just one type. The other concept introduced variance in asset prices as a proxy for risk. Finding variance in asset prices is done through the Sharpe ratio, which measures the performance of an investment compared to a risk-free asset after adjusting for risk.
Despite its wide adoption, Modern Portfolio Theory has received a great deal of criticism. To iterate: in MPT, risk, return, and correlation are each measured by expected values, which means that they are statistical statements about the future. Expected values tend to fail in accounting for new circumstances — also known as a Black Swan event. Additionally, the Sharpe ratio measures risk as a symmetrical distribution, which tends to underestimate volatility and inflate growth of return. True statistical features of risk and return follow highly skewed distributions. Ultimately, Modern Portfolio Theory falls short due to the loss aversion principle, where the intuitive concept of risk is asymmetrical.
Post-Modern Portfolio Theory
In an effort to improve upon Modern Portfolio Theory, software designers Brian M. Rom and Kathleen Ferguson developed Post-Modern Portfolio Theory (PMPT) in 1991 while working for Sponsor-Software Systems Inc.. As stated on Investopedia: “PMPT is a portfolio optimization methodology that uses the downside risk of returns instead of the mean-variance of returns used by the Modern Portfolio Theory” (Chen, 2021). MPT and PMPT both describe how the risk associated with assets should be valued, and how rational investors should use diversification to achieve portfolio optimization. The difference lies in how each theory defines risk, and how that risk influences expected returns. The PMPT uses the standard deviation of negative returns as the measure of risk, while MPT uses the standard deviation of all returns as a measure of risk.
The first element introduced into the PMPT theory was the Sortino ratio, designed to replace the Sharpe ratio as a measure of risk-adjusted return and to improve its ability to rank investment results. The Sortino ratio measures the risk-adjusted return of an asset, portfolio, or strategy. Essentially, the ratio penalizes only those returns that fall below a user-specified target or required rate of return.
In order to model closer to reality in terms of perceived risk, I will be focusing our analysis on the concepts behind Post-Modern Portfolio Theory and the Sortino ratio. Using the Sortino ratio, we are able to differentiate between good and bad variances through the calculation of downward deviation. The Sortino ratio also tends to be useful for retail investors who look to invest with certain defined goals and a target rate of return (or the risk-free return rate).
Risk Profiles
Though the concepts discussed in this study could be applied to any publicly traded stock, in an effort to mitigate the levels of volatility that an individual stock is typically subjected to, I decided to use Exchange-Traded Funds (ETF) for the datasets in this study. I did so because most ETFs are index funds, and hold the same securities in the same proportions as a certain stock market index or bond market index. The most popular ETFs in the U.S. replicate the S&P 500 index, the total market index, the NASDAQ-100 index, the price of gold, the “growth” stocks in the Russell 1000 index, or the index of the largest tech companies.
It should be noted here that each ETF has some level of holding fee that varies depending on which ETF you are invested in. I will not be factoring in the holding fees for the purpose of this study as I am addressing risk in terms of fluctuations in price. I believe this allows for a clearer picture of the notion of uncertainty in this study, though fees will need to be a significant consideration of each investor prior to investing in ETFs.
For the sake of consistency and to reduce the possibility of our results being construed by potential fees, I chose to only utilize Vanguard ETFs. From Vanguard’s website: “All Vanguard clients pay $0 commission to trade ETFs (exchange-traded funds) and stocks online. You also have access to more than 160 no-transaction-fee mutual funds from Vanguard and more than 3,000 funds from other companies” (Vanguard, 2021). Each Vanguard ETF was chosen for each respective risk profile based on their respective levels of risk.
That being said, let’s proceed to the risk profiles:
Risk-Averse
Risk-Aversion in the context of finance is the tendency to prefer outcomes with low uncertainty to those outcomes with high uncertainty, even if the average outcome of the latter is equal to or higher in monetary value than the more certain outcome.
The risk-averse profile consists of the following ETFs:
- Vanguard Total Bond Market Index Fund ETF Shares (BND)
- Vanguard Real Estate Index Fund ETF Shares (VNQ)
- Vanguard Value Index Fund ETF Shares (VTV)
- Vanguard Wellesley Income Fund Investor Shares (VWINX)
- Vanguard Target Retirement 2015 Fund Investor Shares (VTXVX)
Risk-Neutral
Risk-Neutral preferences in the context of finance are preferences that are neither risk-averse nor risk-seeking. A risk neutral party’s decisions are not affected by the degree of uncertainty in a set of outcomes. Thus, a risk neutral party is indifferent to choices with equal expected payoffs even if one choice is riskier.
