Bankruptcy is an inevitable fate of repeated investments with leverage

Due to the globalization and computerization of financial and economic activities, numerous repetitive leveraged investments have become possible in stock markets and currency exchanges. In reality, repeated leveraged investments up to 100 times/day are possible via online access. With computer-aided programs, this repetition number may easily increase 1000 times/day. The possibility of bankruptcy in repeated leveraged investments has never been considered in actual practices because the probability of bankruptcy in a single investment trial is almost negligible. Here, we show that the extremely numerous repetitions have a considerable chance of bankruptcy overall, even if the probability of bankruptcy for a single investment is extremely close to zero. The exact relationship between the repetitions and the probability of bankruptcy is approximated well by n(0.63) = m, where 10ⁿ is the number of repetitions, 10^-m is the bankruptcy probability of a single investment, and n(0.63) is the 63% chance of bankruptcy. Thus, extremely rare events can always lead to bankruptcy in continuously repeated investment, even if the possibility of such an event is almost null. We suggest that the avoidance measure of bankruptcy is necessary in numerous repeated investments even if a single trial is almost certain to win.