Ruin Probabilities for Strategies with Asymmetric Risk

We analyze sequential investment strategies that expose capital to asymmetric payoff structures--scenarios in which losses are small and frequent, while gains are large but rare. This setup generalizes the classical gambler's ruin problem, traditionally framed as a symmetric game, into a framework for modeling repeated financial decisions under uncertainty. Payoff asymmetry is common in domains such as venture capital, tail-risk hedging, and derivative strategies. We consider cases where each investment has positive, zero, or negative expected return, and derive analytic results for ruin probabilities, expected final wealth, and game duration. Our findings show that increasing asymmetry--higher potential rewards but lower success probability--raises the likelihood of ruin in positive-return settings and mitigates it when returns are negative. For zero-return strategies, we establish bounds on ruin probabilities and show that convergence to terminal outcomes is faster when payoffs are skewed. The results have implications for portfolio risk management and capital allocation in repeated-risk environments.