I lost about $20,000 in 2013 because I didn't notice that a company managing some of my retirement funds had helpfully reallocated them from 100% stocks into bonds and real estate, to "avoid risk". My parents are retired, and everyone advising them tells them to put most of their money in "safe" investments like bonds.
Vanguard has an online retirement advisor that asks you 11 questions, most of which are variations of "Will you panic and sell everything if the market falls?" (I was especially amused by one that added, "Tell us what you did, not what you think you would do.") I filled them out for a hypothetical 65-year-old who is retiring immediately and expects to live another 20 years, (answering the "Are you stupid?" questions with "No, I am not stupid.") It advised this hypothetical person to put his money 50% in bonds and 50% in stocks.
Vanguard also has a neat Monte Carlo retirement simulator that runs 5,000 simulations of an initial investment with a given stocks/bonds allocation using random (non-Markovian) historical data, and tells you in what percentage of these simulations you went broke. I put the same numbers into this retirement simulator and ran it with 100% stocks:
Then I ran the same simulation with the recommended 50% stocks, 50% bonds:
So, 100% stocks is 3 times as likely to last for 20 years as the recommended 50% stocks / 50% bonds. (For the Bayesians out there, it gives an odds ratio multiplier over the recommended allocation of 3.5.) It also has a much, much higher expected value. The only sense in which the 50% stocks / 50% bonds allocation provides less risk is that the worst cases--and we're talking about 5% of results--are a few percentage points less bad.
If you play around enough with the simulator, you can find situations in which the 50% bonds allocation gives a smaller chance of running out of money. They're all situations where you have a lot of money and not much time to spend it. One I found is: $1,000,000, spend $48,000 (2015 dollars) per year for 20 years. 50% stocks / 50% bonds allocation leads to 97% survival. For 100% stocks, it's only 90%.
Yet for most people, 100% stocks is far less risky, by which I mean their chances of running out of money are lower, by which I mean they're not 100%. The median retirement savings for Americans age 55 to 64 is $103,000. The recommended "safe" allocation guarantees quick bankruptcy for them.
(That's not to say the safe allocation at this moment is 100% stocks. I just moved money out of stocks. If you're willing to continually change your allocations, the situation is more complex. But the standard retirement fund allocation recommendations aren't based on that; I've never seen them change.)
So what do people mean by "risk"?
I think they can only mean either "variance" or "badness of worst case", and the differences in the worst cases are negligible. It seems people are giving investment advice not to minimize the chances of running out of money, but to minimize the chances of being surprised, even when the surprises are almost always good. Whether that's what the people receiving the advice want, I don't know.