Efficient Markets Revisited

Efficient Markets RevisitedHave 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.



The data is the same as before, pulled from multpl based on Robert Shiller’s stock market data. Let’s cut to the chase.


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:

Efficient Markets Revisited

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:

Efficient Markets Revisited log 1

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:

Efficient Markets Revisited log 2

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.

Efficient Markets Revisited log 3

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:

Efficient Markets Revisited log 4

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:

Efficient Markets Revisited

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.


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:

Efficient Markets Revisited Inefficient Markets

This is the new and improved “efficient” market view:

Efficient Markets Revisited Efficient

And let’s combine the two for a real clusterpuck of lines:

Efficient Markets Revisited together

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.


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.

Efficient Markets Revisited Recessions

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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19 thoughts on “Efficient Markets Revisited

  1. I think the rule of thumb is “don’t go too mathy” 😛 I love a good log function as much as the next weirdo (I actually use them on a daily basis…) 🙂

    1. Haha I’m glad you’re a PhD scientist so you at least nerd out in all this stuff. Eytan and I are really hitting the core demographics with this post – to keep readers like you nerded out and happy 😀

      Hopefully you gleaned something informative/entertaining from this fairly mathy post 😛

  2. Good stuff Steve and Etyan. If we all just focused on the red line, nobody would complain about the market. Unfortunately, investors do the wrong things at the black lines, which leads to them always underperforming the market.

    Also, looking at the last graph, it doesn’t seem as though we are on the cusp of another recession, does it?

    1. One of the most important things I’ve learned about investing is not to try to predict the future.

      Case in point: people have been predicting a market correction/crash since 2011. It hasn’t occurred yet. If you had followed those predictions back in 2014 and stayed out of the market, you would have missed out on all the dividends and cap gains.

      So except for the few rare people who truly are market makers, we mere mortals are best served by buying and holding essentially forever and not concerning ourselves with corrections at all.

      Corrections have happened in the past and they will happen in the future. The only truth is no one knows when the next one will come.

  3. The log plot of long-term market growth is one of my favourite charts.

    To nit-pick, I’m not sure I’d call that perfectly flat trendline the efficient market and everything else an inefficiency — efficient markets can still have volatility. That line would be the perfectly prescient efficient market, looking hundreds of years into the future, forecasting growth, and discounting back.

    1. Hehe I do like the feedback! As a PhD, you must understand the supreme difficulty of A) breaking down complex ideas into simple ideas and B) the difficulty of vocabulary and how words are perceived.

      I totally agree that the linear trend line doesn’t perfectly represent what an efficient market is. I think I was simply trying to say that from the oldest data point to the newest data point (and everything else in between), the market has been amazingly efficient at growing and compounding your invested money. And thus, this was a peek into the idea of the markets being extremely efficient in the long run.

      Wouldn’t there be problems with trying to forecast growth into the future simply because forecasting in and of itself is a problem because they are only assumptions? I mean, past performance is no guarantee of future performance, but at least there are data points to look back on in history and use those to come up with……

      I was going to write a question about forecasting but I think I got caught in an endless academic debate loop with myself on the validity of forecasting vs. using historical data points as a reference for what to expect from the markets in general. Then all the Taleb readings came crashing in and I started doubting whether +100 years of market data is even enough to infer any kind of guess as to what the market “naturally” returns. Oh Taleb… Fooled by Randomness indeed.

    1. I think judging by the amount of comments left on this post vs other less mathy posts, I think you are completely correct to point out that it’s a real snooze fest for those who don’t have the patience 😛

  4. I think that a lot of what you see ia because of all the changes that have taken place after each recession or depression. For instance, it was no problem to finance almost the whole purchase on margin during the 1920’s. If the market went south, there was no point for the owner to pay the margin call because there was no vested interest in keeping the stock. Now rules are way more stringent, maybe not perfect, but it keeps the investor holding some of the bag and allows for more chance to actually pay the margin call. Same for the increases in FDIC insurance earlier this century. There was no insurance when my parents were born. There parents lost everything with the runs on the banks.

    Keep cranking,

    Robert the DividendDreamer

    1. Hmm so the the markets have “improved” and sort-of-kind-of learned their lessons with each recession? I think that’s a good hypothesis based on the what the data and historical records seem to show on that last graph – progress does exist!

  5. Are you guys still with me? I’ll be so mad if you you fell asleep or skipped right to here. Let’s thank Eytan for that invigorating case study in logarithm functions – *the crowd goes wild*.

    Hilarious! If all teachers were like you, 95% of kids would be able to grasp calculus. I know I’ve read a great post when I’ve not only learned something, but also come away smiling.

    1. Thank you for gracing us with your presence! Almost all of the credit goes to Eytan (he’s the aerospace engineer), I just make things pretty to look at 😛

      I’m the Steve Jobs to his Steve Wozniak 😀

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