Risk Is The Standard Deviation of Returns


In this entry we will review some of the different products in which you can invest.  We will explore the concept of risk, some historic data on risk vs. return, and provide an examination of each of these for us as personal investors in the market.


The concept of risk is a challenging one and yet it is the focus of much of what we decide with respect to personal investing.  One interesting way to measure how much risk you are willing to endure, aka your risk tolerance, is to ask yourself how much you would bet on the flip of a coin.  This gives you an idea of your risk threshold.  Although that simple experiment is a classic way to measure risk tolerance,  it is by no means the most complex or comprehensive.


Regardless of how you measure your personal risk threshold, remember:  risk and reward should be closely associated in your mind when it comes to personal investing.  In fact, risk and reward are sort of the flip sides of the same coin.  Risk is what we endure for certain rewards.  Therefore, the reward from a certain investment must be commensurate with the risk we take in putting our money into a certain stock, mutual fund, or other investment vehicle.  Here, we will describe some of the different ways to understand how much risk is inherent in an investment versus its return.  We will examine some classic equity backed securities such as stocks, mutual funds, bonds, and real estate.


One way to conceptualise risk is as the standard deviation of returns.  That is, the risk inherent in, for example, a stock can be considered as its standard deviation of returns where returns are plotted as a histogram.  Therefore, the wider the standard deviation of the amount of return on your money, the riskier the investment. Here, we will cite several distributions for risk from a standard text.


One of the most popular texts that gives us the concept of risk versus return is Burton Malkiel’s A Random Walk Down Wall Street. Page 185 gives us one of the most useful representations of risk inherent in different types of investment opportunities.  For one, consider the distribution of returns from the stock market.  The historic standard deviation of returns from the stock market is approximately 20% as described by Malkiel’s text.  See Figure 1.  Therefore, of the various opportunities we will discuss, investing in the stock market is one of the most ‘risky’.


Figure 1: Histograms with standard deviations of various investment products from Malkiel’s book: A Random Walk Down Wallstreet


This is because, although there is great potential for upside, there is also routine potential for downside. As you probably know, plus or minus one standard deviation on either side of the central tendency of the bell curve contains 66% of values.  Therefore, based on historic data, 66% of the time we invest in the stock market our gain or loss will be between plus 20% or minus 20% with an “average” return of around 11%.  Again, the more narrow the standard deviation in a system the less risk inherent in that system. 


Consider, next, the bond market.  Corporate and municipal bonds are similar in their return.  Corporate and municipal bonds returns approximately 3-4% per year.  (See Figure 1.) There is a much lower standard deviation of returns, historically, in the bond market. Please notice, however, that on an after-tax basis, the corporate and municipal bonds basically give the same return for the same amount of money.  Therefore, corporate bonds are thought to be slightly more risky than municipal bonds in that an individual corporation may default (bankruptcy) and give you no money back.  Therefore, US municipal bonds are felt to be more conservative investment option (the government is unlikely to go bankrupt) for those interested in investing in bonds.


Real estate is another investment option.  The historical return of the real estate market is approximately 2%.  Clearly, just as with stocks & bonds, the real estate market we are talking about is the real estate market as a whole from its inception until now.  There are several things which are not well represented by taking such a global view of the market.  For one, different geographic areas in the country, such as vacation destinations like Hawaii or Florida, may perform better than the market as a whole.  Also, isolated periods of time may buck the trend and outperform history.  The housing bubble, for example, was able to generate significant returns for some participants in the market.  However, in general, real estate is felt to be a secure investment and returns about approximately 2% to personal investors.  Keep in mind, inflation is often higher than this 2% return; however, the real estate can be enjoyed, will generally increase in value with time, and may, depending on the market, generate a nice rental return.  Also, renting a location maybe able to generate some nice passive income.


Did you notice how we considered the real estate market as a whole and then focused on how individual segments with in (as well as isolated periods in its history) may differ? Similarly, the stock market data we cite above is for the market as a whole since inception.  There are important limitations to the model we describe.  For one, it does not highlight crash years such as the great stock market crash or the several significant recessions we have had.  Therefore, some cite that the market has a greater standard deviation of returns than 20%.


