Have you ever been going about your day and an idea slams you in the side of your head like a bag of rusty hammers: “I should revisit that post I wrote a month ago on market efficiency!” I didn’t think so. It doesn’t happen to me either, especially after writing a post a month ago titled Are Markets Efficient? Well not until Eytan – yes the same weirdo who emailed me lessons on advanced lottery probability – hits me up AGAIN to tell me just how wrong my math is. We decided to create a weirdo-tag-team collaboration post revisiting those graphs on market volatility. OH I bet this is going to be fun.
So, I assumed – apparently I should never assume – that it was fairly obvious that charting S&P 500 market volatility based purely on the S&P 500 Real Price by Year would only indicate the changes in price, not the degree of volatility. But, as Eytan so helpfully pointed out, when you see the graph from the original post, you could be misled to think that volatility has increased in the past decade relative to the distant past.
I apparently didn’t point this out very clearly (thanks a lot, Eytan). So, what Eytan has so kindly done is convert the data using the
magic science of log functions so the ups and downs of the market are relative to each other in an absolute sense. I’m probably butchering this explanation up so I’ll just pass it to Eytan to give you the lowdown – and this time I’m not letting you skip the math for pretty pictures!
Hey guys, Eytan here. If you’ve ever looked at the price history of a market index, you may have noticed the following trends: stock prices tend to increase and stock prices have become increasingly volatile. The figure blow shows the historical prices of the S&P Index from 1871 to 2014:
We can see the sub-prime mortgage crisis. We can see the dot-com bubble. But can you see the recession of 1937? Can you see the great depression? Historically, we know the great depression had the most devastating effect on stock prices, but why does it barely register here?
To give you some numbers, during the period from January 1, 1929 to January 1, 1933, the great depression, stock markets fell from $345 to $130, a 62% change decline. On our chart above, this 62% decline only represents an 11% change along the vertical axis. During the sub-prime mortgage crises, prices dropped from $1673 on January 1, 2007 to $975 on January 1, 2009, a 42% decline. However, this 42% decline represents a 32% change along the vertical axis. In both cases, investors lost a crippling amount of money, but the recent recessions are over 3x as noticeable.
The reason is changes in stock prices are an absolute measure of a repeated relative gain. We can see this expressed mathematically with the ubiquitous formula relating the future value of money to the present value of money:
If we now compare the difference between the future value and the present value, we’ll see that it depends not only on the interest rate and the number of years, but also on the present value:
This dependence on the present value makes it very difficult to compare the effects of crashes on the stock market throughout history. However, if we revisit the formula for the future value of money, and apply a mathematical function called a logarithm, we can eliminate the dependence of the present value when calculating changes in the stock markets.
Lets begin by applying the logarithm to the future value of money equation above.
To verify we’re on the right track, lets measure the change Log(FV) – Log(PV) for various interest rates and portfolio sizes. If we’ve removed the dependence of the present value, the value Log(FV) – Log(PV) should only depend on interest rate:
We did it! Now, lets apply the logarithm to our initial data to the historical prices of the S&P and see if the recessions become more visible:
As you can see, we’ve illuminated the historical volatility in the market and have created an even-keel comparison of changes in the stock market.
You can find the original post here.
EFFICIENT MARKETS REVISITED
Are you guys still with me? I’ll be so mad if you fell asleep or skipped right to here. Let’s thank Eytan for that invigorating case study in logarithm functions – the crowd goes wild.
Now, let me revise those graphs I did in that earlier post to incorporate all this
black sorcery math.
This is the new and improved “inefficient” market view with all the volatility:
This is the new and improved “efficient” market view:
And let’s combine the two for a real clusterpuck of lines:
Again, it’s the same takeaway as last time: don’t concern yourself with market “inefficiencies” or volatility. Given enough time, the markets are very efficient. Focus on the red line, not the black ones. The red line is a simple visualization of the markets relentless efficiency. When you make intelligent investments for the long haul, your assets are bound to grow.
VOTING & WEIGHING MACHINES
This point is so important that I am going to restate it word-for-word from the previous post.
Benjamin Graham, the father of value investing, said it best when he stated:
“In the short run, the market is like a voting machine.
But in the long run, the market is like a weighing machine.“
What he meant by this is that in the short term, the markets tally up which companies are popular and unpopular in a given moment. But in the long term, the markets actually separate the wheat from the chaff and accept and discard companies based on their underlying economic and business strength.
Well, I hope you guys learned something about logarithm functions and how you can apply it in the real world. I do have one bonus graph for those of you who trooped it to the end. This is a graph incorporating recessions in the US economy back to 1871 with the graph used in this post on market volatility. The recessions are shaded in light red.
What I found super interesting – note how I said how I found super interesting – was the frequency of recessions have decreased over time. I wonder if this is related to US government providing the Federal Reserve with increased powers after the Great Depression. I guess a lot has been learned and a lot of Fed policy does help alleviate the frequency of recessions experienced in the distant past. Perhaps that’s another topic for a future post.
Disclaimer: Past performance is no guarantee of future performance. Conduct your own due diligence. Learn market history. Your mileage may vary.
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