# 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:

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:

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