Week 40 Day 1: Standard Deviation: The Ruler for Risk

Think of it like weather. City A averages 70 degrees with a 5-degree standard deviation (range: 65-75, very stable). City B averages 70 degrees with a 20-degree standard deviation (range: 50-90, wild swings). Both average 70, but City B requires far more preparation for extremes. Your portfolio is the same: same average return, but higher standard deviation means more extreme years.

Standard deviations of common investments (annualized, approximate): Cash/T-bills: 1-2%. Short-term bonds (BND): 4-5%. Intermediate bonds (VCIT): 6-8%. Balanced portfolio (60/40): 10-12%. U.S. stocks (VTI): 15-17%. International stocks (VXUS): 17-19%. Small-cap stocks: 20-25%. Individual stocks: 25-50%+. Bitcoin: 60-80%+. How to interpret: with VTI's approximately 10% average return and approximately 16% standard deviation, in any given year you should expect returns roughly between -6% (10% - 16%) and +26% (10% + 16%) about 68% of the time (one standard deviation). About 95% of the time (two standard deviations), returns will fall between -22% and +42%. That -22% year will happen -- it is not a surprise, it is within the expected range. The practical rule: if you cannot emotionally tolerate a decline equal to 2x the standard deviation of your portfolio, you have too much risk. If a 32% decline (2 standard deviations for VTI) would cause you to panic-sell, you need more bonds (which lower the portfolio's standard deviation). Match the volatility to your emotional capacity, not to your theoretical risk tolerance.

Standard deviation (sigma) as a risk measure was introduced to finance by Markowitz (1952) in Modern Portfolio Theory (MPT). For a portfolio with returns R_t, sigma = sqrt(E[(R - mu)^2]), where mu = E[R]. Under the assumption of normally distributed returns, sigma fully characterizes the dispersion: 68% of outcomes fall within mu +/- 1*sigma, 95% within mu +/- 2*sigma, and 99.7% within mu +/- 3*sigma. However, stock returns are NOT normally distributed: they exhibit leptokurtosis (fat tails) and negative skewness (more extreme negative returns than positive). Mandelbrot (1963) showed that the actual frequency of 3-sigma events in stock markets is approximately 10x higher than Gaussian predictions. The October 1987 crash (-22.6% in one day) was a 25+ sigma event under Gaussian assumptions (probability approximately 10^-140), but has occurred. This means that standard deviation UNDERESTIMATES tail risk. Despite this limitation, sigma remains the standard risk measure because: (1) it is mathematically tractable (enables portfolio optimization), (2) it is additive under certain conditions (portfolio sigma can be calculated from component sigmas and correlations), and (3) it captures the 'normal' range of variation that drives day-to-day investor experience and behavior. For more accurate tail risk assessment, practitioners supplement sigma with Value at Risk (VaR), Conditional VaR (expected shortfall), maximum drawdown, and semi-deviation (downside deviation only).