# You only live once

You only live once, but if you do it right, once is enough.

— Mae West

Let’s say that you are considering new investment opportunities for your company and that the sales department has guesstimated that the market for one of your products will most likely grow by a little less than 5 % per year. You then observe that the product already has a substantial market and that this in fifteen years’ time nearly will be doubled:

Building a new plant to accommodate this market growth will be a large investment so you find that more information about the probability distribution for the products future sales is needed. Your sales department then “estimates” the market yearly growth to have a mean close to zero and a lower quartile of minus 5 % and an upper quartile of plus 7 %.

Even with no market growth the investment is a tempting one since the market already is substantial and there is always a probability of increased market shares.

As quartiles are given, you rightly calculate that there will be a 25 % probability that the growth will be above 7 %, but also that there will be a 25 % probability that it can be below minus 5 %. At the face of it, and with you being not too risk averse, this looks as a gamble worth taking.

Then you are informed that the distribution will be heavily left skewed – opening for considerable downside risk. In fact it turns out that it looks like this:

A little alarmed you order the sales department to come up with a Monte Carlo simulation giving a better view of the future possible paths of the market development.

The return with the graph below giving the paths for the first ten runs in the simulation with the blue line giving average value and the green and red the 90 % and 10 % limits of the one thousand simulated outcomes:

The blue line is the yearly ensemble averages^{1}; that is the time series of average of outcomes. The series shows a small decline in market size but not at an alarming rate. The sales department’s advice is to go for the investment and try to conquer market shares.

*You then note that the ensemble average implies that you are able jump from path to path and since each is a different realization of the future that will not be possible – you only live once!*

You again call the sales department asking them to calculate each paths average growth rate (over time) – using their geometric mean – and report the average of these averages to you. When you plot both the ensemble and the time averages you find quite a large difference between them:

The time average shows a much larger market decline than the ensemble average.

*It can be shown that the ensemble average always will overestimate (Peters, 2010) the growth and thus can falsely lead to wrong conclusions about the market development.*

If we look at the distribution of path end values we find that the lower quartile is 64 and the upper quartile is 118 with a median of 89:

It thus turns out that the process behind the market development is non-ergodic^{2} or non-stationary^{3}. In the ergodic case both the ensemble and time averages would have been equal and the problem above would not have appeared.

The investment decision that at first glance looked a simple one is now more complicated and can (should) not be decided based on market development alone.

*Since uncertainty increases the further we look into the future, we should never assume that we have ergodic situations. The implication is that in valuation or M&A analysis we should never use an “ensemble average” in the calculations, but always do a full simulation following each time path!*

### References

Peters, O. (2010). Optimal leverage from non-ergodicity. Quantitative Finance, doi:10.1080/14697688.2010.513338

### Endnotes

- A set of multiple predictions that is all valid at the same time. The term “ensemble” is often used in physics and physics-influenced literature. In probability theory literature the term probability space is more prevalent.
An ensemble provides reliable information on forecast uncertainties (e.g., probabilities) from the spread (diversity) amongst ensemble members.

Also see: Ensemble forecasting; a numerical prediction method that is used to attempt to generate a representative sample of the possible future states of dynamic systems. Ensemble forecasting is a form of Monte Carlo analysis: multiple numerical predictions are conducted using slightly different initial conditions that are all plausible given the past and current set of observations. Often used in weather forecasting. [↩]

- The term ergodic is used to describe dynamical systems which have the same behavior averaged over time as averaged over space. [↩]
- Stationarity is a necessary, but not sufficient, condition for ergodicity. [↩]