Why the egoist ought to save the drowning child

A child is drowning. Your cost for saving the child is negligible (let's say just an expensive pair of shoes, or better yet, no cost at all). Are you morally obligated to save the child? At no cost at all, we would want to say 'Sure, it doesn't cost you anything but to not save the child costs the child her life, so employing even the least demanding moral philosophy would require you to save the child, right?' Well, perhaps not- An even less-demanding moral philosophy can be imagined: egoism, which requires the actor to think of no one but herself, not even when the cost to the actor is nothing.

Obligation is a funny word, and in the context of egoism, it's not clear to me that an egoist can even conceive of the concept 'obligation.' (This is a discussion unto itself, so let's put it aside and rephrase the question). If moral obligation is doing what is required of your moral philosophy, and rational egoism requires we do only but always the option that benefits us most, then would an act that increases our 'confidence' or 'self-esteem' enough to continue being rational, cold and calculating in future decisions
be morally required?

I would argue, 'yeah, sure, of course.' Look, egoists can help themselves to the 'egoism' but the 'rationalism' comes at a price. Externally, humans do not behave rationally all the time (that is, they don't always behave in a manner that best matches up their ends with the most efficient mean). If it can be shown that a self-proclaimed egoist often fails to live up to what seems to be a rather emotionally-demanding standard (it's simply not fun to see children drowning, or being ostracized, or think of oneself as 'a drowning-child allower), wouldn't it be in her best interest to maintain a self-image that allows her to avoid such negative emotion?

This study suggest that, to some extent, we are in fact less than altruistic precisely because we think of ourselves as altruistic. If this is true, then it seems in the rational egoist's self-interest to do the occasional good deed (like saving the child) not only to avoid negative reputation, but because a healthy self-esteem allows for callousness.