Stock Returns

Probability · Axiom Academy

Using the normal distribution to understand market risk and the reality of fat tails The Normal Distribution in Finance Financial analysts often model daily stock returns using a normal distribution . This simplification makes risk calculations tractable and is the foundation of modern portfolio theory. Example: The S&P 500 historically has: Mean daily return: approximately +0.03% (about +7% annually) Standard deviation: approximately 1% per day Under the normal distribution assumption, we can write: This model enables calculations of important risk metrics like Value at Risk (VaR) , which tells investors the maximum expected loss over a given time period at a certain confidence level. But here's the catch: Real stock returns have "fat tails" - extreme events happen far more often than the normal distribution predicts. Let's explore this discrepancy and what it means for investors. Value at Risk answers the question: "What's the worst loss I can expect with 95% confidence over the next day?" For a portfolio worth 100,000, we can calculate the 95% VaR using the normal distribution. Interpretation: With 95% confidence, you should not lose more than 1,645 in a single day. However, 5% of days (roughly 1 in 20 trading days) could see larger losses. The Problem with Normal Distributions On October 19, 1987 ("Black Monday"), the S&P 500 fell 20.5% in a single day.

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