Risk of ruin
Risk of ruin is a concept in gambling, insurance, and finance relating to the likelihood of losing all one's investment capital or extinguishing one's bankroll below the minimum for further play. For instance, if someone bets all their money on a simple coin toss, the risk of ruin is 50%. In a multiple-bet scenario, risk of ruin accumulates with the number of bets: each play increases the risk, and persistent play ultimately yields the stochastic certainty of gambler's ruin.
Finance
Risk of ruin for investors
Two leading strategies for minimising the risk of ruin are diversification and hedging/portfolio optimization. An investor who pursues diversification will try to own a broad range of assets – they might own a mix of shares, bonds, real estate and liquid assets like cash and gold. The portfolios of bonds and shares might themselves be split over different markets – for example a highly diverse investor might like to own shares on the LSE, the NYSE and various other bourses. So even if there is a major crash affecting the shares on any one exchange, only a part of the investors holdings should suffer losses. Protecting from risk of ruin by diversification became more challenging after the 2008 financial crisis – at various periods during the crises, until it was stabilised in mid-2009, there were periods when asset classes correlated in all global regions.For example, there were times when stocks and bonds fell at once – normally when stocks fall in value, bonds will rise, and vice versa. Other strategies for minimising risk of ruin include carefully controlling the use of leverage and exposure to assets that have unlimited loss when things go wrong
The probability of ruin is approximately
where
for a random walk with a starting value of s, and at every iterative step, is moved by a normal distribution having mean μ and standard deviation σ and failure occurs if it reaches 0 or a negative value. For example, with a starting value of 10, at each iteration, a Gaussian random variable having mean 0.1 and standard deviation 1 is added to the value from the previous iteration. In this formula, s is 10, σ is 1, μ is 0.1, and so r is the square root of 1.01, or about 1.005. The mean of the distribution added to the previous value every time is positive, but not nearly as large as the standard deviation, so there is a risk of it falling to negative values before taking off indefinitely toward positive infinity. This formula predicts a probability of failure using these parameters of about 0.1371, or a 13.71% risk of ruin. This approximation becomes more accurate when the number of steps typically expected for ruin to occur, if it occurs, becomes larger; it is not very accurate if the very first step could make or break it. This is because it is an exact solution if the random variable added at each step is not a Gaussian random variable but rather a binomial random variable with parameter n=2. However, repeatedly adding a random variable that is not distributed by a Gaussian distribution into a running sum in this way asymptotically becomes indistinguishable from adding Gaussian distributed random variables, by the law of large numbers.