The risk-neutral profile consists of the following ETFs:
- Vanguard Total Stock Market Index Fund ETF Shares (VTI)
- Vanguard Total Bond Market Index Fund ETF Shares (BND)
- Vanguard Growth Index Fund ETF Shares (VUG)
- Vanguard Target Retirement 2015 Fund Investor Shares (VTXVX)
- Vanguard Real Estate Index Fund ETF Shares (VNQ)
Risk-Seeking
The risk-seeking profile in the context of finance is a person who has a preference for risk.
The risk-seeking profile consists of the following ETFs:
- Vanguard Russell 2000 Index Fund ETF Shares (VTWO)
- Vanguard Growth Index Fund ETF Shares (VUG)
- Vanguard Total Stock Market Index Fund ETF Shares (VTI)
- Vanguard Total International Stock Index Fund ETF Shares (VXUS)
- Vanguard Mid-Cap Index Fund ETF Shares (VO)
Data
The structuring of the data in this report was done through two Python Modules: Pandas, which allows us to organize and format complex data into table structures called DataFrames, and Pandas-DataReader, which is used to collect public financial data from the internet and import it into Python as a DataFrame. With these modules, I imported historical closing price data from the last five years directly from Yahoo Finance. Specifically, the five-year timeframe for the data resided between 06–01–2016 and 06–01–2021. This time frame was chosen based on my ability to rule out the influences of market anomalies while remaining obtainable.
The data for each ETF was collected using the ‘Adj Close’ price from the Yahoo Finance website. I collected this data by defining a function called “YahooData” which took in the parameters: dataframe, assets_list, start_date, and end_date. Coincidentally, I created a list with the ticker symbols of each of the assets that I planned on using in the portfolio (assets), established our start (start) and end (end) dates, and created an empty dataframe to be filled by our YahooData function (df_prices). I then ran the YahooData function with the following parameters: df_prices, assets, start, and end; saving it to a dataframe named “df”.
Once my stock data was loaded into df, I calculated a log return of the data to normalize the returns, dropped the first row of the data which was “N/A” after the log calculation, and saved it as “df”. The constants I used in my calculations were “N,” which equaled the number of trading days in the US: 252 and “rf,” which was the risk-free rate of 2%.
Financial Statistics Calculations
The Python code that was written in Jupyter Notebook for the calculations can be found at the following links:
Risk-Averse
Risk-Neutral
Risk-Seeking
To get a better idea of the possible optimized portfolio, I will run a total of 10,000 simulations of different portfolio weight configurations for each risk profile. To do so, I created an empty matrix called num_runs with 10,000 rows, each representing a different iteration. To this matrix, I added five columns to represent each ETF in the portfolio and an additional five columns for each of the calculations that I conducted on the portfolio: “Mean Returns”, “Downside SD”, “Upside SD”, “Volatility Skewness,” and “Sortino.” To finish the structure of the matrix, I filled each spot in the matrix with a zero as a placeholder.
In order to fill the matrix with the result of our calculations, I created a ‘for loop’ using Python that allowed us to iterate through all 10,000 simulations and perform the following calculations:
- Create the weights of each ETF in the portfolio at random, while ensuring that the total weights of the portfolio did not exceed a total of 1
- Calculate the total return of the portfolio based on a given set of weights
- Calculate the annualized mean of each portfolio
- Calculate the annualized downside standard deviation of each portfolio
- Calculate the annualized upside standard deviation of each portfolio
- Calculate the volatility skewness of each portfolio
- Calculate the Sortino ratio
- Populate the result of each calculation into the matrix
See code below for a reference of stated calculations:
From there we converted the matrix into a dataframe and named the columns:
I then utilized the ‘iloc’ command with aggregate functions to find the portfolios with the Maximum Sortino.
Using matplotlib.pyplot, I created a scatterplot with the Downside Standard Deviation on the x-axis, the Return on the y-axis, and designated the Sortino ratio based on a color scale. I then plotted each of the portfolios and designated on the plot where the portfolio with the Maximum Sortino ratio resided with a red star as a marker.
Analysis
The main finding of interest from my analysis shows that the volatility skewness decreased as the portfolio risk level increased. This indicates that as the risk level increases, the volatility skewness becomes more and more asymmetrical. As more downward deviation events are added to a dataset, the Sortino ratio becomes more accurate and allows us to develop more informed forecasts.
