Ah, I believe I understand now. No, I’m afraid this risk is not insurable.
Any insurer would have to guarantee some share of the 1.05x EV per toss, call that share itself X. The insurer would keep the remainder of the EV as premium, call that Y.
X and Y are both positive so it seems at first like you should be able to underwrite this. However, the math will not work out unless you change the dynamics of the model in some way.
The fundamental problem is that this is a model for a sequence of N events, and X (and therefore Y) are exponential functions of N. After some finite N, it’s only the insurer’s most recent guarantee that matters to the total payoffs. No previous events are consequential; the brute force of exponential math says the exponent alone dominates.
So we can just think in terms of x^N. At some point the insurer must pay out x^N in losses from the previous x^(N-1) in gains.
In other words, regardless of the premium charged, or the number of individuals whose risk is pooled, this individual’s status as an insurer doesn’t give them any special exemption from exponential reality that prevents individuals in general from remaining solvent in the limit of this model.
(I haven’t totally worked through the outcome table for the author’s proposed solution— I’d encourage you to do that if you think the solution might be flawed. But this does indeed seem to be a situation where any individual who attempts to capture the EV will fail, and only unconditional sharing can succeed.)
There's nothing magical about the pooling of risk and reward being administered by an entity which is an insurance agent that makes the payoff matrix look different in this model. Yes, this includes the fact that sometimes the syndicate returns a little more to its members than was put in and sometimes a little less depending on who flipped which coin toss.
All of which is moot to my original point which was that the OP's original argument that the mainstream economics profession is focused purely on expected value with no concept of risk is laughably wrong.
They're in fact a few steps ahead of him, because they also assess pooling of earnings in terms of moral hazard and adverse selection, instead of naively assuming that expected return and downside risks are evenly distributed across the population and invariant with respect to wealth pooling. Adverse selection is the actual reason private insurers are unlikely to take on the burden of insuring things like unemployment (people that find it easy to find employment and have lots of savings will rationally avoid participating unless its compulsory, which means more people wanting to claim on it than pay in), but of course introducing variation in likelihood of payoffs to the model leads to potentially very different outcomes...
Any insurer would have to guarantee some share of the 1.05x EV per toss, call that share itself X. The insurer would keep the remainder of the EV as premium, call that Y.
X and Y are both positive so it seems at first like you should be able to underwrite this. However, the math will not work out unless you change the dynamics of the model in some way.
The fundamental problem is that this is a model for a sequence of N events, and X (and therefore Y) are exponential functions of N. After some finite N, it’s only the insurer’s most recent guarantee that matters to the total payoffs. No previous events are consequential; the brute force of exponential math says the exponent alone dominates.
So we can just think in terms of x^N. At some point the insurer must pay out x^N in losses from the previous x^(N-1) in gains.
In other words, regardless of the premium charged, or the number of individuals whose risk is pooled, this individual’s status as an insurer doesn’t give them any special exemption from exponential reality that prevents individuals in general from remaining solvent in the limit of this model.
(I haven’t totally worked through the outcome table for the author’s proposed solution— I’d encourage you to do that if you think the solution might be flawed. But this does indeed seem to be a situation where any individual who attempts to capture the EV will fail, and only unconditional sharing can succeed.)