A Case of Ambiguous Definition

“Managers of government pension plans counter that they have longer investment horizons and can take greater risks. But most financial economists believe that the risks of stock investments grow, not shrink, with time.” – WSJ

This statement mentioned “risks” twice but they actually mean different things. Therefore the second sentence is correct by itself but cannot be used to reject the first one.

The first “risk” is timeless. The way it’s calculated always scales it down to 1 time unit, which is the time interval between any two data points in the sample.

Risk_1 = \sigma^2 = \frac{1}{N} \sum_{i=1}^{N}(R_i - \bar{R})^2

When “risk” is defined this way, a risky investment A and a less risky investment B have their returns look like this:

figure_1

The second “risk” is the same thing but gets scaled for N time units. It’s not how variance is defined but people use it because it has a practical interpolation (adjust for different time horizons).

Risk_2 = N * \sigma^2 = \sum_{i=1}^{N}(R_i - \bar{R})^2

Under this definition, the possible PnL paths for A and B look like this:

figure_2figure_3

A’s Monte Carlo result is wider than B, but both A and B’s “risk” by the second definition increases through time, while by the first definition never changed.

I have intentionally avoided mentioning time diversification because doing so would probably make things more confusing. For more details on this please see Chung, Smith and Wu (2009).

Roy

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A New Post

First of all, apologies to anyone who were expecting new posts or left a comment here but didn’t get a reply from me. There were quite a few changes in my life and I simply had to move my time and energy on blogging somewhere else. Now I’m trying to get back to it.

Because of reasons I will avoid writing about specific investment strategies, factor descriptors or particular stocks. I will write more about my thoughts (or thoughts stolen from people smarter than me) on generic techniques and theories. In an attempt to be rigorous (or more realistically, less sloppy), I will try to stay on one main track: hypothesis -> logical explanations -> supporting data or observations. This time I will use math and programming to make sense of things instead of just getting results on paper.

Roy