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Breakthrough Drugs, Statistics, and Anecdotes: Three Things Every Individual Investor Needs to Know – Statistics and Time Series Data – Part 2 of 3

20 Wednesday Nov 2019

Posted by wmosconi in asset allocation, correlation, correlation coefficient, finance theory, financial advice, Financial Media, Financial News, financial services industry, historical returns, Individual Investing, individual investors, investing, investing advice, investing tips, risks of stocks, standard deviation, stock market, Stock Market Returns, time series, time series data

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correlation, correlation coefficient, Financial Market History, financial markets, Financial Media, Financial News, invest, investing, investing tips, math, mathematics, noise, statistics, time series, time series data

The first article of this three-part series covered the broad strokes of this issues to be aware of in terms of all the “data” and “relationships” that get thrown around by the financial media (print and television).  Most of the discussion uses data points that are not statistically significant to draw any sort of conclusion.  In fact, time series data is notoriously hard to model and predict the future.  Additionally, the specific time series data of stock market returns is even more difficult.

You can refer to the link below to examine the content of the first article:

https://latticeworkwealth.com/2019/11/11/breakthrough-drugs-statistics-and-anecdotes-investing/

The task at hand for the second article is to put some “meat on the bones” of the discussion.  I realize that anything to do with math and statistics is not easy for everyone (or of interest either).  Therefore, I will be writing a supplemental article that covers the mathematics and statistics in more detail.  The goal here is to be able to identify some of the more common errors that you will encounter.

The first item to talk about is any sort of data that has a substantial trend component.  In layman’s terms, there is a data series where the line graph goes up or down in more of a straight-line manner.  You can think of the Gross Domestic Product (GDP) of the United States here.  Every year the GDP figures will generally go up unless there is a recession.  But, even after the recession passes, the trend for GDP will resume upward.  So, where does the problem come in?

I am going to give a contrived example to illustrate why it is dangerous to compare two series that are trending.  The example will consist of two different equations which are trends.  Both have the same trend component and an error term (we will call that eta).  The variables will be exactly the opposite.  More specifically, the two equations we will use are the following:

Trend_1 = Time + 100 + 0.9 * x + eta

Trend_2 = Time +100 – 0.9 * x + eta

Now the x values and eta values were simply generated by selected variables at random between 0 and 1.  The eta values were also selected at random between 0 and 1.  You can think of eta as representing the general “noise” that occurs on a daily basis when observing stock prices in the financial markets.  So, let’s graph the first 100 observations for these two equations:

Trend_Graph_Statistics_Revised

You will notice that the trend component dominates the line graphs.  However, we know by construction that the two equations which produce trend_1 and trend_2 are fundamentally different.  Now the correlation coefficient between those two equations is 0.9984.  A correlation coefficient of 1 means that the two lines move in lockstep.  Why is this important?  Why is it very dangerous?

Well, financial pundits will talk about these types of graphs all the time.  It looks like there is some relationship, but we know there is very little relationship between the two trends.  In fact, we can look at these equations by subtracting the current value from the previous value to see what changes.   Formally, this topic is called first differencing.  It will allow us to see more clearly what we already know.  Here is the graph:

First_Difference_Graph_Statistics_Revised

Now we have a totally different picture.  We can see that at many times the two trend equations are moving in exactly the opposite direction.  In fact, the correlation coefficient for the first-differenced equations is 0.2675.  There is only a slight positive relationship between the two trends.

In the example above, we can see that looking at the two trends is very deceiving.  Remember that I added the eta term to represent “noise” that is always present in financial market data.  So, anytime someone talks to you about the comparison of two trends, you should be very skeptical.  You always want to see first-differenced data or at least a comparison of changes in some manner.  Otherwise, you will mistakenly assume that there is a strong positive or negative relationship between two time series.

The second example that I am going to use is stock market returns for the S&P 500 Index from 1966 through 2018.  Why start at 1966?  Well, the S&P 500 Index started with its current number of component stocks back in 1957, and I would like to show annual stock returns and also ten-year annualized returns.  This particular topic can get messy quite quickly, so I am not going to cover it in a lot of depth with statistical and mathematical jargon.  For those of you who are interested, I had mentioned that it will be contained in a forthcoming supplemental article.

A great many individuals in the financial markets talk about stock market returns in the same breath as the normal distribution.  What is the normal distribution?  It is the old bell curve that you are familiar with.  The normal distribution is symmetrical and tails off at the end as more and more data points are gathered.  Well, stock market returns are anything but strongly normal.

