## Introduction

Recent globalization has had a great impact on economic and financial activities in human society1,2,3,4,5,6,7,8,9,10,11. Until the 1980s, economic activities were mostly limited within regions, such as North America, Europe, and East Asia. Most personal investment activities were further limited to a single country because currency exchanges were also limited by high transaction fees4,7,8,12,13. Similarly, personal investments in stock markets were also highly limited due to high transaction fees for purchases and sales1,7,13,14,15. In the 1980s, these fixed transaction fees were lowered to almost negligible, and the transaction fees became mostly the quantity-based charge for frequent repeated investors4,7,12,13. These repeated investments are leveraged ten-twenty times larger than the transaction deposit. Thus the actual amount of investments (not the transaction deposit) may become larger than the total wealth of an investor. Therefore, the failure of an investment may cause bankruptcy because its debt may be larger than the whole wealth of the investor, although this probability is very small. Many financial funds (companies specializing in investment activities) had been established with the globalization of world economics since the 1980s1,4,14. The historical accounts and negative effects due to the changes in regulations are widely recognized in economics and business [for example, see1,3,5,16,17,18,19,20,21]. Thus, the basis of extremely numerous repeated investments with leverage became practical.

The recent developments in computer technology have had considerable impacts on the international economy1,6. The establishment of international computer networks had been introduced into stock markets and currency change. Together with globalization, the technical developments of computers and the establishment of international computer networks have induced important changes in economic and financial activities. Now, anybody can access any stock market or currency exchange by computer and make a transaction instantly from a distance, thus investment activities around the world being drastically increased.

Another facet of computerization is that stock market and currency exchange transactions can be completely automated by computer programing14,22,23. This automated investment makes it possible to repeat investments in a second interval. It is possible to make 60 transactions in a minute if the program simply states investments repeatedly. In practice, the program makes the investment decision (to send a transaction) when the price of a stock satisfies the condition set by the program. The total repetitions/hour may be ten or less. If the market is open for 10 hours, then the repeated investment occurs 100 times/day or 25,000 times/year. The maximum possible number of repetitions in investments is extremely high because of computerized investment. Established securities companies have shifted to adopt this computerized investment environment by hiring many specialist traders24,25. Many new investment funds or securities companies have also been established recently to specialize in this computerized global investment activity, e.g., long-term capital management (LTCM)2,15,21. The local operator-assisted environments of financial investments have thus shifted into the global computerized systems1.

In this paper, we consider a simple but forgotten feature of repeated investments. This aspect is mathematically trivial but has very profound effects on the investment strategy, once adopted. Traditionally, because the probability of bankruptcy for a one-time investment is nearly equal to zero, this one-time bankruptcy probability is usually treated as zero in current investment strategies. We here assume that this probability is very close to zero but not zero (a positive real number). We derive the mathematical relationship between the number of repetitions and overall bankruptcy in investments. We discuss the impact of this neglected aspect of repeated investments.

## Model and Results

In this repeated investment model, we suppose the situation in which the debts caused by leveraged investment may exceed the current wealth of the investor. The actual investments, e.g., FX investment, become at least ten to twenty times of transaction deposit by leverage and often easily surpass the whole asset (wealth)26,27, see also28,29,30. Assuming the number of repetitions of investment is N, the present wealth after N investments wN is a function of the initial wealth w0 and the multiplicative growth rate fi for i = 0, 1, …, (N − 1), such that $${w}_{N}={w}_{0}{f}_{0}{f}_{1}\ldots {f}_{(N-1)}$$. Note that fi is independent and random. We consider the case of fi = 0, i.e., $${w}_{i+1}={w}_{i+2}=\ldots ={w}_{N}=0$$. This means bankruptcy at the i-th investment. Let this probability be pb. In the presence of leveraged mechanisms, this value (pb) may be significantly small but still a positive number. Then, the total probability of bankruptcy with N-times repeated investments PB is given by

$${P}_{B}=1-{(1-{p}_{b})}^{N}$$
(1)

By transforming Eq. (1), we obtain (Fig. 1):

$$N=\frac{\log (1-{P}_{B})}{\log (1-{p}_{b})}$$
(2)

Figure 1 shows that the overall probability of bankruptcy always converges to unity. It also shows that for a given pb, there is a threshold of repetition N, where PB reaches 0.99 or any given probability (Fig. 1). Expressing pb = 10m and N = 10n, we obtain (Fig. 2):

$$n={\log }_{10}\frac{\log (1-{P}_{B})}{\log (1-{10}^{-m})}$$
(3)

Because pb = m is close to zero, m1. Using the first term of Taylor expansion, we can approximate $$\log (1-{p}_{b})=\,\log (1-{10}^{-m})\cong -{10}^{-m}$$. We obtain

$$n={\log }_{10}(\frac{\log (1-{P}_{B})}{-{10}^{-m}})=m+{\log }_{10}(\log \,\frac{1}{(1-{P}_{B})}).$$
(4)

Therefore, we can express (Fig. 2):

$$n=m+\alpha ({P}_{B}),$$
(5)

where $$\alpha ({P}_{B})={\log }_{10}(\log \,\frac{1}{(1-{P}_{B})})$$ is a measure of bankruptcy in repeated investments with leverage. For example, when PB = 0.99, α = 0.66 (Fig. 2). Furthermore, Eq. (4) is approximated well by a linear relation (Fig. 2):

$$n=m+1.25{P}_{B}-0.79,\,{\rm{for}}\,0.3\le {P}_{B}\le 0.99$$
(6)

Therefore, we can obtain the following inequality:

$$n\ll m+1,\,{\rm{for}}\,\,{P}_{B}\le 0.99$$
(7)

Thus, when the one-time risk of bankruptcy is pb = 10m, we almost certainly face bankruptcy by repeating N = 10m+1 times. In precise, this repetition is approximately 99.995% bankruptcy.

In the case where the probability of bankruptcy, pbi, depends on each investment i, the total probability PB is given by

$${P}_{B}=1-(1-{p}_{b1})(1-{p}_{b2})(1-{p}_{b3})\cdots .$$
(8)

Neglecting the second order terms in pbi, we obtain

$${P}_{B}\approx 1-{(1-\overline{{p}_{b}})}^{N},$$
(9)

where

$$\overline{{p}_{b}}=\frac{1}{N}({p}_{b1}+{p}_{b2}+\cdots ),$$
(10)

is the average of pbi. Therefore, this case is essentially the same as the above, except that pb is replaced with $$\overline{{p}_{b}}$$. It is important to remark that pb or $$\overline{{p}_{b}}$$ is very tiny but a positive number.

## Discussion

The current results indicate two important findings. First, any extremely small risk of bankruptcy can be accumulated to almost certainty (probability one) under repeated investments. Second, the number of repetitions leading to bankruptcy (N or n) is almost determined by the tiny probability of bankruptcy at each investment (pb or m). For example, if the risk of bankruptcy is 10m (m is large), 10m+1 repetition almost certainly leads to bankruptcy (ca. 99.995%), and even 10m repetitions have more than 50% bankruptcy (ca. 63.21%). This quantitative measure of bankruptcy α implies that computerized (programmed) repeated investment is a purely risky gamble, rather than a safe investment, even if a single trial is almost certain to win.