If you read my blog regularly you know I am constantly preaching about diversification. Normally I am talking about incorporating multiple asset classes and style premiums together in order to increase return while reducing volatility. Many people also like to talk about including the length of your investment horizon as an additional diversifier. You hear these comments all the time, “You are young, so you can ride out the ups and downs.” Or, “You won’t need that money for a long time so it’s OK to own 100% stocks. It will eventually work out for you.” These comments imply that given enough time, risky assets will eventually give you the great returns you are hoping for.

These arguments are normally made by presenting data that as you increase your time horizon the standard deviation of compounding time-periods of volatility decreases over time. You might see one of these charts and notice that if an investor holds a risky investment for a long time period the range of possible outcomes of annualized returns is very small. In fact it can lead people to think that all they have to do is hold this risky investment for 30 years and they can’t lose. The problem with this augment, and the reason it is often thought of as an obvious fact, is that the above is absolutely true! Unfortunately it completely misses the point. As investors we are more concerned about total return instead of annualized return. While the standard deviation of compounded annualized returns decreases with time, the standard deviation of total return actually increases with time.

Using the random walk model (see note below), compounded rates of return and portfolio ending values are lognormally distributed and continuously compounded rates of return are normally distributed. The standard deviation of annualized continuously compounded rates of return decreases in proportion to the square root of the time horizon (Like we would have seen in the hypothetical chart above). The standard deviation of total continuously compounded returns increases in proportion to the square root of the time horizon. To visualize last part picture a flashlight facing a wall pointing up at a 45 degree angle. As the beam from the flashlight moves further up the wall it gets wider and wider, both up and down, from the 45 degree angle. This widening beam is your widening future possible total return outcomes as time increases.

From the random walk model we can calculate that if we put our money in stocks there is 30% chance we will lose money over a 1 year period. We can also calculate that over a 3 year period there is only a 19% chance of losing money by investing in stocks. Most people would stop there and conclude that our stock investment is less “risky” over the 3 year period than over a 1 year period. The reason this is faulty logic is that it treats all losses equally. A loss of one dollar is treated the same as losing 50% of your original investment. Obviously we have to take into account the direction and magnitude of a loss (and gain for that matter). In the same example we can calculate that it is 71 times more likely you will lose 50% of your money after 3 years than it is to lose 50% of your money after 1 year. Even though the chance is small we can’t ignore it.

You can see that when you take the magnitude of losses and gains into account it becomes a much more complicated issue to assign something more or less risky over time. To define this risk we must assign greater negative weights to losses of larger magnitude and greater positive weights to gains of larger magnitude. Then we need to somehow take the average of all these weighted probabilities to come up with a fair measure of risk. Then we need to account for the fact that losing, say, $500,000 is more of a bad thing than gaining $500,000 is of a good thing, especially if you only have $550,000 to begin with. To solve this problem we must use something called Utility Theory. In my next post I will discuss Utility Theory and how it relates to time diversification.


Note On The Random Walk Model

The random walk model assumes that returns are independent, identically distributed and have finite variance. All three of these assumptions are flawed.

Returns Are Independent

Actual return data appear to have positive serial correlation in short term returns normally referred to as “momentum”. There is also some evidence (although less convincing) of long term negative serial correlation normally referred to as “reversion to the mean”.

Returns Are Identically Distributed

The expected returns for risky assets like stocks are a premium demanded by investors as compensation for that risk, but there is no reason to think the risk in the financial markets is consistent over time. Standard deviations around the expected return are also not constant, but appear to go through periods of low volatility and periods of high volatility sometimes referred to as heteroskedasticity .

The Variance Of Returns Is Finite

Benoit Mendelbrot and other researchers believe that returns have a “stable Paretan” probability distribution. These distributions have infinite variance (fractals) with “fat tails” (kurtosis), which is more in line with the historical data. Very large losses and very large gains are more likely in these distributions than in a lognormal model. (If you are interested in learning more about this sort of thing I highly recommend the book “The Physics of Wall Street: A Brief History of Predicting the Unpredictable” by James Owen Weatherall)

We should use this model only as a rough approximation when discussing the actual market.