Risk-Averse Portfolio
The Risk-Averse simulation resulted in a portfolio that had a maximum Sortino ratio of .61. This portfolio was composed of the following:
The return that this portfolio would have realized would have been 8.2% on average between 6/1/2016–6/1/2021. The volatility skewness was .726, indicating that the portfolio tended to have greater downside volatility relative to upside volatility. The red star indicates the portfolio with the most optimized Sortino ratio.
Based on these calculations, an individual who is more risk-averse could construct their portfolio with the above ETFs with a similar percentage allocation and under ideal circumstances could possibly expect similar returns.
Risk-Neutral Portfolio
The Risk-Neutral portfolio simulation resulted in a portfolio that had a maximum Sortino ratio of .91. This portfolio was composed of the following:
The return that this portfolio would have realized would have been 14.9% on average between 6/1/2016–6/1/2021. The volatility skewness was .739 which indicated that the portfolio tended to have a greater downside volatility relative to upside volatility. The red star indicates the portfolio with the most optimized Sortino ratio.
Based on these calculations, an individual who is more risk-averse could construct their portfolio with the above ETFs with a similar percentage allocation and under ideal circumstances could possibly expect similar returns.
Risk-Seeking Portfolio
The Risk-Seeking portfolio simulation resulted in a portfolio that had a maximum Sortino ratio of .67. This portfolio was composed of the following:
The return that this portfolio would have realized would have been 8.7% on average between 6/1/2016–6/1/2021. The volatility skewness was .696 which indicated that the portfolio tended to have greater downside volatility relative to upside volatility. The red star indicates the portfolio with the most optimized Sortino ratio. Based on these calculations, an individual who is more risk-averse could construct their portfolio with the above ETFs with a similar percentage allocation and under ideal circumstances could possibly expect similar returns.
Conclusion
The Sortino ratio is a statistical tool that allows us to evaluate the return of an investment for a given level of downside risk. This allows for a more psychologically realistic, asymmetric model of risk into our analysis. However, a single ratio does not give us the full picture and so, should not be the sole method of formulating investment strategies. The limitation of the Sortino ratio indicates that there needs to be enough negative swing volatility events for the calculation of a downward deviation to be statistically significant. If there aren’t enough downward volatility events in the data, there will simply not be enough information to accurately formulate a forecast of the possible outcomes. According to Dheeraj Vaidya, “[The Sortino ratio] is also a better tool for measurement of the performance of a fund manager whose returns are positively skewed as it will ignore all the positive variances while calculating volatility or risk and provide a more appropriate evaluation” (Vaidya, CFA, FRM, 2021). The Sortino ratio becomes a better metric to use under circumstances with a statisically significant downside risk, but cannot be applied in all circumstances. We have indeed come a long way in our ability to analyze data, assess risk, and create forecasts, but we still have a long way to go.
Project Expansion Opportunities
- Construct portfolios based on the correlation of each ETF, then performing the Sortino ratio calculations.
- A distribution curve of each portfolio’s volatility would be interesting to see so that we can visualize the changes in the symmetry as we change levels of risk.
- Comparing the results of the Sortino ratio with the Sharpe ratio. Stagger the analysis data to be 6/1/2016 – 6/1/2020 while creating a forecast over the term 6/2/2020 – 6/2/2021 and compare with real-world results over the same term.
- Create an Efficient Frontier for the scatterplots of each risk profile.
- Add a numpy seed calculation method that locks the random numbers so that the results can be replicable.
Bibliography:
Chen, J. (2021, 2 2). Post-Modern Portfolio Theory. Investopedia. https://www.investopedia.com/terms/p/pmpt.asp
Chen, J. (2021, 3 1). Modern Portfolio Theory (MPT). Investopedia. https://www.investopedia.com/terms/m/modernportfoliotheory.asp
Ganti, A., & Scott, G. (2021, 04 21). Efficient Frontier. Investopedia. Retrieved 05 26, 2021, from https://www.investopedia.com/terms/e/efficientfrontier.asp
Taleb, Nassim Nicholas (2007), The Black Swan: The Impact of the Highly Improbable, Random House, ISBN 978–1–4000–6351–2.
Vaidya, CFA, FRM, D. (2021, 6 7). Sortino ratio. WallStreetMojo. https://www.wallstreetmojo.com/sortino-ratio/
Vanguard. (2021, 6 4). View Benefits at a Glance. Vanguard. https://investor.vanguard.com/investing/benefits/at-a-glance
Wikipedia. (2021, 4 20). Sharpe ratio. Wikipedia. https://en.wikipedia.org/wiki/Sharpe_ratio
Wikipedia. (2021, 6 6). Sortino Ratio. Wikipedia. https://en.wikipedia.org/wiki/Sortino_ratio