Let’s first take a look at one-year stock market returns for the S&P 500 Index.

One Year Returns - Histogram - Non Normal

A useful test to see if a particular distribution is normal is the Jarque-Bera test.  Now it is not necessary to know exactly what is being calculated.  However, you should refer to the bottom of the box that reads “Probability”.  The value of 0.179 is called a p-value.  A p-value less than or equal to 0.10 means that we can reject the hypothesis that the one-year distribution of stock returns is normal.  At a value of 0.179, we would not reject the hypothesis of normality for this distribution.  However, the p-value in our case is not large enough to be totally sure and confident. But what about looking at annualized stock market returns over ten-year periods?

We can look at a similar graph to check to see if stock market returns over longer timeframes are indeed akin to the normal distribution (i.e. the bell curve).  Here is the graph:

Ten Year Returns - Histogram - Non Normal

Looking at the same “Probability” value, we have 0.489.  Therefore, we cannot reject the hypothesis that these stock market returns follow a normal distribution. Looking at ten-year annualized stock market data tells us that we can use the normal distribution as an assumption for calculating statistics.

Now why does this matter?  Well, you will here over and over again statistics that apply only to the normal distribution in relationship to actual, observed stock market returns.  We have just seen that stock market returns over the short-term stock returns weakly follow the normal distribution. On the other hand, long-term stock returns are definitely normal. Now I will not get into the technicalities, but time series data is indeed asymptotically normal.  What?  Say again?

This is just a fancy way of saying that, as the number of data points (sample size) approaches infinitely, the time series will look like the normal distribution.  Pretty much all financial market and economic data have very few data points.  In fact, you usually need several hundred data points prior to making any assumptions and using the statistics related to the normal distribution (think standard deviation or correlation coefficients).

Thus, most of the banter in the financial media is just subjective notions of what is going on in the stock market and the economy. More often than not, an assertion by someone in the financial print or television media is more of an educated guess than based on a solid mathematical foundation. That fact explains why financial pundits hedge their statements. Like I say half-jokingly, “I see the stock market going up in the next several months, but of course it might not resume its uptrend or could even take a leg downward”.

Well yes, I guarantee you that every day stocks will go up, down, or remain unchanged. This type of daily commentary in the financial press about the short-term performance of stocks (or other financial assets) is just not helpful and can be downright distracting you from investing for your long-term financial goals.

I apologize for getting too detailed in certain parts of this article.  What are the key takeaways?  First, you should be extremely leery of drawing any conclusions from the comparison of two or more data series that are trending upward or downward.  Second, you need to have several hundred observations prior to invoking any reference to the normal distribution.  So, what is left after that?  As you might imagine, there are not too many comparisons or studies that pass the muster to give you insights on investing or actionable information to make changes to your investment portfolio.

Don’t focus on the mathematics or statistics.  All you need to remember are the two takeaways above.  And, first and foremost, you should always be skeptical whenever you are presented with comparisons and statistics related to the financial market or the economy as a whole.

How to Become a Successful Long-Term Investor – Summary

30 Monday Sep 2019

Posted by wmosconi in asset allocation, behavioral finance, correlation, correlation coefficient, Dot Com Bubble, Emotional Intelligence, EQ, financial advice, Financial Advisor, financial goals, financial markets, financial planning, financial services industry, historical returns, Individual Investing, individual investors, investing, investing advice, investing information, investing tips, investment advice, investments, market timing, math, personal finance, portfolio, risk, risk tolerance, risks of stocks, S&P 500, S&P 500 historical returns, S&P 500 Index, standard deviation, statistics, Stock Market Returns, stock prices, stocks, volatility

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asset allocation, behavioral finance, education, finance, historical stock returns, investing, investments, math, mathematics, performance, performance monitoring, portfolio, portfolio management, S&P 500, S&P 500 historical returns, S&P 500 Index, statistics, stock returns, stocks, success, successful long term investing, trading, uncertainty

The discussion of how to become a successful long-term investor in my three-part series is now finished.  However, the journey is an ongoing one that takes discipline, constant learning, and monitoring your emotional reactions to fluctuations in the financial markets.  I discussed the history of stock market returns of the S&P 500 Index (dividends reinvested) from 1957-2018, the concept of risk, and also the futility of trying to engage in “market timing”.  But you may be asking yourself, why didn’t you tell me what stocks, bonds, and other assets to buy to build my investment portfolio?  That is a valid question, and there is an extremely important reason why that gets at the very heart of my overall discussion.

