Portfolio risk versus returns: the statistics

Work in process. This article is under review and undergoing a major update. Caveat: This is an advanced investing topic. Past performance does not predict future performance.

Portfolio returns are expected to follow a normal distribution. Deviations from this pattern are a form of investment risk, described below.

Intro
Fund returns are constantly changing. Instead of showing these changes on a performance chart (over time), count the number of times a return falls on each value. Create a plot with returns on the horizontal scale (X-axis), min to max; the vertical scale (Y-axis) with the number of times the return fell on that value. This format is called a distribution plot, as it shows how the values are distributed across the range of returns.

Statistics is the science of the collection, organization, and interpretation of data. Statistical methods use this plot to analyze past performance and to predict future returns. For example; since most returns fall near plot center, it seems likely that future returns will also be near the plot center.

Based on usage, this plot is known as a statistical distribution. For many types of fund returns, the outline of the plot has the shape of a bell (like the shape of the Liberty Bell) and is called a normal distribution.

- insert graphs here - Plot over time, then show distribution
 * Fat Tails ... and how to get 'em (gummy-stuff)

Identify: normal distribution, left side, right side, fat tail, long tail.

The shape of the curve is described using 4 different terms (you need all 4)...

Barbell strategy
Bonds: An investment strategy that concentrates holdings in both very short-term and extremely long-term maturities. When plotted on a timeline, the shape appears as a barbell. The reasoning behind this strategy is that it allows one portion of the portfolio to achieve high yields while the other portion minimizes risk.

Stocks: An investment strategy that concentrates holdings in both large-cap and small-cap funds, which minimizing mid-cap funds. When plotted against market capitalization (market "cap"), the shape appears as a barbell.

Taleb

 * Understanding all the talk about "tails", forum discussion starting point about The Black Swan (Taleb book)
 * White swan
 * Gray swan
 * Black swan
 * How the Finance Gurus Get Risk all Wrong, forum discussion with link to Taleb's (and Mendelbrot's) paper of same title. Tutorial of a sort, emphasizing that use of the Bell curve for risk management can be misleading.

Rough outline

 * Fat tails
 * Long tails
 * Left tale risk
 * Right tale opportunity

The first four moments of a distribution: Investors define volatility to be the standard deviation. For example, William F. Sharpe defines the Return per Unit of Risk (Sharpe Ratio) as some Return divided by the Standard Deviation.
 * Expected (mean) return
 * Standard deviation (volatility)
 * Skewness
 * Kurtosis

Statistics discussions

 * Median of a distribution and how skewness affects the relationship between the mean and the median
 * Correlations and confidence intervals

Log-normal distributions
The proper distribution for a normal model would be a log-normal distribution... The reason for the log-normal distribution is that changes are relative to other changes. The probability that the market drops by 20% in the second half of the year is relatively independent of what happened in the first half of the year, but it is a larger point loss if the first half of the year was a bull market.
 * Why for annual returns the log normal distribution, rather than the normal distribution, should be used