$$m$$ Total invested capital
$$P$$ Rate of investment ($/mo put to investing) $$r$$ Rate of return on invested capital # A brief history of the Dow Jones Industrial Average (DJIA) and the Standard and Poor's 500 (S&P500) index: a group of stocks which are reported on as an indicator of what stock prices are doing in general or in a specific sector of the economy. Weighting schemes vary between indicies. The oldest is the Dow Jones average of the "top" 30 companies in the US going back to the late 1800s. The more modern (and some would say better) is the Standard and Poor's 500 or S&P500. ## Why indicies? A single stock, any single stock, is more volatile in price than a collection of similar stocks because the interstock volatility is averaged out. The total market volatility is not and the indexes of a market (large groups of stocks which are accepted proxies for the whole market or for specific industries within the market) reflect that volatility. To explain why the value investing strategy is not gambling, we need to look at indicies of the US stock market. There are two indicies which purport to model the US markets and which are often reported in the news as evidence of the current economic fortunes of the country - the Dow Jones Industrial Average (the Dow) and the Standard and Poors 500 (S&P500). ## Of historical note Given we are talking about the past, I believe some brief historical context is relevant. The Dow was created by Charles Dow (of the Dow Jones Trading Company and the Wall Street Journal) and his business partner and statistician Edward Jones in 1896. It is the oldest index in the US, giving it historical significance but its antiquated design makes it statistically questionable. Given how well it matches indexes which statisticians find superior, I think it captures the market as intended. It is currently a list of 30 companies which have changed over time and whose individual values contribute to the index directly. Of the original 12 companies which composed the index only 2 remained on the index into the modern era (my lifetime): Tennessee Coal, Iron, and Railroad Company and General Electric which were removed from the Dow in 1991 and 2018, respectively. The S&P 500 is a statistically superior index to the Dow. Standard & Poor's is a financial services company founded by Henry Varnum Poor in the 1860. It has published various stock indicies since 1923 but the first S&P500 was published in March, 1957. ## How should we view an index? The most obvious and common way to display an index is value vs time [see top panel] and it is fairly unhelpful, particularly over long timescales or if we want to compare two or more indicies. Far more helpful is to divide by (or normalize by, in the parlance of engineers) the value of the index today. [see middle panel] This removes the arbitrary nature of the absolute value of the index which we do not care about. We could have normalized by the value at any date; as long as we used the same date for all the indicies, we would arrive at the same plot. Normalizing by today happens to be the easiest choice because every index we might consider is active today. They were not all active at the time the Dow came into existence. Finally, since we are interested in percentage changes which may occur on any order of magnitude, we are interested in a logarithmic y axis. This view clearly shows the major downturns of the 123 year history of the Dow. [see bottom panel] [Caption] [Top] Dow and S&P500 over time from 1896 and 1957, respectively, up to July 2019. [Middle] Dow and S&P500 over time normalized to today. [Bottom] Dow and S&P500 over time normalized to today in logarithmic scale. The Dow has survived as a solid investment choice through a ghastly century of human depravity and economic misfortunes which appeared to be ushering in the end of civilization on more than one ocassion. In its 123 year history, it has watched two world wars kill 2-3% (each) of the world population and destroy unquantifiable amounts of wealth, social revolutions in the US and abroad which ushered in a new era of human rights, dozens of genocides. It has watched nations feud with nuclear weapons for decades and the collapse of several world powers and empires. By many metrics, the Dow has watched the most dynamic, volatile century of the human experience and, over long timescales, it has always survived. Most of the news of the last century did not even register on the Dow. Investing in broadly diversified stocks for the long term is not akin to gambling; it is akin to waiting on a reliable old friend who ocassionally loses his way but always makes it home in time for dinner. The easy and highly flawed but long term way to justify investing in the stock market is to say "In 1925, the Dow was worth a little over 100. Today it's worth 26900. That's a return of 5.95% every year for almost a century!" The problem with this analysis is its dependence on the small noise level fluctuations of the price at the random time selected to buy in. Any investment looks good when the buy in point is selected retrospectively. Depending on when you bought in between 1925 and 1930, you could have computed returns from 4.4% to 6.0% (and over a century those differences make for serious discrepancies). What I want is a model of the Dow which reflects the value of an investment if I had bought a little bit of the index everyday. How would my investment perform if I just bought the Dow? We already have an equation to show how a steady investment performs over time and we want to be able to compare the index to that equation. $$\frac{m(t)}{P} = \frac{1}{r} \left( e^{rt} - 1\right)$$ Now only one one question remains; at what point do we start? The start point with this model is still important but it is less important than in the two point model. If we are looking at one index, the beginning of the index is the logical place. If we are comparing multiple indicies, then we need to start them at the same place so we begin with the start date of the most recent index. [Caption] [Top] Returns of the Dow with continual investment 1896-2019. Continuous investment equation shown with 0.25% increments. At 0% returns,$X per year is $123X, but at the 5.5% return the Dow has averaged in that time,$X per year is worth $17,500X. [Bottom] Returns on the Dow and S&P 500 since the inception of the S&P 500 in 1957. Continuous investment equation shown with 0.33% increments. At 0% returns,$X per year is $62X, but at the 7% return the Dow and S&P500 have averaged in that time,$X per year is worth $$\approx$$ $1,050X. And finally, it may be good to plot these returns in log scale. It is less clear whether log or linear scale is better for visualizing the returns; log scale shows the early returns more clearly and the linear scale shows the late returns better. The graphs below show the identical data over the same range but in a log y axis. [Caption] [Top] Returns of the Dow with continual investment 1896-2019 in log scale. Continuous investment equation shown with 0.25% increments. [Bottom] Returns on the Dow and S&P 500 since the inception of the S&P 500 in 1957 in log scale. Continuous investment equation shown with 0.33% increments. It is important to remember these are not impossible timescales. Many working age adults today are nearly this old, 23% of the workforce is over 55 years old and the US life expectancy is now about 80, higher if you disregard accidents and protracted suicide by lifestyle (smoking, drinking, diet, and physical exercise). The average investing career is therefore 60+ years, the current age of the S&P500. ## What is neglected here? In this analysis we have neglected two major things, inflation and dividends. To a large degree I do not believe that inflation undermines my thesis at all. If your grandfather had$100 in 1910 and he gave it to you in 1980, it would be worth $100 today but that$100 would buy a lot less of any commodity which has not also changed in value significantly. It would be worth millions if it had been invested even if that investment only matched the market. Inflation or not, the hypothetical choice is millions (todays dollars) or \$100 (today's dollars). Inflation makes the numbers a little less attractive but they make holding onto cash less attractive by the same amount so 6 of one, half dozen of the other.

Perhaps more importantly, we neglected dividends. Companies enrich their shareholders in two ways, they appreciate (or grow in price/value, what we looked at here) and they pay dividends. Dividends are not included here for pragmatic reasons - I cannot find historical data on the dividends of the Dow going back to 1896. However, any dividend paid by any company on either index only serves to strengthen my claim - you should be saving and investing.