The best way to answer the question posed above is with an analogy.  Now my international readers will have to indulge me with this example.  My favorite sport is football which is the most popular sport in the world.  Most people in the United States refer to it as soccer and only watch if the men’s or women’s teams are competing in the World Cup.  I could tell you all about the reasons why football clubs rarely use a 4-4-2 formation.  Or I could talk about how the 4-2-3-1 formation has evolved in the Bundesliga.  We also could discuss why goalies now need to be good with their feet in order to pass from the backline.  Finally, I might even be more specific and give my rationale for why Liverpool in the Premier League uses a 4-4-3 formation given their current squad for the 2019-2020 season.

My analogy above relates to long-term investing because I would argue that you should not invest a single dollar in the stock or bond markets without knowing about the history of returns, risks and volatility, and “market timing”.  Most Financial Advisors (FAs), Certified Financial Planners (CFPs), and Registered Investment Advisors (RIAs) jump right into the discussion of how to build an investment portfolio taking into account your financial goals and risk tolerance.  This conversation is directly related to the football analogy above.  Without a firm understanding of investing at a high level (or the general way football is played first), you are likely to fail in your resolve to stick with a long-term focus while investing.  For example, when you are asked if you can tolerate a 20% decline in the stock market, how should you answer?  I would say that, if you do not have some grasp of historical returns and the level of risk, you cannot properly answer.  Remember that we covered how often you will experience negative returns (including 20% declines) in the first article.  You need to understand the “composition of the forest before deciding how to deal with the trees”.

Here are the links to the three articles to have an understanding of first prior to jumping into the mix of long-term investing strategy and building an actual portfolio of investments.

Part 1 – Understanding Historical Stock Market Returns:

https://latticeworkwealth.com/2019/09/23/successful-long-term-investing/

Part 2 – Understanding and Managing Risk:

https://latticeworkwealth.com/2019/09/25/successful-long-term-investor-risk/

Part 3 – Giving up on the Allure of “Market Timing”:

https://latticeworkwealth.com/2019/09/28/successful-long-term-investing-market-timing/

Once you have a firm grasp on these topics, you are ready to get your feet wet in the world of investing.

For those of you wanting a little bit of guidance because your intention is the manage your investments personally, I have written about this topic in the past.  I wrote a two-part series on how to build an investment portfolio and monitor the performance returns of that investment portfolio.  I have included the links below:

Part 1 – Building an Investment Portfolio:

https://latticeworkwealth.com/2013/07/16/how-to-create-an-investment-portfolio-and-properly-measure-your-performance-part-1-of-2/

Part 2 – Monitoring the Performance of an Investment Portfolio:

https://latticeworkwealth.com/2013/07/19/how-to-create-an-investment-portfolio-and-properly-measure-your-performance-part-2-of-2/

Those two articles above will provide you with some ways to go about creating your own investment portfolio without the assistance of a financial professional.  While it does contain a lot of information and suggestions, individual investors who are complete novices may find it easier and less confusing to seek out someone to guide them with investment selection, measuring risk tolerance, and understanding the goals of their financial plan.

In summary, I appreciate you taking the time to read my thoughts in regard to successful long-term investing.  As you can see successful investing has more to do with preparation, setting realistic expectations, and knowing how you personally respond to risk.  These topics need to be studied prior to investing money yourself or before going to seek out investment advice from a financial professional.  If you have any questions, comments, feedback, or disagreements, you can feel free to let me know.

How to Become a Successful Long-Term Investor – Part 2 of 3 – Understanding and Reducing Risk

25 Wednesday Sep 2019

Posted by wmosconi in asset allocation, behavioral finance, Consumer Finance, Education, finance, finance theory, financial advice, Financial Advisor, financial goals, financial markets, Financial Media, Financial News, financial planning, financial services industry, Geometric Returns, historical returns, Individual Investing, individual investors, investing, investing advice, investing tips, investment advice, investments, math, Modern Portfolio Theory, MPT, personal finance, portfolio, risk, risk tolerance, risks of stocks, S&P 500, S&P 500 historical returns, S&P 500 Index, standard deviation, statistics, stock market, Stock Market Returns, Stock Market Valuation, stock prices, stocks, volatility

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academia, academics, behavioral finance, finance, financial, historical stock returns, invest, investing, investments, math, mathematics, Modern Portfolio Theory, MPT, performance, portfolio, portfolio management, S&P 500, S&P 500 historical returns, S&P 500 Index, standard deviation, statistics, stock market, stock returns, stocks, success, uncertainty

Another extremely important part of being a long-term investor is to understand the concept of risk.  Financial professionals define risk in a number of different ways, and we will examine some of those definitions.  The overarching goal is to look at risk from the standpoint of the volatility or dispersion of stock market returns.  Diversification of various investments in your portfolio is normally the way that most financial professionals discuss ways to manage the inevitable fluctuations in one’s investment portfolio.  However, there is another more intuitive way to reduce risk which will be the topic of this second part of this examination into becoming a successful long-term investor.

