If you have been following this blog lately, I have basically been saying that (a) investors try too hard to beat the market by attempting to predict it (b) despite their best efforts, most are doomed to fail.

Some of you may think that I’m too negative about active management (I am). Also, since I used he words “noisy” and “unpredictable” on a number of occasions to describe the zigs and zags of the stock market, you may also have the impression that I view the stock market as nothing more than a casino. That would be incorrect though. In this and the next blog, I will show why I have a high opinion of the stock market as a long-term investment tool. Instead of writing a three-page piece to bring my point across, I will show simply you a picture (below). The picture has an important message. You “read” the story by starting in the top row, left to right, then the bottom row, left to right.

What is this all about? Why is it important?

The picture shows how wealth invested in a stock is created if you **compound its returns **over time. It’s a bit like how you roll over your time deposit after it matures because you want to earn “interest on interest”. Except your time deposit interest rate is fixed but stock returns bump around randomly.

How to model this bumping around random nature of stock prices? Here’s where we turn to statisticians for help. Statisticians use probability distributions, including some clever complicated ones to fit or model stock returns. Fear not. I can tell you what you really need to know using the simplest model.

This is the elegant **binomial model**. “*Bi*” means two. So, I will ask you to imagine a stock whose price evolves each period with just two outcomes: either its price goes up or it comes down compared to the price in the previous period. Since up (u) is one outcome and down (d) is another, we have just modelled a stock using the binomial model! A useful way to think about this model is to imagine the stock price as an expanding tree. At each node of the tree, the stock price splits into an up branch with a certain probability (say p) and a down branch with probability 1-p (these probabilities must add up to one if you recall from high school :))

The picture that you saw simulates a binomial stock price going forward from 2 nodes, to 5 nodes, to 50 nodes and finally 500 nodes. Using the tree analogy, it “grows” the tree from a skinny one with just 2 nodes to a bushy one with 500 nodes. More importantly, the 2-node tree represents the short-term (no compounding yet) while the 500-node tree represents the long-term after many periods of compounding. Instead of plotting all the branches sticking out (ugly!), the picture tallies all the end-period wealth and presents their distribution as a nice, smooth curve. Notice this curve has a long right tail. So it is not quite the normal or bell-shaped curve that most people are familiar with, but its long right-tailed cousin (the exact name for it is the **log-normal distribution**).

What’s the big deal about the log-normal distribution? Three things. First, it shows your mean and median wealth after many periods of compounding (they are indicated by the vertical lines). Half of all your possible wealth will be below the median and half above it. So, the median wealth gives you a “feel” of where you are likely to end up after years of investing. That’s good to know.

Second, the median wealth is lower than the mean. This is a bit unfortunate but is always true of right-tailed distributions.

Third, making up for the fact that the median is less than the mean is the long tail. This feature implies that a long-term investor can look forward to “bonus” lucky draws of very handsome returns. This is good news. Despite all the talk about “black swans”, the reality is that long-term investors are more likely to meet “good” outliers or “white eagles”!

Studies using actual data shows that the log-normal distribution closely model the distribution of long-term wealth derived from the stock market. Hence, we can be confident about its predictions regarding the shape of long-term returns. Now you can see why I do not think that the stock market is merely a casino for speculators. All gambling is a negative sum game (i.e., the average net payoff is negative). True investment is different. In the long run, stocks generally pace the economy and trend upwards. It is most assuredly, a positive-sum game!. See “The Shape of Stock Returns II” coming up next..