To what extent is the crosslinking necessarily a pure trade of tail risk for profit now, and to what extent an actual improvement? Is there an increased reserve requirement which still results in an increased profit but without increasing the tail risk?
Those questions depend on the specific details of the distributions, I think. And also on assumptions about how bad generalised insurance industry collapse is versus localised insurance industry collapse.
Interesting question. It is clear that the probability mass in excess of the reserves is equal in both distributions, yielding identical long-run numbers of industry-defaults-per-year, however the average magnitude of the unrecoverable losses is greater in the no-diversification model.
If you assume a linear cost function for the expected losses, and take the mean of the distribution past a variable reserve level, you will find a reserve level for a unified insurance agent which has the same expected loss-cost, a lower number of absolute industry-loss events, and a lower reserve requirement than the diversified case.
My Wolfram-fu fails me, but you would want to multiply the binomial PDF (or gaussian approximation) by x, and find the integral from y to 100 (or infinity) that is equal to the diverse expected loss, 1*10/200. For binomial distributions, y will be <90, so short answer, ‘yes’.
To what extent is the crosslinking necessarily a pure trade of tail risk for profit now, and to what extent an actual improvement? Is there an increased reserve requirement which still results in an increased profit but without increasing the tail risk?
Those questions depend on the specific details of the distributions, I think. And also on assumptions about how bad generalised insurance industry collapse is versus localised insurance industry collapse.
Interesting question. It is clear that the probability mass in excess of the reserves is equal in both distributions, yielding identical long-run numbers of industry-defaults-per-year, however the average magnitude of the unrecoverable losses is greater in the no-diversification model.
If you assume a linear cost function for the expected losses, and take the mean of the distribution past a variable reserve level, you will find a reserve level for a unified insurance agent which has the same expected loss-cost, a lower number of absolute industry-loss events, and a lower reserve requirement than the diversified case.
My Wolfram-fu fails me, but you would want to multiply the binomial PDF (or gaussian approximation) by x, and find the integral from y to 100 (or infinity) that is equal to the diverse expected loss, 1*10/200. For binomial distributions, y will be <90, so short answer, ‘yes’.