The first part of this series on long-term investing was a look back at the historical returns of the S&P 500 Index (including the reinvestment of dividends).  The S&P 500 Index will again be the proxy used to view the concept of risk.  If you have not had a chance to read the first part of the series, I would urge you to follow the link provided below.  Note that it is not a prerequisite to follow along with the discussion to come, but it would be helpful to better understand the exploration of risk in this article.

The link to part one of becoming a successful long-term investor is:

https://latticeworkwealth.com/2019/09/23/successful-long-term-investing/

But now we will turn our attention to risk.  Risk can be a kind of difficult or opaque concept that is discussed by financial professionals.  Most individual investors have a tough time following along.  Sometimes there is a lot of math and statistics included with the overview.  Although this information is helpful, we need to build up to that aspect.  However, there will be no detailed calculations utilized in this article that might muddy the waters further.  I believe it helps to take a graphical approach and then build up to what some individual investors consider the harder aspects of grasping risk.

Risk related to investing in stocks can be defined differently, but the general idea is that stocks do not go up or down in a straight line.  As discussed in depth in part one, the annual return of the S&P 500 Index jumps around by a large margin.  Most individual investors are surprised at seeing the wide variation.  Ultimately, the long-term historical average of the S&P 500 Index from 1957 to 2018 is 9.8%.  But rarely does the average annual return end up being anywhere near that number.

The first way I would like to look at risk within the context of long-term investing is to go back to our use of “buckets” of returns.  If you have not already read part one, I used “buckets” with ranges of 5% to see where stock returns fit in.  As it relates to risk, we are only going to look at the “bucket” that includes the historical average 7% to 12% and then either side of that “bucket” (2% to 7% and 12% to 17%).  Additionally, we will look at yearly stock returns and then annualized stock returns for three years, five years, and ten years.  Here is our first graph:

Returns on Either Side of Historical Average

The main takeaway from viewing this graph is that, as the length of time increases, more stock market returns for the S&P 500 Index group around the historical average for the index of 9.8%.  Remember that part one covered the useful information that, even though the historical average to be expected from investing in stocks is 9.8%, individual investors need to know that it can take long periods of time to see that historical average.  In fact, if we look only at one-year increments, approximately 33.9% of stock returns will fall into the range of 2% to 17%.    Or, if we use our yearly equivalent, stock market returns will only fall within that range 1 out of every 3 years.  When individual investors see this graph for the first time, they are usually shocked and somewhat nervous about investing in stocks.

The important thing to keep in mind is that as the length of time examined increased many more stock returns fall into this range.  The numbers are 65.0%, 67.1%, and 81.1%, for three years, five years, and ten years, respectively.  Converting those numbers to yearly equivalents we have about 6-7 years out of ten for three years and five years.  And, as one would intuitively suspect, the longest timeframe of ten years will have stock returns falling into the 2% to 17% range roughly 8 in every 10 years.  Now that still means that 20% to 35% of long-term returns fall outside of that range when considering all those time periods.  But I believe that it is certainly much more palatable for individual investors than looking at investing through the lens of only one-year increments.

Another aspect of risk is what would be termed downside protection.  Most individual investors are considered to be risk averse.  This term is just a fancy way of saying that the vast majority of investors need a lot more expected positive returns to compensate them for the prospect of losing large sums of money.  Essentially an easier way to look at this term is that most individual investors have asymmetric risk tolerances.  All that this means in general is that a 10% loss is much more painful than the pleasure of a 10% gain in the minds of most investors.  Think about yourself in these terms.  What would you consider the offset to be equal when it comes to losing and earning money in the stock market?  Would you need the prospect of a 15% positive return (or 20%, 25% and so forth) to offset the possibility of losing 10% of your money in any one year?  Let’s look at the breakdown of the number of years that investors will experience a loss.  To be consistent with my first post, I am going to use the “bucket” of -3% to 2% and work down from there.  Here is the graph:

