Risk Bounds in Finance

Probability · Axiom Academy

REAL WORLD Risk Bounds in Finance How probability inequalities protect investors and stabilize markets The Portfolio Manager's Dilemma You're managing a 100 million investment portfolio. Your client asks: "What's the worst-case scenario for my investment?" You can't predict the exact future returns, but you know the portfolio's historical performance: Without knowing the exact probability distribution of returns, how can you give your client confidence about their risk exposure? This is where probability inequalities become invaluable tools for risk management. Financial regulators and risk managers use three powerful probability inequalities to bound risk. Let's explore how they work with your portfolio: What happens when you increase the volatility from 15% to 25%? Three fundamental probability inequalities provide different levels of protection: Works for any distribution with finite variance Assumes normal distribution of returns Exponentially tight bounds for sums of independent variables Key Insight: Chebyshev works for any distribution but gives conservative bounds. VaR assumes normality but provides tighter estimates. Chernoff gives the tightest bounds but requires specific conditions (independence, moment generating function). How Financial Institutions Use These Bounds During the 2008 financial crisis, many VaR models failed. Why? Example: Setting Margin Requirements

This is the written version of the interactive lesson above. See the full Probability course.