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 .

>Okay, so how do YOU define risk?
I suspect that "risk" is a personal thing (which may explain why there's such a debate over how to measure it).
After all, there are plenty of financial considerations that are associated with "risk", such as:
Sortino Ratios & Downside Risk   Value at Risk   Drawdown Risk   Omega   the Stutzer Index   All Time Periods   ...

>Yeah, yeah, but how do YOU define risk?
Did I mention that it's a personal thing? I've been retired for some fifteen years and MY favourite is:

1. We assume a portfolio with a certain composition (like 25% each of large cap growth & value + 25% small cap value + 25% gov't long bonds).
2. We start with a $1M portfolio and withdraw 5% annually.
3. Our withdrawals increase annually by 3%.
4. We rebalance yearly to maintain the prescribed allocation.
5. We select, at random, annual returns for each asset from historical data from year 1928 to the current year.
6. We apply these returns to our portfolio - and repeat for each of N years.
7. We repeat steps 5 and 6 a jillion times ... a la Monte Carlo.
8. We calculate the percentage of times that our portfolio failed to survive for N years.
9. We define:

   Risk = Percentage of Portfolios that failed to Survive for N years