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’.
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’.