Why are firms taken to be profit-maximizing? Shouldn't that make them risk-neutral? Intro texts normally explain that insurance firms (casinos, etc.) "work" by diversifying risk from many clients. Unsaid, then, seems to be that risk is bad for both firm and client.

But why should a firm even need such diversification, if they merely seek to maximize profits? Likewise why would a firm ever itself be a purchaser of insurance, or spend resources trying to "manage" risk? This seems to be a rather deep oddity in the basic picture we're taught from square one.
Why would a [risk-neutral] firm need to diversify if all it wanted to do was maximise profit?

Suppose there is a risk-neutral firm that has two strategies it could follow: risky and safe. The safe strategy (e.g. holding a diversified portfolio) yields a profit of $1$ every year forever. The risky strategy (e.g. not diversifying) yields a payoff of $2$ with (independent) probability $x$ each year. With probability $1-x$ a risk-taking firm goes bankrupt and earns no more money ever again. The firm discounts the future at rate $\delta$.

The lifetime profit of the safe strategy is
$$\sum_{t=0}^\infty\delta^t=\frac{1}{1-\delta}.$$
The lifetime profit from the risky strategy
$$\sum_{t=0}^\infty 2x^t\delta^t=\frac{2}{1-x\delta}.$$
The safe strategy yields a higher lifetime profit if
$$x&lt;\frac{2\delta-1}{\delta}\iff \delta&gt;\frac{1}{2-x}.$$

In words, even though


the firm is risk-neutral and
the one-shot profit from the risky option (e.g. not diversifying) is higher,


the long-run profit from the safe option (e.g. diversification) can be greater over the lifetime of the firm.

In this example, the safe option maximises lifetime profit when (a) the probability of risky option leading to bankruptcy (and hence the loss of all future profit) is high or (b) the firm values the future highly relative to the present.