What's risk?

Every so often one runs across debates concerning "What's the best measure of risk".

>And you'd like to enter the debate?
Why not?

Suppose we wanted a numerical measure of risk which was determined by stock returns.
What properties would you like this risk number to have?
How about these properties?

1. If the returns were all doubled or tripled then risk should double or triple.
So, if all returns were positive, then multiplying these positive returns by 2 or 3 would increase risk.
2. If 10% were added to all returns, then risk should be unchanged.
So, if my returns were always larger than yours by some positive constant (like 10%), our risk should be the same.
3. Our numerical measure should assign a larger risk to these returns

than to these

If you'd like risk to have these properties, then I'd recommend:
 Risk = Standard Deviation

P.S. In the first chart, there is no uncertainty: the returns are 5%, 5%, 12%, repeated.
So if you wanted your definition to measure "uncertainty", you may not want Risk = Standard Deviation .