Returns Less than 2%

There are 61 years of stock market returns from the S&P 500 Index for the period 1957 to 2018.  If we look at the category of 1 year, stock market returns were 2% or less 38.7% of the time (17 years out of 61 years).  However, if we move to five-year and ten-year annualized returns, there were no observations in the -3% to -8%, -8% to -13%, or less than -13% “buckets”.  When looking at losing money by investing in the stock market, a long-term focus and investment strategy will balance out very negative return years and your portfolio is less likely to be worth significantly less after five or ten years.  Of course, there are no guarantees and perfect foresight is something that we do not have.  However, I believe that looking carefully at the historical data shows why it is important to not be so discouraged by years when the stock market goes down and even stays down for longer than just one year.  Hopefully these figures do provide you with more fortitude to resist the instinct to sell stocks when the stock market takes a deep decline if your investment horizon and financial goals are many, many years out into the future.

The final concept I would like to cover is standard deviation.  The term standard deviation comes up more often than not either in discussions with financial professionals during client meetings or is used a lot in the financial media.  There are many times when even the professionals use the term and explain things incorrectly, but we will save that conversation for another post.  Standard deviation is a statistical term that really is a measure of how far away stock market returns are from the mean (i.e. the average).  It is a concept related to volatility or dispersion.  So, the higher the number is, the more likely it is that stock market returns will have a wide range of returns in any given year.  Let’s first take a look at a graph to put things into context.  Here it is:

Standard Deviation

The chart is striking in terms of how much the standard deviation decreases as the time period increases.  A couple things to note.  First, I do not want to confuse you with a great deal of math or statistical jargon and calculations.  My point is not to obscure the main idea.  Second, the 25-year and 50-year numbers are just included only to cover the entire period of 1957 to 2018 for the S&P 500 Index.  These periods of time are not of much use to individual investors to consider their tolerance for risk and the right investments to include in their portfolios.  And, as one of the most famous economists of the 20th century, John Maynard Keynes, quipped:  “In the long run, we are all dead”.  My only point is that discussion of how the stock market has performed over 25 years or longer is just not relevant to how most individuals think.  It is nice to know but not very useful from a practical perspective.

The main item of interest from the graph above of standard deviation is that you can “lower” the risk of your portfolio just by lengthening your time horizon to make investment decisions on buying or selling stock.  For example, the standard deviation goes down 46.9% (to 8.95% from 16.87%) between one-year returns and three-year annualized returns.  Why do I use “lower”?  Well, the risk of your portfolio will stay constant over time and focusing on longer periods of time will not decrease the volatility per se.  However, most financial professionals tell their clients to not worry about day-to-day fluctuations in the stock market.  Plus, most Financial Advisors tell their clients to not get too upset when reviewing quarterly brokerage statements.  This advice is very good indeed.  However, I urge you to lengthen the period of your concern about volatility in further out into time.  My general guideline to the individuals that I assist in building financial portfolios, setting a unique risk tolerance, and planning for financial goals is to view even one year as short term akin to examining your quarterly brokerage statement.

Why?  If you are in what is termed the “wealth accumulation” stage of life (e.g. saving for retirement), what occurs on a yearly basis is of no concern in the grand scheme of things.  The better investment strategy is to consider three years as short term, five years as medium term, and ten years as the long term.  I think that even retirees can benefit with this type of shift.  Now please do not get me wrong.  I am not advising that anyone make absolutely no changes to his/her investment portfolio for one-year increments.  Rather, annual returns in the stock market vary so widely that it can lead you astray from building a long-term investing strategy that you can stick to when stock market returns inevitably decline (sometimes precipitously and by a large margin).  Note that all the academic theories, especially Modern Portfolio Theory (MPT), were built using an assumption of a one-year holding period for stocks (also bonds, cash, and other investments).  Most individual investors do not fall into the one-year holding period.  Therefore, it does not make much sense to overly focus on such a short time period.

Of course, the next thought and/or comment that comes up is “what if the stock market is too high and I should sell to avoid the downturn?”.  I will not deny that this instinct is very real and will never go away for individual investors.  In fact, a good deal of financial media television coverage and news publications are devoted to advising people on this very topic every single day.   It is termed “market timing”.  In the third and last article in this series on becoming a successful long-term investor, I am going to examine “market timing” with the same stock market data from the S&P 500 Index.  You will clearly see why trying to time the market and buy/sell or sell/buy at the right time is extremely difficult to do (despite what the financial pundits might have you believe given the daily commentary).

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