Now we have a concept of risk as the standard deviation of returns on an investment.  Some other questions include “How we can measure risk for individual stocks in the market?”  There are several nice pieces of data I would like to share which are available for you and are publicly available on Yahoo finance as well as other public finance search engines.


One piece of data you can use to decide whether to invest in an individual stock is the beta value.  The beta is obtained by plotting the stock’s increase or decrease (on the y axis) versus the market’s increase or decrease (x-axis).  That is, a standard stock market index (such as the Dow Jones Industrial average or S&P 500) will be plotted and the stock’s performance will be plotted against this. This XY plot indicates that when the market as a whole moves a certain way the stock moves a certain way.  Next, a regression line model is used so as to generate the equation of a regression line.  The slope of the regression line is the beta value.


Figure 2: Beta is the slope of the regression line that associates S&P 500 return with an individual stock’s return. From Neil Cohen, DBA CFA in Financial Management MBAD 233-EM1, George Washington University Spring 2010


In other words, the beta value is the amount the stock described moves relative to the market as a whole.  (See Figure 2.) Thus, stock with a beta of positive 1 goes up one point when the market increases 1 point.  (It also goes down one point when the market goes down one point.) Therefore, you get a sense of how risky the stock you may like is compared to the market as a whole.  Stocks that have a beta of 2 may increase by 2 when the market goes up and also are more likely to decrease by 2 when the market goes down.  Again, recall this is from a linear regression model.  Therefore, it should not be said that every day the market goes up one point your stock with beta of 2 will go up 2 points.  This is just a general tendency over time.  The beta gives you a sense of how significantly the stock varies when the market varies.  This gives you a concept of risky your stock investment is.


Another useful number is the alpha.  The alpha is a measurement associated with mutual funds.  The alpha indicates the amount of return you can obtain from a mutual fund in excess of the risk inherent in the portfolio that makes up the mutual fund.  (Awesome right?  Who wouldn’t want a return higher than the risk involved!) Now you know why the well-known blog SeekingAlpha.com is called that; who wouldn’t want to seek a return in excess of the involved risk?


As you probably know, a mutual fund is a group of stocks that are maintained in a portfolio by a given company.  You can purchase shares of the fund of stocks.  It is important to know the managerial fees associated with your mutual fund.  Remember, managers maybe paid to help change the stocks which make up the mutual fund so as to trade in stocks that are performing well and decrease stocks that are performing less effectively.  There are some mutual funds which are indexed to the market as a whole.  These funds attempt to reproduce an investment in the market as a whole and reflect the market increases and decreases by being composed of a broad selection of stocks felt to represent the market as a whole.  These allow you to invest in the whole market.


One of the other newer investment vehicles are the derivatives.  The derivatives are so called because they derive from the typical products with which you are now more familiar.  One of the derivative markets is the options market.  Options allow you to purchase or sell a contract that allows someone to buy or sell a stock in the future at a given price.  I will not discuss much about the derivatives market here except to say that it seems as if there is a wider standard deviation of returns for options in general. However there are multiple risk mitigation strategies that allow for some very interesting opportunities in the options market.


In conclusion, we have discussed a definition of risk as it relates to personal investment:  risk may be thought of as the standard deviation of returns on an investment product.  We described one classic test to allow you to get a sense of your risk threshold.  We then highlighted the standard deviation of returns across multiple products including stocks, bonds, and real estate.  Good luck in your personal investment decisions and your attempts to beat inflation.  Remember, a key focus as a personal investor is on the return at the end of the day AFTER taxes, any fees (such as those a mutual fund charges for managing the fund) and inflation.


Please note:  None of the above constitutes investment advice–professional or otherwise.  I’m sharing just a bit of how risk may be conceptualized in complex investment decisions so that you can make your own decision based on your unique situation.


Questions, comments, or feedback?  Feel free to add your thoughts and comments beneath.