# Portfolio risk versus returns: the statistics

Portfolio returns are expected to follow a normal distribution. Deviations from this pattern are a form of investment risk, described below.[1]

## 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.[2] 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.[3]

- insert graphs here - Plot over time, then show distribution

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)...

## 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.[4]

### 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

• Why for annual returns the log normal distribution, rather than the normal distribution, should be used

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.[5]