Would potential immortals be risk-averse?

Anti-aging drugs appear closer on the horizon, but how would a fountain of youth change our behavior? Lawrence Solum asks whether potential immortals would be afraid to ever put their lives at risk:

Given eternally long lives, the real issue concerns taking risks...So the question becomes, "Would it be rational to engage in regular low-risk behaviors (like daily walking), given that over the course of an eternal life, death would be the almost certain consequence?" One might imagine that in the beginning, such behaviors would continue, but that over time one would begin to realize that the odds were catching up with friends, family, and co-workers. Would the loss of eternal life really be a greater cost than the loss of the current human span of several decades? If the answer to this question were yes, then perhaps most humans would began to avoid risk. A very cautious approach to life might add tens of thousands of years to one's anticipated life span. I have an image of fine restaraunts serving only minced food to avoid the small (but statistically significant) risk of choking. No roller-skating, no skiing, no contact sports, no flying on airplances, no boating, no swimming. Would the eternal life lived to minimize risk be a recognizably human life?

Update: An astute reader notes that in contemporary vampire fiction, ancient vampires are generally potrayed as extremely risk averse--employing proxies when personal action in accord with their conception of the good (the bad?) would involve a significant risk to their immortality.

A related question is whether immortals would be less ambitious, since they might always feel they could accomplish their goals in a more distant future. As long as we are citing fiction, I recall seeing a television show about immortal beings. They were content to remain homeless and spent most of their time sitting around a campfire and talking. They accumulated few possessions. They never feared such a course of action would lead to death, and they always held the option of trying to do more.

Solum's query was prompted by a Volokh Conspiracy post of mine. I asked the different question of whether an immortal is necessarily a murderer with a probability approaching one, given the recurring risk of accidents.

Addendum: Read the commentary of Randall Parker.